Tell him to be quite careful. He should not, for example, start thinking about the number of different ways you can pick up 5 golf balls. Because that might get him thinking about the different ways you could scrape 200 insects off a windshield. And so on.
IOW, accepting the constraint that “a number has to represent something that can, at least in the abstract ideal, be counted” will by no means allow you to cap things at 10[sup]900[/sup].
Well, he’s limiting it to physically real things, like atoms, or stars. He rejects things like “the number of subsets of a large set.”
He also rejects “real numbers,” but accepts “rational numbers” within certain ranges. Pi, beyond forty digits or so, stops being meaningful, so he truncates it there.
In practice, there is some real use for such things as the “calculus of finite differences.” You can do some solid and productive work with this kind of material. Engineering generally stops at four or five decimal places.
It just seems strange that this guy would insist on engineering limitations to mathematics. Again, he’s putting ideology first.
Is this at all a “movement” in the philosophy of numbers and arithmetic? Are there other “constructivists” out there?
I don’t know about that. I remarked to one of my (not-very-science-inclined) friends who put about 120,000 miles on his old Jetta that he “drove half-way to the Moon.” He just sort of paused while digesting that notion with a smile, probably thinking of all the years and trips it took just to put that much milage on his car, and thinking he’d have to drive that much more, again, to “reach the Moon.”
What is a number? That is the real question. I am a platonist, which means that I believe there is some etherial domain in which the infinitude of natural numbers “exists”. Another theory is that a number is a social agreement among all the world’s people. Then, there is Alexander Esenin-Volpin (see: https://en.wikipedia.org/wiki/Alexander_Esenin-Volpin) who argued that to exist a number had to be comprehensible and claimed that 1,000,000,000,000 (a trillion) is incomprehensible and doesn’t exist. So you pay your money and you take your choice. Maybe this should be moved to IMHO.
OK, so this guy rejects things like “the number of possible shuffles of a deck of cards” as not “really existing”. And yet, we can use that number to calculate things like “the number of possible poker hands of five cards”, and “the proportion of those hands that are Full Houses”. And we can further observe that a gambler who recognizes the validity of those calculations will consistently take money from a gambler who doesn’t recognize the validity of those calculations. So your friend’s assertion is that, even though those numbers don’t really “exist”, they can still lead to money that presumably does really “exist”.
On the subject of the sheer hugeness of Graham’s Number, note that ordinarily huge numbers can be written as a few layers of exponentiation - that is, a Googol is just 1 followed by a hundred zeroes (10[sup]100[/sup]) and even a Googolplex is 1 followed by a Googol of zeroes (so can be written as 10^10^100 since we can’t cope with showing multi-level exponents directly on the screen). And you can easily imagine ridiculously bigger numbers such as 10^10^10^10^10^10 and so on.
The sheer awfulness of Graham’s number becomes apparent when you grasp that it can’t even be written down as a “tower of exponents” - that is, if you start trying to write it down like I just wrote down five successive exponentiations, you run out of space before you get the number written.
You misspelled awesomeness. But seriously, one of the things I find interesting about really large finite numbers is that they feel much larger than infinite numbers. I can easily get my head around the concept of a line being made of an infinite collection of points, that there are as many even as odd numbers, and the like, but really comprehending something like Graham’s number take a big effort and isn’t something I can casually call back up.
I see what you mean. I can imagine seeing infinitely many things at once (the points on a line segment) but I can’t imagine seeing Graham’s number (or even a much smaller finite number) of things at once.
Assume there are 10^86 elementary particles in the universe. Assume, like the OP, that numbers up to that number are okay, but any greater than that number are basically meaningless. Now tell me why (10^86)+1 is such a horror.
It seems to me that I’ve heard the objections in the OP before and that there’s a school of philosophy of math that explores it. A quick wiki-walk suggests its called Ultrafinitism or at least that’s the closest I could find without spending any amount of time on it.