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7th September 2016, 02:35 PM  #1  
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On number theory and socalled "useless" knowledge
Accordingly, here I would like to discuss the value of the branch of mathematics called number theory, which concerns the properties of and relations between integers, and in particular the final achievement of a fully general proof of Fermat's last theorem, the deceptively simple claim that there are no three positive integers that satisfy the equation a^{n} + b^{n} = c^{n} for any integer value of n above two. A good deal of what we now call number theory goes back to antiquity; the proof that the square root of two is irrational is something a lot of students still go through today and a likely apocryphal legend has it that a member of the Pythagorean sect was drowned out of hatred by other members for having achieving this. It really began to come into its own in the 19th century though, and started to be recognized as a distinct branch of mathematics. In fact, the prolific 19th century mathematician Carl Friedrich Gauss once said: "Mathematics is the queen of the sciences and number theory is the queen of mathematics." However, conspicuously absent during all of this time was much in the way of any applications of number theory. The number theorist Leonard Dickson, whose time came after that of Gauss, but died only a decade after the advent of the digital computer, said: "Thank God that number theory is unsullied by any application." But due to the increasing entrenchment of the digital computer in life in industrial countries, applications of number theory have blown up rapidly in more recent decades for all sorts of calculations. The use of number theory is especially marked in cryptography, and without modern cryptographic methods there wouldn't be many things considered to be of practical value, among them, the entirety of ecommerce. In this way, even accepting the very, very dubious claim that knowledge is of no value for its own sake, all these centuries of work seemed to have paid off after all. One might still consider all this effort a waste and contend that individuals with more "practical" orientations would have come up with these things anyway. There is a very serious issue with that contention though: it's ahistorical. If there is any example of an elite gogetter with both their feet firmly on the ground, focused only on knowledge of "real" value, who ever came up with such advances as the past centuries have seen in number theory, I'd definitely like to see it. I certainly can't think of any myself, and you would think that people with such a superior, focused worldview would be better thinkers overall. Now I'd like to turn my attention to Fermat's last theorem. This was stated in a margin of a copy of an ancient mathematical text by the mathematician Pierre de Fermat. He claimed to have a "marvelous proof" of the same that wouldn't fit in the margin (and is widely considered to have been mistaken), but never wrote it down. Fermat's last theorem was finally proven in a truly general form by Andrew Wiles in 1994, after over 350 years of effort by many others as well. This event is considered a great milestone in mathematical history, but I am not aware of any practical applications of Fermat's last theorem as such, except to other claims in pure mathematics. I am under the impression that, along the way, there were developments in what are called elliptic curves that have applications to cryptography presently, that were vital for the proof. But that doesn't mean that Fermat's last theorem per se has any practical applications. What is the value of such a thing to someone with a "pragmatic" outlook, in scare quotes? Was this 350+ years full of wasted time? If it were some sort of multigenerational business venture to yield practical, profitable outcomes, I can tell you everyone involved would have taken a bath on it, thus far at least. Maybe Fermat's last theorem will have practical applications one day, maybe not. The question of course is whether the value of the theorem is solely contingent on whether these applications ever arise. To draw things to a close, I want to say that even if such things as Fermat's last theorem are "useless" per se, which I of course do not believe, it seems as though it is often the case that the only way to find the practical applications you might want is by not looking for them at all and that people who fancy themselves "pragmatic" when they deride "useless" knowledge are actually just a different word: myopic. 

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Die Mauern wir erklettern, Die Türme wir zerschmettern Und in die Stadt hinein. Wer uns den Lauf will hemmen, Sich uns entgegenstemmen Der soll des Teufels sein. 

7th September 2016, 05:05 PM  #2 
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Geeking out a bit, but basically when I hear stuff like what I'm picking apart in the OP, I can't help but think of SMAC faction leader CEO Nwabudike Morgan and how much he, on balance, irritates me in that game (although this is not an entirely great analogy because some of the things he says are very insightful and what theprestige says, to me anyhow, is really pretty much crapola), more specifically this:
"You ivory tower intellectuals must not lose touch with the world of industrial growth and hard currency. It is all very well and good to pursue these highminded scientific theories, but research grants are expensive and you must justify your existence by providing not only knowledge, but concrete and profitable applications as well." I have a rather different gameplay style and different choice of factions and of course this tends to result in Morgan Industries becoming a subordinate or nonexistent faction. 
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Die Mauern wir erklettern, Die Türme wir zerschmettern Und in die Stadt hinein. Wer uns den Lauf will hemmen, Sich uns entgegenstemmen Der soll des Teufels sein. 

8th September 2016, 05:34 AM  #3 
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8th September 2016, 05:37 AM  #4 
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8th September 2016, 08:57 AM  #5 
Illuminator
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Also, for square root of 1, I think you meant to say, "imaginary".

8th September 2016, 09:55 AM  #6 
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Knowing that we do not know, it does not necessarily follow that we can not know. 

8th September 2016, 04:52 PM  #7 
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8th September 2016, 04:56 PM  #8 
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8th September 2016, 05:37 PM  #9 
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8th September 2016, 06:01 PM  #10 
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BRAINZZZZZZZZ 

8th September 2016, 07:23 PM  #11 
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What is the usefulness of Shakespeare? Or Picasso? Some aspects of mathematics can be treasured for their beauty.

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Beth "You are not the stuff of which you are made." Richard Dawkins, July 2005, 10:45 http://www.ted.com/talks/richard_daw..._universe.html 

8th September 2016, 07:31 PM  #12 
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_{It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong.  Richard P. Feynman} ξ 

8th September 2016, 08:26 PM  #13 
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As Jack by the Hedge said, a number is irrational if it cannot be expressed as the ratio of two integers.
The number 0.333... is rational, since it is equal to 1/3. The square root of two cannot be expressed as a rational number. In fact, any number whose digital expansion is infinite and does not "stabilize" in a sequence of repeating digits is irrational. For a quick proof that sqrt(2) is irrational, let us take as given the following facts. (1) Any ratio a/b of integers is equal to a ratio c/d where c and d have no common factors. (2) If a^2 is even, then so is a. Now, suppose that sqrt(2) is rational, i.e., that there are integers a and b such that 2 = (a/b)^2 Without loss of generality, using (1), assume that a and b have no common factors. It follows that 2 b^2 = a^2 and hence that a^2 is even. Thus, a is even, i.e., a = 2*c for some c. Hence 2 b^2 = (2*c)^2 and hence b^2 = 2 c^2. Therefore, b^2 is even and thus b is even. Here we have a contradiction, since a and b had no common factors and yet both are even (i.e., 2 is a common factor). Thus our assumption that we could write sqrt(2) as a ratio of two integers is false. It is a brilliant little proof, perhaps my favorite. 
9th September 2016, 12:08 AM  #14 
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I don't see how an article of clothing can be indecent. A person, yes.  Robert A. Heinlein If Christ died for our sins, dare we make his martyrdom meaningless by not committing them?  Jules Feiffer If you are going through hell, keep going  Winston Churchill 

9th September 2016, 12:56 AM  #15 
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I don't trust atoms. They make up everything. 

11th September 2016, 09:23 PM  #17 
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Die Mauern wir erklettern, Die Türme wir zerschmettern Und in die Stadt hinein. Wer uns den Lauf will hemmen, Sich uns entgegenstemmen Der soll des Teufels sein. 

11th September 2016, 10:23 PM  #18 
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I also want to point out here that lazily making accusations of some fallacy or another but never detailing how the offending reasoning is an example of such fallacy is the hallmark of a pseudointellectual.

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Die Mauern wir erklettern, Die Türme wir zerschmettern Und in die Stadt hinein. Wer uns den Lauf will hemmen, Sich uns entgegenstemmen Der soll des Teufels sein. 

11th September 2016, 11:05 PM  #19 
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It's worth pointing out that He Who Shall Not Be Named (but still is in one of my posts in this thread) is generally eager to try to point out the perceived absurdity and stupidity of everything I say, but not here.
One wonders why this might be the case. 
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Die Mauern wir erklettern, Die Türme wir zerschmettern Und in die Stadt hinein. Wer uns den Lauf will hemmen, Sich uns entgegenstemmen Der soll des Teufels sein. 

11th September 2016, 11:12 PM  #20 
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Die Mauern wir erklettern, Die Türme wir zerschmettern Und in die Stadt hinein. Wer uns den Lauf will hemmen, Sich uns entgegenstemmen Der soll des Teufels sein. 

12th September 2016, 12:27 AM  #21 
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Die Mauern wir erklettern, Die Türme wir zerschmettern Und in die Stadt hinein. Wer uns den Lauf will hemmen, Sich uns entgegenstemmen Der soll des Teufels sein. 

21st September 2016, 02:07 AM  #22 
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Die Mauern wir erklettern, Die Türme wir zerschmettern Und in die Stadt hinein. Wer uns den Lauf will hemmen, Sich uns entgegenstemmen Der soll des Teufels sein. 

26th September 2016, 02:47 PM  #23 
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Wow, that exchange really rankled you, didn't it?
Anyway, the crux of my objection is that you seemed (still seem) to say that knowledge has inherent value, and I disagree with this position. Knowledge has value only in the value derived from its profitable application, or from the personal pleasure one gets in acquiring it. Rummikub has no intrinsic value, and no practical application, but it's still worth playing with my family and friends. So I say that I play Rummikub because of the pleasure it gives me, in the experience of having played. I don't say I play Rummikub because of the intrinsic value of the game itself. If you're saying that you learn things for the pleasure it gives you, in the experience of learning them and knowing them, you'll get nothing from support from me. If you're saying that knowledge has intrinsic value, and should be learned for this reason alone, then we still disagree. "I study number theory because it's fun; it also happens to be potentially fruitful in practical applications" is a perfectly cromulent attitude, as far as I'm concerned. 
27th September 2016, 12:51 AM  #24 
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I suppose one way to assess whether or not knowledge has an inherent value all of its own is to see whether or not it applies to all knowledge. We could, for example, have a thread on this forum in which each poster posts once a day, describing in great detail (no fewer than 1,000 words per item) what's on the top shelf of their fridge. It could even be in the forum rules that everybody must make this post first thing and include a picture as proof otherwise they are not allowed to make any other posts, and failure to comply with this rule is an instant permaban.
Assuming there wasn't a mass exodus with people forming an offshoot sister site without such a rule, this would quickly become a large repository of knowledge. But not a repository that has any scientific value  it's a selfselecting group, it's selfreporting so people can easily lie, and even if those things weren't true the data as a whole would have little even sociological value as it is, at best, incomplete (both in terms of what's in the fridge as a whole, and in terms of demographical information) and isn't attached to any particular hypothesis. It also seems that it would have limited value as a source of interest  some people may be vaguely interested at first, but I think the novelty would wear off quite quickly, and I think that the number of people who would maintain an interest would be very low. So does this knowledge have inherent value, simply because it's knowledge? If not, how do we determine what knowledge has inherent value, and which doesn't? 
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I don't trust atoms. They make up everything. 

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