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Humans have been doing mathematics for at least 3000 years. In ancient Egypt they did some advanced trigonometry and number theory. Mathematics is thousands of years old, but for some reason it was only in the late 19th (early 20th) century that mathematicians began to ask questions regarding the basic foundation of their discipline.

Euclid, Plato, Aristotle, and a few other great thinkers investigated some questions regarding the foundations of mathematics, but it was nothing like Frege and Russell (as well as Hilbert, Peano and Whitehead). Frege and Russell wanted to derive ALL of mathematics from a few foundational principles, and also demonstrate that there are no contradictions / inconsistencies in the final result. This was an entirely new way of thinking with a new set of questions. There is nothing like Russell's "Pricipia" in ancient or medieval mathematics / philosophy.

It is as if the whole time mathematicians were just doing mathematics without really caring about it's foundations, and then something happened in the early 20th century that made mathematicians ask what is the foundation on which the whole structure rests?

Why did it take so long?

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    The premise of this question is ahistorical. The philosophy of mathematics has been a big part of philosophy since the Pythagorean school, and axiomatic methods have been a big part of mathematics since Euclid. Mar 18, 2023 at 1:58
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    The mathematics from 3000 years ago would have very little to do with some topics in mathematics today. There was a "crisis of the foundations" at the end of the 19th century and early 20th century prompted by new, non-Euclidean geometries. There were also some "foundational" or rigor issues in analysis. I think it's correct that mathematicians cared less about rigor, and foundations before that.
    – Frank
    Mar 18, 2023 at 2:29
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    I don't know, but I suspect the expansion of mathematics from the in-principle empirically verifiable to the abstract mathematics of groups, spaces, structures, etc, had something to do with it.
    – g s
    Mar 18, 2023 at 2:31
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    Euclid’s Elements was all of mathematics in his time. He he secured it with a few dozen definitions, postulates, and common notions. Is it really a new way in thinking or is it the same challenge of securing much larger recent mathematics?
    – J Kusin
    Mar 18, 2023 at 3:48
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    Well, I suspect this a book-level analysis to make any strong claims, but Frege's program is said to have invented the formal system and set analytic philosophy in motion. Formal systems are an abstraction of language, and I suspect it's no coincidence that semiotics was going on about the same time as well as both programmes following up the wide-spread acceptance of non-Euclidean geometries. If axiomatic mathematical methods aren't Laws of Nature and Laws of Thought, then they're just laws, and the question becomes how do we ground them meaningfully?
    – J D
    Mar 18, 2023 at 4:55

3 Answers 3

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At the end of the 19th century there was a crisis in the epistemology of mathematics. Previously, it had always been assumed that mathematics was just obviously and intuitively correct. There was criticism of some mathematical concepts, such as Leibniz's infinitesimal numbers, but for the most part what mathematicians did was taken to be both necessarily true and indubitably knowable.

Then, in the 19th century, along came some problems for this view. Several mathematicians developed non-Euclidean geometries. Gauss appears to have been the first, but he was deterred from publishing his work, apparently because he thought it would damage his repuation if he was interpreted as saying that Euclid was wrong. Later in the 19th century, non-Euclidean geometry became well established and turned out to have useful applications.

Another problem arrived when Russell demonstrated his eponymous paradox in set theory. The crisis was now established. If mathematics was supposed to be intuitively obvious, why was Euclid seemingly wrong and why was there a contradiction in set theory?

The upshot was an attempt by several mathematicians to set mathematics on a sound footing. Whitehead and Russell's Principia Mathematica, building on the logic of Frege, proposed to resolve the issue by attempting to reduce all of mathematics to logic. Brouwer developed intuitionism as an attempt to show that mathematics is founded on what is provable by human cognition. Hilbert challenged mathematicians to undertake a program to formalise all of mathematics into axiomatic theories using finitary methods of proof.

The logicist program is mostly regarded as a bust today. Some of the axioms Russell relied upon, particularly the axiom of reducibility, were widely criticised and Russell himself acknowledged that he could not see a reason to accept it as necessary. Gödel's incompleteness theorems punched a hole in the project. However, there are some neo-logicists who continue to fly the flag for a weaker version of logicism. Intuitionism in mathematics is not highly popular, though it remains an important minority view, and logicians such as Dag Prawitz and Per Martin-Löf have worked on it.

It is debatable whether reducing mathematics to logic really solves the epistemological issue. Arguably, it just pushes the question back a step to how we explain and justify logic itself.

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The mathematics from 3000 years ago would have very little to do with some topics in mathematics today. There was a "crisis of the foundations" at the end of the 19th century and early 20th century prompted by new, non-Euclidean geometries. There were also some "foundational" or rigor issues in analysis. I think it's correct that mathematicians cared less about rigor, and foundations before that, maybe because they didn't venture into domains as advanced as modern mathematics does.

As has been pointed out in comments elsewhere, modern mathematics studies very abstract objects, where intuition might be impossible or very difficult. For those cases, relying on logic and foundations may be the only way to get correct inferences.

It's easy to be convinced by experimentation that the sum of the square of 2 sides of a right triangle equals the square of the other one, but where is the intuition behind "Smooth/étale morphisms can be lifted (Zariski-locally on the source) along closed immersions"? (EGA IV_4, 18.1.1)

When intuition is not available anymore, you can still make progress with foundations and logic.

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The study of the foundations of mathematics is basically the use of modern logic to analyze mathematics, so the work was not possible until the late 19th century when modern logic was invented.

Another factor in inspiring the foundations of mathematics was Kant's distinction between the analytic and the synthetic a priori--a distinction he drew in the eighteenth century. Much of the original work in the foundations of mathematics was aimed at showing that arithmetic is analytic.

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