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I'm currently a math major, but I'm very interested in philosophy of mathematics. I wonder if there are any prerequisites for learning philosophy of mathematics (such as studying metaphysics or epistemology), or if I can learn such things as I go.

If there are indeed some prerequisites, where can I learn them?

Thanks in advance.

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  • I think you need general familiarity with what philosophy of mathematics is (some history, key figures, main problems and areas of study, controversies) more than any particular background from epistemology or metaphysics. That you can pick up as you go along, and it will digest better with specifically mathematical contexts in mind. Shapiro's and Colyvan's texts are good for general orientation.
    – Conifold
    Jun 20 at 6:35
  • It's likely possible that the topics necessary for phil. of math are not that different for any kind of philosophy, such as logic, epistemology, and metaphysics, thus just try to learn as much as possible which may or even may not attract you currently... Particularly for phil. of math, in addition to the usual constructivism/intuitionism, Platonism, you may need to know some structualism, and of course the Godel's incompleteness theorems of phil. of arithmetic... Jun 20 at 19:34
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    Godel's writings on philosophy of mathematics are very interesting and accessible. His "What is Cantor's continuum problem?" for example is a good starting point for understanding his Platonist position about the axioms of set theory. I am also fond of Lakatos's book Proofs and Refutations, which is an easy read but very profound. It advances the thesis that math is irreducibility of grounded in the informal practice of mathematical discourse.
    – Avi C
    Jun 21 at 18:06

3 Answers 3

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Returning...

The available anthologies are replete with excellent papers. Nevertheless, with regard to single authors, next are probably Pierce and Frege. I have not read much Pierce. But, I keep running into papers that discuss his ideas, and, those ideas are significant. It is easy to find collections of translated work from Frege. The overwhelming majority of it has been worth reading.

One thing rarely mentioned about Frege is his retraction of logicism at the end of his career. When he did this, he did not endorse competing foundational paradigms (conventionalism, formalism). Rather, he offered geometry as the only mathematical topic which could provide a homogeneous perspective of mathematics. What is significant here is that modern foundations often repeats folklore about "the crisis in geometry" and "the arithmetization of mathematics" motivating nineteenth century research. Frege clearly understood these trends. His development of arithmetic arose in these contexts. Importantly, then, his retraction of logicism had been much more.

Acknowledging the modern debate between category theory and set theory, let me observe that I have a paper by Reyes explicitly stating a desire to make logic more geometrical as motivating the use of category theory to study logic. The moral: learning foundations will bury you in rhetorical debates which are not mathematical.

A more important aspect to look at is Frege's description theory and the "identity puzzles" which they had introduced. Thomas Morris has written an excellent book, "Understanding Identity Statements," which does a reasonable job of highlighting issues between "equality" as a metalinguistic relation between terms (say, Fregean more or less) and "equality" with an objectual ground (say, Wittgensteinian) corresponding to the commonly used expression "self-identity."

These questions about the use of equality are significant. Tyler Burge has written a paper "Truth and Singular Terms" clearly arguing that Tarski's semantic conception of truth does not entail the necessary truth of reflexive equality statements. His is not the only paper that has looked at this. Dana Scott has also looked at this issue. Why is this significant? The first-order paradigm invokes Tarski's methodology. The first-order inference rules incorporate the necessary truth of reflexive equality statements.

What you will find is that Frege's clarity brought this problem to light and promoters of certain paradigms use rhetoric to obfuscate the issue. What really comes out of it is that theories of truth matter. Aristotle's analysis is an analysis of use cases. The topmost division is between pedagogy (Posterior Analytics) and common belief (Topics). The bivalence of "classical" (Aristotelian notions are now referred to as "traditional") logic involves a theory of truth more relatable to the belief use case. Arguments against intuitionism and other constructive mathematics often rest upon isuues with theories of truth.

The issue of semantics had been a primary motivation for the description theory Russell presented in " On denoting." In some ways, Tarski had to reconcile Russellian ideas that descriptions are a form of interpretable quantifier with the use of models suggested by Padoa. In the first-order paradigm, what is known as Padoa's method is fully realized by Beth's definability theorem.

Descriptions and definitions go hand in hand. One reason for criticizing the first-order paradigm is that the first-order methodology does not seem to "formalize" ordinary mathematical practices in these matters (motivating my comparison of Aristotelian use cases with the distinction between "classical" and constructive mathematics). Analytical philosophers portray definitions as mere substitutions. Whitehead and Russell made these kinds of claims in "Principia Mathematica." You can find a critical analysis of their claims in the book "Definition" by Richard Robinson.

Both Karel Lambert and Abraham Robinson wrote papers on Hilbert-Bernays description theory. The more applicable one is the Robinson paper ("On constrained denotation"), although you might only find it in his collected works. It is "more applicable" because he speaks of using "an existence predicate" (Lambert acknowledges this as a form of free logic) in the last few paragraphs. An appendix of "Algebraizable Logics" by Blok and Pigozzi discusses how the algebraization of first-order logic brings an existentially quantified equality statement into the inference rule for universal elimination. Ultimately, the algebraization by Nemeti which they present does something slightly different -- it includes a non-reflexive equality statement with only one argument quantified and an unquantified reflexive equality statement. Unfortunately, this does not give quite what is needed to apply Robinson.

But, examine the transitivity axiom for equality in Tarski's cylindric algebra. It embeds an existential quantifier into equality statements. This could be used to develop inference rules different from first-order logic related to extant mathematics involved with algebraization.

An interesting thing about the Nemeti axioms is that they can be compared with Russell's discussion of "relative quantity" in Chapter Nineteen of his "Principles of Mathematics." I have not said much about Russell yet. Regardless of what direct applicability it may have to modern approaches, this is an excellent book. It is far more readable than "Principia Mathematica" and much of what he writes is worth reading. On the downside, the passage of time means that he is writing to an audience with a knowledge base lost to us. These kinds of passages can be intractable.

Another noteworthy book by Russell had been his book on the foundations of modern geometry. It is only noteworthy if you recognize that Kant had been anticipating non-Euclidean geometries. This is justifiable given a translation by Ewald ("From Kant to Hilbert" vol. I) in which Kant calls for the development of new geometries in 1747. Russell wrote his book early in his career and suggested that projective geometry might serve as Kant's "Pure Geometry." Given the significance of projective geometry in the explanations of modern physics, Russell may have offered a reasonable suggestion. Poincare's conventionalism for geometry won the day, and, Russell turned to developing logicism.

Wittgenstein had criticized Russell. In "On denoting," Russell basically called for "self-identity" by speaking of "proper logical constants." So, Russellian logicism emphasizes ontology (is "objectual"). Wittgenstein's logicism is "factual." Truth does not arise compositionally from singular denoting terms.

The difference between Russell and Wittgenstein lies with something obfuscated by the dichotomy between syntax and semantics. Following Carnap's popularization of the term, one ought to acknowledge the relationship of language to a language user as "pragmatics." Semantics, then, is more properly understood in terms of what theory of truth is required for a logical calculus to "preserve truth" between syntactic transformations.

Wittgenstein points out that human beings understand both a proposition and its denial. So, he speaks if "states of affairs" under which one or the other obtain. In other words, Wittgenstein's logicism grounds counterfactuality as a particular source of modality different from Leibnizian partiality. By a historical quirk, Carnap folds both into his first attempts at semantics. Carnap failed to accomplish his goal. Kripke would later succeed. What is unfortunate about that success is how the peculiar nature of counterfactuality (say, in relation to bivalence) is obfuscated by Kripke's modal semantics.

For what this is worth, I read a great deal of Wittgenstein (available, purchasable books). The only one I found useful had been "Tractatus Logico-Philosophicus." I have no doubt about the philosophical significance of his work. But, whatever language games mathematicians do play, they must conform with mathematical practice. And, permit them to speak of whatever they choose (so long as they do not claim to speak for all mathematicians as analytical philosophers and logicians seem to do).

Wittgenstein has provided the literature with many useless quotables.

I have said little of Hilbert. His formalism at the time of "Foundations of Geometry" appears different from that later in his career. I found a translated page from Hilbert and Bernays which clearly suggests an understanding of mathematics with wider scope than the modern formalism dominated by the first-order paradigm. To that end, "formalism" should probably be divided into "instrumentalism" and "conventionalism." The modern literature seems dominated by the latter. I tend to think that Hilbert's concern for consistency had simply been motivated by the desire to protect mathematics from irregularities introduced by people applying mathematics. I know that the history of economics includes ardent defenses of misapplied mathematics, and, the modern scientific literature is plagued with misapplied statistics. Whatever Hilbert actually believed, it probably had not been the "game with symbols" so often attributed to him. Hilbert's formalism had probably been instrumentalist.

Now (believing that you could care less), let me tell you why I have taken the time to write these narratives in response to your question.

No young mathematician should waste their time sorting through this garbage. Find something that interests you deeply and study it with the hope of a contribution to its corpus.

If you find that you really must know more, here is the link to the FOM mailing list started by Harvey Friedman and a few others,

https://cs.nyu.edu/pipermail/fom/

Bookmark it, and, read threads you find interesting. Unfortunately, links in older threads are probably obsolete because information technology benefits morons who do not want to pay taxes for respectable libraries. I mean, we could use the Internet for such a purpose. But...

Learn some mathematics and try to ignore this stuff. Good luck in all of your endeavors.

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First, I would strongly advise that you pick up anthologies of translated (or not) original sources. Ewald has a two-volume set, "From Kant to Hilbert." Another is "From Frege to Goedel" by van Heijenoort. Some principals whose works are significant and can be found separately would be Aristotle, Leibniz, Kant, Frege Russell, and Wittgenstein.

In a volume of collected works, Aristotle has three books specifically dedicated to logic. Those would be Prior Analytic, Posterior Analytic, and Topics. The first would be comparable to syntactic analysis. The second would be speaking of pedagogy (and, therefore, epistemology). The third would be speaking of belief (comparable to semantics). There are other books, such as Categories, which occur in proximity to these three books in a common ordering. These additional books are important as well. The further one moves from these early books, the further one moves into a more general philosophy.

Parkinson is one translator of Leibniz. One of his books contains the collected logic papers. Many of these are fragmentary. A few are more relevant. One paper in particular describes how Leibniz chooses to invert the sense of the Aristotelian substance hierarchy. This is important since this is precisely what Frege criticizes when he develops his logical theories. Leibnizian logic focuses on intensional concepts. Fregean logic focuses on extensions of concepts (collections of Aristotelian individual substances). This distinction marks a transition from epistemology to semantics as a primary focus of logicians at the end of the nineteenth century.

Arguably, "philosophy of mathematics" begins with Kant because of how he tried to use mathematics to ground "objective knowledge" in the light of Hume's defensible argument for scepticism. The reaction of many nineteenth century writers had been one of incredulity. In spite of the propensity of analytic philosophers to disparage Kant, more honest authors (such as Boolos) will admit that his critics have enlightened us without actually defeating Kantian views.

"Critique of Pure Reason" is a very hard and time-consuming book. It is accompanied by "Prologomena to Any Future Metaphtsics." Note that the rise of logicism means that one must read what Kant says in "The Pure Ideal of Reason" in addition to what he says about mathematics and logic earlier in the text. Kant effectively recognizes what is today called the principle of indiscernibility of non-existents when noting that "the sum-total of all possibilities" is individuated even though it cannot fall under a concept.

On the one hand, you may compare this with thw single-point compactification yielding a Reimann sphere. On the other, note that recursive function theory has an "identity" for expressions under which all undefined expressions are equated. In first-order logic, every model has an extension. All of these are manifestations of the fact that a notion of "all" requires an individuation external to a system. And, this is in Kant if you know where to look.

I am out of time now. Perhaps I can write more later.

Know that there is no substitute for original sources.

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  • Also, Plato on the teaching of Meno's slave boy and Locke and Berkeley on abstract ideas. Jun 20 at 15:32
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If you are just planning to learn for fun or a class, you can learn as you go. The typical undergraduate first course is billed to both philosophy and mathematics students, and as such is quite gentle on the prereqs. A suitable level of mathematial maturity along with a working knowledge of the historial motivations of say Cantor, Hilbert, Frege will likely suffice. If you wish to jump on ahead, Shapiro 2000 is widely used.

If you plan for further studies, here are some topics I have found useful. In no partcular order, a first course in mathematical logic, category theory, and computational theory as well as a working knowledge of philosophy of language and Platonism/abstract objects.

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