How to Read Mathematics
web.stonehill.edu
web.stonehill.edu
Mathoverflow and math.stackexchange soft-questions tag has lots of mini-essays, e.g. https://mathoverflow.net/questions/143309/how-do-you-not-for...
and https://math.stackexchange.com/questions/617625/on-familiari...
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For proofs, books by Polya, Velleman, Hammack etc
Prof Baez' reading lists for math and physics: http://math.ucr.edu/home/baez/books.html
http://www.staff.science.uu.nl/~gadda001/goodtheorist/ (sidebar for specific areas).
The Language and Grammar of Mathematics, from The Princeton Companion to Mathematics, by Timothy Gowers: http://press.princeton.edu/chapters/gowers/gowers_I_2.pdf
Reading Mathematics, by John Hamal Hubbard: http://www.math.cornell.edu/~hubbard/readingmath.pdf
Has anyone read the author's book[1] and can recommend it (or not)? It only has 2 reviews, a 5-star and a 3-star and the latter review is not helpful.
[0]https://www.edx.org/course/effective-thinking-through-mathem... [1]https://www.amazon.com/Rediscovering-Mathematics-Classroom-R...
He's a superb lecturer, and I owe him a debt of gratitude for his recorded classes.
The most amusing part about him was that he wore those tight biking shorts and shoes while he taught.
Here's an audio clip of Shai introducing Richard Stallman at a talk where Stallman gets pretty annoyed[0]
[0]: http://audio-video.gnu.org/audio/rms-speech-arsdigita2001.og...
"It is an unfortunate fact that proofs can be very misleading. Proofs exist to establish once and for all, according to very high standards, that certain mathematical statements are irrefutable facts. What is unfortunate about this is that a proof, in spite of the fact that it is perfectly correct, does not in any way have to be enlightening. Thus, mathematicians, and mathematics students, are faced with two problems: the generation of proofs, and the generation of internal enlightenment. To understand a theorem requires enlightenment. If one has enlightenment, one knows in one's soul why a particular theorem must be true."
https://www.goodreads.com/book/show/5760666-foundations-of-a...
This guy produces excellent illustrations/explanations of math concepts: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw/vid...
I wish there could be software that could produce these style of explanations.
One thing that novices seem to not get about mathematics, especially those coming from programming languages, is that the syntax of mathematical formulae is not divorced from the natural language that surrounds it. Different authors use different symbols for the same concept or the same symbol with different meanings. That is, mathematical symbols have synonyms and homonyms, so to speak. You can read someone's "accent" when reading mathematics. The French consider 0 to be a positive number; the Americans don't. The Russians call it the Bunyakovsky inequality, most of Western Europe and America calls it the Schwarz or Cauchy-Schwarz inequality. The way they arrange their equations, the particular details of the notation, their preferred term for a particular concept, even tiny things like preferring upright or slant font for dx in an integral (and by the way, often fonts carry semantic meaning, but not in this case); all of these vary from author to author.
Mathematics is a very human activity. It is written by humans who write ambiguous things because it's meant to be read by humans who deal well with ambiguity. Reading human-written mathematics cannot be mechanised, at least, not anymore than reading a natural language can be mechanised.
The bigger problem is that there is often not a simple English description for math terms, whose meaning might involve understanding concepts that take books to explain.
For example, consider the simple sentence "Let x be an arbitrary vector". What is a vector? There is no clear answer to this question. Formally speaking a vector is simply an element of the set V where V is the underlying set for some vector space. You could try to say that V is an ordered pair of n real numbers, which is true up to isomorphism for finite dimensional vector spaces over the real numbers. But, x could also be the function f(x)=x^2, or the matrix [3, 4; 7, 1+i]. In some contexts, the translator might be able to tell from context what vector space is being worked in. But if the source is a general statement about vector spaces (even if made in support of a more concrete point), there is no way to explain it without explaining the concept of a vector space. To explain vector spaces one would need to, at a minimum, explain fields.
There is also a question of which description to use. For example, suppose we needed to define complex numbers. In some contexts, it is most natural to think of them as points on the xy-plane, which is simple enough (although does not give intuition above what multiplication means, which is arguably an essential component of the "true" meaning of complex numbers). In other contexts, it is more natural to think of them as matrices of the form [a -b; b a]. Attempting to show all the (equivelent) meanings would likely overwhelm the intended user of the service.
And in machine learning idiots mix lambda for hyperparameters and eigenvalues....
Consider the "it's easy to see that" from the article. There is no translation of that into a formal language beyond either introducing the claim as an axiom, or actually providing the proof. It also means that every usage of a well known theorem needs to be explicitly mentioned.
The end result would be an incredibly verbose proofs. Moreover, the verbosity of the proof wouldn't relate very well to the complexity of the idea behind the proof. This might bloat rather simple or elegant proofs, and us mathematicians really like our elegant proofs.
Put another way, this would require writing down a lot more than the epiphany that lead to the proof.
In general, it might be nice to somehow make proofs more tree-like rather than linear. That is, show the proof from the highest level, and allow a reader to 'expand' the claims to see their proofs. Down to the 'leaves' which are considered obvious.
Search for his paper "Notation as a tool of thought" (widey available as a Web page or as a pdf).
Handbook for spoken Mathematics (Larry's speakeasy) by Lawrence Chang: http://web.efzg.hr/dok/MAT/vkojic/Larrys_speakeasy.pdf from The Lawrence Livermore Laboratory
A simple example: a superscript can mean exponentiation or an index.
Also, sometimes it is preferred to leave out symbols that occur often, to avoid clutter. One example is dropping the sigmas out of equations, and infer they exist by looking at indices.
Something kind of similar happens in hackerdom, where you may learn to pronounce "/etc" as "slash etsy" or "!" as "bang".
Another problem is that most of the literature and the lectures are in English but you often discuss in another language. So you have to both do the translation from written to spoken mathematics and the translation from English to, in my case, Swedish. This easily leads to half translated English words but I try to avoid that as much as possible.