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Postulate: Mathematics is constructed. We construct the syntax, grammar and assign semantics to mathematical statements artificially.

Lemma: There is no constraint on what constructed mathematical truths should look like. We could define true, false, para-consistent states, real numbers...you name it, and we could define axiomatic systems arbitrarily that would produce different solutions to the same equations.

What makes one axiomatic system yield physically consistent results, but not another? If I could define mathematical systems arbitrarily, what makes one axiomatic system yield the correct equations of motion of a particle and not another?

Are some mathematical systems inherent in nature? Could the necessity of mathematical truths be proved or disproved within a formal self-consistent system?

Do we need to invoke mathematical isomorphism to solve this (kind of) paradox?

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    My interest is exclusively on the first sentence of your question, where it is stated; "Mathematics is constructed.' Could not agree more. An analogy which may or may not be of interest is between Mathematics and Grammar. Whenever a sentence, from either spoken or written language is diagrammed, we do not confuse the structure of the diagram for the meaning of the words which comprise the sentence. So too is it the case with mathematics, when an hypothesis is converted into a mathematical formula, the formula should not be held to be equivalent to the meaning. it is merely a conveyance.
    – user37981
    Jun 10 '20 at 23:19
  • I agree, it is a conveyance, but strictly speaking there is no formalism I know of that constrains what syllogisms following a posited carrier form (equation, diagram, mapping etc...) should be, and that is what I'm looking for. The simplest answer would be mathematical isomorphism i.e. our contemporary notions of mathematics are the only natural possibilities, and any alternative notion could not exist, rendering my question meaningless.
    – Joeseph123
    Jun 11 '20 at 0:11
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    The inference "if something is constructed then there is no constraint on the construction" is demostrably invalid. Bridges are constructed, it neither means that one can "arbitrarily" throw things together and make them a bridge, nor that bridges are "inherent in nature". They are a happy merger of human designs with natural constraints. There is no paradox, and it is unclear what a formal system is supposed to prove here and to what point. If it proves something it has to assume something, and if it proves its own consistency then it is inconsistent, by Goedel's theorem.
    – Conifold
    Jun 11 '20 at 5:48
  • What we would need to do to tackle this properly is find a (meta)physical constraint on what abstract thought could entail. But even then, we could conceive contradictory logical systems without such a constraint anyways. Strictly speaking, Gödel's theorem applies to a restricted set of formal systems and is not necessarily applicable, and secondly assumes a logical framework posited by I think was Frege. It would be reasonable to assume that one could possibly conceive axiomatised systems that do not fit this framework, unless of course this logical framework is absolute.
    – Joeseph123
    Jun 11 '20 at 10:56
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    To call this question off-topic is pretty ridiculous. Maybe it is not clear but certainly very much philosophical.
    – Mr. White
    Jun 11 '20 at 14:52
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Your post seems to involve two question. A first shallower and a second deeper one.

The first question: "Why does the very mathematics work that has been axiomatically developed in the 20th century"?

The semi-obivous answer: Because it has been axiomatized exactly in order to render mathematical results which were in use in the empirical sciencies for hundreds of years. The axioms allow to derive the math developed by Newton based observations of the world.

The second question: "How come any math works at all in the empirical sciences?"

The problem seems to be that if the laws of math are considered an a priori product of our thoughts why should the empirical world care to obey to these laws?

Here different philosophical schools of thought have offered different answers. I give one example: Kant's ingenious move was to argue that math does neither basically deal with the world "as it is" nor with our concepts of thought but with our pure forms of intuition (reine Formen der Anschauung) towards the empirical world.

Hence, according to Kant, the results of math, although a priori (before any specific sensual experience), are synthetical and not analytical. That is, they suit our intuition of the world (synthetical) rather than our mindset of pure reasoning (analytical).

Needless to say the answer of a 20th century empirical holist is different from Kant's answer.

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