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I'm a mathematician, and I'm trying to expand my understanding of the philosophical basis of mathematics. Mathematics is very much taught axiomatically establishing deductive theories, but philosophy seems to suggest that there are additional ways to establish good arguments.

Mathematical methods, especially deductive formal logic, have been used in philosophy. But is it possible to get a probable proof using the method of philosophical argument?

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    What exactly is "the method of philosophical argument"? It might help if you give a concrete example of the sort of application you have in mind.
    – Noah Schweber
    Commented Dec 21, 2021 at 4:56
  • The "method of philosophical argument" is the method of common sense, only more refined and elaborate. That's the only method for making probable proofs about reality we have, only mathematics has the luxury of simply postulating its premises and rules.
    – Conifold
    Commented Dec 21, 2021 at 5:49
  • Your wish of "probable proof" needs clarification, proof is a concept in logic realm without probability modality usually (most inference rules has no probability-mixin), otherwise epistemic closure breaks down easily which sabotages the utility of formal logic. Even fuzzy logic's IF-THEN rules are deterministic. Philosophical argument is nothing but argument with philosophical background knowledge and can be usually formalized if you insist. Sound philosophical argument is conservative extension of math, thus cannot magically prove math theorems "probably" if you cannot do so formally in math
    – Double Knot
    Commented Dec 21, 2021 at 7:02
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    – J D
    Commented Dec 21, 2021 at 15:34
  • philosophy.stackexchange.com/questions/88557/…
    – Wakem
    Commented Dec 21, 2021 at 21:44

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Short Answer

Is it possible to get a probable proof using the method of philosophical argument?

Yes, there are several methods of argumentation that lead to uncertain conclusions such as induction, abduction, and statistical argumentation that are used by philosophers, particularly mathematically savvy ones. Deduction is the gold standard if you are looking for certain conclusions, but human reason is generally informal, and philosophers such as Stephen Toulmin have recognized that most inference is not deductive. (And if you were wondering, mathematicians have a fetish for deduction that might betray the fallibilistic nature of human reasoning generally.)

Long Answer

Welcome! Your profile suggests you are a mathematician getting your feet wet in philosophical thought. Philosophers generally recognize three broad categories of inference, although many philosophers engage in hair-splitting, for instance, distinguishing between abduction and inference to best explanation. Simply put, since Plato's Meno at least, philosophers have tried to show how to overcome uncertainty in reasoning and have arrived at three general classes of argument:

  • Deductive reasoning - Where a conclusion must necessarily follow from sound premises structured as a valid argument.
  • Inductive reasoning - Where a conclusion probabilistically follows from strong premises structured as a cogent argument.
  • Abductive reasoning/IBE - Where a conclusion follows from deductive and inductive reasoning combined with the use of intuition.

These forms of reasoning are not the only that are studied, see Logical reasoning in WP, but these are the three important ones, with abductive reasoning being the most recent model of human reasoning, and particularly important given certain developments in logic such as natural deduction, intuitionistic logic, non-classical logic, and non-monotonic logic. In the philosophy of artificial intelligence, the defeasibility of informal logic is a widely studied topic.

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