Terry Tao mentions that notation can help with error detection. Anyone here aware of some good examples?
One that I like is that in Einstein notation you can't have 3 of the same index, e.g., u_i u_i u_i is invalid.
Terry Tao on some desirable properties of mathematical notation
61–70 of 229 posts
Re: Terry Tao on some desirable properties of mathematical notation
#62Earlier quoted context omitted.
Programming != proofs, or in general communicating abstract mathematical ideas. Writing mathematics is nothing like writing software.
What I am trying to convey is writing software is better than writing maths, just like medieval music notation vs modern notation. Programming is better than proving because most proofs are mere tautologies or artificial constraints. This is why theorem provers in code rely on term rewriting. A triangle has a sum of 180 ? Well how about if you push the triangle inside out. In code you can easily run a more complex si…
Ok, then convince me. Write 'software' of, say, the proof of the the dominated convergence theorem or something else reasonably advanced and let's compare it to the proof in conventional math notation.
Re: Terry Tao on some desirable properties of mathematical notation
#63Earlier quoted context omitted.
> Mathematical notation presumably mostly makes perfect sense to the kind of people who deal with mathematical notation all day long. Maybe the overuse of opaque names leads to self-selection of who becomes a mathematician? Single-letter non-descriptive variable and functions names would “make sense” to programmers who use it all day long too — but that alone doesn’t make it a good idea.
Opaque names have the advantage that they don't trigger any potentially misleading associations and they emphasize the abstract nature of what is being discussed. A variable named x could be anything, even something that the mathematician/programmer didn't anticipate.
I agree with this. Additionally, I think a lot of detractors of single-letter variable names don't realize that it is often the case that the structure of the equation/expression is what's interesting, not what any individual variable represents. That structure is most visible and apparent when the variables don't take up any more space than the operators.
Being able to recognize things like "oh, that's a polynomial!" is extremely useful, especially in the context of learning new math rather than a more applied context. And I think it would be extremely hard to spot an opportunity to do integration by parts when working with Java-style verbose variable names.
Re: Terry Tao on some desirable properties of mathematical notation
#64Imagine I give you a list of words and ask you to remember them. 5 minutes later, I ask you to give me those words in reverse order. Not too hard, right? Now imagine if those words I gave you were in Vietnamese, or some language you don't speak. Suddenly the task becomes much more confusing. You aren't remembering a small handful of objects and ideas, but instead trying to juggle the individual syllables in your head…
I dropped out of college at 19 and attempted to return at 21. My first math course back was discrete math and my teacher was a grad student who very clearly had no interest in teaching and was only there in order to subsidize his "real work". Keep in mind this is a large public university charging $40k/year. Going into his office hours was like going to another country, because his only method to explain math was rea…
> Keep in mind this is a large public university charging $40k/year.
Funny enough, that grad student was almost certainly making less than that. Universities are screwed up.
Re: Terry Tao on some desirable properties of mathematical notation
#65Earlier quoted context omitted.
What I am trying to convey is writing software is better than writing maths, just like medieval music notation vs modern notation. Programming is better than proving because most proofs are mere tautologies or artificial constraints. This is why theorem provers in code rely on term rewriting. A triangle has a sum of 180 ? Well how about if you push the triangle inside out. In code you can easily run a more complex si…
Ok, then convince me. Write 'software' of, say, the proof of the the dominated convergence theorem or something else reasonably advanced and let's compare it to the proof in conventional math notation.
I'm guessing there was a physical intuition behind the theorem, if you can simulate it you will probably do something better than the proof. Now it's your turn to tell me why 1 + 1 = 2.
Re: Terry Tao on some desirable properties of mathematical notation
#66Imagine I give you a list of words and ask you to remember them. 5 minutes later, I ask you to give me those words in reverse order. Not too hard, right? Now imagine if those words I gave you were in Vietnamese, or some language you don't speak. Suddenly the task becomes much more confusing. You aren't remembering a small handful of objects and ideas, but instead trying to juggle the individual syllables in your head…
The frustration is inarguably real, but... Programming language syntax sucks because none of it maps to things that non-programmers know. Every time a new token is introduced, whether it be an ascii symbol or a word it's one more mapping your brain has to create to remember it. And on top of this you have to remember how these actually execute too. Yes I said execute - most programming concepts have so many implement…
Sure, but my point was that good notation takes advantage of the wealth of pre-existing plumbing that exists in our brain.
I think a good example is: https://xkcd.com/936/
"Correct Horse Battery Staple" is easier to remember than "Tl)7x" because the latter maps into plumbing we already have. Likewise, print "Hello world" is easier to remember than printf("Hello World"), which is easier to remember than std::cout Also, my point wasn't about ideas, but rather the notation used to represent these ideas. For example, you might not be able to read "Ăn quả nhớ kẻ trồng cây", but if I tell you it means "When eating a fruit, think of the person who planted the tree" you understand the idea. Notation is just a means of transferring ideas. Having deep complex ideas is not an excuse for poor notation.
Re: Terry Tao on some desirable properties of mathematical notation
#67Earlier quoted context omitted.
Ok, then convince me. Write 'software' of, say, the proof of the the dominated convergence theorem or something else reasonably advanced and let's compare it to the proof in conventional math notation.
I'm guessing there was a physical intuition behind the theorem, if you can simulate it you will probably do something better than the proof. Now it's your turn to tell me why 1 + 1 = 2.
Do you realize that you sound like you have a second-grade math education with your example? Also, do you realize simulation is not feasible for more than a few dimensions, if that?
Re: Terry Tao on some desirable properties of mathematical notation
#68Earlier quoted context omitted.
Ok, then convince me. Write 'software' of, say, the proof of the the dominated convergence theorem or something else reasonably advanced and let's compare it to the proof in conventional math notation.
I'm guessing there was a physical intuition behind the theorem, if you can simulate it you will probably do something better than the proof. Now it's your turn to tell me why 1 + 1 = 2.
Honestly what are you talking about. You can simulate for 100 years without finding a counterexample, but that doesn't make a proof. The whole point of math is to understand why things are true, not to just be satisfied that it seems true.
Re: Terry Tao on some desirable properties of mathematical notation
#69Enumerating what we want from notation helps us understand how far we are from the ideal. The whimsical introduction of Notation to talk about notation makes it practical. Given a domain in mathematics, adding notation (e.g. modulo arithmetic) can make complex notions pretty to express and quick to prove. I used to really enjoy this and tried to redefine notation for each exposition. It's shorter and prettier, but just pushes complexity into the notation :) and teaching people new notations is expensive, actually unless repeatedly used, more expensive than laying out details in a less concise notation.
Programming languages are notations within this framework - and domain specific languages, while much more efficient are unpopular as the costs of changing notation, in terms of training people, are too high.
The cost of communicating the notation is captured in a few of the desiderata (e.g. 1,7) but practically it is most important. If we want to be easily understood we should speak a common language!
Re: Terry Tao on some desirable properties of mathematical notation
#70I found this post a shame. (The post itself, not putting it here; I love seeing math posts on HN, and automatically upvote. Bringing hackers and mathematicians together is highly worthwhile for both.) Usually Tao's posts are so insightful, and crystallise some idea so perfectly that it feels like I was just on the cusp of discovering it myself—a rare talent, and hard to cultivate since it goes against the ego. In thi…
ex-theoretical physicist here. I read Terry's post and I totally saw your point about "Notation^{-1}(C)" being more wit than clarity. Of course, I also share your respect and appreciation for Tao.