[HN Gopher] Mathematics in the 20th century, by Michael Atiyah [...
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Mathematics in the 20th century, by Michael Atiyah [pdf] (2002)
Author : practal
Score : 101 points
Date : 2025-02-09 08:54 UTC (14 hours ago)
(HTM) web link (marktomforde.com)
(TXT) w3m dump (marktomforde.com)
| practal wrote:
| Lots of interesting thoughts and insights in that article. I find
| it especially interesting to relate Geometry to Space, and
| Algebra to Time.
| enriquto wrote:
| intriguing that the word "spacetime" does not appear in this
| text. For sure it wouldn't be related to algebraic geometry!
| bluenose69 wrote:
| An RIS entry: TY - JOUR TI -
| Mathematics in the 20th century AU - Atiyah, Michael
| T2 - Bulletin of the London Mathematical Society AB - A
| survey is given of several key themes that have characterised
| mathematics in the 20th century. The impact of physics is also
| discussed, and some speculations are made about possible
| developments in the 21st century. DA - 2002/01//
| PY - 2002 DO - 10.1112/S0024609301008566 DP -
| DOI.org (Crossref) VL - 34 IS - 1 SP -
| 1 EP - 15 J2 - Bull. Lond. Math. Soc.
| LA - en SN - 0024-6093, 1469-2120 UR - https:/
| /www.cambridge.org/core/product/identifier/S0024609301008566/type
| /journal_article Y2 - 2025/02/09/11:08:49 ER -
| phwlarxoc wrote:
| The most stunning point of this paper is the paradoxical benefit
| of deliberately assuming some kind of "blindness" in one's own
| thinking as a mathematician: in reckoning algebraically we
| proceed eyes closed so to say. What we look at is neither the
| real world nor our own mind but abstract signs on paper. That is
| the algebraic, formal", "symbolic" way of thinking.
|
| Atiyah has this tradition start with Leibniz, and it marks
| exactly his opposition to Newton, the latter being mainly
| interested in physics and therefore restraining math by its
| grounding in the real world, whereas Leibniz would have
| understood the formal nature of the discipline. The antagonism
| re-emerges in the 20th century with Poincare-Arnold on one side
| and Hilbert-Bourbaki on the other.
|
| The point has been aptly made in the polemics of Brouwer against
| Hilbertian formalism, by saying that for the formalist
| mathematical exactness is basically grounded in _paper_ : "Op de
| vraag, waar die wiskundige exactheid dan wel bestaat, antwoorden
| beide partijen verschillend; de intuitionist zegt: In het
| menschelijk intellect, de formalist: Op het papier", see Hermann
| Weyl, Philosophie der Mathematik und Naturwissenschaft, 1927,
| p.49.
|
| I guess a very large majority of people would still think that
| math is the rational, systematic account of what is ("real
| world"), but Atiyah seems to say that from an inner-mathematical
| perspective, the purely formal conception of mathematics
| prevailed. Algebra was the "Faustian offer" handed over to
| mathematicians: in exchange for the formidable machine of
| symbolic reasoning, we would have to sacrifice the meaning of
| what we are dealing with, at leat temporarily.
| jfarmer wrote:
| "It is a profoundly erroneous truism, repeated by all copy-
| books and by eminent people when they are making speeches, that
| we should cultivate the habit of thinking of what we are doing.
| The precise opposite is the case. Civilization advances by
| extending the number of important operations which we can
| perform without thinking about them. Operations of thought are
| like cavalry charges in a battle -- they are strictly limited
| in number, they require fresh horses, and must only be made at
| decisive moments."
|
| -- Alfred North Whitehead, "Introduction to Mathematics" (1911)
| pmdulaney wrote:
| Part of what I take Whitehead to be saying is that the act of
| _truly thinking_ is difficult to the point of being
| psychologically painful. And I believe that fear of this pain
| is at the root of procrastination in the realm of academic
| work.
|
| The American Buddhist Cory Muscara has written:
| _Procrastination is the refusal or inability to be with
| difficult emotions._
| andrewflnr wrote:
| This view presupposes that things "we can perform without
| thinking about them" are done _correctly_. We don 't get
| there without thinking about them first. Doing things without
| thinking about them is, at the level of each individual, a
| luxury we earn by thinking about them really hard at first.
| At the scale of society, well, this is supposedly what school
| is for. But for the society to "not think about" things,
| individuals have to continue thinking about them.
|
| A lot of people don't get that far for a lot of tasks, so
| "think more" is not incorrect advice for them.
| rramadass wrote:
| > I guess a very large majority of people would still think
| that math is the rational, systematic account of what is ("real
| world"), but Atiyah seems to say that from an inner-
| mathematical perspective, the purely formal conception of
| mathematics prevailed. Algebra was the "Faustian offer" handed
| over to mathematicians: in exchange for the formidable machine
| of symbolic reasoning, we would have to sacrifice the meaning
| of what we are dealing with, at least temporarily.
|
| I am not sure that this has been a "good thing" for modern
| mathematics. While symbolic logic is definitely a necessity, it
| has been carried too far in as much as most folks are
| unable/find-it-difficult to model "real world" phenomena.
| Abstraction proceeds from the concrete to the general but if
| one loses sight of this link all symbolic manipulation is mere
| playing games without any understanding.
|
| V.I.Arnold in his essay _On teaching Mathematics_ makes this
| very point - https://www.math.fsu.edu/~wxm/Arnold.htm
|
| Excerpts:
|
| _The scheme of construction of a mathematical theory is
| exactly the same as that in any other natural science. First we
| consider some objects and make some observations in special
| cases. Then we try and find the limits of application of our
| observations, look for counter-examples which would prevent
| unjustified extension of our observations onto a too wide range
| of events.
|
| As a result we formulate the empirical discovery that we made
| as clearly as possible. After this there comes the difficult
| period of checking as to how reliable are the conclusions .
|
| At this point a special technique has been developed in
| mathematics. This technique, when applied to the real world, is
| sometimes useful, but can sometimes also lead to self-
| deception. This technique is called modelling. When
| constructing a model, the following idealisation is made:
| certain facts which are only known with a certain degree of
| probability or with a certain degree of accuracy, are
| considered to be "absolutely" correct and are accepted as
| "axioms". The sense of this "absoluteness" lies precisely in
| the fact that we allow ourselves to use these "facts" according
| to the rules of formal logic, in the process declaring as
| "theorems" all that we can derive from them.
|
| It is obvious that in any real-life activity it is impossible
| to wholly rely on such deductions. The reason is at least that
| the parameters of the studied phenomena are never known
| absolutely exactly and a small change in parameters (for
| example, the initial conditions of a process) can totally
| change the result.
|
| In exactly the same way a small change in axioms (of which we
| cannot be completely sure) is capable, generally speaking, of
| leading to completely different conclusions than those that are
| obtained from theorems which have been deduced from the
| accepted axioms. The longer and fancier is the chain of
| deductions ("proofs"), the less reliable is the final result.
|
| The mathematical technique of modelling consists of ignoring
| this trouble and speaking about your deductive model in such a
| way as if it coincided with reality. The fact that this path,
| which is obviously incorrect from the point of view of natural
| science, often leads to useful results in physics is called
| "the inconceivable effectiveness of mathematics in natural
| sciences" (or "the Wigner principle").
|
| "The subtle poison of mathematical education" (in F. Klein's
| words) for a physicist consists precisely in that the
| absolutised model separates from the reality and is no longer
| compared with it.
|
| nor discussing the danger of fetishising theorems are to be met
| in modern mathematical textbooks, even in the better ones. I
| even got the impression that scholastic mathematicians (who
| have little knowledge of physics) believe in the principal
| difference of the axiomatic mathematics from modelling which is
| common in natural science and which always requires the
| subsequent control of deductions by an experiment._
| Attempts to create "pure" deductive-axiomatic mathematics have
| led to the rejection of the scheme used in physics (observation
| - model - investigation of the model - conclusions - testing by
| observations) and its substitution by the scheme: definition -
| theorem - proof. It is impossible to understand an unmotivated
| definition but this does not stop the criminal algebraists-
| axiomatisators.
| photonthug wrote:
| I've heard of math-envy for physicists or computer scientists
| but physics-envy where mathematicians get riled about
| criminal axiomatizers seems .. less common.
|
| I don't necessarily accept the premise that what's described
| as Definition-theorem-proof is a _hugely_ different from
| observe-model-test, but if I did? Why not both? We only have
| a few ways to manufacture new high-quality knowledge, and I'm
| not sure there is a point to arguing about the "main" or best
| one.
|
| And while it might be impossible to "understand" an
| unmotivated /ungrounded definition, what is really required
| is to _entertain_ it long enough to see where it leads. Some
| gardens will bloom and some will not, based on consistency
| and richness and that's how you know if the definition is
| good. One might call that "testing by observation" also, eh?
| webnrrd2k wrote:
| This whole argument reminds me a bit of the book, Godel-
| Escher-Back, which is (partially) about the interplay of
| formal and informal systems. Both are necessary, and one
| without the other tend to be useless or sterile.
|
| Sometimes, when I've asked for an reason/explanation about
| some physics problem, I get a response like: "because of this
| equation". And in my brain there is a mismatch, were it
| doesn't feel at all like an actual explanation, because
| abstract symbol manipulation feels like, well, symbol
| manipulation.
|
| But I think that's the wrong interpretation, because someone
| who has been really deep in mathematics for a long time feels
| the math at a deeper level than I do. For example, it's like
| explaining something that's true because F=MA. Before I had
| any physics knowledge someone answering a question with "It's
| because F=MA" would have felt off, in the same way. After
| doing a bunch of problems then F=MA is a perfectly fine
| answer. They are just a lot better at getting meaning from
| the symbols and following the manipulations.
| ccppurcell wrote:
| Well to extend the analogy, I close my eyes when I'm listening
| to music. Even more so when I'm playing an instrument. And when
| I'm reading I can't be listening to music, at least not
| actively and nothing too interesting.
|
| I think we move back and forth to improve our understanding.
| The other day I read an article about Handel's messiah. I then
| went and listened to it.
|
| I recently found a family of tilings of the sphere by drawing
| pictures. To convince myself that the tilings really exist and
| I wasn't just tricking myself, I calculated the angles at the
| vertices (I used a computer for this). That in turn told me
| something about the relative size of the tiles. And so it goes.
| jll29 wrote:
| If you want to call it "deliberate blindness", I'll call it
| "focus".
| Jun8 wrote:
| This lecture achieves the astonishing feat of being very
| accessible to non-mathematicians while being deeply insightful.
| The global vs local distinction mentioned, although not the same,
| brought to mind Dyson's distinction between "birds" and "frogs":
| https://www.ams.org/notices/200902/rtx090200212p.pdf
| jll29 wrote:
| Here is Sir Atiyah's Mathematics Geneaology entry:
| https://genealogy.math.ndsu.nodak.edu/id.php?id=30949
|
| Receiving the RSE Fellowship with a handshake from him was one of
| my highlights during my beautiful time in Edinburgh. May he rest
| in peace.
| zyklu5 wrote:
| Atiyah is truly one of the giants of modern mathematics. I
| remember long ago I struggled through a reading course of his and
| Bott's Yang-Mills paper in graduate school. Like many great works
| of math it too had that paradoxical characteristic of
| transforming seemingly 'non-mathematics' into mathematics* by
| reversing the usual direction of application of one to the other,
| in this case, from physics to math. It would start a whole
| movement that'll produce much of modern geometries greatest hits
| like Donaldson's (his student) theorem in 4 manifolds to Witten's
| great papers.
|
| * A reason I think modern LLM architecture as they currently
| stand with their underlying attention mechanisms will not produce
| interesting new mathematics. A few other ideas are going to be
| needed.
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