Are Big Numbers Just a Concept of the Mind or Do They Truly Exist?

[gonzo]

I'm not convinced that big numbers really exist. To convey or represent information requires energy, so if we assume a finite amount of energy in the universe, then that means that the universe can only contain a finite amount of information.

There are other arguments based on the Hubble volume of the universe and how many states it can contain (by calculating the number of particles that can be crammed into that space, that calculation, discussed in a scientific american article was 10^118).

In any case, that means if you pick a sufficiently large number, then you can say that all the number from 1 to that number can not be represented or expressed in this universe. And if there is no way to even theoretically express a range of numbers, how can you say they exist?

 

[A_I_]

do you mean by your post that infinity doesn't exist??

 

[gonzo]

No, not at all. That has nothing to do with it.

What I mean is, pick a big integer, we can say 10^10000 for example. I don't think all integers from 1 to this number exist because they can not even theoretically be expressed.

 

[A_I_]

what you're saying, does it change in the rules that we consider nowadays?

 

[gonzo]

Sorry, your question doesn't make any sense to me. Can you rephrase it?

 

[A_I_]

i mean that if your statement is verified, will it change something in the theories we know.. in maths concepts..??
(i don't know if i made myself clearer :$)

 

[gonzo]

Probably not, at least not directly. Which is why I posted it in the philosophy section. However, it could lead to interesting implications ... the idea that there is a finite amount of information that can be expressed in our universe.

There are more pratical spin offs of the idea that I find strange to think about. For example, assuming your head doesn't keep getting bigger, and assuming that everything you "see" can be expressed as a "brain state" (mostly, but not necessarily limited) to your visual cortex, you get a weird idea. Your head is finite in size, and so there is should be a limited number of states possible inside your head. This means that there is a limit to what you can "see". Which means that if you lived for an incredibly long time and traveled all over the universe, unless you increased your brain compacity somehow, you would eventually see everything you could see ... in other words, new images would look exactly like an old image to you, you would not be able to see the difference, even if they were different.

One can assume given the size of the universe relative to your brain, that there should be far more visual patterns out there than you have possible brain states.

To me it's a weird thought, not necessarily practical in any way, which again is why I posted it in the philsophy section.

 

[A_I_]

while i started to read ur post.. an interesting idea got up to my mind..
well u said that integers from 1 to 10^10000 do not exist..
i guess they do exist but human beings brain is not yet so developed to understand such hard complications..
and then, who told you..
one can spend his lifetime counting from 1 to 100^10000 on sheets of papers..
it can be.. but no one will do it though
my idead opposes yours

 

[gonzo]

But that's the point, it can't be done, not with sheets of paper, not with computers, not with anything even theoretically possible as far as we understand it. The universe can not contain that much information. That's my whole point.

 

[A_I_]

i just said.. that it contains it..
but "we" "human beings" we still don't have enough powers or enough intelligence
to comprehend everything..
don't forget..
we are and we will always be a stain in a huge and wide ocean..

 

[HallsofIvy]

The problem is that gonzo never told us what he meant by "exist". Unless he is using "exist" is some very unusual limited way, like "someone has actually written it down on paper", all numbers exist in the same sense as all ideas exist. Since this is about, presumably, the philosophy of mathematics would say that some number that no has ever even referred to (and you don't have to go to large numbers- there are plenty of numbers no one has ever specifically referred to between 0 and 1!) exists in exactly the same sense that "one" or "two" exist.

 

[Ba]

What I mean is, pick a big integer, we can say 10^10000 for example. I don't think all integers from 1 to this number exist because they can not even theoretically be expressed.

For a number this size, 10^10000 can be expressed in distances for instance 1 meter is 10^9 nanometers c is 3E8 meters, so then there are (10^9 nanometers per meter)x(3x10^8 meters per second)=30x10^18 nanometers per second is the speed of light in a vacuum and the units can go smaller than this. 3.1536x10^7 seconds in a year. There are stars millions of light years away. Even if they are not 10^10000 nanometers away, they are a small part and light can travel back and forth, sooner or later it will reach this distance and every measurable distance in between.

 

[A_I_]

u r talking about the distance..
it is actually a point of view.
can anyone talk about time? ;)

 

[hypnagogue]

I think your use of the word 'exist' is a little fuzzy. What does it mean for a number to exist? Certainly, assuming that the universe is finite, there is some limit to the amount of numbers that can be implemented / represented physically. (Actually, we would probably also have to stipulate that all physical quantities, including time and space, are quantized. Otherwise, for example, for any number n, we could consider any region of space to be composed of n units of volume.)

But it seems that the notion of number is sufficiently abstract that to speak of a number's existence, one needn't refer to physically possible implementations of that number. To take your example, it may be that 10^10000 can't be physically expressed by a counting procedure, but it certainly can be theoretically expressed-- after all, I have just theoretically expressed it using only 8 characters. In the sense that I can speak of it at all, it seems to enjoy some kind of existence, even if it's an abstract one.

One can assume given the size of the universe relative to your brain, that there should be far more visual patterns out there than you have possible brain states.

This is an interesting idea. More relevant than the size of the universe, though, would be the number of retinal images that could impinge upon the retinas during the minimum salient duration of time needed to form a visual image. This gets a bit fuzzy, though.

On the one hand, you probably have significant information loss since the brain probably treats large classes of retinal images as identical. (For example, if you modify your current retinal image by adding a few photons of any visible frequencies, anywhere on the retina, you won't notice a difference. Thus for any given perceived visual images, we can already conclude that there are a large number of slight variations of the retinal image that are perceived identically.) On the other hand, you also have higher order gestalt visual processing which could interpret identical retinal images in a number of different ways. (For example, an ink blot literally looks different once you have decided that it resembles, say, a cow, even though the stimulus has not changed.)

Just for fun, let's estimate a rough upper bound for the number of possible visual conscious states possible in a given time period. One way to tackle this would be to roughly divide the visual field into pixels of minimum visual acuity and multiply the number of pixels by the number of noticeably different colors. But this method seems overly simple, as it neglects important higher order effects like perceptual grouping, foveal vs. peripheral vision, etc.

Another way to do it would be to count brain states, as you suggest. Assuming:
1) Conscious processes are supervenient upon brain processes;
2) A sufficient level of granularity to describe this supervenience is determined by the firing speed of individual neurons;
3) We can't a priori count out any part of the brain as having a potential impact on vision (although the visual cortex is most associated with vision, it seems to be influenced significantly by interaction with at least motor capabilities and higher order thought, and probably other many other capabilities supported in various brain regions);
4) The average human brain contains roughly 10^11 neurons (http://faculty.washington.edu/chudler/facts.html);
5) The maximum firing rate of neurons is 250 Hz (http://64.233.161.104/search?q=cache:zTmPOk-t3hsJ:oxide.eng.uci.edu/publications/C1F05.pdf+%22maximum+neuron+firing+rate%22&hl=en )
6) Neuron firing rates only make a difference to conscious experience in discrete steps of 1 Hz. (There probably is some minimal difference beyond which neural firing rates make identical contributions to consciousness, given that at least some aspects of consciousness, like visual consciousness, apparently comes in discrete steps (eg http://www.nybooks.com/articles/17030), and it's probably lower than 1 Hz-- but our overestimate is going to be large enough as is, and assuming 1 Hz will simplify calculations.)

So if we have 10^11 neurons, each of which can assume 250 causally significant 'states' (firing rates) with respect to consciousness, then there are 250^(10^11) different ways to construct brain processes that are causally distinct with respect to consciousness in the span of one second. Of course, the vast majority of these permutations probably don't result in consciousness at all (ie when all neurons are cycling at 1 Hz), and in principle most of them are also not physically realistic since the cycle of any given neuron will be a function of the cycles of its dendritic neighbors, whereas here I assumed that they are completely independent. But there you have it: we can be reasonably certain that the number of ways to experience consciousness in the span of one second is less than 250^100,000,000,000.

 

[Gokul43201]

To show how your (gonzo) definition of the existence of a number is very selective, and that there do exist "numbers" bigger than any given (real) number, here's a counterexample.

Let the total number of states be N. Then the cardinality of the powerset of this set of states is 2^N, which is bigger than N. This can be extended indefinitely.

 

[Jared Prince]

In fact big numbers such as 10^10000 place limits on our normal reality all the time. For instance, cryptography is largely based on the fact that the space of possible keys is so large that no one can search it. One Time Pad keys can easy have 10^10000 posssible forms. If only small numbers existed you could easily search all the small ones and find the key. And it wouldn't stop at cryptography. Related to the brainstate/retina comment, imagine searching through every possible 100x100 24 bit image. Most of it's just noisy snow, but it also contains a low res version of every image a human could ever see. But then the large number 16777216^10000 steps in and prevents us looking. Even the space of a stark black and white image 10 pixels by 10 pixels (2^100) is too large to search. The same thing happens with potential songs, books and computer programs. If the number of potential patterns wasn't so huge (i.e. if large numbers didn't exist) we could just find all the good ones by scanning every possibility.

On the other hand numbers like 10^10000 don't present any problem when in comes to manipulating them. For instance, I can say that if I divide that number by 10 I get 10^9999. Log math is pretty easy. Does counting up to 10^10000 by lots of 10^9999 sound ok? It's not necessary to count by single units, it can make more sense to talk about bags of rice instead of individual grains, without losing accuracy.

(I made a correction here. I originally said counting by lots of 1^10000. Unfortunately 1^10000=1. Oops. And after me saying log math was easy...)

By the way, in a more general sense of existence, whether any mathematical concept (even just 1 or 3 or 1/2) exists is a hot philosophical topic. Mathematical Platonism says that they do, abstractly and independent of physical existence. Mathematical Formalism says math is squiggley lines on paper (or computer) that we manipulate according to rules. Cultural Relevatism says math is just another cultural concept like Satan or Shiva or The Simpsons. I think there is some value to all these points of view.

 

[ill rich]

that number will eventually exists, in terms of age of the universe

 

[balkan]

i just said.. that it contains it..
but "we" "human beings" we still don't have enough powers or enough intelligence
to comprehend everything..

or enough paper
... but of course the number theoretically exists:
10^10000 <--- there you have it. theoretically... i can even divide it by two:
5*10^9999
the issue is, of course, capacity. what if it was written in atoms? would it fit into space then? if not, then the universe cannot contain this number, if yes, then do'h...
it doesn't seem plausible to me that there arent
10^10000 / 6*10^23 moles of atoms in this univers...

another proof:
kryptographers have discovered a prime number with over 7 million digits... that far surpasses 10^10000...

 

[selfAdjoint]

The representation of a number (say in decimal digits) is not the same as the number.

 

[kreil]

All we need for a number to exist is a representation of it. In theory, every single number imaginable can be expressed, because whenever a system starts to fail in representing numbers, we define a new system.

For example, 1.0 x 10^3 works nicely for representing fairly small large numbers...but
1.0 x 10^ 100000000000000000000000000000000000000000000000000000000...continuing out to 1 million zeroes would be annoying to write down. So what do we do? we define a variable, z, to =1.0 x 10^10000...out to 1 million zeroes. Suddenly, this number is z, not a big long expression. And once z x 10^1000000...out to a million zeroes becomes too hard to handle, we define z' to equal this number...in this manner any number, no matter how large, can be represented.

 

[StatusX]

All we need for a number to exist is a representation of it. In theory, every single number imaginable can be expressed, because whenever a system starts to fail in representing numbers, we define a new system.

For example, 1.0 x 10^3 works nicely for representing fairly small large numbers...but
1.0 x 10^ 100000000000000000000000000000000000000000000000000000000...continuing out to 1 million zeroes would be annoying to write down. So what do we do? we define a variable, z, to =1.0 x 10^10000...out to 1 million zeroes. Suddenly, this number is z, not a big long expression. And once z x 10^1000000...out to a million zeroes becomes too hard to handle, we define z' to equal this number...in this manner any number, no matter how large, can be represented.

It's interesting you bring that up, because I was about to use that argument to show the opposite. Can you imagine a number so big that, no matter what system you use, you could never represent it, not even with all the atoms in the universe. There obviously must be such a number, because, assuming space is quantized, there are only a finite number of states the universe can be in. Therefore, there are only a finite number of numbers that can be represented in it. Try to imagine how big this number is, its disgusting. So, does this number really exist?

 

[kreil]

It's a good point, but at the same time hard for me to comprehend. I'm not sure whether such a number does exist, because the process I described could be repeated over and over and over...however your argument makes sense. This is quite the conundrum.

 

[StatusX]

Before it could be judged true or false, I'd have to be more specific. Say there is only a certain finite number of bits of information that can be encoded in the universe at a given instant of time. This is a reasonable assumption. Now pretend there is another universe, much larger than ours, in which people much smarter than us can take a snapshot of our universe, analyze this data and use it to compute the number. Assume we have some common mathematical language between us. Obviously there is a largest number that can we can encode in such a sense. This number is pretty big. No mathematical operations could be performed on such a number, even in principle.

 

[Jared Prince]

There might be three flaws in that approach.
First, most obvious, is that there's no reason to believe the universe is finite in size. It might be, but it might not. The visible part to us is finite, but that's different.

Second, slightly weirder problem, is one of interpretation. Let's say the universe is finite and can contain a maximum of two bits. To the outside intelligence the universe has 4 states, 00, 01, 10, 11. What do they mean? Is that 0 to 3, or is it -2, -1, 0, 1, with the first bit used to show positive or negative, like on computers. Or is it some other representation, maybe -1, 0, 1 and infinity? Or imaginary numbers?

Third complication would be if the universe is seen as qubits instead of bits.

 

[StatusX]

The universe may be infinite, but I think the vast majority of scientists believe it isn't. Second, you would have to use some of the space available to actually descibe the code. For example, if you were just doing straight binary, you'd have to explain, somehow, that you add the first bit to 2 times the second to four times the thrid, etc. If there is a finite amount of space, then there is a way to maximize the number you can represent by finding the right balance between space to store code and space to store the actual number. But this is still a finite number, and there is no way to represent this number plus one. Third, I don't see the copmlication qubits present, as there would still only be a finite number of them.

 

[Physics_wiz]

Sorry but I don't understand. Why would a number like 10^100000 not exist? because we can't write it on paper? Because we can't make a program that can output that many numbers? Because we don't have any representation for it or we can't comprehend it?

First, if it's because we can't write it on paper think about this:
Can you count all the atoms in the milky way on paper? Does that mean they don't exist?

Second, if it's because we don't have any representation for it or we can't comprehend it think about this:
Do you have a representation for how gravity works? Can you comprehend gravity? Does that mean gravity doesn't exist?

 

[Hurkyl]

Y'all might find the "Busy Beaver Problem" interesting...

 

[Tigron-X]

Well, isn't there really only ten numbers repeated constantly in all directions of dimensions?

And, the idea of finite space doesn't mean infinity does not exist. It only means that a certain relative value will reach a balance of zero within that defined space. Everyone knows that zero is a holding spot until a relative value is assigned.

So, the number 10^100,000 is an object with the value of one that is expressed ten times within geometric planes that only pay attention to one hundred thousand directions of an infinite spectrum. It would take forever to plot each individual point, so logic has fields or dimensions known as matrices. The more complex the system the more individual matrices are created. There is no need to count to 10^100,000 because there are patterns that specify a quantitive amount within an array of functions.

So no matter the biggest number you can't even imagine, somewhere near that number is a prime number that can be assigned a value of one for reference. Calculating prime numbers is much easier than counting by ones, twos, fives, tens, hundreds, and so on.

There's no need to display a number that size, only the path to said number because of numerical patterns consistent in every dimension.

 

[selfAdjoint]

Well, isn't there really only ten numbers repeated constantly in all directions of dimensions?

Only because we use base ten. If we used base 2, like the computers, there would only be two "numbers" (i.e.digits). If we used base 10^10000000, there would be 10^1000000 digits. We could use any whole number as a base.

 

[Tigron-X]

Only because we use base ten. If we used base 2, like the computers, there would only be two "numbers" (i.e.digits). If we used base 10^10000000, there would be 10^1000000 digits. We could use any whole number as a base.

A computer doesn't need 10 base points for reference because it can assign prime numbers a value of 1 and transfer the value to 0. It keeps the functions simple.

 

[selfAdjoint]

A computer doesn't need 10 base points for reference because it can assign prime numbers a value of 1 and transfer the value to 0. It keeps the functions simple.

This is nonsense. You need to study up on base representations AND prime numbers.

 

[ZeAsYn51]

Dear Gonzo

I agree with you but in a different way than you would want me to. I see and understand your argument, but I also see where you fall short in your examples. The brain has a failsafe mechanism that keeps at least some open memory, like a computer that is programmed to always have 10 GB of disk space on a 40 GB hard drive. The mechanism I speak of is dreams. Your brain cycles through unused information when you sleep, so that you would be very unlikely to use all of your mental capacity at anyone given time. I do agree that there is only so much a mind can hold, but arent there other ways to store memories? Photos, movies, drawings, etc. Big numbers exist, they are simply combinations of smaller numbers. 69 is just a 9 stuck onto the butt of 6. They don't have to represent anything. They don't have to be units of something. The only thing they need in order to exist is a system by which to string one number after another. That's why numbers are so beautiful, they only have to be seen in the mind or on the computer screen to exist. Nothing more, nothing less.