Russ Abbott wrote > > > Mathematics is a language of equations and > numbers. Of course equations operate within frameworks, which > themselves involve concepts--such as dimensionality, symmetry, > etc. These are important concepts. But the equations themselves are > conceptless. They are simply relationships among numbers that match > observation. I suspect that this is one of the reasons the general > public is turned off to much of science. The equations don't speak to > them. I would say that the equations don't speak to scientists either > except to the extent that they manage to interpret them in terms of > concepts: this is the strength of this field; this is the mass of this > object; etc. But the concepts are not part of the equations. And > (famously) quantum mechanics has no concepts for its equations! The > equations work, but no one can conceptualize what they mean. So how > should one think about quantum mechanics? As a black box with dials > one can read? What should the public think about quantum mechanics if > that's the best that scientists can do? > > I can think of two > primary goals for science: to understand nature and to give us some > leverage over nature. Equations give us the leverage; concepts give us > the understanding. > > -- Russ > > > > On Sat, Dec 27, 2008 at 7:33
I disagree completely with this. Mathematics is not just about equations, but about concepts and expressing those concepts. The equations are like the letters and words that make up the play Romeo & Juliet. If that is all you see, you miss a fantastic story! Truly, this is important. When I studied linear algebra in first year university, the lecturer could not recommend a single text book. Instead, he taught the concepts of linear algebra, and how one might imagine them in one's mind's eye. (Linear Algebra is basically about rotations and stretching in n-dimensional spaces - we can easily imagine the 3D ones, and handle the other dimensions by analogy. Only infinite dimensional spaces get a little tricky!). Using this technique, significant theorems become obvious. Translating the theorems into algebra often required a page or more of terse equations to express. I once proved a theorem on a necessary condition for "permanence" (an ecological stability concept) in generalised Lotka-Volterra equations one sleepless night using this conceptual way of thinking about linear algebra. In the morning, I translated the proof into algebra, and found it to be correct. Unfortunately, I then discovered that the theorem had been proved and published about 15 years before :(. In quantum mechanics, the concepts are just that of linear algebra (rotations and stretches), complex arithmetic (which are planar rotations and stretches) and Fourier transforms (spectral analysis of a wave form - familiar to all users of "graphic equalisers" in Hi Fi systems.). Cheers ---------------------------------------------------------------------------- A/Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 [email protected] Australia http://www.hpcoders.com.au ---------------------------------------------------------------------------- ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
