I recently had 2 arguments with different people, who took extreme positions on what we mean by truth. One argued that there is no difference between mathematical truths and truths in fiction – in fact, he described mathematics, that is not being ‘applied’, as ‘mathematical fiction’. The other argued that there is no objective truth and everything we claim to know are only ‘beliefs’, including mathematics. When I told her that I know there will always be mathematics that remain unknown, she responded that I ‘believe I know’. I thought that was an oxymoron, but I let it go. The trivial example, that there are an infinite number of primes or an infinite number of digits in pi, should put that to rest, or so one would think.
Norman Wildberger, whom I’ve cited before, says that he doesn’t ‘believe’ in Real numbers, and neither does he believe in infinity, and he provides compelling arguments. But I feel that he’s redefining what we mean by mathematics, because his sole criterion is that it can be computed. Meanwhile, we have a theorem by Gregory Chaitin who contends that there are infinitely more incomputable Real numbers than computable Real numbers. People will say that mathematics is an abstract product of the mind, so who cares. But, as Paul Davies says, ‘mathematics works’, and it works so well that we can comprehend the Universe from the cosmic scale to the infinitesimal.
Both of my interlocutors, I should point out, were highly intelligent, well-educated and very articulate, and I believe that they really believed in what they were saying. But, if there is no objective truth, then there are no 'true or false' questions that can be answered. To take the example I’ve already mentioned, it’s either true or false that we can’t know everything in mathematics. And if it’s false, then we must know everything. But my interlocutor would say that I claimed we’d never know and I can’t say I know that for sure.
Well, putting aside the trivial example of infinity, there are proofs based on logic that says it’s true and that’s good enough for me. She claimed that logic can be wrong if the inputs are wrong, which is correct. In mathematics, this is dependent on axioms, and mathematics like all other sciences never stands still, so we keep getting new axioms. But it’s the nature of science that it builds on what went before, and, if it’s all ‘belief’, then it’s a house built on sand. And if it's a house built on sand, then all the electronic gadgets we use and the satellite systems we depend on could all crash without warning, but no one really believes that.
So that’s one side of the debate and the other side is that truths in art have the same status as truths in science. There are a couple of arguments one can use to counter this, the most obvious being that a work of art, like Beethoven’s 5th, is unique – no one else created that. But Pythagoras’s theorem could have been discovered by anyone, and in fact, it was discovered by the Chinese some 500 years before Pythagoras. I write fiction, and while I borrow tropes and themes and even plot devices from others, I contend that my stories are unique and so are the characters I create. In fact, my stories are so unique, that they don’t even resemble each other, as at least one reader has told me.
But there is another argument and that involves memes, which are cultural ideas, for want of a better definition, that persist and propagate. Now, some argue that virtually everything is a meme, including scientific theories and mathematical theorems. But there is a difference. Cultural memes are successful because they outlive their counterparts, but scientific theories and mathematical theorems outlive their counterparts because they are successful. And that’s a fundamental distinction between truth in mathematics and science, and truth in art.
Addendum: I just came across this video (only posted yesterday) and it’s very apposite to this post. It’s about something called Zero Knowledge Proof, and it effectively proves if someone is lying or not. It’s relevance to my essay is that it applies to true or false questions. You can tell if someone is telling the truth without actually knowing what that truth is. Apparently, it’s used algorithmically as part of blockchain for bitcoin transactions.
To give the example that Jade provides in her video, if someone claims that they have a proof of Riemann’s hypothesis, you can tell if they’re lying or not without them having to reveal the actual proof. That’s a very powerful tool, and, as a consequence, it virtually guarantees that a mathematical truth exists for a true or false proposition; in this hypothetical case, Riemann’s hypothesis, because it’s either true or false by definition.
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