Skip to content
Coordinates of ThoughtHome of Philosophy
Join
Reading Group

Reading group: Benacerraf’s “What Numbers Could Not Be” and “Mathematical Truth”

Derek Hollis Fellow
· 5 replies
Philosophy of mathematicsAnalyticHartry FieldKurt GödelPaul Benacerraf#benacerraf#mathematical-truth#set-theory#structuralism
Follow

Paul Benacerraf wrote two short papers that between them set the agenda for half a century of philosophy of mathematics. Both are in the Benacerraf and Putnam anthology Philosophy of Mathematics: Selected Readings, and both are easy to find.

Plan, three weeks:

  • Week 1. “What Numbers Could Not Be” (1965). Two children are taught arithmetic by reducing numbers to sets, but by different reductions. They agree on every arithmetical fact and disagree on questions like whether 1 is a member of 3. Nothing seems to settle who is right. Benacerraf concludes that numbers can't be sets, and perhaps can't be objects at all.
  • Week 2. “Mathematical Truth” (1973). The dilemma: a semantics that treats mathematical statements like ordinary ones makes numbers objects; an epistemology that treats mathematical knowledge like ordinary knowledge needs causal contact with them. You can't easily have both.
  • Week 3. Responses. Structuralism, Field's fictionalism in Science Without Numbers, and Gödel's defence of mathematical intuition.

I'm a verification engineer who uses set-theoretic encodings every day, so the first paper is personal for me.

LikeReply
Follow5 replies

Join the conversation: members reply here, and joining is free. Reading is always free.

Join free to reply Sign in

5 replies

  1. Tomasz Szewczak

    Fellow

    Week 1, for anyone who wants the example concrete. On Zermelo's encoding, each number is the set containing its predecessor, so 3 is the set whose only member is 2. On von Neumann's, each number is the set of all smaller numbers, so 3 is the set of 0, 1 and 2. Ask “is 1 a member of 3?” and Zermelo says no, von Neumann says yes. Arithmetic itself doesn't care. That's the whole argument in one question.

  2. Ana Clara Ferreira

    Fellow

    Week 1 again. What I find remarkable is the conclusion: arithmetic is the science of a structure, and any system of objects with the right structure would serve. That's the seed of modern structuralism. But structuralists split over whether the structure is itself a kind of abstract object, which brings Platonism back, or whether talk of structure is just talk about all possible systems of that shape. Benacerraf doesn't decide.

    Helpful · 1
  3. Oskar Nyberg

    Fellow

    Week 2. The epistemological horn assumes a causal theory of knowledge, which was fashionable in 1973 and has since been largely abandoned. Does the dilemma survive without it? I think yes, in the form Hartry Field later gave it: whatever the right theory of knowledge, the Platonist owes an explanation of why mathematicians' beliefs reliably match facts about objects that don't interact with them. That reformulation is harder to dodge.

    Helpful · 2
  4. Vikram Sethi

    Fellow

    A teacher's note on week 2. My students never doubt that 7 is prime and they're never told how they could know it by contact with anything. They know it by doing a short calculation. Perhaps the best answer to the epistemological horn is just to look at what proving is. But then I'm not sure what the proof is about, which is the semantic horn again.

    Helpful · 3
  5. Derek Hollis

    Fellow

    Week 3, Gödel. In his essay on Cantor's continuum problem he says we have something like a perception of the objects of set theory, since the axioms force themselves upon us as true. It's a minority view, and it came from the person who knew the foundations best. I came in thinking Field's fictionalism was the obvious answer for an engineer. I leave thinking Benacerraf's first paper is right, that I don't know what my encodings are encodings of, and that I can live with that.