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Beside the foundations and logic, are there fields of mathematics (maybe following the MSC2020 Mathematics Subject Classification) which are currently of interest to philosophers of mathematics?

More generally what is the intersection between "philosophy of mathematics" and "mathematics", or how does "philosophy of mathematics" interact with the various fields and sub-fields of mathematics? What is the "scope" of philosophy of mathematics in mathematics?

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  • This question belongs on a mathematics group. Why would you expect people who follow philosophy to follow mathematics just because the people doing it are also philosophers? Mar 23, 2023 at 20:36
  • To better understand what mathematics is. Mar 24, 2023 at 6:36
  • @jd, he changed the question after I made that comment to make it more about philosophy. Mar 24, 2023 at 9:10
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    Corfield presents a non-orthodox view of such scope in his book Towards a Philosophy of Real Mathematics. It focuses on areas that are most active and considered to be most prestigious by mathematicians themselves (number theory, algebraic and differential geometry, algebraic topology) rather than "foundational" subjects that reflect concerns from half a century ago, but are still favored by academic philosophers. Corfield's approach is a development of Lakatos's adaptation of Kuhn's paradigms to mathematics.
    – Conifold
    Mar 25, 2023 at 3:47
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    @DavidGudeman I have found that most math people tend to reject all philosophy including the philosophy of math as nonsense gibberish. Philosophy of math seeks to verify that the elements of mathematical systems fit together coherently. Math people make sure to avoid looking at that and take these things as "given".
    – polcott
    Apr 23, 2023 at 18:16

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Unsophisticated thinkers who delight in wielding the law of the excluded middle as a cudgel see a fundamental divide in mathematics and the philosophy of mathematics, but for the sophisticated thinker, there is of course no easy way to lump claims as purely non-philosophical and purely philosophical, and they should perhaps be seen categorically as inhabiting two sets of graded and overlapping sets, as should the theories they construct; should the incompleteness theorems not be seen as capable of fueling philosophical discourse regarding the limits of knowledge because they are the conclusions of an axiomatic method as opposed to the presumptions? Balderdash! Of course not.

The same holds true in the relationship between philosophy and scientific practice which following Quine and others should be naturalized. One example of a minority view among math philosophers is Mary Leng who in her Mathematics & Reality explores naturalism, fictionalism, nominalism, and empiricism in regards to the indispensability argument and confirmation holism of mathematical theory, both topics that are also considered in the philosophy of science. (Again, intersectionality pervades real-world language use.)

As to other topics? Philosophers of mathematics have an interest in other topics besides "What is a number?" and "How should we ground mathematics?". The SEP article Philosophy of Mathematics has has a sizeable proportion devoted to the topics of computability and proof. This raises an interest in computability theory, complexity theory, proof and model theory, dynamic systems, and numerical analysis. This should be no surprise, because when Turing posited his formalism for computation, he did so by abstracting from how a mathematician thinks. This of course would be no surprise to John von Neumann either who was, like Turing, a mathematician of quite some competence. von Neumann was very much involved in numerical analysis once WWII broke out given his patriotic contributions to game theory, numerical analysis, and quantum mechanics.

Mathematicians have a special relationship with proof and computation. From the SEP article:

The source of the discomfort that mathematicians experience when confronted with computer proofs appears to be the following. A “good” mathematical proof should do more than to convince us that a certain statement is true. It should also explain why the statement in question holds. And this is done by referring to deep relations between deep mathematical concepts that often link different mathematical domains (Manders 1989).

I'd suggest that abstract algebra is of interest to speculations on mathematical structuralism, and its no surprise that the Curry-Howard Correspondence is sometimes called an isomorphism. And there are philosophical overtones to quantum mechanics in the form of statistics and probability, quantum computation, and quantum logic, all of which overlap. Quantum field theory, a mathematical physics, also raises an interest in differential geometry, algebraic geometry, and tensor fields. Here, physics, mathematics, and philosophy speak to the question "What is space?", which is a continuation of Euclidean geometry giving way to non-Euclidean interpretations.

Mathematical modeling is a practical endeavor too. And according the SEP article on the philosophy of math, sociological, psychological, and pragmatic questions have crept in analogous to the revolution Kuhn brought to philosophy of science. Mary Leng mentioned above fits into this picture. The article again says:

For some decades, the view that the philosophy of mathematics should take a historical and sociological turn remained restricted to a somewhat marginal school of thought in the philosophy of mathematics. However, in recent years the opposition between this new movement of mathematical practice on the one hand, and ‘mainstream’ philosophy of mathematics on the other hand, is softening. Philosophical questions relating to mathematical practice, the evolution of mathematical theories, and mathematical explanation and understanding have become more prominent, and have been related to more traditional themes from the philosophy of mathematics (Mancosu 2008). This trend will doubtlessly continue in the years to come.

So, you can make claims like the difference between actual and potential infinity, a philosophy of mathematics issue, can be better understood by advanced calculus and real analysis which apply limits. You can say that topology is a way to understand deformations of physical objects and therefore gives some insight into the relationship between abstract objects and physical ones. Bayesian probability can be studied to understand epistemological reasoning logically and psychologically, but the notion that mathematical philosophy is confined narrowly to discussion about what numbers are, whether they exist, and how we know that mathematical propositions are true is Schwachsinn.

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  • Some philosophers of mathematics obsessive over the views of mathematics espoused by Frege and Russell seemingly oblivious to the fact that philosophy has advanced quite a bit over the last 75 years. The philosophy of math is far wider than Plato, Frege, Russell, and Whitehead; consider J.R. Lucas's The Conceptual Roots of Mathematics for an overview.
    – J D
    Mar 24, 2023 at 9:14
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Some philosophers work in the field of mathematical logic, but that is not considered philosophy of mathematics. Foundations of mathematics, broadly speaking, has two branches, one of which is philosophical and one of which is mathematical. For example, Dedekind's work on analysis and Zermelo's work on set theory were mathematics because they were working on solving a real problem in mathematics. By contrast, Frege's work on natural numbers and Whitehead's work on geometry were philosophical because there wasn't a mathematical problem that needed to be solved. Some work is a bit hard to qualify as one or the other. For example Peano's work on axiomatizing the natural numbers and Goedel's incompleteness theorem were arguably solving real mathematical problems but also contributed to the philosophical foundations of mathematics.

Generally, what divides mathematics from the philosophy of mathematics is that mathematicians are working out the consequences of mathematical premises, while philosophers are concerned with the ontology of mathematical objects and the epistemology of mathematical objects and propositions. That is, philosophers talk about what kinds of things numbers are, whether they exist, and how we know about them, and also how we know that mathematical propositions are true. Frege, for example, was trying to prove that we know arithmetic is true because it can be reduced to pure logic, and Whitehead was trying to prove that we can know geometry is true because it can be reduced to intuitive knowledge about space.

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