Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

How to Read Mathematics

web.stonehill.edu

Post reply on HN
Print view View original

21–30 of 33 posts

Re: How to Read Mathematics

#21

Post by logicchains »

jordigh wrote: post #9
onuralp wrote: post #5

I think it would be quiet interesting to crowdsource a math-to-english interpreter (a la 'explain shell' [0]) for math equations. [0] https://explainshell.com/

This is not possible, at least not without a very sophisticated AI. 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 ha…

It would certainly be possible if mathematics standardised on writing proofs in a proof assistant like Coq. Then anyone with a basic functional programming background could understand the most complex of proofs, given enough time to step through and understand all steps of the proof in the Coq IDE.

Re: How to Read Mathematics

#22

Post by hackuser »

jordigh wrote: post #9
onuralp wrote: post #5

I think it would be quiet interesting to crowdsource a math-to-english interpreter (a la 'explain shell' [0]) for math equations. [0] https://explainshell.com/

This is not possible, at least not without a very sophisticated AI. 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 ha…

How do you deal with those problems when developing Octave? Do you stick to one 'dialect', for example? Or is it somehow a non-issue?

Re: How to Read Mathematics

#23

Post by keithpeter »

onuralp wrote: post #5

I think it would be quiet interesting to crowdsource a math-to-english interpreter (a la 'explain shell' [0]) for math equations. [0] https://explainshell.com/

Kenneth Iverson had a go at combining the terse nature of mathematical notation and the consistency and universality of code with his apl.

Search for his paper "Notation as a tool of thought" (widey available as a Web page or as a pdf).

Re: How to Read Mathematics

#24

Post by danidiaz »

I like this quote I found in a mathematical analysis book:

"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...

Re: How to Read Mathematics

#25

Post by urathai »

jordigh wrote: post #10
bluenose69 wrote: post #4

I thought this was going to be about how to read Mathematics aloud, e.g. to describe it over the telephone. It seems that even something as common as $f : a \rightarrow b$ might be spoken aloud in different ways.

This is in fact somewhat of a real problem. Mathematics is kind of meant to be learned while interacting other humans, not in pure cold isolation. A lot of what you absorb with other humans is how to pronounce things. A simple example for example is learning to pronounce binomial coefficients as "5 choose 2" or knowing that ŷ is pronounced "y hat". Something kind of similar happens in hackerdom, where you may learn t…

I totally agree with you! Learning the language of mathematics, specifically spoken in this case, is an important part in learning mathematics. I think it is really important to discuss proofs and ideas with your friends because of this.

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.

Re: How to Read Mathematics

#26

Post by guscost »

cosinetau wrote: post #18
bluetwo wrote: post #15

I've always loved math, but as things get more complex I stop thinking in math and start thinking in code.

I think your conflating math as all-descriptive of all properties. It's unfortunately not. Mathematics describes patterns, not procedures. You started thinking in code, because you started thinking about problems as procedures, not patterns.

Maybe this person is thinking in Haskell ;)

Re: How to Read Mathematics

#27

Post by rocqua »

logicchains wrote: post #21
jordigh wrote: post #9

Earlier quoted context omitted.

This is not possible, at least not without a very sophisticated AI. 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 ha…

It would certainly be possible if mathematics standardised on writing proofs in a proof assistant like Coq. Then anyone with a basic functional programming background could understand the most complex of proofs, given enough time to step through and understand all steps of the proof in the Coq IDE.

Most mathematical proofs translate quite badly to formal languages. Their formulation often relies strongly on the peculiarities of language.

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.

Re: How to Read Mathematics

#28

Post by amelius »

I guess how to read mathematics depends strongly on the context or the subfield you are looking into.

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.

Re: How to Read Mathematics

#29

Post by jordigh »

logicchains wrote: post #21
jordigh wrote: post #9

Earlier quoted context omitted.

This is not possible, at least not without a very sophisticated AI. 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 ha…

It would certainly be possible if mathematics standardised on writing proofs in a proof assistant like Coq. Then anyone with a basic functional programming background could understand the most complex of proofs, given enough time to step through and understand all steps of the proof in the Coq IDE.

You would be trying to reverse a couple millenia of mathematical tradition if you wanted to convince all mathematicians to only write machine-readable mathematics. As appealing as the idea may sound to computer people, mathematicians would find it quite repugnant if they were no longer "allowed" to write mathematics in a casual style (as if it were even possible to forbid them from doing so).

Re: How to Read Mathematics

#30

Post by munin »

logicchains wrote: post #21
jordigh wrote: post #9

Earlier quoted context omitted.

This is not possible, at least not without a very sophisticated AI. 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 ha…

It would certainly be possible if mathematics standardised on writing proofs in a proof assistant like Coq. Then anyone with a basic functional programming background could understand the most complex of proofs, given enough time to step through and understand all steps of the proof in the Coq IDE.

That would also require re-doing all of mathematics in a constructive setting, which is a great idea as far as constructivists are concerned but it has both political and technical opposition. It's also the case that a proof in Coq doesn't have to be enlightening, especially if you discharge it with some inscrutable hammer-like custom-written pile of LTAC.
Post reply on HN