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What is Benacerraf’s dilemma, in words a school leaver can follow?

Fajar Nugroho Fellow
· 4 replies
Philosophy of mathematicsAnalyticIslamicPlatonismAl-GhazaliPaul Benacerraf#benacerraf#mathematical-knowledge#numbers#platonism
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I finished senior high school this year and I'm taking a gap year, which my parents call wasting time. I've been reading about whether numbers are discovered or invented, because my maths teacher always said "we discover mathematics, we don't make it up", and I want to know if she was right.

Every article I read mentions "Benacerraf's dilemma" as if everyone knows what it is. I've tried to read the original paper and I got lost in the first few pages.

Part of why I care: last month I read al-Ghazālī's account of his crisis of doubt. He says he lost trust in the senses, but at first he kept trusting necessary truths, like ten being more than three. He was so sure of that one that he says even a miracle couldn't make him doubt it. So it seems mathematical truths were the most certain thing he had. But if numbers aren't anywhere, how did he know anything about them?

Can someone explain the dilemma simply? What are the two horns, and why is it hard to escape both?

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4 answers

Accepted answer

  1. Ana Clara Ferreira

    ✓ Accepted answer Fellow

    Fajar, you've actually stated half of it already in your last question.

    Paul Benacerraf's paper "Mathematical Truth" (1973) says any account of mathematics should meet two demands.

    1. Uniform truth. "There are four prime numbers between 10 and 20" should be true in the same way as "there are four mosques on my street". The second sentence picks out real things. If maths is read the same way, "prime numbers" must pick out real objects too. That pushes you towards platonism.

    2. Possible knowledge. We should be able to explain how we know mathematical truths. The popular picture of knowledge then required causal contact with what is known: light from the mosques reaches your eyes. But numbers, on the platonist view, are outside space and time and cause nothing. So how could we know anything about them?

    The dilemma: views that satisfy demand 1 (platonism) make knowledge mysterious. Views that satisfy demand 2, for example ones that say maths is about rules for manipulating symbols, make it hard to see how mathematical sentences are true in the ordinary way.

    Your al-Ghazālī example is perfect: he was certain, but certainty is not an explanation of where the knowledge came from. Many philosophers now reject the causal theory of knowledge, which softens horn 2, but nobody has a fully accepted replacement.

Other answers

  1. Vikram Sethi

    Fellow

    When I teach this in Delhi I put it as a question to the class: how do you know that 7 is prime? They say "I checked". Then I ask: checked what? You didn't look at the number seven. You looked at marks on paper, or did a calculation in your head. So what connects your checking to the number itself, if the number is somewhere else? The silence after that question is Benacerraf's dilemma.

    Helpful · 1
  2. Emeka Anyaoha

    Fellow

    A footnote from physics: we know electrons exist because they push our instruments around. Numbers don't push anything, but every equation we use to describe electrons is full of them. That's another famous argument, the indispensability argument, which tries to answer Benacerraf by saying we know numbers exist the same way we know electrons do: they are part of our best theories. It has its own problems, but it's worth knowing about.

    Helpful · 2
  3. Fajar Nugroho

    Fellow

    Thank you, this is so much clearer than the paper. So my teacher's "we discover maths" puts her on horn 1, and she owes me an account of how she discovers it. I'm going to ask her that at the next reunion. Emeka, I'd like to read more about the electron argument.

    Helpful · 3