Monday, 13 January 2014

Billions and Billions

Cover Art from Classical Text on Fractal Geometry

Read a quasi-scientific news release from the BBC that reported that scientists have mapped our universe to within one per cent accuracy.  Announced at the 223 meeting of the American Astronomical Society (AAS) were the findings of the Baryon Oscillation Spectroscope Survey (or, BOSS - how nice), in which a new "gold standard" has been set.  To quote from Professor David Schlegel (a professor of physics at Lawrence Berkeley Labs, and no relation to Immanuel Kant):

There are not many things in our daily lives that we know to 1% accuracy. I now know the size of the universe better than I know the size of my house. 
Twenty years ago astronomers were arguing about estimates that differed by up to 50%. Five years ago, we'd refined that uncertainty to 5%; a year ago it was 2%
Don't know how precisely I know the size of my house, but plainly, he is talking about tolerances, and this is an odd sort of comparison.  I would guess he knows the size of his house to within a few square metres, so in one sense, he surely knows its size much better than he does the size of the universe.  Precision as measured by effect size is a bit of tricky sledding.

Still, the announcement and information are remarkable, and provoked a discussion on whether the universe is finite or infinite.  Of course, I had always been of the impression that it was infinite, but it turns out that that may not be accurate. Quoting Professor Schlegel once again:

While we can't say with certainty, it's likely the universe extends forever in space and will go on forever in time. Our results are consistent with an infinite universe.

What would it mean for the universe - which it turns out, is actually quite flat; suddenly, those "flat earth society" jokes aren't quite so funny, huh? - to be flat boggles my mind at least.  Flat and finite. Hmmm....

One of the comments I read said, rather of matter-of-factly, that "of course the universe is infinite.  If it were finite, then one would simply go to the end, and then go a bit further," offering a bit of a pseudo-mathematial inductionist argument.

I'm not so sure that that flies.

I was reminded of my days in high school when the concept of proof by mathematical induction was introduced to me.  (NB:  quickly, the way an inductive proof is made is to take a fixed case, show that the conjecture is true for THAT case, set this case to "N," and then show it to be true for the case of N+1.  The argument is then bootstrapped into the general case).

It was, as I recall, all quite elegant.

A cartoon was used in the book to illustrate the principle, describing a parable of a young boy, who, upon approaching a guru, asked as to how the earth stood "up" in space without falling.  The guru immediately answered that there is a giant, invisible elephant in space holding up the earth.

The boy then asked what held the elephant in space.  Well, of course, another elephant held the first.

After several asked-and-answered questions, the guru tired, and said that the universe itself is simply held by elephants, one holding the next, ad infinitum.

I find these sorts of arguments - the inductive as well as the guru and elephant - less than satisfactory.  Topology does not translate directly and perfectly to the real world.  Reaching the edge of the universe, if it be indeed finite, one cannot simply go beyond.

Topology works in the case of the Heine-Borel theorem.  It may not stand up to the Prime Directive.

Anyways, it's a big breakthrough, and if the universe turns out to be, indeed, finite, then the elephant guru's life just got that much simpler.

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