
A View of Mathematics (2005) [pdf] - kercker
http://www.alainconnes.org/docs/maths.pdf
======
danharaj
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.

~~~
gtani
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...](https://www.amazon.com/All-Mathematics-You-Missed-
Graduate/dp/0521797071)

~~~
hackermailman
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"

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

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devereaux
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
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.

~~~
danbruc
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
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
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.)

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c517402
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"

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iaabtpbtpnn
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?!

~~~
scentoni
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.

~~~
pathsjs
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
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.

~~~
cgmg
> 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.

~~~
Koshkin
> _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...)

~~~
cgmg
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.

