In response to Jaffe and Quinn [math.HO/9307227], the author discusses forms of progress in mathematics that are not captured by formal proofs of theorems, especially in his own work in the theory of foliations and geometrization of 3-manifolds and dynamical systems.
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Fields medalist Terence Tao writes about earlier Fields medalist William Thurston, "In addition to his direct mathematical research contributions, Thurston was also an amazing mathematical expositor, having the rare knack of being able to describe the process of mathematical thinking in addition to the results of that process and the intuition underlying it. His wonderful essay “On proof and progress in mathematics“, which I highly recommend, is the quintessential instance of this; more recent examples include his many insightful questions and answers on MathOverflow."
Besides "On proof and progress in mathematics,"
http://arxiv.org/abs/math.HO/9404236
which I also highly…
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recommend, I especially like Thurston's article "Mathematical Education,"
http://arxiv.org/abs/math/0503081
which is full of good advice about thorough development of budding mathematicians.
From Tao's blog, another link to a remembrance of Thurston:
http://www.math.columbia.edu/~woit/wordpress/?p=5059
William P. Thurston 1994, "On proof and progress in mathematics"
Gives a good amount of insight into how academia works for mathematics, and gives a good contrast with how CS works. Don't be scared by the abstract, it's a completely non-technical paper. The academic/research culture can be more important than the results.
http://arxiv.org/abs/math.HO/9404236
I accept your correction that there is a difference between "promoting progress" and "coming up with ideas". (By coincidence I just reread http://arxiv.org/abs/math.HO/9404236 which argues, among other things, that progress in mathematics can be helped by not proving facts too quickly.)
I agree with you on the intent of the design of the patent system. However in a world filled with overbroad junk patents, where accidental infringement is much better than willful infringement, there are real incentives for innovative people to _not_ read patents. Because reading them opens you up to liability without teaching you anything you couldn't have thought of on your own.
As for…
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the cost of waiting, you are right that the incentive to wait is an inefficiency in my proposal. However the aim here is to achieve a good trade-off. Obviously more detailed analysis is required to find whether it achieves as good a trade-off as I think it does.
However for many years we blindly accepted very large increases in how easy it was to get a patent without substantive debate, and there is now substantial evidence that this has been a bad thing. So it seems to me that it shouldn't be hard to improve on the current state, and I think my proposal would be an improvement.
Finally I have to say that I don't understand what you mean by "presumption of validity". The way things work now is that if IBM wants me to enter into a cross-licensing arrangements, we both know that they can threaten me with a large stack of patents. We both know that most of those patents won't stand up in court, but fighting it will impose serious costs on me and they are likely to get me on something. So even though we both presume the patents are invalid, the threat is still good.
How would you change that dynamic? If your idea is that they spend their money to validate their patent before I spend a dime, how will we know it has been validated to my satisfaction? In theory that kind of validation is the job of the patent office. However regulatory capture has made that review a joke, and would eventually eventually ruin any other level of review you could add. But if I have to spend money in the process of getting the patent validated, then IBM still can threaten me as they would today.
In response to Jaffe and Quinn [math.HO/9307227], the author discusses forms of progress in mathematics that are not captured by formal proofs of theorems, especially in his own work in the theory of foliations and geometrization of 3-manifolds and dynamical systems.
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