First, I take the definition of "trivial" to mean that "a proof immediately springs to mind". For those with a decent topology background, you are correct. For this thread, however, I think you need to tone the condescension down a little.
Second, R, C, and quaternions are the only examples of *associative* finite dimensional division algebras over the reals, but if you drop the associativity condition you need to include the Octonions, also known as the Cayley algebra.
This (the whole set of opinions) is a wonderful collection of ideas from a mathematician who has been arguing the (convincing) idea that in the 21st century real mathematics will be an experimental science, dominated by the study of computer algebra and experimental combinatorics. Zeilberger's computer, named Shalosh B. Eckhad, is actually cited as an author on some of Zeilberger's more computationally intensive papers -- it has been speculated that he has done this because he believes that when computers are our masters, Shalosh's descendants will have mercy on him. I became enamored with Zeilberger's humor when I first read "opinion 1 -- topology, the slum of combinatorics". His April Fool's Day opinions, of which there are several, by his own admission contain some of his best half-formed ideas, including:
1. Resolve the P/NP problem by showing any proof of P != NP would be NP-hard. 2. Storm the gates of MI6 to find Turing's counterexample to the Riemann hypothesis, classified for fifty years by the British government.
To try and stay on topic -- I think that trying to decide the peak age for mathematical creativity and the threshold for the decline of mathematical power is characteristic of the fratboy nature of the community. The concept of generalization as a means to solve a problem (ie, i cant prove A, but maybe i can lift A to some equivalent B that is easy, and includes A as a special case) has snowballed out of control. With few exceptions, contemporary research is so far from the source of mathematical inspiration (ie, physics and computer science) that all direction is lost, and all that matters is that a researcher say something about what someone else has said, not about the core (ie, hardest) problems in the field. So, you have inbreeding, cliques, "hot" fields, "mainstream" mathematics, and concepts like mathematical "talent" which are really quite ridiculous and counterproductive to the science. The young can think quickly, but the old can think deeply -- both are needed to tackle the really important problems, and to push knowledge further. Collaboration, not competition.
Anyway, my original intention is to plug Zeilberger and to let all know that there is a community of mathematicians who are about solving problems, not building castles of abstraction for the sake of giving each other awards. The eminent W.T. Gowers has described it best, so point your browser to:
http://www.dpmms.cam.ac.uk/~wtg10/papers.html
and look for "The Two Cultures of Mathematics". This paper set off a firestorm in the community, but for me it was reassuring and motivating to realize that there are still mathematicians who care about the field, and not about how smart they are or how smart they appear, much less if they are too "washed up" to make an impression on the apprentices.
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First, I take the definition of "trivial" to mean that "a proof immediately springs to mind". For those with a decent topology background, you are correct. For this thread, however, I think you need to tone the condescension down a little. Second, R, C, and quaternions are the only examples of *associative* finite dimensional division algebras over the reals, but if you drop the associativity condition you need to include the Octonions, also known as the Cayley algebra.
according to the good rabbi Dr. Doron Zeilberger:
. ht ml.
http://www.math.rutgers.edu/~zeilberg/Opinion46
This (the whole set of opinions) is a wonderful collection of ideas from a mathematician who has been arguing the (convincing) idea that in the 21st century real mathematics will be an experimental science, dominated by the study of computer algebra and experimental combinatorics. Zeilberger's computer, named Shalosh B. Eckhad, is actually cited as an author on some of Zeilberger's more computationally intensive papers -- it has been speculated that he has done this because he believes that when computers are our masters, Shalosh's descendants will have mercy on him. I became enamored with Zeilberger's humor when I first read "opinion 1 -- topology, the slum of combinatorics". His April Fool's Day opinions, of which there are several, by his own admission contain some of his best half-formed ideas, including:
1. Resolve the P/NP problem by showing any proof of P != NP would be NP-hard.
2. Storm the gates of MI6 to find Turing's counterexample to the Riemann hypothesis, classified for fifty years by the British government.
To try and stay on topic -- I think that trying to decide the peak age for mathematical creativity and the threshold for the decline of mathematical power is characteristic of the fratboy nature of the community. The concept of generalization as a means to solve a problem (ie, i cant prove A, but maybe i can lift A to some equivalent B that is easy, and includes A as a special case) has snowballed out of control. With few exceptions, contemporary research is so far from the source of mathematical inspiration (ie, physics and computer science) that all direction is lost, and all that matters is that a researcher say something about what someone else has said, not about the core (ie, hardest) problems in the field. So, you have inbreeding, cliques, "hot" fields, "mainstream" mathematics, and concepts like mathematical "talent" which are really quite ridiculous and counterproductive to the science. The young can think quickly, but the old can think deeply -- both are needed to tackle the really important problems, and to push knowledge further. Collaboration, not competition.
Anyway, my original intention is to plug Zeilberger and to let all know that there is a community of mathematicians who are about solving problems, not building castles of abstraction for the sake of giving each other awards. The eminent W.T. Gowers has described it best, so point your browser to:
http://www.dpmms.cam.ac.uk/~wtg10/papers.html
and look for "The Two Cultures of Mathematics". This paper set off a firestorm in the community, but for me it was reassuring and motivating to realize that there are still mathematicians who care about the field, and not about how smart they are or how smart they appear, much less if they are too "washed up" to make an impression on the apprentices.