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I remember Dr. Zeillberger from my graduate days at Rutgers. He clearly had a lot of interesting stuff to say which he packages in sensational and emotional for

I remember Dr. Zeillberger from my graduate days at Rutgers. He clearly had a lot of interesting stuff to say which he packages in sensational and emotional form. I found it off-putting then as well as now. But to be charitable, I think it comes from a place of pushing with tremendous force a pendulum of thought back from the axiomatic nature towards more exploratory and relevant mathematics. It reminds me of Howard Zinn's intro to A People's History of the US in which he says that he purposefully did not water down his arguments to be more neutral because of the overwhelming nature of the opposition. The infinity issue is quite interesting to ponder. Obviously with our current understanding of the universe and humans, the number of numbers ever encountered by all humans over all history would seem to be finite regardless of the fundamental nature of the universe itself. There is a very strong notion of our world being discrete in that way. And it is very interesting to pursue these ideas. For example, the proof of infinitely many integers is to assume there is a largest and we add 1. But what if the largest is unknown as is the case much of the time? This is true, for example, with computers (it can be known for the native format, but that can obviously be extended in programming in an arbitrary way) It is a practical question to ask about existence questions that require an axiom (axiom of choice often) which gives no constructive way of doing something. To what extent is that actually useful? And if we had an approximate construction of something without the axiom, but the thing itself may or may not exist depending on the axiom chosen, what would that mean? It is also interesting to think about doing away with infinity and thinking about how annoying it would be to not be able to talk about pi or e or even sqrt(2). There would be no irrational numbers, presumably, in this framework. One can then see the wonders that embracing infinity can lead to as a crutch for dealing with the horrendously messy world of the finite and discrete. Discussing these issues when learning mathematics strikes me as elevating a great deal of bland rule absorption and gives students power over the tools of mathematics. So this debate is useful in my opinion though it would be absurd to try and draw a conclusion. One last note about his discussion of the lack of rigor in QFT and that it was good. Recent work (at Rutgers[0]) suggests that at least some of the infinities and problems come from not appreciating the proper physical picture of what is actually happening. Nonetheless, the standard procedures worked to give us needed answers before such insight was discovered. I think the basic answer should be that formal and exploratory mathematics are both useful and it is important to avoid dogma from either side. [0]: https://arxiv.org/pdf/1703.04476.pdf https://arxiv.org/pdf/1703.04476.pdf

BoiledCabbage 9y ago
> It is a practical question to ask about existence questions that require an axiom (axiom of choice often) which gives no constructive way of doing something. To what extent is that actually useful? I'm by no means a mathematician, but I do sympathize somewhat with this. To put it a bit controversially, the axiom of choice feels to me almost like the string theory of mathematics. There's a lot of really cool stuff you can do with it, but we've left the realm of even potentially useful, and are off approaching a form of rigorous philosophy. I was looking into Banach–Tarski recently. Is it actually that useful to discuss something which almost certainly can never physically exist in our universe? Now of course you could play devil's advocate and say as some point people claimed you could never in the universe have a negative amount of something, so integers aren't useful, or a square root of a negative isn't useful so no imaginary numbers. Things are created all the type in math that we don't find practical applications for for decades or centuries. Yes that's true, but it's not a question of absolute, it's a matter of degree. Has math swung too far into the world non-relevant "play" at the expense of other discoveries? Recently we've just come across one of the most incredible tools of the past 1000 years. Computers! As a programmer aware of Hindley-Milner it boggles my mind the amount of programming people have been doing doing centuries without the benefit of computers to run them on. Now that we have them we can course correct. Previously we shooting in the dark, and were surprisingly accurate. Now that we have computers this utilize them to focus more and prioritize the amazing things that come from powerful computability. Let's explore the systems that can be constructed using these new amazing tools. Infinity in itself is ok as a model common real world applications (pi, e, sqrt 2). Getting out of my knowledge area, here, but what tools would we discover if we were focusing on a math without the Axiom of Choice? What proofs would be reformulated and what tools would we develop to do this work. Again, everyone should be free to think about whatever they like, but as a field it should also be cautious of how it spends it's resources. Particularly it's most valuable resource of intellectual capital.
- fmap 9y ago
  One thing which the author of this essay gets right is that contemporary mathematics is a bit dogmatic when it comes to the logical foundations. We should be looking at the Banach-Tarski theorem as evidence that a particular logical framework for reasoning about volumes/probability is more complicated than strictly necessary. Presumably no non-measurable sets exist in the "real world". At this point there are two possibilities: - A theory with non-measurable sets is much simpler/more expressive than the alternatives and thus is still a useful tool. - There is a simpler theory without this defect, which is at least as expressive when reasoning about real world phenomena. In this case, the latter possibility turns out to be true. It is just unbelievably difficult to convince mathematicians to change the rules of the game, even if you can point to concrete gains.

BoiledCabbage 9y ago

> It is a practical question to ask about existence questions that require an axiom (axiom of choice often) which gives no constructive way of doing something. To what extent is that actually useful? I'm by no means a mathematician, but I do sympathize somewhat with this. To put it a bit controversially, the axiom of choice feels to me almost like the string theory of mathematics. There's a lot of really cool stuff you can do with it, but we've left the realm of even potentially useful, and are off approaching a form of rigorous philosophy. I was looking into Banach–Tarski recently. Is it actually that useful to discuss something which almost certainly can never physically exist in our universe? Now of course you could play devil's advocate and say as some point people claimed you could never in the universe have a negative amount of something, so integers aren't useful, or a square root of a negative isn't useful so no imaginary numbers. Things are created all the type in math that we don't find practical applications for for decades or centuries. Yes that's true, but it's not a question of absolute, it's a matter of degree. Has math swung too far into the world non-relevant "play" at the expense of other discoveries? Recently we've just come across one of the most incredible tools of the past 1000 years. Computers! As a programmer aware of Hindley-Milner it boggles my mind the amount of programming people have been doing doing centuries without the benefit of computers to run them on. Now that we have them we can course correct. Previously we shooting in the dark, and were surprisingly accurate. Now that we have computers this utilize them to focus more and prioritize the amazing things that come from powerful computability. Let's explore the systems that can be constructed using these new amazing tools. Infinity in itself is ok as a model common real world applications (pi, e, sqrt 2). Getting out of my knowledge area, here, but what tools would we discover if we were focusing on a math without the Axiom of Choice? What proofs would be reformulated and what tools would we develop to do this work. Again, everyone should be free to think about whatever they like, but as a field it should also be cautious of how it spends it's resources. Particularly it's most valuable resource of intellectual capital.

fmap 9y ago
One thing which the author of this essay gets right is that contemporary mathematics is a bit dogmatic when it comes to the logical foundations. We should be looking at the Banach-Tarski theorem as evidence that a particular logical framework for reasoning about volumes/probability is more complicated than strictly necessary. Presumably no non-measurable sets exist in the "real world". At this point there are two possibilities: - A theory with non-measurable sets is much simpler/more expressive than the alternatives and thus is still a useful tool. - There is a simpler theory without this defect, which is at least as expressive when reasoning about real world phenomena. In this case, the latter possibility turns out to be true. It is just unbelievably difficult to convince mathematicians to change the rules of the game, even if you can point to concrete gains.