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Assuming that math is the the study of relationships between quantities and sets, why are those entities provable while more qualitative abstract ideas such as beauty or consciousness in philosophy are unprovable? There is also the concept of topology which is considered to be qualitative as well, so where is the line draw for the provability of abstract objects?

Another question is what makes mathematical objects different from other abstract objects?

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    It is not quantity vs quality. Kant's answer was that mathematics deals only with form of things and not with their substance, and forms can be produced in imagination and reasoned about autonomously, under our full control. The modern answer cuts out the imagination and replaces it with formalized systems, but the autonomy and control over form remain. This does not work with substantive abstract concepts that relate to reality. As 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".
    – Conifold
    Aug 7 '20 at 4:12
  • Your response leads me to have a few more questions. Why are these so called substantive abstract concepts not subject to axiomatic methods? Or in other words, what makes the the axioms in mathematics different from potential axioms that could be made about substantive abstractions? Aug 7 '20 at 7:20
  • Nothing. But when something is formalized it becomes part of mathematics, even if called differently. Formalized concepts are no longer substantive, formal system does not know or care where they come from, or what they correspond to, if anything, it is a game of symbols. We can provide proofs for what happens in the game, but what in it resembles reality, and how much, becomes a separate question where the certainty is lost. Substantive concepts are meant to reflect what is out there, not satisfy axioms, formal concepts satisfy axioms, but their relation to out there isn't subject to axioms.
    – Conifold
    Aug 7 '20 at 7:36
  • Substantive concepts may not be intended to satisfy axioms but that isn’t a reason why they are not able to be. They and their mathematical object counterparts are both in the realm of abstraction. Why are substantive concepts held to a different standard of sophistication compared the their quantity and set focused counterparts, in that a formal system is not able to resemble reality in a coherent way like mathematics? Also, by “what is out there” I assume that you mean the truth; however, truth is subject to the deductive reasoning included in a formal system. Aug 7 '20 at 8:16
  • They are not able by definition, reality is not a formalization of itself. That both are abstract is beside the point, and sophistication can be present or lacking in both. There is no different standard, only an empirical fact that the number of well-matched models decreases from physics to biology to sociology to psychology, and there is nothing remotely close for ethics or aesthetics so far. And no, empirical truth isn't subject to deductive reasoning, we can not even deduce that the Sun will rise tomorrow no matter how many times we observe it do so. We can only deduce it for its models.
    – Conifold
    Aug 7 '20 at 8:47
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Mathematical ideas or "objects" are very much simpler than ideas that are handled in philosophy. You only have to look at the concept of Truth. In mathematics the statements can have a Truth value. This word being used in everyday meaning then slides into the realm of reality and philosophy of consciousness, ending up in "What is truth?" (drum roll, bugles etc.).

As I understood the history of philosophy it has been the effort to simplify thought and meaning to where the Truth values of the ideas would become clear and unambiguous, moving the complex human concepts into simple axioms. This has always proved to be more difficult than first considered. There have certainly been longer term goals for the results of such analysis but this has been the normal method. Much discourse moves through the process of showing or convincing the reasonable mind of the clarity of the current case, all with varying degrees of success.

As each age has come along the strangeness of the choices of the earlier generation becomes apparent and the process begins again. I have personally seen little improve over the ages but rhetorical skill, though I do look forward to April.

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The distinction is more vague than exact, and there are philosophers who aim at axiomatic systems (e.g. Spinoza and I think maybe Chalmers? Or Nicholas Rescher at times, iirc, etc.), and Godel's ontological argument is actually looked upon with some favor...

... but ultimately, the difference is external. You can say that the concepts of substance and causation and so on have a definition or demarcation, you can stipulate aspects of word usage in relation thereto, but these approaches advert to the strength of conceptual analysis to be persuasive. So too, Godel argued for "the iterative conception of sets" and thought set theory might be analytically true of this concept. But for many of us, accepting various set/category-theoretic statements can be hypothetically rendered, as in, "If there are infinite sets," or, "If something 'falls into' a category," and this is persuasive enough, as if we (to bring up Conifold's point about game-like formality) played a game of chess and identified the winner according to the rules, only here "winning" is "proving a statement relative to the premises."

By contrast, if we just analyzed arbitrary notions of substance, we might be engaged in deep and elaborate world-building (e.g. there is an author who made up a world where substance is not matter/energy but matter/energy/"investiture", with corollary alternations on thermodynamic conversion) but how rationally persuasive is this when it comes to judging whether there really are substances of a given kind in our world?

Now there was a very influential ethics philosopher who said we should aim for an image of ethics akin to "moral geometry" (Rawls, AToJ), but he did not think he had achieved this goal in his own work and emphasized a method quite unlike a stereotypical axiomatic one, in context.

So we would say: there's nothing wrong with "axiomatizing philosophy" except that no such attempt has ever proven very persuasive beyond hypothetical presumption. Supposing that metaphysical and ethical truths are fundamental and accessible seems to testify against the idea that it should take eons to arrive at such persuasion. Now you might think that such conviction is not attained owing to stubborn/irrational reactions, so that "if people were more honest with themselves in general" they would accept "obvious" axioms; but in philosophy, we focus on the freedom of questions instead of the stricture of deduction (which in context often mutates into ideological certainty, or as Hannah Arendt says, the ideologist is so beholden to "logic" and "inferring one thing from another and another" down through the "whole murderous alphabet" (TOoT, not an exact quote). By the principle of charity (which to be fair looks like an axiom after a fashion!), we try to believe that disagreement is not usually in bad faith; so we try to avoid claiming that our axioms are obvious (indeed, why else would Kant develop the method of transcendental argument do acutely?).

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