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That thing you should've understood...

Post Icon Posted: Submitted by Newtonswig on 19 May 2008 - 2:02am.

Joined: 2006-10-01
Posts: 432

A lack of understanding. Maths is full of it. Like awkwardness and silence at an office party, some vacuum in between the lines pervades our subject.

Every day I nod along sagely and fluff my way through logic's hidden trivialities, blindly groping at concepts I really should have picked up a long time ago.

I could ask. But I don't- I might be exposed as a fraud, beaten mercilessly with a board rubber and banned from maths, doomed to spend my life licking fluff from between David Elworthy's toes as punishment (that's probably how it would happen anyway), instead I burble something incoherent about "let's try drawing a diagram". Confused and fearing toes.

The other week I was in just such a toe fearing position, trying to explain expression of permutations as a product of transpositions to some hapless first year. She was in the middle of expaining to me that drawing a diagram of a permutation probably wouldn't help much, when without warning it clicked. Like some luckless blinking adult actress, I got it suddenly. Egg on my face.

"Bollocks." I thought. "That was obvious." Now I can factorise permutations- piece of p*ss, just like that.

What makes it most embarassing isn't that I didn't get it first time- just that I'd managed to convince myself I'd got it. Like a creationist or an abusive parent I lived a happy lie: "Something fiddly happens there." I'd tell myself, and just keep going- too much of what I was supposed to know depended on it to ever dare give question.

Mathematics is in the unique position of being (at least in theory) B.S.-free: why then do we spend so much time convincing ourselves we understand what we plainly do not? What do we gain?

What's the dumbest thing you've convinced yourself you know? Or convinced someone else?

Ansewrs on a postcard please...

‹ 1000 posts! Discussion group on Thursday ›
Post Icon Posted: 19 May 2008 - 4:13am

Joined: 2007-10-03
Posts: 397

No particular examples spring to mind at the moment, but I remember telling myself "Oh it's fine, I think I understand, I'll just accept the result for now, I'm sure it will become obvious later".

It mostly happens when I'm reading a book and want to get to the more interesting results, but quite luckily I generally end up realising I don't understand things I should and go back and clear things up. Still, I'm sure there are a squazillion results I just accept without really understanding. Though sometimes it's due to unconvincing proofs, that do give the result but fail to really provide insight as to why it is true.

I fully agree that it is really quite bad that we resort to such methods, as it does tend to ruin the appreciation of the subject as a whole.




Ah, now I remember one example, the triple product rule for partial derivatives :
$$\frac{\partial x}{\partial y}\frac{\partial y}{\partial z}\frac{\partial z}{\partial x}=-1$$
and the unconvincing proof that goes along with it :
$$dz = \frac{\partial z}{\partial x}dx + \frac{\partial z}{\partial y}dy$$
Suppose $ dz = 0 $ (not sure why this is possible :D)
$$0 = \frac{\partial z}{\partial x}dx + \frac{\partial z}{\partial y}dy = \frac{\partial z}{\partial x}dx + \frac{\partial z}{\partial y}\frac{\partial y}{\partial x}dx$$
Which then gives
$$-\frac{\partial z}{\partial x}=\frac{\partial z}{\partial y}\frac{\partial y}{\partial x}$$
and
$$\frac{\partial x}{\partial y}\frac{\partial y}{\partial z}\frac{\partial z}{\partial x}=-1$$
(not sure either how $ \frac{1}{\frac{\partial x}{\partial y}}=\frac{\partial y}{\partial x} $, especially what explanation goes with it other than the usual " $ \partial x $ and $ \partial y $ are infinitesimals so we can treat them as numbers" which doesn't really appeal)
Post Icon Posted: 19 May 2008 - 9:42am

Joined: 2007-10-17
Posts: 109

I can't recall anything like that. Perhaps I find it so embarassing I make myself forget. But I recall little things. In A level I was sure that 'as x tends to infinity' (how do you do TeX of the forums?) meant that there was a point at infinity and at that point the function reached its limit. Actually, I understand that can work to some extent but of course that isn't the convention.
Post Icon Posted: 19 May 2008 - 10:05am

Joined: 2006-10-10
Posts: 520

Long division :/
Post Icon Posted: 19 May 2008 - 10:40am

Joined: 2006-10-05
Posts: 699

You got taught about x tending to infinity at A level!?!

I never learned my times tables. If I don't concentrate, I ALWAYS get $ \frac{7}{2}=4.5 $. I have difficulty adding. I can't remember doing the Analysis 3 exam let alone anything in the course. Integration gives me a headache. In addition I have yet to understand a discussion group all the way through (including the one I ran).

How I survive: I like exams and I beat Cosmin at everything other than maths to keep my spirits up.

Seriously though, I've never forgotten a whole subject before.

Colin: For TeX you use dollars ($) just as if you were using LaTeX on the forums.

PS have never convinced someone of finite primes, yet.
Post Icon Posted: 19 May 2008 - 5:26pm

Joined: 2006-11-02
Posts: 1090

How I survive: I like exams and I beat Cosmin at everything other than maths to keep my spirits up.

You wish. :D

Edit: Incidentally, that answers the question as to what the dumbest thing Alex has ever convinced himself of is. :D
Post Icon Posted: 19 May 2008 - 5:42pm

Joined: 2007-02-14
Posts: 105

Well yeah I convince myself that I know stuff that I actually don't all the time, its how I keep going.
And since this seems to have turned into easy stuff that we can't do, integration, I failed integration in foundations, got a poor mark in Geometry and Motion and don't think I evealuated any integrals in Vector Analysis, that may be more a case of convincing myself I can't do something when I can though.
Post Icon Posted: 22 May 2008 - 2:27pm

Joined: 2006-10-01
Posts: 432

Perhaps you're right, as children we learn our place in society by blind imitation, why not so with maths? In some sense, such arrogance as to claim understanding is facet to our universal naivety: the blithe assumption that adulthood can bring a solidarity of mind; that we can become the people we blindly trusted- in control and calculating, when such things do not truly exist.

The desire to be the bigger man, to claim this myth of understanding for our own, is surely fuel to all pretence. And why not? With the sky our target, we may wait forever for foundations, cement will tendril deeper into trivial soil that we may build higher: and we will never lay a brick. So to show foundation, to ourselves and to the world we build half founded, resloving from the first to butress heavy at the joins and stick with tempered faith to our grand and sketched designs. Hoping that a pencil smudge or hurried unruled line won't hide the fault that cause our hearts to topple.

Faith is the price we pay for knowledge. No progress without pretence.

PS

Long division :/

Agreed entirely :p
Post Icon Posted: 22 May 2008 - 3:24pm

Joined: 2006-10-05
Posts: 699

I actually get long division.
Post Icon Posted: 23 May 2008 - 12:06am

Joined: 2006-11-02
Posts: 1090

I'm impressed.
Post Icon Posted: 23 May 2008 - 12:28am

Joined: 2007-10-03
Posts: 397

Makes me think of the famous quote

In mathematics you don't understand things. You just get used to them.
-- John Von Neumann
Post Icon Posted: 23 May 2008 - 1:37am

Joined: 2006-10-18
Posts: 29

Ok so the proof of the triple product thing goes li}ke this, you start with a function $ f:\mathbb{R}^3\to\mathbb{R} $ which is 'nice' enough for the implicit function theorem (I can't remember the exact conditions but...) and imagine the region $ \{(x,y,z)|f(x,y,z)=c\} $.

Consider $ f(x,y,z) $ as $ f((x,y),z) $ and apply the IFT (around $ ((x_0,y_0),z_0) $) to get $ z=z(x,y) $ with $ f(x,y,z(x,y))=c $. Now since $ f(x,y,z(x,y)) $ doesn't change as you move $ x $, we conclude that
\[D_{(1,0,\frac{\partial z}{\partial x})}f(x_0,y_0,z_0)=0\]
and so (evaluating all partials at $ x=x_0,\ y=y_0\ z=z_0 $ as applicable).
\[\frac{\partial f}{\partial x}+\frac{\partial f}{\partial z}\frac{\partial z}{\partial x}=0\qquad(1)\]
similarly
\[\frac{\partial f}{\partial z}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial z}=0\qquad(2)\]
and
\[\frac{\partial f}{\partial y}+\frac{\partial f}{\partial x}\frac{\partial x}{\partial y}=0\qquad(3)\]
We can substitute $ \frac{\partial f}{\partial z} $ from $ (2) $ into $ (1) $ to get
\[\frac{\partial f}{\partial x}-\frac{\partial f}{\partial y}\frac{\partial y}{\partial z}\frac{\partial z}{\partial x}=0\qquad(4)\]
and then substitute $ \frac{\partial f}{\partial y} $ from $ (3) $ into $ (4) $ leaving
\[\frac{\partial f}{\partial x}+\frac{\partial f}{\partial x}\frac{\partial x}{\partial y}\frac{\partial y}{\partial z}\frac{\partial z}{\partial x}=0\qquad(5)\]
divide through by $ \frac{\partial f}{\partial x} $: (of course, if $ \frac{\partial f}{\partial x}=0 $ then we could have done things so as to get a different partial of $ f $ which was non-zero: one of them must be by IFT conds)
\[\frac{\partial x}{\partial y}\frac{\partial y}{\partial z}\frac{\partial z}{\partial x}=-1\qquad\qquad\qquad\qquad\square\]

b.t.w. Sam, I'm sure Wikipedia will accept this as a reference :)
Post Icon Posted: 23 May 2008 - 1:44am

Joined: 2007-10-03
Posts: 397

Yes !
"As posted on the Warwick Mathematics Society forums, 23-05-2008 by Smithers " :p
Post Icon Posted: 23 May 2008 - 1:56am

Joined: 2006-10-18
Posts: 29

Exactly
Post Icon Posted: 23 May 2008 - 2:27am

Joined: 2006-11-02
Posts: 1090

Don't say "exactly", Sam only agreed with everything you said and didn't add anything. That's equivalent to you saying something, someone else agreeing with you, then you saying "yeah, he's right".
Post Icon Posted: 23 May 2008 - 2:37am

Joined: 2006-10-18
Posts: 29

He did add something, he stated how he would cite it, also

don't set something as a quote if it's not
Post Icon Posted: 23 May 2008 - 2:41am

Joined: 2007-10-03
Posts: 397

But do set it as a quote if it actually is one.
Post Icon Posted: 23 May 2008 - 2:46am

Joined: 2006-11-02
Posts: 1090

Check the Top B thread in the publicity subforum. :p
Post Icon Posted: 23 May 2008 - 2:50am

Joined: 2006-10-18
Posts: 29

Well then if you're going to quote from somewhere other than the current thread, make it clear where. It doesn't help that I can't get to the publicity subforum; at least without joining a group or some such which I am too lazy to do. I don't think knowledge of threads in other forums should be assumed, especially when accessing them is non-trivial.
Post Icon Posted: 23 May 2008 - 2:54am

Joined: 2007-10-03
Posts: 397

Join the group now, I'll accept you instantly. It's worth it, at least for the folkore.
Post Icon Posted: 23 May 2008 - 3:36am

Joined: 2006-11-02
Posts: 1090

I know, but it was more of a dig at Alex than anything else. :p
Post Icon Posted: 23 May 2008 - 10:45am

Joined: 2006-10-05
Posts: 699

I'm honoured, the fact people will go to such lengths to remember my words of wisdom. I feel truly accepted.

Also join publicity, you don't have to post or actually do publicity, just Cosmin allows himself to spam it a lot more and I argue back a lot because stuff actually matters. It's hell.
Post Icon Posted: 23 May 2008 - 9:04pm

Joined: 2006-11-02
Posts: 1090

I would have expected you to find a more appropriate word than "spam" by now. I don't remember ever setting up a "che4p m3dz" shop or telling you how to get free mortgage.

Edit: Ok, except that one time. :P
Post Icon Posted: 24 May 2008 - 12:52am

Joined: 2006-10-05
Posts: 699

Ok when I use the word spam I mean posting for the sake of diverting a threads purpose for no apparent reason except to increase your lameness.

PS I'm aware this is spamming, it goes well with eggs.
Post Icon Posted: 24 May 2008 - 12:56am

Joined: 2006-11-02
Posts: 1090

sigh

Starting now I'm not even going to bother replying to this type of post. :P
Post Icon Posted: 24 May 2008 - 9:13pm

Joined: 2006-10-10
Posts: 520

RABU RABU CHU CHU RA RA RA CHU CHU
JUNPEI II NO HAPPIIIIII

Everyone be happi ._.
Post Icon Posted: 24 May 2008 - 9:25pm

Joined: 2006-11-02
Posts: 1090

Word.
Post Icon Posted: 24 May 2008 - 10:53pm

Joined: 2007-02-14
Posts: 105

Wat
Post Icon Posted: 24 May 2008 - 11:11pm

Joined: 2007-10-03
Posts: 397

son ?
Post Icon Posted: 25 May 2008 - 1:05am

Joined: 2007-02-14
Posts: 105

Elementary my dear
Post Icon Posted: 7 July 2008 - 10:21am

Joined: 2007-10-17
Posts: 109

Is six weeks of idleness a bit too much for me to get away with bumping a thread? This isn't strictly a "maths" thing which I've only just understood, but I mentioned recently that I've been mucking about with a Rubick's cube. You know that they say the centre cubes are fixed? It only just dawned on me that they are fixed in position, but not in orientation. I only realised that because my cube has "Rubick's cube" printed on the white centre cube. I scrambled it, solved it, then spotted that that cube had swivelled round.

In retrospect I've no excuse at all for not having figured that out earlier. Some weeks back Chris Smith was playing about with his cube in the workroom. He was picking various combinations of twists and finding out how many repetitions of those combinations it took to return the cube to its identity state (the solved state). If I remember rightly, one of those combinations, consisting of consecutive quarter turns of adjacent faces, had order 63. 63 isn't divisible by two, let alone four, so you'd think I'd have noticed what must have happened to the cubes in the centre of those faces. But no, it had to wait until there was physical evidence in front of me before it became clear to me that this could happen. Thick or what?

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