## Wednesday, December 28, 2011

### 117: A Question Of Assumptions

We commonly view mathematics as a process of starting with simple
axioms, based on commonsense notions of what the universe must be
like, and then building up from them to theorems that show their
various consequences.  You are probably familiar with the most
classical example of this, Euclidean geometry, where simple notions
about points and lines build up into surprising conclusions like the
existence of only five regular polyhedra.  You may also recall from
earlier podcasts the concept of Non-Euclidean geometry.  By
slightly modifying Euclid's basic assumption known as the Parallel
Postulate, which specifies that only one parallel can be drawn to a
line through an external point, we are able to come up with different
geometries that are just as self-consistent, but don't happen to
describe our typical notions of the universe.  But despite having
originally begun as intellectual exercises, sometimes these
non-Euclidean geometries turn out to have real applications.  In fact,
one of the surprises of Einstein's theory of relativity was that our
universe is not truly Euclidean, and one of these alternate geometries
is actually a better description.
Of course, there are many other mathematical models of our
universe besides Euclidean geometry.  One of the most important is
what modern physicists call the "Standard Model", a set of
descriptions of elementary particles and their interactions, along with
19 related constants, that seems to be an excellent description of the
behavior of subatomic particles in our universe.  The details are a
little complex to describe in a podcast, but there's a link in the
show notes if you want to delve into them in more depth.  Like the
parallel postulate of Euclidean geometry, the many seemingly arbitrary
constants of the Standard Model have been disturbing to physicists.
Is there some reason the numbers have to work out exactly this way?
Some subscribe to the "anthropic principle", the idea that there
probably are many universes with many different values for these
fundamental constants, but the ones we observe in our universe are the
ones that could enable the set of phenomena the lead to intelligent
life-- otherwise we wouldn't be here to observe them.  Kind of like
the old paradox about the tree falling in the woods:  if a physical
constant assumes a value and there's no one there to see it, does it
make a sound?
This has led to some interesting lines of speculation in recent
years:  could there be alternative values for some of these
fundamental constants that might also lead to life?  Martin Rees of
the University of Cambridge has theorized that there are many
inhabitable 'islands' of life-supporting physical laws in the
multiverse.  Since string theory supports the existence of 10^500
different universes, this doesn't seem that implausible.  To make this
more complete, a trio of physicists named Roni Hanik, Graham Kribs,
and Gilad Perez attempted to analyze a universe with one specific
change:  turning off the 'weak nuclear force', one fundamental force
of the standard model.
The 'weakless universe' they describe is different in some basic
ways from our own.  The primary nuclear reactions that fuel our stars,
hydrogen fusing into helium, cannot happen, but with slightly more
deuterium in its starting state, other types of star-fueling reactions
could occur.  A type of supernova would still be possible, a
critical factor since these are what synthesize and disperse the
heavier elements needed for life, though few elements heavier than
iron would be likely to appear.  Stars would be much smaller and
shorter-lived, but some about 2% the size of our sun could survive for
the billions of years needed to evolve life.    Since the stars would
be small and cool, planets would have to orbit very close to them.
The plate tectonics of these planets would be much calmer than
Earth's, since much of our volcanic activity is ultimately fueled by
the decay of heavy elements deep within the planet.  To inhabitants of
weakless planets, their sun would appear gigantic in the sky, but the
night sky would be nearly empty due to distant stars being so dim.
And this is not the only alternate universe.  Anthony Aguierre of
the University of California discovered another possibly
life-supporting universe by varying a different constant, the number
of photons per baryon, and a recent issue of Scientific American
discusses other possibilities resulting from assuming a different mass
for quarks.  So, is there an answer to the original question, of why we
have our particular constants in our universe?  Maybe the anthropic
principle is still the answer, and we are residents of a small subset
of universes lucky enough to have life-supporting constants.  Maybe
there is something fundamental that has not been discovered yet, and
our laws of physics really are the only ones possible.  To some
extent, the question is in the realm of pure mathematics, since
however much fun it is to speculate about them, nobody knows how we
could ever observe one of these alternate universes in any case.
And this has been your math mutation for today.

• Non-Euclidean Geometry at Wikipedia
• Weakless Universe at Wikipedia
• Standard Model at Wikipedia
• Scientific American article on alternate universes
• New Scientist article on alternate universes