a number that exceeds the total number of atoms in the universe

I have asked this question before, this type of question but now I have a more detailed way of asking it. If a number that exceeds the total number of atoms in the universe, is that number “real” or is it just made up simply for mathematicians to talk about?

The general question is math something humans invented or is it distinct and separate from humans?
2 rocks plus 2 rocks makes 4. You can count that. Rocks exist therefor we didn’t create math.

Well, the number of atoms in the universe is estimated at to contain between 10^78 and 10^80 atoms (e.g. 10 followed by 80 zeros!). If you type that number out it doesn’t even fill up half a sheet of paper. The newest prime number, however, is so large that it fills up 3 notebooks when you type it out!!!

As of January 2016 , the largest known prime number is 2 to 74,207,281 − 1, a number with 22,338,618 digits. (three notebooks)

To use the analogy, if a number is larger than all of the known rocks in the whole universe, ie, if it can’t actually be proven to exist via counting, how doe we know that number exists?

I number is just a concept.
It isn’t a physical thing, which means it can be as big (or bigger) than we can imagine.

Well, what if you wanted to number the number of ways atoms could be arranged with each other? So if you have 2 atoms they can be arranged 2 ways. 3 atoms can be arranged 6 ways. 4 atoms can be arranged 24 ways. 5 atoms can be arranged 120 ways. 6 atoms can be arranged 720 ways.

With 100 atoms there are 9.33 x 10^157 combinations.

So we very quickly start using numbers much much larger than the number of atoms in the universe, just to describe permutations of 100 things. There are 52 cards in a deck, how many possible poker hands are there? There are 64 squares on a chessboard, how many possible chess games are there? There are more possible chess games than there are atoms in the universe. So there you go.

I realize that.

But, you don’t think the rocks/countable issue affects it in any way? Even for the basis of discussion???

Ah, ok, good point.

But do you think the numbers you listed are as large as the newest prime number?

For the purposes of this discussion, you’ll have to define what is meant by saying that a number “exists.” Once we understand what you mean, we can then argue about whether any given number exists.

IDK, it was more a reaction to seeing how large the prime number was…

I would define “exist” as something that can be observed/counted… as opposed to an idea is simply inside the head of one or more people.

Why pick on atoms? Why not limit the “real” numbers to the number of electrons in the universe, or photons, or something else. It’s not like the atom is the smallest unit.

Yes, I agree, I almost put that stipulation in my OP… but I have no idea how many sub atomic particles there would be.

We’ve had threads on this question before. Here’s one, whose OP links to several others:
Math was discovered, not created

You’re going down a very bumpy road. Can you show me that a square root “exists”? That -53 “exists”? That the square root of -53 “exists”? That matrixes, or quaternions, or null sets “exist”? That countable or uncountable infinities “exist”? That the stuff in a whole library of math textbooks “exists”?

Thousands of years of teaching and learning go into mathematics, all of it, every single bit, dedicated to saying that countable things is not mathematics and should not limit our understanding of the world. Every concept in mathematics is just as real and exists just as surely as five does.

The dark side of the moon existed before we sent a satellite to photograph it, too.

Thank you

Yeah, I’m not actually saying that largest prime number doesnt exist… I just think it is an interesting conversation.

It’s a trap!

Wait until they discover the next largest prime number …

If you’re having problems conceiving this as a quantity, I’m right there with you … few of us can really wraps our brains around what it means to be $20 trillion in debt or how many atoms of Hydrogen exists in 1 little gram. There’s 324 possible chess positions after just one move.

Aside from the philosophical basis of mathematics and numbers (which others can discuss better than I), I’d point out that neither you or I can count the number of atoms in a rock, let alone the universe.

Even if we limited discussion to counting the grains of sand on one beach, I think most of us would happily embrace a more theoretical notion of numbers (even at the cost of very big ones) after counting for a week or two. Hopefully before we tried counting sand grains at the water’s edge.

Lol, I agree with you both! :smiley:

You mean “the newest largest known prime number.” “Newer” prime numbers are discovered constantly, whenever anyone generates a large prime pair for RSA encryption, for example. But these aren’t close to the largest. (But still are far, far larger than the number of atoms in the known Universe.)

And no, none of the numbers Lemur866 gave are remotely anywhere near as large as the newest Mersenne prime.

BTW, if the Universe is infinite, there are an infinite number of atoms in it. Usually we take statements about the Universe in such as context as meaning “known Universe” in some way, shape or form. But still there’s a lot of ways of defining that.

It is impossible to demarcate what part of Mathematics is “real”/“useful”, etc. Take GH Hardy’s famous line about his work in Number Theory: “Nothing I have ever done is of the slightest practical use.”

But, Number Theory is the basis for the above mentioned RSA encryption which is incredibly practical. And other uses for it continue to be found.

Even Combinatorial Topology was the basis for a very important result concerning a type of decision making among computer processes a bit back.

You just never know.

There is an apple on the table and another apple on the table. The notion that these are a set, numbered one and two, is just an idea in your head.

However, the apples and the table are also just ideas in your head.

That conversation is interesting in the same way that hitting yourself in the head with a hammer is interesting because it feels good when you stop.