Mere Efficient or inefficient maths?

Main Question or Discussion Point

We hear phrases like beautiful or elegant maths/theorem or ugly maths although more the former since it's mostly the good that is talked about. But in most cases is it just efficient and inefficient maths? We label efficient maths as beautiful/elegant and the inefficient as ugly?

Efficient basically being maximal result and minimal work.

The same thing goes for physical theories. The beautiful theories tend to be the efficient ones with minimal neat equations (although may be very complex) but maximal information and generality. Dirac said "It must be beautiful" but does he really mean it must be efficient?

When I say efficient/inefficient I mean efficiency to the human mind and not to a computer or anything else. i.e. One may write a really efficient program for a computer to calculate aspects of nature or maths but the code will not look efficient to a human so is not beautiful.

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morphism
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It's just a matter of taste..............

Sure but I am trying to make it impersonal and rationalise our emotions if you like. I want to get down to the core of this just like what maths tries to do in the first place, to draw an analogy.

Gib Z
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Think about the words that describe it : Beautiful, elegant, ugly etc etc.

These are all subjective words, it depends on taste. But obviously mathematicians tastes will be similar in certain ways.

It is not always the most efficient form that makes it beautiful.

Eg Most mathematicans prefer $e^{i\pi} + 1=0$ to the obvious alternative.

Elegant/Beautiful may mean different things to a mathematician and a physicist.

I can't speak for the physicists, but i personally think elegance is not always about the final form but more about the method in proving it. An elegant proof has intuitive and amazing leaps of understanding, whilst an ugly proof has lots of calculations and conditions. Proof By exhaustion or similar methods are ugly. Proof's by contradiction can be ugly, but some are good. In my opinion the whole idea of induction was clever, but generally proofs which use it aren't to my liking, I find it too repetitive using the same method and pattern of proof over and over.

personally i like long proofs, especally those by contradiction, it may sound a little wierd but i dunno why i like these kind of proofs. I do agree with Gib Z that an elegant proof has intuitive and amazing leaps of understanding.

But there seems to be some mathematics that is unversally considered to be beautiful. What properties do they have?

If you include 0 to Euler's equation then there is actually more information and you could claim that this equation contains all the big features of maths.

JasonRox
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Gold Member
I can't speak for the physicists, but i personally think elegance is not always about the final form but more about the method in proving it. An elegant proof has intuitive and amazing leaps of understanding, whilst an ugly proof has lots of calculations and conditions. Proof By exhaustion or similar methods are ugly. Proof's by contradiction can be ugly, but some are good. In my opinion the whole idea of induction was clever, but generally proofs which use it aren't to my liking, I find it too repetitive using the same method and pattern of proof over and over.
How can induction be ugly when it's practically require in most cases that we use it.

Proof by contradiction isn't always the greatest, but a lot of times the problem posed leads us to solve it that way. The only way around it sometimes is to use the contra-positive.

Personally, I think beautiful mathematics is not about the proofs all the time. It's also about the results, and the work leading up to it and such. Like, Group Theory. The mere idea of that coming around is just jaw dropping. Or the idea of uncountably infinite. Sure the proof is nice, but that's not what makes my jaw drop. The idea itself does. Or the proof that every second countable normal space is metrizable is probably nice (I haven't read it yet), but the idea itself that you can construct the metric made my jaw drop. The mere idea that you can actually do that is just... WOW! I don't give a rats ass how you do it, whether using contradictions, conditions, induction and so on. The bottom line is... you freaking did IT! Or, the proof of Fermat's Last Theorem. Who cares what it looks like, you freaking did it! Like, WOW! Blows my mind. Sure, making a nice proof is fantastic, but that's not what really gets me going. Even the equation you pointed out, that's the idea of it is beautiful. The proof is just like a normal one. It's the IDEA that just like... dang who would of thought of that.

Nice proofs are nice, but that's not what really gets me going. What gets me going is that someone saw a connection and saw a finish line (a proof) and made it to that finish line.

Gib Z
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It was my Opinion, JasonRox. Don't chastise me for it. I said GENERALLY the proofs which use it aren't to my liking, because GENERALLY Induction is used as the easy way out when there are other ways. This is a case where the shortest way may not always be the most elegant. Induction Rarely shows an understanding in the concept, just shows that it works. Other methods of proof, which contain certain bouts of logic or intuition, shows that the concept is actually understood!

JasonRox
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It was my Opinion, JasonRox. Don't chastise me for it.
I barely criticized your opinion. I was writing my opinion and also declared that it was and wrote it explicitly. I don't see where you think I'm criticizing your opinion that badly.

Gib Z
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Then I apologize, please forgive me.