Pure Mathematics
Pure MathematicsMathematics for the sake of mathematics, as opposed to mathematics for the sake of applications.
It is occasionally remarked upon that even though the problems are posed, worked on and even sometimes solved purely because the mathematicians find them interesting, still it happens that results from obscure areas turn out to be useful. Lead times on the order of 50 to 100 years have been suggested, although that seems to be dropping.
Examples welcome
Number theory and factorization spring immediately to mind. Originally regarded as the purest of the pure, these two (related) topics now forming the basis of most modern cryptography.
Another example is non-Euclidean geometry. Type theory (defunct as math) might be another example. Set theory? Mathematical logic?
And nowadays, one must consider the case of the theory that used to be superstrings. It's a case where the traditional relation between mathematics and physics has been reversed. Physicists have created new math in order to keep up with the physics. Lead time is negative and guesses range from -20 to -50 years. Why so large? The mathematics is not particularly profound.
One theory is that a mathematician's concept of "interesting" is necessarily shaped by their own existence in "the RealWorld" and so even though there is no attempt to be "relevant", nevertheless anything that diverges so far as to be completely pointless is dismissed as "uninteresting."
My opinion on this subject has long been that physics is the mathematics of our universe while mathematics is the physics of all possible universes. This opinion was only reinforced by studying mathematical logic and Willard Quine's formal definition of 'existence'.
The only quantitative difference between 'physical reality' and a mathematical theory is that the theory called 'physical reality' is the one that includes all of our perceptions. The qualitative differences between 'physical reality' and other mathematical theories are explained by 1) the former being intractable and 2) our being creatures embedded in an intrusive sensorium. That is, we can't embrace the whole of reality in one go and even if we could, mere ideas are less compelling to us than raw percepts. (This second point holds even for the rare humans who, unlike 90+% of the population, do not value experiences over thoughts.) To an ArtificialIntelligence that can solve M-theory equations in the blink of an eye (our mathematics isn't advanced enough to express those equations) and can shut off its senses in favour of its imaginings, the whole of physical reality is just so much math.
However, I wasn't thinking of this at the time I wrote the above. Instead, I had some vague thoughts that the newly found LimitsOfMathematics have downgraded the special status of mathematics. We now know that nearly all mathematical facts have no relation to any other mathematical facts. However, we perceive overwhelming order in mathematics despite the fact that the subject has statistically infinitesimal order. This must be because mathematicians are picking and choosing "interesting" mathematical facts on the basis of subconscious impressions from our physical reality. That is, all theorems and all facts ever discovered by mathematicians are either taken verbatim from our reality or are direct extensions thereof. All known mathematics is just rehashing of physical reality. -- RichardKulisz
Can one have 'fact A is unrelated to fact B' when that very statement relates them, albeit somewhat trivially? Also, 'direct extensions thereof' sounds good, but I doubt if it could be given a precise definition. How does 'rehashing' differ from 'extending'?
Something doesn't sound quite right in the above, RK. Sure, most mathematical facts have no relation to other mathematical facts, which I'm taking to mean a random set of axioms has probability one of being consistent for nice weightings. So you're saying that mathematicians are picking and choosing those few things that happen to show relationships to "physical axioms", and so often with each other, or something else?
Yup, that's it. To be more precise, I should actually read The Limits Of Mathematics sometime.
That's fine, but it's to be expected that axioms are not usually, if at all, selected at random - I suggest that there is a sensible, conscious choice (sometimes related to physical reality), and see no evidence of 'subconscious impressions', especially ones based on our physical reality, having anything to do with it.
Then what do you call the basis upon which mathematicians decide that something is "interesting"? The decision that some math facts are interesting is intuitive and not something that is arrived at by formal reasoning based on meta-mathematical axioms.
But many mathematical facts do have relation to other mathematical facts - Elliptic Curves in the solution to FermatsLastTheorem, CalabiYau? manifolds in StringTheory, knots in LoopQuantumGravity?. One can imagine a graph with 3 nodes - the RealWorld, PureMathematics and AppliedMathematics?. There is a constant flow between the 3, also loopback arcs from each node to itself. Color each node separately, e.g., red, blue, yellow; perhaps degree of intensity could represent discrete branches in each node. The RealWorld node then represents the set of all non-mathematical concepts and would have more of a gradient but there is a constant flow between the 3 and within themselves, albeit at different rates. In the RealWorld, people use metaphor and AnalogyFest all the time but within AppliedMathematics? analogies are exact - Electrical, Mechanical, Thermal even Sociological systems having the same differential equations. One could even animate it; perhaps there is an underlying theory that would describe the flow, a velocity of concepts. Some might say CategoryTheory. The rates of flow and sizes of arcs different for each node (constrained by the number of KnowledgeWorkers? in each field - i.e. it seems there are 1000 string theorists compared to 100 LQG practitioners, so the flow of ideas within that "channel" would be faster. Not passing judgment on which is right or wrong, just trying to envisage the overall dynamics at a high level.
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