It's axiomatic

I found high school geometry unsatisfying.  I liked mathematics in general, and my grades in geometry were fine.  But the subject was somewhat irritating for a number of reasons, one of which was the concept of "axioms" and "postulates."

"Axiom" is generally given two definitions.  The first, and more traditional, according to Wikipedia: "an axiom is a premise so evident as to be accepted as true without controversy."  To me, nothing was without controversy. The second, which is that used in modern logic: "an axiom is simply a premise or starting point for reasoning."  (A "postulate" in traditional geometry was an "axiom" applicable by its terms expressly to geometry; I'll use the two terms interchangeably.)

I don't know how geometry is taught today.  But we were given Euclid's axioms, and from those axioms were taught to prove a large number of theorems.  The axioms and the theorems focused, of course, on the properties of geometrical figures.  A typical axiom, Euclid's first, was "It is possible to draw a straight line from any point to any other point."  The axioms were supplemented by a list of "common notions" used in development of proofs: e.g., "Things that are equal to the same thing are equal to each other."

The proof of theorems -- while not as congenial to me as, say, manipulating trigonometric functions -- was interesting and satisfying.  But if we were going to work so hard to prove theorems that seemed intuitively obvious anyway, why were we exempted from providing proofs for the axioms (and "common notions")?   I asked my teacher that question, and he (who spent most of his time as "boys' counselor") replied -- rather impatiently -- that we accepted them because they were "obvious." The "traditional" -- and, to me, irritating -- definition.

I would have been happier if I had been taught and had understood the second definition.  Geometry had its origins in the efforts of early mathematicians to describe real shapes in the real world -- and has its uses even now in so doing (it's always nice to know that A=πr²).  But what we are really doing when we study geometry formally is playing a game with arbitrary rules, not working scientifically from empirical data.  We are proposing that certain concepts be accepted as "true" -- the axioms -- and discovering what we can prove from those concepts using only logic.  It's like chess.  We don't ask why it should be so that a bishop can move only diagonally; we instead see what we can accomplish given that arbitrary limitation. 

We can -- and mathematicians do -- propose different axioms (ones less intuitively related to the world as we perceive it), and see where those axioms logically lead.

I've been lead to these rambling thoughts by a book review in Sunday's New York Times.  The book -- Time Reborn, by Lee Smodin -- apparently is too complex to lend itself to an intelligible one-page review, even a review written by a highly regarded physicist and writer .  (At least, I found myself unable to understand the general argument of the book from reading the review.)  But the review does present some of the author's more interesting conclusions. 

Smodin is a physicist, and his book discusses cosmological questions such as "what is time" and "where did the universe come from."  He has concluded, apparently, that black holes beget new universes, and through "natural selection" of such universes we have finally arrived at our own universe -- a universe that possesses the peculiarity that the constants upon which its physical laws depend are precisely that narrow set of constants that life required in order to have developed and to continue in existence.  With "time" in his title, it's clear that Smodin believes that this discussion is somehow related to what we mean by "time," and its passage -- and the review touches on this relationship without really conveying Smodin's argument.

The review ends with the following summary, which brings me back to my discussion of geometry:

Putting aside the sensational ideas proposed in "Time Reborn," it is a triumph of modern physics that we are even asking such questions as what determined the initial conditions of the universe.  In previous centuries, these conditions were either accepted as given or attributed to the handiwork of the gods.  A triumph, and also possibly a defeat.  For if we must appeal to the existence of other universes -- unknown and unknowable -- to explain our universe, then science has progressed into a cul-de-sac with no scientific escape.

In a sense, the reviewer shares my frustration with geometry as a kid -- if our universe sprang forth from a black hole in another universe, like Athena from the forehead of Zeus, we have reached a starting point beyond which we can probe no further.  To ask whether the Big Bang was "really" generated by that black hole, or by a Word from the Voice of God, or by some Child from a Superior Race's having turned on his celestial Gameboy, is scientifically meaningless.  Information as to what  happened before the Big Bang is scientifically unrecoverable.  Empirical data take us back no further than the Big Bang itself -- not only the data we have now, but also any data conceptually obtainable in the future.

Rather than despairingly view the Big Bang as a dead end, a scientific "cul-de-sac," we should consider it as a cosmological axiom of the "game" that is our universe.  We accept the Big Bang -- that the universe began as an infinitesimal point of infinite density -- as our starting point, just as we accept Euclid's axioms -- for example, that one can always draw a straight line between any two points -- as the starting point in geometry.  We can't prove scientifically -- and will never prove in the future -- why the Big Bang exploded or whence burst all that energy out of an infinitesimal point, any more than we can prove logically Euclid's first axiom.  All we can do is speculate, as has Smodin, apparently -- with no empirical support. 

Admittedly, speculation is fun.  But it's not science; it's science fiction. In so speculating, we're like chess pieces trying to understand why bishops, unlike rooks, can move only diagonally.