I recently read about an interesting proof (I think it was in Contact by Carl Sagan, a much better
book than the film). It posits that all numbers are interesting through a proof by contradiction. It
goes as follows:
1) Given, there are a set of uninteresting numbers.
2) Within the range of these numbers, there exists one such number which is lower than all the others.
3) This number is the lowest of all uninteresting numbers, therefore it is interesting.
What can mathematicians say about this sort of mathematical thinking? It seems very different
to me than the basic maths I've studied.
For reference: http://en.wikipedia.org/wiki/Interesting_number_paradox
The problem is that you end up with a large set of numbers who are interesting only because they are the smallest interesting number, and if several numbers share this property, all of them cease to have this property.
And the circular reasoning begins like a bad time paradox..
I like the dynamic aspect vs static expression explanation for why this paradox doesn't really work.
Also curious what you think of "the smallest positive integer not defineable in under eleven words"...
Originally posted by geepamoogle I like the dynamic aspect vs static expression explanation for why this paradox doesn't really work.
As I said (or tried), my math background is pretty weak. I went through multivariable calculus
in my last college math course twelve years ago and, with a little effort, can do most basic
calculus problems. This more abstract math is novel to me.
I don't know what dynamic/static aspect explanations are. Do you care to elaborate?
Originally posted by geepamoogle Dynamic refers to those things that can change with time.
What interests you now may not interest you in the future so "interesting" is a dynamic term in a sense.
Static is something which remains constant. Some of the wording is static in nature, even though the property is dynamic, or so I understand it.
Maybe it's just cause it's late at nig ...[text shortened]... tic nature of the presentation of the problem (wherein lies the humor of the paradox).[/i]
I think it's referring to the dynamic solution in that, you must change parameters in order to classify a number as interesting, yet, the problem is static where the need is to find the proof that ALL numbers are interesting simultaneously.
the problem lies in the fact that interesting is not defined. in the set "all not interesting numbers" the least "not interesting number" [b]cannot[\b] be interesting by definition of the set regardless of how you define interesting. e.g.: you cannot say 7 is the most interesting number in the set of irrational numbers because 7 is not in the set.
In order to show that a set has no elements you have to define the properies of the set; then show that there are no numbers that have those properties. the argument has not done the first step.
Originally posted by deriver69 I still think the second lowest non interesting number is not interesting. I would think the lowest one is interesting though by being the lowest one.
But now by your reasoning given that the least interesting number became interesting the second lowest one is in fact the least interesting number. So now it too becomes interesting. So after exausting this reasoning you have 2 numbers. Well the least interesting of them is interesting. So now we have only one non interesting number and it seems that this reasoning can't be applied anymore. But just think about the fact of having only one non-interesting number. That makes it a pretty interesting number doesn't it?