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I’m going to go with a few assumptions here:

a) You don’t do this full time.

b) By “bottoms up” you just mean “with firm grasp on fundamentals”, not logic/set/category/type theory approach.

c) You are skilled with programming/software in general.

In a way, you’re ahead of math peers in that you don’t need to do a lot of problems by hand, and can develop intuition much faster through many software tools available. Even charting simple tables goes a long way.

Another thing you have going for yourself is - you can basically skip high school math and jump right in for the good stuff.

I’d recommend getting great and cheap russian recap of mathematics up to 60s [1] and a modern coverage of the field in relatively light essay form [2].

Just skimming these will broaden your mathematical horizons to the point where you’re going to start recognizing more and more real-life math problems in your daily life which will, in return, incite you to dig further into aspects and resources of what is absolutely huge and beautiful landscape of mathematics.

[1] https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...

[2] https://www.amazon.com/Princeton-Companion-Mathematics-Timot...




The Princeton Companion to Mathematics is a good resource consisting of a huge collection of detailed articles on many mathematical subjects by knowledgeable contributors. It requires no specialized background and is curated by Fields Medalist Tim Gowers. Whoever reads it from cover to cover is my hero, but failing that there's always an interesting article to jump to.

Don't just be a consumer but write something as soon as you're inspired. I wish there were more emphasis on writing mathematics in school prior to the graduate level. Leslie Lamport says if you're thinking but not writing you're not really thinking; you only think you're thinking. For Feynman the act of discovery wasn't complete until he had explained it to someone. There's also the rule of thumb that if you can't explain a mathematical concept to a ten year old, you don't understand it yourself.

Edit: typo


> The Princeton Companion to Mathematics is a good resource...

I think Princeton Companion to Physics curated by Frank Wilczek, a Nobelist, is due to be published this year.

> Whoever reads it from cover to cover is my hero...

Yeah, I'd die an accomplished man if I would grok just a few books I treasure, amongst which are TPCTM and MICMAM.

> Don't just be a consumer but write something as soon as you're inspired.

Absolutely. That's why I recommend just a small amount of comprehensive resources. It's hard to get motivated by a pile of books complemented with synthetic problems related to a particular chapter. The idea is to just go about your daily life and start to slowly see more and more math problems everywhere around you; it does wonders to motivation.


What is MICMAM?

Whitepapers, lectures, and speech transcriptions are also good motivation, and useful resources. Sometimes overwhelming, especially if reading mathematical text is as a foreign language. And sometimes it takes you down a rabbit hole.

My biggest block for learning math has really been all the unlearning. After a while ideas like negative numbers and zeros and processes like addition and subtraction stop making as much sense as I thought.

Here is my favorite rabbit hole:

http://www.turingarchive.org/viewer/?id=465&title=01

leads to:

http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002278499...

leads to:

http://www.jamesrmeyer.com/pdfs/godel-original-english.pdf

Where to go from there - philosophy or computation? Lambda calculus is only a couple clicks away. Lisp papers, perhaps?


> What is MICMAM?

Check my [1] at root.

> Where to go from there - philosophy or computation?

For me, there's plenty of fun in mathematics without venturing even near the edges of it. Maybe one day I'll grow bored of it, who knows - it's a lifelong process.


I didn't mean to insinuate that the process I describe is one to be taken out of boredom. Let me try to explain what I am thinking:

I have been studying lisp and wanted to understand more about the origins. So I went back to the beginning of the language and read the various McCarthy papers. But what he was thinking is not entirely clear to me. So I wonder, what papers was he studying himself when he wrote this? That is easy to answer as he put the references right there in the back of the paper for me to track down. So I start reading papers written by Church and Godel. I repeat this process recursively while looking for shared references. That network of interconnected papers is a treasure trove of useful information. Reading the same papers an author was reading during their writing process is a valuable way to expand your understanding of their work.


Can you please elaborate more how to be a producer rather than just a consumer in maths? Advice like to do more maths rather than read it is clear and a regular undergrad can follow it. Producing in maths sounds like writing papers to me that I can hardly imaging as an undergrad student. It is actually a problem for me as for a software engineer: in programming I can produce rather early and it creates a motivational feedback plus helps learing things better. With maths I cannot get the feeling of creating anything, the only pleasure is in solving problems from books.


>Don't just be a consumer but write something as soon as you're inspired. I wish there were more emphasis on writing mathematics in school prior to the graduate level. Leslie Lamport says if you're thinking but not writing you're not really thinking; you only think you're thinking. For Feynman the act of discovery wasn't complete until he had explained it to someone. There's also the rule of thumb that if you can't explain a mathematical concept to a ten year old, you don't understand it yourself.

Fantastic quotes and points, thanks for sharing.


>Another thing you have going for yourself is - you can basically skip high school math and jump right in for the good stuff.

I'd strongly disagree with this. To the mathematically literate, concepts like "imaginary numbers", "prime numbers" and "logarithms" are just simply understood things which are familiar and have always been a part of your lexicon. These are actually wildly complex, abstract ideas which take years to fully grasp as an adult being first exposed to the material. Developing a mathematical intuition to the level of an advanced high schooler is no small feat for an adult with zero mathematical training. I'd strongly suggest anyone actually starting from zero mathematical knowledge to go back and spend time doing basic remedial math courses from the point of simple algebra and arithmetic with a good teacher to truly understand numbers first.


I actually think I agree with everything you've said, but here's why I think it's moot - majority of visitors here have finished high school and I wouldn't be surprised that majority have at least started on tertiary education. Numbers are pretty high worldwide too. [1]

So, terms like "mathematically [i]literate", "adult with zero mathematical training", taken at face value, don't apply to most of us in the world, and almost certainly not to the OP either.

[1] https://ourworldindata.org/primary-and-secondary-education#c...


What's your opinion of Mathematics for computer science?


Have no opinion - haven't read it and Amazon has no preview for it.


If we're talking about the same book, it is available for free: https://courses.csail.mit.edu/6.042/spring17/mcs.pdf

I bought the book for sale on Amazon. The printed version seems like a print-on-demand copy of the free PDF. The paper size is 8.5x11 and the layout is the same. I'm a little suspicious of the publisher.

I have only used the book as a reference for a few sections. The style is very approachable.




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