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Logic and First Principles, 15: On the architecture of being. Or, are certain abstract entities (“abstracta”) such as numbers, natures, truth etc real? If so, how — and where?

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For some weeks now, an underlying persistent debate on the reality of numbers has emerged in several discussion threads at UD. In part, it has been cast in terms of nominalism vs platonic realism; the latter being the effective view of most working mathematicians. Obviously, this is a first principles issue and is worth focussed discussion.

Now, No. 14 in this series, on objectivity of aesthetics principles as canons of beauty, begins by pointing to an underlying challenge:

We live in a Kant-haunted age, where the “ugly gulch” between our inner world of appearances and judgements and the world of things in themselves is often seen as unbridgeable. Of course, there are many other streams of thought that lead to widespread relativism and subjectivism, but the ugly gulch concept is in some ways emblematic. Such trends influence many commonly encountered views, most notably our tendency to hold that being a matter of taste, beauty lies solely in the eye of the beholder.

Of course, F H Bradley, long since pointed out that to claim the un-know-ability of things in themselves is already to claim a major point of just such knowledge. So, this is self-referentially absurd. Wisdom, then, is to acknowledge that we can and do err but even that is a point of undeniably certain knowledge therefore we can and do confidently know some aspects of reality as it is, not just as it appears to us. Reality is in part intelligible, it is not utterly inscrutable. Already, this is a hint that there is a rational . . . a logical . . . structure to being which rational creatures may seek to understand, succeeding part of the time. Where of course aspects of that structure will be quantitative.

Let me highlight the core argument (and pardon the inflexibility of the new block style WP is using):

>>to assert that in effect conceptualism about abstracta is true, one relies on abstracta being in reality, e.g. here that a description or assertion can hold a relationship of accurate description with things as they are. Absent the reality of such a relationship independent of our individual or collective concepts, truth is meaningless. If only the concrete exists in reality, truth, an abstract relationship using symbolic representation (other abstracta!) is a case of non-being, illusion. Actually, illusion is another abstract relationship. Meaninglessness is next up, but this too is an abstract state of affairs. The infinite regress of abstracta begging to be acknowledged as real yawns open.

The reality of core abstracta is inescapably the case, i.e. it is necessarily true on pain of not being able to think, communicate conceptually, reason [implication is abstract], speak truth, demonstrate, warrant, know etc.

The serious issue then follows: in what way are such things real?

The best I can answer for now is that such abstracta are connected to the logic of being for worlds or things in the world. They are logically relevant characteristics of being, which in many cases are shared across beings as archetypes that are in-common, or even are in-common across possible worlds. In some cases such as numbers they are in common to all possible worlds as part of the fabric of any distinct possible world.

We may recognise or discover them and try to identify what they precisely are, but in many cases they defy particular definition in words.

Where do they come from, where are they? They come from the logic of being and are embedded as constraints on being. For instance, no entity E is such that it has two core characteristics x and y where y = ~x.

That is why square circles are impossible of being. Regardless of how we may form a fuzzy imagination that oscillates between the shapes or may try to superpose and blend the two.

There are squares, there are circles but no square circles

Thus, abstracta are part of the distinct identity, nature and being of any particular entity. That is, the principle of distinct identity has ontological, not just conceptual, significance. That’s why we recognise it as a first principle of right reason.

So, not a spooky, mysterious, metaphysical world of forms, just the architecture of — rational principles or “logic” of — being or possible being (and of impossibility of being). Where of course a considerable part of that embedded architecture of being is structural and quantitative. That is, Mathematical. Mathematics has in key part ontological import. Hence, Wigner’s point on its astonishing power. The music of the spheres is written in the language of mathematics, with — I daresay — Fourier leading the charge.>>

Fourier in action:

And again (a mechanical implementation in our hearing . . . relevant to octaves and fifths in music etc):

Let me then set it in the context of an ongoing exchange in the thread on beauty, and I take liberty to headline comment 390:

KF, 390: >>H, Let’s roll the tape a bit:


H, 377: >>kf writes,
What happens in the world is independent of [–> antecedent to and insofar as it is intelligible, influences] our thoughts about it [which thoughts in many cases may and do accurately describe reality, concrete and abstract.”
I’ll agree that the world is antecedent to our thoughts: we experience the world and then form thoughts about.
I’ll agree that “insofar as it is intelligible, [the world] influences our thoughts about it, which thoughts in many cases may and do accurately describe reality, concrete and abstract” [sorry, WP suppressed strike-throughs]>>

KF, 378: >>H, that apparent rejection of the reality of certain abstracta, if so, is fatally self-referential for much the same reason as nominalism (which is a form of such rejection) fails.>>

H, 379: >>I’ve explained my position, and see nothing “fatally self-referential” in it. The world is intelligible, and we are intelligent, so our understandings provide reasonably accurate maps of the world. We use abstractions to describe the world, but the world itself is “concrete” in the sense that it is its behavior which we observe that is the source of the material for our abstractions.
Probably no need for you (or me) to repeat ourselves again (although I do have a new thought on the matter that I may share later in the day when I have some time.)>>

KF, 380: >>when an objective matter is on the table, agreement or disagreement is immaterial. Just to make statements you have had to repeatedly rely on abstracta being the case not just perceptions. Indeed, truth is an abstract relationship of statements to what is the case, belief or disbelief, agreement or disagreement too. The reality of core abstracta is inescapable.>>

H, 381: >> I have clearly said that we use abstractions – we have to – just to talk about the world, so of course I agree with you when you write, “Just to make statements you have had to repeatedly rely on abstracta being the case not just perceptions.” Perceptions of the world bring in the data from which we create our abstractions, but abstractions are a necessary, central aspect of our ability, as rational, logical creatures, to understand the world.
Is this the point upon which you think my position is “fatally self-referential”?, because if so it misrepresents me. Perhaps you could explain more about your “fatally self-referential” statement.>>

KF, 382: >>this begins to approach the inescapability of the laws of thought, which embed cases in point. To attempt to deny one is forced to accept implicitly. For instance, you are affirming or implying that somethings are true, are accurate descriptions of reality, which is itself an abstract relationship, indeed the words and what they represent involve abstract relations. That is telling us something — we are at a start-point.>>

H, 383: >>Yes, I have continually said that we use abstract concepts to make statement about reality that are, to various degrees, accurate descriptions.>>

KF, 386: >>we cannot escape core abstracta and they are inescapably true or real as appropriate.>>

H, 389: >>kf writes “that apparent rejection of the reality of certain abstracta, if so, is fatally self-referential.” I accept the reality of the abstract concepts we create that describe the reality we experience. How is that “fatally self-referential”? I don’t see how you have explained that.>>

Notice, how you repeatedly affirm certain things to be true, i.e. to actually accurately describe real states of affairs? That is itself an abstract relationship, which must be real albeit abstract or discussion collapses. Likewise, the Mobius strip’s behaviour pivots on how it has ONE edge, ONE surface, etc. So if by cutting we introduce one or two further edges, it will form a longer loop or two interlocked loops. One-ness, two-ness, three-ness and consequences on the logic of being are abstract but take effect in space and bodies. It does so independent of our thoughts, concepts, expectations, as the relevant abstract properties are part of its core characteristics.

Above, at 375, I again laid out a demonstration as to why numbers are necessary entities that will manifest in any possible world, antecedent to our thoughts about a world. We are contingent beings within an already formed world.

Going back to the self-reference, to assert that in effect conceptualism about abstracta is true, one relies on abstracta being in reality, e.g. here that a description or assertion can hold a relationship of accurate description with things as they are. Absent the reality of such a relationship independent of our individual or collective concepts, truth is meaningless. If only the concrete exists in reality, truth, an abstract relationship using symbolic representation (other abstracta!) is a case of non-being, illusion. Actually, illusion is another abstract relationship. Meaninglessness is next up, but this too is an abstract state of affairs. The infinite regress of abstracta begging to be acknowledged as real yawns open.

The reality of core abstracta is inescapably the case, i.e. it is necessarily true on pain of not being able to think, communicate conceptually, reason [implication is abstract], speak truth, demonstrate, warrant, know etc.

The serious issue then follows: in what way are such things real?
The best I can answer for now is that such abstracta are connected to the logic of being for worlds or things in the world. They are logically relevant characteristics of being, which in many cases are shared across beings as archetypes that are in-common, or even are in-common across possible worlds. In some cases such as numbers they are in common to all possible worlds as part of the fabric of any distinct possible world.
We may recognise or discover them and try to identify what they precisely are, but in many cases they defy particular definition in words.
Where do they come from, where are they? They come from the logic of being and are embedded as constraints on being. For instance, no entity E is such that it has two core characteristics x and y where y = ~x.
That is why square circles are impossible of being. Regardless of how we may form a fuzzy imagination that oscillates between the shapes or may try to superpose and blend the two.

Thus, abstracta are part of the distinct identity, nature and being of any particular entity. That is, the principle of distinct identity has ontological, not just conceptual, significance. That’s why we recognise it as a first principle of right reason.

So, not a spooky, mysterious, metaphysical world of forms, just the architecture of — rational principles or “logic” of — being or possible being (and of impossibility of being). Where of course a considerable part of that embedded architecture of being is structural and quantitative. That is, Mathematical. Mathematics has in key part ontological import. Hence, Wigner’s point on its astonishing power. The music of the spheres is written in the language of mathematics, with — I daresay — Fourier leading the charge.

Speaking of architecture, that does point to architect. But that is an onward discussion tied to the necessary being root of reality. >>

So, what is now on the table is the architecture of — i.e. rational principles or “logic” of — being or possible being or even impossibility of being. Which, in part we may tabulate:

Where also, it is worth the effort to also headline from 375:

KF, 375: >>[W]e can show that key abstract elements of structure and quantity are necessary aspects of the logic of being a distinct possible world.

Consider a distinct possible world, W which is distinct from near neighbours (say W’, W’) by having some aspect of core characteristics A, unique to itself. Were there no A, the world would be indistinguishable from near neighbours and we would recognise that distinct labels have been attached to the same underlying possible world. Such allows us to view W as a structured set:

W = {A|~A}

Now, nothing is in W that is not in A or else ~A, the dichotomy is empty and there is no x in W but not in A or else ~A. This is the quantitative property, nullity; thus zero is present, {} –> 0. Likewise, A is a distinct thing, a unit. Unity is present, so one. Following von Neumann, {0} –> 1, where also A manifests unity. In a different sense, ~A is a complex unity, collecting many other things, pointing to collectives, to systems, to organisation, to function based on organisation etc. For our purposes, ~A is a unit but one different from A, so we need to recognise duality, two-ness, thus two: {0,1} –> 2. Obviously, such succession continues without limit and manifests the naturals, also implying the transfinite ordinals on the premise of order type {0,1,2 . . . } –> w (omega).

Likewise, we may contemplate an inverse such that -x + x –> 0, which is a vector of one dimension. We now have integers. Ratios of integers gives rise to rationals and convergent sums yield the rest of the reals. This gives us continuum. From this, the vector rotation operator i*x repeated twice to give – x allows us to have 2-d vectors in a continuum, a plane. An abstract plane that we may contemplate but which pervades any possible world. Where such a world is sufficiently spatially extended and actualised, we may observe continua, dimensions, vectors, rotations, trajectories etc.

So, we see where any possible world, simply on being distinct, manifests directly 0,1,2 and by extension on the logic of being, N, Z, Q, R, C. The vector phenomenon captured from Z on, allows us to extend the abstract continuum to arbitrarily many dimensions. (Notice the distinction between world manifestations and our extension to n-dimensional entities, n arbitrarily high.In physics we speak of 10^22 degrees of freedom routinely, for statistical thermodynamics, just for a reasonably accessible case.)

Our world manifests three spatial dimensions on the macro scale, and we can observe things like Mobius strips etc.

The underlying point is, that we see intelligible, abstract, necessary, structural and quantitative entities as part of the fabric of any distinct world, part of its framework, part of the logic of its being as a distinct possible world.

In that context, we may identify certain facts of structure and quantity that necessarily obtain.

For instance consider five distinct units and how they may be partitioned into a pair and a triple: ||||| –> || + |||. Obviously, this can be reversed, || + ||| –> |||||. Addition and subtraction have a natural sense of partitioning and combining units. Multiplication and division are extensions as are many onward operations, relations and functions. And so forth.

The point is, that there are abstract, structural and quantitative entities that are intelligible on logic of being which are necessary corollaries of any distinct possible world. These abstracta, we recognise and observe through the effects of the logic of being, we do not invent. They are not merely concepts and constructs we invent and project to a world of things in themselves. That, being in reality just an inner game on the appearances we have and imagine as reflecting the outer world. No, the Kantian ugly gulch fails and we have no good reason to imagine the behaviour of a Mobius strip is some sort of contemplative inner dream. Such dreams we could modify at will, the logic of being is far less yielding than that.

So, we need to frame an understanding of Mathematics that recognises that we may study the logic of structure and quantity, but this is not isolated from the intelligible substance of structure and quantity manifest in the world. Yes, our sense of being and of cause needs to adapt to the logic of being that involves necessary albeit abstract entities. For instance, nullity, the empty set, zero are manifest in a myriad circumstances, indeed in any possible, distinct world. But as {} is indistinguishable from {} there is good reason to see that it is one and the same common entity. Which is a characteristic shown by many abstract entities. >>

So, now, let us further reflect. END

183 Replies to “Logic and First Principles, 15: On the architecture of being. Or, are certain abstract entities (“abstracta”) such as numbers, natures, truth etc real? If so, how — and where?

  1. 1
    kairosfocus says:

    Logic and First Principles, 15: On the architecture of being. Or, are certain abstract entities (“abstracta”) such as numbers, natures, truth etc real? If so, how — and where?

  2. 2
    hazel says:

    See here for my last reply to kf.

  3. 3
    kairosfocus says:

    F/N: I transfer H at 392 in the beauty thread, in order to respond, using the edit feature, hoping this captures all she wrote:

    H, 392: >>kf, I understand your position. However, I see nothing where you actually try to explain why my position is “fatally self-referential”.

    at 379, I wrote,

    I’ve explained my position, and see nothing “fatally self-referential” in it. The world is intelligible, and we are intelligent, so our understandings provide reasonably accurate maps of the world. We use abstractions to describe the world, but the world itself is “concrete” in the sense that it is its behavior which we observe that is the source of the material for our abstractions.

    I accept that we have a difference philosophical position, and I appreciate the way you have distinguished your view from a Platonic one, “a spooky, mysterious, metaphysical world of forms”, but rather “the architecture of — rational principles or “logic” of — being.” That’s a ood distinction.
    This paragraph might illuminate the difference:

    Going back to the self-reference, to assert that in effect conceptualism about abstracta is true, one relies on abstracta being in reality, e.g. here that a description or assertion can hold a relationship of accurate description with things as they are. Absent the reality of such a relationship independent of our individual or collective concepts, truth is meaningless.

    Perhaps the issue here is truth. Perhaps this is an accurate paraphrase of your view: that if there is to be true truth about the world, the abstract concepts in our minds must have a valid and complete relationship with corresponding abstractions in the world. To me, however, truth is always provisional and partial, because the abstractions only exist in our minds, and can never correspond to reality completely. Even if our abstractions seem to correspond exactly, that is only because they are abstractions which leave out the huge multiplicity of details which are not being abstracted.

    So perhaps you think my position is “fatally self-referential” because it can’t lead to Truth, with a capital T, and if so, I stand guilty as charged. However, I don’t see that as “fatal”: I see it as realistic about the nature of human knowledge.>>

    I will comment later, RW still calls.

    KF

    PS: Yes, it captures it all.

  4. 4
    kairosfocus says:

    F/N: For the now, I simply point out from H:

    >>Perhaps the issue here is truth. >>

    1: That is a key abstract entity, a relationship between symbolically expressed ideas and reality as it is.

    >>Perhaps this is an accurate paraphrase of your view: that if there is to be true truth about the world, the abstract concepts in our minds must have a valid and complete relationship with corresponding abstractions in the world.>>

    2: No one said that THE abstractions as we conceive “must have a valid and complete relationship with corresponding abstractions in the world.”

    3: I am simply alluding to Aristotle’s apt summary: truth says of what is that it is and of what is not that it is not. Thus, there are multiple abstracta at work (symbolisation, representation etc) and an abstract structure of correspondence where a true proposition [abstractum] accurately [abstractum] describes [abstractum] some facet of reality.

    4: The pervasive nature of implicit but very real abstracta is plain.

    >>To me, however, truth is always provisional and partial,>>

    5: This conflates knowledge of truth, knowledge being warranted, credibly true belief.

    6: there may be truths we do not know, but only believe or are dismissive of or are utterly ignorant of.

    >> because the abstractions only exist in our minds,>>

    7: The conceptualism, which here asserts to be truth or at least believed truth.

    8: So, again and again we see the implicit use of abstract realities

    >> and can never correspond to reality completely. >>

    9: Our lack of knowledge and proneness to error has nothing to do with the reality of truth and its knowability in key cases such as error exists.

    >>Even if our abstractions seem to correspond exactly, that is only because they are abstractions which leave out the huge multiplicity of details which are not being abstracted.>>

    10: A particular truth is so independent of the degree of our ignorance of truth in general.

    Later.

    KF

  5. 5
    hazel says:

    Kf writes, “That (truth) is a key abstract entity, a relationship between symbolically expressed ideas and reality as it is.”

    I agree. Of course the issue, as you have highlighted in your title, is in what ways are abstract entities real. I maintain that truth, both as a general concept and in reference to particular statements, is an abstract concept in our minds.

    I wrote, in an effort to make our differences clear, that “perhaps this is an accurate paraphrase of your view: that if there is to be true truth about the world, the abstract concepts in our minds must have a valid and complete relationship with corresponding abstractions in the world.>>

    In response, you wrote a number of things that I don’t understand

    You wrote, “No one said that THE abstractions as we conceive “must have a valid and complete relationship with corresponding abstractions in the world.”.

    I don’t understand why you capitalized THE.

    You wrote, “I am simply alluding to Aristotle’s apt summary: truth says of what is that it is and of what is not that it is not.”

    I don’t think this sentence, which you commonly quote, says anything useful. Of course the aim of truth is to accurately say something about what is, but that in itself gives us no guidance in establishing what is in fact true.

    You wrote, “ Thus, there are multiple abstracta at work (symbolisation, representation etc) and an abstract structure of correspondence where a true proposition [abstractum] accurately [abstractum] describes [abstractum] some facet of reality. … The pervasive nature of implicit but very real abstracta is plain.”

    I don’t know why you keep pointing out that the concepts we use to discuss these things (true proposition, accurate, describes, etc.) are abstractions. Yes, abstractions are real, and we can’t think or talk without them. Again (again!) the issue is in what ways are abstract entities real: as concepts in the minds of human beings (my position) or in some other way.

    When I wrote. “To me, however, truth is always provisional and partial”, you wrote, “This conflates knowledge of truth, knowledge being warranted, credibly true belief.”

    Did you mean “knowledge with truth?

    If so, once again, the issue is what is “credibly true belief”. Your common use of the word “warrant” implies your belief that some truths about the world are certain, and can be shown to be so.

    You write, “So, again and again we see the implicit use of abstract realities.”

    Yes, and again and again I point out that this isn’t implicit: I explicitly state that abstract concepts are the means by which we understand the world: we can’t think about or talk about the world without using the abstract concepts that are in our minds.

    I’ll end by pointing out that I don’t see how what you wrote establishes, or even addresses, your idea that my position is “fatally self-referential”.

    P.S. It would be easy to follow your posts if you used quotation marks and not double brackets when you quoted people.

  6. 6
    daveS says:

    KF,

    Based on this post, I take it that you believe actually infinite sets, including uncountable sets, do exist. Is that correct?

  7. 7
    Brother Brian says:

    DaveS

    Based on this post, I take it that you believe actually infinite sets, including uncountable sets, do exist. Is that correct?

    They exist in an abstract sense, but cannot exist in a real physical sense. 🙂 What do I win?

  8. 8
    hazel says:

    The thread?

  9. 9
    daveS says:

    The thread?

    The special boxed set of the entire 15-part series of posts. Narrated by Liam Neeson, with a guest appearance by Richard Lewontin.

  10. 10
    hazel says:

    LOL and multiple 🙂 ‘s

  11. 11
    ET says:

    Infinite sets cannot exist for the simple fact that sets are collections of things.

    Just sayin’…

  12. 12
    kairosfocus says:

    DS (et al), the abstract quantity of endlessness of succession per von Neumann exists as part of the logic of quantity. Any particular finite natural you can state or represent as say k exists and is exceeded k+1, k+2 etc almost as if k were 0, these are part of the logic of quantity expressed in numbers; ties to the distinct identity required for any particular possible world. Thus, we recognise that the transfinite exists as an abstract domain of quantity in any possible world. Also, take a look at the y = ln x curve as x –> 0. This gives good reason to recognise infinitesimals and transfinites, ie. we see a natural catapult function in action, it is the mirror to y = e^x in the line y = x, i.e. in effect an exponential growth downwards approaching 0y and exploding in magnitude as x –> 0. Similar for y = 1/x. Can we have a MATERIAL entity built of atoms that is infinite? No, any particular number of atoms, say u, will be finite. KF

  13. 13
    Brother Brian says:

    DaveS

    The special boxed set of the entire 15-part series of posts. Narrated by Liam Neeson, with a guest appearance by Richard Lewontin.

    I’m saving up for the director’s cut, with the episode containing agit-prop flaming strawmen and colorful herring set in Plato’s cave. 🙂

  14. 14
    kairosfocus says:

    ET, we cannot exhaustively list an infinite set but can indicate it. (0,1] an interval, is a continuum and is the mirror to x GRT 1 under y = 1/x, i.e. it contains an infinity of values (again, abstract). {0,1,2 . . . } is likewise countably endless, given unlimited succession, such are part of the logic of quantity for any particular distinct world. KF

  15. 15
    kairosfocus says:

    H, you continue to try to confine abstracta to our minds. However, just in making that as a truth claim, you inescapably infer to what obtains as reality beyond our minds, involving various abstract structures, e.g. the correspondence of proposition to things in themselves that allows such to be true or else false. This is one aspect of self-reference that continues to show how the attempt to confine fails. And of course Mobius strips and many other phenomena likewise demonstrate how the architecture of our world embeds key structural and quantitative abstracta. KF

    PS: Note, Mathematical Platonism, per IEP, showing that it is not the same as Plato’s world of the forms (certainly as popularly thought of):

    Mathematical platonism is any metaphysical account of mathematics that implies mathematical entities exist, that they are abstract, and that they are independent of all our rational activities. For example, a platonist might assert that the number pi exists outside of space and time and has the characteristics it does regardless of any mental or physical activities of human beings. Mathematical platonists are often called “realists,” although, strictly speaking, there can be realists who are not platonists because they do not accept the platonist requirement that mathematical entities be abstract.

    You will recall, my comment (cf. OP):

    The reality of core abstracta is inescapably the case, i.e. it is necessarily true on pain of not being able to think, communicate conceptually, reason [implication is abstract], speak truth, demonstrate, warrant, know etc.

    The serious issue then follows: in what way are such things real?

    The best I can answer for now is that such abstracta are connected to the logic of being for worlds or things in the world. They are logically relevant characteristics of being, which in many cases are shared across beings as archetypes that are in-common, or even are in-common across possible worlds. In some cases such as numbers they are in common to all possible worlds as part of the fabric of any distinct possible world.

    We may recognise or discover them and try to identify what they precisely are, but in many cases they defy particular definition in words.

    Where do they come from, where are they? They come from the logic of being and are embedded as constraints on being. For instance, no entity E is such that it has two core characteristics x and y where y = ~x.

    That is why square circles are impossible of being. Regardless of how we may form a fuzzy imagination that oscillates between the shapes or may try to superpose and blend the two.

    Thus, abstracta are part of the distinct identity, nature and being of any particular entity. That is, the principle of distinct identity has ontological, not just conceptual, significance. That’s why we recognise it as a first principle of right reason.

    So, not a spooky, mysterious, metaphysical world of forms, just the architecture of — rational principles or “logic” of — being or possible being (and of impossibility of being). Where of course a considerable part of that embedded architecture of being is structural and quantitative. That is, Mathematical. Mathematics has in key part ontological import.

    PPS: Lost in the laugh is a fail.

  16. 16
    kairosfocus says:

    F/N: The following are available for the low, low price of a click. Caution, they may require re-thinking so those not inclined to re-think need not apply:

    Logic and principles series links:

    1 Logic & first principles, 1: Analogy, Induction and the power of the principle of identity (with application to the genetic code) https://uncommondescent.com/the-design-of-life/logic-first-principles-analogy-induction-and-the-power-of-the-principle-of-identity-with-application-to-the-genetic-code/

    2 Logic and First Principles, 2: How could Induction ever work? (Identity and universality in action . . . ) https://uncommondescent.com/intelligent-design/logic-and-first-principles-how-could-induction-ever-work-identity-and-universality-in-action/

    3 Logic & First Principles, 3: The roots of right reason and the power/limits of entailment https://uncommondescent.com/mathematics/logic-first-principles-3-the-roots-of-right-reason-and-the-power-limits-of-entailment/

    4 Logic & First Principles, 4: The logic of being, causality and science https://uncommondescent.com/mathematics/logic-first-principles-4-the-logic-of-being-causality-and-science/

    5 Logic and first principles, 5: The mathemat-ICAL ordering of reality https://uncommondescent.com/philosophy/logic-and-first-principles-5-the-mathemat-ical-ordering-of-reality/

    6 Logic and First Principles, 6: Reason/Rationality and Responsibility (i.e. moral government) are inextricably entangled https://uncommondescent.com/logic-and-first-principles-of-right-reason/logic-and-first-principles-6-reason-rationality-and-responsibility-i-e-moral-government-are-inextricably-entangled/

    7 Logic and First Principles, 7: The problem of fallacies vs credible warrant https://uncommondescent.com/logic-and-first-principles-of-right-reason/logic-and-first-principles-7-the-problem-of-fallacies-vs-credible-warrant/

    Oh well, I was never a salesman

    Cont’d:

  17. 17
    kairosfocus says:

    Continuing:

    7a SM: Is the slippery slope argument ALWAYS fallacious? https://uncommondescent.com/logic-and-first-principles-of-right-reason/sm-is-the-slippery-slope-argument-always-fallacious/

    8 Logic & First Principles 8: Bridging the Wigner MATH-PHYSICS GAP (with help from phase/ configuration/ state space) https://uncommondescent.com/physics/logic-first-principles-8-bridging-the-wigner-math-physics-gap-with-help-from-phase-configuration-state-space/

    9 Logic and First Principles, 9: Can we be “certain” of any of our views or conclusions? https://uncommondescent.com/logic-and-first-principles-of-right-reason/logic-and-first-principles-9-can-we-be-certain-of-any-of-our-views-or-conclusions/

    10 Logic and First Principles, 10: Knowable Moral Truth and Moral Government vs. Nihilistic Manipulation https://uncommondescent.com/ethics/logic-and-first-principles-10-knowable-moral-truth-and-moral-government-vs-nihilistic-manipulation/

    11 Logic and First Principles, 11: The logic of Ultimate Mind as Source of Reality https://uncommondescent.com/mind/logic-and-first-principles-11-the-logic-of-ultimate-mind-as-source-of-reality/

    12 Logic and First Principles, 12: The crooked yardstick vs plumb-line self-evident truths https://uncommondescent.com/logic-and-first-principles-of-right-reason/logic-and-first-principles-12-the-crooked-yardstick-vs-plumb-line-self-evident-truths/

    13 Logic and First Principles, 13: The challenge of creeping scientism (and of linked nominalism) https://uncommondescent.com/intelligent-design/logic-and-first-principles-13-the-challenge-of-creeping-scientism-and-of-linked-nominalism/

    –> 14 is active.

    Again, I never was a salesman.

    KF

  18. 18
    hazel says:

    Kf, you write, “H, you continue to try to confine abstracta to our minds.”

    This is true. That is the main point I am making.

    The you write the following, with edits as you sometimes do, although I think this makes it hard to read.

    In making that as a truth claim, you inescapably infer to what obtains as reality beyond our minds [-> absolutely, there is a reality beyond our minds], involving various abstract structures [-> that is the point I don’t think is true], e.g. the correspondence of proposition to things in themselves [-> such propositions are never an exact correspondence] that allows such to be true or else false [-> as a provisional but never perfect description, map, or model of reality.]

    .

    You write, “This is one aspect of self-reference that continues to show how the attempt to confine fails.”

    And exactly how do we have self-reference here?

    You write, “ And of course Mobius strips and many other phenomena likewise demonstrate how the architecture of our world embeds key structural and quantitative abstracta”

    34, and counting.

    You add,

    Mathematical platonism is any metaphysical account of mathematics that implies mathematical entities exist, that they are abstract, and that they are independent of all our rational activities. For example, a platonist might assert that the number pi exists outside of space and time and has the characteristics it does regardless of any mental or physical activities of human beings.”

    Yes, and as I’ve engaged in these conversations over the last few months, I have decided that I am not a mathematical Platonist.

  19. 19
    kairosfocus says:

    F/N: When Wikipedia does a job reasonably well (at least in the introduction) it is worth noting that. Here, on truth — as a point of context for further discussion:

    Truth is most often used to mean being in accord with fact or reality, or fidelity to an original or standard.[1] Truth is also sometimes defined in modern contexts as an idea of “truth to self”, or authenticity.

    Truth is usually held to be opposite to falsehood, which, correspondingly, can also suggest a logical, factual, or ethical meaning. The concept of truth is discussed and debated in several contexts, including philosophy, art, theology, and science. Most human activities depend upon the concept, where its nature as a concept is assumed rather than being a subject of discussion; these include most of the sciences, law, journalism, and everyday life. Some philosophers view the concept of truth as basic, and unable to be explained in any terms that are more easily understood than the concept of truth itself. To some, truth is viewed as the correspondence of language or thought to an independent reality, in what is sometimes called the correspondence theory of truth.

    Various theories and views of truth continue to be debated among scholars, philosophers, and theologians.[2] Language is a means by which humans convey information to one another. The method used to determine whether something is a truth is termed a criterion of truth. There are varying stances on such questions as what constitutes truth: what things are truthbearers capable of being true or false; how to define, identify, and distinguish truth; what roles do faith and empirical knowledge play; and whether truth can be subjective or is objective: relative truth versus absolute truth.

    Okay, a place to begin: truth says of what is that it is and of what is not that it is not, per Ari in Metaphysics 1011b. Propositions make assertions as to what is the case or is not the case of affairs and are identifiably either true or false. They are the focus of logic, Mathematics, Science and a great deal of philosophy.

    Let me pull up Copi again:

    A. Propositions
    Propositions are the building blocks of our reasoning. A proposition asserts that
    something is the case or it asserts that something is not. We may affirm a propo-
    sition, or deny it—but every proposition either asserts what really is the case, or
    it asserts something that is not. Therefore every proposition is either true or false.

    There are many propositions about whose truth we are uncertain. “There is
    life on some other planet in our galaxy,” for example, is a proposition that, so far
    as we now know, may be true or may be false. Its “truth value” is unknown, but
    this proposition, like every proposition, must be either true or false.
    A question asserts nothing, and therefore it is not a proposition. “Do you
    know how to play chess?” is indeed a sentence, but that sentence makes no claim
    about the world. Neither is a command a proposition (“Come quickly!”), nor is
    an exclamation a proposition (“Oh my gosh!”). Questions, commands, and excla-
    mations—unlike propositions—are neither true nor false.
    When we assert some proposition, we do so using a sentence in some lan-
    guage. However, the proposition we assert is not identical to that sentence.
    This is evident because two different sentences, consisting of different words
    differently arranged, may have the same meaning and may be used to assert
    the very same proposition.

    Key concepts.

    KF

  20. 20
    kairosfocus says:

    H, are you making statements that assert that certain things are or are not the case? If so, you are making propositions, which are automatically linked to reality by the relationship of truth or falsity (as opposed to our ability to be certain of truth or falsity, cf. Copi). Thus, in asserting truth claims, you are self-referential. In this context, necessarily involving the abstract relationship to reality of claimed truth which may be so or not so. If that bridge is not real in some relevant sense, it reduces truth claims to meaninglessness. KF

  21. 21
    hazel says:

    As a digression, I just finished a book on loop quantum gravity, which posits that even space and time are quantized, in which case the continuum of the abstract number system would not exist in the real world.

  22. 22
    daveS says:

    KF,

    Is that a yes or a no?

    It doesn’t look like you are willing to say that, for example, the set we normally refer to as [0, 1] actually exists. Likewise for any other actually infinite (abstract) set.

    But you do talk about the real numbers frequently. Are you a nominalist about infinite sets?

  23. 23
    kairosfocus says:

    DS, why do you think I have not said a definitive yes that there are transfinite sets, such being integral to the logic of a possible world, including infinitesimals? I for cause hold that structure and quantity are primary, labels are secondary. Remember, I have tied N, Z, Q, R, C to the distinct identity of any particular possible world, i.e. such abstracta are necessary to reality. That we call such domains sets is just a way to refer collectively and definably. KF

    PS: Notice my comment to ET on the continuum.

  24. 24
    hazel says:

    Kj writes, “H, are you making statements that assert that certain things are or are not the case?”

    Sure,( although as I’m pretty sure I’ve said 🙂 ), such statements (if they are about the world as opposed to a formal logical system) are provisional, and are accurate only to a certain varying degree. But we are certainly justified in saying, with appropriate disclaimers if needed, that certain propositions about the world are true. That doesn’t mean Certainly True, but it certainly means true enough to add into a functioning knowledge system about the world.

    You write, “ If so, you are making propositions, which are automatically linked to reality by the relationship of truth or falsity (as opposed to our ability to be certain of truth or falsity, cf. Copi).”

    Not sure what you mean by that. Propositions about the world are “linked to reality” by assessing whether they accurately describe reality. Whether they do are not is sometimes easy to show, and sometimes quite difficult. Not sure who Copi is: I think you mentioned him once before and I agreed with what you quoted, but I can’t remember and could be wrong about that.

    But anyway, as I said before as far as propositions about the world go, we can not be Certainly Sure about their Truth.

    You write,

    “Thus, in asserting truth claims, you are self-referential [You still haven’t explained what you mean by this.]. In this context, necessarily involving the abstract relationship to reality of claimed truth which may be so or not so. If that bridge is not real in some relevant sense, it reduces truth claims to meaninglessness.

    Sure, “the bridge is real”. I just explained, again, that we test our propositions about the world and come to a conclusion as to how valid they are: our abstractions are testable models that describe observable facts about the world.

    I’ll repeat something I wrote on the other thread, because I think it is a fairly succinct and accurate statement of my position these days:

    I’ve explained my position, and see nothing “fatally self-referential” in it. The world is intelligible, and we are intelligent, so our understandings provide reasonably accurate maps of the world. We use abstractions to describe the world, but the world itself is “concrete” in the sense that it is its behavior which we observe that is the source of the material for our abstractions.

    P.S. I see now that there was a quote from Copi in a post that I didn’t see when I read 20 by kf, but I think my reply covered the points he raised anyway.

    And as always, we have to distinguish propositions about the world from propositions about the elements of formal logical systems.

  25. 25
    daveS says:

    KF,

    DS, why do you think I have not said a definitive yes that there are transfinite sets, such being integral to the logic of a possible world, including infinitesimals?

    I don’t know (and that is a confusing sentence). I can think of two reasons not to give a definitive yes or no answer:

    1) You don’t know (which is a respectable position)

    2) You prefer to keep your position vague, because it’s easier to defend.

  26. 26
    kairosfocus says:

    H, please read Copi again, as a standard reference on a proposition: “A proposition asserts that something is the case or it asserts that something is not. We may affirm a propo-sition, or deny it—but every proposition either asserts what really is the case, or it asserts something that is not. Therefore every proposition is either true or false.” Our opinion or whether it succeeds or fails notwithstanding, a proposition asserts that X is really the case or X is not the case. So, it asserts a truth claim and thus a structured relationship between assertion and reality. If the claim is successful, it is truth. We here see the abstract relationship, truth. Which has to obtain independent of our views, opinions, degree of confidence, warrant etc.It is independent of the degree to which we may err, so long as this does not entail that we always err (which would be self-referentially absurd by being self-refuting and would at once end serious discussion: the proper end of mind, reason and discussion is truth). Similarly, we find that in implications, only true consequents follow properly on a true proposition, another tie-in between proposition and reality, another sign of ontological import. KF

  27. 27
    kairosfocus says:

    DS, really! I have clearly stated that there are structures and quantities built into reality, indeed I showed — cf OP — that N, Z, Q, R and C follow on distinct identity of any possible world. Each of these lays out a transfinite range of quantities in the relevant sense: any finite count of numbers of each type can be exceeded, so the numbers of each type are without limit: immeasurably large. For N, this points to the recognition of the type of quantity, the order type of naturals, a transfinite. Obviously, c for continuum is beyond that. I have pointed to how the term, set, is denoting an identifiable collection and is secondary to the quantities and their structural relationships which are part of the architecture of any particular possible world. As you know from three years past was it, I am noting the proper sense of transfinite: beyond any particular finite measure. For instance for any natural k, we may exceed k+1, K+2 etc, this obviously being recursive. KF

  28. 28
    daveS says:

    KF,

    Ok, this is fairly clear, I think. You do not accept that the real numbers comprise an actually infinite set. Or at least you will not comment on the matter (which would be completely fair, IMHO).

  29. 29
    hazel says:

    kf, I think 24 responds to Copi.

    Also, as an example, for many years the standard model was a Cartesian 3-D space and a separate entity Time, both of which were continuous in a linear fashion, and were the background structure of the world in which particles moved. This model is still the beginning framework for physics.

    But we now know that that model is false: it is a workable abstract concept for many purposes, but it doesn’t accurately describe reality at either the quantum level or in respect to relativity. And, as I noted, the hypothesis of quantum loop gravity, which may have some truth that will eventually be supported (or not) posits that neither space nor time are continuous.

    So the abstraction of the continuum as represented by the real numbers is perhaps only approximately a “true” model of reality in respect to space and time, and the old abstract model of space and time as continuous containers for the activity of the elements of the world is now outdated, or at least seen as less true than it used to be.

    This supports my general point, I think.

  30. 30
    kairosfocus says:

    DS, has it registered that I identified what a transfinite set is, and how this applies to N, Z, Q, R and C? I find your reaction and line of questions strange at best. They are also quite obviously tangential. KF

  31. 31
    kairosfocus says:

    H, tangential at best. What a proposition is, the tied centrality of truth-claims, the inherently abstract relationship of truth have all been identified. Notice, in talking about school level physics, you spoke to falsity, which is a related abstract relationship. Similarly, distinct identity of a world implies structure and quantity of abstract character being embedded: N, Z, Q, R, C. Where, such coupled with relevant onward structures will describe any particular form that space-time manifests in this world. That is separate from the abstract reality of C as a guaranteed flat domain which is an abstract space present in the logical framework of any and every possible world — the best way to get “the” plane that is so often discussed. We can extend to ijk as spanning a similar abstract space of three dimensions. And much more. Further to such it has long been pointed out that truth or falsity are independent of how well we know such. KF

  32. 32
    hazel says:

    Dave’s question is not tangential, as it pertains to what is the nature of abstractions: are they real, and “If So, How — And Where?”

    My points about the nature of time and space is not tangential also. These are good examples of abstract concepts which are elements of our logical symbolic number system and which provide useful approximations as part of our models of reality at times, but which do not match some real property of the world.

  33. 33
    daveS says:

    KF,

    I find your answers confusing as well, FWIW. Every time, you seem to stop just short of affirming that the real numbers comprise an actual infinite set (actual as opposed to potential). I recall that in the past, the issue of actual vs. potential infinities has caused us some difficulties.

    But I do believe this is relevant to the discussion over nominalism. You seem to accept the validity of Cantor’s argument showing that the set [0, 1] is uncountable, yet the proof depends on the existence of an actual infinite set.

  34. 34
    math guy says:

    DS
    Give KF a break! OF COURSE uncountable sets exist once we accept ZFC (in fact, only ZF is needed). The natural numbers are a consequence of Peano arithmetic (a strict subsystem of ZF) and the Power Set Axiom applied to the natural numbers yields an uncountable set, courtesy of Georg Cantor.

    Hazel can assure you that all the logical consequences of a formal system exist as plainly as the axioms themselves. On the other hand, I doubt that an uncountably infinite set can be physically realized in our universe, but that was not the question.

  35. 35
    math guy says:

    H
    Stop being so difficult. KF has shown how you have painted yourself into a corner, so to speak. In a nutshell, your claim that abstracta exist only in minds is a logical proposition (call it P) about the universe which has a truth value of T or F. The population of planet Earth has various opinions about the truth value of P, but that is irrelevant. The point is that the proposition itself is an abstract object.

    If P is true, then it only exists within minds and is not the universal truth that P claims. So if we actually adhere to logic and Law of Excluded Middle, then P must be false.

  36. 36
    daveS says:

    Math Guy,

    IIRC, KF has questioned the existence of the actual infinite, even in abstract settings. We had several long threads on this some time ago.

    I recall going over a simple proof (written by Terence Tao, I believe) of the proposition “all natural numbers are finite”, and I was not able to convince him it was correct.

    I’m not saying this to insult anyone; some brilliant mathematicians have different views on these matters.

  37. 37
    hazel says:

    MG, my belief that abstractions exists only in minds is not a “logical proposition” in the same sense that, for instance, “five is a prime number” is a logical proposition. It is a proposition about the world in which the terms, the meaning, and the evidence for or against are all beyond the realm of pure logic, and subject to a lack of specificity compared to what we expect in purely logical statements. As such, it doesn’t boil down our being able to say whether it is purely True or purely False.

    However, I’ve offered a number of arguments to support my view. Kf has offered evidence to support his. Kf might be right, but he hasn’t convinced me. I might be right. Mostly likely the truth, at a metaphysical level, goes beyond what either of us can conceive.

    You offer a reason why you think my position is “fatally self-referential” (although you didn’t use that phrase). However, to think all of this can be boiled down to a “logical proposition” confuses the power of pure logic with the search for knowledge about the real world.

  38. 38
    math guy says:

    H
    “Mostly likely the truth, at a metaphysical level, goes beyond what either of us can conceive.”

    The “truth” of your declaration is an abstract object. Your insinuation that there is a T/F value of your declaration makes it a proposition.

  39. 39
    StephenB says:

    Hazel

    Of course the issue, as you (KF) have highlighted in your title, is in what ways are abstract entities real. I maintain that truth, both as a general concept and in reference to particular statements, is an abstract concept in our minds.

    Truth is more than an abstract concept in your mind. It is about the *correspondence* of the abstract concept in your mind with a worldly concrete truth (like the *whatness of a cat or dog – i. e. its identity) or with a worldly abstract truth (like the *definition* of the Pythagorean Theorem).

    Of course the aim of truth is to accurately say something about what is, but that in itself gives us no guidance in establishing what is in fact true.

    If you (the thinking subject) understand the Pythagorean Theorem (the object of your thought) then you apprehend the truth in that context. That is all the guidance that you need.

    Yes, and again and again I point out that this isn’t implicit: I explicitly state that abstract concepts are the means by which we understand the world: we can’t think about or talk about the world without using the abstract concepts that are in our minds.

    You seem to ignore extra-mental abstract realities. This is the question: Do your mental concepts conform to the abstract or concrete truths about the real world. If so, then you grasp the truth, if not, then you don’t.

  40. 40
    kairosfocus says:

    DS, what does “infinite” or “transfinite” mean? The answer actually lurks in the terms: not-finite, that is beyond any finite bound. In the case of N, the von Neumann implementation of succession is just that, limitless. We cannot complete the set in any finite stage stepwise succession. If we symbolise any particular value say k, it is at once exceeded by k+1. k+2 etc which can be put into 1:1 correspondence with 0, 1, 2 etc thus it is as good as starting from 0 again. Onward we could use the evens and set them into 1:1 correspondence, or the odds etc. This is of course a definition: a set is of infinite character if a logically proper subset can be placed in such correspondence with it, demonstrating that it can be transformed into the set and implying the limitlessness of the set. The cardinality of the set is not a finite particular value but a scale index of transfinite character; for the naturals, aleph-null. For the continuum — notice the interval from 0 to 1 can correspond to the range from 0 onward without bound — is c, a dispute being on which member of the aleph series of transfinite power set scaled numbers is represented by c, where it exceeds aleph null. By such reasoning, we arrive at a truth of understanding that we cannot directly produce, much like a Turing Machine on a non-halting case. We recognise the truth of an abstract, fresh class of quantity, the transfinite, sometimes called the infinite. The first such is omega [for convenience, w] the order type of the naturals. But notice, the example of k above shows that we cannot even represent a non-finite member of N. Similarly, every such k — and yes this is recursive — implies a bound to a finite subset, it is itself finite, a finite order type in the succession. The transfinite property, proper, resides in the lack of limit, not in the concept that there is an infinite number of successive naturals, every one of which is finite. By the nature of finitude, every natural we can specify or simply represent is finite and bounds a finite subset which can be exceeded onwards. It is the beyond finite limits property which confers to the naturals as a whole their transfinite character. KF

    PS: Notice the power of symbolic representation and the implied propositions which refer to abstract entities and truths, also the use of implication, another abstract relationship. Where an implication of a truth will also be true.

    PPS: Notice, above, I suggested the observed universe has u atoms at some time. U is a particular number and will be finite. The structure of numbers, say the surreals, is transfinite. Such are connected to distinct identity of any particular world. They are not concrete, they are abstract but real. Indeed through the logic of being, they frame how a world can be.

  41. 41
    kairosfocus says:

    MG & SB, thanks for the intervention. This thread has been sadly eye-opening for me. It seems there is something worldview threatening about the idea that there are abstract entities, facts, structures, relationships of rational, intelligible character. Worse, that some such are embedded in our world — as the practical example of the Mobius strip demonstrates, among many others. Worse yet, that some such are embedded in any particular distinct possible world. That truth is a structure of abstract relationship by which assertions accurately describe reality independent of our finite, fallible minds, likewise seems a flash-point. And yet, propositions indubitably assert truth claims, which in some cases are manifestly correct. For example, as you pointed out years past SB it is undeniably true for a self-aware, conscious entity that he is conscious. Likewise, as I have pointed out ever so often following Royce and Trueblood, that error exists is another undeniable, self evident truth. So, it matters not that we often err, the very fact that we know that error exists undeniably is itself a self-evident truth. Such are plumb-line test cases that expose crooked yardsticks by being naturally straight and upright. That seems to be the underlying problem, there are crooked yardstick standards that are manifesting their failure to be straight and upright. KF

  42. 42
    kairosfocus says:

    MG, 35 (attn, H): Maybe that is a problem, there are people who reject the first principles of right reason. If one rejects the principle of dichotomy which is what excluded middle is about then one has a yet deeper problem than the reality of abstract entities. Indeed, as this is a tightly coupled corollary to distinct identity and to non-contradiction, also holding ontological import, one has problems with first steps of reasoning. And reality. To all such, I point out, just to communicate and think conceptually (thus, abstractly), you inescapably rely on distinct contrasting symbols, whether vocal sounds or alphanumeric glyphs or pictures or whatever. Such a person would be well advised to rethink. KF

    PS: I again point to a comment in a classical work, St Paul in correcting the irrational, disorderly behaviour of the Corinthians:

    1 Cor 14: 7 Yet even lifeless things, whether flute or harp, when producing a sound, if they do not produce distinct [musical] tones, how will anyone [listening] know what is piped or played? 8 And if the [war] bugle produces an indistinct sound, who will prepare himself for battle? 9 So it is with you, if you speak words [in an unknown tongue] that are not intelligible and clear, how will anyone understand what you are saying? You will be talking into the air [wasting your breath]! 10 There are, I suppose, a great many kinds of languages in the world [unknown to us], and none is lacking in meaning. 11 But if I do not know the meaning of the language, I will [appear to] be a [c]foreigner to the one who is speaking [since he knows exactly what he is saying], and the one who is speaking will [appear to] be a foreigner to me. [AMP]

    PPS: That H thinks Aristotle’s summary on truth is of little value is likely diagnostic. His point was that truth says of what is that it is and of what is not that it is not. Truth has ontological reference and import. Propositional assertions, if true, pass the Aristotle test. They claim to be true by their nature, and that truth claim is already an abstract tie to the reality of what is, what exists concretely or as abstract structures and entities tied to the at- least- partly- intelligible- to- us architecture (organising principles and structures) of a world. Where, a cosmos as opposed to a chaos is an ordered system of reality.

  43. 43
    daveS says:

    KF,

    Thanks for posting #40—Math Guy should find that summary helpful.

    Here is my question: How do you know that the two sets [0, 1] and N cannot be put into one-to-one correspondence?

  44. 44
    ET says:

    If a set is defined as a collection of things then it is obvious there can never be a set with infinite elements. No one can collect an infinite number of anything. With a collection you know every element in that collection.

    Now, if you want to redefine the set, then that is something different.

  45. 45
    hazel says:

    kf writes,

    Maybe that is a problem, there are people who reject the first principles of right reason. If one rejects the principle of dichotomy which is what excluded middle is about then one has a yet deeper problem than the reality of abstract entities. Indeed, as this is a tightly coupled corollary to distinct identity and to non-contradiction, also holding ontological import, one has problems with first steps of reasoning. And reality. To all such, I point out, just to communicate and think conceptually (thus, abstractly), you inescapably rely on distinct contrasting symbols, whether vocal sounds or alphanumeric glyphs or pictures or whatever. Such a person would be well advised to rethink.

    I am utterly baffled. I don’t reject the laws of logic, and I have repeatedly pointed to the importance of our verbal and written symbols systems as the way we represent and communicate our abstract understandings. A person who thinks that I “would be well advised to rethink” would be well advised to actually try to remember and understand what other people say. Over and out.

  46. 46
    kairosfocus says:

    DS, this is not a mathematics inquisition. That the counting numbers and the continuum are of different transfinite cardinalities is a well known commonplace result. It does not become suspect because I report it. The point being even more tangential, I consider these comments sufficient. KF

  47. 47
    kairosfocus says:

    ET, it is a definable collection of entities, which could be concrete or abstract. Definable means, we can determine what’s in and as relevant, what is not in. In the case of N, as the successor operation is unlimited, we may properly term it transfinite. The elements of this set are {} –> 0, {0} –> 1, {0,1} –>2, etc without limit. It contains abstract entities, numbers. Such are not made of atoms or other materiel substance, and so there is no limit on the logic involved, unlike how many atoms are in the universe. KF

  48. 48
    daveS says:

    KF,

    It is indeed a commonplace result, but the proofs use methods which are not available in your system, I believe.

    You’re welcome to show I’m wrong by posting a proof of this fact without referring to actual infinite sets or actual infinities in general. Consider that a challenge.

  49. 49
    kairosfocus says:

    H,

    I responded specifically to MG and his mention of the LEM. In that context, I spoke to the consequences of such. Recall, personalities and their particular views are secondary to the primary issue of objective warrant. Though of course it is relevant to the arguments on the table.

    Notice, MG’s summary:

    In a nutshell, your claim that abstracta exist only in minds is a logical proposition (call it P) about the universe which has a truth value of T or F. The population of planet Earth has various opinions about the truth value of P, but that is irrelevant. The point is that the proposition itself is an abstract object.

    If P is true, then it only exists within minds and is not the universal truth that P claims. So if we actually adhere to logic and Law of Excluded Middle, then P must be false.

    The point on LEM here is that any proposition is either true or else false, independent on our knowledge or doubts etc. P claims a certain thing, that “that abstracta exist only in minds” which “is a logical proposition (call it P) about the universe” which per being a proposition “has a truth value of T or F.” he then goes on to highlight the issue of the ontological import of truth, i.e. that it accurately describes reality. Thus, he brings up: “the proposition itself is an abstract object” so “If P is true, then it only exists within minds and is not the universal truth that P claims.” That is it has no ontological import, no reference outside the world of ideas, perceptions etc. Thus, as truth is about accurate ontological import, “So if we actually adhere to logic and Law of Excluded Middle, then P must be false.”

    Thus, the issue pivots on, what is truth.

    You have dismissed Aristotle, that truth says of what is that it is and of what is not that it is not. You have objected regarding diversity of opinions proneness to error and challenges of warrant. But as MG says, “The population of planet Earth has various opinions about the truth value of P, but that is irrelevant.”

    If truth is not the accurate description of reality, concrete and abstract, then what is it.

    KF

  50. 50
    kairosfocus says:

    DS, again tangential. Kindly see the OP. KF

    PS: BTW, just for record, kindly explain to me what {0,1,2 . . . } –> w is but a transfinite sequence, one that cannot be exhausted in steps. Perhaps, the nature of the set as structured into a sequence that continues beyond any finite k should be noted. This sequence, of course being an abstract entity. Then, ponder similar {b_0, b_1, b_2 . . . } binary sequences, each distinct from the other, that may be counted s_0, s_1, etc of arbitrary character and proceed to change the appropriate digits, constructing some T not in the collection. I would think we may infer from this that T is not in the enumeration of sequences. We cannot exhaustively list any sequence but the logic of structure and quantity leads to T not in it without needing to actually exhaust any sequence.

  51. 51
    ET says:

    We will just have to agree to disagree. My point is that no one can ever collect an infinite number of anything.

  52. 52
    daveS says:

    KF,

    If you write out that argument more precisely, you will see that you never have a T that you know is not in the original set of sequences.

  53. 53
    kairosfocus says:

    ET, an abstract collection is not made of things gathered one by one, it is delivered at once by a logical principle. KF

  54. 54
    ET says:

    Logically infinite elements cannot be delivered at once. Logically, infinity is a journey. A never-ending journey.

  55. 55
    kairosfocus says:

    ET, observe, N = {0,1,2, . . . } KF

  56. 56
    ET says:

    Yes, I observe there isn’t a last number in that alleged set.

  57. 57
    hazel says:

    This looks at lot more recreational and fun than our past topics.

    ET, are there an infinite number of real numbers in the set of all real numbers between 0 and 1?

  58. 58
    ET says:

    What’s an “infinite number”?

    There are infinite real numbers between any two integers. My point is that no one can collect infinite elements. But you can give it a try.

  59. 59
    ET says:

    Trying to clarify:

    Infinity is not a number. Infinity is a journey.

    My ONLY disagreement comes from the fact that a set is defined as a collection of something (elements) and no one can collect infinite elements.

  60. 60
    hazel says:

    Related to our discussion: Feynman on law of physics

  61. 61
    kairosfocus says:

    ET, it is the ability to decide Y/N that allows us to include any particular member and often to construct a way to build out the set, so we may infer to transfinite character once we see the no particular bound effect; which does mean, one cannot exhaustively list, but one can identify a decision rule that allows the distinction of identity: IN or OUT. That runs into problems that forced a tightening up of the set concept, hence talk of a former naive theory. We see here a sort of irreducible complexity. Where, as one consequent, a proper subset can be matched 1:1 with the whole set. (The last is the usual choice on identifying the transfinite.) For N, the von Neumann construction will do. KF

    PS: Notice, we are on tangents relative to the OP.

  62. 62
    kairosfocus says:

    DS, notice my exchange with ET. KF

    PS: I observe the tangential nature of comments. That may imply something significant.

  63. 63
    kairosfocus says:

    H, there are many ways to skin a catfish, or to look at a physical, structural, quantitative phenomenon. He did not discuss the import of the field view, that the object of interest influences or warps space so that there is a disturbance which in case of motion can be viewed as propagating a wave. Likewise, how that view can be interpreted as a flux. Similarly, the least action approach is suggestive and leads into variational principles. And more. All of these interact, as he actually notes. None of this changes the point that structure and quantity, often abstract, pervade the architecture of our world. KF

    PS: Ac 17 has a lesson for those who seek intellectual entertainment at expense of those viewed as the equivalent of the spermologos. The future did not belong to those who tried to dismissively laugh.

  64. 64
    kairosfocus says:

    F/N: Recall, MG at 35:

    [To H:] In a nutshell, your claim that abstracta exist only in minds is a logical proposition (call it P) about the universe which has a truth value of T or F. The population of planet Earth has various opinions about the truth value of P, but that is irrelevant. The point is that the proposition itself is an abstract object.

    If P is true, then it only exists within minds and is not the universal truth that P claims. So if we actually adhere to logic and Law of Excluded Middle, then P must be false.

    KF

    PS: Note the power set chain from Aleph Null. The first is usually accepted as the cardinality of c.

  65. 65
    daveS says:

    KF,

    DS, notice my exchange with ET. KF

    PS: I observe the tangential nature of comments. That may imply something significant.

    The comments are quite central in my view, directly addressing the question Are Certain Abstract Entities (“Abstracta”) Such As Numbers, Natures, Truth Etc Real? If So, How—And Where?

    ET apparently does not believe that infinite sets are real, and explains why.

    I have asked you how you know uncountable sets are real.

  66. 66
    ET says:

    My issue comes from the definition of a set:

    In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

    Note the bolded word “collection”.

  67. 67
    daveS says:

    And every collection must have a collector … 😳

  68. 68
    ET says:

    That’s where we come in. 😎

  69. 69
    hazel says:

    I’m not sure how one “collects” numbers.

    And I agree with Dave. It would seem that the question of whether an infinite set exists, and how and where, would be very much on the topic of the thread.

  70. 70
    kairosfocus says:

    DS & ET, and it is where necessary being mind at the root of reality would come in. Where, note, the issue is first the architecture — structure, elements, organisation and linked quantitative elements of a world. I note that just simply the distinct identity of any particular world implies some aspect A that is unique to it as opposed to near neighbour possible worlds. This directly leads to partition and structure W = {A|~A} with direct manifestation of the quantitative structural elements: A — unity, ~A — complex unity, partition on LEM — nullity (likewise surrounding A and ~A). these are already world framework abstracta. They lead to N, Z, Q, R, C and many other things. Note a possible world is a sufficiently complete description or specification, implying propositions. Such are actualised in at least one case and are also present in models and particularly in Mathematical Systems. As I have pointed out, abstracta beyond our concepts are inescapable, they express the structure of reality tied to the logic of being. KF

  71. 71
    kairosfocus says:

    H, cf OP. Start:

    {} –> 0
    {0} –> 1
    {0,1} –> 2

    . . .

    {0,1,2 . . . } –> w, omega.

    KF

  72. 72
    hazel says:

    re 71: ??? I have no idea what point kf is making to me, or to what he is responding. ???

  73. 73
    Brother Brian says:

    Hazel

    I have no idea what point kf is making to me, or to what he is responding. ???

    You are in good company.

  74. 74
    kairosfocus says:

    H, BB: collecting numbers. KF

  75. 75
    hazel says:

    I see. You were really responding to ET: he’s the one who “collects numbers.”

    Up above, you wrote to ET “ET, an abstract collection is not made of things gathered one by one, it is delivered at once by a logical principle.”

    I agree with this.

    However, I’m pretty sure you have also said that if you try to traverse an infinite set one element at a time, you can’t “come to an end”, which I also agree with, of course.

    So we can abstractly grasp the concept of an infinite set by understanding the principle which defines it even though we can’t enumerate the entire set. Does that sound correct to you?

    My question to you, then, is (because I’m not clear although you may have mentioned it), is the set N = {1, 2, 3… } real? If so, how and where? What is your answer to this question?

    Another way, perhaps, to ask the same question: does every natural number exist? If so, how and where?

  76. 76
    kairosfocus says:

    H, kindly see the OP, it is highlighted in red. KF

  77. 77
    hazel says:

    OK, I read that and see you addressed that.

    So the key idea is the “logic of being”. Also, you use the phrase “shared across beings as archetypes that are in-common”, but I’m not sure what that means. You also write, “So, not a spooky, mysterious, metaphysical world of forms, just the architecture of — rational principles or “logic” of — being .”

    Can you explain more about how the “architecture of being” (which is a nice phrase) is different from a Platonic metaphysical world of forms. And can you explain more by what you mean by archtypes, and how the differ from a metaphysical world of forms? Is the “architecture of being” metaphysical?

  78. 78
    kairosfocus says:

    H,

    compare objects in computing. They have a pattern of derivation from “ancestral” structures through inheritance and adaptation. They show a “family resemblance” through that balance between the in-common and the different that gives particular, distinct identity and function. Obviously, such a pattern of attributes or characteristics is in part at least intelligible. Objects fall into classes, with an architecture that is intelligible in part, with a pattern of the in-common and the distinct. That is, beings have distinct identity, being is being as of type X with distinguishing feature Y. This is close to the concept of definition by genus and difference, also to inferring such a pattern on ostensive definition by exemplar and family resemblance.

    The archetypes would be shared patterns, notice how possible worlds are treated as having a structure of the in common and the distinctive attribute, A marking them apart from near neighbour possible worlds.

    Blend with the concept of a candidate entity or being. Given square circles, some are not possible of being precisely because proposed core characteristics stand in mutual contradiction. For, squarishness and circularity inescapably clash. We here see how first principles of logic tied to distinct identity can and do have existential import.

    We may contrast possible being and see here that such would be feasible of being in at least one possible world. (And yes, suggested PW’s with incoherent core characteristics are not possible of being.)

    If also, there is at least one PW in which a candidate being would not exist, it is contingent. Closely connected ponder two neighbour PW where in W, candidate B exists but in W’ it does not. The distinguishing characteristic A in W but not W’ is, logically, causally connected to and enabling of B’s existence in W. B is an example of a contingent being and we see here a natural basis for cause to be accepted as a reasonable concept.

    There is another logical possibility, a possible being that is present in all possible worlds, a necessary being. Such would logically be part of the framework for a world to be. We saw (cf. OP) that for a distinct world W to be just that, it has some attribute A that marks it apart from close neighbours, so we see how core first principles of being are structurally — architecturally — woven into the framework of being any distinct possible world (and are attributes in common):

    W = {A|~A}

    This instantly brings with it the triple first principles of right reason, marking core logic of being for worlds, LOI, LNC, LEM are not just laws of how we are forced to think but are structurally connected to worlds that may be. We may have further phenomena, say connected to superposition and fuzzy characteristics etc, but those are further attributes, they do not undermine this core. The core laws of logic in part have their power as they are directly connected to the framework of a possible world. Thus also this actual one we inhabit.

    Similarly, the same structure of a distinct PW exhibits, inescapably, quantitative aspects: A is a unit, ~A is a complex unit (which already shows how the subtle in-common unifies, here: part of the characteristics of W but not A). The two units together show duality. The sharpness of the partition and the exhaustion of what is W implies nullity. From this, we see that structure and quantity are part of the architecture of any distinct possible world.

    Wikipedia on systems architecture, is useful:

    A system architecture or systems architecture is the conceptual model that defines the structure, behavior, and more views of a system.[1] An architecture description is a formal description and representation of a system, organized in a way that supports reasoning about the structures and behaviors of the system.

    A system architecture can consist of system components and the sub-systems developed, that will work together to implement the overall system. There have been efforts to formalize languages to describe system architecture, collectively these are called architecture description languages (ADLs)

    This can be extended to PW’s in the wider context of reality.

    From this also, we may extend N, Z, Q, R, C.

    Other aspects such as topology connect. And in general, we here see the in part intelligible substance of structure and quantity that inhere to all PW’s, as connected to the framework of being a distinct PW. This gives us confidence in exploring that substance through logically guided study. Which, we term the discipline of Mathematics.

    Thus, we see frameworks, in common, distinct aspects, logic, intelligibility to rational creatures, though I do not doubt that our bounded rationality limits our ability to see. Though we see enough to know it is inevitably true that error exists. Being self-aware, that is undeniably true, and more.

    Now, on Plato, an article from Philosophy Now by David Macintosh is a handy start-point:

    https://philosophynow.org/issues/90/Plato_A_Theory_of_Forms

    Plato was influenced by a tradition of scepticism, including the scepticism of his teacher Socrates, who is also the star of Plato’s dialogues. What was obvious to many of the early Greek philosophers was that we live in a world which is not an easy source of true, ie, eternal, unchanging knowledge. The world is constantly undergoing change. The seasons reflect change. Nothing is ever permanent: buildings crumble, people, animals and trees live, and then die. Even the present is deceiving: our senses of sight, touch and taste can let us down from time to time. What looks to be water on the desert horizon is in fact a mirage. Or what I think of as sweet at one time may seem sour the next. Heraclitus, a pre-Socratic philosopher, claimed that we can never step into the same river twice.

    In his Socratic dialogues Plato argues through Socrates that because the material world is changeable it is also unreliable. But Plato also believed that this is not the whole story. Behind this unreliable world of appearances is a world of permanence and reliability. Plato calls this more real (because permanent) world, the world of ‘Forms’ or ‘Ideas’ (eidos/idea in Greek). But what is a Platonic Form or Idea?

    Take for example a perfect triangle, as it might be described by a mathematician. This would be a description of the Form or Idea of (a) Triangle. Plato says such Forms exist in an abstract state but independent of minds in their own realm. Considering this Idea of a perfect triangle, we might also be tempted to take pencil and paper and draw it. Our attempts will of course fall short. Plato would say that peoples’ attempts to recreate the Form will end up being a pale facsimile of the perfect Idea, just as everything in this world is an imperfect representation of its perfect Form. The Idea or Form of a triangle and the drawing we come up with is a way of comparing the perfect and imperfect. How good our drawing is will depend on our ability to recognise the Form of Triangle. Although no one has ever seen a perfect triangle, for Plato this is not a problem. If we can conceive the Idea or Form of a perfect triangle in our mind, then the Idea of Triangle must exist.

    The Forms are not limited to geometry. According to Plato, for any conceivable thing or property there is a corresponding Form, a perfect example of that thing or property. The list is almost inexhaustible. Tree, House, Mountain, Man, Woman, Ship, Cloud, Horse, Dog, Table and Chair, would all be examples of putatively independently-existing abstract perfect Ideas.

    Plato says that true and reliable knowledge rests only with those who can comprehend the true reality behind the world of everyday experience. In order to perceive the world of the Forms, individuals must undergo a difficult education . . . We must be taught to recall this knowledge of the Forms, since it is already present in a person’s mind, due to their soul apparently having been in the world of the Forms before they were born. Someone wanting to do architecture, for example, would be required to recall knowledge of the Forms of Building, House, Brick, Tension, etc. The fact that this person may have absolutely no idea about building design is irrelevant. On this basis, if you can’t recall the necessary knowledge then you’re obviously not suited to be an architect, or a king. Not everyone is suited to be king in the same way as not everyone is suited to mathematics. Conversely, a very high standard in a particular trade suggests knowledge of its Forms. The majority of people cannot be educated about the nature of the Forms because the Forms cannot be discovered through education, only recalled.

    To explain our relationship to the world of the Forms, in the Republic Plato uses the analogy of people who spend their whole lives living in a cave [see Allegory of the Cave]. All they ever see are shadows on the walls created by their campfire. Compared with the reality of the world of the Forms, real physical objects and events are analogous to being only shadows. Plato also takes the opportunity to use the cave analogy as a political statement. Only the people who have the ability to step out into the sunlight and see (recall) the true reality (the Forms) should rule. Clearly Plato was not a fan of Greek democracy. No doubt his aristocratic background and the whims of Athenian politics contributed to his view, especially as the people voted to execute his mentor Socrates.

    Plato leaves no doubt that only special people are fit to rule. Who are the special people who can recognise the Forms? For Plato the answer is straightforward: the ideal ruler is a philosopher-king, because only philosophers have the ability to discern the Forms.

    A lot more comprehensive and subtler than we tend to imagine. I particularly recall the idea of being reminded of the forms as the heart of education and finding it repulsive, not just a matter of disagreement. I think you know that the parable of the cave is connected to my tendency to question dominant narratives and more. By contrast, I suggest that our rationality allows us to explore reality by experience, observation, reflection, introspection, discussion, debate, formation of disciplined bodies of objective knowledge etc, however much residual uncertainty obtains. Where, we then are forced to seek a reasonable faith point, with finitely remote first plausibles that are addressed through comparative difficulties. And Godel haunts my thought at all times.

    In that context, I suggest that seeing the architecture of being and linked logic, including on structure and quantity is part of the disciplined study we need. I know, such can be hard to study and may cut across our habits of thought, sometimes to the point that it seems more a threat than an achievement. Yes, it may require existential crisis to open up to change. But I reject philosopher kings in favour of the humility of wisdom in fear of God.

    I can see God as world-framer who contemplates all things, and so holds the architecture of the world in mind, but that architecture of rationality is also necessary to any world, in part. This is the context in which the concept of God as reason himself is relevant. But that is a separate though related subject.

    Likewise, we may ponder divided and polarised, manipulative masses with scheming ruthless elites vying for the top jobs, the ones with power. If Plato’s linked parable of the ship does not haunt your nightmares, it is because it haunts your waking hours. And yet, we know just how dangerous an elite unified in evil can be. So, we see also that we need enough diversity, freedom and government by reason and truth (with plumb lines) to restrain the suicidal tendencies of societies. Again, a linked but separate subject. One link: to what extent are we under the spell of the shadow-shows? To what extent do the mutineers hold the bridge on the ship of state? Are the competent navigators on the bridge, respected or are they mocked and derided as the mutineers do as they please rather than what is soundly advised. (My pessimism on the path of our civilisation peeks out, I had better pull back a bit. I think, there is some hope but much peril.)

    The focus here is on an embedded organised structure that frames not only the world but entities in it. A structure we can explore and at least in part confidently learn and know. That is, it is significantly intelligible. (And having some exposure to Q-mech, I would not dare to suggest that we may grasp it all.)

    KF

  79. 79
    daveS says:

    Regarding what is “real”, I think many mathematicians believe that as long as an axiom system has not been shown to be inconsistent, then whatever mathematics they produce from that system is “real enough”, that is, worthy of study.

    I posted a link to a paper earlier which shows that the 8000th busy beaver number is independent of ZFC. This naturally leads to the question of what axiom system(s) are strong enough to allow one to (in principle) calculate this number. This is a question in “reverse mathematics”.

    Once you’ve answered that question, then you naturally ask what systems are strong enough to determine the 8001st busy beaver number, the millionth one, etc., farther and farther along the sequence.

    Since any particular axiom system is strong enough to determine only a finite number of busy beaver numbers, you are led to ponder an infinite sequence of stronger and stronger axiom systems.

    Now the busy beaver function is relatively “concrete” and down-to-earth, so this is not empty speculation without practical application. That indicates to me that all these axiom systems in this infinite sequence are just as real as the rest of the mathematics we are talking about.

  80. 80
    hazel says:

    DaveS writes, “Regarding what is “real”, I think many mathematicians believe that as long as an axiom system has not been shown to be inconsistent, then whatever mathematics they produce from that system is “real enough”, that is, worthy of study.”

    Nice sentence.

  81. 81
    kairosfocus says:

    DS & H: axiomatisations set up logic model possible worlds. They in turn are shaped by the constraints of pre-existing core mathematical facts and the state of the discipline. In many cases the core facts are about necessary entities that help to frame any possible world. That also indicates that freshly discovered necessary entities will be present in any world. In any case, within a world one is looking at implications, whereby once a certain p is true, q follows, where p and q describe real or possible states of affairs. BTW, this requires connexions of sufficiency and necessity to be real. KF

  82. 82
    daveS says:

    Thanks, hazel.

  83. 83
    kairosfocus says:

    H, does implication exist? As what? KF

  84. 84
    hazel says:

    re 83: ??? This appears to be a response to kf’s own post at 81, but I’m not sure what he is asking, or why. Perhaps he will explain more???

  85. 85
    kairosfocus says:

    H, what is implication, the logical connective between propositions used to construct theorems? What is its nature, and how does it relate to our conceptions? KF

  86. 86
    hazel says:

    I still don’t understand what you are trying to get at. In logical systems, we use conditionals to create chains of valid reasoning: If a triangle is isosceles, its base angles are equal; in triangle ABC, AB = BC; therefore angle A = angle C. Every element of logic is an abstract concept in our minds, expressed in the physical world in written or verbal form. We’ve been over this many times, so I don’t know why you are bring up conditionals specifically right now.

    In the real world we also use conditionals, but they are subject to the many qualifiers that have to do with empirical knowledge.

    That’s all pretty obvious, isn’t it?

  87. 87
    math guy says:

    In response to 71, 83 & 85, it would appear that Hazel the math teacher is feigning ignorance.

    @71 is the Von Neumann recipe for constructing the integers from the empty set, using recursion and the axiom of specification. The process can be stopped at the first infinite ordinal omega, which is what KF does.

    @83 The answer to “do implications exist?” is: YES.

    @85 The answer to “what is implication?…..what is its nature?” is: AN ABSTRACT OBJECT

  88. 88
    hazel says:

    MG, I’ll note your ungenerous interpretation, and move on.

    You write,

    @71 is the Von Neumann recipe for constructing the integers from the empty set, using recursion and the axiom of specification. The process can be stopped at the first infinite ordinal omega, which is what KF does.

    I know that. But I didn’t know why he posted it. It turns out he was replying to ET’s idea of “collecting numbers”, but it wasn’t obvious that that was what kf was doing. kf has posted that same type of thing countless times before, by the way.

    You write,

    @83 The answer to “do implications exist?” is: YES.

    Yes, but the whole issue in this thread is “how and where” they exist. I didn’t know why he all of a sudden chose implications to ask about, as if that somehow changed the nature of the discussion. Implications exist in logic, and they are used when we talk about the real world. There is nothing new in that that we have discussed across multiple threads and multiple weeks, and maybe months.

    You write,

    @85 The answer to “what is implication?…..what is its nature?” is: AN ABSTRACT OBJECT.

    And at 86 I wrote, “Every element of logic is an abstract concept in our minds, expressed in the physical world in written or verbal form.” The phrase “abstract object” still/again doesn’t address where and how those things exist.

  89. 89
    kairosfocus says:

    H (& MG):

    Let’s roll a tape:

    H, 69 (responding to DS & ET): “I’m not sure how one “collects” numbers . . . ”

    KF, 71, responding to H: “H, cf OP. Start:

    {} –> 0
    {0} –> 1
    {0,1} –> 2

    . . .

    {0,1,2 . . . } –> w, omega.”

    Yes, ET earlier said “My issue comes from the definition of a set: In mathematics, a set is a collection of distinct objects, considered as an object in its own right. Note the bolded word “collection”.” and DS responded “And every collection must have a collector …”

    It is specifically H’s comment at 69 on HOW one collects numbers that drew my focus, and I therefore highlighted that the OP was not silent on the matter and laid out exactly how a relevant collection happens.

    This is connected to the implications of a distinct particular possible world as we ponder what sets it apart from close neighbours. Namely, some attribute A allowing a partition of characteristics:

    W = {A|~A}

    Such then leads to recognising A as a unit, ~A as a complex unit [think: a bunch of grapes], the duality of having these distinct units, and the nullity of a crisp separation so that no x in W is between A and ~A or outside them, any x in W is in A X-OR ~A. Thus, nullity. We see the pattern established, 0, 1, 2 and the principle of distinct units so we can freely continue per von Neumann. The implied endlessness then allows us to recognise the order type of the naturals [even just using an ellipsis for a succession we cannot complete as any k has k+1 etc following] as a transfinite, omega, the first transfinite ordinal. This is now beyond N, there is a qualitative difference.

    So, just from context it seems clear that the meaning of 71 should have been clear.

    While, feigning is a strong word, it does highlight a point of concern on apparent rhetorical strategy. In the real world, intentionally or inadvertently loaded Socratic questions or responses can carry a penumbra of passive aggression. Even in Plat’s dialogues, this is quite clear. A caution we should duly note.

    KF

  90. 90
    kairosfocus says:

    H,

    Your response on implications overlooks a key factor: in implication reasoning, we appeal to the grounds on which the TRUTH of an antecedent p, provides sufficient warrant that a consequent q would then also be TRUE.

    That is, p is sufficient for q and q is necessary for p.

    Thus we reason p => q, p so q or ~q so ~p. Where of course p may be simple or compound, in effect the cumulative result of a chain of reasoning so far, leading to next step q.

    Implication rests on the nature of propositions as assertions that make TRUTH claims. So, a proposition p is either true or else false, antecedent to whether we have evidence or reason to hold it true or false. Truth is ontological, it says of what is that it is, and of what is not that it is not. This is antecedent to and independent of questions as to whether we have good warrant to accept or reject the truth claim p.

    Your rhetorical suggestion that I injected an irrelevant matter, implication is falsified.

    It seems that a key point between us is that you consistently inject the epistemological concern on warrant in a context where the ontological question is prior and focal.

    In that context, implication is an abstract logical relationship between propositions. Proposition p claims that some state of affairs S is the case in reality. Taking that for the moment, it carries with it something else about reality, q, which is necessary for p to be the case, i.e, if p holds, q is also the case. For example, a certain fire exists. For that to be so, it requires fuel, oxidiser, heat and an uninterfered with combustion chain reaction [this last points to how halon extinguishers work]. Once a fire obtains, those elements are in place in combination. To prevent or stop the fire, knocking out any one of these enabling conditions would be enough. As firemen know very well.

    Here, fire implies presence of the combination cluster. Not just as a concept-game but as a matter of the logic of being in the real world.

    We therefore see one way in which an implication is an abstract relationship grounded and even embedded in reality. Not just in our thinking.

    So also, if we reject such abstract structures, or if we isolate abstracta to our heads, if we hold that only concrete entities are real, we undermine our own thoughts. For truth is an abstract relationship that bridges mind and world and implication is a further relationship that bridges truths that are linked together and the real world.

    Indeed in decision making it is reasoning about implications that leads us to decide which p to make into the case on the ground. As the Spartans highlighted, if is a very big word.

    KF

  91. 91
    kairosfocus says:

    MG, prezactly. Implications (often causal) are abstract relationships that do not just obtain in our thoughts but in reality. The idea that reasoning, mathematics etc reduce to a game in our heads so that abstracta obtain only in thought, the real world only has the concrete undermines ability to reason about reality and possible reality. I have therefore repeatedly highlighted how the Mobius strip empirically demonstrates the embedding of abstract realities through the logic of being as a plumb-line test case: contrast an untwisted loop, then what happens when one cuts around in the middle vs 1/3 way in from an edge. The rhetoric of response has been by turns evasive, then dismissive, now it seems oh you keep saying something that is off the table. In short, a direct demonstration that abstract structure and quantity is embedded in physical reality regardless of our concepts is not triggering the re-thinking that is warranted. That is a sign. KF

  92. 92
    daveS says:

    KF,

    oh you keep saying something that is off the table.

    Or ‘tangential’?

  93. 93
    daveS says:

    KF,

    While, feigning is a strong word, it does highlight a point of concern on apparent rhetorical strategy.

    Or perhaps a better explanation is that hazel truly didn’t understand your questions? Just as Math Guy apparently misunderstood hazel’s issue with your post describing the von Neumann ordinals.

    I’ve had that experience many times where someone posts a reply to me and I initially have no idea what they are getting at. Sometimes it’s my fault, sometimes it’s not.

  94. 94
    kairosfocus says:

    DS, the issue is not tangential, it is a plumb-line test case on abstract structure and quantity manifestly, objectively in real space and bodies independent of our thought. Your objection fails. KF

  95. 95
    daveS says:

    KF,

    I was referring to your own use of the word ‘tangential’. That word seems to appear when certain pointed questions which require precise answers are raised.

  96. 96
    kairosfocus says:

    DS, I have spoken of tangents that occur when there is persistent subject switching. Such as this right now. KF

  97. 97
    daveS says:

    OT: progress on the sum of three cubes problem.

  98. 98
    math guy says:

    H @ 88
    In light of DS @ 93, I should grant you benefit of doubt regarding my assumptions about your understanding. I will attempt to do so.

  99. 99
    hazel says:

    Thanks, and thanks to Dave’s for his comment.

  100. 100
    math guy says:

    Recap:
    1) We agree that correct use of classical logic is necessary (although not sufficient) for understanding science, cause & effect, mathematics, and just plain everyday life in the human world. (Roger Bacon taught us that pure logic without empirical evidence is insufficient for an accurate model of physical reality.)

    2) We agree that there are known truths about the physical world: for instance KF’s example of combustion requiring an oxidant among other factors. There are also known truths about the human world, such as “error exists”.

    3) KF argues that implication is manifested in the actual physical universe. The denial of this leads to absurdity. Besides, how did platonists get the idea about universals, like the existence of abstract implication, other than through numerous examples seen in the physical universe? Please note the difference between “manifested in the physical universe” and “exists independent of the physical universe”. The latter assumes the platonist viewpoint, the former merely says that we (the only observers in the neighborhood) recognize instances of implication in the world (like “what goes up must come down”, etc.)

    4) So let us apply 1) to the statement P: “Abstracta exist only within minds”. For simplicity it can be assumed that “minds” mean “human minds” in this context. Some of us believe that P has truth value T, others say F. For some* who claim T, this is a non-negotiable universal truth, else the platonist school of thought has a foot in the doorway, so to speak.
    (Aside: the Wittgenstein/Ayers school of philosophy will dodge the issue by saying that P has no meaning. To my way of thinking, that denies 1). Selective non-use of logical reasoning is rigging the game, putting one’s thumb on the scale, or any other euphemism for cheating.)

    5) Statement P is self-referential: P itself is an abstract proposition. If P is true, then P exists only within minds. Therefore P is not universally true in the physical universe, contradicting the assumption that P is true.

    6) Since P cannot be universally true, it should be modified to a weaker version P’ that avoids the contradiction above. I look forward to logically examining such a P’.

    *In staying consistent with post 88, I am not insinuating that any particular UD commentators take the viewpoint that P is a universal truth. But my aim is to show that such a viewpoint is contradictory.

  101. 101
    hazel says:

    re 100, to MG

    Even though I know you don’t have time for very many, or frequent, posts, I’ll comment

    Points 1-3 have been discussed at length in previous threads. Point 3 is particular is the heart of the matter in respect to my thoughts about abstraction, but I’m not going to repeat things I’ve said multiple times about that.

    I don’t understand point 4. First, I don’t know what “non-negotiable” means here. What is even being negotiated? My belief that abstract concepts exists only in human minds is partially based on the evidence I see, and partially based on some philosophical perspectives related to point 3. Some people strongly disagree with me about some of of this, and some people seem to agree. We are sharing thoughts, but I don’t see how we are negotiating anything.

    My beliefs are not set in stone, and unlike some* (I’ll name no names 🙂 ), I don’t at all believe that my views have some special warrant to truth. I may be wrong, although as I have said many times, I think the “real” nature of the reality of both the world and minds is probably forever beyond our comprehension.

    I also don’t know what you mean by a “universal proposition” in this regard. I have pointed out that “abstract concepts exist only in human minds” is not a purely logical proposition, but rather a belief about the world and human beings, with all the attendant possible disclaimers and provisions that go along with such beliefs. It includes various concepts, including the concept of abstract concepts themselves, as well as “exists” and “human minds”, all of which are full of problematic and unknown elements. To say P must be either true or false is to treat it as purely logical proposition, which it is not.

    The two other alternatives that have been offered are a Platonic one, in which abstractions exist in some non-material realm independent from human beings, or one in which abstractions exist embedded in the physical world. I see problems with both those views, which I’ve been discussing for several months, so on balance I’m currently thinking that the belief I’ve been stating makes the most sense to me.

    YMMV

  102. 102
    daveS says:

    MG & hazel,

    Regarding (5), I wouldn’t expect to see implication (a relation between abstract entities) in the physical universe. Causation, yes.

  103. 103
    hazel says:

    I agree, Dave. This is the main point for me. Yes, we live in a world that is causally connected, but we create abstract propositions that “if this then” that by generalizing from what we observe. Conditional propositions are descriptions of the world, but they themselves are not in the world: they are not part of causality itself.

  104. 104
    math guy says:

    “Non-negotiable” is a euphemism for “absolute” and was not intended to imply actual negotiations.

    Regarding point 3), how do we generalize implication to the formal P —> Q if not from real world examples? If I toss a rock into the air, then it returns to the surface. Is this not a special case of implication? Historians of science note that application of logic to understanding and predicting physical phenomena did not really work in ancient China, say, because of philosophical beliefs regarding the chaos of nature. Modern science depends on the assumptions of underlying laws that have logical consequences. Point 3) does not assume KF’s posts about logic being a foundation of underlying reality (although it is consistent with KF’s posts).

    ….I have pointed out that “abstract concepts exist only in human minds” is not a purely logical proposition….

    On the contrary, P (as stated in quotation marks) is in fact a logical proposition by definition since there is an actual T/F assignment to it.

    It seems that what you are actually proposing as your belief is an unstated variant P’ that contains quantifiers regarding minds and abstracta. (Please don’t try to redefine “exists”. Recall that all formal systems have undefined terms as their starting points. Arguing that something depends on the meaning of “is” doesn’t even get out the starting gate.)

    If P’ seems too complicated to express exactly, perhaps use some universal quantifiers.

  105. 105
    hazel says:

    editing …

  106. 106
    hazel says:

    MG our posts crossed in the mail, so to speak.

    You write, “Regarding point 3), how do we generalize implication to the formal P —> Q if not from real world examples?”

    That is what we do: we observe things happen and then generalize them into abstractions: that is what I said in 103.

    You write, ” If I toss a rock into the air, then it returns to the surface. Is this not a special case of implication?”

    No, there is an event in the world, which is one of many which leads to an abstract conclusion “If I throw …, etc.”, but there is no actual abstraction conditional proposition out there when the thrown object goes up and down.

    As I discussed when we were discussing monkeys (I have no idea whether you followed that or if it meant anything to you), we can’t talk about, or even conceive of, the world without using our abstract terms and models, so it is easy to believe that our abstractions have a “life of their own” in the world, so to speak. However, my belief (and I have had some interesting psychological experiences in regards to this) is that it is possible to experience the world with a direct immediacy that clearly separates that existence of the world as it is from our thinking about the world.

    P.S. I see that by non-negotiable you meant absolute. I think I’ve already addressed the fact that I am not claiming absolute certainty about my beliefs, nor claiming that they have special claim to deductive truth. See post 101 for more on that.

  107. 107
    kairosfocus says:

    MG (& attn H and DS):

    Yesterday was an emergency, given regional legal developments involving a Chief Justice who set out to single-handedly amend a Constitution from the judge’s bench. So, pardon effective absence and focus elsewhere.

    MG, your argument is correct in the main, though I would not use the what goes up must come down case, as there is such a thing as escape velocity and the onward premise that in principle one could climb a ladder to the moon. Escape velocity is a threshold whereby “all at once” an object would have enough kinetic energy to escape a gravitational potential well. There is a reason why Copi and others use combustion as the classic empirical example of causality, it demonstrates how a cluster of enabling conditions are each necessary and are together jointly sufficient for an effect to occur. This is thus a key example or case study on implication logic and its link to causality. Which, of course, was challenged by Hume et al. (Itself a warning.)

    We also observe in reasoning, that there is a general condition of implication, whereby p is sufficient for q, and q necessary for p; in key part captured by the material implication truth table. For simple example, if Tom is a cat, then Tom is an animal. It is sufficient for Tom’s animality that he is a cat, and animality is necessary for Tom to be a cat. However as there are other ways to be an animal, animality is not sufficient for Tom to be a cat, he could be a boy or a pig. That is, implication is not equivalence, that requires double-implication as a further component. (However, [p => q AND q => p] is not yet sufficient to establish identity of p and q, we need to move beyond mutual presence to identity, i.e. there are no distinctives so p and q are different labels for the same thing.) And notice the reflexivity here, I used implication in discussing implication.

    Implication is in fact yet another inescapable truth of reasoning, as H would discover were she to draw out the chain by which she has yet again injected the epistemology of potential error — yet another inference turning on implication logic — into a matter of logic of being and first principles of reason which are antecedent to our problems with knowing what is so. Let’s observe:

    [H, 106:] I see that by non-negotiable you meant absolute. I think I’ve already addressed the fact that I am not claiming absolute certainty about my beliefs, nor claiming that they have special claim to deductive truth. See post 101 for more on that . . . . [101:] I don’t at all believe that my views have some special warrant to truth. I may be wrong, although as I have said many times, I think the “real” nature of the reality of both the world and minds is probably forever beyond our comprehension.

    I also don’t know what you mean by a “universal proposition” in this regard. I have pointed out that “abstract concepts exist only in human minds” is not a purely logical proposition, but rather a belief about the world and human beings, with all the attendant possible disclaimers and provisions that go along with such beliefs. It includes various concepts, including the concept of abstract concepts themselves, as well as “exists” and “human minds”, all of which are full of problematic and unknown elements. To say P must be either true or false is to treat it as purely logical proposition, which it is not.

    Notice, the chaining of argument and implicit propositional claims(such as the potential for error) as well as explicit ones?

    As a Mathematical practitioner and/or educator, H knows that reasoning from givens or axioms and established results requires chained implications. She knows that such implications are inextricably embedded in Mathematics and that such mathematics is often manifestly relevant to the real world (including of course the Mobius strip snip around examples I have so often referenced to hammer home a plumb-line test case). She knows that this extends to reasoning in general, she must know the material implication truth table and its application to real-world examples such as a fire, where the Chemistry of combustion chain reactions now undergirds the longstanding empirical facts that are of such great importance to civilisation.

    How does she handle this, let’s see:

    [H, 106:] we observe things happen and then generalize them into abstractions: that is what I said in 103.

    You write, ” If I toss a rock into the air, then it returns to the surface. Is this not a special case of implication?”

    No, there is an event in the world, which is one of many which leads to an abstract conclusion “If I throw …, etc.”, but there is no actual abstraction conditional proposition out there when the thrown object goes up and down.

    Of course, inductive inference is not merely by blind psychological association, but pivots on awareness of cause as a case of implication. Similarly, we infer on tested reliable signs [such as Hippocrates’ facies of death] precisely because we are aware of causal connexions and associated logic of implication. This is part of the inescapable first principles of right reason. That is, our concept-forming abstraction on seeing a fire or (absent awareness of escape velocity etc) of projected rocks falling etc is due to a prior logical structure. But if such is kept implicit and hedged about with the epistemology of potential error, especially on induction, it can be evaded.

    That evasiveness must cease.

    Do we or do we knot rely on logically complelling inference from premises to conclusions, not only in Mathematics but generally? Are we or are we not able to escape using such, i.e. starting from givens p then drawing out consequents q?

    I put it, that we cannot escape such in reasoning, life and disciplined study, in general.

    Thus, we are looking at an inescapably true first principle of right reason connected to the logic of being. Cat-ness entails animality as part of identity and core characteristics, so if Tom is a cat, that is sufficient to establish animality; which can be independently recognised. Likewise, man-hood implies mortality, so id Socrates is a man, it is only a reminder that men are mortal which then leads to closing the syllogism, Socrates is mortal. Similarly, given addition as clustering and partition as subtraction, we observe both:

    ||||| –> || + |||

    AND

    || + ||| –> |||||

    That is if we partition a five set by separation of a pair, we are left with a trio, necessarily based on what cardinality of a five set means. Similarly, clustering a two-set with a three set yields a five set. These are implications, simple and self-evident ones.

    We know the inescapability of implication, to the point of its being a self-evident, undeniable truth that is just as certain as that error exists. Indeed, let us use E to symbolise this proposition. Given that this is a proposition — an assertion that is either true or else false [notice, this too has been obscured under a fog of epistemological doubts] — we can freely contrast ~E. But we see — by direct implication — that ~E has a meaning: it is an error to assert that error exists. So, patently, E is undeniably so as its attempted denial immediately implies its truth. And yes, one may argue about a self-evident truth even though its truth is primarily seen based on understanding what the claim means i/l/o sufficient experience and insight to recognise its force. In particular, such arguments typically draw out the absurdity of the attempted denial.

    This SET is of course directly relevant to H’s persistent objection on the possibility of error by way of mismatch between our abstractions and the world of things in themselves. The very claim that error exists is itself a certain truth, undeniably true and thus bridging our inner world of contemplated abstracta and the external world of an ordered, intelligible in part cosmos. Yes, we may and do err but even this is an illustration that often we do know the truth, sometimes to incorrigible certainty. But of course yet another evasive problem is, what is truth.

    Though it has been challenged as unhelpful, that perception seems to be an error regarding Aristotle: truth says of what is, that it is; and of what is not, that it is not. Truth accurately describes reality, things in themselves, on particular points and aspirationally on the whole. Where, on logic turning on inescapably true first principles of reason, observation and insight, we may so warrant certain truth-claims [that is, propositions] that they are credibly deemed knowledge. This being warranted, credibly true (and so, reliable) belief. And yes, there is room for provisionality in this definition, though certain truths can be known to certainty.

    Maybe, Locke has something to teach us, if we will listen:

    [Essay on Human Understanding, Intro, Sec 5:] Men have reason to be well satisfied with what God hath thought fit for them, since he hath given them (as St. Peter says [NB: i.e. 2 Pet 1:2 – 4]) pana pros zoen kaieusebeian, whatsoever is necessary for the conveniences of life and information of virtue; and has put within the reach of their discovery, the comfortable provision for this life, and the way that leads to a better. How short soever their knowledge may come of an universal or perfect comprehension of whatsoever is, it yet secures their great concernments [Prov 1: 1 – 7], that they have light enough to lead them to the knowledge of their Maker, and the sight of their own duties [cf Rom 1 – 2, Ac 17, etc, etc]. Men may find matter sufficient to busy their heads, and employ their hands with variety, delight, and satisfaction, if they will not boldly quarrel with their own constitution, and throw away the blessings their hands are filled with, because they are not big enough to grasp everything . . . It will be no excuse to an idle and untoward servant [Matt 24:42 – 51], who would not attend his business by candle light, to plead that he had not broad sunshine. The Candle that is set up in us [Prov 20:27] shines bright enough for all our purposes . . . If we will disbelieve everything, because we cannot certainly know all things, we shall do muchwhat as wisely as he who would not use his legs, but sit still and perish, because he had no wings to fly.

    KF

  108. 108
    kairosfocus says:

    H, kindly observe 107. Neither you nor I nor anyone else can escape implicit reliance on implication, even to infer oh errors exist, I am error prone so in my thought I may err so I hedge around my conclusions with that possibility and build a scheme of thought on it. By contrast, it is a plumb-line truth that Error exists, an undeniably certain and self evident truth. Thus, not all truths are prone to error, there are specific, necessary truths that are inescapably true, relatively few but vital, truths that apply to reality and are not just locked up inside the circle of our error-prone minds; indeed, truths that express logic of being, here being an unavoidable part of the nature of causes and effects. This includes the logic of implication which governs inference and particularly Mathematics. Which, is an application of the principle that to prove that logic is necessary or valid requires the use of logic, i.e. it is inescapable. KF

  109. 109
    kairosfocus says:

    H, is it just in our heads that we hold an abstract notion that men exist as a class and that being human entails mortality as a key characteristic? Is it just an abstract notion with no reliable reference to reality that Socrates of Athens is a man? Is it just a mere matter of a logic-game in our heads with no reliable reference to reality that Socrates is therefore mortal and thus predictably prone to the effects of judicially imposed Hemlock in an “appropriate” dose? Is it just an abstraction that the executioner said, no one may not pour out a libation from the fatal dose in the cup of Hemlock, as it is exact? Was it just a coincidence unconnected to real-world causality that on drinking said cup of Hemlock, Socrates proceeded to die under sentence of the Areopagus? KF

  110. 110
    daveS says:

    MG, KF, hazel,

    The more I look at this argument, the more I am convinced it has serious problems:

    5) Statement P is self-referential: P itself is an abstract proposition. If P is true, then P exists only within minds. Therefore P is not universally true in the physical universe, contradicting the assumption that P is true.

    Let me try varying it slightly. Let P be the proposition “Propositions do not exist in the physical universe”. Surely P is true, right? Unless someone here has converted to physicalism.

    I don’t know what to make of the next step, but I presume MG would say that “P is not universally true in the physical universe” just as before. Hence we conclude that P is false. That is, some propositions do exist in the physical universe, which is absurd.

    Therefore I don’t think this argument succeeds.

  111. 111
    hazel says:

    Kf writes, “Implication is in fact yet another inescapable truth of reasoning, as H would discover ….”

    True. I accept logical reasoning. I can’t imagine why you think I don’t.

    No sense in continuing to respond to kf’s repetition of what I see as a confusion between what the world is and what we think about the world.

    I will note that he repeats the 2 + 3 = 5 example, which I addressed a while back in thinking about a monkey observing the five pebbles.

    But one last example: he writes,

    H, is it just in our heads that we hold an abstract notion that men exist as a class and that being human entails mortality as a key characteristic? Is it just an abstract notion with no reliable reference to reality that Socrates of Athens is a man?

    The answer to the first question is “yes”: the abstract notion that men exist as a class and that we have observed and generalized that all men die exists only in our head minds.

    The answer to the second question is of course our abstract notions have reference to reality. The fact that kf continues to make statements like that shows that he does not understand my position.

    I know he and I disagree. What is frustrating is that he doesn’t understand what we disagree about, and continually attributes things to me that I don’t believe.

  112. 112
    kairosfocus says:

    DS, propositions do not exist materially (they are not equal to glyphs in a chain or brain wiring etc), but if true the substance they refer to can be and is manifest in reality. Socrates is a man is not the glyphs nor brain electrochemistry but a truth, a reality accurately described, which may then be logically contemplated. KF

  113. 113
    daveS says:

    KF, agreed.

  114. 114
    kairosfocus says:

    H, implication is a relationship, which reflects a logic of being or even potential being. If it is so that Socrates is a man . . . member of a class — abstract relationship, and that men are mortal — characteristic of members of said class [another abstract entity], then it is implied — abstract logical relationship — Socrates (as a man) is a mortal. My point is, that you cannot escape using implication, regardless of whether or not you hold opinions pro or con on the matter. This is a proper first principle of right reason. Note, appeal to dismissed claim does not undermine its point, and is here incorrect: I here point out that partition and union of sets of discrete units is connected to and an implication of their cardinality. That is, implication is here embedded in reality as an abstract relationship and was so before there ever were humans to name and contemplate it. You also fail to see that the reality of man-ness, of mortality of members of that class and the implied conclusion that Socrates — a man — is antecedent to our contemplations. Men have a common substantial nature with characteristics including mortality, which carries with it the implication on logic of being that a particular man is mortal. Substantial reality that we discover and contemplate, through the power of truth, not just notions swirling around in our heads that we invent or may not invent. KF

  115. 115
    hazel says:

    Another pointless comment: kf writes, “My point is, that you cannot escape using implication, ”

    OF COURSE WE CAN’T. HUMAN BEINGS HAVE TO USE THE ELEMENTS OF LOGIC TO THINK, TALK, AND DESCRIBE THE WORLD.

    Maybe shouting will help? 🙂

    Another pointless comment: “Substantial reality that we discover and contemplate, through the power of truth, not just notions swirling around in our heads that we invent or may not invent.”

    YES, REALITY IS SUBSTANTIAL, AND YES WE DISCOVER THINGS ABOUT IT. WE DON”T JUST MAKE STUFF UP!

    The things we discover we express with abstractions and logic. The world is intelligible, and we are intelligent, so we create abstractions which can, to various degrees, accurately model the reality we observe. (I got tired of typing in all caps.)

  116. 116
    daveS says:

    hazel,

    Maybe shouting will help?

    It’s worth a try, I guess. On the other hand, this.
    🙂

  117. 117
    hazel says:

    Thanks, Dave. I’ll hyper down. 🙂

  118. 118
    daveS says:

    hazel,

    Just to be clear, my post was not intended to dissuade you from “shouting” or anything else. Just a humorous commentary on our situation.

  119. 119
    hazel says:

    I understand completely, Dave. I appreciate lightening the atmosphere.

  120. 120
    StephenB says:

    Hazel

    “yes”: the abstract notion that men exist as a class and that we have observed and generalized that all men die exists only in our minds.

    This is an error and I think it is the basis for your misunderstanding with KF. If an abstract reality, such as the Pythagorean Theorem, can be discovered, it must exist first as an extra-mental reality in order to be discovered. For some reason, you refuse to address this issue.

    You have two choices: Either you think that abstract realities exist only in the mind and are therefore, “made up,” or you think that they exist as abstract realities outside you mind and are, therefore, “discovered.” You cannot have it both ways.

    The things we discover we express with abstractions and logic. The world is intelligible, and we are intelligent, so we create abstractions which can, to various degrees, accurately model the reality we observe.

    Forming models of the world can be useful in some cases, but they have nothing to do the way we know things at the most basic level. I do not need a model of the world in order to know that 2+2 = 4, or that evil exists, or that a dogs nature is different from that of a cat. I did not “create” these abstractions, I “discovered them.”

  121. 121
    hazel says:

    StephenB writes, “If an abstract reality, such as the Pythagorean Theorem, can be discovered, it must exist first as an extra-mental reality in order to be discovered.”

    No. I have addressed that multiple times, but you have to read a bit to find the relevant portions of the threads.

    StephenB writes, “Either you think that abstract realities exist only in the mind and are therefore, “made up,” or you think that they exist as abstract realities outside you mind and are, therefore, “discovered.” You cannot have it both ways.

    You have created a dichotomy where one half is a strawman: I don’t think that abstract realities are just “made up.”

    And you write, “I do not need a model of the world in order to know that 2+2 = 4.”

    Yes, in fact you do use a model of the world to know that 2 + 2 = 4, because the abstractions about quantity, summation, and equality are in fact abstractions in your mind. Without the abstractions, you would be like the monkey I discussed a ways back.

  122. 122
    StephenB says:

    SB: If an abstract reality, such as the Pythagorean Theorem, can be discovered, it must exist first as an extra-mental reality in order to be discovered.”

    Hazel

    No. I have addressed that multiple times, but you have to read a bit to find the relevant portions of the threads.

    If you had covered this ground before, you would likely have provided evidence of that fact rather than ask me to chase it down. It would be more honest for you to say that you have no answer to my challenge rather than to pretend that you have already addressed it.

    SB: Either you think that abstract realities exist only in the mind and are therefore, “made up,” or you think that they exist as abstract realities outside you mind and are, therefore, “discovered.” You cannot have it both ways.

    You have created a dichotomy where one half is a strawman: I don’t think that abstract realities are just “made up.”

    You appear not to understand the implications of your own philosophy. If, as you claim, all abstract realities exist in the mind, and no abstract realities exist outside the mind, it also means that all abstract realities originate in the mind, which is the same as saying that they are made up and are not discovered.

    In truth, abstract realities must exist outside the mind in order to be discovered and understood by the mind. If you discover or apprehend the Pythagorean Theorem, for example, it can only be because its truth as an extra-mental reality preceded your discovery of it. In spite of your claims to the contrary, you have not addressed this point.

  123. 123
    hazel says:

    I wasn’t pretending anything, Stephen. There have been multiple threads with extended discussions about the nature of mathematical truths, and I don’t think you have been a part of them. I don’t want to go track them down, and I don’t want to start a new conversation with you about them.

  124. 124
    StephenB says:

    Hazel

    I wasn’t pretending anything, Stephen.

    Yes, you were. You were pretending to have already addressed a point that you didn’t really address in an attempt to evade my argument @122.

    There have been multiple threads with extended discussions about the nature of mathematical truths, and I don’t think you have been a part of them.

    This thread is about extra-mental abstract realities, which you claim do not exist, and their correspondence to our mental formulations about them (mental abstract realities). What you had to say about mathematics on other threads is irrelevant to the philosophical errors that you are making now.

    I don’t want to go track them down, and I don’t want to start a new conversation with you about them.

    Fine, I don’t want to go back and track them down either. So stay with the present and don’t allude to your unspecified past comments on other threads – which cannot be verified – as evidence for something that never happened.

  125. 125
    StephenB says:

    After rereading my comments at 120 and 124, I have decided that I went too far with my critical comments to Hazel and I also failed to nail down the definitions of terms that I was using.

    Under the circumstances, it would be impossible for readers (including Hazel) to fully comprehend what I was saying, possibly because I didn’t differentiate, among other things, between universals (things that individual things have in common, such as class) and abstract realities, such as non material existence.

    When I return next, I will begin by defining my terms so that everyone can understand my arguments and can discern what I mean and what I don’t mean. As it is, I don’t think my arguments are fully comprehensible and I hold myself, and no one else, responsible for that situation.

  126. 126
    kairosfocus says:

    H,

    I am busy elsewhere so will make only a brief remark.

    It is quite clear to me that by virtue of refusal to acknowledge and accept implications and import of the significance of things like actually seeing what happens when say a Mobius strip is snipped around in the middle vs 1/3 way from its edge, you are in the position of refusing to recognise evidence that there is embedded in reality — not just our minds — a pattern of intelligible (in part) abstract principles that help to structure the world and its contents.

    Things, which are antecedent to us and our thoughts.

    They include that various things have distinct natures pivoting on core characteristics that must be compossible — a square circle cannot exist precisely as core characteristics are incoherent. This manifests the logic of being. So, we see that squares and circles are distinct, that men and dogs and mountains and stars are distinct. Yet, we have things in common, i.e. there is a blend of in-common characteristics and in-difference characteristics in the nature of an entity.

    Likewise, there are intelligible abstract principles that are embedded in reality, e.g. a suggested being is only possible if its core characteristics are mutually coherent — proof that there is a logic of being. Similarly (and often connected), distinct things must have distinct characteristics, up to possible worlds . . . otherwise we have different labels for the same thing.

    Likewise again, implication is an intelligible abstract relationship embedded in the world, often tied to the nature and identity of beings and classes: men are mortal so as Socrates of Athens is a man, he will be mortal.

    Likewise yet again, as a man, the in-common nature to other men implied that there was a particular minimum dose of Hemlock required for his execution as is revealed in the dialogue on his death. This manifests one form of implication, cause-effect links. Such can be sufficiently known that we may reliably use them, as we do when we cross the road and avoid getting knocked down by a vehicle.

    Similarly, being a right angle triangle implies that the other two vertices must have complementary angles leading to a specific relationship between the sine and cosine function — which are ways we summarise ratios of sides relative to a vertex of interest and its angle. Tied, is the Pythagorean relationship on squares on sides. There are many other cases which we discover rather than invent.

    Underlying all of this is the abstract nature of propositions as assertions as to what is or is not the case in reality and their capability to bridge our inner world of thought to reality by being either true or else false.

    A true proposition asserts an accurate description of some facet of reality, both the proposition and its truthfulness being abstract.

    Further to this certain propositions imply certain others, and if the first are true the second will be true.

    The denial or dismissal or reinterpretation as an inner mind exercise of intelligible — so discoverable — abstracta embedded in reality through the logic of being (and which are antecedent to as well as independent of our concepts and contemplations but may be objectively known by us) fails.

    KF

  127. 127
    Eugen says:

    Hazel

    “I presented my arguments elsewhere…”

    Hic Rhodos, hic salta

  128. 128
    hazel says:

    I don’t want to take the time to do that for people who have not been invested in the ongoing conversation, which has been going on for a number of months.

  129. 129
    Brother Brian says:

    For the record, I followed Hazel’s previous discussions about this and, occasionally made a comment. With the caveat that I am far from being a mathematical expert, I believe that Hazel’s arguments are far more compelling than those offered by KF.

  130. 130
    kairosfocus says:

    BB, they will be to you. The problem is, there are worldview commitments involved and a little matter of self-referential incoherence and direct counter example. On the latter, kindly tell us whether you have made an ordinary paper loop and two Mobius loops. Then, did you cut around the first two in the middle and the other M-strip 1/3 way in from an edge? Kindly explain how the difference in outcome does not demonstrate world embedded substance and properties of structure and quantity independent of our concepts etc: ______ There are of course many other mathematical cases and a world of manifest, observable lawlike structural and quantitative order studied in physics and other sciences. Explain to us, then, how empirically observed laws and constants of nature reduce to our [uncertain] concepts: ____ KF

  131. 131
    kairosfocus says:

    PS: Cf here https://uncommondescent.com/philosophy/logic-spaghetti-who-created-god/#comment-675056 in answer to 3 there, as a direct illustration of my worldviews concerns as just noted.

  132. 132
    daveS says:

    Is it true that all abstract objects which are logically possible actually do exist independently of our minds?

  133. 133
    hazel says:

    Dave, that is the thought that started to lead me away from Platonism.

  134. 134
    Brother Brian says:

    KF

    On the latter, kindly tell us whether you have made an ordinary paper loop and two Mobius loops. Then, did you cut around the first two in the middle and the other M-strip 1/3 way in from an edge? Kindly explain how the difference in outcome does not demonstrate world embedded substance and properties of structure and quantity independent of our concepts etc: ______

    We are all familiar with Möbius strips. Why you continue to raise them as some sort of proof to your point is what is not understood. Unless you can explain why this is proof of the inherent existance of mathematics outside of the human mind, all you have is s parlor trick.

  135. 135
    hazel says:

    Agreed, BB. No one here denies that the physical world behaves as it does, whether it be drawing a circle in the sand with a string and a stick, as the Greeks did when they first started formalizing geometry, or cutting strips of looped paper. The issue is whether the abstract generalized mathematical descriptions of the world that exist in our minds, and which are expressed in our symbol systems, also exist anywhere else. The world is what it is, as a concrete reality*, and we describe it with abstractions.

    Continually pointing to the Mobius strip adds nothing to any argument about the nature of abstractions.

    *And of course I know the QM nature of the world shows that there is nothing “concrete” about it, but the point still stands that whatever it is, it is each and every moment of what it is, not an abstraction of our about anything.

  136. 136
    StephenB says:

    Hazel

    The issue is whether the abstract generalized mathematical descriptions of the world that exist in our minds, and which are expressed in our symbol systems, also exist anywhere else. The world is what it is, as a concrete reality*, and we describe it with abstractions.

    You are taking a great many things for granted here. What is the origin of these concepts? Are they derived from some extra-mental source, such as the object of sense experience, or do they simply form in the mind? Does our knowledge begin with sense experience, or does it begin with mental models? Can our senses be trusted to inform us about extra-mental reality, or do they mislead us in serious ways? You are silent on all these critical issues.

    I say that the issue is the *source* of those mathematical descriptions and abstractions, which do not exist in our minds. What, for example, is the source of the Pythagorean Theorem and in what way was it discovered? Did it exist before it was discovered? What is the pathway from our experience of the concrete world to the conceptual representation of it in our minds, which tells us what a thing *is,* including its properties and qualities, as opposed to a mere mathematical *description* of it.

    Abstract realities are more about conceptual *representations” of concrete reality and their origins than they are about mathematical *descriptions.* Yet you emphasize the latter and completely ignore the former. KF’s original question was about abstract entities (“abstracta”) such as [numbers, natures, truth etc real?] In your attempt to narrow the discussion unreasonably, you define the problem solely in terms of mathematics, and even there, you don’t address the truly important questions, as I have already indicated.

  137. 137
    kairosfocus says:

    BB, familiarity with M-strips does not equate to adequately addressing the embedded structure and quantity that they manifest independent of our concepts. Where — on fair comment, the rhetoric of, we have evaded or ignored or dismissed several times so how dare you raise it again, fails. Fails the plumb-line test case challenge . . . I can safely say that you have never addressed cogently the implications of having one edge and of how depending on where one cuts [already a quantitative, spatial structural matter] one sees two edges or three edges emerging (the latter for two interconnected loops, one a narrower m-strip). One, two and three at work, embedded in a body in space, antecedent to and independent of our minds and concepts, labels or contemplations and calculations . . . which, here can and do accurately and reliably refer to extra-mental realities, demonstrating another abstract entity, the relationship we term truth. Indeed, implication also lurks, where we cut around and cylinder vs m-strip imply the result that will follow. Embedded structure and quantity in bodies in space, showing abstracta at work through the logic of being. Similarly — and as I noted in another thread — in any distinct possible world there will be two-sets and three sets that clustered yield five-sets, an abstract framework relationship showing how numbers here are necessary abstract beings in any possible world and how certain necessary relationships obtain independent of our thought about them. || + ||| –> ||||| never began, cannot cease, is necessarily true everywhere at all times and is intimately connected to entities in any world. This is part of the logic of being of any world. KF

  138. 138
    hazel says:

    Stephen you write, “You are silent on all these critical issues.”

    You are expecting an unreasonable amount of content for one short forum post. We have been discussing all those issues, but it would take a multitude of posts, or a substantial essay, to address them all.

  139. 139
    hazel says:

    Stephen, above Dave asked a question,

    Is it true that all abstract [mathematical] objects which are logically possible actually do exist independently of our minds?

    (Note that I added the word “mathematical” to narrow the scope of the question, which may or may not have been Dave’s intent.)

    What is your answer to this question?

  140. 140
    daveS says:

    hazel,

    Dave, that is the thought that started to lead me away from Platonism.

    Yes, it seems quite extravagant to insist that all possible abstract objects do exist. And once you are committed to the existence of *some* abstract objects, I don’t know what the rationale would be for not accepting the existence of all of them.

  141. 141
    kairosfocus says:

    DS, a possible entity would exist in at least one world, were it instantiated. That is what possible means. Contingent entities would occur in some worlds but not others (connected to cause), necessary ones are part of the fabric of any possible world and would exist in all worlds. These include for example numbers. When we turn to propositions, let us note that a possible world is a sufficient set of propositions that specify a given world. So, in effect if a state of affairs (a world) is possible, or actual, it can be described accurately. Where also any proposition carries with it as a shadow its denial. KF

  142. 142
    daveS says:

    PS: At the end of the last sentence in #140: All logically possible abstract objects, that is.

  143. 143
    hazel says:

    Dave, this pertains to your question, and it is something for Stephen to consider if he decides to answer your question.

    John Conway’s Game of Life

    John Conway’s Game of Life, henceforth Life, is not a game: it is a example of a type of mathematical system called a cellular automaton. John Conway invented it about 1970. There are other similar systems, and the whole field of iteration from a beginning configuration is an important topic in mathematics.

    Life is played on an infinite grid, in which each cell is either on (alive) or off (dead). You start with a beginning configuration of live cells, which is generation 0. Then a set of rules are applied to each cell, so that some live cells die and some dead cells come alive, creating the next generation. This process is continued forever, although in practice most configurations come to some kind of end, dying off completely or stabilizing in a static pattern or one that oscillates regularly. However, some patterns create figures that move off into space, so to speak, and continue for an infinite number of generations.

    There is a great simulation of Life, along with a description of the rules here. If you are not familiar with Life, I encourage you to go there and play a bit. There is small dropdown menu to pick a few interesting configurations to start with, or just make up your own. (See especially the Gospel Glider Gun!) You can also read about Life at Wikipedia.

    Now here are some points:

    1. This is an axiomatic mathematical system. Once a beginning configuration is chosen, the succeeding generations are logically determined.

    2. There is no algorithm for figuring out what the state of a particular generation will be without just stepping through the intervening generations. There is no way to discover (note that word) the state of a particular generation without moving through the logical chain of preceding generations.

    3. There are an infinite number of possible logical situations, as there is an infinite plane upon which beginning configurations can be set, and at least some configurations continue indefinitely.

    And some philosophical questions and comments:

    4. In what way, if any, would we say that every one of the infinite number of logical possibilities in Life exist?

    5. Does every logical possibility exist outside of and independently of the human minds which have devised and expressed this system using symbols and abstract concepts?

    6.If the answer to 5 is “Yes”, have they existed eternally even though the game wasn’t invented until 1970?

  144. 144
    daveS says:

    KF,

    Is your answer to my question “yes”?

  145. 145
    daveS says:

    hazel,

    Coincidentally, while on a walk about an hour ago, I was pondering very similar questions about the games Go and chess. Conway’s game of life is probably better because it has infinitely many states. I haven’t made much progress on those questions though.

    It does occur to me that if abstract objects “exist” if and only if they are logically possible, then perhaps that’s a very weak notion of existence.

  146. 146
    hazel says:

    Totally off-topic: Go is a fascinating game. I haven’t played for years, but enjoyed playing it in college many years ago, and then taught my son to play. He bought me a nice Japanese set one time, which is very aesthetically pleasing.

  147. 147
    math guy says:

    DS @ 132 and 145
    The philosophy that all logical possibilities exist is called “full-blooded Platonism”.

  148. 148
    math guy says:

    DS @ 110
    Let P denote “Propositions do not exist in the physical universe”. KF might argue P is false based on necessary conditions for any universe, but let us assume it is true. Then applying this to the specific proposition P, implies that P does not exist in the physical universe. I fail to see any contradiction. Surely you are not going to equate the sentence written above in English that symbolizes P with the abstract content/meaning of P as a logical proposition? That is a category error similar to equating the numeral 5 with the abstract idea of cardinality of five.

  149. 149
    StephenB says:

    Dave

    Is it true that all abstract [mathematical] objects which are logically possible actually do exist independently of our minds?

    Hazel injected the word “mathematical” into your definition, but I will assume that you meant exactly what you said [abstract objects] and not what she said you meant [abstract mathematical objects]. Even so, it is probably not a deal breaker either way.

    I suspect that a Platonist would say yes to your question. However, I am an Aristotelian, so I would put it this way: Abstract objects that are logically possible, but do not exist, exist potentially; abstract objects that do, in fact, exist, exist actually.

    The second order question about *where* the abstract object can exist, (inside or outside the mind) is a different matter and can be answered only on condition that you (or someone) rigorously define the meaning of an “abstract object” or “abstract mathematical? object.” Mathematical models are of no help in that context. I don’t think that your broader question can be answered definitively with the limited information that you and Hazel are providing.

    One solution would be for Hazel to answer my questions, or for you to answer them in her name, so that a true dialogue could emerge. So far, no effort has been put forth in that direction.

    Meanwhile, I will provide a partial answer: Clearly, there are extra-mental, non-material, truths about the nature of reality that can be identified and represented through mental concepts. Mathematics can describe them, but it cannot represent them. Only a sound philosophical approach can do that. This, I think, is what KF has in mind.

  150. 150
    Brother Brian says:

    KF

    in any distinct possible world there will be two-sets and three sets that clustered yield five-sets, an abstract framework relationship showing how numbers here are necessary abstract beings in any possible world and how certain necessary relationships obtain independent of our thought about them.

    Worlds certainly exist without numbers or an abstract mathematical framework. It is the human (or alien) mind that provides this abstract framework as a tool to model the world.

    Planets are roughly spherical because of the interactions of matter, energy and gravity. Not because Pi exists as a number inherent to the universe.

  151. 151
    kairosfocus says:

    H, Conway’s game is a logical structure, a model world. Its potential outcomes on given start-points are just that, that is they specify possible worlds evolving across time. These can be accurately described, in principle just as the set of counting numbers, stepwise; with a set-building process. That potential for description implies the use of assertions as to what states of affairs are. That is, propositions. The set of accurate descriptions also carries with them a shadow of inaccurate descriptions, false propositions. So the propositions exist as part of the abstract logical structure involved. It is not that they can only exist as coded in some description language and coded in some machine somewhere, requiring actually infinite physical storage, they are there, implicit in the logic of possible worlds. So long as the states of the game may be accurately described, the propositions are there along with the logical possibilities. A playing out of the game in real time is exploring the chain from one start-point it is not inventing them out of nothing, the possibilities were there all along constituting a possible albeit restricted world. Further to all this (and though these days it is pushed outside the window of “reasonable” matters to mention — a bad sign), arguably there is a possible mind that can contemplate all of these Possible Worlds, God’s mind. Where God is at minimum a serious candidate necessary being. KF

    PS: The above exchanges on Platonism underscore why though I will accept that the label Mathematical Platonism is valid (the view that key mathematical entities are abstract but real and antecedent to our thoughts) the term is confusing as it tends to be conflated with general Platonism in a sense of imagining a world out there serving as repository for the famous forms. Of course, the mind of the infinite, all-pondering God would fill such a bill, except that Plato was not a theist in the modern sense. That grand expansion of a world of forms out there somewhere is not what Mathematical Platonism is about or the logic of relevant being. I think a better description is that there is a substance of structure and quantity inherent in any distinct possible world, starting with N. That logical structure is real, is at least in part intelligible to us and is embedded in the framework for any particular possible world. Such carries with it a panoply of descriptive assertions, propositions that accurately specify the world or are implicit in it or .fail to do so. Such propositions are true or false and this abstract relationship is also a necessary part of the fabric of any possible world. The cluster of specifying propositions [in a mathematical world, axioms or the like] carries implications that play out into subsequent propositions, another abstract relationship that is real. We come along and explore, perhaps creating such a world on a computer or writing it down in a book or setting up a board game etc, but it is all one and the same. Our unfolding discovery of implications or successive stages etc is a playing out of possibilities and their equally possible accurate descriptions that are antecedent to our thinking about them. We do not freely, arbitrarily invent logical chains — we explore them stage by stage.

  152. 152
    kairosfocus says:

    F/N: Massimo Pigliucci on Mathematical Platonism may be helpful, in issue 84 of Philosophy Now — where, remember, it is a general rule of philosophy that every alternative answer to hard questions bristles with difficulties so the issue is to compare and see which difficulties one will live with, why:

    https://philosophynow.org/issues/84/Mathematical_Platonism

    There is a difference between general Platonism and the mathematical flavor. For Plato, each apple, say, is but an imperfect example of the absolute (and perfect) Idea of an apple. But as Aristotle quickly realized, Plato has it exactly backwards: we arrive at the general idea of ‘apple’ by mentally abstracting a set of characteristics we think common to all actual apples. It is we who conjure the ‘perfect’ idea from the world, not the world copying the concept.

    But now contrast the idea of an apple with the idea of a circle. Here Aristotle’s approach becomes more problematic, as we don’t find any true circles in nature. No natural object has the precise geometric characteristics of a circle, and in a very strong sense we can also say that the circles we draw are but imperfect representations of the perfect idea of a circle. Ah – but whence does such a perfect idea come from?

    Consider another way to put the problem. One major difference between science and technology is that science discovers things, while technology is about human inventions. We discover the law of gravity; but we invent airplanes to allow heavier-than-air flight despite the law of gravity. But where do mathematical objects, like circles and numbers, or mathematical theorems like the Pythagorean one, or Fermat’s Last one, come from? Are they inventions of the human mind, or are they discoveries?

    I hope you’re beginning to feel as queasy as I did when I started to take the matter seriously, because contrary to Aristotle’s approach to knowledge, my gut feeling was that mathematicians discover things, not invent them. This was a huge paradigm shift from my days as a scientist. Of course, one can reasonably argue that if there were no mathematically-inclined minds around, nobody would be able to think about Fermat’s theorem, while gravity would still exist. True, but nobody would be able to conceive the law of gravitation either – and that doesn’t imply the law itself wouldn’t exist, yes? As Brown puts it in his book: “The thought, for example, which we express in the Pythagorean theorem, is timelessly true, true independently of whether anyone takes it to be true. It needs no bearer. It is not true for the first time when it is discovered, but is like a planet which, already before anyone has seen it, has been in interaction with other planets.” [ –> Cf. James Robert Brown’s Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures (Routledge, 2008)]

    Perhaps nobody should be particularly surprised by this, as, after all, the laws of nature physicists acknowledge also seem to be timelessly true independently of whether anyone takes them to be true. Where do they come from? Since there are somewhat mundane interpretations of what laws of nature are – including the possibility that they are accidental generalities valid in this particular universe and/or within a certain time-span – the case posed by mathematical constructs seems to be even more clear and powerful. Math, like diamonds, truly seems to be forever.

    If one ‘goes Platonic’ with math, one has to face several important philosophical consequences, perhaps the major one being that the notion of physicalism goes out the window. Physicalism is the position that the only things that exist are those that have physical extension [ie, take up space] – and last time I checked, the idea of circle, or Fermat’s theorem, did not have physical extension. It is true that physicalism is now a sophisticated doctrine that includes not just material objects and energy, but also, for instance, physical forces and information. But it isn’t immediately obvious to me that mathematical objects neatly fall into even an extended physicalist ontology. And that definitely gives me pause to ponder.

    KF

    PS: I suggest, that apples exhibit apple-ness in the network of possible or actualised entities. That is, there are in-common archetypal qualities with other fruit (or, fruit-like) objects, all the way back to objects and coming back to apples having what is only in-common to apples and which reflects a known genetically coded pattern expressed by way of a certain kind of tree, the apple-bearing tree. Indeed, by their fruit and other manifestations shall ye know them. That is the genus-difference pattern obtains, likely with overlapping families of inheritance/resemblance, suppression and over-riding etc as the objects paradigm and related ideas explore in computing. Familiarity with such deep patterns allowed field biologists of old to instantly recognise say a cichlid on sight before working out details.

  153. 153
    kairosfocus says:

    PPS: Brown argues onward, distinguishing two kinds of abstracta, one a collective or archetypal, the other a particular but not one that is concrete, arguably, we can make out that relationships, mathematical operations and functions in the operational sense [y = e^-x] may be something distinct, blending aspects of the two:

    Mathematical entities are abstract in one sense, but not in another The
    term ‘abstract’ has come to have two distinct meanings. The older sense per-
    tains to universals and particulars. A universal, say redness, is abstracted from
    particular red apples, red blood, red socks, and so on; it is the one associated
    with the many. The notions of group, or vector space perhaps ?t this pattern.
    Numbers, by contrast, are not abstract in this sense, since each of the integers is
    a unique individual, a particular, not a universal.

    On the other hand, in more current usage ‘abstract’ simply means outside
    space and time, not concrete, not physical. In this sense all mathematical
    objects are abstract. A simple argument makes this clear: There are in?nitely
    many numbers, but only a ?nite number of physical entities; so most math-
    ematical entities must be non-physical.

  154. 154
    kairosfocus says:

    BB: Again, we see the regrettable failure to attend to what has been shown or on the table, here due to conflating substance with study of structure and quantity:

    Worlds certainly exist without numbers or an abstract mathematical framework. It is the human (or alien) mind that provides this abstract framework as a tool to model the world.

    By way of correction, I again note, excerpting the OP on why certain structures and linked quantities are necessarily present in any distinct possible world as part of its framework, antecedent to our thinking about such or study:

    [W]e can show that key abstract elements of structure and quantity are necessary aspects of the logic of being a distinct possible world.

    Consider a distinct possible world, W which is distinct from near neighbours (say W’, W’) by having some aspect of core characteristics A, unique to itself. Were there no A, the world would be indistinguishable from near neighbours and we would recognise that distinct labels have been attached to the same underlying possible world. Such allows us to view W as a structured set:

    W = {A|~A}

    Now, nothing is in W that is not in A or else ~A, the dichotomy is empty and there is no x in W but not in A or else ~A. This is the quantitative property, nullity; thus zero is present, {} –> 0. Likewise, A is a distinct thing, a unit. Unity is present, so one. Following von Neumann, {0} –> 1, where also A manifests unity. In a different sense, ~A is a complex unity, collecting many other things, pointing to collectives, to systems, to organisation, to function based on organisation etc. For our purposes, ~A is a unit but one different from A, so we need to recognise duality, two-ness, thus two: {0,1} –> 2. Obviously, such succession continues without limit and manifests the naturals, also implying the transfinite ordinals on the premise of order type {0,1,2 . . . } –> w (omega).

    Likewise, we may contemplate an inverse such that -x + x –> 0, which is a vector of one dimension. We now have integers. Ratios of integers gives rise to rationals and convergent sums yield the rest of the reals. This gives us continuum. From this, the vector rotation operator i*x repeated twice to give – x allows us to have 2-d vectors in a continuum, a plane. An abstract plane that we may contemplate but which pervades any possible world. Where such a world is sufficiently spatially extended and actualised, we may observe continua, dimensions, vectors, rotations, trajectories etc.

    So, we see where any possible world, simply on being distinct, manifests directly 0,1,2 and by extension on the logic of being, N, Z, Q, R, C. The vector phenomenon captured from Z on, allows us to extend the abstract continuum to arbitrarily many dimensions. (Notice the distinction between world manifestations and our extension to n-dimensional entities, n arbitrarily high.In physics we speak of 10^22 degrees of freedom routinely, for statistical thermodynamics, just for a reasonably accessible case.)

    Our world manifests three spatial dimensions on the macro scale, and we can observe things like Mobius strips etc.

    The underlying point is, that we see intelligible, abstract, necessary, structural and quantitative entities as part of the fabric of any distinct world, part of its framework, part of the logic of its being as a distinct possible world.

    In that context, we may identify certain facts of structure and quantity that necessarily obtain.

    For instance consider five distinct units and how they may be partitioned into a pair and a triple: ||||| –> || + |||. Obviously, this can be reversed, || + ||| –> |||||. Addition and subtraction have a natural sense of partitioning and combining units. Multiplication and division are extensions as are many onward operations, relations and functions. And so forth.

    The point is, that there are abstract, structural and quantitative entities that are intelligible on logic of being which are necessary corollaries of any distinct possible world. These abstracta, we recognise and observe through the effects of the logic of being, we do not invent. They are not merely concepts and constructs we invent and project to a world of things in themselves. That, being in reality just an inner game on the appearances we have and imagine as reflecting the outer world. No, the Kantian ugly gulch fails and we have no good reason to imagine the behaviour of a Mobius strip is some sort of contemplative inner dream. Such dreams we could modify at will, the logic of being is far less yielding than that.

    So, we need to frame an understanding of Mathematics that recognises that we may study the logic of structure and quantity, but this is not isolated from the intelligible substance of structure and quantity manifest in the world. Yes, our sense of being and of cause needs to adapt to the logic of being that involves necessary albeit abstract entities. For instance, nullity, the empty set, zero are manifest in a myriad circumstances, indeed in any possible, distinct world. But as {} is indistinguishable from {} there is good reason to see that it is one and the same common entity. Which is a characteristic shown by many abstract entities.

    I can also add that the concept that our minds can accurately perceive, describe and quantify implies two further abstracta as real and framework to a world with minded creatures or beings: truth as a relationship between propositions [oops, a third!] and the relationship of observable reliability even where utter truth may not apply. Where, observation of course is an operation and its results rely on the relationship, truth.

    The reality of abstracta embedded in the world is inescapable.

    KF

    PS: I note you have no answer on the Mobius strip. I draw the conclusion, that the plumb-line test case succeeds and that your scheme has failed. The M-strip demonstrates that structure and quantity are embedded in the world antecedent to and independent of our thinking on the matter. For that matter, so is the Pythagorean relationship for right angle triangles, i.e. triangularity exists, it has the angle sum triangle relationship, right angle triangles exist and in existing, their sides are forced to fulfill a particular relationship, c^2 = a^2 + b^2, c being the hyp, where also c can be seen as the diameter of a semicircle and the right angle vertex will then lie along its arc, equally by necessity. The old Egyptian 12-segment rope trick that sets up a 3-4-5 triangle used in setting up right angles is a particular application. (I also knew of an old contractor who would set up three timbers in a 6-ft, 8-ft, 10-ft pattern to make sure of squareness of buildings.) Go get some Lego bricks of square form and set up the 3-side, 4-side and 5-side square then make the triangle. Or you could use graph paper. Do that, set up a compass on the 5-side, strike the arc and see how the right angle is on it, at a specific, predictable point. Ponder the linked trig relationships. Was that fitting together of so many things in perfect coherence our invention and concept or was it there, a facet of the world that was discovered then analysed. Think about how we get to the Euler expression 0 = 1 + e^ i*pi and the deep, broad coherence of whole domains of mathematics it sums up. Ask yourself — really ask yourself, did such structures and quantities not exist until we came along and invented systems of measurement tied to length and angle etc? Not at all, patently. This is discovery and discovery that it holds in the general case — a universally applicable relationship for a class of objects of transfinite size — through the power of the logic of implication, which again is abstract but real. Ponder, what you would do away with as a recognised truth, even while you are forced to use such to reason and live. The inescapably true is true and accurately points to realities that are objectively there. Even if these entities, being abstract, force us to reconceptualise what it means for something to be real, to exist; thus to shape a deeper view on the logic of being and possible vs impossible being.

  155. 155
    kairosfocus says:

    PPS: An odd further result, at least as it seems on a thought or two; I share. Having struck the arc on the 3 side, the radii at the right angle and on the hyp are 2.5 units each as the 5-side is the diameter. The radii sit on each end of the 3-side forming a squat isosceles triangle. Strike the perp bisector of the triangle, we see symmetrical 1.5, 2.5, 2 triangles i.e. similar triangles to the original. Fractal self similarity chains onward with ever reducing similar triangles. Going the other way, we can keep on scaling up. What happens on the 4-side, with two 2.5 arcs and a similar bisector, yielding 2, 2.5, and demanding 1.5 as the remaining side?

  156. 156
    kairosfocus says:

    MG, that propositions are non-physical in themselves, i.e. are abstract, does not mean that they do not refer (accurately or falsely) to states of affairs that may be physical, or that they may not specify a possible world, which then embeds the substance they state or imply and if actualised would shape the logic of that world’s being and operations, e.g. no world may embody a square circle. Thus, the propositions would be integrated into the fabric of the world through the specified and/or implicit realities and may be partly intelligible through discerning patterns while being essentially antecedent to and independent of our particular error-prone conceptions. Weird but eye opening on what reality of abstracta means. KF

  157. 157
    daveS says:

    Math Guy,

    Let P denote “Propositions do not exist in the physical universe”. KF might argue P is false based on necessary conditions for any universe, but let us assume it is true. Then applying this to the specific proposition P, implies that P does not exist in the physical universe. I fail to see any contradiction. Surely you are not going to equate the sentence written above in English that symbolizes P with the abstract content/meaning of P as a logical proposition? That is a category error similar to equating the numeral 5 with the abstract idea of cardinality of five.

    I don’t believe I am making this particular error. However, your question here might shed some light on the last step of your argument, which I am having difficulty with.

    Would you mind expanding on this:

    … P exists only within minds. Therefore P is not universally true in the physical universe, contradicting the assumption that P is true.

    In particular, what does it mean for the proposition “abstracta exist only within minds” not to be universally true in the physical universe?

  158. 158
    kairosfocus says:

    DS, see 154 above. KF

  159. 159
    daveS says:

    KF,

    Thanks for the assistance. Can you help me understand this specific case? What is the answer to the question I ask in the last sentence of my post?

  160. 160
    hazel says:

    MG at 147 writes,

    The philosophy that all logical possibilities exist is called “full-blooded Platonism”.

    MG, do you think full-blooded Platonism is true? Or do you yourself hold to some other version of Platonism?

  161. 161
    hazel says:

    Stephen writes,

    One solution would be for Hazel to answer my questions, or for you to answer them in her name, so that a true dialogue could emerge. So far, no effort has been put forth in that direction.

    Hmmm. I have been fairly continually posting here in various dialogues for about 3.5 months. Also, I’ve explained above that I would have to re-write a substantial amount of material in response to your questions, which I haven’t felt like doing.I haven’t felt I have the time for. However, I have more time today, so here is response.

    You write,

    The second order question about *where* the abstract object can exist, (inside or outside the mind) is a different matter and can be answered only on condition that you (or someone) rigorously define the meaning of an “abstract object” or “abstract mathematical? Object.”

    Back at 125, you wrote,

    When I return next, I will begin by defining my terms so that everyone can understand my arguments and can discern what I mean and what I don’t mean. As it is, I don’t think my arguments are fully comprehensible and I hold myself, and no one else, responsible for that situation.

    So perhaps you could tell us what your rigorous definition of an “abstract object” or “abstract mathematical object.” is?

    You write,

    Mathematical models are of no help in that context.

    I don’t know what you mean by that.

    You write,

    Meanwhile, I will provide a partial answer: Clearly, there are extra-mental, non-material, truths about the nature of reality that can be identified and represented through mental concepts. Mathematics can describe them, but it cannot represent them. Only a sound philosophical approach can do that.

    It is not clear to me that this is true. This is the subject under discussion.

    To help explain my position, I’ll address your question about the Pythagorean Theorem.

    At 120, you wrote “ If an abstract reality, such as the Pythagorean Theorem, can be discovered, it must exist first as an extra-mental reality in order to be discovered”, which I disagreed with,

    Human beings create abstract models of the world we experience, and we express those abstract models with symbols systems (words, pictures, and mathematical symbols). We use these physical forms of the models to both manipulate the systems logically for our own understanding and to share our understandings with others. However, underlying these outward expressions of our abstractions is the content of our rational, logical minds. I don’t know what our minds are, in what fundamental ways our thoughts exists in our minds, or how the mind interacts and interfaces with the body, and thus the physical world, to create the outward physical expressions of our thoughts–and I don’t think anyone does.

    One such abstract logical model is geometry, which models the spatial world we live in. We start with certain undefined terms and ideas which represent the model, but are ideal in a way the world is not. We start with a point, which is dimensionless, even though no such object exists in the physical world . We then include lines, planes, the concept of straight line and distance, etc. All of these concepts are in our minds and the symbol systems that we use to represent them: they are abstract ideal concepts which model aspects of the physical world, but those things in the physical world are not perfect, and they are not abstract.

    Now, if we assume the parallel postulate that creates flat, Euclidean space (as opposed to the other two possible versions of the postulate which produce the two non-Euclidean geometries), and if we introduce some other ideas (angles, triangles, etc.) we can logically prove that within that logical system the Pythagorean Theorem is true. The Pythagorean Theorem is an abstract idea in our minds, expressed externally with the physical symbols we use.

    The key idea is that we discovered the Pythagorean Theorem as a logical consequence of the few beginning concepts that are the foundation of the system. We did not have to look “outside of our minds” to discover the Pythagorean Theorem.

    The Pythagorean Theorem accurately models the physical world when considered as (modeled as) a flat space, but only approximately, just as a point models the physical world, but only approximately. The abstractions are in our minds: the physical world is not an abstraction.

    As I have quoted before, Einstein said, “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”

    The Game of Life post above is meant to illustrate this fundamental idea that the abstract mathematical fact we discover are contained within abstract system in our minds. They may or may not be applicable to the physical world (the Pythagorean Theorem is and the game of Life is not, at least not directly), but that is an issue of modeling.

    In none of this is some “extra-mental” existence of the mathematical ideas necessary to explain our experience of how mathematics works, including the discovery of additional mathematical truths through logical manipulation of our abstract concepts.

    There is some substantial response, Stephen. I look forward to your continued dialogue.

  162. 162
    kairosfocus says:

    DS, have you done the mobius strip exercise yet? I did it today, publicly. The impact of where it was cut around was significant. KF

  163. 163
    kairosfocus says:

    H, have you done the Mobius strip exercise? Does its result depend on our thoughts or models, and if you think so, why. If not, does it not demonstrated empirically that substance of structure and quantity are embedded in space and in bodies in space? KF

  164. 164
    daveS says:

    KF,

    Yes, I just did. The result was as expected.

  165. 165
    hazel says:

    kf writes,
    “H, have you done the Mobius strip exercise? Does its result depend on our thoughts or models, and if you think so, why”

    I have NEVER, EVER, EVER, EVER said that the results of anything that happens in the physical world “depends on our thoughts or models,” and it boggles my mind that you would even write that sentence. The world does what the world does irrespective of what we think about it, or whether we understand it, or even whether we exist.

    Could you please explain why you wrote that sentence: what part of what you think I believe led you to even ask if I think what happens in the world “”depends on our thoughts or models.”?

  166. 166
    StephenB says:

    Hazel

    So perhaps you could tell us what your rigorous definition of an “abstract object” or “abstract mathematical object.” is?

    I assume that you want me to define the objective component since the meanings of mathematical and abstract are evident. By “object” I mean the object of the thought, or the thing being referred to. I am making a distinction similar to the one made by Math Guy – the distinction being between the existence of a mental concept and the substance of the mental concept.

    SB: Mathematical models are of no help in that context.

    Hazel

    don’t know what you mean by that.

    I mean that mathematics describes things in quantitative terms and has nothing to say about what they are or what qualities they possess. That is the business of sound philosophy.

    SB: Meanwhile, I will provide a partial answer: Clearly, there are extra-mental, non-material, truths about the nature of reality that can be identified and represented through mental concepts. Mathematics can describe them, but it cannot represent them. Only a sound philosophical approach can do that.

    It is not clear to me that this is true. This is the subject under discussion.

    Perhaps a few examples would help. The laws of non-contradiction, identity, and causality are all extra-mental (and mental) truths that exist as abstract realities. We can say the same thing about the law-like regularities of the universe, or the natures of various animals and many other things. Why would this not be clear to you since it is obviously true?

    To help explain my position, I’ll address your question about the Pythagorean Theorem.

    OK. Good.

    You wrote, If an abstract reality, such as the Pythagorean Theorem, can be discovered, it must exist first as an extra-mental reality in order to be discovered”, which I disagreed with,

    And I can’t imagine why you would disagree with that. (I should offer a mild apology, though, because I wrote “the Pythagorean Theorem” when I should have written, “the abstract extra-mental truth from which the PT is derived.”

    Human beings create abstract models of the world we experience, and we express those abstract models with symbols systems (words, pictures, and mathematical symbols).We use these physical forms of the models to both manipulate the systems logically for our own understanding and to share our understandings with others. However, underlying these outward expressions of our abstractions is the content of our rational, logical minds. I don’t know what our minds are, in what fundamental ways our thoughts exists in our minds, or how the mind interacts and interfaces with the body, and thus the physical world, to create the outward physical expressions of our thoughts–and I don’t think anyone does.

    Sometimes, humans create abstract models to understand and share their experiences, but usually they don’t. I don’t need to create an abstract model to know that a giraffe is a giraffe. I understand that all giraffes are individuals and also that they all have something in common (their class), which I express as an abstraction in the form of a definition, which is all about *what* a thing is. I get that information directly through the senses by way of mental abstraction (abstracting the non-physical form from the physical matter.) If its essence or whatness wasn’t there, we couldn’t know *what it is.* Certainly, no mathematical model would help. Matter is physical, but forms are not.

    One such abstract logical model is geometry, which models the spatial world we live in. We start with certain undefined terms and ideas which represent the model, but are ideal in a way the world is not. We start with a point, which is dimensionless, even though no such object exists in the physical world . We then include lines, planes, the concept of straight line and distance, etc. All of these concepts are in our minds and the symbol systems that we use to represent them: they are abstract ideal concepts which model aspects of the physical world, but those things in the physical world are not perfect, and they are not abstract.

    Yes, but as I have pointed out above, mathematical models of the real world do not provide the same information as philosophical representations of the real world. Mathematics (or mathematical models) can provided quantitative information about a giraffe, but it cannot tell us what a giraffe is or how it differs from a goat. The second point is much more informative than the first and it can be summed up neatly with an abstract definition.

    Now, if we assume the parallel postulate that creates flat, Euclidean space (as opposed to the other two possible versions of the postulate which produce the two non-Euclidean geometries), and if we introduce some other ideas (angles, triangles, etc.) we can logically prove that within that logical system the Pythagorean Theorem is true. The Pythagorean Theorem is an abstract idea in our minds, expressed externally with the physical symbols we use.

    Yes, but don’t forget that your entire formulation is grounded in an extra-mental abstract truth, which we describe as the law of non-contradiction. It is only because a thing cannot be true and false at the same time and in the same sense that a syllogism, given a true premise, can produce an infallibly true conclusion. The discovery of the Pythagorean theorem depends on an extra-mental abstract truth.

    The key idea is that we discovered the Pythagorean Theorem as a logical consequence of the few beginning concepts that are the foundation of the system. We did not have to look “outside of our minds” to discover the Pythagorean Theorem.

    The law of non-contradiction applies both to our minds (psychologically and logically) and to the world outside of our minds (ontologically). That is why the Pythagoreon theorem, (our abstract concept) reflects an abstract truth about the real world. They can correspond with one another only on condition that both exist. The truth about the PT existed long before humans discovered it. And let’s not forget about the earlier noted examples of extra-mental abstract truths, such as the existence of natures, physical laws, and the basic laws of thought.
    .

    There is some substantial response, Stephen. I look forward to your continued dialogue.

    Yes, it was substantial. Thank you. I hope I provided an adequate response.

  167. 167
    hazel says:

    Stephen, you start by defining object in reference to an abstract mathematical object, and you write,

    By “object” I mean the object of the thought, or the thing being referred to. I am making a distinction similar to the one made by Math Guy – the distinction being between the existence of a mental concept and the substance of the mental concept.

    And later,

    Clearly, there are extra-mental, non-material, truths about the nature of reality that can be identified and represented through mental concepts. Mathematics can describe them, but it cannot represent them. Only a sound philosophical approach can do that.

    I understand that this is your view, but it is the point at which we disagree.

    Our mathematical mental concepts aren’t about something else outside of our mind. They are about their own existence: they have abstract content, and in doing so become an abstract object that we can tie together with other abstract concepts. Thus we build a mental world full of intertwined concepts, tied together with our rational and logical capabilities, and used to understand the world and share our understandings with others. None of this requires any extra-mental existence to which our concepts refer.

    The abstract concept of a dimensionless point in geometry is what the mathematical object “point” is. The Pythagorean Theorem is a mathematical object expressing a quantitative relationship among the sides of a right triangle in Euclidean 2-dimensional space. These objects exist in the mind that understand them, and are expressed in the physical world with sounds, writing, and drawings. I see no reason to think that there has to be some Pythagorean Theorem beyond our minds. The Pythagorean Theorem is a logical consequence, in symbolic form, that followings from more primitive concepts.

    By the way, perhaps it would help for you to respond to what I wrote about the Game of Life. Do you also see a mathematical system such as the Game of Life as being about some “extra-mental truths about reality”?

    You write,

    The laws of non-contradiction, identity, and causality are all extra-mental (and mental) truths that exist as abstract realities. We can say the same thing about the law-like regularities of the universe, or the natures of various animals and many other things. Why would this not be clear to you since it is obviously true?

    I accept that the laws of logic are a fundamental aspect of our rationality, and that we follow them as we manipulate abstract concepts. We can’t talk rationally about the world without using them. But that is a feature of human minds: my previous statement about the lack of a need for those to be extra-mental apply.

    I think that if you think that it is “obviously true” that the natures of various animals (which you and I have discussed before) are extra-mental truths that exist as abstract realities, then our views are too far apart to even admit of discussion.

    So I understand your philosophical point of view, I think, and I’m satisfied for myself (although of course there is no way I could satisfy you) that I find no reason to believe as you do, either practically or philosophically. Therefore, I am going to agree with myself that we will need to stand in disagreement.

  168. 168
    kairosfocus says:

    H, it continues. The Mobius strip directly demonstrates how structures and quantities are embedded in space, independent of our concepts or explanations or models but in ways that we can (and historically did) access. That is, this is a case of discovery that transcends oh we build mental models. We analyse what we see, and find that an architecture of structure and quantity is part of the fabric of our world. Abstracta, beyond our thought life, in a physical world. Just an observation for the moment. And BTW, the Pythagorean result was evidently known empirically from particular cases before an axiomatic generalisation was deduced from a very carefully thought through axiomatisation. KF

  169. 169
    hazel says:

    You didn’t answer my question, kf. Why did you ask me if I thought that what happens in the world “depends on our thoughts or models”?

    And yes, I know the history of the Pythagorean Theorem. The 12 knot system was known as a concrete, practical technique for finding a right angle good enough for good construction. The abstract relationship was discovered as part of the development of formal geometry.

  170. 170
    kairosfocus says:

    H, I am extra busy elsewhere and can only respond as time and energy permit. I do note that the idea has been put on the table that abstracta are only in our minds and not in the physical world. There has been talk of models. The bridge to reality is a central question and that is a key reason why I have again highlighted a specific case which allows us to see abstract structure and quantity at work in the world antecedent to our thinking, modelling or whatever but clearly intelligible and amenable to analysis starting with propositions that refer to reality, i.e. are true. . In the OP I have suggested some of how that could be. KF

  171. 171
    hazel says:

    But you still didn’t answer my question. Do you really believe that I, our anyone, think that what happens in the world “depends on our thoughts or models”?

  172. 172
    hazel says:

    P.S. I understand being busy, and priorities, so I don’t mean to imply that answering my question should be pressing at all in comparison to your real world responsibilties. But I will remain curious about why you wrote what you did.

  173. 173
    StephenB says:

    Hazel

    Our mathematical mental concepts aren’t about something else outside of our mind. They are about their own existence: they have abstract content, and in doing so become an abstract object that we can tie together with other abstract concepts. Thus we build a mental world full of intertwined concepts, tied together with our rational and logical capabilities, and used to understand the world and share our understandings with others. None of this requires any extra-mental existence to which our concepts refer.

    Our mathematical concepts exist largely for the purpose of evaluating things outside of the mind and are possible only because of the extra-mental (and abstract) law of non-contradiction. Without the law of non-contradiction, there is no mathematics, just as, without the law of causality, there is no physics. Both sciences depend on higher abstract truths.

    The Pythagorean Theorem is a mathematical object expressing a quantitative relationship among the sides of a right triangle in Euclidean 2-dimensional space. These objects exist in the mind that understand them, and are expressed in the physical world with sounds, writing, and drawings. I see no reason to think that there has to be some Pythagorean Theorem beyond our minds. The Pythagorean Theorem is a logical consequence, in symbolic form, that followings from more primitive concepts.

    The Pythagorean Theorem is a mental concept (construction), yes, but it is also a symbolic expression of something on the outside, a rational universe structured, in part, on the basis of geometric truths, Geometric truths can be discovered only because they existed prior to their discovery. This is basic logic. You cannot discover something that doesn’t already exist in some form or in some way.

    By the way, perhaps it would help for you to respond to what I wrote about the Game of Life. Do you also see a mathematical system such as the Game of Life as being about some “extra-mental truths about reality”?

    Not as far as I can tell. It’s an admirable exercise in thought stimulation, but it seems related to ideas like emergence, self-organization, or even survival of the fittest, all of which would seem to militate against the idea of transcendental truths or a designed universe.

    I accept that the laws of logic are a fundamental aspect of our rationality, and that we follow them as we manipulate abstract concepts. We can’t talk rationally about the world without using them. But that is a feature of human minds: my previous statement about the lack of a need for those to be extra-mental apply.

    The law of contradiction, which is primarily about our minds, is connected to the Law of Identity, which is primarily about things outside the mind. Indeed, they are the same law containing two aspects: The LNC is the psychological and logical component, and the LOI is the ontological component. That is why we can reason about the world outside of our minds with our minds. It is because the logic of our minds corresponds perfectly with the logic of extra-mental reality. If there were no extra-mental truths, there would be nothing for our internal logic to correspond with. Again, this is basic logic. When you think about the law of identity, think about what that means: The identity of a cow cannot also be the identity of a bird, which also means that the nature of a cow cannot also be the nature of a bird. This is an abstract truth about the real world, not a mere mental exercise.

    I think that if you think that it is “obviously true” that the natures of various animals (which you and I have discussed before) are extra-mental truths that exist as abstract realities, then our views are too far apart to even admit of discussion.

    Perhaps that is because you don’t appreciate the significance of the argument. A cow’s nature is not something that is made of matter. If it is not made of matter, then it is a non-material, abstract reality that transcends our mental operations. Surely, you are not going to argue that a cow does not really have a nature, or that it is not different from that of a bird, or that my assertion of that fact is not true, or that it doesn’t apply to the real world

    So I understand your philosophical point of view, I think, and I’m satisfied for myself (although of course there is no way I could satisfy you) that I find no reason to believe as you do, either practically or philosophically. Therefore, I am going to agree with myself that we will need to stand in disagreement.

    No one can stop you from believing what you want to believe.

  174. 174
    hazel says:

    Stephen, you write,

    ,

    Geometric truths can be discovered only because they existed prior to their discovery. This is basic logic. You cannot discover something that doesn’t already exist in some form or in some way.

    Yes, we have been discussing that: Mathematical facts exist as logical consequences within the logical symbol system in which they are discovered. The history of math is about one discovery after another, not because the discoveries exists someplace else, but because the process of following the logical implications of the system leads to things we didn’t know.

    And I don’t think you can dismiss the Game of Life: its rules lead to logical discoveries just as surely, and in the same way, as those of geometry do. As Dave wrote at 140,

    Yes, it seems quite extravagant to insist that all possible abstract objects do exist. And once you are committed to the existence of *some* abstract objects, I don’t know what the rationale would be for not accepting the existence of all of them.

    If you think the Pythagorean Theorem exists in some extra-mental way, then you also, I think, need to admit that every possible generation of every starting configuration in Life also exists in some extra-mental way. The fact that some mathematical facts have a direct (albeit approximate) relationship with events in the physical world and some don’t doesn’t change the nature of the facts as existing as logical consequences in a logical system.

    And, FWIW, the Pythagorean Theorem isn’t true in either spherical or hyperbolic geometry. The Pythagorean Theorem is true only given certain assumptions about parallel lines, and the assumption that through a point not on a line one and only one parallel line exists is not known to be true of our universe at either large or small scales. The whole idea upon which classical 2-dimensional geometries are based, that of existing in an infinite and independent spatial structure, is an out-dated model in physics.

    So, just to summarize, I think the math we discover exists as logical consequences of the symbol systems in which they are found, and that we apply math for building models which describe reality in testable ways.

    But it all start with our abstractions. Take the simple idea of a point. In geometry those are “infinitely small”, with no size and no dimensions. In the abstract sense there are no points in the real world. People who first formalized geometry took things like stakes in the ground as a stimulus to create that abstraction, just as they took a string between two stakes as the stimulus for a line, etc. But the abstractions are in our mind, and real things, which are not abstractions, are in the physical world.

    And last, you write,

    No one can stop you from believing what you want to believe.

    True, and the same for you. However, I take seriously that I want to believe things that seem most likely to be true, and I’m sure you feel the same way about yourself.

  175. 175
    kairosfocus says:

    H, you also left off material context, from 163, which shows that exclusive, exhaustive alternatives are in view:

    have you done the Mobius strip exercise? Does its result depend on our thoughts or models, and if you think so, why. If not, does it not demonstrated empirically that substance of structure and quantity are embedded in space and in bodies in space?

    I take it from your reaction that you reject the first. That leaves the second on the table: the Mobius strip result is independent of and antecedent to our thoughts on the matter.

    Now, as you will readily be able to confirm, cutting around in the middle yields a longer strip with more twists and two edges. Cutting at 1/3 way in results in a narrower Mobius strip interlocked with a longer multiple twist loop. This directly demonstrates that structural and quantitative properties are embedded in the body, which is in 3-d space. These properties exhibit a logic of being pattern such that essentially the same exercise has dramatically different results based on where in the width one cuts. And an ordinary cylindrical loop will separate into two when so cut, reflecting its lack of a twist and orientable surface.

    The presence or absence of a twist in the loop changes core characteristics, thus the type of body and topology at work, yielding drastically divergent results in the three cases. The ordinary loop has two edges and sides, and the cut injects two more edges. The Mobius strip cut has just one edge and just one side (it is a non-orientable surface). That difference is manifested in a concrete entity, but that concrete entity, depending on the twist, has different structural and quantitative properties and characteristics.

    These are abstract, though they manifest concrete results, meaning that abstract structure and quantity have obvious logic of being effects. Such effects also carry implications for what happens if and as the loop is cut around in the middle or 1/3 way from an edge. The architecture of being at work has causal consequences, manifesting an abstract implication. In effect, intelligible natural laws of structural and quantitative form are at work in such bodies. Implications, causes, laws are abstract relationships, but are manifestly present — and are intelligible.

    We come along and do the exercise. We are at first puzzled, then as we study relevant logic of structure and quantity, we come to better understand and form mathematically shaped descriptions of what is the case. That is, we here form abstract propositions that describe states of affairs: a Mobius strip has one edge and one side, as opposed to a cylinder. Propositions exist as what sentences assert, and can have the abstract relationship to external reality of being true, with the further power that implications of true propositions will also be true. Such pivots on an abstract, logic of being structure: p => q means that p is sufficient for q and that q is necessary for p. Perhaps, q may arise in some other way, but once p is present and true, q must also be present and true. This is relevant to cause seen as an implication, and to explanations that are true.

    So, abstract archetypal characteristics that make a Mobius strip different from an ordinary cylindrical loop manifest how when an object or entity x is, it has a distinct identity, what it is. That what-ness involves in-common characteristics with other entities (which allows us to classify at ever higher levels: Tom is a domestic cat of male character, a cat, an animal, an embodied entity etc), and specific ones: Tom is a particular concrete cat, not like Tab, his mate sitting next to him. Distinct identity has ontological, logic of being import. Import that embeds abstract characteristics that give what-ness and that can and do imply many other things such as what happens if a loop is cut around in the middle or 1/3 way across.

    In short, abstract structure and quantity are objectively real and embedded in our world and in bodies etc in the world. Which is the substantial claim on the table. A claim often termed “Mathematical Platonism.”

    I again note IEP, which takes on far more powerful effect given concrete cases that manifest what is being discussed:

    Mathematical platonism is any metaphysical account of mathematics that implies mathematical entities exist, that they are abstract, and that they are independent of all our rational activities. For example, a platonist might assert that the number pi exists outside of space and time and has the characteristics it does regardless of any mental or physical activities of human beings. [–> pi is a particular number manifest in the structural relationships of a circle such that it has a definite centre and magnitude of radius as one sweeps around the figure, circumference and diameter.] Mathematical platonists are often called “realists,” although, strictly speaking, there can be realists who are not platonists because they do not accept the platonist requirement that mathematical entities be abstract.

    Mathematical platonism enjoys widespread support and is frequently considered the default metaphysical position with respect to mathematics. This is unsurprising given its extremely natural interpretation of mathematical practice. In particular, mathematical platonism takes at face-value such well known truths as that “there exist” an infinite number of prime numbers, and it provides straightforward explanations of mathematical objectivity and of the differences between mathematical and spatio-temporal entities [–> i.e. bodies and physically extended worlds]. Thus arguments for mathematical platonism typically assert that in order for mathematical theories to be true their logical structure must refer to some mathematical entities [–> truth is accurate description of what is, involving propositions, reference and reality], that many mathematical theories are indeed objectively true, and that mathematical entities are not constituents of the spatio-temporal realm.

    This should also show a bit more on why I posed the two alternatives.

    KF

    PS: It is not just priorities but emergencies, which just compounded overnight.

    PPS: The following exchange with SB in 173, is illustrative:

    H: Our mathematical mental concepts aren’t about something else outside of our mind. [–> this suggests they are isolated from essentially accurate reference to extra-mental realities in a spatially extended world with bodies like M-strips, they can only accidentally and unreliably/provisionally model] They are about their own existence: they have abstract content, and in doing so become an abstract object that we can tie together with other abstract concepts. Thus we build a mental world full of intertwined concepts, tied together with our rational and logical capabilities [–> which are?], and used to understand the world and share our understandings with others [–> the other is extra-mental, and requires communication with a common reference]. None of this requires any extra-mental existence to which our concepts refer.

    [–> This is a pivotal and questionable claim, on logic of being; for one, a necessary being/ entity/ relationship that is discovered in one possible world will . . . by virtue of being part of the framework for any possible world . . . be a part of the structure of every possible or actual world. This was shown for numbers, starting with N, then going to Z, Q, R, C. That is, every mental model, every inner logic-model world we ponder will necessarily embed and entail numbers and the vast network of structures which are manifest in these nested sets. Thus while SOME aspects of an inner, contemplated logic model may have no external reference, other aspects can, do and even must by force of the logic or architecture of being. See the OP and above in the thread. That is, there is on the table a direct demonstration of why this last claim is false. In addition, given the genus-difference, class membership vs individual distinctives pattern tied to the law of identity, certain contingent properties can also be shared in common between worlds: our logic models may have particular features that are contingent but are in common with our spatially extended world also. This has long been discussed and highlighted. ]

    SB: Our mathematical concepts exist largely for the purpose of evaluating things outside of the mind [–> minds exist towards truth, truth being accurate description of reality]and are possible only because of the extra-mental (and abstract) law of non-contradiction [–> which is part of the logic of being and a corollary of the law of identity]. Without the law of non-contradiction, there is no mathematics, just as, without the law of causality, there is no physics. Both sciences depend on higher abstract truths.

    H: The Pythagorean Theorem is a mathematical object expressing a quantitative relationship among the sides of a right triangle in Euclidean 2-dimensional space [–> or 3-dimensional]. These objects exist in the mind that understand them, and are expressed in the physical world with sounds, writing, and drawings. I see no reason to think that there has to be some Pythagorean Theorem beyond our minds. The Pythagorean Theorem is a logical consequence, in symbolic form, that followings from more primitive concepts.

    SB: The Pythagorean Theorem is a mental concept (construction), yes, but it is also a symbolic expression of something on the outside, a rational universe structured, in part, on the basis of geometric truths, Geometric truths [–> which refer to Geometrical realities of an abstract Euclidean world and apply to our world under certain limited conditions] can be discovered only because they existed prior to their discovery. This is basic logic. You cannot discover something that doesn’t already exist in some form or in some way.

  176. 176
    kairosfocus says:

    DS, kindly see the just above on the significance of how Mobius strips behave, comparing also the OP esp what is highlighted in red. KF

  177. 177
    daveS says:

    KF,

    Thank you; noted.

  178. 178
    hazel says:

    kf, you write, “H, you also left off material context, from 163, which shows that exclusive, exhaustive alternatives are in view.”

    No, the two things you mentioned are not “exclusive, exhaustive alternatives.”

    You still haven’t answered my question: do you really think my view is that I, our anyone, think that what happens in the world “depends on our thoughts or models”?

  179. 179
    kairosfocus says:

    H, I disagree for cause. If something is antecedent to and/or independent of our particular thoughts (thus, “models”), it holds an objective character. In the case of the Mobius strip, it has properties that are structural and quantitative, leading to behaviours when cut around that are different from those of an ordinary cylindrical loop. The properties are demonstrably antecedent to our thoughts. The single edge and surface are manifestly observable, are structural, are quantitative, the effect when cut around in the middle vs 1/3 way from an edge are implied from the properties of the strip. Those properties and implications are abstract, independent of our thoughts on models, are observable and intelligible. They are in the strip, i.e. are embedded — as was asked. KF

  180. 180
    hazel says:

    We are using the world “models” in a different sense, it seems. I am talking about models we create using the abstract concepts in our minds, and expressed via written, verbal, and pictorial symbols. These are not “antecedent to and/or independent of our particular thoughts.” They are a result of our describing, as best we can, the world we experience.

  181. 181
    StephenB says:

    Hazel

    Mathematical facts exist as logical consequences within the logical symbol system in which they are discovered.The history of math is about one discovery after another, not because the discoveries exists someplace else, but because the process of following the logical implications of the system leads to things we didn’t know.

    You seem to be conflating the process of discovery with the substance of what is being discovered. Naturally, if one begins the process of discovery with a true premise, one can arrive at new truths through the process of deduction with no additional input from the outside. However, that doesn’t mean that the mathematical truths that are discovered exist separately from real world truths, On the contrary, mathematical truths are also real world truths. Recall that the logic of our minds corresponds perfectly with the logic of the universe. That correspondence is possible only if both realms exist. Because mathematical truths are also real world truths, engineers know that they can use sound physical and mathematical principles to design bridges that don’t collapse.

    And I don’t think you can dismiss the Game of Life: its rules lead to logical discoveries just as surely, and in the same way, as those of geometry do

    You asked me if I thought the Game of Life reflected the idea that abstract mathematical truths exist. I said that it didn’t and I explained why. Now you are asking a different question as if it was your original question. We should strive to be consistent whenever possible.

    If you think the Pythagorean Theorem exists in some extra-mental way, then you also, I think, need to admit that every possible generation of every starting configuration in Life also exists in some extra-mental way.

    Sorry, but that doesn’t follow,

    And, FWIW, the Pythagorean Theorem isn’t true in either spherical or hyperbolic geometry. The Pythagorean Theorem is true only given certain assumptions about parallel lines, and the assumption that through a point not on a line one and only one parallel line exists is not known to be true of our universe at either large or small scales. The whole idea upon which classical 2-dimensional geometries are based, that of existing in an infinite and independent spatial structure, is an out-dated model in physics.

    There is such a thing as truth in context. But metaphysical truths are broader – they rise above context because they are true in all possible worlds. Not all mathematical truths are metaphysical truths.

    So, just to summarize, I think the math we discover exists as logical consequences of the symbol systems in which they are found, and that we apply math for building models which describe reality in testable ways.

    If the mathematical concepts that are discovered do not reflect the truth, then no discovery has been made. In order to discover the truth by deductive means alone, one must begin with a true premise. Pythagoras, for example, began his analysis by acknowledging, what for him, was a truth about the real world, saying that the universe was built on mathematical principles. If that statement is true, then any concept derived from that premise (except in the case of faulty reasoning) will also be true. The Pythagorean Theorem is one such concept. It reflects a truth about mathematics and, of course, the real world. Do you agree with Pythagoras that the universe was built on mathematical principles? It appears that you do not.

    But it all start with our abstractions. Take the simple idea of a point. In geometry those are “infinitely small”, with no size and no dimensions. In the abstract sense there are no points in the real world. People who first formalized geometry took things like stakes in the ground as a stimulus to create that abstraction, just as they took a string between two stakes as the stimulus for a line, etc. But the abstractions are in our mind, and real things, which are not abstractions, are in the physical world.

    Each discipline uses its own methodology for arriving at the truth, even if that truth us provisional. It is the same with the study of mathematics. That has nothing to do with the fact that some mental concepts are true and some are false. Your insinuation (if not your explicit claim) that the real world is limited to the physical realm (if, indeed, that is your assumption), will compromise your search for the truth. If the universe is solely physical (or material), there can be no abstract truths at all, not even in the realm of mathematics.

    However, I take seriously that I want to believe things that seem most likely to be true, and I’m sure you feel the same way about yourself.

    Yes, that would be the point of any serious investigation – to find the truth. The first order of business, then, would be to acknowledge the fact that abstract truths (as opposed to mere abstract concepts), do, in fact, exist.

  182. 182
    kairosfocus says:

    H, my intent was that models are specifically subsumed under our thoughts above: ” If something is antecedent to and/or independent of our particular thoughts (thus, “models” [–> such is included under thoughts]), it holds an objective character.” The behaviour and properties of a mobius strip are antecedent to our thoughts and are independent of them. Our alternatives are, in our thoughts or in the bodies. Not in thoughts, in the actual bodies, say the ones I publicly cut yesterday and the ones DS cut yesterday. That embeds numbers, spatial location, edges, surfaces, interaction of cutting around and the properties etc, cause in action, implication, etc. KF

  183. 183
    kairosfocus says:

    SB (attn H):

    You seem to be conflating the process of discovery with the substance of what is being discovered. Naturally, if one begins the process of discovery with a true premise, one can arrive at new truths through the process of deduction with no additional input from the outside. However, that doesn’t mean that the mathematical truths that are discovered exist separately from real world truths, On the contrary, mathematical truths are also real world truths. Recall that the logic of our minds corresponds perfectly with the logic of the universe. That correspondence is possible only if both realms exist. Because mathematical truths are also real world truths, engineers know that they can use sound physical and mathematical principles to design bridges that don’t collapse.

    Yes, and in the case of the Mobius strip:

    https://www.smithsonianmag.com/science-nature/mathematical-madness-mobius-strips-and-other-one-sided-objects-180970394/

    The Mathematical Madness of Möbius Strips and Other One-Sided Objects
    The discovery of the Möbius strip in the mid-19th century launched a brand new field of mathematics: topology

    ou have most likely encountered one-sided objects hundreds of times in your daily life – like the universal symbol for recycling, found printed on the backs of aluminum cans and plastic bottles.

    This mathematical object is called a Mobius strip. It has fascinated environmentalists, artists, engineers, mathematicians and many others ever since its discovery in 1858 by August Möbius, a German mathematician who died 150 years ago, on Sept. 26, 1868.

    Möbius discovered the one-sided strip in 1858 while serving as the chair of astronomy and higher mechanics at the University of Leipzig. (Another mathematician named Listing actually described it a few months earlier, but did not publish his work until 1861.) Möbius seems to have encountered the Möbius strip while working on the geometric theory of polyhedra, solid figures composed of vertices, edges and flat faces . . . .

    August Möbius’s discovery opened up new ways to study the natural world. The study of topology continues to produce stunning results. For example, last year, topology led scientists to discover strange new states of matter. This year’s Fields Medal, the highest honor in mathematics, was awarded to Akshay Venkatesh, a mathematician who helped integrate topology with other fields such as number theory.

    Note, discovery and properties are antecedent to study and led to study.

    KF

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