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As a PhD student in mathematics, I have to say that history is valuable in mathematics because it explains why people are interested in certain problems. When

As a PhD student in mathematics, I have to say that history is valuable in mathematics because it explains why people are interested in certain problems. When I was a master level student, I always got annoyed by advanced courses in algebra (things like homological algebra) where it was often unclear why one should be interested in the problems stated there. Prompting the profs to give some short historical overview can be very enlightening. I do believe you can leave the vast majority of the history out. Advances in notation etc. were made for a reason. But a little bit of history can be quite important for context.

derleth 14y ago
> I have to say that history is valuable in mathematics because it explains why people are interested in certain problems. Motivation is essential, but there are motivations other than the ones that historically lead to the creation of the field. In number theory, one of my favorite examples, the motivations that lead people to be interested in it now come from modern cryptography, which flatly didn't exist when the field was founded but provides more interesting and relevant examples than the obsessions of century-old mathematicians.
btilly 14y ago
History may be valuable in mathematics, but it is demonstrably skipped. Few know much about it. When I was in grad school in math, I made the interesting discovery that if p and q are polynomials over a commutative ring, any polynomial that is symmetric in the roots of p and q is actually a polynomial in the original ring in the coefficients of p and q. (The construction works whether or not the ring can be embedded in a field where said roots actually exist.) Using this observation it is trivial, for instance, to write down in fully factored form a polynomial that has sqrt(2) + cube_root(3) as a root. This construction was news to various mathematicians that I talked to, including a combinatorics prof who studied symmetric polynomials and a number theorist who worked on stuff related to the algebraic integers. Then finally I talked to a very old mathematician with an interest in history. He told me that I had rediscovered an old way to do things. At his encouragement I went to the library, and picked up an algebra book from the 1800s. My construction was taught, and there was a whole chapter full of problems where students were expected to use it to come up with polynomials with specific roots. Furthermore as I looked into it both of the professors that I mentioned before worked in areas whose history dated back to the observation that I mentioned. It was used in the original proof that the algebraic integers form a ring, and that construction was the original reason that people were interested in symmetric polynomials. For another demonstration of how little of their own history mathematicians know, ask anyone why the notation for the second derivative is d^2y/dx^2. Then ask them where the f' notation comes from. Then ask them what Cauchy was trying to do that lead to Cauchy sequences. Most will draw a blank on all three. Don't read on until you're satisfied that you don't know the answers. In the original infinitesmal notation, d was an operator. It could be defined by d(y) = y(x + dx) - y(x). And you'd calculate a slope as dy/dx (drop any infinitesmal bits). Well when you work out d(dy/dx)/dx it turns out that you get d(d(y))/(dx * dx) which is more compactly written d^2y/dx^2. The f' notation was introduced by Lagrange in an attempt to get rid of infinitesmals by defining differentiation as a formal algebraic operation on polynomials and power series. This fell apart when Fourier demonstrated that apparently well-behaved power series could be used to construct pathological things like step functions. Cauchy came up with Cauchy sequences while attempting to define infinitesmals rigorously. His approach fell apart on the seemingly trivial example of how you rigorously prove the chain rule when the derivative of the inner thing is 0. (He was trying to avoid 0/0, but in that special case you get 0/0 all over the place.)

derleth 14y ago

> I have to say that history is valuable in mathematics because it explains why people are interested in certain problems. Motivation is essential, but there are motivations other than the ones that historically lead to the creation of the field. In number theory, one of my favorite examples, the motivations that lead people to be interested in it now come from modern cryptography, which flatly didn't exist when the field was founded but provides more interesting and relevant examples than the obsessions of century-old mathematicians.

btilly 14y ago

History may be valuable in mathematics, but it is demonstrably skipped. Few know much about it. When I was in grad school in math, I made the interesting discovery that if p and q are polynomials over a commutative ring, any polynomial that is symmetric in the roots of p and q is actually a polynomial in the original ring in the coefficients of p and q. (The construction works whether or not the ring can be embedded in a field where said roots actually exist.) Using this observation it is trivial, for instance, to write down in fully factored form a polynomial that has sqrt(2) + cube_root(3) as a root. This construction was news to various mathematicians that I talked to, including a combinatorics prof who studied symmetric polynomials and a number theorist who worked on stuff related to the algebraic integers. Then finally I talked to a very old mathematician with an interest in history. He told me that I had rediscovered an old way to do things. At his encouragement I went to the library, and picked up an algebra book from the 1800s. My construction was taught, and there was a whole chapter full of problems where students were expected to use it to come up with polynomials with specific roots. Furthermore as I looked into it both of the professors that I mentioned before worked in areas whose history dated back to the observation that I mentioned. It was used in the original proof that the algebraic integers form a ring, and that construction was the original reason that people were interested in symmetric polynomials. For another demonstration of how little of their own history mathematicians know, ask anyone why the notation for the second derivative is d^2y/dx^2. Then ask them where the f' notation comes from. Then ask them what Cauchy was trying to do that lead to Cauchy sequences. Most will draw a blank on all three. Don't read on until you're satisfied that you don't know the answers. In the original infinitesmal notation, d was an operator. It could be defined by d(y) = y(x + dx) - y(x). And you'd calculate a slope as dy/dx (drop any infinitesmal bits). Well when you work out d(dy/dx)/dx it turns out that you get d(d(y))/(dx * dx) which is more compactly written d^2y/dx^2. The f' notation was introduced by Lagrange in an attempt to get rid of infinitesmals by defining differentiation as a formal algebraic operation on polynomials and power series. This fell apart when Fourier demonstrated that apparently well-behaved power series could be used to construct pathological things like step functions. Cauchy came up with Cauchy sequences while attempting to define infinitesmals rigorously. His approach fell apart on the seemingly trivial example of how you rigorously prove the chain rule when the derivative of the inner thing is 0. (He was trying to avoid 0/0, but in that special case you get 0/0 all over the place.)