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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 15th 2021
    • (edited Mar 15th 2021)

    I gave the reference

    the following commentary-line:

    for historical context see Hitchin 20, Sec. 8

    I admit that I only fully grasp this now, that Jaffe-Quinn’s article was in response to Atiyah’s influence on mathematics in whose wake Witten received the Fields medal.

    diff, v4, current

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeMar 16th 2021
    • (edited Mar 16th 2021)

    This article emphasizes that Jaffe-Quinne article was hi-eyebrow sanitizing view from rigorous school while the Wittten’s school was more inclusive. I was graduate student soon after that time (late 1990s) at the University of Wisconsin where mathematical physics was around local minimum (what changed in 2000s), a couple of mathematical physics by vocation both not active any more and I tried to learn developments between conformal field theory, quantum groups and string theory on my own from papers and it was impossible. Papers written by Witten’s school at that time had many silent conventions and jargon which you could learn in Princeton or Cambridge but not in Wisconsin. This changes dramatically in few years, and already the appearance of di Francesco’s book on CFT a couple of weeks before my preliminary was a huge step, the earlier books on the subject being incomplete and unclosed. I recall several colloqiua where lots of propaganda on 4-manifolds would be given and when you ask the speaker they wave hands, not willing to give you a definition. Harvey came from Chicago talking some relations to moonshine and so on and talking Lorentzian lattices. I ask what is a Lorentzian lattice. Answer I am a physicist and can not answer. I try to help. A lattice is a finite rank Abelian subgroup of Rn or Cn, so what additional requirements or properties you require to call it Lorentzian. You certainly have a criterium how to use the notion. And no wish to answer again. Another mathematician, a group theorist, on the way out of the room after the colloqium told me, that he was so unhappy that colloquia are inaccessible and people unwilling to help and that he was lost almost every week.

    A very loud person in the discussion after the Jaffe-Quinn article was Thurston. He said we experts do not need to say all the detail we know when we feel that we can complete the argument in principle and this is all superfluous. This was equally hi brow as the attitide of Jaffe and Quinn and frustrating for us not being at centers like Princeton where you have an expert next door to help. Today things are different, so much access, communication over internet, things like this nlab and so on, and young generation may get false impression if not knowing of the kind of problems I discuss. You see, in Wisconsin we graduate students organized a seminar for couple of years on Griffiths-Harris book on algebraic geometry, mainly introductory chapters to help each other to get through at least some background, it was so hard…

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 16th 2021

    It’s become clear that proceeding on implicit and alleged knowledge is bound to break beyond some point, also for the inner circle.

    As in historical examples:

    At first this did not matter too much, as Enriques’s intuition was so good that essentially all the results he claimed were in fact correct, and using this more informal style of argument allowed him to produce spectacular results about algebraic surfaces. Unfortunately, from about 1930 onwards…

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeMar 16th 2021

    Thank you Urs for quoting a beautiful analogy. It is in fact happy fact that there are so many styles in mathematics and that eventually most important progress gets organized in fairly natural way. And that most mathematicians (including you) are enthusiastic and honest to aid the process.

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 17th 2021

    This is one of the cases that Jaffe & Quinn mention

    More recently in this century the “Italian school” of algebraic geometry did not avoid major damage: it collapsed after a generation of brilliant speculation. See[EH, K] for discussions of the difficulties and the long recovery. In 1946 the subject was still regarded with such suspicion that Weil felt he had to defend his interest in it; see the introduction to [W].

    I spent quite a bit of time with this debate back in the 90s. It was a great eruption of people reflecting on how to practice at a time when this was immensely rare.

    In retrospect, the positive points of each contributor – rigour is important to maintain, only claim results when you have a proper proof, intuition should be conveyed, physical insight is feeding very important ideas into mathematics, etc. – were all well made. The internet has changed so much.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2021
    • (edited Mar 18th 2021)

    I had not actually seen this in Jaffe&Quinn, thanks for the pointer! It might be worth extracting their comment and whatever else might be found in the references they point to, in an entry on “Italian school of algebraic geometry” (or maybe some adjusted title – though that seems to be how they have entered posterity already).

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMar 18th 2021

    or maybe some adjusted title

    Maybe Enriques’ school of algebraic geometry.

    By the way, I think the whole debate is resolved not by weighing in for or against more or less rigorous arguments but by having a culture of proper citation, including proper self-citation: Always indicate your source, and be it “divine inspiration”!

    For example and regarding “Witten’s school”, the blame is much on people not citing him properly. A stark example is the paramount claim that there are non-abelian gauge fields on coincident D-branes:

    Witten stated this originally and explicity as an “obvious guess” (! p. 8 in arXiv:hep-th/9510135), but the next author and all their colleagues cited this as an established fact right away.

    This example is rather amazing if you know and realize the size of collapse the would occur should this guess one day turn out wrong.

    (The latest references that I am aware of that are trying to actually check this are listed here. I don’t doubt that it’s true, but it gives me a feeling of vertigo to think about this situation.)

    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 18th 2021
    Re #7: Guido Castelnuovo had Enriques and Zariski as students.

    Enriques has only 3 students listed on Math Genealogy, is it really accurate to list him as the main figure?
    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMar 18th 2021
    • (edited Mar 18th 2021)

    is it really accurate to list him as the main figure?

    Oh, you seem to be getting me wrong. I was thinking about how to properly attribute the collapse aspect.

    These days “Italian school” seems to become the chiffre for a major blunder in the history of mathematics. If the Wikipedia entry is at all accurate, then this is not only politically incorrect but outright incorrect.

    • CommentRowNumber10.
    • CommentAuthorzskoda
    • CommentTimeMar 18th 2021
    • (edited Mar 18th 2021)

    Enriques has only 3 students listed on Math Genealogy, is it really accurate to list him as the main figure?

    Often the pinacle of the Italian school of algebraic geometry (and yes, this is the usual way people refer to this tradition and period, but one could say classical Italian school of algebraic geometry, to be sure) is (also) associated with Severi.

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