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Human understanding is critical. See what Bill Thurston said:"The rapid advance of computers has helped dramatize this point, because computers and people are v

Human understanding is critical. See what Bill Thurston said:"The rapid advance of computers has helped dramatize this point, because computers and people are very different. For instance, when Appel and Haken completed a proof of the 4-color map theorem using a massive automatic computation, it evoked much controversy. I interpret the controversy as having little to do with doubt people had as to the veracity of the theorem or the correctness of the proof. Rather, it reflected a continuing desire for human understanding of a proof, in addition to knowledge that the theorem is true." From Bill Thurston: “On Proof and Progress in Mathematics”, Bulletin of the American Mathematical Society, 30(2), 161-177 https://arxiv.org/pdf/math/9404236 https://arxiv.org/pdf/math/9404236

deleted 6y ago
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dwheeler 6y ago
Some mathematicians think that human understanding of every step is critical. That doesn't actually make it critical. At one time it was very important to have humans weave cloth. Now we generally expect machines to do that. Computers will not eliminate the need for human mathematicians. It will merely change what they do. In many ways it will be more freeing, because no one will need to review a proof to see if its steps are correct. Mathematicians will still need to develop definitions and determine what is important to prove, and at least in the short-term they will need to do much of the proving since machines are not yet very good at that. Many other fields have changed with computers, there's no reason to believe mathematics cannot be changed.
- raincom 6y ago
  Well, if you think knowledge is same as truth, computer verifications can do. Today, at least the consensus in philosophy of science is not centered on truth. It is true that a falsity can't be knowledge. There is more to knowledge than truth.
  - dwheeler 6y ago
    But that is the problem: we are claiming that we "know" things are absolutely true, but we are using methods that are known to be inadequate for that purpose. Mathematicians are smart, but they are human. They make mistakes and miss others'. Math has increasingly become too complex to entrust mere human verification as being adequately trustworthy. Human verification has already been repeatedly demonstrated as inadequate. We must choose between truth or easier-to-understand claims that are sometimes false because we depend on fragile human verification. I think we should choose truth. Where we can get human understandability as well as truth, great! But formal verification should become the minimum over time, because truth should be the minimum. We can't do that all at once right now. But knowing the goal is the first step.

deleted 6y ago

[deleted]

dwheeler 6y ago

Some mathematicians think that human understanding of every step is critical. That doesn't actually make it critical. At one time it was very important to have humans weave cloth. Now we generally expect machines to do that. Computers will not eliminate the need for human mathematicians. It will merely change what they do. In many ways it will be more freeing, because no one will need to review a proof to see if its steps are correct. Mathematicians will still need to develop definitions and determine what is important to prove, and at least in the short-term they will need to do much of the proving since machines are not yet very good at that. Many other fields have changed with computers, there's no reason to believe mathematics cannot be changed.

raincom 6y ago
Well, if you think knowledge is same as truth, computer verifications can do. Today, at least the consensus in philosophy of science is not centered on truth. It is true that a falsity can't be knowledge. There is more to knowledge than truth.
- dwheeler 6y ago
  But that is the problem: we are claiming that we "know" things are absolutely true, but we are using methods that are known to be inadequate for that purpose. Mathematicians are smart, but they are human. They make mistakes and miss others'. Math has increasingly become too complex to entrust mere human verification as being adequately trustworthy. Human verification has already been repeatedly demonstrated as inadequate. We must choose between truth or easier-to-understand claims that are sometimes false because we depend on fragile human verification. I think we should choose truth. Where we can get human understandability as well as truth, great! But formal verification should become the minimum over time, because truth should be the minimum. We can't do that all at once right now. But knowing the goal is the first step.