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

Terry Tao on some desirable properties of mathematical notation

mathoverflow.net

Post reply on HN
Print view View original

201–210 of 229 posts

Re: Terry Tao on some desirable properties of mathematical notation

#201

Post by lambdatronics »

dopu wrote: post #91

Is it just me, or does probability theory in general have fairly terrible notation? Ambiguity between random variables and their distributions because of them simply being distinguished by being upper-case or lower-case, writing likelihood functions alternatively with an L() or p(), and using p() (with different arguments) to refer to different probability distributions. Perhaps I'm just having such a difficult time…

It's a real mess. I ran into an issue recently because I'm dealing with probability distributions in terms of several sets of general curvilinear coordinate systems. In a context like this, the usual abuse of notation in which the function is identified by which arguments go into it just doesn't work. I have a probability density in Cartesian coordinates expressed as a function of (say) elliptical coordinates -- which differs by the Jacobian from the probability density in elliptical coordinates.

Re: Terry Tao on some desirable properties of mathematical notation

#202

Post by lambdatronics »

rytill wrote: post #79
Koshkin wrote: post #18

Difficulties, if any, perceived or real, arising in connection with notation, are usually incomparably smaller than those presented with the subject itself. (Personally, I only wish mathematical notation were better integrated with software in general and programming languages in particular.)

Not true at all, there are several times I've attempted to read through a textbook only to be stopped by notation because something was introduced prior to being referenced, or notation is overloaded with multiple meanings. I consistently have run into "perceived or real" confusing mathematical notation as an impediment to learning in a way that programming languages have never, ever caused me. Does no one else feel…

Now imagine being able to 'recompile' a mathematical text as an AST for the derivation, so that you could dive down and see where all the terms and definitions were coming from, instead of having to try to search a PDF for some symbol you can't ctrl-F for.

Re: Terry Tao on some desirable properties of mathematical notation

#203

Post by rimliu »

tmpz22 wrote: post #16
pubby wrote: post #9

Imagine 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…

It will not help you, but generally, you should go to a university with an intent to learn, not to be taught.

Re: Terry Tao on some desirable properties of mathematical notation

#204

Post by jayjader »

madcaptenor wrote: post #180
jimhefferon wrote: post #155

Earlier quoted context omitted.

When I teach the intro to proofs class I require that they learn LaTeX. Some students like the availability of symbols so much that they go a bit nuts. Something like this sentence: $\forall x\in\mathbb{R}$ $\exists y$ that is $>$ the number $x+1$. Sigh.

This isn't exclusive to LaTeX; I think students in general just think that the symbols make things "more mathematical". I remember feeling this way briefly when I was first exposed to things like $\forall$ and $\exists$, and it wore off. This was in the late nineties; in theory I could have had access to a typesetting system but I was writing things by hand.

As a former maths student, I loved the "compressibility" of using symbols for things like "forAll" and "thereExists". They're quicker to write and allow fitting in more information in less space, two qualities which become highly useful when taking tests in limited amounts of time on sheets of paper with limited space (albeit needing more paper to fit your test answers on is more arguable in it's negativity).

For a (even more) subjective point of view, as part of my studies I had oral practicals every 2 weeks or so where I was basically standing at a whiteboard along with 2 classmates, each solving problems given to us by an examiner for ~1 hour. It's simply easier for a lot of students to draw a symbol that's legible from 5 meters away than some words without making them huge.

Re: Terry Tao on some desirable properties of mathematical notation

#205

Post by cryptica »

joshuaissac wrote: post #153
cryptica wrote: post #142

Earlier quoted context omitted.

Math symbols are a minor issue for me. What confuses me the most are descriptions of mathematical concepts. For example, Wikipedia describes a 'field' like this: "In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do." It doesn't make sense to me. What does it mean if an operation 'is defined…

> What does it mean if an operation 'is defined' on a set? Does it mean that any 2 elements combined together using that operation always need to output an element which is also in the same set? For a binary operation f to defined on a set, f(x,y) must exist for every x and y in the set. There is no requirement that f(x,y) itself is in the set. Adding that requirement would mean that the set is "closed" under the ope…

This is a proper definition, thanks. The first sentence here is about the same length as the one on Wikipedia but it fully encapsulates the meaning without ambiguity.

Re: Terry Tao on some desirable properties of mathematical notation

#206

Post by cryptica »

omaranto wrote: post #186
cryptica wrote: post #142

Earlier quoted context omitted.

Math symbols are a minor issue for me. What confuses me the most are descriptions of mathematical concepts. For example, Wikipedia describes a 'field' like this: "In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do." It doesn't make sense to me. What does it mean if an operation 'is defined…

If informal descriptions confuse you, skip them and read actual definitions instead. I actually like informal definitions a lot. I think they serve two different purposes: 1. For beginners they usually soften the blow of a fully rigorous definition, letting them get an idea of the concept before getting it exactly. 2. For experts they can often suggest what the exact definition is faster than it would be to read a pr…

It's not about 'informal definitions'. I also really like informal definitions but not the way that most mathematicians currently tend to write them.

Re: Terry Tao on some desirable properties of mathematical notation

#207

Post by cryptica »

gspr wrote: post #177
cryptica wrote: post #142

Earlier quoted context omitted.

Math symbols are a minor issue for me. What confuses me the most are descriptions of mathematical concepts. For example, Wikipedia describes a 'field' like this: "In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do." It doesn't make sense to me. What does it mean if an operation 'is defined…

It sounds very much to me like you would like mathematicians to change our notation to accommodate someone who has not put in the effort to learn mathematics. Do you wish the same from structural engineers? Programmers? Physicists? Medical doctors? Musicians? > "In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on ratio…

The user joshuaissac gave a very good description of a 'field' as a response to my comment and it was about the same length as the definition on Wikipedia. It shows that it's possible.

I don't see why certain knowledge should be out of reach of those who are not involved directly in that field. I could explain complex software engineering concepts to a layman. They wouldn't be able to use that knowledge to implement the software themselves, but they would be able to use the knowledge to make good high level decisions about it; for example to decide which of two solutions is better given a specific problem.

Re: Terry Tao on some desirable properties of mathematical notation

#208

Post by jhanschoo »

impendia wrote: post #168
cryptica wrote: post #142

Earlier quoted context omitted.

Math symbols are a minor issue for me. What confuses me the most are descriptions of mathematical concepts. For example, Wikipedia describes a 'field' like this: "In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do." It doesn't make sense to me. What does it mean if an operation 'is defined…

I'm a math researcher, and I'll explain why I like these sorts of definitions. In the first place, what you quoted is not a formal, precise definition; it is not a substitute for such a definition, nor is it intended to be one. The Wikipedia page you mention has a precise definition further down the page. So what, then, is the purpose of the description you quoted? Why include it at all? Because it's how mathematicia…

I'd also like to point out that it is also frequent that one encounters equivalent but different formal definitions for the same mathematical structures, and this is why the informal descriptions are important as well.

Re: Terry Tao on some desirable properties of mathematical notation

#209

Post by gspr »

foobar_ wrote: post #197
gspr wrote: post #195

Earlier quoted context omitted.

> The way I see it ... Most mathematicians nowadays use mathematica or matlab or even python, proving my point. Yes. But most of us don't use those to prove anything; rather, a lot of us use it to implement computations based on those proofs (and do some exploratory "could this possibly be tru?" kind of work). Useful tools, for sure, but not something that remotely proofs your point. Most mathematicians also eat brea…

I'm not insane ... you are just the type of person who will defend roman numerals. Maybe you just have OCD. 1. Socrates is mortal 2. Mortals die 3. Socrates dies Deduction is really like amazing. Holy shit we really proved something spectacular here. I guess you would be really impressed if I used tau and sigma and defined death with vietnamese alphabet. Almost the entirety of calculus was derived from problems relat…

> you are just the type of person who will defend roman numerals.

In the face of what? No system for writing numbers? Sure. Arabic numerals? No way, they're far superior to Roman ones.

> Almost the entirety of calculus was derived from problems related to physics.

No it is not. It is/was largely motivated by problems in physics. One may obtain a lot of intuition about calculus from physical intuition, but one does not obtain calculus from it.

> Volumes were calculated for doing engineering.

Yes. And?

> Mathematics != Thinking.

I don't see anyone in this thread claiming that.

> The last time I checked both logic and critical thinking were branches of philosophy.

I thought we were talking about math?

> All good mathematicians are physicists or engineers.

This is patently not true. Tell me how many Abel prize winners of Fields medalists are physicists or engineers. There are indeed some, but they are a minority – your claim is absurd.

> Heck some even learnt maths on their own.

Absolutely. Can you elaborate on how this is relevant?

> All mediocre mathematicians write textbooks and hide behind notations.

What are you on about? As a mediocre mathematician myself, I must admit I have never written a textbook.

> Most abstract mathematics is like the definition of protocols/interfaces and other platonic garbage.

You have made it abundantly clear that you haven't that slightest grasp of abstract mathematics. Would you at least humor me as to provide a few examples?

Re: Terry Tao on some desirable properties of mathematical notation

#210

Post by gspr »

cryptica wrote: post #207
gspr wrote: post #177

Earlier quoted context omitted.

It sounds very much to me like you would like mathematicians to change our notation to accommodate someone who has not put in the effort to learn mathematics. Do you wish the same from structural engineers? Programmers? Physicists? Medical doctors? Musicians? > "In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on ratio…

The user joshuaissac gave a very good description of a 'field' as a response to my comment and it was about the same length as the definition on Wikipedia. It shows that it's possible. I don't see why certain knowledge should be out of reach of those who are not involved directly in that field. I could explain complex software engineering concepts to a layman. They wouldn't be able to use that knowledge to implement…

> The user joshuaissac gave a very good description of a 'field' as a response to my comment and it was about the same length as the definition on Wikipedia. It shows that it's possible.

Sure, his definition is also a good one. It leaves out a lot, though. Which is fine, if one can assume the reader knows the context. My definition, too, leaves out a lot (it assumes set theory), and rests on the informal language known as English.

> I don't see why certain knowledge should be out of reach of those who are not involved directly in that field.

It isn't. The content of mathematics research papers may well be out of reach, but that's quite natural, don't you think? I, as a mathematician, do not expect to be able to read research papers on chemistry without putting in a lot of work.

> I could explain complex software engineering concepts to a layman.

OK. It does not follow from that that everything can be explained to a layman. Some things are easier to explain with layman analogies and mental images than others. However, to keep this fair, I think you should see how many laymen can follow Wikipedia articles on complex software engineering topics with ease! That is, afterall, where we started this discussion.

Post reply on HN