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Reading group: Dummett’s ‘The Philosophical Basis of Intuitionistic Logic’

Ana Clara Ferreira Fellow
· 5 replies
Philosophy of mathematicsAnalyticModern EuropeanL. E. J. Brouwer#anti-realism#excluded-middle#intuitionism#meaning-theory
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I keep noticing that people defend intuitionistic logic for two very different reasons. Brouwer's reason was a picture of mathematics as mental construction. Dummett's reason, in his 1973 paper "The Philosophical Basis of Intuitionistic Logic", was a theory of meaning. I'd like to read the paper slowly with others, especially people who don't already agree with it.

It's reprinted in Dummett's Truth and Other Enigmas and in the Benacerraf and Putnam anthology Philosophy of Mathematics, so most libraries will have it.

Plan (three weeks, about a third of the paper each):

  • Week 1: the opening. Dummett separates two routes to intuitionistic logic, one from a metaphysics of mathematical objects and one from meaning, and says only the second can work. Is he fair to Brouwer?
  • Week 2: the core argument. Meaning must be fully manifest in use. We can't manifest a grasp of truth conditions that might transcend any possible proof. So the meaning of a mathematical statement must be given by what counts as a proof of it.
  • Week 3: the consequences. Why that undermines excluded middle, and whether the argument stays inside mathematics or spreads to statements about the past, other minds and so on.

Post notes under the week you're on. Objections more welcome than summaries.

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5 replies

  1. Tomasz Szewczak

    Fellow

    Week 1. What struck me is how quickly Dummett sets Brouwer aside. Brouwer's view, that mathematics is an activity of the mind prior to language, could make intuitionistic logic look like a report about one man's mental life. Dummett wants a justification anyone who speaks a language would have to accept. It's a generous move toward intuitionism and a rather ungenerous one toward Brouwer.

    Helpful · 3
  2. Vikram Sethi

    Fellow

    Week 1, from a schoolteacher. I can't judge the metaphysics, but the idea that you understand a statement when you know what would count as proving it matches how my students actually learn. Nobody in Class 12 understands "there are infinitely many primes" by picturing an infinite set. They understand it when they can see the proof that, given any list of primes, there is always another.

  3. Oskar Nyberg

    Fellow

    Week 2. The manifestation argument is elegant, but I think it proves too much. We manifest our understanding of undecidable sentences in all sorts of ways: by how we draw inferences from them, what we'd count as evidence for them, how we'd revise if a proof turned up. Why must manifestation be "recognising a proof when presented with one"?

    And the argument doesn't obviously stop at mathematics. Dummett himself took it to the past ("The Reality of the Past" is the place), and a lot of readers get off the train there.

    Helpful · 1
    1. Ana Clara Ferreira

      Fellow

      Oskar, that's the objection I worry about most, and I don't think the paper fully answers it. Dummett's reply, as I read him, is that inferential use alone can't fix a determinate truth condition beyond proof, so the extra content is idle. Whether that's convincing depends on how much you trust holistic meaning.

      Week 3 practical note: proof assistants make this vivid. Coq's core logic is constructive, and classical axioms have to be added on purpose. Lean's big mathematics library, by contrast, uses classical reasoning freely. Same mathematics, two attitudes, both working.

      Helpful · 2
  4. Derek Hollis

    Fellow

    Week 3. In hardware verification the payoff of constructive proofs isn't philosophical, it's practical: a constructive proof that a value exists can be turned into a program that computes it. The Curry–Howard correspondence makes that literal. When I read Dummett I hear a meaning theory that happens to match how our tools already behave. Whether that vindicates his view or just shows his logic is useful is, I suppose, the question for another thread.

    Helpful · 3