Theorems every mathematician should know

[discrete*]

You'll think differently when somebody asks you to solve a nonhomogeneous partial differential equation.

While nonhomogeneous PDEs certainly satisfy the "hard" criterion, I wouldn't say they're particularly satisfying, but that's just me.

 

[usljoo]

yes but when you work harder you then get more pleasure out of it and i know math very well because i AM a mathematician and i don't see where the fun is in it, but physics really goes beyond your imagination and opens whole new views of the world and not views of some imagined set of objects witch has some rules that go with it. and another thing is that math would never evolve anywhere without the physics and the concepts developed there.

it is true that group theory for example evolved from solving polynomial equations but without physics it would be the only field where it would have been applied ... its just boring pfff

 

[disregardthat]

yes but when you work harder you then get more pleasure out of it and i know math very well because i AM a mathematician and i don't see where the fun is in it, but physics really goes beyond your imagination and opens whole new views of the world and not views of some imagined set of objects witch has some rules that go with it. and another thing is that math would never evolve anywhere without the physics and the concepts developed there.

it is true that group theory for example evolved from solving polynomial equations but without physics it would be the only field where it would have been applied ... its just boring pfff

You are arguing against yourself here. First you state that mathematics could not evolve without the physics to develop new concepts. Second, you give a prime example of a whole new area of mathematics that evolved without motivation from physics, but curiously then you dismiss it as boring, and wrongfully state that it cannot be applied anywhere else without physical motivation, which it obviously can. Your point seems to have been lost somewhere along the lines.

The matter of the fact is that much mathematics has evolved historically and still does without physical motivation. The applications tends to come in turn.

 

[discrete*]

yes but when you work harder you then get more pleasure out of it and i know math very well because i AM a mathematician and i don't see where the fun is in it, but physics really goes beyond your imagination and opens whole new views of the world and not views of some imagined set of objects witch has some rules that go with it. and another thing is that math would never evolve anywhere without the physics and the concepts developed there.

it is true that group theory for example evolved from solving polynomial equations but without physics it would be the only field where it would have been applied ... its just boring pfff

If, in fact, you are a mathematician, than I must ask: why? If you don't find mathematics interesting, than why is it your chosen field? For you to come into the math boards and refer to mathematics as "the easiest discipline" and imply that mathematicians are lazy is just a discredit to yourself (albeit a fallacy), if you truly are a mathematician.

Also, I'll say again -- everything you've said is a matter of opinion. You seem to think that only mathematics with application are interesting. That may be true in your case, but it's not true in all cases, especially mine.

You also seem to describe both mathematics and physics in an immature and unsophisticated manner. The two disciplines are completely separate entities; they differ in methodologies, conceptually, and in the goals that they set and accomplish. In my opinion, physics and mathematics ought not be compared to one another, it's simply comparing apples and oranges.

 

[gb7nash]

While nonhomogeneous PDEs certainly satisfy the "hard" criterion, I wouldn't say they're particularly satisfying, but that's just me.

Point well made.

 

[Char. Limit]

This is getting off topic. Therefore, I'm going to submit Fubini's Theorem.

 

[Mathitalian]

Banach fixed point theorem. I love this theorem :)

 

[Curl]

Fundamental theorem of line integrals.

Clairaut's theorem

 

[discrete*]

How about the always overshadowed Fermat's Little Theorem.

 

[discrete*]

Banach fixed point theorem. I love this theorem :)

Nice one. And because you mentioned Banach, how about the Banach-Tarski Paradox. I have been fascinated by this theorem for years.

 

[fourier jr]

How about the always overshadowed Fermat's Little Theorem.

or euler's theorem! one of the first things I thought of were the isomorphism theorems, especially the first one.

 

[jasonRF]

How about Cauchy's theorem from complex analysis, as proven by Goursat.

 

[LCKurtz]

Stone-Weierstrass theorem:

If X is any compact space, let A be a subalgebra of the algebra C(X) over the reals R with binary operations + and ×. Then, if A contains the constant functions and separates the points of X, A is dense in C(X) equipped with the uniform norm.

 

[nonequilibrium]

\forall metric spaces \exists metric space such that he can say "you complete me"

 

[Landau]

I feel like people are responding to either 'what is your favourite theorem?' or 'name a random theorem'. Of course it's quite subjective which theorems every mathematician "should" know, but perhaps we could try to give arguments?

For example, I can imagine a number theorist or logician never have to use or know Fubini's theorem. So why should he know it?

 

[micromass]

I feel like people are responding to either 'what is your favourite theorem?' or 'name a random theorem'. Of course it's quite subjective which theorems every mathematician "should" know, but perhaps we could try to give arguments?

For example, I can imagine a number theorist or logician never have to use or know Fubini's theorem. So why should he know it?

Well perhaps the question isn't well-frazed. The way I interpret the question is "What theorem do you want every math student to know". And certainly Fubini's theorem is something that every math student should have heard about. Maybe they will never use it later on, but I think they should still know it as a form of general culture.

Another theorem I would like to nominate is Taylor's theorem. It's importance is well-established. I have used it in analysis, probability theory, number theory,... Moreover, you can use the theorem to give approximations to a variety of functions. And a lot of useful inequalities are coming from the theorem. I don't think I could call anybody a mathematician if they have never heard of Taylor's theorem...

 

[discrete*]

I feel like people are responding to either 'what is your favourite theorem?' or 'name a random theorem'. Of course it's quite subjective which theorems every mathematician "should" know, but perhaps we could try to give arguments?

For example, I can imagine a number theorist or logician never have to use or know Fubini's theorem. So why should he know it?

Perhaps the best way to avoid the differences of opinions between different types of mathematicians is to state what theorems everyone should know based on their historical significance and general impact.

That's the basis of my argument for Godel's Incompleteness Theorems. They may not have particular utility for many (most) mathematicians, however the impact that Godel made was meteoric.

Of course, theorems that come into use a lot, like the aforementioned Taylor's Theorem have a particular utility that automatically earns them a spot on such a list.

 

[micromass]

Perhaps the best way to avoid the differences of opinions between different types of mathematicians is to state what theorems everyone should know based on their historical significance and general impact.

That's the basis of my argument for Godel's Incompleteness Theorems. They may not have particular utility for many (most) mathematicians, however the impact that Godel made was meteoric.

Of course, theorems that come into use a lot, like the aforementioned Taylor's Theorem have a particular utility that automatically earns them a spot on such a list.

Yes, I completely agree with Godel's Incompleteness Theorems in that respect. Another theorem that is important for the same reason is Cohen's result that the continuum hypothesis and the axiom of choice is independent of ZF. I have never used this result, but I think the significance of the theorem is huge!

 

[Landau]

That the axiom of choice is independent of ZF: agreed, significant! But the continuum hypothesis? Nice to know, but I don't see the significance... The difference is that Choice leads to many important and useful results (hahn-banach, existence of maximal ideals, etc.), while I don't know any 'use' of the continuum hypothesis.

 

[micromass]

Well, a theorem is important not only because it is useful. A theorem can also be important because it has been an open question for a long time, or because it has made an impact on the mathematical world.

I think the continuum hypothesis satisfies that. When studying countability and uncountability, students of mathematics naturally come up with the continuum hypothesis. In fact, every first-year student of mathematics is confronted with the continuum hypothesis in some way. And it's the first statement that is shown to be independent of ZFC, something which is often hard to grasp for students. I don't think there is any mathematician out there which has never heard of the continuum hypothesis, and regardless of it's usefulness, that implies that the theorem is important.

And then there's the fact that Cohen won the Fields medal for his work, which means that the question must have had some importance. The continuum hypothesis was also one of Hilbert's millenium problems, which further adds to it's significance.

There are another set of results that satisfy thesame criteria: the insolvability of the quintic, the parallel postulate, the transcendence of pi and e,...
While these may not seem to be important, their historic significance is overwhelming!

 

[discrete*]

Well, a theorem is important not only because it is useful. A theorem can also be important because it has been an open question for a long time, or because it has made an impact on the mathematical world.

I think the continuum hypothesis satisfies that. When studying countability and uncountability, students of mathematics naturally come up with the continuum hypothesis. In fact, every first-year student of mathematics is confronted with the continuum hypothesis in some way. And it's the first statement that is shown to be independent of ZFC, something which is often hard to grasp for students. I don't think there is any mathematician out there which has never heard of the continuum hypothesis, and regardless of it's usefulness, that implies that the theorem is important.

And then there's the fact that Cohen won the Fields medal for his work, which means that the question must have had some importance. The continuum hypothesis was also one of Hilbert's millenium problems, which further adds to it's significance.

There are another set of results that satisfy thesame criteria: the insolvability of the quintic, the parallel postulate, the transcendence of pi and e,...
While these may not seem to be important, their historic significance is overwhelming!

Well said. I agree with every bit.

 

[discrete*]

That the axiom of choice is independent of ZF: agreed, significant! But the continuum hypothesis? Nice to know, but I don't see the significance... The difference is that Choice leads to many important and useful results (hahn-banach, existence of maximal ideals, etc.), while I don't know any 'use' of the continuum hypothesis.

Why does the Continuum Hypothesis have to have a "use" for it to be of importance?

You say that the AoC is important -- which we can all agree upon -- because it leads to important results. How can you deny that of the Continuum Hypothesis? It may not lead to important results outside of the Foundations (which is probably arguable), but it has weight none-the-less.

 

[adoado]

The prime number theorem!

 

[adoado]

But the continuum hypothesis? Nice to know, but I don't see the significance...

Surely even the new insight itself brought by the theorem is enough? 'Usefulness' in terms of practicality and extension is great, but sometimes the theorem itself is just beautiful for it's intricacies and logical outcome. Maybe not every person will agree, but any theorem that has these qualities I personally feel I need to know..

 

[disregardthat]

My opinion is that a prerequisite for a theorem that "every mathematician should know" should be a broad scope of application throughout different fields in mathematics. The independence of the Continuum hypothesis from the axioms of set theory is hardly relevant to any field outside the study of formal set theory, and independence results came before that. If you want to go down that road, I believe that Gödels incompleteness theorems are way more important.

I can't think of a theorem more satisfactory to this criterion than Zorn's lemma. It is essential to great many vitally important theorems throughout mathematics, something which can be said for few other theorems that still are non-trivial.

And how are first-year students confronted with the continuum hypothesis?

 

[micromass]

And how are first-year students confronted with the continuum hypothesis?

When I was a first-years student, I was confronted with the continuum hypothesis while I was learning basic set theory. I was immediately intrigued by the theorem and shifted my entire world-view of mathematics. Before, I thought everything can be proven by math, but thanks to (CH) I realized that this is not true, and that it's the choice of axioms that matter.

It's not only the mathematics consequences that matter to me, it's also the philosphical consequences. And that's why I think (CH) is quite important. In fact, I don't think any mathematician has never heard of the continuum hypothesis. Not because it is important, but because it has a lot of consequences about how you think of math.

 

[fourier jr]

definitely the 1st isomorphism theorem. It almost always comes in handy in two fairly common situations
1. showing a subset is a normal subgroup (or ideal or submodule, etc) by showing that it's the kernel of a homomorphism
2. showing two things are isomorphic

 

[fourier jr]

the chinese remainder theorem is another good one imho

 

[Robert1986]

So, I think that the title of this thread is kind of silly. IMO, there are very few theorems that EVERY mathematician should know. However, I think that these are very important in combinatorics:

Dilworth's Theorem and its dual

In all likelihood, only a Combinatorialist would REALLY need to know this; nonetheless, it is vitally important.

 

[disregardthat]

So, I think that the title of this thread is kind of silly. IMO, there are very few theorems that EVERY mathematician should know.

There are many theorems every mathematician must know. Basic knowledge of analysis is for example always necessary.