What is the most beautiful definition you've encountered?

[quasar987]

I don't know if I find the definition of a topological space beautiful in itself, but the whole abstractization of the "theory of the neighborhood" (i.e. freeing it from epsilons!) is certainly a beautiful feat of the mind. Although yes, since the definition of a topological space captures the essence of the idea of a neighborhood and all its power in 3 simple yet mysterious axioms, then I can definitely see how it has some beauty in itself! I guess it's my favorite definitions too then.

Not a mathematical dfn but it appears in an old absolutely beautiful philosophy book attempting to explain the human mind by starting from a handful of definition and axioms about the most elementary but fundamental concepts about the universe and working upwards by means of the "Lemma, Theorem, Corolary" formula. This is one of the first definitions of the book I think:

"Per aeternitatem, intelligo ipsam existensiam" (By eternity, I mean existence itself)

:!)

 

[Psycopathak]

I don't know if this is ok, but It's just amazing

Euler's Identity:

e^i*pi = -1

It relates the exponential base which is found with calculus, the imaginary unit which literally has to be made up to solve functions where there are no real solutions, and pi, which links all of geometry together. And it all equals -1!

It relates algebra, geometry and calculus to the most basic number.

 

[mathwonk]

i agree that the free basis definition is typical of the most beautiful definitions, namely definitions by universal mapping properties.

 

[tgt]

i agree that the free basis definition is typical of the most beautiful definitions, namely definitions by universal mapping properties.

It took ages to understand them. Do you know who invented them?

 

[mathwonk]

i believe peter freyd, in his book on categories and functors, credits maclane with this type of definition in a paper on groups.

 

[olliemath]

Continuity in the way I first heard it:

A funtion $$f:A\subseteq\mathbb{R}\rightarrow\mathbb{R}$$ is continuous if, for each $$x\in A$$ and each $$\varepsilon>0$$, we can find some $$\delta>0$$ such that $$|f(x)-f(y)|<\epsilon$$ whenever $$|x-y|<\delta$$.

I know a lot of people tend not to like it when they first see it, but it was the first time I saw a mathematician take something so intuative then transform it into a solid mathematical form. That gave me a love of analysis/topology that I still hold. An alternative for me would possibly be the definition of the fundamental group.

 

[mathwonk]

what about the definition where f^-1(U) is open whenever U is open.

 

[olliemath]

I like the classical form (I think because I have the memory "wow, maths can be beautiful" associated to it, rather than its intrinsic genius) but I can see why you like the more topological version.

 

[mutton]

compactness: every open cover has a finite subcover

 

[EternalVortex]

I like the definition of NP the best. Easy to verify, hard to solve.

 

[CRGreathouse]

I like the definition of NP the best. Easy to verify, hard to solve.

That's a good one.

 

[tim_lou]

All those cohomology business. It's hard to believe how much antisymmetry (of simplexes, tensor products..etc) gives you... Stokes theorem, De Rham's theorem and what not...even though I haven't fully understood them yet.

 

[tim_lou]

The definition of Lebesgue integral as well. To understand how Riez representation theorem falls right out of it is amazing.