[gbjw]

Well, based on a quick Google, there are approximately 10^186 Planck-length volumes in the known universe. So 10^241 is not physical in the sense that there is no way to 'realize' a set of 10^241 things.

Another related notion is that there is no way to 'realize' an irrational 'number' -- sqrt(2) can be defined but we can never draw a length of sqrt(2).

[anon291]

It really depends what you consider 'real'. It is actually pretty straightforward to realize a representation of 10^241 (you just did). Why does that make it any less real than the digit 2, which represents two things, but is actually just itself one thing.

> Another related notion is that there is no way to 'realize' an irrational 'number' -- sqrt(2) can be defined but we can never draw a length of sqrt(2).

Again, it's unclear what anyone means by things like 'sqrt(2)'. Why is drawing a thing of length sqrt(2) any different than drawing a thing of length 1? If I draw a thing of length 1 and say it's a line of length 1, then why is that different than my drawing the same line, claiming its length is sort(2) and then pointing out that it's actually now impossible to mark where 1 would appear along its length?

[gbjw]

> It really depends what you consider 'real'. It is actually pretty straightforward to realize a representation of 10^241 (you just did). Why does that make it any less real than the digit 2, which represents two things, but is actually just itself one thing.

One is a map, and the other is the territory. Both 'real' in some sense but a map without the territory feels less 'grounded' (pun?).

> Again, it's unclear what anyone means by things like 'sqrt(2)'. Why is drawing a thing of length sqrt(2) any different than drawing a thing of length 1? If I draw a thing of length 1 and say it's a line of length 1, then why is that different than my drawing the same line, claiming its length is sort(2) and then pointing out that it's actually now impossible to mark where 1 would appear along its length?

Perhaps a more precise way to describe the situation is that one can define one or the other as a base 'unit' but you can never get one from the other (they are 'incommensurate', as the greeks would say). Irrational numbers can be defined but not 'realized' (or 'constructed') in the same way that the rationals can.