Comment author:[deleted]
14 July 2010 12:27:44AM
5 points
[-]

One of the things I've always disliked about mathematical culture is this taboo against making allowances for human weakness on the part of students (of any age.) For example, the reluctance to use "plain English" aids to intuition, or pictures, or motivations. Sometimes I almost think this is a signaling issue, where mathematicians want to display that they don't need such crutches. But it seems to get in the way of effective communication.

You can go too far in the other direction -- I've found that it can also be hard to learn when there's too little rigorous formalism. (Most recently I've had that experience with electrical engineering and philosophy.) There ought to be a happy medium somewhere.

Comment author:JoshuaZ
14 July 2010 12:45:35AM
*
4 points
[-]

Sometimes I almost think this is a signaling issue, where mathematicians want to display that they don't need such crutches.

This isn't really a signaling issue so much as a response to the fact that mathematicians have had centuries of experience where apparent theorems turned out to be not proven or not even true and the failings were due to too much reliance on intuition. Classical examples of this include how in the 19th century there was about a decade long period where people thought that the Four Color Theorem was proven. Also, a lot of these sorts of issues happened in calculus before it was put on a rigorous setting in the 1850s.

There may be a signaling aspect but it is likely a small one. I'd expect more likely that mathematicians err on the side of rigor.

ETA: Another data point that suggests this isn't about signaling; I've been too a fair number of talks in which people in the audience get annoyed because they think there's too much formalism hiding some basic idea in which case they'll ask questions sometimes of the form "what's the idea behind the proof" or "what's the moral of this result?"

Comment author:[deleted]
14 July 2010 02:28:28AM
4 points
[-]

Just to be clear: I'm not against rigor. Rigor is there for a good reason.

But I do think that there's a bias in math against making it easy to learn. It's weird.

Math departments, anecdotally in nearly all the colleges I've heard of, are terrible at administrative conveniences. Math will be the only department that doesn't put lecture notes online, won't announce the correct textbook for the course, won't produce a syllabus, won't announce the date of the final exam. Physics, computer science, etc., don't do this to their students. This has nothing to do with rigor; I think it springs from the assumption that such details are trivial.

I've noticed a sort of aesthetic bias (at least in pure math) against pictures and "selling points." I recall one talk where the speaker emphasized how transformative his result could be for physics -- it was a very dramatic lecture. The gossip afterwards was all about how arrogant and salesman-like the speaker was. That cultural instinct -- to disdain flash and drama -- probably helps with rigorous habits of thought, but it ruins our chances to draw in young people and laymen. And I think it can even interfere with comprehension (people can easily miss the understated.)

Comment author:SilasBarta
14 July 2010 12:53:42AM
2 points
[-]

Over 99% of students learning math aren't going to be expected to contribute to cutting-edge proofs, so I don't regard this as a good reason not to use "plain English" methods.

In any case, a plain English understanding can allow you to bootstrap to a rigorous understanding, so more hardcore mathematicians should be able to overcome any problem introduced this way.

Comment author:JoshuaZ
14 July 2010 12:56:55AM
1 point
[-]

I agree that this is likely often suboptimal when teaching math. The argument I was presenting was that this approach was not due to signaling. I'm not arguing that this is at all optimal.

Comment author:SilasBarta
14 July 2010 03:37:43PM
*
1 point
[-]

I don't think this problem is limited to math: it's present in all cutting-edge or graduate school levels of technical subjects. Basically, if you make your work easily accessible to a lay audience[1], it's regarded as lower status or less significant. ("Hey, if it sounds so simple, it must not have been very hard to get!")

And ironically enough, this thread sprung from me complaining about exactly that (see esp. the third bullet point).

[1] And contrary to what turf-defenders like to claim, this isn't that hard. Worst case, you can just add a brief pointer to an introduction to the topic and terminology. To borrow from some open source guy, "Given enough artificial barriers to understanding, all bugs are deep."

## Comments (606)

BestOne of the things I've always disliked about mathematical culture is this taboo against making allowances for human weakness on the part of students (of any age.) For example, the reluctance to use "plain English" aids to intuition, or pictures, or motivations. Sometimes I almost think this is a signaling issue, where mathematicians want to display that they don't need such crutches. But it seems to get in the way of effective communication.

You can go too far in the other direction -- I've found that it can also be hard to learn when there's too little rigorous formalism. (Most recently I've had that experience with electrical engineering and philosophy.) There ought to be a happy medium somewhere.

*4 points [-]This isn't really a signaling issue so much as a response to the fact that mathematicians have had centuries of experience where apparent theorems turned out to be not proven or not even true and the failings were due to too much reliance on intuition. Classical examples of this include how in the 19th century there was about a decade long period where people thought that the Four Color Theorem was proven. Also, a lot of these sorts of issues happened in calculus before it was put on a rigorous setting in the 1850s.

There may be a signaling aspect but it is likely a small one. I'd expect more likely that mathematicians err on the side of rigor.

ETA: Another data point that suggests this isn't about signaling; I've been too a fair number of talks in which people in the audience get annoyed because they think there's too much formalism hiding some basic idea in which case they'll ask questions sometimes of the form "what's the idea behind the proof" or "what's the moral of this result?"

Just to be clear: I'm not against rigor. Rigor is there for a good reason.

But I do think that there's a bias in math against making it easy to learn. It's weird.

Math departments, anecdotally in nearly all the colleges I've heard of, are terrible at administrative conveniences. Math will be the only department that doesn't put lecture notes online, won't announce the correct textbook for the course, won't produce a syllabus, won't announce the date of the final exam. Physics, computer science, etc., don't do this to their students. This has nothing to do with rigor; I think it springs from the assumption that such details are trivial.

I've noticed a sort of aesthetic bias (at least in pure math) against pictures and "selling points." I recall one talk where the speaker emphasized how transformative his result could be for physics -- it was a very dramatic lecture. The gossip afterwards was all about how arrogant and salesman-like the speaker was. That cultural instinct -- to disdain flash and drama -- probably helps with rigorous habits of thought, but it ruins our chances to draw in young people and laymen. And I think it can even interfere with comprehension (people can easily miss the understated.)

Over 99% of students learning math aren't going to be expected to contribute to cutting-edge proofs, so I don't regard this as a good reason not to use "plain English" methods.

In any case, a plain English understanding can allow you to bootstrap to a rigorous understanding, so more hardcore mathematicians should be able to overcome any problem introduced this way.

I agree that this is likely often suboptimal when teaching math. The argument I was presenting was that this approach was not due to signaling. I'm not arguing that this is at all optimal.

*1 point [-]I don't think this problem is limited to math: it's present in all cutting-edge or graduate school levels of technical subjects. Basically, if you make your work easily accessible to a lay audience[1], it's regarded as lower status or less significant. ("Hey, if it sounds so simple, it must not have been very hard to get!")

And ironically enough, this thread sprung from me complaining about exactly that (see esp. the third bullet point).

[1] And contrary to what turf-defenders like to claim, this isn't that hard. Worst case, you can just add a brief pointer to an introduction to the topic and terminology. To borrow from some open source guy, "Given enough artificial barriers to understanding, all bugs are deep."