Intuition in Mathematics (2012) [pdf] | Hacker News Reader

Intuition in Mathematics (2012) [pdf]

as.miami.edu

119 pointsjimsojim34 comments

A quote from Von Neumann comes to my mind:

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

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

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.

> 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.

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 :).

> 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...

What do you mean by AC is provably false in intuitionistic logic? Intuitionistic logic + AC has as models ordinary models of classical logic since intuitionistic logic is a refinement of classical logic.

Do you mean [Diaconescu's theorem](https://en.wikipedia.org/wiki/Diaconescu's_theorem) which shows that AC implies excluded middle?

Maybe you mean there are models of intuitionistic logic where AC (and EM) is false?

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), 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.

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."

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

I hate to say it but I don't know grammar rules well enough to explain it. The best I can offer is something that sounds better to me. First example is the paper, second example is one possible rewrite:

"The first order of business will be to draw some distinctions between these notions and pick an appropriate focus for our present inquiry. That is the aim of section 1."

"Section 1 draws distinctions between these notions, one of which will be the focus of our inquiry."

You're probably mostly objecting to the passive voice, which is a reasonable stand to take in scientific writing. He uses passive voice three times in these two sentences alone.

Using "to be" isn't necessarily the passive voice. The author doesn't use the passive voice in those sentences.

http://writingcenter.unc.edu/handouts/passive-voice/

You're absolutely right, I misread those.

Passive voice would be "to be drawn", "to be read", etc.

Thank you.

It's always either redundant or indicative of a problem elsewhere. Either section 1 has a title/thesis that makes it obvious what it's about and so the restatement is unnecessary, or you need to say it because you haven't already made it obvious. Instead of saying "This is the aim of section 1", just make the point of the section apparent.

This is a class of stylistic issue that's common in professional writing by those new to it and usually stems from a lack of confidence in the topic or in one's self.

Giving "overview of the rest of the paper" that points out what will be happening in future sections is quite common in academic literature - there are better ways to phrase it (and many people writing academic papers, myself included, try to word it so that it does not sound that silly), but this wording (even if grammatically quite clumsy) sounds pretty standard to me - I would not overthink it as a "lack of confidence".

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

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.

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..".

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!

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.

There is nothing wrong with it. Rather question the judgment of someone who has used the words "witting skills".

> The problem here seems to be witting skills.

Heheheh... ;-)

It's also annoyingly and unnecessarily double spaced. While decent leading (line-height) is an important typographical element for legibility, double spacing is just frustratingly inefficient for reading.

You don't have to be a type setter or poet to care enough to get the basics right for your "paper", learn by example at the least.

In the publishing industry, many require you to submit double spaced - they will change it to single spaced before the final publication.

I agree the final one posted to the web should be single spaced, but perhaps he is merely uploading the document he submitted somewhere.

Let me guess: You're one of the "serious papers should be written in the passive voice" people?

I don't think any modern academic writing guide recommends this; almost every piece of advice I've seen tries to emphasize writing in the active voice.

You don't have to write in passive voice to be rigorous; I'm guessing the OP was talking about this lack of rigour.

I thought the tide had shifted towards active voice as well. However, a year or two ago I looked it up and some well known journals still preferred passive voice.

Here's an interesting recent paper that found that roughly 30% of papers are still in the passive voice (admittedly, I was expecting a higher percentage).

I would say this is relatively typical. (Background: I did a masters in philosophy at Pitt).

Edit: I would also say that it's unfortunate that parts of philosophy are this hand-wavy. For instance, at one point in the paper, the author describes hallucinating that you see a letter in your mailbox as being phenomenologically just like actually seeing one. Daniel Dennett has a well known discussion in the first chapter of Consciousness Explained arguing that hallucinations are phenomenologically quite different from normal perception. Perhaps the author would say it doesn't matter, that's just a contingent fact that doesn't undermine the specific point being made, but 1) I'm not confident that argument works, and 2) either way, it should be stated explicitly.

I'm on the other side of the fence when it comes to writing style, especially in published research papers in the hard sciences. I can count the number written in this style on one hand.

When there is passion in someone's writing voice, it infects me. One of the reasons I didn't pursue a scientific career is that I write in a somewhat whimsical style and would feel my soul being crushed if my career was spent churning out papers devoid of any passion or love for the field

Funny related reading: Imre Lakatos - Proofs and refutations

Freely available version: https://math.berkeley.edu/~kpmann/Lakatos.pdf

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.

Also: George Polya, "How to solve it"

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