On large numbers, by ultrafilter, Dec 2002.
From The Ackermann Function:
Bolding mine.
Hmm, I’m surprised that no one has really ventured into the philosophical yet. I know the OP didn’t mean it that way, but asking whether number “X” exsists is a pointed question. Forget about the size of googol for a moment. Let’s say that the OP had asked if the number 5 exists. How would you answer that?
It’s not like the number 5 is floating out there in space somewhere to be discovered. To phrase it differently - what is math, anyway? Does it exist apart from human conception? Certainly we use it describe real things, but do the descriptions themselves have some kind of “reality”? Or put another way - does the universe itself posses some sort of “mathiness” that we discovered, or is math purely an invented language, and now we’re trying to pound the universe into a shape that will fit our pre-conceived methods?
I’ve always taken the view that math is a language, specifically one that’s geared towards very high precision. That’s it’s chief strength, but it’s also a blind spot. It’s almost comical to watch how utterly unequiped it is to deal with the imprecise realities it comes into contact with. Irrational numbers are a good example. It’s all well and good to say that pi is defined by the ratio of the radius of a circle to it’s circumfrence or area, but when it comes down to evaluating it numerically, mathemetics just kind of craps out after a while. It’s that nagging little pebble in your shoe that just won’t quite go away. Or witness how often you have to juggle an equation around to get rid of infinities and division by zero. Both represents concepts which are probably essential to understanding reality, and yet mathemeatics often have a terrible time coping with them.
So, back to the original question - does a googol exist? Well, that depends on what you mean by that. If math is just a language, it’s a very unique one. In almost every spoken language, there are “nonsense” words that don’t really have any meaning. But those languages are equiped to deal with the meaningless and imprecise - it’s understood that they have no meaning and that they aren’t precisely “real”. But googol can be very precisely defined and has a very real meaning. But it’s meaning is such that we’ll probably never find anything real to describe with it. I’m not sure that there’s any applicable analogy to that. Perhaps it’s like a mythical beast, the proverbial griffin - everyone knows what it is, but what it is is a figment of our imagination.
sure it exists its 216.239.41.100 !!
ba dum dum ching
Actually, they round-robin their DNS servers. Every few seconds, it comes up with a difference answer when you ask for google.com’s IP address.
At the moment, it’s 216.239.39.99
Countless metaphysical essays, books, and discussions centering around just this idea. Of course there are many different theories, but the three most popular are Platonism, Nominalism, and Conceptualism. You can look up these ideas in relation to numbers on Wikipedia.org or other websites that deal with Philosophy.
Oh yeah, that particular branch of study is usually referred to as “Number Theory”.
If 0 exists, and whenever n exists, n + 1 exists, then any finite number exists. The axioms of standard set theory guarantee that those two conditions hold, so we’re golden on a googol.
Achernar was kind enough to link to my old thread, so I’ll just say that if you can write it with exponents, it’s not big.
btw, I would never read a^b^c as anything other than a[sup]b[sup]c[/sup][/sup].
Because that is the road to madness. 
Isn’t there an ancient debate on this very subject between the “Platonists” and the, well, I guess non-Platonists they’d be called? Did these two ever kiss and make up?
As a practical sort of person, I guess all I can say is that the number 5 has the same solidity as a googol, however strong or weak that shared solidity might be. But let me argue that their existence is pretty solid…
Well, hmmm. Would you agree that a water molecule implicitly specifies the integer 2, the number of hydrogen atoms it has, and 10, the number of protons it has, and perhaps even specifies an irrational number: the least-energy angle between the hydrogen atoms as measured from the center of the oxygen atom? (I think it’s about 100 degrees, but it’s not a simple constant.)
My point being, physical reality seems to provide a clear basis for (A) the whole numbers, because matter comes in aggregations of discrete particles, and (B) geometry, because the arrangement and motion of these particles is influenced by distances and directions to other particles. Or anyway, much of our universe becomes comprehensible and predictable if you assume that whole numbers and geometry are real properties, that they’re not just an artifact of arbitrary human whim.
On the other hand, a mathematician named Kronecker once said, “God gave us the integers; all else is the work of man.” So maybe geometry is really just a hoax.
Let’s assume for a moment that, say, Maxwell’s equations of electromagnetism — which involve concepts of calculus, not just geometry and arithmetic — are the Truth, that at least in this one small area of physics, we humans are dead right. Assume then that these equations genuinely and accurately describe some of the rules which govern the physical universe.
If so, this would presumably mean that even if we humans went extinct tomorrow, and another intelligent species eventually arose in our place, that they would discover the exact same electromagnetic relationships that Maxwell’s equations describe — though presumably they wouldn’t use the same notation when writing them.
If their brains were congenitally incapable of understanding calculus, then I suppose they’d never discover Maxwell’s equations. If their brains were more mathematically capable than ours, then presumably they could discover physics equations that we are blind to. That second one’s a guess; maybe there are no such higher-level rules.
Not that all the rules that govern our universe must be mathematical. But a whole hell of a lot of them sure seem to be.
I vote for the first one. Consider an interesting, and I hope relevant, or at least entertaining, case: the extinction of the passenger pigeon. The last passenger pigeon, “Martha”, died in 1914, when she threw herself on a grenade in the trenches on the Western Front.
No wait, that’s not what happened. Turns out she died in captivity without reproducing — that in fact, the whole last gaggle of passenger pigeons died without reproducing, despite everyone’s then desperate efforts to encourage it — because the birds wouldn’t breed without the presence of their fellow birds in large numbers. (Cite.) In other words, at some point before 1914, a vicious spiral of doom had begun for the passenger pigeon, when its population slipped silently below the magic threshold — a fuzzy one I’m sure — at which the species could perpetuate itself.
Of course we humans were responsible for killing off 99.99% of the initial population, thereby setting up the spiral of doom. But the same thing could have happened without our help, by a plague or a widespread natural disaster. In any case, I assume the pigeons had no concept of integers, yet they were affected by them nevertheless.
My point here? I suppose I’m saying that numbers matter, that the quantity and geometry of objects result in real physical events taking one path and not another — and that this goes on regardless of whether we humans are here to observe it or understand all the rules behind it. So yes, the universe has a certain “mathiness” to it that we discovered, and did not invent.
Well, I would say that π has a perfectly well-defined value, and can be represented easily enough by a precise point on a number line — but that our decimal notation system simply requires an infinite number of digits to express the value. In fact you can never express π in any positional notation system (except base π perhaps, though that seems like cheating), but that doesn’t mean mathematics is “crapping out” on you.
I don’t mean to seem disagreeable in this article by the way. I think you raise a very good philosophical question: in what sense do numbers exist, as distinguished from the Moon, or electrons, whose physical existence is presumably beyond question, and which we understand innately.
Oh god - not another of those crazy people who think the moon really exists!
But that really doesn’t solve the associative problem, ya see? 'Cause it’s still possible to see it as “Compute a[sup]b[/sup] and raise that to c.”
I accept that exponentiation is right-associative. I don’t want to drag this out. But simply putting in superscripts instead of carats (’^’, which is the Basic standard) or doubled-asterisks (’**’, which is the Fortran standard) doesn’t solve anything.
The only ways to solve associative problems are to make things explicit. In an algebraic language, that means introducing parenthesis. If you don’t want to do that, you can switch to a postfix system (where a^b^c becomes a b c ^ ^) or a prefix system (where a^b^c becomes ^ a ^ b c).*
*(But most prefix-notation systems are Lisp-based and therefore make all implied parenthesis explicit. So my ^ a ^ b c becomes (^ a (^ b c)), which is bleedingly obvious to even a CS prof. :))
Super. However, since they aren’t a googol of anything in existance, is it really worthwhile to have such a number? It seems kind of pointless. The number which, cannot even be written out since it would take up more space than there is space in existance. Now if it pushed mathematics in a new direction like the realization to use the number 0, then I could understand. But if it doesn’t, then I support striking it from the lexicon. And don’t even get me started on googolplex… the nerve of those mathemeticians (and their kin)… to name a number so utterly useless!
It (“googol”) exists but it isn’t a dignified name to give a number, nor is it a mathematically interesting number.
A better name for it would be “ten duotrigintillion”. Which perhaps gives you a quick idea of why it isn’t mathematically interesting. It isn’t a “nice round number” the way that a million is and a billion is but 1 billion 110 million isn’t.
We cluster digits in clumps of three columns, like so:
123,456, 789
A one followed by a hundred zeros may sound like a nice round number but it’s actually ten of a named “clump” (the duotrigintillions). Furthermore, once you get high ehough up there (i.e., beginning with the decillions), you get nominal “superclumps” – if you think of the decillion clump as the tenth series, the clumps that follow (undecillion, duodecillion, tredecillion, etc), being the eleventh, twelfth, thirteenth, etc., series, are at least in name “of” the tennish superclump in the way that 11, 12, 13, etc., are “in the teens”. The second superclump would be the vigintillions (twentieth nominal series). So the duotrigintillions is the non-milestone-ish 32nd series.
So the “googol” is just ten of the 32nd-series clump. BFD. Not to mention a silly and ugly name.
10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
(Real enough, though).
There is no associative problem, exponents aren’t associative.
a^b != b^a
a[sup]b[/sup] != b[sup]a[/sup]
Furthermore, you always do the innermost work first, so you would always do b[sup]c[/sup] before looking at a.
Derleth, all the points you raise are valid, but there’s no confusion as to what the expression a + b * c means. Why not? Because we have a convention as to what precedence each operator takes. Similarly, we have a convention that ^ is right-associative. So there’s no problem.
dakravel, that’s the commutative property. If exponentiation were associative, then a^(b^c) would equal (a^b)^c. It doesn’t, so we need to agree on what a^b^c means. By convention, it means a^(b^c).
For an interesting compendium of interesting numbers both large and small, I recommend http://www.mrob.com/pub/index.html.
That URL inks to the guy’s home page; the sub-links you want are in the middle cell of his table, labeled “numbers” and “large numbers”.
Some of his other pages are also interesting in a very geeky (but nice) way.
Sorry for resurrecting this thread, and sorry about that thing I posted up there. I always get those association and commutation confused, but luckily the result was the same. Thanks ultrafilter.
The chance of you correctly guessing the outcome of rolling a 10 sided dice 100 times in a row would be exactly 1 in a googol. This is a relatively simple demonstration of how the number does exist in reality.
As for a visceral way of imagining a googolplex, the best example I have heard was that imagine you had a hypothetical box in which you could fit all the atoms of the universe. If you shook the box so that all the particles were in completely random positions going at random velocities, the chance of you looking inside the box and seeing an exact replica of the universe at this very instance is approximately 1 in a googolplex.