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Ask HN: Math books that made you significantly better at math?

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Re: Ask HN: Math books that made you significantly better at math?

#141

Post by time_to_smile »

I highly recommend working through Claude Shannon's Mathematical Theory of Communications [0]. It's originally a paper but was later restructured as a book, in either form it works quite well.

The reason I recommend it is because it shows mathematical reasoning that is easy to follow and relevant to your daily life. It's real math, but very easy to read through and understand. If your unfamiliar this paper is where the very idea of "bits" comes from.

One of the most important things in the paper for non-mathematicians to see is that the definition Information Entropy is derived simply from the mathematical properties Shannon desires it to have.

This is important because I find that one of the biggest questions people ask about mathematical formula and idea is "What does this mean? Why is it this way?" without realizing that math is really not engineering nor physics. When deriving his definition of Information, Shannon simply states that information should have the following x,y... properties and then goes on to show that the now standard definition of information meets all these criteria.

In mathematics it is very often the case that only after an idea is created to we start realizing the applications. This is quite different than science where a model is only adopted if it correctly describes a physical process.

Work through the paper and you will have worked through the mathematical underpinnings of the information age and will likely have understood most of it pretty well.

0. https://people.math.harvard.edu/~ctm/home/text/others/shanno...

Re: Ask HN: Math books that made you significantly better at math?

#143

Post by time_to_smile »

fouronnes3 wrote: post #55
binarymax wrote: post #42

“Mathematical Notation: A Guide for Engineers and Scientists”[0] really changed my abilities with being able to read papers and decipher what was going on. I had university math experience but it was a long time ago. When I started reading papers for algorithms later in my career I couldn’t get past the notation. Once the symbols are explained, as a programmer, I was able to grok so much more. This should be on every…

As a programmer I really wish math notation was more rigorous: less ambiguity, more explicit typing, no implicit variables, etc. So much of it would never pass code review. We programmers figured out that code should be optimized for readability, not writtability ; I wish mathematicians did too.

I find one of the biggest mistakes programmers have about mathematical notation is that it's somehow just a terse, badly implemented programming language. But this is a very poor understanding of what mathematical notation is doing.

I think this error in thinking comes from the fact that Sigma notation can often be trivially implemented as a for loop.

Programming languages are designed to describe a specific computation, whereas mathematical notation is typically trying to describe an idea (one that might not even have a implementation!) Notation only sometimes and coincidentally describes computation as well.

The ambiguity, implied variables etc are an essential part of mathematical notation in the same way it is in common spoken language. Mathematical notation exists to help abstract and work out very hairy ideas, and often that ambiguity is necessary to show connections.

> code should be optimized for readability, not writtability

Mathematical notation is readable if you're literate in it. It takes lots of practice to become fluent in it, but once you become more familiar it's much easier to read than text (which is why it's used in the first place). Mathematical notation is an extension of mathematical writing, not computational implementation.

Reading mathematical notation is much closer to reading poetry than reading code.

Re: Ask HN: Math books that made you significantly better at math?

#144

Post by JoelMcCracken »

vippy wrote: post #62

Velleman's How to Prove It greatly helped my ability to construct set theoretic proofs, which better prepared me for Spivak's calculus and Baby Rudin. Hamkins' Proof and the Art of Mathematics is designed as a a good, less set-theory heavy, introduction to proof writing that leads more naturally to analysis. OpenStax books are FREE.

I have been going through Velleman, and it has been significantly helping me understand various CS papers and books, for example, I struggled understanding through proof outlines in PFPL, but working through just part of this book has helped.

I have had life things interfere with my learning now for the past month or so, but I hope to get back to it soon.

Re: Ask HN: Math books that made you significantly better at math?

#145

Post by tdba »

yig wrote: post #83
fouronnes3 wrote: post #55

Earlier quoted context omitted.

As a programmer I really wish math notation was more rigorous: less ambiguity, more explicit typing, no implicit variables, etc. So much of it would never pass code review. We programmers figured out that code should be optimized for readability, not writtability ; I wish mathematicians did too.

Our project IHeartLA is a language with syntax designed to closely mimic conventionally-written linear algebra, while still ensuring an unambiguous, compilable interpretation: https://iheartla.github.io/

This is amazing, thank you! The symbol replacement seems really slick https://github.com/iheartla/iheartla/wiki/Keywords-And-Speci...

Re: Ask HN: Math books that made you significantly better at math?

#147

Post by blevin »

User23 wrote: post #61
fouronnes3 wrote: post #55

Earlier quoted context omitted.

As a programmer I really wish math notation was more rigorous: less ambiguity, more explicit typing, no implicit variables, etc. So much of it would never pass code review. We programmers figured out that code should be optimized for readability, not writtability ; I wish mathematicians did too.

It can be made to be. Dijkstra came up with a nice and rigorous notation he used for his own proofs[1]. That page also includes some slightly spicy takes on why things are as they are. I agree that this is an area where the broader mathematical field has much to learn from computing science. The unforgiving nature of computing automata really drove that innovation. Meanwhile one can afford to be sloppy when one is tr…

I did not expect that to read so similarly to a good doc on coding conventions as it did.

Re: Ask HN: Math books that made you significantly better at math?

#148

Post by macrolocal »

dfan wrote: post #20

This is different from the other answers, but it does answer your question: When I was a kid I had tons of math and logic puzzle books. Two I remember specifically are "Aha! Insight" and "Aha! Gotcha" by Martin Gardner. Decades later, when a math problem comes up in my work, I have an apparently unusual ability to cut to the heart of it ("by symmetry, we must have X" or "looking at this extreme case, we must have Y"…

A Mathematical Mosaic is a little-known gem here.

Re: Ask HN: Math books that made you significantly better at math?

#149

Post by ska »

javajosh wrote: post #133
college_physics wrote: post #123

Math always seemed a bit arbitrary to me. Why and how did all those fields and branches develop, why are some so much more intuitive than others etc. What helped me cope with that challenge (and ultimately be a better learner / user of mathematics) was digging into the history of mathematics. Many great books in that genre but a very influential one for me was the Concise History of Mathematics by Dirk Jan Struik

>Math always seemed a bit arbitrary to me. That's because math is fundamentally arbitrary . This realization came to me late in life. Math was always presented as some aspect objective reality. However over time I've come to understand it as software for human brain. Using this math or that to describe something is often simply a matter of taste! Very similar to programming, in fact. Your study of history gives conte…

I think "arbitrary" gives the wrong flavor here. Math is fundamentally curated.

It's easy to create something new, because you get to play with the rules - but unless it is in some sense deep, effective, and usually elegant [1] it won't stick around.

[1] this is a bit acculturated.

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