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.

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