A View of Mathematics (2005) [pdf] | Hacker News Reader

A View of Mathematics (2005) [pdf]

alainconnes.org

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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.

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.

Gowers also has an entry in the Very Short Introduction series, which you might like:

https://global.oup.com/academic/product/mathematics-a-very-s...

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/d...

The revised Stuart edition nothing is changed except Stewart has added a preface, and a 37 page chapter "Recent Developments".

John Stillwell also has a good survey book "Elements of Mathematics From Euclid to Gödel"

Stillwell is a great author. Naive Lie Theory is a great read, with a big section on quaternions.

also for all in 1 math for physics books, there's a bunch: Arfken et al., Mary Boas, Riley/Hobson /Bence etc: https://www.amazon.com/Mathematical-Methods-Physicists-Seven...

and http://www.goldbart.gatech.edu/PostScript/MS_PG_book/bookmas...

These are sort of orthogonal to the OP's link, they're not covering analysis, abstract algebra and topology, instead they're covering dif eq, spectral analysis, probability/stats, linear algebra.

We should also mention Penrose's "The Road To Reality" in this subthread. While it is essentially insane as a commercial/popular publishing project, I suspect it is also one of the best attempts to survey all of maths and mathematical physics in one location. Unless you have a PhD in mathematical physics, you will not be able to read all of it profitably. However, I suspect it might be a useful set of signposts for someone attempting to learn mathematical physics, to come back to again and again (over the course of, say, a lifetime). I'm sure it has some idiosyncracies (his graphical tensor notation I don't think has caught on that much in the field, and is likely to cause some surprise to anyone who hasn't seen it before) and controversial treatments of some areas, but in general I think (as someone firmly in early undergrad math territory) it is a fantastic book. The illustrations. Those have to be the best collection of informal attempts to illustrate physics-oriented maths that exist.

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.

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.

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?

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.

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?

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.)

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.

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

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"

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?!

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).

Also the random latin phrases and obscure words. Why is 'fecund' is used 3 times?

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.

Interesting, I have communicated in writing with many French people without encountering this. Maybe it's a more traditional thing. But thanks for explaining!

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 ;-)

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.

> pounds and feet remain just as convenient as kilograms and meters

You're probably American and therefore accustomed to it. If it really is 'just as convenient', why is metric the standard in scientific practice?

> the power-of-ten based metric system is not very efficient when it comes to doing calculations on computers anyway

It's more efficient than the Imperial system.

> why is metric the standard in scientific practice?

It is merely a historic artifact that became a tradition. With the advent of computers it has lost its objective significance as a "preferred system".

(BTW I find all these "kilo-nano" prefixes to be rather ugly...)

Do you really think a system where 1 mile = 1760 yards = 5280 feet = 63360 inches is 'just as convenient'? I think you're not being honest.

The metric system is standard in scientific practice because it is more sane than the imperial system.

My comment was a little tongue in cheek, but it seems I was not able to convey this. Anyway, roman numerals do just fine to number centuries, which are rarely used to do computations.

My point is that it all amounts to familiarity, and while people complain about someone mentioning the XIXth century, apparently everyone forgets that someone across the ocean may have the same difficulties hearing about someone being 6ft tall.

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