Mathematics poll

[ciphergoth]

Inspired by a similar poll in palmer1984's journal.

About the "proof" question below: examples of the kind of proof I mean would be a proof that there are infinitely many primes, or that the square root of two is irrational, or of Pythagoras's Theorem. A proof in computer science counts too. By "know a proof off by heart" I mean that you'd be able to convince someone of it at a party, if they had the background to follow the proof.

If you know lots of proofs, feel free to choose one you particularly like in the last question...

[Poll #1351621]

Comments

[alextiefling.livejournal.com]

How would you prove Pythagoras' Theorem? Isn't it really a metric axiom rather than a genuine theorem?

[wildeabandon.livejournal.com]

You can start with Euclid's axioms and from there derive Pythagorus' Theorum - I think that counts as a proof.

[fizzyboot.livejournal.com]

Fuck Euclid. The best way to do geometry (and geometrical proofs) is cartesian coordinates.

[ciphergoth.livejournal.com]

Depends on whether you use R² or Euclid's axioms to define the plane I guess. The proof I have in mind is the variant on proof #4 here, for example.

[skx.livejournal.com]

Cute page - the one I knew is listed as #3.

[adjectivegail]

I just read proofs 1 and 2 and found them interesting, right up until the author said "obviously the resulting shape..." Because, no, it's not obvious, some of us actually have to sit and think a while before we realise that it's the case. And this is My Huge Button about maths, which has grown ever-larger through the years as people have said "but it's obvious that..." about a million different things, none of which have been obvious.

It puts me right off, as evidenced by my immediately clicking away from the page and going to do something more fun instead.

[ciphergoth.livejournal.com]

Try Proof 9, which is almost entirely visual.

The thing that makes the proof work is that in both cases, the side of the big square is the long and short sides of the triangle put together. The four triangles are just moved from one place to another between the two, so they cover the same area. That means the area remaining - blue in one case, red in the other - is the same. The blue area is exactly the square on the hypotenuse, and the red area is the sum of the squares on the other two sides.

Is that any clearer?

[valkyriekaren.livejournal.com]

To clarify: I find maths awful in both th old and new senseof the term. I'm moderately discalculate which means I struggled with maths at school (though fortunately being fairy bright and knowing my way around a calculator meant that I didn't suffer academically because of it), and while now I'm older I can understand how powerful and awe-inspiring the equations that keep the universe ticking are, the idea of working them out gives me the cold sweats!

[seph-hazard.livejournal.com]

Yes, this - including the dyscalculia. Still if someone asks me a very simple sum I freeze up, stutter, stammer, feel three inches high and extremely stupid and run away.

[elsmi.livejournal.com]

For whatever it's worth, that's also roughly the reaction of most mathematics grad students I've known.

[ciphergoth.livejournal.com]

BTW your example theorem was the same as mine, I think!

[djm4]

I can also prove that root 2 is irrational. It's a proof I'm particularly partial to as Pythagoras allegedly proved it and then tried to suppress the knowledge, as it offended his mystic sensibilities of the Universe.

[drdoug.livejournal.com]

I like this tale too. I have a dim memory of a smart alec going on a boat trip with Pythagoreans, and casually asking them how long the diagonal of the unit square is. They took it in good humour and merely drowned their impious interlocutor.

[ex-pipistre.livejournal.com]

I thought one of his students proved it and he tried to suppress it by getting his minions to drown the upstart!

Damn, I forgot to put that one on my list.

[aegidian]

There's an interesting subset of your friendslist!

[ali-in-london.livejournal.com]

By "know a proof off by heart" I mean that you'd be able to convince someone of it at a party, if they had the background to follow the proof.

Now I want to organise a "bring a bottle and a theorem" party.

[simont]

You'd have to do something about the considerable risk of two people turning up wearing the same theorem.

(... and also, at the beginning of the party, sort the proofs in descending order of complexity, so that anyone wanting to listen to a complicated one can do so while still relatively sober ...)

[keirf.livejournal.com]

It wouldn't matter if two people turned up with the same theorem, provided they each brought along a different proof.

[valkyriekaren.livejournal.com]

Also, sort the alcohol by proof.

[sphyg.livejournal.com]

I enjoyed double A-Level maths but have unfortunately forgotten most of it.

[ajva.livejournal.com]

I suggest that the proof I've nominated is the only one worth taking to parties if you want to talk maths to non-mathmos, thus getting your maths fix in but also avoiding getting stuck in the corner all night discussing group theory or diagonal proofs. :o)

[hairyears.livejournal.com]

I enjoyed mathematics at school, but the teaching of mathematics had effectively collapsed by the time I went on to Sixth form, and I missed the opportunity to cultivate a skill and a field of study which would be both profitable and fascinating to me today.

As for proofs: I can follow a simple proof in geometry, with some effort. It needs training.

[ergotia.livejournal.com]

I enjoy mental arithmetic on a simple level i.e. multiplying in hundreds is my limit and I enjoy the proofs I can understand, which are few indeed :) But generally maths has been torture all my life!

[ergotia.livejournal.com]

Although on further thought I could not walk into a party right now and explain that proof - would have to revise it first.

[lizw.livejournal.com]

I used to know some proofs off by heart when I was doing the German equivalent of A-levels, which rather scarily involved an oral examination. I've forgotten them all now, but can still follow things of a similar level if I encounter them in a book or an article.

[battlekitty.livejournal.com]

I definitely knew some, and could probably reconstruct them if I sat and thought about it, but as it's not something I've done for donkey's years I couldn't tell you which.

I'm pretty sure one of the gas laws (Boyle's Law, etc) I knew at some point (that'd be the one that was easy...!) and a few other physics ones and several geometric ones and the like are among them. I definitely knew the conics ones.

Bah, must be getting old!

plums and maths, a mini saga

[plumsbitch.livejournal.com]

artremis had to explain to me that 'a proof' was a specific mathematical usage, so er, no, I don't know any. ;-)

The 'have you ever' qu, I ticked other because:

-In primary school I loved what was taught as maths (coz as artremis usefully pointed out, what 'maths' is changes hugely depending on level/style of education) *and* was in a class of three people who had extra maths lessons because we whizzed through everything too quickly.

Somewhere by my teens however, I grew an utter fear of/boredom with 'maths' (whatever it had come to represent by then) and thought (erroneously - I wasn't brilliant, but certainly wasn't awful) that I was utterly and completely useless at it/it exemplified Me Being Stupid and/or Failing. (partly this might be to do with going to same school as sister who was mathsy and sporty(maths/further maths/maths degree), and being expected to be same)

Recently, a friend who is a maths academic has explained various mathsy things in ways which I have
a)understood to some degree, this as asssessed by him in our discussions and
b)found very interesting.

So maths and me have an interesting history.

[wight1984.livejournal.com]

I knew the proofs for Gödel's first incompleteness theorem well enough to write an exam paper detailing them and their implications whilst at university. Doubt that whatever I could explain now would sound very convincing though :oP

I always liked maths as a child but grew board of the teaching environment for it at about 14-16 so let myself down a lot and lost all chances of further study. Enjoyed symbolic logic at uni though :oD

[xquiq.livejournal.com]

I did first and second year maths at university, but I've possibly forgotten more than a remember, it seems since I no proofs sprang to mind other than those you've mentioned.

I'm now thinking I'll have to dig into my old maths books. I do recall spending a large amount of time messing around with Taylor series, so presumably I must've been doing it for a reason, but can't for the life of me remember what!

I often wish I'd stuck with it, rather than switching to Economics. There's definitely something pleasing about the rigor required in mathematics.

[fizzyboot.livejournal.com]

I have a rather unconventional proof, that there is a mapping that converts a number (fraction, irrational, complex, etc) into an integer.

[ciphergoth.livejournal.com]

The mapping f(x) = 0 does that - what other property does your map have?

[fizzyboot.livejournal.com]

Sorry, I should have added that different numbers get converted to different integers.

And it works for all numbers.

[djm4]

I'm clearly missing something here, because at first sight that would make the irrationals countable. And I have this here proof by Cantor which says they're not.

[fizzyboot.livejournal.com]

Yet my mapping exists. Tell you what, you give me a number -- it can be irrational if you like -- and I'll give you the integer for it according to my mapping.

[djm4]

That's merely an assertion that you can do that for all numbers I could give you. I have a mapping that does that - I start a counter from 1, and for every subsequent different number I'm given, I increment it.

For many irrational numbers (an uncountable infinity of the things, actually), the only way for me to express them would be to recite an infinite string of numbers at you. I mean, sure, there are a few that can handily be expressed as 'Pi' or 'root 2', but if your mapping can only cope with those, then it doesn't work for all numbers. And if your 'proof' of your mapping is to have me tell you a number and you give me the integer for it, then you've restricted yourself to proving the mapping for any numbers small enough for me to be able to give them to you. Which isn't very many, and doesn't prove a thing.

You did say it worked for all numbers. Not just 'all numbers you'd be able to give me'.

[fizzyboot.livejournal.com]

That's merely an assertion that you can do that for all numbers I could give you. I have a mapping that does that - I start a counter from 1, and for every subsequent different number I'm given, I increment it.

That's not what I'm thinking of; my mapping is independent ogf the order in which the numbers are presented to it. (The output depends only on the input; my function has no internal state).

For many irrational numbers (an uncountable infinity of the things, actually), the only way for me to express them would be to recite an infinite string of numbers at you. I mean, sure, there are a few that can handily be expressed as 'Pi' or 'root 2', but if your mapping can only cope with those, then it doesn't work for all numbers.

You catch on a good deal quicker than the last people I presented this to! Yes, indeed, my mapping only works with those numbers for which there is a finite symbolic expansion (I use the ascii values of the characters, raised to powers of 256 to depending on the place-order of each character.)

I submit that any entities that cannot be represented in this way are not numbers; indeed they are not anything at all, and certainly not anything mathematical.

The whole point of maths is that it involves arbitrary manipulations of arbitrary symbols on pieces of paper (and lately on computers). Except they're not arbitrary, because they're constructed in such a way that you can make valid inferences about the real world from them. (And that's the whole point of maths: it's intensely useful such that modern society couldn't exist without it).

If maths is about manipulating arbitrary groups of symbols, then anything that cannot -- not even in principle -- be represented as an arbitrary groups of symbols, isn't part of maths.

I realise that the way I think about maths isn't the way most most mathematicians do; although I suspect my way is equally valid.

You did say it worked for all numbers. Not just 'all numbers you'd be able to give me'.

If you can't give me something, not even in principle, and nor can anyone else, it doesn't exist.

[ciphergoth.livejournal.com]

You're going to find a lot of math awfully hard to do if you can't think about any uncountably large collections.

[fizzyboot.livejournal.com]

You're going to find a lot of math awfully hard to do if you can't think about any uncountably large collections.

The phrase "uncountably large collections" isn't uncountably large. Your brain is a finite object or finite size and there is a limit to the complexity of the things it can think about. The concept of "uncountably large" is not sich a complex thing.

I don't really do maths, but I do do programming. For example, I am designing/implemententing a programming language ATM. If I were to try to enumerate all the programs that could be written in my language, it would take up more than all the matter in the universe. However that does not stop me from reasoning about it, nor does it dtop my computer from running a compiler and interpreter for that language.

If my language could not be espressed in a finite set of symbols, it couldn't exist in any real sense.

[ciphergoth.livejournal.com]

So, numbers that can't be expressed don't "exist" but you're prepared to treat them as if they do "exist" if, for example, you're talking about measure theory. I think you think you're doing something terribly important and clever, but from here it looks more like theology than mathematics.

[fizzyboot.livejournal.com]

So, numbers that can't be expressed don't "exist",

Putative symbol-lists (whether you want to think of them as numbers or otherwise) that cannot in principle be expressed, are not expressed anywhere. In that sense they clearly do not exist.

A movie on my hard disk takes up about 1 GB, i.e. it's an 8,000,000,000 digit long binary number. Do all such numbers "exist"? I submit that the only ones that exist, are those that actually have physical representations somewhere; to say otherwise is to say that a movie exists before filming has started on it, which IMO is ridiculous.

I think you think you're doing something terribly important and clever

No, I think it's bloody obvious.

[ciphergoth.livejournal.com]

Naah, it's pretty much a mathematician's green ink letter...

[elsmi.livejournal.com]

> The phrase "uncountably large collections" isn't uncountably large. Your brain is a finite object or finite size and there is a limit to the complexity of the things it can think about. The concept of "uncountably large" is not sich a complex thing.

Quite so. And one of the inferences we can make in doing that non-infinite reasoning is that, using the relevant senses of the words, there is no mapping with the property you propose. You can't just retreat behind puns when someone calls you on that...

But, you know, room for all kinds. I hope you and Brouwer have a nice beer together.

[djm4]

You are Ludwig Wittgenstein and I claim my £5. ;-)

[ciphergoth.livejournal.com]

OK, please stop being coy and tell us what you're on about.