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Comment by keiferski | Hacker News Reader

I would not call myself great at math – I struggled with it in school, in fact – but in recent years I’ve begun “correcting” my lack of mathematical knowledge. The single best decision I’ve made is to first start with the philosophy of mathematics. Maybe it’s because my background is in philosophy, but I also think that for certain people like myself, understanding what math is makes me far more interested in understanding how it works, rather than just doing context-less calculations using formulas I don’t know the history or deeper purpose of. When I learned math in school, it was entirely cut off from any of these deeper questions.

Here’s a good starting point for philosophy of mathematics :

https://plato.stanford.edu/entries/philosophy-mathematics/

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Reading Euclid's Elements and Newton's Principia really helped me get an intuitive feel for geometry and calculus. They may not be entirely easy (at least the second) without some commentary, but well worth the study.

While it's laudable that you sought those texts and profited from them, I worry about what others might take away from this. When I was young I knew some geniuses who highly spoke of Principia and how it gave them great insights. And the teenager me said, okay cool, I'll have a go!

The problem is that it's in Latin and quite impenetrable.

We have some geniuses here and they would no doubt be able to take away a lot from these texts, but for you normals out there: don't optimize too much, you're quite alright in taking the normal approach of just taking a class at a community college, doing the exercises the teacher assigns, etc.

It might make sense to read a translation in a language you understand. Many of the books that are considered classics are specifically because they ARE accessible. That doesn't necessarily mean that they are easy, but there is a big difference between reading Euclid and learning how to create mathematical proofs, and taking a class focused on calculating the area of various shapes or determine angles.

I haven't read Mathematical Principles of Natural Philosophy (the English title) but I have read Euclid and it definitely doesn't require a genius to understand. here is an online edition with great illustrations:

https://www.c82.net/euclid/

From what I've heard, Euclid is fairly accessible and was for centuries the standard geometry textbook for children; the Principia is incredibly daunting, and Newton even admitted that he made it extra confusing on purpose to deter readers who weren't already experts.

I've been flirting with the idea of working through all of Leonard Euler's publications (as a life goal). Many of them are still not translated from Latin, so there's a possibility I may have to learn it.

Anyone knows how long it would take to learn enough latin to undertake such a task?

If you're trying to understand a domain work (such as Euler's) you could probably get a working knowledge in a month of strong study, a year of off-and-on.

I bet you could start this week if you used machine translations as a crutch.

I'd start working with a publication that exists in Latin and a good translation, so you can compare your work.

Thank you! Sounds much less intimidating.

My fear with machine translations is that subtle errors here and there might throw me off in something like math where things are precisely stated.

A year of part time study sounds doable though

That's the advantage of it being a particular mathematic domain, you'll learn the terms relatively quickly and be able to catch errors in the math parts; the prose is where you will need the machine.

In fact, you'll find that many philosophers will just use the Latin words directly, and not bother translating them - Latin qua jargon if you will.

Once you've learned the various forms of "is" (sum, very irregular) you can kinda survive reading without conjugations, just like this sentence can be worked out:

he to go to store and to buy cheese yesterday

Use the book "Lingua Latina per se illustrata" to learn Latin. It's quite magical, you just start reading Latin which is comprehensible due to similarities to English and it stacks on this without using anything but Latin. It's also much faster and more thorough than other textbooks.

Physics for Mathematicians by Spivak is basically the Principia updated to modern prose and rigor.

I didn't read Euclid in Greek or Newton in Latin; there are quite good translations available - even free!

In general I find that if someone is insisting that you study the philosophy of someone in their original language, they don't have a good enough translation yet.

This is sort of like recommending the art of computer programming as a way to learn how to code, isn’t it? Starting very far down the stack if you’re working through a 2000 year old book in Ancient Greek!

Elements was a school textbook for 2000 years, up until ~100 years ago. It's a fine book to use for self-study.

Edit: Also, to state the obvious, it's been translated into English

Perhaps, but the Elements is quite understandable, and surprisingly enough it's been translated - https://www.barnesandnoble.com/s/Euclid%20elements

Some schools still teach geometry from the Elements. It doesn’t matter how old the book is. Mathematics is timeless.

Ditto for Euclid. Doing this early in life pays huge dividends.

History of mathematics as well, it will give you a very subtle entry into the minds of mathematicians and the motivation behind their theorems.

This will surely make you more appreciative of subjects and concepts you are learning.

Absolutely, and there are some really interesting personalities in the history of mathematics. Newton and Galois come to mind.

Gödel springs to mind.

See https://www.newyorker.com/magazine/2005/02/28/time-bandits-2

In that vein I highly recommend Foundations and Fundamental Concepts of Mathematics by Howard Eves. I think it might be a little dated, but it gives an amazing overview of the most important developments in mathematics that were relevant at the time. It's less focused on practice (though there are some problems) and more on the history and motivation behind the ideas. This book introduced me to axiomatics, non-Euclidean geometry, quaternions, and abstract algebra in my senior year of high-school.

I have a very similar background, did my undergrad in Philosophy and feel that I need to learn some basics. Do you have any pointers on where to move after this?

I just started with that SEP article and then googled around for some other books and videos. There are some excellent lectures on YouTube, this one for example:

https://youtu.be/UhX1ouUjDHE

Also, you might find that symbolic logic is a good introduction to thinking mathematically. I used Klenk’s Understanding Symbolic Logic for a course a decade ago and really enjoyed it.

For actual mathematics lessons, Khan Academy is pretty solid.

Is that entire course just wading through the wreckage wrought by Gödel?

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