Which proof is better?

[madnak]

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I hardly see how proof 2 is 'more powerful'.

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Proof Two is more extensible and more intuitive, and therefore has a more powerful impact on the understanding. I read Proof One, I learn that 2 is irrational. I read Proof Two, I learn why 2 is irrational. I learn about other roots, I can even extend some of the understanding to exponential functions, to discrete math, to combinatorics... Proof Two is eye-opening.

[bunny]

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Proof Two is more extensible and more intuitive, and therefore has a more powerful impact on the understanding. I read Proof One, I learn that 2 is irrational. I read Proof Two, I learn why 2 is irrational. I learn about other roots, I can even extend some of the understanding to exponential functions, to discrete math, to combinatorics... Proof Two is eye-opening.

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This is why I prefer two as well - it leaves you with some insight into why the result holds, rather than with just a fact about a number. I do think it is harder to understand (although this is partly because it requires more complicated notation which doesnt translate to the internet so well - on paper the notation is not so clumsy imo).

My rather limited survey (around 50 mathematicians) shows a pretty clear (and predictable) distinction between pure mathematicians who prefer the second and applied mathematicians who prefer the first. There have been some exceptions, but not very many.

[CityFan]

Count me as another exception then. I like proof one. I can extend it to other roots fairly easily if I like, too.

I like generality, but if you want to be general then prove a general result.

[holmansf]

I am rather surprised that so many people favor proof 1. I think proof 2 is way better. I think folks are just scared of the "fundamental theorem of arithmetic," even though it is used implicitly in proof 1.

[bunny]

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I am rather surprised that so many people favor proof 1. I think proof 2 is way better. I think folks are just scared of the "fundamental theorem of arithmetic," even though it is used implicitly in proof 1.

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Well, of course it hinges on what people think makes a good proof which was intentionally left unspecified. I'm more surprised that there arent more "neither" answers, since I had kinda expected this forum to be full of "maths is just a language game - premises to conclusions according to set rules, it doesnt matter how you get there" type people.

[bunny]

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Count me as another exception then. I like proof one. I can extend it to other roots fairly easily if I like, too.

I like generality, but if you want to be general then prove a general result.

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You dont think proof two generalises more readily? Just replace "2" with "p for any prime p". I think this is its strength - generalising proof one is not so straightforward (although clearly possible since they ultimately depend on the same thing).

[bigpooch]

Technically, Proof One DOES NOT use the fundamental theorem
of arithmetic, but rather uses the result that the square of
an odd integer is odd (or equivalently, if a square of an
integer is even, then the integer is even).

[bigpooch]

Not just that, but from Proof Two, the following proposition
can be readily proven:

For any positive integer, its kth root (where k is an
integer >=2), on the positive real line, is either a
positive integer or irrational (there is no such result that
is rational but not an integer).

[bigpooch]

So, because I studied both pure and applied mathematics,
would you say that I was an applied mathematician or an
exception? [img]/images/graemlins/smile.gif[/img]

[FortunaMaximus]

lol, depends on the rule. [img]/images/graemlins/smirk.gif[/img]