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This got me thinking about human cognition and the meaning of mathematical truths, or even how does one know that something is true (and I assume that probably

This got me thinking about human cognition and the meaning of mathematical truths, or even how does one know that something is true (and I assume that probably none of this is original). At which point would Mochizuki’s work be considered proof? If another person regarded as an authority would claim to fully understand it (but couldn't explain it better to their peers)? If only 10% of mathematicians specialising in this field could understand it? If 40%? What if only a few percent understood the proof, but managed to "prove" to others that they understood it by using this knowledge to unlock other difficult problems? This seems to point that even in mathematics we can never claim to know what is true. At most, we can hold a belief, an assessment based on some parameters that increases our degree of belief that something is true. In fact, there's no need for a controversial proof to show this - it occurs with the simplest of mathematical proofs - after all, everything we believe in is based on the ability to detect consistency or lack thereof, which in itself is based on the ability of a system (in this case, our brains) to have a memory and notice where something happening now is consistent or inconsistent with something stored in our memory. Now this is quite problematic in terms of proof, because how can we trust our memory with a 100% certainty? The answer is that we can't. We attribute a high probability that what we know is true based on the consistency of other things we remember, but we can never prove anything .

Certhas 9y ago
"What if only a few percent understood the proof, but managed to "prove" to others that they understood it by using this knowledge to unlock other difficult problems?" Check the Terrence Tao comment below the post (also posted by someone else in here). I think if those who claim to understand the proof could use the techniques to proof other things, or even to provide new proofs of known results, the level of trust would immediately go up dramatically.
- deleted 9y ago
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tel 9y ago
While on one hand your "degrees of uncertainty" approach here is a pretty nice and valid one most of the time, I'd like to present another idea of mathematical proof. Something cannot be proven to a collection of people, but it can be proven to an individual. You can further say that a proof is generally true if, in principle, it can be proven to any given individual under mild assumptions of their mathematical experience. To prove something to an individual is to actually walk them through each step and show that they follow one after another. Actual mathematical proofs rarely follow such a simplified flow, but again in principle they _could_ and the missing steps are elided under the assumption that someone could in principle reconstruct them whenever they like (this time perhaps making slightly stronger assumptions about their background and skill). So, for all people the ABC conjecture will only become a theorem when they personally have walked through the proof. It's very expensive for any person to do this for any proofs they want to own, but they should do so for core ones and should feel comfortable that they could in principle do it for any that they take faith on. Thus, the ABC conjecture may fail to be a proof practically for at least two reasons: (1) it's just factually wrong, it does not exist as a script for making step-by-step deduction to the conclusion or (2) there is no pathway for a reasonable person to obtain confidence that they could in principle verify it. For instance, it could fail to be a proof for reason (2) entirely because IUTT is extremely complex and totally useless for all things besides the ABC conjecture. It would never be in the best interest of any person to obtain enough familiarity with IUTT to make the conjecture available to "in principle" verification. It seems like (2) is sort of the case as well. So, you can again boil this down to degrees-of-uncertainty and betting odds if you want, but proof has some interesting other dynamics: - the notion that it's about communication of an idea - the notion that it's dependent upon each individual's verification - the notion of "in principle" verifications - the notion of personal expense In particular, it creates a divide between some sort of Platonic reality and a person's belief but guards it by expense of consuming/integrating information instead of expense of gathering data. For instance, assume a completely trusted (99.99%) third party reads the proof and commits that it is correct. Bayesian reasoning would suggest that even if the person is normally trustworthy they could still be fallible and would still lend a lot of doubt to the truth of their words. But no number of tests could make this more "proven" to you if it's still just not worth the effort for anyone else to ever read the paper and internalize it. You'd never be able to justify the costs of those other tests because it's just so impractical as to never really reach the status of "universally accepted as true".

Certhas 9y ago

"What if only a few percent understood the proof, but managed to "prove" to others that they understood it by using this knowledge to unlock other difficult problems?" Check the Terrence Tao comment below the post (also posted by someone else in here). I think if those who claim to understand the proof could use the techniques to proof other things, or even to provide new proofs of known results, the level of trust would immediately go up dramatically.

deleted 9y ago
[deleted]

tel 9y ago

While on one hand your "degrees of uncertainty" approach here is a pretty nice and valid one most of the time, I'd like to present another idea of mathematical proof. Something cannot be proven to a collection of people, but it can be proven to an individual. You can further say that a proof is generally true if, in principle, it can be proven to any given individual under mild assumptions of their mathematical experience. To prove something to an individual is to actually walk them through each step and show that they follow one after another. Actual mathematical proofs rarely follow such a simplified flow, but again in principle they _could_ and the missing steps are elided under the assumption that someone could in principle reconstruct them whenever they like (this time perhaps making slightly stronger assumptions about their background and skill). So, for all people the ABC conjecture will only become a theorem when they personally have walked through the proof. It's very expensive for any person to do this for any proofs they want to own, but they should do so for core ones and should feel comfortable that they could in principle do it for any that they take faith on. Thus, the ABC conjecture may fail to be a proof practically for at least two reasons: (1) it's just factually wrong, it does not exist as a script for making step-by-step deduction to the conclusion or (2) there is no pathway for a reasonable person to obtain confidence that they could in principle verify it. For instance, it could fail to be a proof for reason (2) entirely because IUTT is extremely complex and totally useless for all things besides the ABC conjecture. It would never be in the best interest of any person to obtain enough familiarity with IUTT to make the conjecture available to "in principle" verification. It seems like (2) is sort of the case as well. So, you can again boil this down to degrees-of-uncertainty and betting odds if you want, but proof has some interesting other dynamics: - the notion that it's about communication of an idea - the notion that it's dependent upon each individual's verification - the notion of "in principle" verifications - the notion of personal expense In particular, it creates a divide between some sort of Platonic reality and a person's belief but guards it by expense of consuming/integrating information instead of expense of gathering data. For instance, assume a completely trusted (99.99%) third party reads the proof and commits that it is correct. Bayesian reasoning would suggest that even if the person is normally trustworthy they could still be fallible and would still lend a lot of doubt to the truth of their words. But no number of tests could make this more "proven" to you if it's still just not worth the effort for anyone else to ever read the paper and internalize it. You'd never be able to justify the costs of those other tests because it's just so impractical as to never really reach the status of "universally accepted as true".