Your favorite mathematical theorems

[micromass]

What are some examples of the proofs that you don't consider rigourous?

Maybe I should say what I mean with a rigorous proof. A rigorous proof starts from certain axioms and then proves the theorem using certain inference rules. However, you will never see things proven that way in textbooks. However, the proofs in textbooks always make it clear that it can be done in principle, if you want to.

So take a look at Proof 97, which is a proof without words:

I do not see at all how to reformulate this proof into a rigorous proof starting from the Euclidean axioms. It seems to me that it is very difficult to do so. So I don't consider this a rigorous proof.

However, I would have no problem believing the theorem to be true if I saw the above proof. And I would have no problem presenting this proof in certain classes. It certainly does make the statement believable, so it's not a bad proof.

 

[atyy]

However, I would have no problem believing the theorem to be true if I saw the above proof. And I would have no problem presenting this proof in certain classes. It certainly does make the statement believable, so it's not a bad proof.

For a proof of the Pythagorean theorem, is it permissible not to start from the Euclidean axioms? Could one start from a Riemannian metric which is Euclidean? Or is that cheating - almost proof by definition?

 

[micromass]

For a proof of the Pythagorean theorem, is it permissible not to start from the Euclidean axioms? Could one start from a Riemannian metric which is Euclidean? Or is that cheating - almost proof by definition?

I would be inclined to say that they are different theorems. Sure they're both called the Pythagorean theorem since it states the same result. But part of a theorem are also the axioms. So a theorem is a statement "If these axioms are true, then blablabla". As such, the Pythagorean theorem starting from the Euclidean axioms is a very different statement than one starting from a Riemannian metric.

Also (and this has nothing to do with the mathematics but more with the motivation), I don't see how you would motivate a Euclidean metric without the Pythagorean theorem. Sure, you can introduce it, but I see no real reason that it would be particularly interesting. The Euclidean axioms on the other hand are very intuitive statements about the world (which happen to not describe the world we live in due to GR).

 

[atyy]

I would be inclined to say that they are different theorems. Sure they're both called the Pythagorean theorem since it states the same result. But part of a theorem are also the axioms. So a theorem is a statement "If these axioms are true, then blablabla". As such, the Pythagorean theorem starting from the Euclidean axioms is a very different statement than one starting from a Riemannian metric.

Ah thanks! I always thought a true mathematician would see it that way. I tend to think it's true because my little patch of the world is a little patch of the GR universe!

Also (and this has nothing to do with the mathematics but more with the motivation), I don't see how you would motivate a Euclidean metric without the Pythagorean theorem. Sure, you can introduce it, but I see no real reason that it would be particularly interesting. The Euclidean axioms on the other hand are very intuitive statements about the world (which happen to not describe the world we live in due to GR).

The funny thing is that even though it is not true in GR, it seems difficult to motivate the Riemannian metric and pseudo-Riemannian metrics without the Pythagorean theorem.

 

[atyy]

Maybe the reason Euclidean geometry is a good motivation even for GR is that Euclid was both a physicist and a mathematician? Mathematically, because of the duality between lines and points, for many purposes we can take Euclidean points as a model for physical lines. But Euclid specified a point not as an abstract object, but as something physical "that which has no parts". Similarly, the Euclidean straight line is a model of a rigid body - the straight edge. So he incorporated spacetime and condensed matter into his axioms.

 

[zoobyshoe]

But Euclid specified a point not as an abstract object, but as something physical "that which has no parts". Similarly, the Euclidean straight line is a model of a rigid body - the straight edge.

How do you figure these aren't abstractions? A thing with no parts has only one property left: location. The point of a point is to make a perfect specification of a location. If that location has any length, width, or height associated with it, it becomes ambiguous. A "part" is a fraction. A thing with no parts is a thing that can't be subdivided into any equal parts. Because, if it could, the location we're trying to specify would become ambiguous. If we could cut a point in half we would no longer be sure in which half we should locate the end of our line or the intersection of two lines, etc. Anyway, there is no physical object that conforms to that. It's an abstraction. And a straight line is a series of points with length only and no width or height. It, too, is abstract. There is no physical entity with length but without width and height.

I thought the "rigid body" of SR was required to have three dimensions and time. The whole point is to see how length and time are altered by great velocity relative to the observer.

 

[atyy]

How do you figure these aren't abstractions? A thing with no parts has only one property left: location. The point of a point is to make a perfect specification of a location. If that location has any length, width, or height associated with it, it becomes ambiguous. A "part" is a fraction. A thing with no parts is a thing that can't be subdivided into any equal parts. Because, if it could, the location we're trying to specify would become ambiguous. If we could cut a point in half we would no longer be sure in which half we should locate the end of our line or the intersection of two lines, etc. Anyway, there is no physical object that conforms to that. It's an abstraction. And a straight line is a series of points with length only and no width or height. It, too, is abstract. There is no physical entity with length but without width and height.

I thought the "rigid body" of SR was required to have three dimensions and time. The whole point is to see how length and time are altered by great velocity relative to the observer.

What you are saying makes sense, but the reason I think they aren't mathematical abstractions is that mathematicians seem not to accept these parts of the axioms any more. If I understand mathematicians correctly, one must always start with fundamental undefined objects and relations in the axioms that mutually define each other. Since Euclid never used "that which has no parts" in future derivations, here he must (according to me) have been doing physics, not maths. Or at least that would explain how even in a structure like GR in which Euclidean geometry is false, the Pythagorean theorem seems to remain a major motivation.

Here is the famous quote from Hilbert: "One must be able to say at all times (instead of points, lines, and planes) tables, chairs, and beer mugs." http://www3.canyons.edu/faculty/matsumotos/MathQuotes.htm

 

[zoobyshoe]

What you are saying makes sense, but the reason I think they aren't mathematical abstractions is that mathematicians seem not to accept these parts of the axioms any more. If I understand mathematicians correctly, one must always start with fundamental undefined objects and relations in the axioms that mutually define each other. Since Euclid never used "that which has no parts" in future derivations, here he must (according to me) have been doing physics, not maths. Or at least that would explain how even in a structure like GR in which Euclidean geometry is false, the Pythagorean theorem seems to remain a major motivation.

Here is the famous quote from Hilbert: "One must be able to say at all times (instead of points, lines, and planes) tables, chairs, and beer mugs." http://www3.canyons.edu/faculty/matsumotos/MathQuotes.htm

Okay, then:

''Mathematicians are like Frenchmen: whatever you say to them, they translate it into their own language and forthwith it is something entirely different.'' -Goethe

 

[jostpuur]

You need to look at better sources, Lagrangian mechanics has a rigorous framework.

Blatant to you maybe. I certainly know that it's rigorous enough, and I don't think I'm alone in this.

I'll show you how I see rigorous Lagrangian mechanics. Let's see what happens.

We fix a dimension N=1,2,3,\ldots. We fix some function L:\mathbb{R}^{1+2N}\to\mathbb{R}, and assume that all its second order partial derivatives exist, and are continuous. We agree on a custom, that the parameter of L is often denoted as (t,x,\dot{x}), where t\in\mathbb{R}, x\in\mathbb{R}^N and \dot{x}\in\mathbb{R}^N. We assume that spacetime points (t_A,x_A) and (t_B,x_B)\in\mathbb{R}^{1+N} are fixed. We define sets \mathcal{X} and \mathcal{X}_0 in the following way:

<br /> \mathcal{X}= \big\{x\in C^2([t_A,t_B],\mathbb{R}^N)\;\big|\; x(t_A)=x_A,\; x(t_B)=x_B\big\}<br />

<br /> \mathcal{X}_0 = \big\{x\in C^2([t_A,t_B],\mathbb{R}^N)\;\big|\; x(t_A)=0,\; x(t_B)=0\big\}<br />

Now \mathcal{X}_0 is a vector space, while in most cases \mathcal{X} will not be, at least not with the ordinary addition and scaling. By using the ordinary Euclidean norm in \mathbb{R}^N we make \mathcal{X} into a metric space by setting

<br /> d(x,y) = \sqrt{\int\limits_{t_A}^{t_B} \|x(t)-y(t)\|^2dt} + \underset{t_A\leq t\leq t_B}{\textrm{max}} \|\dot{x}(t) - \dot{y}(t)\|<br />

for all x,y\in\mathcal{X}. We make \mathcal{X}_0 into a norm space by setting

<br /> \|x\| = \sqrt{\int\limits_{t_A}^{t_B} \|x(t)\|^2dt} + \underset{t_A\leq t\leq t_B}{\textrm{max}} \|\dot{x}(t)\|<br />

for all x\in\mathcal{X}_0. Now these metric and norm spaces are related in an obvious way. For example d(x,x+h)=\|h\| holds for all x\in\mathcal{X} and h\in\mathcal{X}_0. Then we define a mapping S:\mathcal{X}\to\mathbb{R} by setting

<br /> S(x) = \int\limits_{t_A}^{t_B} L\big(t,x(t),\dot{x}(t)\big)dt<br />

Now S is a continuous mapping. (A non-trivial claim! The continuity is considered with respect to the defined metric, of course.) For all x\in\mathcal{X} there exists a continuous linear mapping DS(x):\mathcal{X}_0\to\mathbb{R} and a real coefficient M(x)\in\mathbb{R} such that

<br /> \big|S(x+\eta) - S(x) - DS(x)\eta\big|\leq M(x)\|\eta\|^2<br />

holds for all \eta\in\mathcal{X}_0 with the additional condition \|\eta\|\leq 1. With a fixed x this linear mapping is unique. (This means that the mentioned inequality does not hold with any other linear mapping even if the coefficient M(x) was replaced with a new coefficient too.) If x is a local minimum or a local maximum of S, then DS(x)=0. (A non-trivial claim! The locality is considered with respect to the mentioned metric.) Also the relation

<br /> DS(x)=0\quad\Longleftrightarrow\quad D_t\nabla_{\dot{x}} L\big(t,x(t),\dot{x}(t)\big) = \nabla_x L\big(t,x(t),\dot{x}(t)\big)\quad\quad\forall\; t_A\leq t\leq t_B<br />

holds.

How do you like this theorem? Not very elegant, but I like this. I could mention this as one of my answers to the opening post's question.

I have few questions to those who claim that mainstream Lagrangian mechanics is rigorous. Firstly, have you seen anything like this anywhere ever? I know that this kind of mathematics can be found from some mathematics books, but they are not books on physics. I have never seen a book on physics or mechanics that would give results like this. Have you? Another thing is that do you think that I'm making things too complicated here? Do you think that Lagrangian mechanics can be made rigorous more easily, without these kind of function spaces? I would be interested to hear more.

 

[Enigman]

http://d2tq98mqfjyz2l.cloudfront.net/image_cache/137408946864354.jpg​

More seriously- the Divergence theorem.
(Stokes' is taken)

 

[Pond Dragon]

Particularly beautiful? How about de Rham's theorem?

 

[Jorriss]

Stone-Weierstrass theorem(s)

 

[atyy]

More seriously- the Divergence theorem.
(Stokes' is taken)

I think most people included the Divergence theorem when they said Stokes's theorem. I did. I'd be interested to know if that was indeed what people meant (probably you too, but you were joking:)

Here is an interesting blog post which talks about the Stokes's theorem in a way I think about it - the way in Spivak's Calculus on Manifolds: http://owl-sowa.blogspot.com/search/label/Endre Szemerédi: "Let me clarify how I understand the term “conceptual”. A theory is conceptual if most of the difficulties were moved from proofs to definitions (i.e. to concepts), or they are there from the very beginning (which may happen only inside of an already conceptual theory). The definitions may be difficult to digest at the first encounter, but the proofs are straightforward. A very good and elementary example is provided by the modern form of the Stokes theorem. In 19th century we had the fundamental theorem of calculus and 3 theorems, respectively due to Gauss-Ostrogradsky, Green, and Stokes, dealing with more complicated integrals. Now we have only one theorem, usually called Stokes theorem, valid for all dimensions. After all definitions are put in place, its proof is trivial. M. Spivak nicely explains this in the preface to his classics, “Calculus on manifolds”."

The funny thing is at the end he says Euclidean geometry is dead. But as were discussing, although I usually think of geometry as Riemannian geometry just out of naive physics habit, it doesn't seem possible to motivate Riemannian geometry without Euclidean geonetry. So in that sense, how could Euclidean geometry be dead, even though it is true that metric geometry is the form which seems more alive now. What do others think?

 

[Demystifier]

The proof that there are more reals than integers, by Cantor diagonalization.

 

[slider142]

Well I have never seen rigor Lagrangian mechanics either, except in my own notes. The physicists talk about minimizing the Lagrangian, but they have no clue of what kind of function space the domain is, or what metric it would have. Of course the domain space must have some metric, because otherwise the Lagrangian couldn't have local minima. I mean how could a mapping

<br /> L:?\to\mathbb{R}<br />

have a local minima, if nobody knows what the domain is, or what metric it would have?

Hmm. I believe pages 4-5 of Quantum Mechanics for Mathematicians provides the necessary rigor. Walter Thirring's Classical Mathematical Physics also goes through the requisite differential geometry. The formalism is that of finding the stationary point of a functional using generalized derivatives, such as the Gateaux and Frechet derivatives, so a tiny bit of functional analysis is required.

As for myself, I always loved the generalized Stokes Theorem, as many others have already mentioned.

 

[Hologram0110]

Although not theorems, an engineer I've always been blown away by Fourier series & transform. So incredibly useful for a wide range of applications. While it seems obvious that you should be able to use different function spaces to simplify problems in this case it just works out so elegantly.

As for real theorems, I'm going to go with Residue theorem or Stokes theorem.

 

[atyy]

The proof that there are more reals than integers, by Cantor diagonalization.

When I first read the proof the Goedel-Rosser incompleteness theorem, I read a proof with an explicit costruction of an undecidable statement, and it seemed very mysterious. Later, I learned that there was a later proof, by Tarski, based on Cantor's diagonalization.

I'm still trying to understand this method based on diagonalization. Dan Hathway has a write-up here based on Hinman's book: http://danthemanhathaway.com/MathNotes/Incompleteness.pdf.

 

[Demystifier]

When I first read the proof the Goedel-Rosser incompleteness theorem, I read a proof with an explicit costruction of an undecidable statement, and it seemed very mysterious. Later, I learned that there was a later proof, by Tarski, based on Cantor's diagonalization.

I'm still trying to understand this method based on diagonalization. Dan Hathway has a write-up here based on Hinman's book: http://danthemanhathaway.com/MathNotes/Incompleteness.pdf.

I think the original Goedel's proof is also based on Cantor's diagonalization, as is any proof in abstract logic based on self-reference.