What should a person interested in the philosophy of mathematics know?

Question

What philosophy should a person interested in the philosophy of mathematics know, at a minimum?

Having delved into the subject, it looks like there are things you need to know.

Answer 1

From my viewpoint as a logician, here is the bare minimum that you need to know.

On the mathematics side (with the full technical details and proofs):

1. Basic FOL (first-order logic), including a full deductive system.
2. The (semantic) completeness theorem for FOL (e.g. from Hannes' notes or Rautenberg's text).
3. The generalized (syntactic) incompleteness theorems.
4. The formal systems PA and ZFC, and the best proofs (i.e. using least assumptions) of important theorems (e.g. Zorn's lemma and the well-ordering theorem within ZC, and transfinite induction along any well-ordering within Z).
5. Basic computability theory (including programs, oracles).
6. Some amount of reverse mathematics.

Without knowing these, one may easily get fooled by false or misleading claims from people about the incompleteness theorems or about platonism or formalism or practically anything related to logic. And there are tons of such bad information out there. The only way one can sieve out the correct claims from among the trash is to fully understand the actual mathematics on one's own. =)

On the philosophy side:

1. The notion of ontology.
2. The notion of extensionality.
3. The philosophical notion of reification, meaning the act of taking a definable predicate/function and claiming the existence of an object corresponding to that. (See this and this.)
4. The various kinds of predicativism. (For a little taste see this and this.)
5. Non-classical logic and the notion of truth-value gaps, including 3VL (Kleene's 3-valued logic). (3VL allows us to cleanly handle common logical 'paradoxes', as well as Russell's paradox.)
6. Some amount of science (e.g. relativity, quantum mechanics).

I wish to emphasize that it is extremely dangerous to read the philosophical stuff before you have understood the mathematics, because many papers on the philosophy of mathematics are actually clearly wrong to any professional logician. (See here for some details.)

And why science? Because if we do not know that there are physical limits to what we can do in the real world, we might believe that certain mathematical concepts have a real-world embedding even though there is scientific evidence against it (that we are unaware of).

Answer 2

It is not what philosophy one needs to know as much as the historical trends that have been significant. Currently, formalism as evolved from Hilbert's program to use logic as a means to ensure the consistency of mathematics dominates. Mathematicians are less likely than philosophers to assert the idea that mathematical necessity follows from stipulated rules. But, that idea has generated a plurality of logics. And, one may easily understand how no single logic characterizes ordinary mathematics. Material implication admits irrelevant antecedents. Most mathematicians would object to this, and, proofs with fewer assumptions are often sought.

But, because of Hilbert's program you need good sources in first-order logic like Shoenfeld or Mendelsohn.

Truth in mathematics is problematic. One would look to model theory because it employs truth in a natural way. In fact, however, Hilbert's earlier work in geometry convinced him of an intimate relationship between logic and arithmetic. So, Hilbert's metamathematics is arithmetical. The reason for this is historical. Academics used to believe that Euclidean geometry served as a faithful representation of spatial experience. With the development of non-Euclidean geometries, geometry had been abandoned for analytical methods using number systems. But, this belief in arithmetic had been undermined by Goedel.

Incompleteness is often portrayed as exposing the relationship between proof and truth. Its importance in the philosophy of mathematics lies in the proof theory originating with Hilbert. One must understand truth with respect to the encodings of formal systems used in first-order logic. And since truth is indefinable in this paradigm, it must be understood in terms of a hierarchy of formal systems providing truth predicates for its immediate predecessor in the hierarchy.

Smullyan has a book on recursion theory for metamathematics. He refers to a prior book on Goedel's incompleteness theorems. It is not required as a prerequisite, but, it often helps to read a single author because of notational usage.

An important aspect of recursion theory in this paradigm is the verification that a given formula is an axiom. First-order theories are based upon languages obtained from structural recursion. Because individuals are represented by singular terms, a first-order theory often includes infinite schemes of formulas. Peano arithmetic, for example, is finitely described in most mathematics books. But, it first-order axiomatization has infinitely many axioms. In order for "mathematics" to be definite, one must be able to "know" that the formulas being used as axioms are, in fact, members of an infinite schema.

The structural recursion used for generating first-order languages is dictated by the need to apply Tarski's semantic conception of truth. So, the use of recursion for verification is essential.

One can read a great deal about mathematics in the philosophical literature. And, one can dispute the legitimacy of formalism as it has developed out of the Hilbert program. But, if one wishes to study the philosophy of mathematics with any understanding, one needs to understand the consequences of the incompleteness theorems for proof theory.

So, why first-order logic? Read about Bradley's regress on the Stanford Encyclopedia of Philosophy and read a copy of "On denoting" by Russell. While "foundations" with respect to mathematics may mean different things to different people, "foundationalism" as it is used with reference to Munchhausen's trilemma means that one must begin without circularity and without infinities. So, one obtains Russell's identification of individuals with terms. And, one obtains the "self-identity" that a term has uniquely with itself in justification of ontology. And one obtains the syntactic category of individual constants and variables in the specification of a first-order language.

One subtly here is that first-order inference rules mandate the necessary truth of reflexive equality statements. This is, in fact, not a consequence of Tarski's semantic conception of truth. Formalism cannot be absolute because substitutivity in a logical calculus requires a warrant. Hilbert's explicit presuppositions for studying consistency emphasize given domains over existential considerations. So, "self-identity" as an ontological presupposition is presumed by the inference rules.

Once you have some basics from some modern presentations such as those mentioned earlier, Kleene's "Introduction to Metamathematics" is a good historical reference.

Answer 3

Probably the biggest issue is whether mathematics is a branch of philosophical logic, or logic is a branch of mathematics. Most authors will take a strong stance one way or the other, depending on their personal background. Don't believe philosophers when they criticise mathematicians, or mathematicians when they criticise philosophers.

Plato proposed a world of ideals which mathematical objects (among others) inhabited. His Republic and other works are where it all started.

Roger Penrose is a mathematician and Nobel prize-winning physicist. His The Road to Reality is heavyweight in both meanings of the word, but Chapter 1 especially also covers many relevant issues, such as Plato's ideal world, in clearer language.

Russell and Whitehead's Principia Mathematica is a more directly philosophical work and set the tone for the twentieth century's love affair with set theory (Russell was a philosopher, Whitehead a mathematician). However their ideas are somewhat dated by modern standards.

Imre Lakatos; Proofs and Refutations attends to the more practical issue of what "doing maths" is, as opposed to the ontology of the maths itself.

Ernest Nagel and James R. Newman; Gödel's Proof examines in plain language his famous incompleteness theorem for certain logical systems, and some of its consequences. Bear in mind that he also proved a contrasting completeness theorem for a rather different principle of logical (in)completeness. (Sorry I do not have a good suggestion for that).