Thoughts about a number much larger than Graham's number [closed]

Question

This question is not really a mathematical question but rather a philosophical one.

I know that Graham's number is so huge that humans just can't imagine how large it really is. It is at position 64 (g64) on a list of ever increasing numbers (starting with g1).

Imagine continuing this list beyond g64. g65, g66, g67... until the index reaches Grahams number itself.

What would you think about this number – a number that is not at position 64 but at position G (Graham's number)? This number would be so much larger than Graham's number itself.

But would it make any difference to our human minds? Could anybody tell the difference in magnitude between these numbers? Or are numbers beyond a certain boundary just unimaginably huge, hence basibally of the same magnitude?

Thanks for your thoughts.

Answer 1

One of the advantages of mathematical thinking is that it allows us to deal rationally with things that we cannot imagine or visualize. That includes not only extremely large finite numbers (starting with Archimedes' "Sand Reckoner" if not earlier, and continuing with things like Graham's number and the Ackermann function) but also infinite sets and their combinatorial properties.

So my answer to your question is: Yes, the difference between G(64) and G(G(64)) makes a difference to our human minds, because our human minds are capable not only of imagination and visualization but also of mathematical understanding of things beyond the scope of our visualization.