Reddit reviews Conceptual Mathematics: A First Introduction to Categories
We found 17 Reddit comments about Conceptual Mathematics: A First Introduction to Categories. Here are the top ones, ranked by their Reddit score.
Cambridge University Press
"Without a bunch of jargon" is your interpretation of what Dr. Feynman said. If you read Six Easy Pieces and Six Not-So-Easy Pieces, you'll find that Dr. Feynman's actual teaching is loaded with physics jargon, and a big part of his genius lay in offering supporting intuitions for the very precise terminology physics uses.
Actual computer science—as opposed to the dressed-up vocational training that calls itself "computer science," not that there's anything wrong with vocational training—is the same way. It's mathematical, painstakingly precise, and the terminology shows it. With that said, there are Feynmans out there to help lead us through it. For example, Conceptual Mathematics: A First Introduction to Categories is a text suitable for a bright high-schooler that nevertheless will have you understanding the terms "monad," "monoid," "category," and "endofunctor" by the time you're through it, should you choose to work through it.
Without knowing much about you, I can't tell how much you know about actual math, so apologies if it sounds like I'm talking down to you:
When you get further into mathematics, you'll find it's less and less about doing calculations and more about proving things, and you'll find that the two are actually quite different. One may enjoy both, neither, or one, but not the other. I'd say if you want to find out what higher level math is like, try finding a very basic book that involves a lot of writing proofs.
This one is aimed at high schoolers and I've heard good things about it, but never used it myself.
This one I have read (well, an earlier edition anyway) and think is a phenomenal way to get acquainted with higher math. You may protest that this is a computer science book, but I assure you, it has much more to do with higher math than any calculus text. Pure computer science essentially is mathematics.
Of course, you are free to dive into whatever subject interests you most. I picked these two because they're intended as introductions to higher math. Keep in mind though, most of us struggle at first with proofwriting, even with so-called "gentle" introductions.
One last thing: Don't think of your ability in terms of your age, it's great to learn young, but there's nothing wrong with people learning later on. Thinking of it as a race could lead to arrogance or, on the other side of the spectrum, unwarranted disappointment in yourself when life gets in the way. We want to enjoy the journey, not worry about if we're going fast enough.
Best of luck!
I'm not so sure this is a fundamental difference, so much as a distinction in who is looking at each field. For the most part, category theory is studied by those who are looking to make advances in knowledge. Sure, the things researchers are looking at can be complex. But if you look at current research in abstract algebra, it's equally difficult to get up to speed and comprehend. The reason abstract algebra can be seen as simpler is that there is also introductory material, aimed at undergraduates, and even the general population.
Is it fundamentally impossible to produce such introductory material in category theory? Of course not! Several people have made serious and credible attempts. For example, here and here
Conceptual Mathematics by Lawvere and Schanuel is a good low level introduction to category theory (and a bit of set theory) if you are feeling shaky on those grounds. From there lots of books open up to you.
The best books I know on how to "think" like a functional programmer are all written by Richard Bird. http://www.amazon.com/gp/product/1107452643/ref=pd_lpo_sbs_dp_ss_1?pf_rd_p=1944579842&pf_rd_s=lpo-top-stripe-1&pf_rd_t=201&pf_rd_i=0134843460&pf_rd_m=ATVPDKIKX0DER&pf_rd_r=090NKMWKY6078Z0WPCTW http://www.amazon.com/Pearls-Functional-Algorithm-Design-Richard/dp/0521513383
Not much is available in book form, especially that I can recommend on the FRP front.
Dependent types is a broad area, you're going to find yourself reading a lot of research papers. You might be able to get by with something more practical like Chlipala's Certified Programming with Dependent Types, but if you want a more theoretical treatment then perhaps Zhaohui Luo's Computation and Reasoning might be a better starting point.
There are essentially "two types" of math: that for mathematicians and everyone else. When you see the sequence Calculus(1, 2, 3) -> Linear Algebra -> DiffEq (in that order) thrown around, you can be sure they are talking about non-rigorous, non-proof based kind that's good for nothing, imo of course. Calculus in this sequence is Analysis with all its important bits chopped off, so that everyone not into math can get that outta way quick and concentrate on where their passion lies. The same goes for Linear Algebra. LA in the sequence above is absolutely butchered so that non-math majors can pass and move on. Besides, you don't take LA or Calculus or other math subjects just once as a math major and move on: you take a rigorous/proof-based intro as an undergrad, then more advanced kind as a grad student etc.
To illustrate my point:
Linear Algebra:
Linear Algebra Through Geometry by Banchoff and Wermer
3. Here's more rigorous/abstract Linear Algebra for undergrads:
Linear Algebra Done Right by Axler
4. Here's more advanced grad level Linear Algebra:
Advanced Linear Algebra by Steven Roman
-----------------------------------------------------------
Calculus:
Calulus by Spivak
3. Full-blown undergrad level Analysis(proof-based):
Analysis by Rudin
4. More advanced Calculus for advance undergrads and grad students:
Advanced Calculus by Sternberg and Loomis
The same holds true for just about any subject in math. Btw, I am not saying you should study these books. The point and truth is you can start learning math right now, right this moment instead of reading lame and useless books designed to extract money out of students. Besides, there are so many more math subjects that are so much more interesting than the tired old Calculus: combinatorics, number theory, probability etc. Each of those have intros you can get started with right this moment.
Here's how you start studying real math NOW:
Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers. Essentially, this book is about the language that you need to be able to understand mathematicians, read and write proofs. It's not terribly comprehensive, but the amount of info it packs beats the usual first two years of math undergrad 1000x over. Books like this should be taught in high school. For alternatives, look into
Discrete Math by Susanna Epp
How To prove It by Velleman
Intro To Category Theory by Lawvere and Schnauel
There are TONS great, quality books out there, you just need to get yourself a liitle familiar with what real math looks like, so that you can explore further on your own instead of reading garbage and never getting even one step closer to mathematics.
If you want to consolidate your knowledge you get from books like those of Rodgers and Velleman and take it many, many steps further:
Basic Language of Math by Schaffer. It's a much more advanced book than those listed above, but contains all the basic tools of math you'll need.
I'd like to say soooooooooo much more, but I am sue you're bored by now, so I'll stop here.
Good Luck, buddyroo.
This:
http://www.amazon.com/Conceptual-Mathematics-First-Introduction-Categories/dp/052171916X
is a good book as an introduction for a math student!
well my favorite subjects in graduate school were differential geometry, particularly the theory of smooth manifolds, and functional analysis, in particular distributions. once i got a job writing software and dealing with hardware systems, i tried to keep up with my math (a losing battle somewhat) to see what connections i could make, and i eventually found the book conceptual mathematics: a first introduction to categories. i was also at the same time trying to pick up haskell, so between haskell and the category theory book and my job and my mathematics background, i started to realize that there are some connections between what people do in software and systems and the math. then i came across the book/paper category theory for scientists.
i'm now convinced that category can serve as a fantastic foundation for applied mathematics. when people think of applied mathematics, they immediately think discrete, combinatorial mathematics or throw differential equations at whatever problem is at hand. but i think there's a lot of the more abstract mathematics that can be applied, and i think (or at least agree with the authors of the materials i linked to) that category theory can help with this. you should also take a look at the work of robert ghrist as well, who applies algebraic topology to many engineering problems.
http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178/ref=sr_1_1?ie=UTF8&qid=1394386195&sr=8-1&keywords=pinter+abstract+algebra)
There are a couple of easy-ish sources on category theory that are good to have under your belt.
Category Theory for Programmers is available for free: https://github.com/hmemcpy/milewski-ctfp-pdf
It's not amazing, but it's good for programmers who want to start having basic intuitions about category theory.
Lawvere's Conceptual Mathematics is enjoyable and accessible
https://www.amazon.com/Conceptual-Mathematics-First-Introduction-Categories/dp/052171916X/ref=mp_s_a_1_1?keywords=conceptual+mathematics&qid=1568389352&s=gateway&sr=8-1
To answer your general question: in my experience, your question is less about math and maybe more about chasing something you think has the answers. You'll meander as long as you feel like something is lacking.
I've seen this a lot with people who have massive textbook collections. A massive collection of textbooks is debt, and it provokes anxiety. You may have to figure out some squishy human stuff in addition to the technical math stuff.
There is this online Category Theory book (PDF). Also, the book Conceptual Mathematics has been well recommended as an introduction to CT starting from the basics.
For me, it was Conceptual Mathematics: A First Introduction to Categories. Today, I might suggest How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics or Category Theory in Context to start.
Category theory is an overkill. If you think you're gonna have an easier time with it, you're mistaken. Category Theory is an extreme generalization of abstract math. Although, there's a very nice intro that you can get started with: Conceptual Mathematics: A First Introduction to Categories by Schanuel and Lawvere. It's accessible to most high school students.
What you are trying to understand is trivial. Most any intro to proofs/higher math book has an explanation of the subject.
In general, you need to learn how to think logically because the way you're going right now won't get you anywhere.
Again, read a book on the very basics of logic and sets. It would contain everything you need to know. For example,
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers.
> ... relation between finite and infinite.
...relation between finite sets and infinite sets. Just about everything in math is a set. There are many different types of relations. Some are functions, some are equivalence relations, some are isomorphisms.
> Just because something is an adjective or property does not mean it can't be negated.
Ok. Opposite of infinite is finite. In fact, we can say that a set is finite if it is not infinite. But limit is a number and infinity is not. You can't compare apples to oranges.
> In fact almost everything has an inverse.
Relations and special kind of relations called functions have an inverse. Also, operations can be inverse.
Yeah, I've just never been shown a problem where this stuff gives deep insight, and until I see one and understand it these are just gonna be arbitrary definitions that slide right out of my brain when I'm done reading them. I'll definitely give the book a look - is it motivated with examples?
The only book I have on category theory is Conceptual Mathematics: A First Introduction to Categories, and I must say, I'm not a fan of it - too intuitive, not detailed enough, not well organized, not formal enough - should have gone for MacLane instead.
Here are some great books that I believe you may find helpful :)
and last but definitely not least:
Later on:
I don't claim to know Category Theory, but I came across it when doing exercises in the beginning part of Chapter 0 by Aluffi. It was very terse, but still understandable. The video seems to be much more relaxed in comparison. It is even more relaxed than Awodey's book which is a much better intro to CT than Aluffi's Chapter 0. In short, it reminds me of Conceptual Mathematics: A First Introduction to Categories by Lawvere/Schnauel a little.
I love this book, personally.
I've always wanted to recommend this book to someone who knows no math. I find the writing infuriating. It is a dialog but this approach to dialog totally sucks. On the other hand, this is a stunning introduction to categorical logic. It will not help you solve problems etc. but I can guarantee that this book will change your entire outlook on the world.
http://www.amazon.com/Conceptual-Mathematics-First-Introduction-Categories/dp/052171916X/ref=sr_1_1?ie=UTF8&qid=1320560710&sr=8-1