Humans and Really Really Large Numbers

The following conundrum can be found at Overcoming Bias:

3^^^3 is an exponential tower of 3s which is 7,625,597,484,987 layers tall.  You start with 1; raise 3 to the power of 1 to get 3; raise 3 to the power of 3 to get 27; raise 3 to the power of 27 to get 7625597484987; raise 3 to the power of 7625597484987 to get a number much larger than the number of atoms in the universe, but which could still be written down in base 10, on 100 square kilometers of paper; then raise 3 to that power; and continue until you've exponentiated 7625597484987 times.  That's 3^^^3.  It's the smallest simple inconceivably huge number I know.

Now here's the moral dilemma.  If neither event is going to happen to you personally, but you still had to choose one or the other:

Would you prefer that one person be horribly tortured for fifty years without hope or rest, or that 3^^^3 people get dust specks in their eyes?

I'm reminded of the well-known saying “Extraordinary claims require extraordinary evidence.” A ratio of 3^^^3 is an extraordinary claim and I'm not sure the evidence needed to establish it can be stuffed into human minds as presently constituted.

Given that any such ratio is currently dubious, one part of the claim or the other is bound to be less certain. I can think of two possible back stories for the decision:

• We can establish the Union of Dust-Free Socialist Republics that will prevent a calculated 3^^^3 dust specks if we only torture just one dissident for fifty years.

• We have an anti-dust vaccine ready, but an article in The Journal of Really Dubious Medical Research claims that after it has been used 3^^^3 times, an Alien Space Bat will be attracted to human civilization. It will then kidnap somebody to torture for fifty years.

In the first case, we should stop the torture and in the second case, we should stop the dust.

The really disturbing part

The really disturbing part of the above reasoning (to this Platonist mathematician who's unwilling to be expelled from the Cantorian paradise) is what it implies about axioms of infinity. If human beings are unable to think properly about very large finite numbers, consider how much more we are unable to comprehend infinite numbers …

I'm reminded of the saying “This is not mathematics. This is theology.” (about the more abstract parts of mathematics). There's also the following passage from The Hitch Hiker's Guide to the Galaxy:

The car shot forward straight into the circle of light, and suddenly Arthur had a fairly clear idea of what infinity looked like.

It wasn't infinity in fact. Infinity itself looks flat and uninteresting. Looking up into the night sky is looking into infinity—distance is incomprehensible and therefore meaningless. The chamber into which the aircar emerged was anything but infinite, it was just very very big, so that it gave the impression of infinity far better than infinity itself.

On the other hand, if there's anything to Roger Penrose's speculations, maybe we are able to think in terms of infinity … which doesn't explain why thinking in terms of very large finite numbers is so difficult. Maybe the MIPS are infinite but the IO is limited. Maybe after the Singularity, we'll be able to absorb the data needed to make a decision. Maybe I've been using the term “maybe” too often in this paragraph.

One possible way to rehabilitate axioms of infinity is to think of them as establishing the consistency of finite mathematics. A consistent system cannot yield its own consistency but it can prove the consistency of simpler axiom systems, especially when a model of the simpler system is embedded. So the axiom of infinity will establish the consistency of finite mathematics and the power-set axiom will establish the consistency of finite mathematics plus the assumption of the consistency of finite mathematics, etc.

Addendum: If something similar to the above defense of the theological type of mathematics were applied to religion itself, I'm sure it would be regarded as a threadbare excuse at Overcoming Bias.