How to understand numbers that become really large?

Question

If we begin with a notion of number N that we denote F(N) as a function of time, can a decidable procedure exist on definability of the growth of numbers? Inspired by Tipler's Omega point and Thomson's lamp, what would be the bound when definability cease to have meaning?

Prologue: It all started after reading The Unimaginable Mathematics of Borges' Library of Babel and the review of it here. The problem arises when one sets to catalog the books as the number of the different books become approximately 10^10^6 (yet smaller than googoolplex), justifying the term "unimaginable". Susan Stepney points out in the review that when one wants to catalogue the number of books in Library:

[...] the problem of finding a "short" description of the book to put in the catalogue: there are not enough short descriptions. For the Vast majority of the books in the Library, the shortest description (that distinguishes it from other books) is the book itself. Most books cannot be "compressed" to a short description.

And then comes the punchline:

Or,as Bloch puts it, the Library is its own catalogue.

This brings to my thought experiment:

Thought Experiment: Suppose I type a single digit '1' and then I die with my thumb 'forever' locked on '0'. Is it possible that when the number

1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000...

keeps increasing some interesting change happen to our understanding and philosophy of number system?

Recent edit: More specifically two points: 1) just like the Library becomes it's "own catalogue", if a number becomes inaccessibly large can there be self-reference leading to paradoxes? 2) what are then the implications of Poincare recurrence theorem? [latter being clarified already by Robert Munafo about non-literal meaning]

Background: This is related to my earlier question on Kunnen inconsistency in Math.SE. However, I am still having trouble grasping the behavior of large numbers even consulting a definitive website here on large numbers

I was reading a paper by Douglas Hofstadter on large numbers On Number Numbness, but again the argument veered towards philosophical interpretation.

Question: How to understand the behavior of large numbers? My motivation is from perspective of Poincare recurrence theorem a la Don Page's alternate universe count, or Skewes' number. Does logic as we know it 'break down'?

EDIT:

Here is the relevant portion from On Number Numbness that I had in mind while formulating OP:

If, perchance, you were to start dealing with numbers having millions or billions of digits, the numerals themselves (the colossal strings of digits) would cease to be visualizable, and your perceptual reality would be forced to take another leap upward in abstraction-to the number that counts the digits in the number that counts the digits in the number that counts the objects concerned. Needless to say, such third-order perceptual reality is highly abstract. Moreover, it occurs very seldom, even in mathematics. Still, you can imagine going far beyond it. Fourth- and fifth-order perceptual realities would quickly yield, in our purely abstraCt imagination, to tenth-, hundredth-, and millionth-order perceptual realities. By this time, of course, we would have lost track of the exact number of levels we had shifted, and we would be content with a mere estimate of that number (accurate to within ten percent, of course). "Oh, I'd say about two million levels of perceptual shift were involved here, give or take a couple of hundred thousand" would be a typical comment for someone dealing with such unimaginably unimaginable quantities. You can see where this is leading: to multiple levels of abstraction in talking about multiple levels of abstraction. If we were to continue our discussion just one zillisecond longer, we would find ourselves smack-dab in the middle of the theory of recursive functions and algorithmic complexity, and that would be too abstract. So let's drop the topic right here.

Relevant portion highlighted.

Answer 1

I have revised my answer somewhat, condensing in some places and adding other ideas, in response to your revision of your question.

Consider first, numbers involving one "conceptual shift" as alluded to in the quotation by Hofstadter. You may well complain that while there least can manifestly be 300 of something (e.g. markings of hollow circles), the best of our knowledge there can never be measurably 10³⁰⁰ of anything. Let's grant for the sake of argument that the latter claim is true. All this means is precisely what is implicitly taught to students of physics in secondary school: that we have mathematical models of the world which, because they are simple, fail to capture complications which are inherent in the world. Just as we contemplate perfectly rigid spherical cows falling in vacuum under the influence of a perfectly uniform gravitational field, we can concieve of a number of objects which is so large that we cannot actually imagine concretely what such a collection of objects would even look like, and which are unlikely ever to represent phenomena that we will ever encounter. The reason for both is because of the simple formulation of the models in both cases: Newtonian mechanics on the one hand, arithmetic on the other.

The notion of piling on layers of conceptual shifts, as suggested by Michael Dorfman in the comments to his own answer, is akin to Knuth's up-arrow notation. But the crux of this, and even of our familiar Indo-Arabic numeral system, is that we only deal with numbers through representations of them (even if those representations are through visual images of objects such as apples). A googol is, in very ruggedly practical terms, unimaginably large (in that a googol of objects is not something you can really imagine), and a googolplex is unimaginably larger than that (in that it is not really possible to imagine how many boxes of a googol objects each would suffice to make a googolplex). But the fact that we can represent them by 10¹⁰² and 10¹⁰¹⁰² means we can still talk about them, and somehow conceive of the numbers abstractly.

Is being able to write such absurdly large numbers in such a way cheating — does it hide the fact that we cannot fully grok the significance of these numbers, somehow? Well yes, it does perhaps hide the fact that we don't really understand these numbers except to recite their names, to point out trite things such as that they are multiples of 2 and 5, and are prefect squares, etc. But this isn't cheating; we also understand numbers such as 300 less perfectly than we do the number 3, and use the same extension of our cognitive powers to try and come to grips with 300 by imagining three groups of ten groups of ten. Nearly all mathematics, even arithmetic, is indirect in this respect, and while some people may be able to grasp more numbers somewhat directly, we ultimately rely on highly compressed descriptions of numbers to reason about quantity. As such, we are limited in our contemplation of numbers to those which we can easily describe somehow; and we can only reason about those numbers as well as our representations allow. Multiplication was hard in ancient times for those relying on Roman numerals; and similarly our representation of a googolplex gives us little intuition as to e.g. what the next largest prime after a googolplex is.

As with Borge's library, "most" numbers don't have a simple representation; and even those that do may have superficially similar representations which make them hard to meaningfully distinguish. In fact, if a 'simple' representation has to be of at most some length, then all but a finite number of numbers are beyond all human ability to reason. Does this mean they escape logic? Well, it certainly means that we can't reason with them; but it also means that we will never have to worry (or more to the point, we are unable to worry) about their properties in any productive fashion. Again, as with the books in Borge's library, most numbers are gibberish; they have no particular importance to us.

If you suppose that 'logic' is a human concern about the structure of the world, and that reality simply 'is', then worrying about the potential illogicality of numbers which are so large that they cannot be represented in reality is to worry about a counterfactual, and so not of any importance except how much we are entertained by the question.

Answer 2

I am going to state a more precise form of this question after corresponding off-line with the OP. I hope this still captures the intent of the question:

In a perfect world, where I will never grow old or go hungry, I am watching a giant computer display screen with enough space to show trillions, or quadrillions, or even centillions of characters or digits.

I have set up the computer to show "1" for a moment, then "10" for a moment, and then "100", and then "1000", and so on. Every moment (perhaps once per second) another 0 appears. Every time a 0 appears, there is a new number being shown.

Can I watch this "forever", and perceive a new number every time a 0 is added? Or is there a limit to my ability to perceive, comprehend, or remember what I am seeing? To what extent does this limit how we as humans can understand numbers and number systems?

I believe that there is a limit to the ability of human beings to perceive, comprehend what they are seeing, and remember what they have seen.

Every time a new "0" appears, it is clearly different from what was there a moment ago. I also know that every number I am seeing is different from each of the numbers I saw before. But as time goes on, I will repeatedly experience the feeling of "what I am seeing is very large, and I have been watching a very very long time". That feeling will be more and more common as time goes on, and eventually I'll be in exactly the same mental state that I was in at some earlier time.

Suppose I try to keep count of how many zeros there are? I can train myself to remember lots of facts, things that can be written out in letters and words.

The mind can remember a lot of information. Perhaps you have enough space in your mind, that if it were all written out it would take a billion = 10^9 letters. That means you can have about 26^(10^9) distinct mental states, because there are that many different combinations of a billion letters with a 26-letter alphabet.

With my mental capacity of 26^(10^9), I am "counting" the 0's as they get displayed, and I keep track of it with my mental state. When there are 876 zeros, I have the number "876" in my mind. There are about 10^3 zeros on the huge computer screen, and 3 digits in my mind. Since I can hold "about a billion letters" in my mind, that means I can "count" the 0's until there are about 26^(10^9) zeros on the screen. Then, because of the limited capacity of my mind, I must lose count. Beyond that, any perception of exactly how big the number is, must be subjective. I will eventually have the same "really big" mental state twice. The largest number I can comprehend, without being confused that it was some other number, is less than 10^(26^(10^9)).

This is like the "Poincare Recurrence Theorem" that the OP linked to, applied to minds. It is one of the natural limits that affect how well we can think about large numbers. I am not speaking of the literal Poincare theorem, which is very precise and mathematical. I merely use it as a metaphor: if a field is of limited finite size, and you walk around in the field indefinitely, you will eventually step on a spot where you have stepped before.

In our off-line discussion, the OP suggested that we can get larger numbers by programming the computer to display 2, then 2^2, then 2^2^2, then 2^2^2^2, or (using words) it could display "zwei", then "zweizenzic", then "zweizenzizenzic", and so on (see http://en.wikipedia.org/wiki/Zenzizenzizenzic ). The screen fills up with 2's, or with the letters "zenzi". Once again, there will come a point where I am no longer able to see anything changing, or perhaps I'll see it changing but my state of mind will eventually wander back to a point where it was at some time earlier. I know it is getting bigger every moment, but even that state of knowledge will eventually recur in exactly the same form.

We can so the same thing with any mathematical notation, like g(1), g(g(1)), g(g(g(1))), ... where g(N) is the "g function" described on wikipedia's "Graham's_number" page. This time instead of squaring each time, the numbers are getting bigger in a much faster way. Perhaps I have trained myself to understand what this means. If so, I could then watch the computer screen display "g(1)", then "g(g(1))", then "g(g(g(1)))", and so on... but again, eventually my mind would reach its "Poincare recurrence".

No amount of effort using more sophisticated or elaborate notation, or methods of abstraction and understanding, will overcome the finite limit of the human mind to perceive, comprehend, and remember.

This is all very similar to what my "Superclass 6" is about, near the end of my Large Numbers discussion: http://www.mrob.com/pub/math/largenum-4.html#superclass

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EDIT: I added a simple analogy for the "Poincare" reference, and pointed out that the mathematical Poincare theorem is not relevant. This is about the concept of re-visiting the same spot in a finite space.

Added the "To what extent..." bit at the end of the restatement to try to encompass more of the original question.

Answer 3

Your thought experiment has a simple answer: It's not a number until you take your thumb off the keyboard. Up until then it is just a string of digits. The placement of the "1" (and thus its meaning) cannot be interpreted until then.

Notice that this is different than the case where you being by typing a decimal place; in that case, the ongoing series of digits serves as an approximation of the intended number, because each digit stays in its proper place.

Answer 4

Borges' intuition is formalized by the notion of nonstandard integer in Abraham Robinson's nonstandard analysis. Such integers are bigger than any naive counting number like 1,2,3, and in this sense are infinite. The more appropriate technical term is unlimited.

Such numbers are naturally useful in applications. For example, if one wishes to model a market economy with a vast number of agents, one would develop an idealizing model with a nonstandard number of agents. An analysis of the idealized model helps understand market economy. Many applications of this type are covered in the book by Loeb and Wolff:

Nonstandard analysis for the working mathematician. Second edition. Edited by Peter A. Loeb and Manfred P. H. Wolff. Springer, Dordrecht, 2015. xv+481 pp.

Answer 5

The point I want to clarify is what does Mahmud mean by Number? Does he mean the usual integers with their arithmetic properties?

ie (N,+) that is N=0,1,2,3,4,... with addition as the the operation allowed. This is the earliest arithmetic we're introduced to at school.

Later we're told that we can have (N,+,x) that is N=0,1,2,3,... with operations of addition & multiplication.

It was understood much later, or perhaps much earlier (when the decimal system was invented in India & earlier in China) that the representation of the integers do not have to be in base 10, it could be in base 3, or 123, or most commonly now, but hidden away from us in base 2. ie 0,1,10,11,100,...

That is the representation of an integer is not the integer itself.

Although, I've said (N,+) is the first number system we're introduced to, this in fact is not quite correct. Infants at six months, can distinguish very small numbers (KHH). (Their experiment carefully distinguishes a group of two apples and three apples as a naming act, to understanding 1,2 or 3). That is then they are beginning to appreciate the truncated N=0,1,2,3.

But Number is also Quantity, that is it has magnitude; when do infants acquire this knowledge?

According to (MLF) they acquire this knowledge at by the time they're four years old. (Personally, I think their methodology is faulty; they discount the infants ability to distinguish on the basis of length, which is also pure magnitude, and given the relationship between numbers & geometry - the real line - makes perfect sense. I'd suspect the ability to distinguish comes much earlier than this, an infant at less than four years can surely distinguish which he prefers - two sweets or four sweets. They don't need to formally count, they can simply see the magnitudal difference; and I don't think this should be discounted - or at the very least distinguished)

So our earliest pre-school understanding is N, and then to (N,<).

Now Cantor generalised Number in this sense - the sense of magnitude - that is he extended (N,<) to sets, and hence invented cardinals. That there is a kind of arithmatic of cardinals is a by product. So to understand large numbers we go into the so-called transinfinite realm.

But modern set theorists have invented large cardinal axioms, and they can be ordered by consistency strength. So far there isn't a generally widely accepted theory of large cardinals although Shelah speculates 'is our vision more uniform than we suspect'.

I suppose there are less than a hundred or so, arranged in order, so in a sense we're back to the beginning when at a year old we could count upto, say a hundred...(perhaps an example of Nietsches eternal return in the platonic realm).

I'm offering a different perspective to what I understand some of the posters are doing here, which is discussing compact means of representing 'large' numbers taken from the finite realm (which is a matter of notation), and what they may mean if we can't concretely express them. ie as lots of apples. Perhaps this isn't quite answering Mahmuds question on his own terms...

notes

KHH: Kobayashi T, Hiraki K, Hasegawa T. Auditory-visual intermodal matching of small numerosities in 6-month-old infants.

XA: Xu F, Arriaga RI. Number discrimination in 10-month-olds

MLF: Muldoon K, Lewis C, Francis B. Using cardinality to compare quantities

All in Numerical Knowledge in Early Childhood, Catherine Sophian, PhD

Answer 6

The human brain is finite in it's ability to compress and abstract over numbers. I have done some work with large ordinals, but even though I have good intuitions, it gets muddy out about Googols, Googolplexes, the Ackermann function applied to arguments both larger than 6, Graham's ordinal and the like. There are some ways to notationally encode incredibly large numbers in very short strings, but they don't really help, because those are some endearingly short formulas and thus confusing.

Humans are made of brains, brains are finite, there is a largest number the human mind is capable of imagining in it's extent QED.

PS. Really big numbers aren't even that interesting IMHO.