I wanted to take a sounding of the Ars Mathematica readership, if for no other reason than me being nosy. It seems to me that we have a pretty broad readership (from Ph.D.’s to high school students), but I for one would like to get a better idea of the distribution.
Let me be the first to divulge info.
Walt, Robbie and I all went to grad school at the University of Washington – Robbie does/did Ergodic Theory, I am somewhere in the intersection of algebraic topology, combinatorics and logic, and – as far as I have been able to ascertain – Walt knows everything.
So sound off if you would…
I realize that I have been a bit out of touch recently, but I feel ashamed that I just realized that my mathematical great-grand parent Raul Bott passed away Dec 20th.
Vis – a – vis the comments in sigfpe’s last post, Krzystof Burdzy has a booklet online: Probability is Symmetry. On Foundations of the Science of Probability which introduces, discusses and critiques the Frequentist and Subjectivist foundational positions of Probability Theory. For any of our readers unfamiliar with either the terms or the arguments behind these philosophies of Probablility, this book forms an excellent primer on the subject, in addition to arguing for Burdzy’s own interpretation.
MOND, the acronym for MOdified Newtonian Dynamics is a theory put forth by Moti Milgrom in 1983 to resolve problems with galaxy kinematics without resorting to dark matter. The theory decouples inertial and gravitational mass (breaking the equivalence principle) positing that at very small accelerations (those below an observationally determined constant a0), the gravitation force felt by a body is actually smaller than the force predicted by the famous Newtonian equation F = ma. Aside from the bizzare nature of the change, the theory has several things going for it, not the least of which is that it made predictions that were later verified (namely the existence of low surface brightness galaxies).
Another point in its favor is that it can explain the Pioneer anomoly. When I first heard of the theory and the problems it was meant to solve, I thought that the theory was crazy, but that it might just be crazy enough to be true. I even bandied about the idea of writing a popular science book about it which itself would have several things going for it:
- MOND could actually be correct – such a book would be early to the game
- While there are many science popularization books written, there certainly aren’t many speculative hard science popularizations written – unless you count all the string theory stuff – it could jump start a whole category!
Some final food for thought; I haven’t done the calculation myself, but apparently constant acceleration at a0 for our best guess at the age of the universe produces a velocity of – wait for it – the speed of light. I don’t know if this new world would be cool enough for Walt to have to put on shades before he glanced at it, but quite a few textbooks would need to be rewritten
And some MOND links:
Since most posts don’t get many comments, I thought I would make one the required audience participation. The subject is “fundamental” theorems in the various subjects. What I am going for is hard to actually describe, but encapsulates a theorem being fundamental, its utility, its depth. It is the result in the subject that would hurt the most not to have, but does not have to be the putative “fundamental theorem of X”
For example, my votes for a few subjects:
Calculus: Mean Value Theorem.
Probability: Linearity of expected value.
Model Theory: The compactness theorem.
Allen Hatcher of Cornell appears to be undertaking the quixotic goal of writing accessible “introductory” textbooks for the entirety of Algebraic Topology. The first volume is already the definitive introductory work in the subject, covering the Fundamental Group, Homology, Cohomology, and Homotopy and is available online.
The style is geometric, so those whose nascent views on Algebraic Topology are functorial may be better served by Rotman’s book, but it is hard to recommend highly enough since Hatcher is an excellent expositor, and the book is clearly written for students rather than a vanity piece aimed at colleagues, a major pet peeve of mine.