Category Theory for Dummies

It seems that category theory is the new hype — almost nobody actually understands what it is about, or, more importantly, what it is for. Let me tell you what it is for — it’s an important technical tool in mathematical research, which gives you new, coherent language, sometimes provides you with an additional insight in the structure of the stuff you are researching and makes it easier to notice and classify similarities between different kind of structures. Unfortunately, it is almost completely useless and uninteresting by itself — because, well, what’s interesting in objects and arrows anyway?

What make category theory interesting are its connections with various field[s] and [in] math and computer science. That’s why introducing “category theory for dummies” makes completely no sense — it’s like following Erlangen program to teach kids about points, lines and circles on a plane. The need and the significance of Erlangen program arise when you learn about many different geometries, notice what they have in common and what they do not, and try to find out what the geometry is all about. Without it, the Erlangen program is all about abstract bullshit, and the situation is completely the same with category theory. But nobody writes or posts Erlangen program for dummies on HN. Why? “General theory of everything” hype, that’s why. Erlangen program is “general theory of geometry”, but geometry seems a bit pale when compared to everything.

If you really want to understand the significance of category theory, then learn set theory, then algebra, then topology, then algebraic topology and algebraic geometry, or take abstract programming languages theory path. If you don’t care about all this stuff, because you’re hyped on the category theory, then you’re missing the point — it’s like you wanted to learn about algebraic topology, but did not care about algebra or topology.

Also anything that has “for dummies” in title should invariably remind you of Norvig’s essay (google Peter Norvig 21 days).

— xyzzyz 171 days ago

— Hacker News

2012.03.14 Wednesday ACHK

Category Theory | Lisp, 4

這段改編自 2010 年 3 月 6 日的對話。

我在前晚發現,我一直在研究的兩個課題,原來在某個角度之下,是同一樣東西。

(安:哪兩樣東西呢?)

一個就是 Paul Graham 強烈推介的 Lisp programming language(Lisp 編程語言)。另一個就是 John Baez 強烈推介的 Category Theory(範疇論)。

利用「範疇論」,我們可以將「相對論」的數學語言和「量子力學」的數學語言,統一成一套數學語言。留意,我不是指統一「相對論」和「量子力學」本身,而是指統一它們的數學語言。「相對論」所用的數學語言是 differential geometry(微分幾何)。「量子力學」所用的數學語言是 representation theory(表示論)。

Lisp 和「範疇論」都是我幾年前就開始關心的課題。最近才發現,它們竟然是「同一樣」東西,令我十分驚奇。幾天前,我還正正考慮著,好不好暫時放棄其中一門,因為我沒有時間同時深入研究 Lisp 和「範疇論」。現在,問題突然消失。

彷彿是同時愛上了兩位女仕,最終要二擇其一。正當面臨痛苦決擇之際,發現她們,竟然是同一個人。

— Me@2011.08.03

2011.08.03 Thursday (c) All rights reserved by ACHK

Quantum gravity 4

Ultimate goal is to merge:

quantum theory (representation theory)

&

general relativity (differential geometry)

by combining notions of

quantum state & geometry of space (spin network)

quantum history & geometry of spacetime (spin foam)

— John Baez

2010.07.25 Sunday ACHK

Mathematics, physics, logic and philosophy

So, we can only hope that in the future, more interaction between mathematics, physics, logic and philosophy will lead to new ways of thinking about quantum theory — and quantum gravity — that take advantage of the internal logic of categories like Hilb and nCob.

— Quantum Quandaries: A Category-Theoretic Perspective, John C. Baez

2010.05.19 Wednesday ACHK

Category Theory | Lisp

My current self-study programme includes

1. read almost everything written by Paul Graham

public domain image
Wikipedia

2. learn Lisp programming language (Common Lisp/Scheme) by reading Paul Graham’s textbooks

3. learn Category Theory so that I can read almost everything written by John Baez

public domain image
Wikipedia

Yester-night and tonight, I discovered that they are related:

“Lists, and recursive operations on them, are an excellent case in point. But the path connecting them to their mathematical underpinnings is a long and winding one, which lays in the realm of Category Theory.” — jao

I had never expected that. They are one thing.

Paul Graham –> Lisp –> Category Theory –> John Baez

I have been learning Lisp since 2000 (Machine Intelligence course), since 2006 (Structure and Interpretation of Classical Mechanics, Structure and Interpretation of Computer Programs), since 2010. I have been learning Category Theory since 2006-2008 (John Baez), since 2008 (Sets for Mathematics), since 2010. I had never expected that they are just two different languages of the same thing.

— Me@2010.03.04

2010.03.05 Friday (c) All rights reserved by ACHK

Quantum Gravity

Here I would like to propose another possibility, namely that quantum theory will make more sense when regarded as part of a theory of spacetime. Furthermore, I claim that we can only see this from a category-theoretic perspective — in particular, one that de-emphasizes the primary role of the category of sets and functions.

— Quantum Quandaries: A Category-Theoretic Perspective, John C. Baez

2010.02.23 Tuesday ACHK