The most common argument that mathematical truth is not provability uses Tarski’s indefinability of truth theorem or Goedel’s first incompleteness theorem. But while this is a powerful argument, it won’t convince an intuitionist who rejects the law of excluded middle. Plus it’s interesting to see if a different argument can be constructed.
Here is one. It’s much less conclusive than the Tarski-Goedel approach. But it does seem to have at least a little bit of force. Sometimes we have experimental evidence (at least of the computer-based kind) for a mathematical claim. For instance, perhaps, you have defined some probabilistic setup, and you wonder what the expected value of some quantity Q is. You now set up an apparatus that implements the probabilistic setup, and you calculate the average value of your observations of Q. After a billion runs, the average value is 3.141597. It’s very reasonable to conclude that the last digit is a random deviation, and that the mathematically expected value of Q is actually π.
But is it reasonable to conclude that it’s likely provable that the expected value of Q is π? I don’t see why it would be. Or, at least, we should be much less confident that it’s provable than that the expected value is π. Hence, provability is not truth.
2 comments:
Is this case different in important ways from, say, empirically verifying Goldbach's conjecture for the first billion cases and concluding that it is reasonable to think that Goldbach's conjecture is true?
It's different, but it may not be different in important ways. Maybe one reason it's different is that it's a bit easier to imagine reconstructing my case as a piece of good Bayesian reasoning. It could be like this: On reasonable priors, if E[Q]=pi, it's pretty likely that we'd get an average near pi. On reasonable priors, if E[Q] is not pi, then our hypotheses about what E[Q] is will be uniformly distributed in some range, say between 0 and 5. Then we get Bayesian confirmation that E[Q]=pi by finding the average is near pi. On the other hand, to reconstruct a Bayesian argument for Goldbach's conjecture in a Bayesian way is a bit more difficult, I feel. But of course Bayesian reasoning about necessary truths is always at least a bit dubious.
Anyway, I chose the example to have two properties. First, it's a case where it sure seems reasonable on empirical grounds to conclude the conjecture is true. Second, the empirical grounds are not grounds for thinking the conjecture is provable. (Compare the case where the four-color conjecture is proved in a way where a computer is used to check a lot of cases. The empirical evidence of the computer saying that it checked all the relevant cases is not only evidence that the conjecture is true, but is evidence that the conjecture is provably true.)
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