Will learning category theory lead to a better and clearer understanding of mathematics?

Question

I read the first chapter on a book about category theory Conceptual Mathematics:A first introduction to categories.In the preface the authors say: It has been the good fortune of the authors to live in these interesting times, and to see how the fundamental insight of categories has led to clearer understanding, thereby better organizing, and sometimes directing, the growth of mathematical knowledge and its applications.

The introduction was about how the flight of a bird is a function from time to space.It was a very neat explanation.Is category theory about stuff similar to this ?

So my question is: Will learning category theory lead to a better and clearer understanding to mathematics and are there any prerequisites to learning category theory ?

I wouldn't say there are prerequisites to start learning the basics, but there are many, many interesting examples that would require some background in algebra, topology, geometry, logic, etc. — José Siqueira, Apr 29 '14 at 22:57
It's kind of a chicken and the egg problem; category theory is helpful, but unless you have an idea how and where, it may be difficult to comprehend and appreciate. — Marcin Łoś, Apr 29 '14 at 23:05
@VladimirPetrov Note that you can notify users by putting an @ and typing in their username. — , Apr 29 '14 at 23:16
Not necessarily. Further, there are many fantastic research mathematicians who know nothing about category theory. And knowing category theory wouldn't help them be any better. — Potato, Apr 29 '14 at 23:18
@VladimirPetrov That's why I've put this under comments, and not as a proper answer. — José Siqueira, Apr 29 '14 at 23:20
related: What are the prerequisites for learning category theory? Is category theory worth learning for the sake of learning it? — Grigory M, Apr 29 '14 at 23:48
'clearer understanding to mathematics' is very broad and vague (what mathematics, on which level, what do you know already?..) If you expect some kind of magic that will make just everything completely clear, obviously you're going to be disappointed... — Grigory M, Apr 29 '14 at 23:54

score 16 · Answer 1 · answered Apr 29 '14 at 23:37

A physicist, a biologist, and a chemist are all trying to study a particular differential equation. Each of these differential equations are mathematically identical but full of specialized constants and terms related to each field. The physicist has an equation about mechanical stress and the biologist has an equation about the movement of cells. Each of the three scratch their heads: how can one solve such a difficult equation with dozens of components and factors?

They go to the mathematics dept. to speak to somebody who works on differential equations. When the mathematician sees these equations she is considerably less frightened. Because she does not know about cells or metabolic rates or chemical reactions, she just sees an equation full of arbitrary constants, and is not confused by the unnecessary information about what those constants mean. She recognizes all three problems as a certain sort of differential equation that she has studied in the past, and solves them quickly using the general theory she has developed.

In this analogy, physics, chemistry, and biology are different branches of math. Sometimes these fields have specific problems phrased in different languages which, on some intrinsic level, are logically identical. In certain situations this extra information only confuses us and is not necessary for solving the problem. When we show this problem to category theory, it does not see the specialized details but only the inherent underlying problem, and we can then apply the broad framework of category theory to tackle all these issues at once.

May I quote "physics, chemistry, and biology are different branches of math." without context? :-] — Martin Brandenburg, Apr 30 '14 at 06:17

score 0 · Answer 2 · answered Dec 16 '20 at 14:16

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I believe it will. Category theory barely existed before Stefan Banach passed in 1945, but this quote makes me think of category theory:

"...the ultimate mathematician is one who can see analogies between analogies."

The reason I think Banach may have considered category theory relevant to his quotation is that the mathematicians who invented it (e.g., Sanders Mac Lane) are the type who see analogies between analogies. It gives you a wider perspective.

answered Dec 16 '20 at 14:16

Hank Igoe

1,450

And as to the question of prerequisites: I'm working through the same book myself, and although I have had a course in abstract algebra and one on proof techniques, neither would be strictly necessary for a patient and motivated student who is willing to do the work. – Hank Igoe Dec 16 '20 at 14:20
And on that note ... – Noah Schweber Dec 16 '20 at 16:45
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Not so much an analogy between an analogies as much as a meta-analogy, but I recall seeing a quote along the lines of "An analogy is like a fletcher's bow; if one stretches it too far, it breaks" by Aristotle - if I recall right. Unfortunately, I couldn't find a reference. – Hank Igoe Dec 17 '20 at 17:35

Will learning category theory lead to a better and clearer understanding of mathematics?

2 Answers2