[math-fun] Mathematical surprises
http://divisbyzero.com/2010/08/18/mathematical-surprises/ Im interested in compiling a list of mathematical surprises. The best possible example would be a mathematical discovery that no mathematician saw coming, but after it was discovered it changed mathematics in some fundamental wayCantors discovery of the nondenumerability of the continuum is such an example. But Ill settle for any surpriseAndrew Wiles surprised everyone with his proof of Fermats Last Theorem, the solution of the Monty Hall problem surprised many capable mathematicians, etc. Ive spent a couple days brainstorming and Ive come up with the following list. Some are better than others, and theyre listed in no particular order. Please add your surprises in the comments below! ... --- co-chair http://ocjug.org/
Virtually every major advance in mathematics "surprised" mathematicians, so it is difficult to find "unsurprising" advances. Here's my list: Symbolic calculus & eventually the Risch algorithm for closed form integration. Lie groups for solving ordinary differential equations. Topological "monsters" in general. They fell out of the "cracks" when making the foundations of math more precise. The incredible difficulty of the 3-body problem. In retrospect, the fact that 2 bodies produce a closed ellipse is the miracle, because virtually everything else about orbital mechanics is chaotic! The elegance of logarithms & slide rules. Complex numbers & algebraic closure. Singularities off the real axis can affect the region of real convergence -- you can no longer ignore complex numbers. The beauty of complex analysis. Fourier Transforms & especially FFT's. Matrices/determinants/noncommutative algebra. The simplex algorithm for "linear programming". The incredible simplicity of _universal_ computing machines. How could such simple machines be equivalent to complex computers? The incredible complexity of Conway's "Life". Bennett's proof that computation doesn't require energy (still being digested). The theory of "distributions". The fact that the non-rigorous elimination of so many infinities from modern physics could ever be repaired surprised a lot of mathematicians! (Have they all been repaired, even today?) The theory of real closed fields (e.g., classical algebra & geometry) is decidable, but provably extremely difficult in practise. The Turing Bombe/ULTRA decryption of ENIGMA? It surprised the heck out of the Nazis... RSA algorithm? Shor's quantum factorization algorithm? (still being digested) The surprising difficulty of P=NP? At 02:25 AM 9/17/2010, Ray Tayek wrote:
http://divisbyzero.com/2010/08/18/mathematical-surprises/
IÂm interested in compiling a list of Âmathematical surprises. The best possible example would be a mathematical discovery that no mathematician saw coming, but after it was discovered it changed mathematics in some fundamental wayÂCantorÂs discovery of the nondenumerability of the continuum is such an example. But IÂll settle for any surpriseÂAndrew Wiles surprised everyone with his proof of FermatÂs Last Theorem, the solution of the Monty Hall problem surprised many capable mathematicians, etc.
IÂve spent a couple days brainstorming and IÂve come up with the following list. Some are better than others, and theyÂre listed in no particular order. Please add your surprises in the comments below! ...
--- co-chair http://ocjug.org/
You should look at mathoverflow http://mathoverflow.net/questions/14574/your-favorite-surprising-connections.... I asked a similar question about 7 months ago, and got lots of responses. Victor On Fri, Sep 17, 2010 at 1:07 PM, Henry Baker <hbaker1@pipeline.com> wrote:
Virtually every major advance in mathematics "surprised" mathematicians, so it is difficult to find "unsurprising" advances. Here's my list:
Symbolic calculus & eventually the Risch algorithm for closed form integration. Lie groups for solving ordinary differential equations.
Topological "monsters" in general. They fell out of the "cracks" when making the foundations of math more precise.
The incredible difficulty of the 3-body problem. In retrospect, the fact that 2 bodies produce a closed ellipse is the miracle, because virtually everything else about orbital mechanics is chaotic!
The elegance of logarithms & slide rules.
Complex numbers & algebraic closure. Singularities off the real axis can affect the region of real convergence -- you can no longer ignore complex numbers. The beauty of complex analysis.
Fourier Transforms & especially FFT's.
Matrices/determinants/noncommutative algebra.
The simplex algorithm for "linear programming".
The incredible simplicity of _universal_ computing machines. How could such simple machines be equivalent to complex computers?
The incredible complexity of Conway's "Life".
Bennett's proof that computation doesn't require energy (still being digested).
The theory of "distributions".
The fact that the non-rigorous elimination of so many infinities from modern physics could ever be repaired surprised a lot of mathematicians! (Have they all been repaired, even today?)
The theory of real closed fields (e.g., classical algebra & geometry) is decidable, but provably extremely difficult in practise.
The Turing Bombe/ULTRA decryption of ENIGMA? It surprised the heck out of the Nazis...
RSA algorithm?
Shor's quantum factorization algorithm? (still being digested)
The surprising difficulty of P=NP?
At 02:25 AM 9/17/2010, Ray Tayek wrote:
http://divisbyzero.com/2010/08/18/mathematical-surprises/
I’m interested in compiling a list of “mathematical surprises.” The best possible example would be a mathematical discovery that no mathematician saw coming, but after it was discovered it changed mathematics in some fundamental wayCantor’s discovery of the nondenumerability of the continuum is such an example. But I’ll settle for any surpriseAndrew Wiles surprised everyone with his proof of Fermat’s Last Theorem, the solution of the Monty Hall problem surprised many capable mathematicians, etc.
I’ve spent a couple days brainstorming and I’ve come up with the following list. Some are better than others, and they’re listed in no particular order. Please add your surprises in the comments below! ...
--- co-chair http://ocjug.org/
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I found the mathoverflow list to be disappointing; the Baker & Tayek lists are more to my taste. Rich ------ Quoting Victor Miller <victorsmiller@gmail.com>:
You should look at mathoverflow http://mathoverflow.net/questions/14574/your-favorite-surprising-connections.... I asked a similar question about 7 months ago, and got lots of responses.
Victor
On Fri, Sep 17, 2010 at 1:07 PM, Henry Baker <hbaker1@pipeline.com> wrote:
Virtually every major advance in mathematics "surprised" mathematicians, so it is difficult to find "unsurprising" advances. Here's my list:
Symbolic calculus & eventually the Risch algorithm for closed form integration. Lie groups for solving ordinary differential equations.
Topological "monsters" in general. They fell out of the "cracks" when making the foundations of math more precise.
The incredible difficulty of the 3-body problem. In retrospect, the fact that 2 bodies produce a closed ellipse is the miracle, because virtually everything else about orbital mechanics is chaotic!
The elegance of logarithms & slide rules.
Complex numbers & algebraic closure. Singularities off the real axis can affect the region of real convergence -- you can no longer ignore complex numbers. The beauty of complex analysis.
Fourier Transforms & especially FFT's.
Matrices/determinants/noncommutative algebra.
The simplex algorithm for "linear programming".
The incredible simplicity of _universal_ computing machines. How could such simple machines be equivalent to complex computers?
The incredible complexity of Conway's "Life".
Bennett's proof that computation doesn't require energy (still being digested).
The theory of "distributions".
The fact that the non-rigorous elimination of so many infinities from modern physics could ever be repaired surprised a lot of mathematicians! (Have they all been repaired, even today?)
The theory of real closed fields (e.g., classical algebra & geometry) is decidable, but provably extremely difficult in practise.
The Turing Bombe/ULTRA decryption of ENIGMA? It surprised the heck out of the Nazis...
RSA algorithm?
Shor's quantum factorization algorithm? (still being digested)
The surprising difficulty of P=NP?
At 02:25 AM 9/17/2010, Ray Tayek wrote:
http://divisbyzero.com/2010/08/18/mathematical-surprises/
I’m interested in compiling a list of “mathematical surprises.” The best possible example would be a mathematical discovery that no mathematician saw coming, but after it was discovered it changed mathematics in some fundamental wayCantor’s discovery of the nondenumerability of the continuum is such an example. But I’ll settle for any surpriseAndrew Wiles surprised everyone with his proof of Fermat’s Last Theorem, the solution of the Monty Hall problem surprised many capable mathematicians, etc.
I’ve spent a couple days brainstorming and I’ve come up with the following list. Some are better than others, and they’re listed in no particular order. Please add your surprises in the comments below! ...
--- co-chair http://ocjug.org/
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Could you provide a link for the "Baker & Tayek lists" my attempts to find it (them?) with google failed. On Tue, Sep 21, 2010 at 1:03 PM, <rcs@xmission.com> wrote:
I found the mathoverflow list to be disappointing; the Baker & Tayek lists are more to my taste.
Rich
------ Quoting Victor Miller <victorsmiller@gmail.com>:
You should look at mathoverflow
http://mathoverflow.net/questions/14574/your-favorite-surprising-connections.... I asked a similar question about 7 months ago, and got lots of responses.
Victor
Never mind, I now see you were just referring to the items they mentioned in this mailing list. On Tue, Sep 21, 2010 at 3:25 PM, James Buddenhagen <jbuddenh@gmail.com> wrote:
Could you provide a link for the "Baker & Tayek lists" my attempts to find it (them?) with google failed.
On Tue, Sep 21, 2010 at 1:03 PM, <rcs@xmission.com> wrote:
I found the mathoverflow list to be disappointing; the Baker & Tayek lists are more to my taste.
Rich
------ Quoting Victor Miller <victorsmiller@gmail.com>:
You should look at mathoverflow
http://mathoverflow.net/questions/14574/your-favorite-surprising-connections.... I asked a similar question about 7 months ago, and got lots of responses.
Victor
I learned about and started participating in the mathoverflow list 16 days ago (it tells me), and I have found it very stimulating and interesting. There's of course many things I'm uninteresting to me, but there is a high density of things I find interesting and engaging. I've learned a number of things that are not just fun, but helpful for mathematical topics I was pursuing. They have a system for ratings and reputation and moderation (closing or deleting questions on occasion) that may seem cheesy, but it does help let interesting material bubble to the top, and it keeps the forum from being overrun by fluff, student-level questions and poorly thought out questions. Bill On Sep 21, 2010, at 2:03 PM, rcs@xmission.com wrote:
I found the mathoverflow list to be disappointing; the Baker & Tayek lists are more to my taste.
Rich
------ Quoting Victor Miller <victorsmiller@gmail.com>:
You should look at mathoverflow http://mathoverflow.net/questions/14574/your-favorite-surprising-connections.... I asked a similar question about 7 months ago, and got lots of responses.
Victor
On Fri, Sep 17, 2010 at 1:07 PM, Henry Baker <hbaker1@pipeline.com> wrote:
Virtually every major advance in mathematics "surprised" mathematicians, so it is difficult to find "unsurprising" advances. Here's my list:
Symbolic calculus & eventually the Risch algorithm for closed form integration. Lie groups for solving ordinary differential equations.
Topological "monsters" in general. They fell out of the "cracks" when making the foundations of math more precise.
The incredible difficulty of the 3-body problem. In retrospect, the fact that 2 bodies produce a closed ellipse is the miracle, because virtually everything else about orbital mechanics is chaotic!
The elegance of logarithms & slide rules.
Complex numbers & algebraic closure. Singularities off the real axis can affect the region of real convergence -- you can no longer ignore complex numbers. The beauty of complex analysis.
Fourier Transforms & especially FFT's.
Matrices/determinants/noncommutative algebra.
The simplex algorithm for "linear programming".
The incredible simplicity of _universal_ computing machines. How could such simple machines be equivalent to complex computers?
The incredible complexity of Conway's "Life".
Bennett's proof that computation doesn't require energy (still being digested).
The theory of "distributions".
The fact that the non-rigorous elimination of so many infinities from modern physics could ever be repaired surprised a lot of mathematicians! (Have they all been repaired, even today?)
The theory of real closed fields (e.g., classical algebra & geometry) is decidable, but provably extremely difficult in practise.
The Turing Bombe/ULTRA decryption of ENIGMA? It surprised the heck out of the Nazis...
RSA algorithm?
Shor's quantum factorization algorithm? (still being digested)
The surprising difficulty of P=NP?
At 02:25 AM 9/17/2010, Ray Tayek wrote:
http://divisbyzero.com/2010/08/18/mathematical-surprises/
I’m interested in compiling a list of “mathematical surprises.” The best possible example would be a mathematical discovery that no mathematician saw coming, but after it was discovered it changed mathematics in some fundamental way Cantor’s discovery of the nondenumerability of the continuum is such an example. But I’ll settle for any surprise Andrew Wiles surprised everyone with his proof of Fermat’s Last Theorem, the solution of the Monty Hall problem surprised many capable mathematicians, etc.
I’ve spent a couple days brainstorming and I’ve come up with the following list. Some are better than others, and they’re listed in no particular order. Please add your surprises in the comments below! ...
--- co-chair http://ocjug.org/
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (6)
-
Bill Thurston -
Henry Baker -
James Buddenhagen -
Ray Tayek -
rcs＠xmission.com -
Victor Miller