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Can I prove a philosophical theory mathematically? If yes? How? For example, can the theory of materialism be proved mathematically?

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    No, not even scientific theories can be proved mathematically. Only mathematical theorems can be, and only after some axioms and inference rules are taken for granted. – Conifold Sep 4 '19 at 10:16
  • No, you cannot. A Mathematical proof is a proof made using rules of logic starting from Mathematical axioms. Every philosophical theory must relies on (at least some) philosophical axiom (like e.g. God exists). – Mauro ALLEGRANZA Sep 4 '19 at 10:51
  • Indeed you cannot prove a philosophical theory with mathematics. But you can disprove philosophical theories with mathematics. Alas, that is not always accepted by philosophers. This is one of the main differences between philosophy and mathematics. – Oбжорoв Sep 4 '19 at 12:14
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Can I prove a philosophical theory mathematically? If yes? How? For example, can the theory of materialism be proved mathematically?

The crucial issue would be the standard of proof. The standard of proof in physics is not the same as the one in mathematics. In mathematics, you don't prove anything true, you prove them valid or not valid. In physics, scientists probably really believe not only that the speed of light is c, but that there is a scientific proof that it is true that it is c. It is conceivable that there might be a mistake somehow, and that therefore the speed of light might not be c after all, but most scientists are probably very certain that it is c and proven to be c.

What standard of proof would be expected in philosophy? Arguably, philosophy is somewhere between mathematics and physics. Philosophy takes into account the empirical facts of the world, but definitely not with the same rigour that science does it. Philosophy, at least analytical philosophy, is also mostly logical, though, again, definitely not with the same rigour as mathematics or physics.

Proving any philosophical claim seems beyond our capabilities, both in terms of the lack of any adequate theoretical framework and in terms of computational power.

The most sustained effort in the direction of proving philosophical statements has been analytical philosophy. It started out with an ambitious perspective:

From about 1910 to 1930, analytic philosophers like Russell and Ludwig Wittgenstein emphasized creating an ideal language for philosophical analysis, which would be free from the ambiguities of ordinary language that, in their opinion, often made philosophy invalid. This philosophical trend can be termed "ideal-language analysis" or "formalism". During this phase, Russell and Wittgenstein sought to understand language (and hence philosophical problems) by using formal logic to formalize the way in which philosophical statements are made. https://en.wikipedia.org/wiki/Analytic_philosophy

However, today, analytical philosophy seems to have renounced the aim of creating "an ideal language for philosophical analysis". Instead, it emphasises a much more modest but ideal of conceptual analysis and rigour:

Scott Soames agrees that clarity is important: analytic philosophy, he says, has "an implicit commitment—albeit faltering and imperfect—to the ideals of clarity, rigor and argumentation" https://en.wikipedia.org/wiki/Analytic_philosophy

Leibniz already had worked on a similar project:

Leibniz believed that much of human reasoning could be reduced to calculations of a sort, and that such calculations could resolve many differences of opinion: "The only way to rectify our reasonings is to make them as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate, without further ado, to see who is right. It is obvious that if we could find characters or signs suited for expressing all our thoughts as clearly and as exactly as arithmetic expresses numbers or geometry expresses lines, we could do in all matters insofar as they are subject to reasoning all that we can do in arithmetic and geometry. For all investigations which depend on reasoning would be carried out by transposing these characters and by a species of calculus." https://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz

He didn't succeed in his effort, but there does not seem to be any reason that it couldn't be done in principle. Rather, the effort to develop the proper theoretical framework seems for now beyond our reach. And the computational power required to carry out the calculations would probably exceed the capacity of our current technology, and by probably several orders of magnitude.

Also, many philosophical questions seem intractable because philosophers themselves disagree as to what may be the relevant empirical facts. For example, such a simple statement as Descartes' Cogito, "I think, therefore I am", is still subject to a controversy, some claiming to have falsified it, other claiming it to be nonsense, and still others accepting it as simply obviously true. The disagreement seems to come from our inability to agree on what is the relevant empirical evidence, and no amount of mathematics will ever solve that difficulty.

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If you mean, can a philosophical theory be proven exclusively by mathematics, then the answer is no. This is because the two bodies of knowledge cover potentially overlapping, but distinct domains of discourse. Often times arguments in philosophy, mathematics, and science overlap to some degree, so it is possible that some aspect of a theory might be supported by argumentation. For instance, in the philosophy of mathematics there is a great degree of overlap with mathematics proper since the ontology of mathematics is derived from metaphysical presumption and the arguments in the philosophy of math. Science and mathematics are very intertwined (see Wigner and WP's entry on mathematical physics), but here too, there are different domains of discourse, so strictly speaking, science is more than just mathematical reasoning. Any further reflection on how philosophy and mathematics relate should also visit how the terms 'philosophy' and 'mathematics' are used as labels for categories that the mind imposes on experience.

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It depends on the theory. The best definition of mathematics I've ever heard was "The study of precisely defined ideas". In mathematics, you have a set of axioms, which are just precisely defined statements you assume to be true, and you use logical inference to draw conclusions that are necessarily true given the axioms. There's nothing in principle that prevents us from doing this with philosophical theories, but in practice, those aren't defined sufficiently rigorously to permit proving theorems about them.

Let's take ethics, for example. Most ethical theories state (to first approximation) the conditions that make an action good. But this isn't really a claim to be proven; rather, they're defining the term good. But suppose that we took a step back and stated that we want to define a "goodness" value for each action, such that if humans select actions in proportion to their goodness, then the resulting world will have certain properties. Then we could propose ethical frameworks, and prove whether or not they would result in a world with these properties if they were universally followed. Alternately, we could take our existing ethical theories, assume them to be universally followed, and prove what properties the resulting world would have.

In practice, that isn't happening; the world is too large, and there are too many distinct actions and combinations thereof (infinite in some cases), and not enough ways to group them into manageable classes. But if we operated over a finite world and set of actions, we could derive a function that assigns a goodness value to each action, and then prove that this function optimizes the probability that the resulting world will have the desired properties. A few ethical theories aren't too far off from this approach: utilitarianism defines an end goal of maximizing happiness (although it doesn't define that), and the categorical imperative has an end goal of internal consistency. It's the size of the problem space that prevents us from proving theorems about these, rather than an inherent incompatibility between philosophy and mathematics.

Of course, given how often mathematicians in antiquity were also philosophers, one could argue (as a very loose approximation) that mathematics started as the branches of philosophy in which the ideas could be sufficiently well defined that we are able to reason this way in practice.

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