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Preface: Kant's assertion is rebutted by Prof David Joyce who references non-Euclidean geometry and by the last sentence on Sparknotes which states that 'empirical geometry is synthetic, but it is also a posteriori'. So I explain why maths appears a posteriori to me using high school mathematical examples that should be easy enough for Kant.

[Source :] For Kant, mathematical judgments have an intrinsic connection to space and time. He thinks of math as involving geometry and arithmetic, and the basis of geometry being the quantity we apprehend as extension in space while the basis of arithmetic is the quantity we apprehend as extension in time. Accordingly, for Kant the question about the nature of math's bases becomes the question about the nature of our apprehension of the quantities of spatial and temporal extension.

So, on the basis of taking space and time to have an a priori source he infers that mathematics has an a priori source. But the nature of this a priori source, on his view, is not merely one of recognizing the content of concepts we already possess (like when we judge that a bachelor is unmarried), but rather has its basis in our capacity to synthesize spatial or temporal extension in order to arrive at propositions describing geometric or arithmetic quantities. So, by taking mathematical judgments to be acts of syntheses involved our apprehension of space and time, he takes them to be synthetic a priori.

Understanding and so not challenging that

  1. maths is synthetic (eg: Can anyone solve the cubic equation at first sight without doing any algebra?)

  2. and elementary school maths appears a priori to an adult,

I challenge only that maths is a priori at a high-school and university level.

Suppose that a maths student can correctly prove or quantify a concept (eg: the Möbius strip (picture), Principal Component Analysis (picture) or an equation that can be proven visually), but pictures or intuitive explanation enriches this knowledge to the next level. Then all such students learn maths only AFTER exposure to these intuitive explanations and visualisations, and so maths must sometimes be a posteriori. Correct?

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    I think this question has a frequent misunderstanding of the term a priori involved: This means in no sense that one precedes the other in time, but moreso that it precedes the other logically. All Kant is actually saying here is that mathematics are or at least can be (I take this to be the main point) constructed out of its principles (axioms) without the need of corresponding intuitions, i.e. unlike empirical concepts, which work with reflective judgement and imagination. That there have been axioms found very late by e.g. Kolmogorov supports this claim. – Philip Klöcking Mar 9 '16 at 23:47
  • Students learn mathematics from experience; but once they learn it they recognise it's apodictic certainty; and generally the nature of this certainty is taken as a priori - how can it be otherwise; how can 1+1 be anything other than two; or how can the angles of the triangle add upto anything greater or less than 180 degrees? – Mozibur Ullah Mar 10 '16 at 7:23
  • But Kant says that one cannot from the mere definition of the triangle deduce that it's angles must add upto 180 degrees - that is, it is not an analytic proposition; given that Gauss read Kant it strikes me as not implausible that the invention of non-Euclidean geometry, concretely, had some connection with Kant. – Mozibur Ullah Mar 10 '16 at 7:27
  • @PhilipKlöcking Thanks for the elucidation whence I benefited. Just to clarify: I was not basing my last paragraph on the order of time; I was basing it on order of logic: the pictures and intuition that I referenced are NOT logical arguments, and so do not engage any logic; BUT these a posteriori experiences do contribute, if not generate, the a priori part that confuses students. – Greek - Area 51 Proposal Mar 10 '16 at 20:45
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    The question of the Kantian status of mathematics as "synthetic a priori" is, as far as I know, very complicated and controversial. While I cannot contribute without a bit of work, I do think the comments and answers so far are not satisfactory. It is not clear, for starters, that geometry and arithmetic can be treated the same way in Kant. One can say that geometry entails "a priori intuition," though in some readings of Kant this would be contradictory. – Nelson Alexander Mar 11 '16 at 3:48
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The idea of mathematics being a priori has nothing to do with the difficulty in learning it or the amount of experience a mathematician might require in order to master a given discipline. The question has to do whether it depends upon experience or not:

"Thus, moreover, the principles of geometry—for example, that 'in a triangle, two sides together are greater than the third,' are never deduced from general conceptions of line and triangle, but from intuition, and this a priori, with apodictic certainty." [A25/B39]

Mathematical truth is completely independent of experience. It doesn't depend on social conventions, and it is not possible that someday new evidence will overthrow what we know to be mathematical truth. It's rooted in logic, which is something that Kant understood extremely well.

The argument that non-euclidean geometry somehow refutes Kant's position on this demonstrates a misunderstanding of what he was saying. When Kant spoke in terms of Euclidean geometry, he wasn't asserting that it was the only possible geometry. Rather, he was asserting that our representations and how we experience reality is limited to three-dimensional space:

"We never can imagine or make a representation to ourselves of the non-existence of space, though we may easily enough think that no objects are found in it. It must, therefore, be considered as the condition of the possibility of phenomena, and by no means as a determination dependent on them, and is a representation a priori, which necessarily supplies the basis for external phenomena...." [A23/B37]

What's ironic about this is that even mathematicians when they are speaking of alternative geometries describe those geometries in terms of Euclidean geometry. When they speak of curved space, for example, the idea of the curvature of space is presented relative to Euclidean geometry. It is curved in relation to Euclidean straightness. In so doing, they are actually bearing witness to the fact that Euclidean geometry serves as the basis of our experience.

When Gauss was trying to illustrate the lack of necessity in non-Euclidean geometry, he drew pseudo-Euclidean figures which were sometimes inconsistent with his descriptions. How would you, for example, draw an arc with two different radii: one finite and the other infinite? Of course it's not possible. He was trying to represent objects which are inconsistent with experience as if they were. Not to detract from his work as a mathematician, but he wasn't talking about the same thing as Kant. Kant was interested in objects of experience, and Gauss' extra-experiential entities did nothing to diminish our certainty with respect to Euclidean geometry being determinate of such experience.

  • When Kant writes "In a triangle, two sides are greater than the third, are never drawn from general conceptions of line and triangle" surely he is showing that this proposition can't be analytic; an analytic proposition has no more content than its subject, and here Kant is denying explicitly by "are never drawn from general conceptions of line and triangle" that it can be analytic; if it's not analytic, then it's synthetic. – Mozibur Ullah Mar 11 '16 at 10:30
  • And this ties in with Kants manoeuvre to show that geometry and arithmetic, along with space and time are synthetic a priori propositions. I suggested that Gauss might have been inspired by Kant - given that he had read him; but this inspiration need not be direct; and if this is true, then it's ironic given how often the fact of non-Euclidean geometry, either geometrically or physically is used to suggest Kant was wrong about how space is experienced. – Mozibur Ullah Mar 11 '16 at 10:35
  • For sure, Kant and Gauss are 'talking about different things'; but this doesn't undermine the possibility of inspiration, especially given Kant phrasing. – Mozibur Ullah Mar 11 '16 at 10:40
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    My impression is that Gauss didn't fully appreciate what Kant was saying. It's not important that Kant be 100% correct in his account of geometry. What is important is that there is no substitute for the function that it fulfills as a form of intuition. There is no such thing as an empirical source for apodictic certainty. – user3017 Mar 11 '16 at 11:29
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The reason math has to be a priori is that we assume that all humans will agree ultimately upon the same mathematical truths.

This is not true of any other domain. We presume that our physics is moderated by our experience, but not our math. Equally competent and intelligent physicists of every generation have disagreed, even with access to the same data. Likewise for biology, ethics, law, etc. But mathematicians, once given proofs, expect not to disagree. If there is no consensus, we must presume the flaw is in the proof -- it is in some way incomplete.

So the truth value is set outside the individual, irrelevant of experience. It may not yet be 'synthesized' by exposure to the stimuli that make it relevant. But it is already formed, or it would ultimately vary between individuals.

A materialist way of framing a priori thought would be that it is at least phylogenetic: All humans agree on it, and once they form the concepts, it never changes for them. We cannot know whether non-humans would, but by this argument Kant suggests that they will do so, unless their perception of space and time is entirely different, sharing no common basis with our own.


Appendix

To answer @Conifold's objection: In order to combine experiences and derive general principles at all, there has to be a mechanism to do so -- experience does not naturally correlate itself into rules -- we do that to it. Kant proposes the Categories, which are a bit audacious in their detail and specificity.

In a more materialist vein, I would propose that mechanism is the inborn subjective emotional feeling of 'clarity'. There are is a kind of combination that is most clear, across the species, and the result is a given shared substrate of assumptions that underly and become logic and mathematics. (The feeling that this basis is shared, and that we should delve into the shared aspects of it is most obvious in our experience of musical melody.)

This includes two deeply shared core sets of intuitions:

  1. our shared stereoscopic model of space which:

    • is common across people, even with many senses impaired
    • is very independent of actual views, or even potential ones -- consider out-of-body experience
  2. the experiences of continuity and separability of moments we experience as time (a la Brouwer's analysis in Intuitionism) which:

    • base our notions of discrete and continuous -- including their basic paradoxical failure to properly combine, and the weird, flawed notions of infinity and negation that ultimately result
    • create the impulse to count and measure, via rhythm and tempo, that we extrapolate into mathematical notions of numbers
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    I presume that mathematics is moderated by our experience just as physics is, although it may be less transparent. If our sense organs were different we would structure geometry differently, if common objects behaved differently we'd have different arithmetic. Many humans, including mathematicians, disagree even about basic mathematical "truths", such as existence of infinite sets or natural numbers, not to mention axiom of choice and its consequences despite their proofs. "All humans" do not even agree on logical laws. Universal experience independent pure mathematics is a pure fantasy. – Conifold Mar 10 '16 at 22:47
  • @Conifold. Pure math may be a fantasy, but I am not so sure about universal experience. In any case, I am confused about your response to the question, which is quite fundamental. To say that people do not agree about "this or that" hardly answers Kant's premise that such disagreements are only possible "a priori" in a common discursive "space." Which is... "space," for lack of a better term. Geometry is precisely the "a posteriori" scientific exploration of this "a priori" state, is it not? – Nelson Alexander Mar 11 '16 at 4:01
  • Then mathematics, as a discipline simply does not exist -- geometry is physics, arithmetic is simply an aspect of logic, a subdomain of linguistics, etc. In which case the question has no meaning whatsoever, Kant cannot be right or wrong about a domain with no contents. But the fact is that we do agree, at base, about the things we can agree are proven. We may have different standards of proof, but that is beside the point, we end up agreeing on content in a way we do not agree about physics. Math may be a matter of mere psychology, but that psychology is common. – user9166 Mar 11 '16 at 4:02
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    @Nelson I think Kant's premise was rather that Knowledge (in his maximalist sense) is possible, and common a priori of experience are a condition of its possibility. It is hard to maintain today that his premise holds. As for the deflated knowledge we do have Wittgenstein for example outlined how it can emerge from communal practice along with common "discourse", a reified language game. Other a priori-less accounts of intersubjectivity are also available, e.g. Husserlian ones. What unites them is the agreement that assuming our "common ground" to be conceptual is The Error of rationalism. – Conifold Mar 12 '16 at 1:08
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    @PédeLeão It seems better to cite the succeeding sentence of Felix Klein's book: Here he conceded an a priori truth only to arithmetic, placing geometry on the same level as mechanics, as empirical science. – Yai0Phah Aug 12 '16 at 16:02
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I have a different understanding of mathematics than the one visible in the interesting contribution https://philosophy.stackexchange.com/a/32859/40722. I will provide some reasons here.

I disagree with the assumption that all humans will agree ultimately upon the same mathematical truths as there is no such thing like mathematical truth. There are, however, certain sets of axioms with certain consequences which can be derived by mathematical reasoning.

Argument 1: The choice of the axioms is not obvious. Would you admit Zorn's lemma and the axiom of choice in your set theory or not?

Argument 2: The choice of reasoning and derivation mechanisms is not obvious. How would you treat double negation? Would proves have to be constructive? Are transfinite mechanisms allowed?

Argument 3: Reasonably complex axiom sets suffer from (Goedel) incompleteness. So, for a specific axiomatization of arithmetic you would be able to find numerous formulae X which cannot be derived and for which you have a choice to add X or non-X to the axiom set.

Argument 4: You may use what is known as internal set theory to describe what is known as non-standard analysis. So, what is the "true" analysis now? Traditional analysis? Non-standard analysis? Traditional analysis without Zorn's lemma restricted to intuitionistic proofs? Or some other choice?

Argument 5: Contrary to common belief, mathematics is empirical with a notion of finding truth in the lab. The lab is the human brain. I come up with some axioms, check the consequences, realize that they do not adequately model the domain in question and thus adjust my axioms.

However there is a property of our mind , very strong, making us believe that many things are a priori. Particularly good candidates are logic, geometry and counting. That's why most of my arguments appeared only quite recently in mathematical and logic research and stirred up confusion in the field.

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In Thomas Vincis Kant, Geometry and Space, he writes:

The Second Geometrical Argument requires Kant to derive geometrical theorems from the principles of his doctrine of mathematical method and to demonstrate that they have the status of a priori synthetic propositions - something the first argument assumes.

That this is not an easy task is what leads Kant to say in the introduction of the CPR and the Prologemena

B19: How is it possible for human reason to produce mathematical judgements that are synthetic a priori

Synthetic means the truth of proposition lies outside the subject or the grammar of the proposition, whilst a priori suggests the reverse since it is before all possible experience, and so relies on pure cognition; hence asking for such a proposition is almost if one is looking for a kind of dialethic truth, since the two terms are opposites.

He goes on to say that:

philosophical cognition is rational cognition from concepts, mathematical cognition that from the construction of concepts.

Hence, possibly constructivism...

But to construct a concept is to exhibit a priori the intuition corresponding to it.

Therefore

For the construction of a concept, a non-empirical intuition is required ...

If it is a priori it must be non-empirical

Thus I construct a triangle by exhibiting an object corresponding to this object, either through mere imagination, in pure intuition; or in paper, as empirical intuition; but in both cases completely a priori without having to borrow the pattern for it from any experience.

He explains why the empirically drawn figure can serve as a priori:

The individual drawn figure is empirical, and nevertheless serves to express the concept, without damage to its universality.

Since

For in the case of this empirical intuition we have only taken into account the action of constructing this concept, to which many determinations e.g. those of the magnitude of the sides and angles are entirely indifferent.

And

thus we have abstracted from these differences, which do not alter the concept of a triangle.

This the picture I have in my mind when I think of a triangle, is as though I drew before me a triangle whose sides and angles are not labelled with particular numbers, but with letters to express - with a sign - that I'm indifferent to their actual magnitude, but that they are neccessary.

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I can't for the life of me remember who originally argued this or find the article through Google search, but @Conifold hinted at it above: mathematics is inextricably related to the physical world we inhabit and thus is not necessarily a priori true.

Imagine a world where all matter behaved like some sort of fluid, down to a molecular level. Assume the physical laws of this universe are drastically different. Would inhabitants of this world hold the same truths that we hold about math without rigid shapes or strictly defined objects? Would they have a priori knowledge of polygons? Would triangles ever even cross their minds? It even seems dubious that without the nifty feature where matter clumps together in our universe that we'd even have the same understanding of how numbers work.

Food for thought, I guess.


As for your thought experiment, I don't find it particularly motivating. By asking me to "assume that math cannot be fully understood without external input", you're assuming the conclusion to your argument that mathematical knowledge is not necessarily a prior.

Once you've sat down with a pencil and paper and actually proved the theorem yourself there's nothing else that can "deepen" your understanding: you already know it through and through. Maybe your understanding can be "broadened" by interpretation or visualization, but even then, these graphs are just visual representations of the logic contained in the math, not akin to how experiments relate to science.

  • There would still be separate, and countable, groups of fluid. Problem resolved. Math achieved a priori. – ErikE Mar 13 '16 at 2:00

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