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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

Comments
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. KFkairosfocus
April 3, 2019
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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. KFkairosfocus
April 3, 2019
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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.StephenB
April 3, 2019
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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.hazel
April 3, 2019
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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. KFkairosfocus
April 3, 2019
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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”?hazel
April 3, 2019
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KF, Thank you; noted.daveS
April 3, 2019
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DS, kindly see the just above on the significance of how Mobius strips behave, comparing also the OP esp what is highlighted in red. KFkairosfocus
April 3, 2019
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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.
kairosfocus
April 3, 2019
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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.hazel
April 2, 2019
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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.StephenB
April 2, 2019
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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.hazel
April 2, 2019
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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”?hazel
April 2, 2019
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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. KFkairosfocus
April 2, 2019
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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.hazel
April 2, 2019
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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. KFkairosfocus
April 2, 2019
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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.hazel
April 2, 2019
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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.StephenB
April 2, 2019
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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."?hazel
April 2, 2019
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KF, Yes, I just did. The result was as expected.daveS
April 2, 2019
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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? KFkairosfocus
April 2, 2019
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DS, have you done the mobius strip exercise yet? I did it today, publicly. The impact of where it was cut around was significant. KFkairosfocus
April 2, 2019
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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.hazel
April 2, 2019
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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?hazel
April 2, 2019
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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?daveS
April 2, 2019
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DS, see 154 above. KFkairosfocus
April 2, 2019
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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?daveS
April 2, 2019
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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. KFkairosfocus
April 2, 2019
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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?kairosfocus
April 2, 2019
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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.kairosfocus
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