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Habits of highly mathematical people

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Re: Habits of highly mathematical people

#91

Post by woopwoop »

There is an interesting Math Overflow thread called "Mathematical habits of thought and action which would be useful to non-mathematicians". The answers there all interesting (and sometimes conflicting), but my favorite (part of an) answer comes from Terence Tao:

>Equivalence. Basically, the idea that two things can be functionally equivalent (or close to equivalent) even if they look very different (and conversely, that two things can be superficially similar but functionally quite distinct). For instance, paying off a credit card at 10% is equivalent (as a first approximation, at least) to investing that money with a guaranteed 10% rate; once one sees this, it becomes obvious why one should be prioritising paying off high-interest credit card debt ahead of other, lower-interest, debt reduction or investments (assuming one has no immediate cash flow or credit issues, of course). Not understanding this type of equivalence can lead to real-world consequences: for instance, in the US there is a substantial political distinction between a tax credit for some group of taxpayers and a government subsidy to those same group of taxpayers, even though they are almost completely equivalent from a mathematical perspective. Conversely, the mistaking of superficial similarity for functional equivalence can lead to quite inaccurate statements, e.g. "Social Security is a Ponzi scheme".

[1]http://mathoverflow.net/questions/74707/mathematical-habits-...

Re: Habits of highly mathematical people

#93

Post by johncolanduoni »

srtjstjsj wrote: post #78
johncolanduoni wrote: post #39

Earlier quoted context omitted.

This is not true at all. I know many people in mathematics that are below average when it comes to arithmetic, some to the point of suffering from dyscalcula. I myself am probably in at least the bottom quartile of arithmetic ability, and there have been quite a few historic examples of top mathematicians with a similar "problem", Hilbert being a common example. There's a huge difference between reasoning about abstr…

Hilbert's supposed dyscalcula is a myth..unless you have a source we haven't seen? It's similar to the Grothendeick story -- a mathematician making a simple mistake -- that get exaggerated to "mathematicians can't do arithmetic." https://en.wikipedia.org/wiki/57_(number) Another common myth was that Einstein was bad at math, because he (and others) had trouble in school because the material was too easy and he quarre…

You're right, it looks like that is a myth (I remember seeing it on this math stackexchange question[1], which is actually a great example of some mathematicians' opinions on the matter).

There's a wide margin between "couldn't pass high school" and "had difficulty with math in high school". And I've known plenty who confessed that were quite explicit that was not what they meant (e.g. by comparing themselves unfavorably with their children).

[1]: http://math.stackexchange.com/questions/551074/are-all-mathe...

Re: Habits of highly mathematical people

#94

Post by CarolineW »

unfamiliar wrote: post #86
redipiny wrote: post #52

Earlier quoted context omitted.

One of the problems of philosophical discourse, imo, is that it seems to be impossible to give mathematically rigorous definitions of concepts like 'good' or 'moral' or even 'knowledge' that somebody won't be able to disagree with by counter example. What you sometimes see among philosophers is that even definitions that are the result of many iterations are still treated more like rules of thumb than precise definit…

> that somebody won't be able to disagree with by counter example I really don't understand this. You give a counterexample to a theorem; you don't give one to a definition. If someone disagrees with the definition and wants to use a different one, then they are talking about a different thing (even if they want to use the same English word to refer to each of them) and their conclusions can not be compared in any me…

  >> that somebody won't be able to
  >> disagree with by counter example

  > I really don't understand this.
  > You give a counterexample to a
  > theorem; you don't give one to
  > a definition.
In a sense you do. Often when we are giving a definition we are intending to capture a particular idea. Sometimes the definition we give captures too little, or too much. If you find that out early enough then you can change your definition to better match what you intend.

The classic example is "connected" from topology. Speaking very loosely, a set is "disconnected" is there is a "disconnection", which is a separation of the set into two pieces which are contained in disjoint open sets. Basically, the set is disconnected if it's made up of two (or more) pieces that can be divided by a "surface". A set is "connected" if there is no disconnection.

The problem is that there are sets we want to think of as not connected, and yet which satisfy the definition of connected as given above. Here's an example:

  X = { (x,sin(1/x)) : x in R, x>0 } u { (0,y) : -1
So the net result is that we have the two definitions: "connected" and "pathwise connected."

If that example had been thought of earlier, it's possible that the definition of "connected" might have been fixed earlier. So in a sense, X is a counter-example to the definition of connected.

Further reading:

http://planning.cs.uiuc.edu/node140.html

https://en.wikipedia.org/wiki/Connected_space

https://en.wikipedia.org/wiki/Connected_space#Path_connected...

Re: Habits of highly mathematical people

#95

Post by gunnihinn »

woopwoop wrote: post #90
gunnihinn wrote: post #88

Earlier quoted context omitted.

> for example, that the set of all vector spaces is itself a vector space Hmm... under what operations? It's a semigroup under direct sum, and probably something under the tensor product, but I'm having trouble imagining what the scalar multiplication should be. (Or was that a "fictional" example? :)

Unless I'm mistaken, The totality of vector spaces is not even a set.

There's one for every cardinality (up to isomorphism). I forget my set theory, but if the cardinalities form a set, then the totality of vector spaces does too. If not, we can amuse ourselves by looking at the set of finite-dimensional vector spaces.

Re: Habits of highly mathematical people

#96

Post by wolfgke »

srtjstjsj wrote: post #80
wolfgke wrote: post #8

Earlier quoted context omitted.

> The precision you hold yourself to through a rigorous approach to a problem is also very valuable and is the reason I think calculus is very important. What property concerning precision does calculus have that doesn't hold for any topic in mathematics?

Infinity (and infinitesimals) is intuitively east to use but easy to misuse. Arithmetic isn't so easy to misuse. Students usually see infinity for the first time in calculus (or maybe when working with infinite series) These are the situations where we commonly see people being confident in their incorrect answers (which is worse than being unconfident and unable to get correct answers) Difficulty wrangling infinity…

> Students usually see infinity for the first time in calculus (or maybe when working with infinite series)

Students also see groups or R-modules the first time in abstract algebra. So what.

> These are the situations where we commonly see people being confident in their incorrect answers (which is worse than being unconfident and unable to get correct answers)

> Difficulty wrangling infinity comes up a lot on Hacker News, even https://hn.algolia.com/?query=infinite%20series&sort=byPopul....

I rather the reason why "infinity" is misused so often, but, say, groups or R-modules, not so lies rather in the fact that too most math instructors too much to appeal to intuition in calculus, but not in abstract algebra. Thus mathematics should be taught in a much more abstract way where you are not misled by your bad intuition because you simply aren't able to formulate wrong thoughts in the abstract framework (that's why the abstractions and formalism was invented).

Re: Habits of highly mathematical people

#98

Post by mfn »

It might also be that individuals who have these traits self-select into studying math. So we can't easily conclude that studying math leads to the development of these "highly mathematical habits", because the causation could also flow the other way.

Re: Habits of highly mathematical people

#99

Post by ktRolster »

Kenji wrote: post #43

Mathematicians need logical precision because they work in the realm of things which can be definitively proven or disproven. Not really, by Gödel's incompleteness theorems, they are always working in a realm where some things cannot be definitely proven or disproven.

Your quote didn't say that all things can be disproven or proven. You need to think mathematically!

Re: Habits of highly mathematical people

#100

Post by JohnStrange »

unfamiliar wrote: post #37
raldu wrote: post #18

Earlier quoted context omitted.

Nice spot on. The described qualities would also apply to philosophers, logicians, etc. I think a broader term would be, habits of highly "analytical" people.

I strongly disagree with the claim that philosophers fit this description. As a mathematician, it often seems to me that 90% of philosophy is spent arguing about things with extremely loose definitions. As a result, you can argue from the same starting point and come to completely different conclusions (as separate philosophers often do), because the starting point was already self-contradictory for some interpretati…

I'm a philosopher and agree. There are some philosophers who make sufficiently precise definitions and use reasonable formal method - where certainly not all formal philosophy is useful, though. They are in a small minority. I'd even give the same ad hoc figures, it's about 90% trash, 10% worth reading and then a few gems, so you have to be careful which authors you read and what topics to study in philosophy.
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