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Onward exchange on the objectivity of Mathematical Knowledge

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Over recent days, there has been an exchange at UD on the objectivity vs subjectivity of mathematical knowledge. This is relevant to our understanding of knowledge, and to our recognition of the credibility of Mathematical findings on debated matters. This instantly means that the specific concern and the penumbra of generalised perceptions of Mathematics, Science and objectivity of knowledge are relevant to the ID debate. So, it is appropriate to clip from the discussion in the axioms of math thread.

First, BO’H and his suggestion that he and I actually in the end agree:

BO’H, 34: >>[to:] EricMH – I believe that mathematics, in different respects, is both subjective and objective.

[to:] kf – yes, some parts of mathematics are objective, I don’t think anyone would disagree with that. But that doesn’t stop subjective choices being made, only that some subjective choices are then (objectively) shown to be a bad idea.

Even if the whole logical structure of mathematics is a coherent whole, the choice of what parts we look at, and how we express that, are (at least in part) subjective. Fermat’s Last Theorem isn’t objectively that important in mathematics, but we know a lot about it because of a subjective feeling that it was important. Thus branches of mathematics that are related to Fermat’s last theorem are relatively well sketched out.>>

I responded in 35, but I think it is appropriate to note, on the framework/ structure of mathematical knowledge:

KF, 24: >>The first locus of warranted, accountable truth in a novel domain of structure and quantity (i.e. Mathematics, thus defined insofar as succinct definition is possible) will be the logical import of the framework so laid out in axioms, elaborated in proofs of key theorems and derivation of onward results in accord with logical necessity:

If [AXIOMS] Ax1, Ax2 . . . Axn then perforce

[THEOREMS:] Th1, Th2 . . . Thm.

Per Axi and Thj, Th k etc then also

[RESULTS OF INTEREST/ APPLICATIONS:] Res1, Res2 . . . Resq (which may be of practical interest)

and so forth.

This already imports a huge body of controlling, structuring, accountable facts, principles, patterns and results from the long historical and logical core of mathematics (and logic).

All of this, in a context where numbers, structures, frameworks from across the domains and the raw reality of what is logically possible constrain the work to conform to the states of affairs in physically actualised and abstract worlds defined by things such as key sets [cf. above on numbers great and small], algebraic structures [classically, groups, rings, fields . . . which have such technical meanings that one has to be careful not to invoke them unintentionally in the discussion] and entire sub-disciplines of Mathematics.

If such were not so, we would not have a discipline, we would have a chaos of incoherent, incompatible, unreliable claims, counter-claims and much more.>>

Where, too, I note on the meaning of objective vs subjective:

KF,  PPS to OP: >>Collins English Dictionary is helpful:

1. belonging to, proceeding from, or relating to the mind of the thinking subject and not the nature of the object being considered
2. of, relating to, or emanating from a person’s emotions, prejudices, etc: subjective views.

1. (Philosophy) existing independently of perception or an individual’s conceptions: are there objective moral values?.
2. undistorted by emotion or personal bias>>

The triple first principles of right reason, connected to distinct identity: LOI, A is itself (per core characteristics that permit recognition as distinct), LNC: no x in World W will be A AND ~A, LEM: any y in W will be A X-OR ~A (one or the other but not both or either)

Mathematicians are subjects, but the discipline and its body of knowledge will be objective through having accountable, logically driven warrant and the need to connect even novel axiomatic systems to the core body of facts on structure and quantity, starting with numbers.

It will be helpful to throw a side-light from Wikipedia (speaking against ideological tendency), on Groups:

KF,  13: >>To give an idea, let me select a typical algebraic structure:

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, applies much more widely [–> there is a back-trail and pattern of structure right there, and it goes back beyond axiomatisation to an ancient pool of knowledge of mathematical facts etc]. It allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.[1][2] [–> notice the role of coherence, the group ties a lot of things together]

Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; Poincaré groups, which are also Lie groups, can express the physical symmetry underlying special relativity; and point groups are used to help understand symmetry phenomena in molecular chemistry.

The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. [–> again, reaching back to roots] After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870 [–> notice, the pattern came first and led the structuring definitions] . Modern group theory—an active mathematical discipline—studies groups in their own right.>>

15: >>[To illustrate:] For integers, closure: a + b is an integer too, say c. There is an identity element 0 such that a + e = a. Likewise associativity: (a + b) + c = d = a + (b + c). For each a there is a b so that invertibility: a + b = 0, i.e. b is -a. This pattern can be generalised, and it opens up a new world. But that new world will have links tracing back into the logical and often the historic core of mathematics. Also, there are accountable principles of reasoning that allow reliable argument and accurate conclusions that are in principle accessible to anyone.>>

With such background in hand, we may now consider:

KF, 35: >>I begin to think we are thinking of different things; and that the story of the blind men and the elephant or the metaphor of climbers attacking a new mountain may speak.

Explorers are subjects, knowers are subjects (one has to believe to know, it is the warrant that lends credibility). Likewise, the choice of angle of attack may reflect individual, institutional or cultural process, though the pattern that many separate workers pick up a fresh area in a given time [Quaternions are a case in point, and the handing on to the somewhat derivative vector approach] also suggests that opportunity reflects presence of a common reality.

Blind men and an elephant. Notice, the mental conceptions respond to an actual reality and a sufficient experience will provide an objective picture.

On the other hand, the elephant and the mountain — here, the substantial domain of quantity and structure, the effect of logic and the effect of connectivity back to the heart of mathematical reality — are very much out there and have properties of their own.

Then, there is the impact of Godel’s incompleteness, which shows that there are truths in an axiomatic domain that are not reachable from a given set of axioms. This directly implies that there are structures, quantities and properties that are independent of our conceptualisation and exploration.

So, when axiomatisation sets out a new field of exploration, it is not like writing a new work of say Sci Fi literature that is then studied as a creation of an author. Different authors create different “universes” and even if other authors join in, some effort has to be made to keep things coherent and realistic. Others will naturally create very different universes, as is a very familiar pattern from the many series of novels that are out there. A domain that is at heart creative will naturally be divergent.

By contrast, explorers of a new mountain or territory find that they converge on a consistent map. The blind men come to a consistent pattern about the elephant as they move about the beast and share then unify perspectives. (The story deliberately exaggerates the differences due to partial exploration.)

Now, too, we live in a day and age in which subjectivism and relativism are dominant. There is a natural tendency to exaggerate the significance of subjectivity of persons (where, ironically evolutionary materialism actually undermines the responsible rational freedom that lies at the heart of being a subject). There is a tendency to reduce just about everything to a narrative of power and marginalisation or oppression, with a myth of fundamentally socialistic, statist liberation. (Ironically, the powerful state is perhaps the single greatest threat to freedom.) And lurking deep underneath is the Kantian ugly gulch beteweeen the inner phenomenal world and the outer, extra-mental world of things in themselves. Where, it is commonly perceived that we project a creative rationalisation to the outer world, but cannot actually come to know it as it is in any material degree.

The last is perhaps the most subtle influence, but its worldview shaping power is enormous. That is why I have taken time to repeatedly share F H Bradley’s opening remarks from his Appearance and Reality. The very name of the book speaks.

The essential point is that there is a fatal self-referentiality in making a Kantian ugly gulch claim. To claim to know the un-knowability of external reality as it actually is (as opposed to how it appears) is to claim to know just such an external reality. Namely, the alleged un-knowability characteristic property. So, we have self-referential incoherence and self-falsification. And indeed, we also see a case where, never mind whatever distortions our organs of sensing and perceiving may impose, we also have access through logical analysis to key, powerful truth that connects to reality through a contemplative abstract process of insightful reasoning.

A sounder approach, then, is to recognise with Josiah Royce that error exists — and, that this is not merely a perception or a consensus but that it is undeniably, self-evidently true. To try to deny it rapidly turns around and demonstrates the truth of the claim — to suggest it is an error to imagine that errors exist is obviously self-defeating logically (but may happen psychologically or rhetorically).

It is a point of certain knowledge about our interaction with the realities of our world.

Thus, schemes that deny certain knowledge and the possibility of truth as accurately describing reality fall to the ground. Their name is legion.

Instead, we can accept that we have perspectives which are error prone but can also have well warranted access to the credible, reliable truth about much of reality. In some cases to undeniable certainty. The triple first principles of right reason that turn on properties of distinct identity are a crucial further case in point.

A sketch map of the surreal numbers

As was outlined in the OP that act of recognition of distinct identity immediately leads to two-ness, thence the natural numbers. In 18 above, further points were outlined, to show how we can get to the domain of numbers using several approaches and even complementary definitions, but we also see that numbers all the way to continuum, transfinites, infinitesimals, as well as complex numbers, vectors and similar structures that span abstract spaces in multiple dimensions, are tied to logic and form the substantial heart of Mathematics.

Thus emerges the property of mathematics that it is naturally the logic of structure and quantity leading to abstract logically structured model worlds; so that through coherence and common facts and findings Mathematical reality — yes, an abstract reality starting with numbers — has a substantial objectivity. In exploring it, we may have perspectives and some freedom to set out start points and exploratory projects, but that does not change the underlying common reality that we tend to converge upon and find to have a coherent and consistent unity that is powerful and functional.

Indeed, Fermat’s last theorem, so-called, was like an irritating bit of sand that sparked worlds of exploration.

Wiki is a useful source for a summary:

In number theory, Fermat’s Last Theorem (sometimes called Fermat’s conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known to have infinitely many solutions since antiquity.[1] [–> such is of course an answer to an inviting onward question from Pythagoras’ theorem.]

This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The proof was described as a ‘stunning advance’ in the citation for his Abel Prize award in 2016.[2] The proof of Fermat’s Last Theorem also proved much of the modularity theorem and opened up entire new approaches to numerous other problems and mathematically powerful modularity lifting techniques.

The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof, it was in the Guinness Book of World Records as the “most difficult mathematical problem”, one of the reasons being that it has the largest number of unsuccessful proofs.[3]

Notice that studied, fence-sitting word, development. Development by inner subjectivity or development by collaborative exploration of a space of abstract but very real possibilities, entities, quantities and structures with substantial properties of their own?

The evidence from the at long last proof, is that the properties inhere in the logic of structure and quantity.

Where, too, there is an implication that the intersection of such abstract links of logical necessity with the physical world leads to the astonishing power of mathematics in the sciences.

Which again underscores that we are grappling with realities that are objective and largely independent of our own subjectivity. Never mind that which projects we pick to explore, how we happen to explore such and the subsequent history and body of credible knowledge will reflect such acts of subjects.

We must not conflate or confuse the two.>>

Food for thought. END

PS: I think it is also helpful to clip 18 and flesh out with some illustrations, as numbers great and small are the utter core of Mathematics, which I understand to substantially be — compact definition — the logic of structure and quantity, which we then create a discipline by studying as best we can. Where it is that substance and that logic of warrant that confers objectivity to the knowledge involved:

KF, 18: >>I am thinking, let me start with numbers per the von Neumann construction and show how the core quantitative structure emerges through logical connectivity and then extends without upper limit, implying w as order type of the natural succession of counting numbers, :

The set that collects nothing is {}

Now, we assign:

{} –> 0

However, this is now a distinct albeit abstract entity (and one that exists in any possible world) so:

{0} –> 1

We can now continue collecting, where the RHS of the arrow is a numeral, a name for a number which designates the order type of the set on the LHS:

{0,1} –> 2

{0, 1, 2} –> 3

. . .

{0, 1, 2 . . . k} –> k+1

. . .

That is, without limit.

We may now recognise a new type of quantity, limitless countable succession, the first transfinite ordinal:

{0, 1, 2, . . . k, . . . } –> w [omega]

Omega has cardinality aleph-null.

By using a different and richer strategy, we may lay out the surreal numbers [a representation of which I will now add to the OP], which allows us to extend to numbers that are partly whole and partly fractional, capturing first the rationals then by extension the continuum so the reals. along the way we incorporate negatives as additive inverses a + (-a) = 0. In simple terms if you owe $a and pay $a you clear a debt, you do not have a positive value.

Also, we may freely extend the transfinites and speak to hyperreals which involve infinitesimals, numbers arbitrarily close to 0 in the first instance and which can be seen as reciprocals of numbers that exceed any finite real value. Such form a cloud that can by addition surround any specific real number.

The complex come in as disguised vectors, where we suggest a rotation such that i*a is orthogonal to the real line, then that i*i*a is -a, so that i*i = -1. This opens up a powerful onward world. Including, that we can define angles in the plane. A three dimensional extension opens up models of 3-d space, and concepts of temporal succession and inertia open up model worlds that can map to the physical one we experience.

So now, we have a system of numbers and open the way to further structures such as vectors [and phasors, rotating vectors], matrices, power series, functions, operations [including of course Calculus] and transformations and much more.

The above process is accountable, logically connected, opens up successive logical model worlds and shows true connexions.

Let me note a key logical property of deductive chains, the weakest link principle:

a => b => . . . f, where f is false or self contradictory shatters the chain. We then have to find the key failure point and fix it. And yes, this is used heavily in the reductio ad absurdum proof technique. Systems are only accepted into the fold if they stand up to this.

Where also, let us recall the double-edged sword of implication: a true proposition will only properly imply a true consequent that it is a sufficient condition for, but a false antecedent indiscriminately implies true and false consequents. And also, implication is not equivalence, that requires double implication, often represented by IFF for if and only if.

(Two linked fallacies are affirming the consequent and denying the antecedent. Just because p => q and q is so does not mean p is so unless q also implies p. Likewise, if p => q and p is false does not entail that q must be false as some true r may be such that r => q. And in any case, q’s truth is a matter of accurate description of some relevant reality, which can be logical, relational, structural or quantitative, not just physical. I add: reality is the state of affairs that exists across actualised and abstract worlds, truth accurately describes some targetted facet of reality to which it refers. Truth says of what is that it is, and of what is not that it is not.)

On these and similar factors, the overall system, axiomatisation and all, holds objective truth. Truth on the logic of model worlds, tied back one way or another to historic core schemes, principles and facts tied to clear realities.

Then now, we have in effect a paradigm, which we can extend to connected model worlds that use sets of postulated start-points, axioms and are used to elaborate systems of thought that need not connect to physically observed realities (but often turn out to be surprisingly relevant). Such systems however will cohere through requisites of inner self-consistency and connexions to the core model worlds, force of logic applied to structure and quantity and patterns set by paradigms.

This is the context in which objective truth first speaks to the accountable logical ties, then also to the connexions into the core of established facts and systems. Much of which actually antedates and is materially independent of axiomatisation schemes — we accept certain key axiomatisations in the first instance because they sufficiently comprehend significant domains and give them credible though not certain — Godel counts here — coherence; though we obviously face undecidables and the premise that a system that addresses a complex enough domain comparable to Arithmetic and captures all true statements will be incoherent. (Actually, existence of truths unreachable by axiomatisation schemes that are coherent and perforce limited actually substantiates independent reality, that we are looking at real albeit abstract entities! For, these are obviously not whims and fancies tossed up by our fevered imaginations.)

We have objectivity and warranted, credible reliable truth as an overall pattern in the system. Where, we may fork possible model worlds such as Euclidean/ non Euclidean, or even ZF + C or ZF with something other than C etc.

Mathematics exponentiates its power through its complex, coherent interconnectivity AND its power to lay out and explore abstract, logical model worlds.

(And BTW, I suspect we are here seeing some of the roots of trouble with the idea of fine tuning of the observed cosmos, as a lot of that is explored through what if sensitivity analysis of model worlds initially developed by exploring the dynamics of the observed cosmos.)>>



12 Replies to “Onward exchange on the objectivity of Mathematical Knowledge

  1. 1
    kairosfocus says:

    Onward exchange on the objectivity of Mathematical Knowledge

  2. 2
    kairosfocus says:

    F/N: Avi Sion has a very useful discussion of a phenomenological abstract logical model world in which we can address axiomatisation and conception of Euclidean Geometry, tying it to the coherence and objectification issues I have highlighted.

    Let me make some clips:


    >>The assault on reason throughout the 20th Century has also had its effects on the way philosophers of mathematics understood the developments in that subject. Having a different epistemological background, I can propose alternative viewpoints on certain topics, even while admitting great gaps in my knowledge of mathematics . . . .

    The idea that mathematical systems such as Hilbert’s[2] are “axiomatic” – that is, pure of any dependence on experience is a recurring myth, which is based on an erroneous view of how knowledge of this field has developed. I have discussed the source of this fallacy at length in my Future Logic (see chapter 64, among others); here I wish to make some additional, more specific remarks.

    I do not deny that Hilbert’s postulates are mutually consistent and by themselves sufficient to develop geometrical science. My objection is simply to the pretentious claim that his words and propositions are devoid of reference to experience. We need only indicate the use of logical expressions like “exists,” “belonging,” “including,” “if – then –,” etc., or mathematical ones like “two,” “points” “line,” etc., to see the dependence.

    Take for example the concept of a group (to which something “belongs” or in which something is “included”). The concept is not a disembodied abstract, but has a history within knowledge. The idea of grouping is perhaps derived from the practice of herding animals into an enclosure or some such concrete activity. The animals could all be cows – but might well be cows mixed with goats and sheep. So membership in the group (presence in the enclosure) does not necessarily imply a certain uniformity (a class, based on distinctive similarity – e.g. cows), but may be arbitrary (all kinds of animals, say). Thus, incidentally, the word group has a wider, less specific connotation than the word class (which involves comparison and contrast work). Without such a physical example or mental image of concrete grouping, the word would have no meaning to us at all. So, genetically, the word grouping – and derived expressions like belonging or including, etc. – presupposes a geometrical experience of some sort (a herding enclosure or whatever). We cannot thereafter, after thousands of years of history of development of the science of geometry, claim that the word has meaning without reference to experience. Such a claim is guilty of forgetfulness, and to claim that geometry can be built up from it is circular reasoning and concept-stealing . . . .

    Geometrical “axioms” are thus not absolutes somehow intuited ex nihilo, or arbitrary rules in a purely symbolic system[4], but hypotheses made comprehensible and reasonable thanks to experience. That experience, as I argue below, need only be phenomenal (it does not ultimately matter whether it is “real” or “merely illusory”) but it needs to be there in the first place.

    [–> thus, abstract, model worlds that are tied to our experience and understanding of reality, and our longstanding body of mathematical knowledge, experiences and facts, forcing accountability on both axiomatisation and warrant of onward knowledge claims per “proofs”]

    That experience does not have to give us the axioms ready-made – they remain open to debate – but it gives us the concepts underlying the terms we use in formulating such axioms. In this sense, geometry – and similarly all mathematics – is fundamentally empirical (in a phenomenological sense) – even if much rational work is required beyond that basic experience to express, compare and order geometrical propositions.

    It is futile to attempt to avoid this observation by talking of succession of symbolic objects, A, B, C. Even here, I am imagining the symbols A, B, C in my mind or on paper as themselves concrete objects placed in sequence next to each other! I am still appealing to a visual – experiential and spatial – field. Thus, any claim to transcend experience is naïve or dishonest. Experience is evidently a sine qua non for any axiomatization, even though it is clearly not a sufficient condition. The experiences make possible and anchor the axioms, but admittedly do not definitely prove them – they remain hypotheses[5]. Geometry is certainly not as some claim a deductive science, but very much an inductive one, and the same is true of other mathematical disciplines . . . .

    The experiential aspect of geometrical belief is that there seems to be points, lines (straight or curved), surfaces (flat or warped) and volumes (of whatever shape) in the apparently material world we sense around us as well as in the apparently mental world of our imaginings. This seeming to be is enough to found a perfectly real and valid geometry. The justification of geometry is primarily phenomenological, not naturalistic!

    Seeming is (I remind you) the appearance, or (in this case) phenomenal, level of existence, prior to any judgment as to whether such phenomenon is a reality or an illusion. In other words, geometrical objects do not have to be proven to be realities – in the sense of things actually found in an objective physical nature – they would be equally interesting if they were mere illusions! Because illusions, too, be they mere ‘physical illusions’ (like reflection or refraction) or mental projections, are existents, open to study like realities.

    The study of phenomena prior to their classification as realities or illusions is called phenomenology. At the phenomenological level, ‘seeming to be’ and ‘being’ are one and the same copula. Only later, on the basis of broad, contextual considerations, is a judgment properly made as to the epistemological status of particular appearances, some being pronounced illusions, and the remainder being admitted as realities[9]. If, therefore, geometrical science has a phenomenological status, i.e. if it is a science that can and needs be constructed already at the level of phenomena, it is independent of ultimate discoveries about the physical world.

    The mere fact, admitted by all, including radical critics of geometry, that we get the impression, at the human everyday level of perception, that a table has four corners and sides and a flat top, suffices to justify geometry. This middle-distance depth of perception, even if it is ultimately belied at the microscopic level of atoms or the macroscopic level of galaxies, still can and has to be considered and analyzed. A science of geometry only requires apparent points, lines and surfaces.

    And even if this last argument were rejected, saying that the points, lines and surfaces we seem to see in our table are just mental projections by us onto it, we can reply that even so, mental projections of points, lines and surfaces are themselves real-enough objects existing somehow in this world. They may be illusions, in the sense that they wrongly inform us about the external world, they may be purely internal constructs, but they still even as such exist. A subjective existent is as much an existent as an objective one – in the sense that both are equally well phenomena.

    [–> he is here discussing abstracta which need not have a more or less directly corresponding extra-mental source or referent]

    The mental matrix of imagination, at least, must therefore be capable of sustaining such geometrical objects. And if this restricted part of the world – our minds – displays points, lines and surfaces – then geometry is fully justified, even if the rest of the world – the presumed material part – turns out to be incapable of such a feat and geometry turns out to be inapplicable to it.

    But the latter prospect thus becomes very tenuous! As long as geometry could be rejected in principle, by the elusiveness of its claimed objects under the microscope, there was a frightening problem. But once we realize that the very existence of Geometry requires the possibility somewhere of the concretization of its objects – even if only as a figment of our imaginations – the problem is dissolved. In short, our very ability to discuss geometrical objects, if only to doubt their very existence, is proof of our ability to at least produce them in the mind, and therefore of their ability to exist somewhere in this world. And if all admit that geometrical objects can exist in some part of the world (the mental part at least), then it is rather inductively difficult and arbitrary to deny without strong additional evidence that they exist elsewhere (in the material part). The onus of proof reverts to the deniers of material geometry.

    3.5 The conceptual aspect of geometrical belief must however be emphasized, because it moderates our previous remarks concerning the experiential aspect. Conceptualization of geometrical objects has three components, two positive ones and a negative one.

    a) The primary positive aspect of geometrical conception consists of rough observation, abstraction and classification, (i) refers to the above mentioned concrete samples of points, lines, surfaces and volumes, apparent in the material and mental domains of ordinary experience – this is phenomenological observation; and (ii) observes their distinctive similarities (e.g. that this and that shape are both lines, even though one is straight and short and the other is long and curved, say) – this is abstraction; and (iii) groups them accordingly under chosen names – this is classification.

    b) The negative aspect of geometrical conception is the intentional act of negation, reflecting the inadequacy of mere reference to raw experience. Unlike their empirical inspirations, a theoretical point has no dimension (no length, no breadth, no depth); a theoretical line is extended in only one dimension – it has no surface; a theoretical surface in only two dimensions – it has no volume. Each theoretical geometrical object excludes certain empirical extensions. It is thus an abstraction (based on concretes, of course) rather than a pure concrete . . . .

    c) Another, more daring positive conceptual act may be called assimilation, which we can broadly define as: regarding something considerably different as considerably similar. This a more creative progression by means of somewhat forced simile or analogy, through which we expand the senses of terms . . . .

    Another example is the evolution from Euclidean geometry, the first system that comes to mind from ordinary experience (and in the history of geometrical science), to the later Non-Euclidean systems. A shape considered as “curved” in the initial system is classed as “straight” or “flat” in another system. We have to assimilate this mentally – i.e. say to ourselves, within this new geometrical system, straightness or flatness has another concrete meaning than before, yet the role played by these previously curved shapes in it is equivalent to that played by straight lines or flat surfaces Euclidean system . . . [and much more]>>

    Again, food for thought.


  3. 3
    kairosfocus says:

    F/N2: Karlis Podnieks of Latvia has an interesting remark:


    What is Mathematics? (My Main Theses)

    I define mathematical theories as stable self-contained (autonomous?) systems of reasoning, and formal theories – as mathematical models of such systems. Working with stable self-contained models mathematicians have developed their ability to draw a maximum of conclusions from a minimum of premises. This is why mathematical modeling is so efficient (my solution to the problem of “The Incomprehensible Effectiveness of Mathematics in the Natural Sciences” (as put by Eugene Wigner).

    For me, Goedel’s results are the crucial evidence that stable self-contained systems of reasoning cannot be perfect (just because they are stable and self-contained). Such systems are either non-universal (i.e. they cannot express the notion of natural numbers: 0, 1, 2, 3, 4, …), or they are universal, yet then they run inevitably either into contradictions, or into unsolvable problems.

    For humans, Platonist thinking is the best way of working with imagined structures. (Another version of this thesis was proposed in 1991 by Keith Devlin on p. 67 of his Logic and Information.) Thus, a correct philosophical position of a mathematician should be: a) Platonism – on working days – when I’m doing mathematics (otherwise, my “doing” will be inefficient), b) Formalism – on weekends – when I’m thinking “about” mathematics (otherwise, I will end up in mysticism). (The initial version of this aphorism was proposed in 1979 by Reuben Hersh (picture) on p. 32 of his Some proposals for reviving the philosophy of mathematics.)

    Next step

    The idea that stable self-contained system of basic principles is the distinctive feature of mathematical theories, can be regarded only as the first step in discovering the nature of mathematics. Without the next step, we would end up by representing mathematics as an unordered heap of mathematical theories!

    In fact, mathematics is a complicated system of interrelated theories each representing some significant mathematical structure (natural numbers, real numbers, sets, groups, fields, algebras, all kinds of spaces, graphs, categories, computability, all kinds of logic, etc.).

    Thus, we should think of mathematics as a “two-dimensional” activity. Sergei Yu. Maslov could have put it as follows: most of a mathematician’s working time is spent along the first dimension (working in a fixed mathematical theory, on a fixed mathematical structure), but, sometimes, he/she needs also moving along the second dimension (changing his/her theories/structures or, inventing new ones).

    Do we need more than this, to understand the nature of mathematics?

    BTW, he clips from von Neumann: “In mathematics you don’t understand things. You just get used to them.”

    FFT, again.


    PS: Some useful books on Founds of Math:





  4. 4
    Origenes says:


    Do you agree with the definition of “subjective” offered by the Collins English Dictionary?

    1. belonging to, proceeding from, or relating to the mind of the thinking subject and not the nature of the object being considered
    2. of, relating to, or emanating from a person’s emotions, prejudices, etc: subjective views.

    It seems to me that here subjective is defined as “wrong.” Subjective does not relate to “the nature of the object being considered.” I have a profound problem with this definition. I do not see why subjective is necessarily wrong.

  5. 5
    Origenes says:

    // follow-up #4

    If we have a scenario where a certain alleged real objective thing, in fact, does not exist — e.g. the non-existence of real objective moral laws, as posited by subjective moralists —, then, of course, necessarily all subjective perceptions of a real objective morality are illusionary and thus wrong.

    However, if there is both an objective reality and a subjective perception of that reality, then, IMHO, it makes no longer sense to speak of “subjective” as necessarily wrong as the Collins English Dictionary does.

  6. 6
    kairosfocus says:

    F/N 3: A note from Wiki as a convenient summary on using possible worlds concepts and the logic of being to set up statements that carry truth or falsity in those worlds:

    Possibility, necessity, and contingency
    Further information: Modal logic § The ontology of possibility

    Those theorists who use the concept of possible worlds consider the actual world to be one of the many possible worlds. For each distinct way the world could have been, there is said to be a distinct possible world; the actual world is the one we in fact live in. Among such theorists there is disagreement about the nature of possible worlds; their precise ontological status is disputed, and especially the difference, if any, in ontological status between the actual world and all the other possible worlds. One position on these matters is set forth in David Lewis’s modal realism (see below). There is a close relation between propositions and possible worlds. We note that every proposition is either true or false at any given possible world; then the modal status of a proposition is understood in terms of the worlds in which it is true and worlds in which it is false. The following are among the assertions we may now usefully make:

    True propositions are those that are true in the actual world (for example: “Richard Nixon became president in 1969”).
    False propositions are those that are false in the actual world (for example: “Ronald Reagan became president in 1969”). (Reagan did not run for president until 1976, and thus couldn’t possibly have been elected.)
    Possible propositions are those that are true in at least one possible world (for example: “Hubert Humphrey became president in 1969”). (Humphrey did run for president in 1968, and thus could have been elected.) This includes propositions which are necessarily true, in the sense below.
    Impossible propositions (or necessarily false propositions) are those that are true in no possible world (for example: “Melissa and Toby are taller than each other at the same time”).
    Necessarily true propositions (often simply called necessary propositions) are those that are true in all possible worlds (for example: “2 + 2 = 4”; “all bachelors are unmarried”).[1]
    Contingent propositions are those that are true in some possible worlds and false in others (for example: “Richard Nixon became president in 1969” is contingently false and “Hubert Humphrey became president in 1969” is contingently true).

    The idea of possible worlds is most commonly attributed to Gottfried Leibniz, who spoke of possible worlds as ideas in the mind of God and used the notion to argue that our actually created world must be “the best of all possible worlds”. Arthur Schopenhauer argued that on the contrary our world must be the worst of all possible worlds, because if it were only a little worse it could not continue to exist.[2]

    Scholars have found implicit earlier traces of the idea of possible worlds in the works of René Descartes,[3] a major influence on Leibniz, Al-Ghazali (The Incoherence of the Philosophers), Averroes (The Incoherence of the Incoherence),[4] Fakhr al-Din al-Razi (Matalib al-‘Aliya)[5] and John Duns Scotus.[4] The modern philosophical use of the notion was pioneered by David Lewis and Saul Kripke.

    We are in effect dealing with roots of Mathematics and objectivity of Mathematics here.


    PS: More from Wiki on Lewis’ thought:

    Today, possible worlds play a central role in many debates in philosophy, including especially debates over the Zombie Argument, and physicalism and supervenience in the philosophy of mind. Many debates in the philosophy of religion have been reawakened by the use of possible worlds. Intense debate has also emerged over the ontological status of possible worlds, provoked especially by David Lewis’s defense of modal realism, the doctrine that talk about “possible worlds” is best explained in terms of innumerable, really existing worlds beyond the one we live in. The fundamental question here is: given that modal logic works, and that some possible-worlds semantics for modal logic is correct, what has to be true of the world, and just what are these possible worlds that we range over in our interpretation of modal statements? Lewis argued that what we range over are real, concrete worlds that exist just as unequivocally as our actual world exists, but that are distinguished from the actual world simply by standing in no spatial, temporal, or causal relations with the actual world. (On Lewis’s account, the only “special” property that the actual world has is a relational one: that we are in it. This doctrine is called “the indexicality of actuality”: “actual” is a merely indexical term, like “now” and “here”.) Others, such as Robert Adams and William Lycan, reject Lewis’s picture as metaphysically extravagant, and suggest in its place an interpretation of possible worlds as consistent, maximally complete sets of descriptions of or propositions about the world, so that a “possible world” is conceived of as a complete description of a way the world could be – rather than a world that is that way. (Lewis describes their position, and similar positions such as those advocated by Alvin Plantinga and Peter Forrest, as “ersatz modal realism”, arguing that such theories try to get the benefits of possible worlds semantics for modal logic “on the cheap”, but that they ultimately fail to provide an adequate explanation.) Saul Kripke, in Naming and Necessity, took explicit issue with Lewis’s use of possible worlds semantics, and defended a stipulative account of possible worlds as purely formal (logical) entities rather than either really existent worlds or as some set of propositions or descriptions.

    FWIW, my position is, a possible world is a logical, coherent, materially sufficient description of a way the physical or a physical or an abstract world may be or actually is. That is, I am not locking down on how many worlds are, nor do I require maximal sets of propositions once one has enough for relevant purposes. The Euclidean abstract world is good enough for certain purposes never mind debates about exactitude of planes, points, lines etc, for one instance. And the von Neumann world of successive sets suffices to show a necessity of being that establishes foundational, framework necessity of numbers and linked things in any world.

  7. 7
    kairosfocus says:

    O, I would rather suggest that “subjective” is phenomenological, having to do with how things appear or seem to a subject or a circle of subjects. Such appearances may or may not correspond to concrete or abstract realities, to physically actual or possible worlds, or to the sort of abstract model logic- worlds and associated necessities of being and truth that are at stake for Mathematics. In other words, the subjective is prone to error and so must face warrant as an accountability over credibility of truth and so also of reliability. I do not think CED implies that the subjective is automatically more or less erroneous, but that may well be inferred from how they frame it. KF

    PS: Recall, Objective is also possibly partly or wholly in error but there is warrant of credible truth. Only the necessarily true is beyond rational doubt so. And, something subjective may in fact be absolutely true: the truth, the whole relevant truth, nothing but the truth that says of what is that it is not and of what is not that it is not.

  8. 8
    Origenes says:

    Kairosfocus @

    Thank you for addressing my concerns with a clear response. I think we are in agreement that these terms should be used with great care and can mean different things depending on scenario’s.

  9. 9
    kairosfocus says:

    O, it is apparent that a whole new front has opened up, for what truth is in mathematics is under question — with far reaching implications. I intend to further explore in this thread, for starters. Right now, I gotta go deal with a nailed tyre and other RW issues. Later. KF

  10. 10
    kairosfocus says:

    F/N: As it seems there is a lack of familiarity with the context in which I have spoken to the interconnectedness of Mathematics and how axiom systems are constrained by a core, let me add a cite from Wilder’s The Foundations of Mathematics, p. 5:

    There exists a large literature devoted to the discussion of the nature of axioms and postulates and their philosophical background. Most of this is in?uenced by the fact that only within comparatively recent years have axioms and postulates been very generally employed in parts of mathematics other than geometry. Even though the method popularized by Euclid is acknowledged now as a fundamental part of the scientific method in every realm of human endeavor, our modern understanding of axioms and postulates, as well as our comprehension of deductive methods in general, has resulted to a great extent from studies in the ?eld of geometry. And since geometry was conceived to be an attempt to describe the actual physical space in which we live, there arose a conviction that axioms and postulates possessed a character of logical necessity. For example, Euclid’s ?fth postulate (the “parallel postulate”) was “Let the following be postulated that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.”T Proclus (A.D. 410-485) described vividly in his writings the controversy that was taking place in connection with this postulate even in his time; in fact, he argued in favor of the elimination “from our body of doctrine this merely plausible and unreasoned statement.”’l’ With the renewal of interest in Greek learning during the Renaissance, controversy in regard to the ?fth postulate was renewed. Attempts were made to prove the “parallel postulate,” often from 1ogical—-non-geometrical—principles alone. Surely, if a statement is a “logical necessity” the assumption of its invalidity should lead to contradiction—such was the motivation of much of the work on the postulates of geometry. With the invention of non-euclidean geometries the futility of such attempts became clear.

    1.3 The development of the non-euclidean geometries was evidence of a growing recognition of the independent nature of the ?fth postulate; that is, this postulate cannot be demonstrated as a logical consequence of the other axioms and postulates in the euclidean system. By a suitable replacement of the ?fth postulate, we may obtain the alternative and logically consistent geometry of Bolyai, Lobachevski, and Gauss in which the ?fth postulate of Euclid fails to hold. In it appears, for example, the proposition that the sum of the interior angles of a triangle is less than two right angles. Riemann in 1854 developed another non-euclidean geometry, likewise composed of a non-contradictory collection of prop-ositions, in which all lines are of ?nite length and the sum of the interior angles of a triangle is greater than two right angles. The invention of the non-euclidean geometries was only part of the rapidly moving developments of the nineteenth century that were to lead to the acceptance of formal geometries apart from those that might be regarded as constituting de?nitive sciences of space or extension.

    Here, we see recognition of a forking in Math, based on a set of more or less neighbouring possibilities that give rise to different possible geometric model worlds. But at the same time there is much commonality and a unifying core.


  11. 11
    StephenB says:

    Origenes, subjective morality is wrong because, among other things, it claims that morality comes from the *subject* (the investigator), while objective morality is right because it comes from the *object* of the investigation (God, nature, and the truths arrived at through faith and reason)

  12. 12
    kairosfocus says:

    F/N: Continuing on objectivity of math facts and of axiomatised knowledge domains.

    You will observe, I start with math facts, such as of course the existence of the natural counting numbers from zero and their endlessness, which implies a new class of quantities, the transfinite. Additive inverses also arise as math facts (originally seen from Accounting and the meaning of money owed). Fractions arise from sharing or parts and wholes, and ratios allow representation. Mixed numbers then come in as wholes and parts. We then can standardise on fractional powers of ten (as the main case) and define place value notation decimal numbers. These turn out to be compressed power series. We have gone to rational numbers.

    The big bridge is the irrationals and the continuum, which was seen in antiquity. The side and diagonal of the square are incommensurate. And, we are looking at a bridge to the new province, Geometry. Going forward we can use trigonometry and co-ordinate Geometry to bridge the arithmetic and the Geometric. Then also, symbolising and variables gets us to our first Algebra [there are many Algebras such as Boolean and Matrix etc].

    Complex numbers viewed as rotations allow us to bring space within the ambit so far. Vectors on ijk as orthogonal units — I skip Quaternions — allow us to factor in 3-d space, time gives us the fourth dimension. Vectors, Matrices and Tensors follow as framing new domains of structure and quantity.

    Of course, from variables we go to relationships, mappings and functions. Calculus comes in as we look at rates and accumulations of change in space, time, value etc. The concept of a physical measurement as an extended ratio to a standard amount of a quantity allows us to represent scaled phenomena using techniques of Coordinate Geometry and to access Calculus.

    All of this is before axiomatisation, set theory and the like.

    When the non-Euclidean Geometry breakthrough happens, and axiomatisation is gradually generalised, standardised and established as gold standard, it does so i/l/o a thousands of years old cumulative body of facts, phenomena, reliable methods and more. That axiomatisation and non Euclidean Geometries then feed back into Physics with the General Theory of Relativity.

    I suggest, that were axiomatisations put on the table that were not compatible with the body of established facts and knowledge, they would not have been taken seriously.

    So, again, we see how objectivity pervades the discipline, including when axiomatisation enters as a means of wide but post Godel, not universal, unification and coherence.

    Mathematics is not arbitrary or a mere matter of personal unconstrained choices.

    Of course, such axiomatisation also ties in to the possible worlds frame, as we see that we are exploring abstract logic model worlds of possibilities. Arguably, they have a real albeit abstract existence.

    Indeed, through the significance of distinct identity, we see how numbers become part of the framework for any possible world, and also how logic is inextricably entangled in both mathematics and in an actualised physical world.

    That is a powerful result.

    All of this ties into logic of being also, hence an exploration of wider reality through possible worlds analysis.

    All of this gives us reason to value and prudently use the power of thought.

    And along the way, we can see a reason why the summary that Mathematics is the logic of structure and quantity is also credibly significant.


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