5 ms·

Intuition in Mathematics (2012) [pdf]

119 points by jimsojim 10y ago

satai 10y ago
Funny related reading: Imre Lakatos - Proofs and refutations
- Etheryte 10y ago
  Freely available version: https://math.berkeley.edu/~kpmann/Lakatos.pdf https://math.berkeley.edu/~kpmann/Lakatos.pdf
  - satai 10y ago
    Only a part. It was published several times (as a paper, thesis and a book at least and the book is much longer). Anyway this part looks good enough to catch some points.
- HarryHirsch 10y ago
  Also: George Polya, "How to solve it"
deleted 10y ago
[deleted]
Etheryte 10y ago
While the author has ample experience in the field of philosophy and has spent considerable time studying intuition (curriculum vitae: http://www.as.miami.edu/personal/echudnoff/CV.pdf http://www.as.miami.edu/personal/echudnoff/CV.pdf), the writing style seems too informal and personal for my taste. Without looking up the author, I would not have dared to guesstimate that the author is an associate professor, but rather a hobbyist. Perhaps such a style is common in the field (I rarely read philosophy papers), but coming from a natural sciences background it seems very odd.
- WhitneyLand 10y ago
  I don't think informality is what's objectionable, I think it's good when papers are more accessible without sacrificing precision or expression. The problem here seems to be witting skills. For example, this should never be a sentence: "That is the aim of section 1."
  - handcreme 10y ago
    Non-native speaker here. What is wrong with this sentence and how would you improve it?
    - WhitneyLand 10y ago
      I hate to say it but I don't know grammar rules well enough to explain it. The best I can offer is something that sounds better to me. First example is the paper, second example is one possible rewrite: "The first order of business will be to draw some distinctions between these notions and pick an appropriate focus for our present inquiry. That is the aim of section 1." "Section 1 draws distinctions between these notions, one of which will be the focus of our inquiry."
      - jnbiche 10y ago
        You're probably mostly objecting to the passive voice, which is a reasonable stand to take in scientific writing. He uses passive voice three times in these two sentences alone.
        jackcarter 10y ago
        Using "to be" isn't necessarily the passive voice. The author doesn't use the passive voice in those sentences. http://writingcenter.unc.edu/handouts/passive-voice/ http://writingcenter.unc.edu/handouts/passive-voice/
        jnbiche 10y ago
Koshkin 10y ago
Intuition in mathematics is just as important as it is in any other human endeavor. On the other hand, there are many results and methods that are deemed non-intuitive - by students and (consequently) by instructors. Some of the results are so non-intuitive that they may end up being labeled "pathological" - despite being, in fact, completely logical consequence of the assumptions, of which one of the most well-known is Axiom of Choice. Another example is that differential forms are often seen by students as very non-intuitive compared to vectors, even though they are much more useful and can be efficiently used in more general settings than vectors can.
- justinpombrio 10y ago
  > Some of the results are so non-intuitive that they may end up being labeled "pathological" - despite being, in fact, completely logical consequence of the assumptions, of which one of the most well-known is Axiom of Choice. That's not true. The axiom of choice is actually famous for being independent of ZFC, meaning that it cannot be proven or disproven using the usual axioms of set theory. That's why it's an axiom: you need to assume its truth without grounds. Also, in intuitionistic logic, the axiom of choice is provably false.
  - waqf 10y ago
    Firstly, the sentence you quoted, while unfortunately ambiguous, is correct if you take "of which" as picking up "assumptions" instead of "results". That is, the assumptions, of which a famous one is AC, have counterintuitive consequences, of which the original poster does not give an example but Banach–Tarski is a commonly cited one. Secondly, the axiom of choice may be independent of ZF (depending on whether ZF is consistent) but it is certainly not independent of ZFC :).
    - justinpombrio 10y ago
      > Firstly, the sentence you quoted, while unfortunately ambiguous, is correct if you take "of which" as picking up "assumptions" instead of "results". Oh, that's what Koshkin was trying to say. > Secondly, the axiom of choice is not independent of ZFC, it's independent of ZF (assuming ZF is consistent) :). Ah, the C stands for choice...
xwat 10y ago
A quote from Von Neumann comes to my mind: "Young man, in mathematics you don't understand things. You just get used to them."
- deleted 10y ago
  [deleted]
- gamma_function 10y ago
  The same might be true of electrical engineering. I find it quite difficult sometimes to develop intuition about how electricity moves in analog circuits.
ihaveahadron 10y ago
The stupid paper doesn't make any sense. It says when you think of a line you think of the ends ending. That would be a LINE SEGMENT, a line goes in infinite directions. Fucking christ

119 points by jimsojim 10y ago

satai 10y ago

Funny related reading: Imre Lakatos - Proofs and refutations

Etheryte 10y ago
Freely available version: https://math.berkeley.edu/~kpmann/Lakatos.pdf https://math.berkeley.edu/~kpmann/Lakatos.pdf
- satai 10y ago
  Only a part. It was published several times (as a paper, thesis and a book at least and the book is much longer). Anyway this part looks good enough to catch some points.
HarryHirsch 10y ago
Also: George Polya, "How to solve it"

deleted 10y ago

[deleted]

Etheryte 10y ago

While the author has ample experience in the field of philosophy and has spent considerable time studying intuition (curriculum vitae: http://www.as.miami.edu/personal/echudnoff/CV.pdf http://www.as.miami.edu/personal/echudnoff/CV.pdf), the writing style seems too informal and personal for my taste. Without looking up the author, I would not have dared to guesstimate that the author is an associate professor, but rather a hobbyist. Perhaps such a style is common in the field (I rarely read philosophy papers), but coming from a natural sciences background it seems very odd.

WhitneyLand 10y ago
I don't think informality is what's objectionable, I think it's good when papers are more accessible without sacrificing precision or expression. The problem here seems to be witting skills. For example, this should never be a sentence: "That is the aim of section 1."
- handcreme 10y ago
  Non-native speaker here. What is wrong with this sentence and how would you improve it?
  - WhitneyLand 10y ago
    I hate to say it but I don't know grammar rules well enough to explain it. The best I can offer is something that sounds better to me. First example is the paper, second example is one possible rewrite: "The first order of business will be to draw some distinctions between these notions and pick an appropriate focus for our present inquiry. That is the aim of section 1." "Section 1 draws distinctions between these notions, one of which will be the focus of our inquiry."
    - jnbiche 10y ago
      You're probably mostly objecting to the passive voice, which is a reasonable stand to take in scientific writing. He uses passive voice three times in these two sentences alone.
      - jackcarter 10y ago
        Using "to be" isn't necessarily the passive voice. The author doesn't use the passive voice in those sentences. http://writingcenter.unc.edu/handouts/passive-voice/ http://writingcenter.unc.edu/handouts/passive-voice/
        jnbiche 10y ago

Koshkin 10y ago

Intuition in mathematics is just as important as it is in any other human endeavor. On the other hand, there are many results and methods that are deemed non-intuitive - by students and (consequently) by instructors. Some of the results are so non-intuitive that they may end up being labeled "pathological" - despite being, in fact, completely logical consequence of the assumptions, of which one of the most well-known is Axiom of Choice. Another example is that differential forms are often seen by students as very non-intuitive compared to vectors, even though they are much more useful and can be efficiently used in more general settings than vectors can.

justinpombrio 10y ago
> Some of the results are so non-intuitive that they may end up being labeled "pathological" - despite being, in fact, completely logical consequence of the assumptions, of which one of the most well-known is Axiom of Choice. That's not true. The axiom of choice is actually famous for being independent of ZFC, meaning that it cannot be proven or disproven using the usual axioms of set theory. That's why it's an axiom: you need to assume its truth without grounds. Also, in intuitionistic logic, the axiom of choice is provably false.
- waqf 10y ago
  Firstly, the sentence you quoted, while unfortunately ambiguous, is correct if you take "of which" as picking up "assumptions" instead of "results". That is, the assumptions, of which a famous one is AC, have counterintuitive consequences, of which the original poster does not give an example but Banach–Tarski is a commonly cited one. Secondly, the axiom of choice may be independent of ZF (depending on whether ZF is consistent) but it is certainly not independent of ZFC :).
  - justinpombrio 10y ago
    > Firstly, the sentence you quoted, while unfortunately ambiguous, is correct if you take "of which" as picking up "assumptions" instead of "results". Oh, that's what Koshkin was trying to say. > Secondly, the axiom of choice is not independent of ZFC, it's independent of ZF (assuming ZF is consistent) :). Ah, the C stands for choice...

xwat 10y ago

A quote from Von Neumann comes to my mind: "Young man, in mathematics you don't understand things. You just get used to them."

deleted 10y ago
[deleted]
gamma_function 10y ago
The same might be true of electrical engineering. I find it quite difficult sometimes to develop intuition about how electricity moves in analog circuits.

ihaveahadron 10y ago

The stupid paper doesn't make any sense. It says when you think of a line you think of the ends ending. That would be a LINE SEGMENT, a line goes in infinite directions. Fucking christ