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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

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.

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.