Fermat

rugbug86

ok, maybe i'm not normal but has anyone read the book by simon singh about fermats theorem? its great stuff.

for instance, did you know that (for the majority of rivers) if you get a map and measure the distance from source to mouth directly and then measure the distance taking in all the bends and stuff, the one with the bends will be approx 3.14 times the length of the straight one. thats pi!

i had to get a pen when i was reading the book and i started making notes and stuff in the borders, and when my mom read it she did the same!

anyone else read it? or am i just a loser??!!!

David19

Yeah i read that book a couple of years ago. It is a great read, especially for people who don't have a lot of maths(i was in 6th year when i read it). Im planning on reading it again soon, along with Singh's code book which I have never read.

Another book along the same lines is 'Dr. Riemann's Zeros: The Search for the $1million Solution to the Greatest Problem in Mathematics' by Karl Sabbagh.
Here's a link: http://www.amazon.com/exec/obidos/tg/detail/-/1843541009/qid=1110760833/sr=1-1/ref=sr_1_1/102-5434754-9693749?v=glance&s=books

rugbug86

my next one is "Surely you're joking Mr. Feynman" but i'll look into Riemanns Zeros

Singhs code, think ive heard of it and was going to buy it but was going on holiday and wanted light reading!

ecksor

rugbug, there's a few books reviewed in the stickies at the top of the maths forum. I've read "Surely you're joking ...", it's a good book.

Pherekydes

I read Fermat's Last Theorem, too. It's a little bit unsatisfactory, in that Wiles' solution is not exactly what Fermat intended. Still a good book. Anyone read "The music of the primes", another in the same vein?

Delphi91

Slow coach wrote:

...It's a little bit unsatisfactory, in that Wiles' solution is not exactly what Fermat intended...

How do we know what Fermat intended? Do we actually know what Fermat's solution was? All we have is a note scribbled in a margin in which he "claims" to have a solution but the margin is too small a space in which to write it.

On the subject of the book, I found it a fantastic book. Simon Singhs "The Code Book" is also highly recommended - a fascinating insight into cryptography.

Feyman's two books - "Surely you're joking, Mr Feynman" and "So what do you care what others think" - are EXCELLENT books. They give a fascinating insight into a brilliant mind.

Mike

Pherekydes

Delphi91 wrote:

How do we know what Fermat intended?

We know what he didn't intend. Because much of what Andrew Wiles produced was new Mathematics.

Delphi91 wrote:

Do we actually know what Fermat's solution was?

Obviously not. Otherwise mathematicians wouldn't have spent significant parts of their careers trying to look for it.

Delphi91

Slow coach wrote:

...Obviously not. Otherwise mathematicians wouldn't have spent significant parts of their careers trying to look for it.

Which suggests that in all likely hood that he didn't have a solution at all!

Pherekydes

Delphi91 wrote:

Which suggests that in all likely hood that he didn't have a solution at all!

Agreed, it does seem strange that the world's greatest minds haven't found such a "trivial" solution.

Fermat might have been bluffing...

ecksor

Slow coach wrote:

We know what he didn't intend. Because much of what Andrew Wiles produced was new Mathematics.

So? Considering that Fermat considered his alleged proof to be "astounding", then if he had a proof then it was likely to have been a new idea.

Slow coach wrote:

Agreed, it does seem strange that the world's greatest minds haven't found such a "trivial" solution.

Fermat might have been bluffing...

I doubt he was bluffing. I'd imagine he just had a proof that was flawed. But perhaps he did have a proof and we'll never know what that was. Either way, if you suspect that he didn't actually have a proof then what's so unsatisfying about Wiles' proof?

mikemac

Original poster has an interesting point.
If you want to follow that up read Dan Browns books.
I'm not sure which one(Probably the Da Vinci Code) but a lot of examples are given about how Pi has applied itself in loads of situations.
Check it out

Delphi91

micmclo wrote:

Original poster has an interesting point.
If you want to follow that up read Dan Browns books.
I'm not sure which one(Probably the Da Vinci Code) but a lot of examples are given about how Pi has applied itself in loads of situations.
Check it out

On a related point (sort of!!), yesterday was "Pi Day" - March 14th, or
3.14(.05) if youo write the date in the American way!

Mike

ecksor

PI is approximately 3.14159265358979, so I think we'll have to wait 10 or 11 years for a better approximation to PI day.

Pherekydes

ecksor wrote:

Either way, if you suspect that he didn't actually have a proof then what's so unsatisfying about Wiles' proof?

I didn't say I suspected he didn't have a proof. I said it was strange that the top guys hadn't found it, if it was so obvious to Fermat.

ecksor

What do you think he might have been bluffing about then?

Pherekydes

ecksor wrote:

What do you think he might have been bluffing about then?

Again I didn't say I think he was bluffing. Don't take everything so literally. It's just another possibility.

ecksor

Well, when you say that you find it strange that the greatest minds in mathematics have struggled to find something so trivial and then offer the explanation that Fermat might have been bluffing then the obvious reasoning to me is that he may not have actually had a proof. There are superficially correct proofs that have been offered that have turned out to be incorrect after all.

As for my interpretation, you "agreed" (your word) with the suggestion "in all likely hood that he didn't have a solution at all", so I don't think it's unreasonable to interpret what you said in the way I did.

Pherekydes

ecksor wrote:

Well, when you say that you find it strange that the greatest minds in mathematics have struggled to find something so trivial and then offer the explanation that Fermat might have been bluffing then the obvious reasoning to me is that he may not have actually had a proof. There are superficially correct proofs that have been offered that have turned out to be incorrect after all.

As for my interpretation, you "agreed" (your word) with the suggestion "in all likely hood that he didn't have a solution at all", so I don't think it's unreasonable to interpret what you said in the way I did.

I'm not certain as to the truth either way. It's possible Fermat had a proof, which he neglected to write down; it's possible he had no proof, but just said he had to impress some people; it's possible the proof is not based on "new" mathematics (for the time); it's possible that the proof is still there to be produced. Have I covered all the bases? Anyway, I'm off to bed (to dream about Fermat's proof). Nighty night.

ecksor

Have I covered all the bases?

Not really. I'm not trying to be long winded, but the original point was that I don't see how you can find Wiles' proof so unsatisfying given that we haven't a bloody clue what Fermat may or may not have had, which was the question I posed to you. Am I to guess that you would find an "elementary" proof of the theorem more satisfying for some romantic notions? I suppose most people would.

I will conjecture that if Fermat had a proof and it has been missed for the last 350 odd years then we can pretty much discount the possibility that it wasn't based on some new idea that he had.

(I personally can't find Wiles' proof particularly satisfying or unsatisfying since I can't understand the damn thing).

dudara

Wiles proof is awkward and inelegant. But maybe that's the proof required. It's understandable to want a simple elegant proof to such a simple equation. But I do believe, that if Fermat had a simple solution we would have found it by now.

I remember reading about Fermat before, correct me if I'm wrong here, but didn't he come up with incorrect proofs a few times himself.

ecksor

My understanding is that most or all of his proofs wouldn't be considered rigourous today, and some wouldn't have been accepted at the time. What you're saying rings a bell with me, but I'm struggling to remember or find the example if I have seen one.

Capt'n Midnight

dudara wrote:

But I do believe, that if Fermat had a simple solution we would have found it by now.

wasn't one of Eculids "axioms" proved relatively recently, something about six points and a triangle and a one page proof means it's now a theorm.

so it's unlikely but possible that he had a correct proof of that famous one.

carl_

Capt'n Midnight wrote:

wasn't one of Eculids "axioms" proved relatively recently, something about six points and a triangle and a one page proof means it's now a theorm.

I doubt it but I'd be interested to hear more.

dudara wrote:

Wiles proof is awkward and inelegant.

Is it the sheer size of the proof that makes it awkward or is it something else?

Incidentally, the book "The World's Most Famous Math Problem" by Marilyn Vos Savant sparked an interesting debate. It can be found here (mathpages.com).

ecksor

Capt'n Midnight wrote:

wasn't one of Eculids "axioms" proved relatively recently, something about six points and a triangle and a one page proof means it's now a theorm.

I don't think so. Link to what you're talking about?

Even so ...

so it's unlikely but possible that he had a correct proof of that famous one.

I don't see how the above relates to this?

Professor_Fink

rugbug86 wrote:

for instance, did you know that (for the majority of rivers) if you get a map and measure the distance from source to mouth directly and then measure the distance taking in all the bends and stuff, the one with the bends will be approx 3.14 times the length of the straight one. thats pi!

I think thats more to do with sqrt(g). Where g is the acceleration due to gravity (9.807 ms^-2 approx on earth). So sqrt(g) is about 3.13 (very close to pi) on earth. But I could be wrong.

Now if only we had some rivers on Mars, we could check it out...

Oh well.

spacetweek

Professor_Fink wrote:

I think thats more to do with sqrt(g). Where g is the acceleration due to gravity (9.807 ms^-2 approx on earth). So sqrt(g) is about 3.13 (very close to pi) on earth. But I could be wrong.

Now if only we had some rivers on Mars, we could check it out...

Oh well.

This couldn't possibly be the case as sqrt(g) is a value that relies on the metre and the second - it's not unit-independent.

I heard that it was pi. Pi is a value independent of human measurement units as it is a fundamental property of all circles.

Professor_Fink

It doesn't have to be dimensionless to be proportional to it. I'm not sugesting that its sqrt(g) on its own, because, as you say, the ratio is dimensionless. But a function of velocity, g and density could easily be dimensionless.

By the way the density of water in those units it 1 kg/litre.

As I said, I don't know that this is definitely the case, but I'm more inclined to believe it is dependant on gravity,velocity and density, since thats what determines the pressure on the banks. The velocity of the water is determined by the geometry of the landscape and gravity. So I dont think its unreasonable to think that the shape of the river depends on the landscape, the density of water and g.

Capt'n Midnight

ecksor wrote:

I don't think so. Link to what you're talking about?
Even so ...
I don't see how the above relates to this?

IIRC it was in that book by the girl who won the young scientist by doing cryptography using the non linearity of matrix multiplication. About half way though with a triangle on the right hand side page..
Eculids book was essential reading for most top mathematicians for about two thousand years so it's not as if it was an obscure problem. It shows that even the best minds can overlook something obvious. [inset quote from Billy Connoly about how a genius is not someone who design complex things but someone who makes a fortune from something so simple that you could have thought of it yourself - like "paperclips" or "who wants to be a millionare"]

Capt'n Midnight

Professor_Fink wrote:

I think thats more to do with sqrt(g). Where g is the acceleration due to gravity (9.807 ms^-2 approx on earth). So sqrt(g) is about 3.13 (very close to pi) on earth. But I could be wrong.

Isn't the PI figure (or whatever) for meandering rivers on flat plains so I'd doubt it's that, also that would imply that rivers on high gravity planets would meander more and up on the moon we'd have straighter ones. What if the rivers were of molten lava (density guestimate 2.5) or mercury ??
One limiting effect is the formation of Ox Bow lakes so rivers can't meander too much without cutting off parts or crossing itself.

Random walks though get you a distance of sqrt(N) after N unit steps in a random from your starting point (on average)

breadmonkey

Random walks though get you a distance of sqrt(N) after N unit steps in a random from your starting point (on average)

Do you mean that the displacement from your original position is sqrt(N)?

Professor_Fink

Capt'n Midnight wrote:

Isn't the PI figure (or whatever) for meandering rivers on flat plains so I'd doubt it's that, also that would imply that rivers on high gravity planets would meander more and up on the moon we'd have straighter ones.

I hate to point it out, but rivers flow down hill, and even flowing across a flat surface gravity still plays a role in the pressure felt by the banks.

Have you ever seen rivers on other planets or the moon? I know there are thing which look like they may be dry river beds on Mars, but I've never heard of anyone meassuring their lenght relative to their distance from source. And then they may turn out never to have been rivers.

Lava flow is not river like at all. For a start it doesn't errode the banks, but rather can build then up. Lava also has a different viscosity, which also affects its behaviour.

Also, the Navier-Stokes equations are density dependant and gravity, so I don't think its unreasonable to think that these play a role in the shape of a river.

Capt'n Midnight

breadmonkey wrote:

Do you mean that the displacement from your original position is sqrt(N)?

yeah on average - it's a know meandering mechanism that involves a square root.

Professor_Fink wrote:

I hate to point it out, but rivers flow down hill,

D'oh
/me hastily rubs out a minus sign.

The lava in hawaii is quite different and should at times reach thermal equillibrium with it's surroundings, so it would be interesting to see how it behaved out side it's tunnels, but yeah it's erosion mechanisms are different.

ecksor

Capt'n Midnight wrote:

Eculids book was essential reading for most top mathematicians for about two thousand years so it's not as if it was an obscure problem. It shows that even the best minds can overlook something obvious.

It sounds like you might be talking about Pappus' theorem or Pascal's theorem, but it's hard to tell if you don't describe it clearly. However, those weren't axioms. The only axiom due to euclid that was controversial as far as I know was the parallel axiom, but that issue was settled approximately a couple of hundred years ago with the invention/discovery of non-euclidean geometry (specifically hyperbolic geometry I think).