9 ms·

What Is Mathematics and What Should It Be? [pdf]

107 points by santaclaus 9y ago

openfuture 9y ago
My formulation would be something like this: Mathematics is how you discover further truths about worlds that have objective truths. We don't know if the world "we live in" is such a world but it contains a lot of those other worlds =)
- pdfernhout 9y ago
  Brilliant!
tpeo 9y ago
Reading the word "pseudo-problems" sets me off to a rough start. But calling problems stupid merely because they're unsolvable is a mortal sin, specially if there's a non-trivial reason why we cannot mathematically go from this to that. That the impossibility of squaring the circle is due to it's equivalence with the transcendentality of π is in itself an interesting piece of mathematics.
- marcosdumay 9y ago
  He didn't call them stupid because they were unsolvable. He called them stupid because they didn't exist on any practice. And, of course, the knowledge that solved them also become essential to other areas of Math, so I also don't like his framing.
110011 9y ago
Nothing is preventing you from using computers cleverly to gain a better understanding of mathematical problems you are facing. But abandoning or discounting the importance of rigor is down right stupid. The great achievement of mathematics is the vast repository of irrefutable statements which will stand for eternity. And this comes from rigor, which helps and does not hinder. Going back to his example, how does it help if we all collectively agree that, yes, there must be infinitely many twin primes because of computational evidence X, Y and Z. Nothing more than agreement has been achieved in such a case. And we are none the wiser in the way of insight or explanation besides the computations we already had...
- trentmb 9y ago
  > But abandoning or discounting the importance of rigor is down right stupid. https://en.wikipedia.org/wiki/Italian_school_of_algebraic_geometry#Collapse_of_the_school https://en.wikipedia.org/wiki/Italian_school_of_algebraic_ge...
WiltedSpinach 9y ago
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
jhanschoo 9y ago
It is unclear to me why this rant was necessary. The field of applied mathematics and other mathematical modeling sub-fields seems to be what the author means when he talks of mathematics as science and of mathematical truths. His criticism would have been much more salient if applied mathematics weren't thriving today more than ever--with the explosion of computational power every field now seeks to make evermore complex mathematical models of the problems they study and solve.
nippples 9y ago
We are already suffering greatly in computing because reduction of rigor. People who can't handle rigor in mathematics, the tool that every freaking scientist in STEM should seek a different, more enjoyable, field.
- kwoff 9y ago
  offer some proof of this?
0x264 9y ago
Mathematics is the art of highlighting the necessary consequences of a situation. That's all. The situations that we mathematicians mostly consider are axiomatic constructs, because it's easier to then unambiguously establish necessity. That this art, its methods, techniques and tools, happens to be so useful to other sciences (and in fact most of human knowledge) is an interesting phenomenon...
- amelius 9y ago
  Mathematics is also used for purely abstract entertainment. Also, your definition applies to physics as well.
  - 0x264 9y ago
    > Also, your definition applies to physics as well. Um.... I think that it looks like that a lot of what maths does is shared with physics, but I think (I could be wrong) that it's mostly because physics has adopted the language of mathematics. That having been said, there is an aspect where they are definitively different. Physics is motivated by the understanding of the physical world. For instance a physics theory gets dropped when we discover that we mis-observed whatever it was meant to explain. This doesn't happen in maths. The theories in maths (here defined as axiomatic structures) do not need to align with the natural world and get studied for other reasons than because they would increase the understanding of the natural world. Such understanding may eventually happen, but was not the motivation factor. What do you think ? :)
    - amelius 9y ago
      In my opinion, math is: - an extension of pure logic - not a science (since it cannot be falsified) - only indirectly concerned with observations (because we don't know the universe) - used to predict, but also to reason about the past (big bang theories) - a tool - a game (puzzle)
    - Klockan 9y ago
      Maths is logic with axioms that are intuitively true. Physics is maths with the addition of experimentally verified axioms. Chemistry is Physics with the addition of heuristics that explain phenomena which are too complex to express using casual logic. Every step after chemistry just add additional heuristics.
ldp01 9y ago
From an engineering perspective... Mathematics looks like an add-on to natural language designed to express precise concepts in the simplest possible way. (Simple != easy, of course). In a scientific or engineering context mathematics evolves naturally whenever the requirement for precision exceeds the capability of the available language and you have the right folks around to develop it.
- auggierose 9y ago
  Very nice description. This is how I feel about mathematics as well. This is also why I have no problems with infinity: In a lot of situations it describes problem and solution just in the most succinct way.
- majewsky 9y ago
  As evidence, look at how math looked before we formalized notation. For example, the algebra book by al-Khwarizmi (the guy whom algorithms are named after). Wikipedia has a commented excerpt: https://en.wikipedia.org/wiki/Muhammad_ibn_Musa_al-Khwarizmi#Algebra https://en.wikipedia.org/wiki/Muhammad_ibn_Musa_al-Khwarizmi...
- marcosdumay 9y ago
  I wonder if all engineers are algebrists. I also wonder what is the motivation of the other factions, since I was never able to understand them.
danielam 9y ago
Mathematics is a formal science, not an empirical science. In other words, both Zeilberger and those he criticizes (as he characterizes them) are wrong. Some interesting books on the subject... "An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure" by James Franklin "German Science" by Pierre Duhem
ithinkso 9y ago
That was very unpleasant to read but what he basically proposes is to change "don't know if true until proven true" to "true until proven false" which is ridiculous.
fmap 9y ago
> In particular, one should abandon the dichotomy between conjecture and theorem. Wasn't that the status quo before the 20th century? It's strange to suggest that working with infinite (or rather, ideal) objects is stupid. The sheer amount of progress in 20th century mathematics provides incontrovertible evidence that ideal objects are a useful reasoning tool. This is even true in combinatorics: working with generating functions is working with an algebraic structure on infinite streams... This essay is written to incite... there is so much in there that invites comment from everyone who has ever spent five minutes thinking about these things. At the same time it is lacking in examples for ways in which a mathematical world without rigor would be better than what we have today. So instead of getting worked up about the essay itself, does anybody here have concrete examples where a lack of rigor lead to faster progress?
- OJFord 9y ago
  It seems strange to me for the author to write that, and then to end with: > But I do believe that it is time to make [mathematics] a true science. Surely any 'true science' has a dichotomy between conjecture or hypothesis, and theorem or accepted fact?
hardlianotion 9y ago
I can't say I like the essentially manufactured grievance against Hardy and his young man's game quote, completely out of context. It is a cheap shot and has nothing to do with his argument.
pflats 9y ago
I am truly confused by this screed. >• Stupid Question 2: Trisect an arbitrary angle only using straight-edge and compass. >Many, very smart people, tried in vain, to solve this problem, until it turned out, in the 19th century, to be impossible >• Stupid Question 3: Double the cube. >Ditto, 2^(1/3) is a cubic-algebraic number. >• Stupid Question 4: Square the Circle. >Many, very smart people, tried (and some still do!) in vain, to solve this problem, until it turned out [...] to be impossible. >Today’s Mathematics Is a Religion >Its central dogma is thou should prove everything rigorously. To me, that's the entire point. You prove things rigorously so that (among many other reasons) when someone considers trying to square the circle, you can point them to the proof that the circle cannot be squared.
- ldp01 9y ago
  >Its central dogma is thou should prove everything rigorously. >To me, that's the entire point. You prove things rigorously so that (among many other reasons) when someone considers trying to square the circle, you can point them to the proof that the circle cannot be squared. This. Those defending mathematical rigour are a minority almost everywhere except a school of mathematics. There is no shortage of people trying all the slap-dash ways of doing things.
  - protonfish 9y ago
    Then why do I frequently get confronted by the religion of "inherent truth in math" and we keep teaching our children math made overly convoluted and unintuitive by the "rigorous" crowd? Pretending that this is only a minority is not helpful in quashing the damage done by magical thinking in mathematics. An example in computer science is the whole P = NP nonsense. We have never found any evidence that P = NP, nor has anyone discovered a compelling reason why it should be. Yet, the first sentence on its Wikipedia page is: > The P versus NP problem is a major unsolved problem in computer science. "Unsolved?" Only in the perspective of the "rigorous proof" crowd which you imply are an unimportant minority. The truth is that P = NP is just another, as Dr. Zeilberger calls it, "stupid question."
    - 9y ago
deleted 9y ago
[deleted]
roenxi 9y ago
This gentleman needs to watch Feynman talking about the difference between maths and physics. Science is all about gathering evidence and the scientific method focuses very heavily on observation and testing. Basically, it is impossible to conduct science without data. Maths is data independent. More data or less data doesn't influence what maths is. Maths links the axiom and the result - once a result has been prooven to follow from an axiom, the data is irrelevant. No amount of observation or testing will change the value of Pi. https://www.youtube.com/watch?v=obCjODeoLVw https://www.youtube.com/watch?v=obCjODeoLVw
- empath75 9y ago
  I wouldn't say it's data independent. Look at goldbach's conjecture. While there is no rigorous proof, it seems almost impossible to be the case that it is not true. A lot of conjectures like that are based on data gathered first.
  - roenxi 9y ago
    You are referring to an as yet unsolved problem. The maths is as yet un-done! Sure, there is data that suggests it is probably true. The scientific can say there is enough data to be almost certain it is true and move on. The mathematical community cannot say for certain it is true because it is not yet prooven. Which is why they are still working on it.
    - empath75 9y ago
      I would suggest that formulating conjectures based on data is part of the mathematical process, would you not?
      - roenxi 9y ago
        I would suggest that formulating conjectures based on data is specifically not maths. Some mathematicians start with data, and no doubt about it data is effective. But I'm specifically saying that that is employing a tool of the scientific community (see [1]) as a starting point before then proceeding to do some actual mathematical work. This distinction is why maths is often classified in the Arts rather than the sciences. But if you can point to a maths textbook that teaches someone "general theories" by printing 10,000 data points followed by a QED then I'd suggest it is a pretty extraordinary theory. [1] https://en.wikipedia.org/wiki/Scientific_method https://en.wikipedia.org/wiki/Scientific_method EDIT I'll throw in an example; a software consultant might be involved in invoicing for a project. The invoicing is still accounting work, even if it is being done for a software project.
jcranberry 9y ago
I hate combinatorics. Number theory and algebra are so beautiful and profound and I feel combinatorics just sucks out all the soul. Also strange that he cites Hersch, who, if I remember correctly became relevant during the post-modernism vs science wars, unless this is from that particular conflict--in which case I wasted my time reading this at all.
tempodox 9y ago
Just consider that academic math refused for over 300 years to acknowledge the existence of complex numbers, leaving the problem to the engineers. No scientist could afford this substitution of dogma for reality.
- tgb 9y ago
  Do you have a citation for that? Wikipedia's history of complex numbers makes no mention of this. What uses for complex numbers did engineers have prior to Gauss that weren't anticipated by mathematicians? https://en.m.wikipedia.org/wiki/Complex_number#History https://en.m.wikipedia.org/wiki/Complex_number#History
- j2kun 9y ago
  This is a straw man and does a huge disservice to the complicated history behind complex numbers. [1] In particular, mathematicians came up with it first (the Italians, Cardano et al., if you don't count the greeks who did something fuzzily approaching complex numbers), but it doesn't matter anyway because back then there were no clear lines between the sciences. Mathematicians were physicists, engineers, philosophers, astronomers, and whatever else. The "refuse to acknowledge the existence" line is also nonsense (ignoring for the moment that "existence" is not a useful term for an abstraction), because Euler made "i" widespread by including it in his famous textbook, the complex plane was invented as early as the 1600s, Gauss used it for some of his most impressive discoveries (and he lived at the same time as Fourier, who, let's be real, was also a mathematician), and their reservations were more along the lines of "guarding against its misuse, "which still occurs today in algebra classes all around the world, and having doubts about its "metaphysics." But I understand the desire to glorify engineers. It does often happen that practical concerns drive great mathematical discoveries. Just... be humble and do your research :) [1]: https://en.wikipedia.org/wiki/Complex_number#History https://en.wikipedia.org/wiki/Complex_number#History
- protonfish 9y ago
  I don't have a problem with the pure math crowd doing their own thing. They absolutely discover some amazing things using their techniques. What I have a problem with is that these people write the math textbooks. Why are we bamboozling our poor children with this impractical, esoteric, overcomplicated, pseudo-religious, gobbledygook? Primary and secondary school mathematics need to change focus to applied and computational math.
Ceezy 9y ago
"Its central dogma is thou should prove everything rigorously" I'm really worried that peoples who have knowledge in computer science doesn't get that from a false statement you can prove ANYTHING. And yes literally ANYTHING. Should we accpet models of computer that can compute faster than speed of light....?
- kwoff 9y ago
  I'm not convinced that you're really worried about that. You seem fairly dramatic, to be honest.
  - Ceezy 9y ago
    That's some undergraduate logic and it was my first weeks in college so yes i'm concern of what people get from expensive computer science degree.
aqsalose 9y ago
>[..] Since slaves did all the manual labor, the rich folks had plenty of time to contemplate their navels, and to ponder about the meaning of life. Hence Western Philosophy, with its many pseudo-questions was developed in the hands of Plato, Aristotle, and their buddies, and ‘Modern’ pure mathematics was inaugurated in the hands of the gang of Euclid et. al. Some call it "contemplating their navels", some other call it curiosity. I'm reminded of this other essay [1] that was discussed on HN a week ago: >[..] throughout the whole history of science most of the really great discoveries which had ultimately proved to be beneficial to mankind had been made by men and women who were driven not by the desire to be useful but merely the desire to satisfy their curiosity. Regarding the rest of the Zeilberger's essay ... while it is intuitively obvious that there something odd going with the assumptions in problems like the Zeno's paradox (that's why we call them paradoxes), it is certainly not intuitively obvious what that something exactly is, because it took us a good amount of principled application of curiosity (and quite while of time) to see why and how. What you'd call a mind that does not yearn to know the details but instead yells and lavishly prefers to drop "that's stupid" in boldface and with an exclamation point? I'm not sure, but "uninteresting" comes to mind. In general, it's easy to see that ancient Greeks were mistaken on some issue or their ideas on some other ones were downright silly now that we have spent over two millenia improving on them and developing the ideas and tools such as "experimental science". [1] Abraham Flexner, "Usefulness of Useless Knowledge", Harpers 179, 1939, https://library.ias.edu/files/UsefulnessHarpers.pdf https://library.ias.edu/files/UsefulnessHarpers.pdf HN discussion: https://news.ycombinator.com/item?id=14558775 https://news.ycombinator.com/item?id=14558775

107 points by santaclaus 9y ago

My formulation would be something like this: Mathematics is how you discover further truths about worlds that have objective truths. We don't know if the world "we live in" is such a world but it contains a lot of those other worlds =)

pdfernhout 9y ago
Brilliant!

tpeo 9y ago

Reading the word "pseudo-problems" sets me off to a rough start. But calling problems stupid merely because they're unsolvable is a mortal sin, specially if there's a non-trivial reason why we cannot mathematically go from this to that. That the impossibility of squaring the circle is due to it's equivalence with the transcendentality of π is in itself an interesting piece of mathematics.

marcosdumay 9y ago
He didn't call them stupid because they were unsolvable. He called them stupid because they didn't exist on any practice. And, of course, the knowledge that solved them also become essential to other areas of Math, so I also don't like his framing.

110011 9y ago

Nothing is preventing you from using computers cleverly to gain a better understanding of mathematical problems you are facing. But abandoning or discounting the importance of rigor is down right stupid. The great achievement of mathematics is the vast repository of irrefutable statements which will stand for eternity. And this comes from rigor, which helps and does not hinder. Going back to his example, how does it help if we all collectively agree that, yes, there must be infinitely many twin primes because of computational evidence X, Y and Z. Nothing more than agreement has been achieved in such a case. And we are none the wiser in the way of insight or explanation besides the computations we already had...

trentmb 9y ago
> But abandoning or discounting the importance of rigor is down right stupid. https://en.wikipedia.org/wiki/Italian_school_of_algebraic_geometry#Collapse_of_the_school https://en.wikipedia.org/wiki/Italian_school_of_algebraic_ge...

WiltedSpinach 9y ago

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

jhanschoo 9y ago

It is unclear to me why this rant was necessary. The field of applied mathematics and other mathematical modeling sub-fields seems to be what the author means when he talks of mathematics as science and of mathematical truths. His criticism would have been much more salient if applied mathematics weren't thriving today more than ever--with the explosion of computational power every field now seeks to make evermore complex mathematical models of the problems they study and solve.

nippples 9y ago

We are already suffering greatly in computing because reduction of rigor. People who can't handle rigor in mathematics, the tool that every freaking scientist in STEM should seek a different, more enjoyable, field.

kwoff 9y ago
offer some proof of this?

0x264 9y ago

Mathematics is the art of highlighting the necessary consequences of a situation. That's all. The situations that we mathematicians mostly consider are axiomatic constructs, because it's easier to then unambiguously establish necessity. That this art, its methods, techniques and tools, happens to be so useful to other sciences (and in fact most of human knowledge) is an interesting phenomenon...

amelius 9y ago
Mathematics is also used for purely abstract entertainment. Also, your definition applies to physics as well.
- 0x264 9y ago
  > Also, your definition applies to physics as well. Um.... I think that it looks like that a lot of what maths does is shared with physics, but I think (I could be wrong) that it's mostly because physics has adopted the language of mathematics. That having been said, there is an aspect where they are definitively different. Physics is motivated by the understanding of the physical world. For instance a physics theory gets dropped when we discover that we mis-observed whatever it was meant to explain. This doesn't happen in maths. The theories in maths (here defined as axiomatic structures) do not need to align with the natural world and get studied for other reasons than because they would increase the understanding of the natural world. Such understanding may eventually happen, but was not the motivation factor. What do you think ? :)
  - amelius 9y ago
    In my opinion, math is: - an extension of pure logic - not a science (since it cannot be falsified) - only indirectly concerned with observations (because we don't know the universe) - used to predict, but also to reason about the past (big bang theories) - a tool - a game (puzzle)
  - Klockan 9y ago
    Maths is logic with axioms that are intuitively true. Physics is maths with the addition of experimentally verified axioms. Chemistry is Physics with the addition of heuristics that explain phenomena which are too complex to express using casual logic. Every step after chemistry just add additional heuristics.

ldp01 9y ago

From an engineering perspective... Mathematics looks like an add-on to natural language designed to express precise concepts in the simplest possible way. (Simple != easy, of course). In a scientific or engineering context mathematics evolves naturally whenever the requirement for precision exceeds the capability of the available language and you have the right folks around to develop it.

auggierose 9y ago
Very nice description. This is how I feel about mathematics as well. This is also why I have no problems with infinity: In a lot of situations it describes problem and solution just in the most succinct way.
majewsky 9y ago
As evidence, look at how math looked before we formalized notation. For example, the algebra book by al-Khwarizmi (the guy whom algorithms are named after). Wikipedia has a commented excerpt: https://en.wikipedia.org/wiki/Muhammad_ibn_Musa_al-Khwarizmi#Algebra https://en.wikipedia.org/wiki/Muhammad_ibn_Musa_al-Khwarizmi...
marcosdumay 9y ago
I wonder if all engineers are algebrists. I also wonder what is the motivation of the other factions, since I was never able to understand them.

danielam 9y ago

Mathematics is a formal science, not an empirical science. In other words, both Zeilberger and those he criticizes (as he characterizes them) are wrong. Some interesting books on the subject... "An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure" by James Franklin "German Science" by Pierre Duhem

ithinkso 9y ago

That was very unpleasant to read but what he basically proposes is to change "don't know if true until proven true" to "true until proven false" which is ridiculous.

fmap 9y ago

> In particular, one should abandon the dichotomy between conjecture and theorem. Wasn't that the status quo before the 20th century? It's strange to suggest that working with infinite (or rather, ideal) objects is stupid. The sheer amount of progress in 20th century mathematics provides incontrovertible evidence that ideal objects are a useful reasoning tool. This is even true in combinatorics: working with generating functions is working with an algebraic structure on infinite streams... This essay is written to incite... there is so much in there that invites comment from everyone who has ever spent five minutes thinking about these things. At the same time it is lacking in examples for ways in which a mathematical world without rigor would be better than what we have today. So instead of getting worked up about the essay itself, does anybody here have concrete examples where a lack of rigor lead to faster progress?

OJFord 9y ago
It seems strange to me for the author to write that, and then to end with: > But I do believe that it is time to make [mathematics] a true science. Surely any 'true science' has a dichotomy between conjecture or hypothesis, and theorem or accepted fact?

hardlianotion 9y ago

I can't say I like the essentially manufactured grievance against Hardy and his young man's game quote, completely out of context. It is a cheap shot and has nothing to do with his argument.

pflats 9y ago

I am truly confused by this screed. >• Stupid Question 2: Trisect an arbitrary angle only using straight-edge and compass. >Many, very smart people, tried in vain, to solve this problem, until it turned out, in the 19th century, to be impossible >• Stupid Question 3: Double the cube. >Ditto, 2^(1/3) is a cubic-algebraic number. >• Stupid Question 4: Square the Circle. >Many, very smart people, tried (and some still do!) in vain, to solve this problem, until it turned out [...] to be impossible. >Today’s Mathematics Is a Religion >Its central dogma is thou should prove everything rigorously. To me, that's the entire point. You prove things rigorously so that (among many other reasons) when someone considers trying to square the circle, you can point them to the proof that the circle cannot be squared.

ldp01 9y ago
>Its central dogma is thou should prove everything rigorously. >To me, that's the entire point. You prove things rigorously so that (among many other reasons) when someone considers trying to square the circle, you can point them to the proof that the circle cannot be squared. This. Those defending mathematical rigour are a minority almost everywhere except a school of mathematics. There is no shortage of people trying all the slap-dash ways of doing things.
- protonfish 9y ago
  Then why do I frequently get confronted by the religion of "inherent truth in math" and we keep teaching our children math made overly convoluted and unintuitive by the "rigorous" crowd? Pretending that this is only a minority is not helpful in quashing the damage done by magical thinking in mathematics. An example in computer science is the whole P = NP nonsense. We have never found any evidence that P = NP, nor has anyone discovered a compelling reason why it should be. Yet, the first sentence on its Wikipedia page is: > The P versus NP problem is a major unsolved problem in computer science. "Unsolved?" Only in the perspective of the "rigorous proof" crowd which you imply are an unimportant minority. The truth is that P = NP is just another, as Dr. Zeilberger calls it, "stupid question."
  - 9y ago

deleted 9y ago

[deleted]

roenxi 9y ago

This gentleman needs to watch Feynman talking about the difference between maths and physics. Science is all about gathering evidence and the scientific method focuses very heavily on observation and testing. Basically, it is impossible to conduct science without data. Maths is data independent. More data or less data doesn't influence what maths is. Maths links the axiom and the result - once a result has been prooven to follow from an axiom, the data is irrelevant. No amount of observation or testing will change the value of Pi. https://www.youtube.com/watch?v=obCjODeoLVw https://www.youtube.com/watch?v=obCjODeoLVw

empath75 9y ago
I wouldn't say it's data independent. Look at goldbach's conjecture. While there is no rigorous proof, it seems almost impossible to be the case that it is not true. A lot of conjectures like that are based on data gathered first.
- roenxi 9y ago
  You are referring to an as yet unsolved problem. The maths is as yet un-done! Sure, there is data that suggests it is probably true. The scientific can say there is enough data to be almost certain it is true and move on. The mathematical community cannot say for certain it is true because it is not yet prooven. Which is why they are still working on it.
  - empath75 9y ago
    I would suggest that formulating conjectures based on data is part of the mathematical process, would you not?
    - roenxi 9y ago
      I would suggest that formulating conjectures based on data is specifically not maths. Some mathematicians start with data, and no doubt about it data is effective. But I'm specifically saying that that is employing a tool of the scientific community (see [1]) as a starting point before then proceeding to do some actual mathematical work. This distinction is why maths is often classified in the Arts rather than the sciences. But if you can point to a maths textbook that teaches someone "general theories" by printing 10,000 data points followed by a QED then I'd suggest it is a pretty extraordinary theory. [1] https://en.wikipedia.org/wiki/Scientific_method https://en.wikipedia.org/wiki/Scientific_method EDIT I'll throw in an example; a software consultant might be involved in invoicing for a project. The invoicing is still accounting work, even if it is being done for a software project.

jcranberry 9y ago

I hate combinatorics. Number theory and algebra are so beautiful and profound and I feel combinatorics just sucks out all the soul. Also strange that he cites Hersch, who, if I remember correctly became relevant during the post-modernism vs science wars, unless this is from that particular conflict--in which case I wasted my time reading this at all.

tempodox 9y ago

Just consider that academic math refused for over 300 years to acknowledge the existence of complex numbers, leaving the problem to the engineers. No scientist could afford this substitution of dogma for reality.

tgb 9y ago
Do you have a citation for that? Wikipedia's history of complex numbers makes no mention of this. What uses for complex numbers did engineers have prior to Gauss that weren't anticipated by mathematicians? https://en.m.wikipedia.org/wiki/Complex_number#History https://en.m.wikipedia.org/wiki/Complex_number#History
j2kun 9y ago
This is a straw man and does a huge disservice to the complicated history behind complex numbers. [1] In particular, mathematicians came up with it first (the Italians, Cardano et al., if you don't count the greeks who did something fuzzily approaching complex numbers), but it doesn't matter anyway because back then there were no clear lines between the sciences. Mathematicians were physicists, engineers, philosophers, astronomers, and whatever else. The "refuse to acknowledge the existence" line is also nonsense (ignoring for the moment that "existence" is not a useful term for an abstraction), because Euler made "i" widespread by including it in his famous textbook, the complex plane was invented as early as the 1600s, Gauss used it for some of his most impressive discoveries (and he lived at the same time as Fourier, who, let's be real, was also a mathematician), and their reservations were more along the lines of "guarding against its misuse, "which still occurs today in algebra classes all around the world, and having doubts about its "metaphysics." But I understand the desire to glorify engineers. It does often happen that practical concerns drive great mathematical discoveries. Just... be humble and do your research :) [1]: https://en.wikipedia.org/wiki/Complex_number#History https://en.wikipedia.org/wiki/Complex_number#History
protonfish 9y ago
I don't have a problem with the pure math crowd doing their own thing. They absolutely discover some amazing things using their techniques. What I have a problem with is that these people write the math textbooks. Why are we bamboozling our poor children with this impractical, esoteric, overcomplicated, pseudo-religious, gobbledygook? Primary and secondary school mathematics need to change focus to applied and computational math.

Ceezy 9y ago

"Its central dogma is thou should prove everything rigorously" I'm really worried that peoples who have knowledge in computer science doesn't get that from a false statement you can prove ANYTHING. And yes literally ANYTHING. Should we accpet models of computer that can compute faster than speed of light....?

kwoff 9y ago
I'm not convinced that you're really worried about that. You seem fairly dramatic, to be honest.
- Ceezy 9y ago
  That's some undergraduate logic and it was my first weeks in college so yes i'm concern of what people get from expensive computer science degree.

aqsalose 9y ago

>[..] Since slaves did all the manual labor, the rich folks had plenty of time to contemplate their navels, and to ponder about the meaning of life. Hence Western Philosophy, with its many pseudo-questions was developed in the hands of Plato, Aristotle, and their buddies, and ‘Modern’ pure mathematics was inaugurated in the hands of the gang of Euclid et. al. Some call it "contemplating their navels", some other call it curiosity. I'm reminded of this other essay [1] that was discussed on HN a week ago: >[..] throughout the whole history of science most of the really great discoveries which had ultimately proved to be beneficial to mankind had been made by men and women who were driven not by the desire to be useful but merely the desire to satisfy their curiosity. Regarding the rest of the Zeilberger's essay ... while it is intuitively obvious that there something odd going with the assumptions in problems like the Zeno's paradox (that's why we call them paradoxes), it is certainly not intuitively obvious what that something exactly is, because it took us a good amount of principled application of curiosity (and quite while of time) to see why and how. What you'd call a mind that does not yearn to know the details but instead yells and lavishly prefers to drop "that's stupid" in boldface and with an exclamation point? I'm not sure, but "uninteresting" comes to mind. In general, it's easy to see that ancient Greeks were mistaken on some issue or their ideas on some other ones were downright silly now that we have spent over two millenia improving on them and developing the ideas and tools such as "experimental science". [1] Abraham Flexner, "Usefulness of Useless Knowledge", Harpers 179, 1939, https://library.ias.edu/files/UsefulnessHarpers.pdf https://library.ias.edu/files/UsefulnessHarpers.pdf HN discussion: https://news.ycombinator.com/item?id=14558775 https://news.ycombinator.com/item?id=14558775