Does 5332^(4726^(9402^(6681^8390))) have an actual numerical representation? If it is impossible for a human to actually identify such a number, but instead to be forced to use an approximation, is it proper to say that there is a specific number which represents this phrase?

We assume that the set of numbers is infinite (via the Axiom of Infinity). But when we attempt to access numbers which have no analog in the natural universe, then why do we accept the validity of the axiom of infinity?

It is simple to generate an algorithm that could calculate extremely large finite numbers, but we can only do so in theory. In practice, we cannot write down extremely large finite numbers.

If a number is inaccessible to human use, but exists purely hypothetically, then why is it even considered to exist? Just because communism works in theory, does not mean it works in practice. Similarly, just because a number exists in theory, does not mean it exists in practice.

Another analogy would be that it is similar to the existence of items past the event horizon at a black hole. Can an object truly be said to exist if it is past the event horizon of the black hole, where it is by definition impossible to verify its existence?

I think that equals 11

Someone give Rockman a beer.

... or a hobby. ;)

Reading math books and doing crazy things like questioning the validity of the axiom of infinity is a hobby.

the topic title is flawed. this is not a math question :P enjoy philosophy

Mathematics and Philosophy are often the same thing.

No, they are never the same thing unless you count ghost acres. No matter how big the number is that you cannot grasp, all I have to do is multiply it by 1/X and I can bring it right back to paper. That's the beauty of math, it works regardless of scale.

I disagree... it's not a hobby, it's an affliction! ;)

How about a number that would require, in binary, the same number of bits as their are quarks in the universe?

=D

I'd say every number that can be represented by the regular physical and quantum state of all particles that can have a quantum state/location/temperature/etc. is valid, because techically we could make all of reality a representation of one big-ass number. Which would be quite big (at least larger than what you wrote up there I think)

Though large numbers are useful, for things like statistical mechanics.

Like, say there is a finite probability of something happening to a particle, like decaying from a neutron to a proton spontaneously. This can be expressed via some number; now say you want to check what the probability of this happening to *TWO* neutrons, or *EVERY* neutron in a GALAXY (neutrons that aren't free will decay to protons with a half-life on the order of the life of the universe; protons can "decay" to neutrons by emitting an anti-electron and neutrino...), and you start to need the size of numbers you're talking about... nobody really cares what these numbers *are* specifically, more they care about the relative sizes...

444^(3NA^(796^(55l^6H2))) is at least shorter... (Base 36)

So in theory, there is a maximal number that can be represented by the regular physical and quantum state of all particles. If such a number existed, then to add one to that number would be an incomprehensible action, as it would be devoid of meaning connected to the reality of this physical universe. And if you cannot add one to the number, that means that the number has no successor, thus negating the axiom of infinity.

5332^(4726^(9402^(6681^8390)))

Isn't that itself a representation of a number?

of course the number exists.

Not being able to perceive something does not mean it lacks existence.

Just because I only have 10 fingers and 10 toes doesn't mean I'm limited to a subset of mathematics only involving them.

oh... i thought you only have 9? :D

Prove that the universe is finite.

The answer is 42.

This.

Besides, we don't have to be able to present the number in base 10 format to represent the number. "Google" is a representation of a very large number. So is 3.65879x10^752. As long as we can formulate a way to represent the number we can make use of it. No matter how high the number goes, you can always make it higher by saying "x2."

join imaginary numbers. at imag we have all the numbers we even have some in number jail. as they represent treehugggers

Shhh mmixx, on the interwebs I can pretend like I have all 10 and that I'm a MMA champion. don't take that away from me.

@vic, I believe it is "googol" which is a very large number. "Google" is of course a popular website.

Isnt that in and of itself a representation of the number?

^That is representing the number. =P

But is it a numerical representation?

I think the thought has some merit, but that would mean that the finite universe would expand and contract due to our knowledge alone. It really is a point of proving that it doesn't exist rather that proving it does exist in this case I think. If we continue down the path of increased acceleration of knowledge, would we not be able to define that number some day as a finite and move the scale of what is finite and what is not further down the line?

I always have fun reading your stuff Rockman, sometimes I can wrap my head around it, sometimes not.

Someone said before, but...

The answer is 42.

philosophy, not math. Math says that it does have a numerical representation, just like pi, 2/3rds and sqrt(-1) :P

42

You didn't have to be a fluff about it.

My question was whether or not the number can ever actually be written out, and that is what the debate in this thread has been about.

I lean toward the belief that it cannot be written out, not in a manner that humans can verify is complete and correct. Computing power at some point may be able to calculate the exact value of this number and have sufficient memory to store this number and be able to calculate it in a finite period of time which is not too incomparable to a human lifetime. But then the question is, how would it display it in a manner viewable to humans?

Additionally, I never questioned whether or not it was useful, merely whether there was a numerical representation (and not a representation using a combination of numbers and symbols like the one I gave) of this number.

Please provide proof of your assertion that it can never be written out.

And thank you for proving that your hatred of me allows you to embarrass yourself by trying to be a douchebag to me about it, when you apparently lack the basic reading skills to understand my initial post and the questions I was raising.





It is a representation, but it is a representation using both numbers and symbols.

Does it need to be written out? I can understand 10^100 as a number, I don't need to write it out... in fact, writing it out would make it pointlessly unclear, as you'd have to count the zeroes heh; that can be extended to numbers like the above.


Kindof like PI

Why don't we write out PI "in full (say to i dunno 1 billion digits) every time we use it in a mth text book? because we have a BETTER representation of it (the pi symbol) that is not only more accurate, but more understandable, and less wasteful /pointless

For the axiom of infinity to be valid, I would say that it does.

Without writing out a numerical representation of the number absent from symbols or other abbreviations, determining the order of various extremely large numbers might be impossible. But if the natural numbers are an ordered set, and every natural number has a successor, then we must be able to identify successors to all numbers, and to take any set of numbers and place it in order.

We have easy algorithms for taking natural numbers and either identifying a successor, or putting them in order, but this algorithm is dependent upon using the numerical representation of that number.

If I took two strings of 1000 4 digit numbers, and in each case, formed the nested string where if (a,b,c,d ... ) was the string, and for each formed the natural number represented by a^(b^(c^(d^( ... etc., and asked you to identify which number was smaller, would that task be possible?

If it is not possible, that would mean that we've identified a set of two natural numbers for which the Well Ordering Principle is not demonstrable.

In the opinion of some mathematicians, the Well Ordering Principle would still hold, it would just be undemonstrable. In the opinion of an intuitionist, the inability to demonstrate which of the two numbers is smaller means that the Well Ordering Principle did not hold, because it is impossible to identify the smallest element of the set.

usually you can use a few tricks to determine which is smallest :)

Yes, but the Well-Ordering Principle doesn't say that a non-empty set of natural numbers 'usually' has a smallest element :)

What you want to say is, I believe, that if something exists in theory that doesn't really mean that it has a practical usefullness. If the theory undoubtly proves that something works, it will work in practice just as theorised. If you bother to try.

PS: Communism doesn't "work". Neither in theory nor in practice. Not even Marx or Lenin or Stalin claimed that. What they said is that everyother society "will/should/we-will-help-you-if-you-can't-evolve-as-profeciesed" and naturally become extinct and eventually converge to communism. A society where the means of production are based on collective property, end products are divided equally among the commune members, social classes are nonexisting and the "state" loses its meaning. Nirvana. They even invented a scientific background to "prove" that theory, called "dialectical and historical materialism".

Thing is that, according to the same marxit-leninist-stalinist doctrine, ironically, such a society already existed: they called it "the primitive commune". Homo neanderthalensis lived in it. So I guess, yeah, communism can work afterall. Badly, but works. In other words, we're fluffed. Dialectically and historically and scientific socialism. According to the theory, which you said it "works" :P

i don't see anything that demonstrates that it's not possible for that number to be written down. might take some time though.

This isn't a Mathematics question - it's just a philosophical question in guise.

Why, don't we scrap the numbers and question existence itself? How do you know there exists a black-hole? Have you ever seen one? No, but you're using it in your argument - comparing it to a number you've never seen "written out".

wouldn't happen to have an equation that might state how many digits i'd need to store to be able to print it out, would you?

It is a mathematics question. The philosophy of mathematics is a field in both mathematics and philosophy.

We know a black hole exists because we have seen the gravitational force it exerts on nearby physical objects, as well as the bending of light that passes near a black hole. What we call a black hole is that area which exerts the gravitational force. I have not physically seen a black hole myself, but there are presumably scientists who have looked through a telescope at the area where a black hole exists and seen the absence of light at that place in space, and the point in space with the absence of light is what we call the black hole.

So just as being in a cave with all the lights out allows you to have 'seen total darkness', so have scientists seen black holes.

Black holes exist because they are a name given to a phenomena that has been observed.

The answer is "27" I know because I read it in the "Hitchhiker's Guide to the Universe".

Cerberus of the MI

a black hole is something that can infinitely digest the mass of things around it without exploding in a burst of fusion and hence producing light? would that mean that gravity isn't actually based on mass? oh well, guess i should figure out how many pieces of paper i'll need.

5332 raised to the power of
4726 raised to the power of
9402 raised to the power of
6681 raised to the power of 8390

so, i have 12 raised to the power of 2.
equals 24. number of digits is same as the base, but 1 bigger than the exponent.

so, i have 12 raised to the power of 12.
equals 144. number of digits is 1 bigger than the base and the exponent.

so, i have 12 raised to the power of one-hundred eleventy.
where's my calculator...
6.1534448313282326574929132187923e+119

lotta farking digits. don't think windows calculator is going to cut it.

should probably look up the meaning of exponent and base, and call martian a bunch of names just for the fluff of it.

just because you cant write it down does not mean it does not exist.

maths is pure, phylosophy is fluffy

To add a different slant to this question , i googled a bit of info the other day . you know how the bigwigs think the universe is 20 odd billion years old . and you know how they think it all spread out from the original big firecracker , these same bunch of thinkers tell us that nothing can move faster than light dont they ? if thats the case , how come the universe is over 150 billion light years in diameter . last i checked my maths thats distance of 75 odd billion light years from the middle so , foolish me is thinking , if they arent making mistakes in lots of places that means the bits that are 75 billion light years from where they where 20 billion light years ago , moved at about 3 and 3/4's times the speed of light to get there .

anyone care to correct my assumptions please .

One of your incorrect assumptions is that the speed of light is constant.

Another incorrect assumption is that your information that the universe is 150 billion light years in diameter is incorrect. The universe is not a sphere, even when the bending of space due to gravity is accounted for. The phrase 'diameter' has no meaning when referring to the universe. The universe is like the surface of a balloon, where everything is expanding and getting farther away from each other. But where the surface of a balloon is 2 dimensional, the surface of the universe is 3 dimensional, and scientists have figured out from the relative movements of different stars and galaxies that the universe is not the most basic 4 dimensional circle, but they are unsure what it actually is.

If you are on the surface of a balloon, there is no center of the balloon on the surface of that balloon. Similarly, there is no middle of the universe.

@arthog: It is space between the objects that is stretching.

hehe. do they have any idea what the balloon is filling up with to make it expand? :-P

Its not filling up with anything. The big bang threw the galaxies away from each other with such violence that the gravitational force was insufficient to prevent them from moving away from each other. With more force pushing them apart than pulling them together, the push for expansion comes from the surface of the balloon itself, not from the interior of the balloon.

not to be an ass dibs... but your math was wrong from the point you said 12^2 = 24.

and I'd rather enjoy this conversation if there were more mathematicians around... but I'll still add my 2 cents.

Your logic, Rockman, from the first post...



Impossible for a human, currently. Yes, it is a bit pretentious to say that we will evolve to the point that we or some alien race (that we will inevitably encounter; re: Stephen Hawking) will be capable of computing such calculations and assigning some purpose to the output of that statement. This is only because, by your logic, if we were a simple human race capable only of comprehension and identification of numbers up to let's say 5 decimal places (so that a number like 59500 "existed", but not 595005) and created simple computer systems in binary based upon our already limited use of our numbers and numerical representations, we'd be able to discount the existence of any number beyond that value. As it stands, in the 'natural universe' there are applications to a seemingly infinite amount of numbers (re: e, which works quite effectively for this argument, but others as well, natural logarithms, etc) and we can currently see that were we simpler, we'd be incorrect in assuming that to be reality. Just as we'd be unfathomably in the wrong to conclude that the axiom of infinity doesn't hold simply because we can't 'see' those numbers or apply them to a known natural event.




To state that all that we know now is all that we'll ever know is simplistic and in effect the opposite train of thought (required) to philosophize and examine theories of numbers.

Its not stating that its all we'll ever know, but its stating that all we know is all that we can consider to be true. We cannot consider something to be true merely because there is a chance we may at some point have the knowledge to know that it is true.

Then your argument, as others have stated lies more in philosophy than in mathematics. Yes, we can both assert that the two are more intertwined than not, but for the sake of this discussion, it must be recognized that in mathematics, very little is stated as 'truth' or 'fact' or 'axiom' for that matter... Every axiom our textbooks ever discussed was always prefaced with the knowledge that we could be wrong, but to philosophize, we'd need to accept them as true. One of mine made 0 a positive integer, another made 0 a non-negative integer. One of mine defined 0 beforehand, another had 0 defined through a series of proofs with more basic axioms.

So for philosophy, change the title of the thread.