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If numbers aren’t physical, how could we ever know anything about them?

Hetal Shah Fellow
· 4 replies
Philosophy of mathematicsAnalyticGreek & RomanPlatonismGottlob FregePaul BenacerrafPlato#abstract-objects#benacerraf-dilemma#mathematical-knowledge#platonism
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I count tablets all day. Three tablets, thirty tablets, a strip of ten. The tablets are physical: I can drop them, and I do. But the three is not in any of them. You can't drop a three.

My son's maths teacher told the class that numbers exist “out there”, independently of us, like Plato's forms. My son came home delighted. I came away confused. If numbers are not in space and time, and they don't push or pull anything, how does any of us, my son, the teacher, the woman at the pharmacy counter, come to know facts about them? I can check the tablets by looking. How would anyone check a number?

I asked a cousin who studied engineering and he said “don't think about it, just use them”, which is what my father said about honey and onions too, and I have been thinking about those for forty years.

Is there a name for this problem, and what are the main answers?

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4 answers

Accepted answer

  1. Ana Clara Ferreira

    ✓ Accepted answer Fellow

    There is a name: Benacerraf's dilemma, from Paul Benacerraf's paper “Mathematical Truth” (1973).

    He noticed that a philosophy of mathematics has to do two things at once. First, give a semantics: “there are three primes below ten” looks exactly like “there are three tablets on the counter”, so the natural reading treats numbers as objects, which leads toward Plato. Second, give an epistemology: we know most things through some causal contact, seeing, touching, being told. Abstract objects have no causal powers. So if numbers are Platonic objects, it's mysterious how anyone knows about them. Your question, exactly.

    The main answers: Platonists bite the bullet and offer a special faculty of mathematical intuition; Gödel held a view like this. Structuralists say mathematics is about patterns, and a number is a position in a pattern, and we know patterns because we meet instances of them, including in your strips of ten. Fictionalists say mathematical statements are strictly false but enormously useful, like a well-run story. Formalists say it's rule-governed symbol manipulation. Intuitionists say numbers are mental constructions.

    None commands agreement. Structuralism is probably the most popular today, partly because it softens the dilemma. Whether it escapes the problem or just moves it is still argued.

Other answers

  1. Vikram Sethi

    Fellow

    There's also the empiricist answer, which children seem to start with. John Stuart Mill thought arithmetic is made of very general facts about the world, learned from experience with collections like your tablets. Frege mocked this as pebble-and-gingerbread arithmetic: what experience of piles would teach you about a number with a million digits, or about zero? Most philosophers think Frege won that round, but every year I watch eleven-year-olds learn “three” from tablets, sweets and fingers before anything else.

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  2. Hetal Shah

    Fellow

    Thank you, both. I grew up Jain, so I can't help asking: is there anyone who says in one respect numbers are discovered and in another respect invented, and that both are true? It sounds like a question my grandfather would have answered with a seven-fold syādvāda table.

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    1. Ana Clara Ferreira

      Fellow

      Closer than you'd think, Hetal. Mark Balaguer argued in the late nineties that a careful Platonism and a careful fictionalism are each defensible and that there may be no fact of the matter which is right. Not quite syādvāda, but your grandfather would recognise the spirit.

      Helpful · 3