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Dr. Zeilberger writes a rather pleasant rant. He appears frustrated with the lack of imagination in mathematicians which occurs when they focus too strongly on

Dr. Zeilberger writes a rather pleasant rant. He appears frustrated with the lack of imagination in mathematicians which occurs when they focus too strongly on the axiomatic nature of mathematics, and seems to wish that we take a step back from the powerful axiomatic tools we have developed, so that we may search for what is 'really true about the world' rather than 'playing a mathematical game/believing in a mathematical religion'. Three specific complaints he has involve a belief in infinity (and limits), which he asserts do not exist in the real world; a delay in the publication of a pair of papers on which he worked, due to a variety of circumstances; and a "pernicious" influence of axiomatic mathematics which leads to "stupid" questions such as Hilbert's Second Problem. The failed publication of one paper is particularly notable, as it claimed a counterexample of Fermat's Last Theorem (according to Dr. Zeilberger's "Opinion 123", on his Rutgers website), and (ibid) was recognized by Andrew Wiles himself as one of three possible counterexamples: the reason given for this oversight was in fact the acceptance of the decidedly non-rigourous statement "it is easily seen ..." (ibid). This particular instance seems to contradict the main thrust of Dr. Zeilberger's rant against a mathematics overburdened by rules. Unfortunately, little example of what mathematics SHOULD look like is offered - beyond a statement that 'obvious things should be treated as such' and an assertion that infinities and continuities do not occur in real life/nature/the universe. While such a statement may be understandable coming from a respected (and clearly accomplished) combinatorist such as Dr. Zeilberger, physics has yet to demonstrate conclusively that space and time are discrete - undisproven interpretations of quantum mechanics exist which allow for continuities, and 'the size of the universe' is defined as that which we can see (ie, within ~13.8 billion light-years of Earth/Sol) so that actual infinities are ruled out only because of our inability to perceive them. Perhaps Dr. Zeilberger needs to think outside his own discrete box.

Baeocystin 9y ago
A slight aside- due to the expansion of space, we can see objects that are currently further away than the simple amount of time it took the light to reach us. GN-z11 is one of the most distant objects imaged, has a light-travel distance of 13.4 billion light-years, but a proper distance of little more than double , 32 billion light-years. https://en.wikipedia.org/wiki/GN-z11 https://en.wikipedia.org/wiki/GN-z11
hvidgaard 9y ago
Re: infinities and continuities, to my understanding, it has been proved than interaction with the world is inheriently discrete determined by Planks constant - I am not a physicist, so may be wrong. If we cannot interact with arbitrarily precision and can only ever observe a finite universe, does it make sense to even ask if the it is any other way? Should it be figure out how to interact with the world using some new fundamental mechanic, other than a wave bound by C and h, it opens the question.
- mtzet 9y ago
  Even if the world is fundamentally discrete, that does not mean that continuous models are not valuable. They are. We are used to doing discrete approximations to continuous problems, but the other way around works too. For example, evaluating the sum 1 + 1/4 + ... + 1/n^2 is quite hard. On the other hand, we may approximate by the integral over x^2, which remarkably has an easy formula. Continuous and discrete models complements each other, they are not mutually exlusive.
  - hvidgaard 9y ago
    I wasn't trying to say that continues models are not valuable, only that the question "is the world continues?" does not make sense if we can only interact with it in a discrete way.
    - tome 9y ago
      We don't interact with complex numbers but "Are quantum 'probabilities' complexL" is still a meaningful question.
    - hardlianotion 9y ago
      That is simply a rewording of the question: is it useful to use continuity to model the world.
      - hvidgaard 9y ago
        This is getting quite philosophical now. It depends on the intrepetation of "world". If we mean the universe then no, it is to our current understanding of it, not useful to model it's entirety using continuity, if we can only messure it discretely. If by world we mean a subset of the universe, then it makes perfect sense, because the world might be on the macro level where this "discretness" is far smaller than we ever want to go.
jostylr 9y ago
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
thetwiceler 9y ago
Zeilberger's Opinion 123 is an April Fool's Day joke.

Baeocystin 9y ago

A slight aside- due to the expansion of space, we can see objects that are currently further away than the simple amount of time it took the light to reach us. GN-z11 is one of the most distant objects imaged, has a light-travel distance of 13.4 billion light-years, but a proper distance of little more than double , 32 billion light-years. https://en.wikipedia.org/wiki/GN-z11 https://en.wikipedia.org/wiki/GN-z11

hvidgaard 9y ago

Re: infinities and continuities, to my understanding, it has been proved than interaction with the world is inheriently discrete determined by Planks constant - I am not a physicist, so may be wrong. If we cannot interact with arbitrarily precision and can only ever observe a finite universe, does it make sense to even ask if the it is any other way? Should it be figure out how to interact with the world using some new fundamental mechanic, other than a wave bound by C and h, it opens the question.

mtzet 9y ago
Even if the world is fundamentally discrete, that does not mean that continuous models are not valuable. They are. We are used to doing discrete approximations to continuous problems, but the other way around works too. For example, evaluating the sum 1 + 1/4 + ... + 1/n^2 is quite hard. On the other hand, we may approximate by the integral over x^2, which remarkably has an easy formula. Continuous and discrete models complements each other, they are not mutually exlusive.
- hvidgaard 9y ago
  I wasn't trying to say that continues models are not valuable, only that the question "is the world continues?" does not make sense if we can only interact with it in a discrete way.
  - tome 9y ago
    We don't interact with complex numbers but "Are quantum 'probabilities' complexL" is still a meaningful question.
  - hardlianotion 9y ago
    That is simply a rewording of the question: is it useful to use continuity to model the world.
    - hvidgaard 9y ago
      This is getting quite philosophical now. It depends on the intrepetation of "world". If we mean the universe then no, it is to our current understanding of it, not useful to model it's entirety using continuity, if we can only messure it discretely. If by world we mean a subset of the universe, then it makes perfect sense, because the world might be on the macro level where this "discretness" is far smaller than we ever want to go.

jostylr 9y ago

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

thetwiceler 9y ago

Zeilberger's Opinion 123 is an April Fool's Day joke.