This is the third and last summary of the Saturday breakfast meetings of an engineer (my Canadian friend Gerry, who died last month) and an auditor (me, alive but moving gingerly).
Setting one morning’s agenda, I asked: Does infinity exist, outside high-school maths and the circumference of a circle? The Meeting agreed unanimously that the scope of the debate be limited to logic alone – no physics, no astronomy, no obscure scientific theories of any kind. We were two old codgers at a buffet breakfast, not a pair of academics trying to score points by quoting peer reviews.
Being the only professing linguistics expert present, I ruled that we must accept the literal definition of “infinity” – namely, “without end or boundary”. My ability to spell correctly and to remember Latin roots put me in the Chairman’s seat. No argument there, then.
Gerry argued that there could be different sizes of infinities. An infinite number of cows would weigh more than an infinite number of mice. Hmf! I pointed out that the weights could be measured only when the counting stopped, and that “infinity” meant it never would stop. As a backup, I pointed out that infinity is beyond numbers; it isn’t itself a number.
Next: can parallel straight lines ever meet? Again we argued the definition, noting that “define” comes from the same root as “finite”. As an engineer, he couldn’t conceive of a straight line that never ended, never mind two of them in parallel. Maybe the creator of the universe wasn’t an engineer, Gerry; maybe it was a philosopher, or an artist. For Gerry, those were even less conceivable than an endless line.
Is the universe itself finite (very, very big) or infinite (endless)? Is it eternal, or will it just last for a long, long, time? How relevant is the Big Bang theory? In an infinite universe a Big Bang might be a single pulse-beat in an eternity of pulse-beats. In a finite universe, it would have to have been an act of magic. The only astronomers who get worked up about it must be those whose universe is finite.
Some day, they believe, they will discover one or more of the finite number of boundaries to the universe. Some day, their space-probes will go over the edges. Or, the probes might knock up against an invisible wall, like Jim Carrey did in his boat in that movie. [The Truman Show]
But if the universe is truly infinite, we speculated, our Earth might well be just one of an infinity of Earths created every split second since time began. (Not that it ever did begin, of course...) If so, which Earth are we living on? One of the infinity of Earths on which Jesus died for our sins, or one of the other Earths, where he didn’t? We’ll never know – at least, any time soon.
If the universe is truly infinite, then time travel will be feasible – once we redefine “time travel”. The problem of what’s called “the grandfather paradox” is immediately resolved, because the grandfather you shoot exists on a different Earth, and it doesn't matter that you don't get to be born. It's not your this-Earth grandfather, and the man with the gun is not your this-Earth you.
Gerry and I struggled to stay on the path. It was, and is, easier by far to be like the astronomers and base our speculations on a finite universe in all four dimensions – just a very big place that would last a very long time, with only one Earth and only one future, and with alternative universes left to the pages of science fiction.
Who cares if the value of pi can be measured only to ten gazillion decimal places? Does it matter that the area of a circle can’t be calculated exactly? Not really. The world has gotten along pretty well with approximations, so far. A finite universe may mean that the decimal places of pi will come to an end, some day. If not, then maybe infinity really doesn’t exist outside high-school maths and the radius or circumference of a circle. That seems a bit limited, for a limitless function.