10 ms·

A View of Mathematics (2005) [pdf]

179 points by kercker 9y ago

devereaux 9y ago
I am not a mathematician but I found that an interesting read. I am not sure how rigorous all this is, and even whether I understood the material completely. It just tickles the imagination. So here is my summary, in the hope it will tickle your imagination too. (If a mathematician can read it, please correct my layman mistakes) TLDR: after applying some special re-normalization to results from quantum field theory, coefficients of different models match and we can even get ratio of integers instead of bizarre numbers. It implies quantum field theory is not so special: we just don't understand some of its features, that seem to come from geometric symmetries. If I'm not too wrong, it also implies everything is discrete in the universe.
- mysterypie 9y ago
  You have definitely intrigued me enough that I want to read it too. May I ask how long it took you to read it? It's 36 pages not including the references, and it's only been up on Hacker News for a short time. As a non-mathematician were you able to read that quickly? It doesn't look dense with proofs but it does look like something I'd print out and read over coffee for several hours. I guess I'm surprised that some people can read (and understand/appreciate) math and physics papers so quickly.
  - devereaux 9y ago
    I am happy I intrigued you enough that you decided to read it too! That's exactly why I posted a TLDR. I usually skim through comments before reading long articles, to check whether the article is interesting and worth my time. There was no comment so I made a bet and gave the article a try, as a payback to all the times other HN comments guided me to very interesting reads. About the time it took me to read the article, I read it during a break while working on some R code this afternoon. Based on the time of my commits, I can tell you it took me about 20 minutes. But I just went through the reasoning. I did not go into it very deeply or check the equations because mathematics is not my domain. I am in economics and IT (but going into statistics as it just seems more fun) I will also try to go over this article in more detail with a coffee in a few days as I found it very inspiring. Hopefully there will be more comments here to share some insights and act as a guide. Like, maybe a mathematician can spot errors in my summary, and suggest us something to read before this article to understand it better?
  - danbruc 9y ago
    I skimmed it in a bit less than an hour, like reading every second paragraph. If you are not a mathematician and physicist, it will probably take you weeks or months to follow the argument in any detail and even then you will still be far away from really understanding it. That is some seriously advanced stuff and it touches on a quite broad spectrum of ideas. I also tend to question what devereaux thinks is the essence of it. I would say it is mostly a presentation of the evolution of mathematical ideas with some focus on ideas applicable to physics. I did not notice any new physical ideas, at best new mathematical ideas applied to existing physical theories and even that only on the last couple of pages. Also to the best of my knowledge our current physical theories are heavily constrained by very general principles, at times up to uniqueness. As a simple example, the Poincaré group of special relativity is the unique solution if you make some quite basic assumptions like homogeneity and isotropy of space. That conflicts with saying that quantum theory is not special. Also the sentence mentioning renormalization does not really sound like it was written with an understanding of what renormalization in this context means. I hope I do not sound to harsh, I am far from an expert myself, but that TL;DR seems way of in my opinion.
    - devereaux 9y ago
      It is not hash, I welcome criticism! This is how we improve!! And yes, my TLDR is mostly based on the last pages. The rest just seems to present the ideas that helped reach this conclusion. Regarding renormalization, there are many things I (wrongly?) call a renormalization - even doing a PCA and dropping the components that add little to the variance of the data. This destructs some signal, unless you believe that is just noise and the data should only be analyzed along say the first 3 PCs. I'd call that a renormalization in a 3d space. Maybe this is an improper use of the word in this context?
      - auntienomen 9y ago
        In QFT, renormalization is a way of mapping the behavior of a QFT at one distance scale onto its behavior at a different distance scale. Long story short, there's an equivalence between changing distance scales and changing the coupling constants in the theory. Once you get this mapping right, you discover the infinities present in the naive computations have vanished. (There is actually a quantity which has its normalization reset in this mapping, but that's actually the least interesting part of the story. It's somewhat odd that it's become the name for the whole mapping.)
- Retra 9y ago
  It doesn't imply everything is discrete. That is a rather meaningless phrase: language is already discrete, so everything that can be described with language can be said to be discrete in that sense, including anything you'd think is not discrete. For instance, the topological notion of continuity is about preserving nearness which wouldn't say anything about a non-discreteness of elements considered. I'm struggling to come up with a proper notion of non-discreteness in the first place, since there are plenty of things known to be both discrete and continuous at the same time. I would argue instead that the "discreteness of the universe" is not even a very interesting question to resolve. And in physics, it would need to have scientific verification. If you say space is discrete, then where is your minimum wavelength? How does the discretization of position relate to the discretization of angle? Etc.. It's not really about mathematics at that point, but about finding a physical phenomena for which this (vague) notion of total discreteness is relevant.
- MaysonL 9y ago
  From the preface to his current book (on the web[1]): "However, there are two fundamental problems whose difficulty is a clear reminder of our limited knowledge, and whose solution would require a more sophisticated understanding than the one currently within our immediate grasp: • The construction of a theory of quantum gravity (QG) • The Riemann hypothesis (RH) The purpose of this book is to explain the relevance of noncommutative geometry (NCG) in dealing with these two problems. Quite surprisingly, in so doing we shall discover that there are deep analogies between these two problems which, if properly exploited, are likely to enhance our grasp of both of them." The connecting of quantum gravity and the Riemann hypothesis blows my mind. I may never get any great understanding of it, but I have to try. [1]http://www.alainconnes.org/docs/bookwebfinal.pdf http://www.alainconnes.org/docs/bookwebfinal.pdf
iaabtpbtpnn 9y ago
The use of Roman numerals for centuries in this paper is obnoxious. I already have to mentally subtract 1... now I have to translate number systems too?!
- Koshkin 9y ago
  I have not noticed this as I am probably too used to it, but I tend to agree that using Roman numerals in this day and age doesn't make much sense and should be avoided (outside art anyway).
- enthdegree 9y ago
  Also the random latin phrases and obscure words. Why is 'fecund' is used 3 times?
- scentoni 9y ago
  The author is French. Using Roman numerals for centuries is the standard in Romance languages. And fecund is less obscure if your language derives from Latin. Not disagreeing, just explaining.
  - iaabtpbtpnn 9y ago
    Interesting, I have communicated in writing with many French people without encountering this. Maybe it's a more traditional thing. But thanks for explaining!
  - pathsjs 9y ago
    Absolutely common in Italy as well. If someone doesn't like this convention... well, this is more or less what we think when we meet articles measuring weight in pounds or height in feet ;-)
    - Koshkin 9y ago
      While nobody in their right mind would use Roman numerals to teach or do arithmetic today, pounds and feet remain just as convenient as kilograms and meters. Also, the power-of-ten based metric system is not very efficient when it comes to doing calculations on computers anyway.
danharaj 9y ago
This is a nice overview of some of the fundamental aspects of modern mathematics and how they fit together. It's definitely targeted at an advanced reader though. A book describing the large-scale structure of modern mathematics and its fundamental concepts at an undergraduate level is Mathematics, Form and Function by Saunders Mac Lane. If you like this paper but feel like it's currently out of your reach, you might like this book. Something deep about mathematics that I feel has only begun to be explored in the last few decades or so is how intimately connected computation/logic is to algebra and geometry. Connes doesn't really touch on that here but research programs like Geometry of Interaction and Geometric Complexity Theory are really exciting to me.
- Myrmornis 9y ago
  Looks really interesting, I'm going to try to read it, and thanks for your suggestion of the MacLane book. Another really excellent resource along those lines is the Princeton Companion to Mathematics, in particular the introductory essay by Tim Gowers (who is also the editor of the book). By along those lines, I mean pointing to advanced mathematics, from the point of view of someone who is kind of OK with introductory undergrad math.
  - stablemap 9y ago
    Gowers also has an entry in the Very Short Introduction series, which you might like: https://global.oup.com/academic/product/mathematics-a-very-short-introduction-9780192853615 https://global.oup.com/academic/product/mathematics-a-very-s...
- gtani 9y ago
  some other survey /panorama books for various levels: What is Mathematics, Courant and Robbins (I haven't read the 2nd ed, edited by Ian Stewart) https://www.amazon.com/All-Mathematics-You-Missed-Graduate/dp/0521797071 https://www.amazon.com/All-Mathematics-You-Missed-Graduate/d...
  - hackermailman 9y ago
c517402 9y ago
If you liked this essay, you might also like the book by Marcel Berger, Geometry Revealed: A Jacob's Ladder to Higher Geometry. A quote from the cover: "The underlying motivating concept for the present book is that it offers readers the elements of a modern geometric culture by means of a whole series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction. Starting with such natural, classical objects as lines, planes, circles, spheres, polygons, polyhedra, curves, surfaces, convex sets, etc., crucial ideas and above all abstract concepts needed for attaining the results are elucidated. These are conceptual notions, each built "above" the preceding and permitting an increase in abstraction, represented metaphorically by Jacob's ladder with its rungs"

179 points by kercker 9y ago

I am not a mathematician but I found that an interesting read. I am not sure how rigorous all this is, and even whether I understood the material completely. It just tickles the imagination. So here is my summary, in the hope it will tickle your imagination too. (If a mathematician can read it, please correct my layman mistakes) TLDR: after applying some special re-normalization to results from quantum field theory, coefficients of different models match and we can even get ratio of integers instead of bizarre numbers. It implies quantum field theory is not so special: we just don't understand some of its features, that seem to come from geometric symmetries. If I'm not too wrong, it also implies everything is discrete in the universe.

mysterypie 9y ago
You have definitely intrigued me enough that I want to read it too. May I ask how long it took you to read it? It's 36 pages not including the references, and it's only been up on Hacker News for a short time. As a non-mathematician were you able to read that quickly? It doesn't look dense with proofs but it does look like something I'd print out and read over coffee for several hours. I guess I'm surprised that some people can read (and understand/appreciate) math and physics papers so quickly.
- devereaux 9y ago
  I am happy I intrigued you enough that you decided to read it too! That's exactly why I posted a TLDR. I usually skim through comments before reading long articles, to check whether the article is interesting and worth my time. There was no comment so I made a bet and gave the article a try, as a payback to all the times other HN comments guided me to very interesting reads. About the time it took me to read the article, I read it during a break while working on some R code this afternoon. Based on the time of my commits, I can tell you it took me about 20 minutes. But I just went through the reasoning. I did not go into it very deeply or check the equations because mathematics is not my domain. I am in economics and IT (but going into statistics as it just seems more fun) I will also try to go over this article in more detail with a coffee in a few days as I found it very inspiring. Hopefully there will be more comments here to share some insights and act as a guide. Like, maybe a mathematician can spot errors in my summary, and suggest us something to read before this article to understand it better?
- danbruc 9y ago
  I skimmed it in a bit less than an hour, like reading every second paragraph. If you are not a mathematician and physicist, it will probably take you weeks or months to follow the argument in any detail and even then you will still be far away from really understanding it. That is some seriously advanced stuff and it touches on a quite broad spectrum of ideas. I also tend to question what devereaux thinks is the essence of it. I would say it is mostly a presentation of the evolution of mathematical ideas with some focus on ideas applicable to physics. I did not notice any new physical ideas, at best new mathematical ideas applied to existing physical theories and even that only on the last couple of pages. Also to the best of my knowledge our current physical theories are heavily constrained by very general principles, at times up to uniqueness. As a simple example, the Poincaré group of special relativity is the unique solution if you make some quite basic assumptions like homogeneity and isotropy of space. That conflicts with saying that quantum theory is not special. Also the sentence mentioning renormalization does not really sound like it was written with an understanding of what renormalization in this context means. I hope I do not sound to harsh, I am far from an expert myself, but that TL;DR seems way of in my opinion.
  - devereaux 9y ago
    It is not hash, I welcome criticism! This is how we improve!! And yes, my TLDR is mostly based on the last pages. The rest just seems to present the ideas that helped reach this conclusion. Regarding renormalization, there are many things I (wrongly?) call a renormalization - even doing a PCA and dropping the components that add little to the variance of the data. This destructs some signal, unless you believe that is just noise and the data should only be analyzed along say the first 3 PCs. I'd call that a renormalization in a 3d space. Maybe this is an improper use of the word in this context?
    - auntienomen 9y ago
      In QFT, renormalization is a way of mapping the behavior of a QFT at one distance scale onto its behavior at a different distance scale. Long story short, there's an equivalence between changing distance scales and changing the coupling constants in the theory. Once you get this mapping right, you discover the infinities present in the naive computations have vanished. (There is actually a quantity which has its normalization reset in this mapping, but that's actually the least interesting part of the story. It's somewhat odd that it's become the name for the whole mapping.)
Retra 9y ago
It doesn't imply everything is discrete. That is a rather meaningless phrase: language is already discrete, so everything that can be described with language can be said to be discrete in that sense, including anything you'd think is not discrete. For instance, the topological notion of continuity is about preserving nearness which wouldn't say anything about a non-discreteness of elements considered. I'm struggling to come up with a proper notion of non-discreteness in the first place, since there are plenty of things known to be both discrete and continuous at the same time. I would argue instead that the "discreteness of the universe" is not even a very interesting question to resolve. And in physics, it would need to have scientific verification. If you say space is discrete, then where is your minimum wavelength? How does the discretization of position relate to the discretization of angle? Etc.. It's not really about mathematics at that point, but about finding a physical phenomena for which this (vague) notion of total discreteness is relevant.
MaysonL 9y ago
From the preface to his current book (on the web[1]): "However, there are two fundamental problems whose difficulty is a clear reminder of our limited knowledge, and whose solution would require a more sophisticated understanding than the one currently within our immediate grasp: • The construction of a theory of quantum gravity (QG) • The Riemann hypothesis (RH) The purpose of this book is to explain the relevance of noncommutative geometry (NCG) in dealing with these two problems. Quite surprisingly, in so doing we shall discover that there are deep analogies between these two problems which, if properly exploited, are likely to enhance our grasp of both of them." The connecting of quantum gravity and the Riemann hypothesis blows my mind. I may never get any great understanding of it, but I have to try. [1]http://www.alainconnes.org/docs/bookwebfinal.pdf http://www.alainconnes.org/docs/bookwebfinal.pdf

iaabtpbtpnn 9y ago

The use of Roman numerals for centuries in this paper is obnoxious. I already have to mentally subtract 1... now I have to translate number systems too?!

Koshkin 9y ago
I have not noticed this as I am probably too used to it, but I tend to agree that using Roman numerals in this day and age doesn't make much sense and should be avoided (outside art anyway).
enthdegree 9y ago
Also the random latin phrases and obscure words. Why is 'fecund' is used 3 times?
scentoni 9y ago
The author is French. Using Roman numerals for centuries is the standard in Romance languages. And fecund is less obscure if your language derives from Latin. Not disagreeing, just explaining.
- iaabtpbtpnn 9y ago
  Interesting, I have communicated in writing with many French people without encountering this. Maybe it's a more traditional thing. But thanks for explaining!
- pathsjs 9y ago
  Absolutely common in Italy as well. If someone doesn't like this convention... well, this is more or less what we think when we meet articles measuring weight in pounds or height in feet ;-)
  - Koshkin 9y ago
    While nobody in their right mind would use Roman numerals to teach or do arithmetic today, pounds and feet remain just as convenient as kilograms and meters. Also, the power-of-ten based metric system is not very efficient when it comes to doing calculations on computers anyway.

danharaj 9y ago

This is a nice overview of some of the fundamental aspects of modern mathematics and how they fit together. It's definitely targeted at an advanced reader though. A book describing the large-scale structure of modern mathematics and its fundamental concepts at an undergraduate level is Mathematics, Form and Function by Saunders Mac Lane. If you like this paper but feel like it's currently out of your reach, you might like this book. Something deep about mathematics that I feel has only begun to be explored in the last few decades or so is how intimately connected computation/logic is to algebra and geometry. Connes doesn't really touch on that here but research programs like Geometry of Interaction and Geometric Complexity Theory are really exciting to me.

Myrmornis 9y ago
Looks really interesting, I'm going to try to read it, and thanks for your suggestion of the MacLane book. Another really excellent resource along those lines is the Princeton Companion to Mathematics, in particular the introductory essay by Tim Gowers (who is also the editor of the book). By along those lines, I mean pointing to advanced mathematics, from the point of view of someone who is kind of OK with introductory undergrad math.
- stablemap 9y ago
  Gowers also has an entry in the Very Short Introduction series, which you might like: https://global.oup.com/academic/product/mathematics-a-very-short-introduction-9780192853615 https://global.oup.com/academic/product/mathematics-a-very-s...
gtani 9y ago
some other survey /panorama books for various levels: What is Mathematics, Courant and Robbins (I haven't read the 2nd ed, edited by Ian Stewart) https://www.amazon.com/All-Mathematics-You-Missed-Graduate/dp/0521797071 https://www.amazon.com/All-Mathematics-You-Missed-Graduate/d...
- hackermailman 9y ago

c517402 9y ago

If you liked this essay, you might also like the book by Marcel Berger, Geometry Revealed: A Jacob's Ladder to Higher Geometry. A quote from the cover: "The underlying motivating concept for the present book is that it offers readers the elements of a modern geometric culture by means of a whole series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction. Starting with such natural, classical objects as lines, planes, circles, spheres, polygons, polyhedra, curves, surfaces, convex sets, etc., crucial ideas and above all abstract concepts needed for attaining the results are elucidated. These are conceptual notions, each built "above" the preceding and permitting an increase in abstraction, represented metaphorically by Jacob's ladder with its rungs"