
Intuition in Mathematics (2012) [pdf] - jimsojim
http://www.as.miami.edu/personal/echudnoff/Intuition%20in%20Mathematics.pdf
======
xwat
A quote from Von Neumann comes to my mind:

"Young man, in mathematics you don't understand things. You just get used to
them."

~~~
gamma_function
The same might be true of electrical engineering. I find it quite difficult
sometimes to develop intuition about how electricity moves in analog circuits.

------
Koshkin
Intuition in mathematics is just as important as it is in any other human
endeavor. On the other hand, there are many results and methods that are
deemed non-intuitive - by students and (consequently) by instructors. Some of
the results are so non-intuitive that they may end up being labeled
"pathological" \- despite being, in fact, completely logical consequence of
the assumptions, of which one of the most well-known is Axiom of Choice.
Another example is that differential forms are often seen by students as very
non-intuitive compared to vectors, even though they are much more useful and
can be efficiently used in more general settings than vectors can.

~~~
justinpombrio
> Some of the results are so non-intuitive that they may end up being labeled
> "pathological" \- despite being, in fact, completely logical consequence of
> the assumptions, of which one of the most well-known is Axiom of Choice.

That's not true. The axiom of choice is actually famous for being
_independent_ of ZFC, meaning that it cannot be proven or disproven using the
usual axioms of set theory. That's why it's an axiom: you need to assume its
truth without grounds.

Also, in intuitionistic logic, the axiom of choice is provably false.

~~~
waqf
Firstly, the sentence you quoted, while unfortunately ambiguous, is correct if
you take "of which" as picking up "assumptions" instead of "results". That is,
the assumptions, of which a famous one is AC, have counterintuitive
consequences, of which the original poster does not give an example but
Banach–Tarski is a commonly cited one.

Secondly, the axiom of choice may be independent of ZF (depending on whether
ZF is consistent) but it is certainly not independent of ZFC :).

~~~
justinpombrio
> Firstly, the sentence you quoted, while unfortunately ambiguous, is correct
> if you take "of which" as picking up "assumptions" instead of "results".

Oh, _that 's_ what Koshkin was trying to say.

> Secondly, the axiom of choice is not independent of ZFC, it's independent of
> ZF (assuming ZF is consistent) :).

Ah, the C stands for choice...

------
Etheryte
While the author has ample experience in the field of philosophy and has spent
considerable time studying intuition (curriculum vitae:
[http://www.as.miami.edu/personal/echudnoff/CV.pdf](http://www.as.miami.edu/personal/echudnoff/CV.pdf)),
the writing style seems too informal and personal for my taste. Without
looking up the author, I would not have dared to guesstimate that the author
is an associate professor, but rather a hobbyist. Perhaps such a style is
common in the field (I rarely read philosophy papers), but coming from a
natural sciences background it seems very odd.

~~~
WhitneyLand
I don't think informality is what's objectionable, I think it's good when
papers are more accessible without sacrificing precision or expression.

The problem here seems to be witting skills. For example, this should never be
a sentence: "That is the aim of section 1."

~~~
handcreme
Non-native speaker here. What is wrong with this sentence and how would you
improve it?

~~~
thatcat
Remove it. If you've stated your goal, that should be clear. There is no
reason to state that you've stated your goal.

~~~
HarryHirsch
There are undertones here. What the section really says is that mathematical
intuition has different aspects that are very distinct, and that a discussion
of some of these will follow. Which ones these are and why these were chosen
will be discussed in Section 1.

The task of the editor is never to impose their style upon the author. There
are different writing styles even in academic writing, a wordy one, one that
chooses outrageous words, a very terse one, and the awful MBA-style kind.

What's worst is an editor that sees a sentence that he does not understand,
and edits that, thinking it's just style but completely changes the meaning.

~~~
thatcat
The editors job is to improve the writing by making it easier for the intended
audience to read and comprehend.

In this case, the sentence can be easily worked into the preceding sentence
which it echos - by changing the place holder clause "The first order of
business is.." to "The aim of section one is..".

~~~
HarryHirsch
That version loses the emphasis on recognizing that there are distinct aspects
of mathematical intuition. Congratulations! You successfully changed the
meaning of the sentence! The intended audience is mathematicians and
philosophers, these people can be expected to have full grasp of the English
language and to engage with the paper.

I once was working with a horrible boss like that. I'd write a draft of the
paper, she would not understand subject or prose and change some sentences
into something else, based on her limited understanding of the subject and the
prose. It was excruciating. You'd read her draft full of MBA-speak and
couldn't make out the meaning of some of it!

~~~
thatcat
First off all, the meaning is equivalent and this sentence is completely
unnecessary. Repeating your intents in sequential lines doesn't add meaning.

"The first order of business will be" is a placeholder used in speech, to me
it has the equivalent meaning of "Uhm, I will be". Care to elaborate further
on what you think the special emphasis of this phrase is in this context?

>Congratulations!

Not to edit your style bro, but the condescension is a bit much. Seems like
you have just soured on editors due to some bad personal experiences. Editing
is suppose to be a two way conversation in which the editor marks up all
possible errors and you accept the changes that you think add clarity or just
rewrite it yourself knowing that your previous version was unclear to at least
1 person.

That said, I don't really see the grey area in this case.

------
satai
Funny related reading: Imre Lakatos - Proofs and refutations

~~~
Etheryte
Freely available version:
[https://math.berkeley.edu/~kpmann/Lakatos.pdf](https://math.berkeley.edu/~kpmann/Lakatos.pdf)

~~~
satai
Only a part. It was published several times (as a paper, thesis and a book at
least and the book is much longer).

Anyway this part looks good enough to catch some points.

