Thread: Does the real number line contain "dark matter"?

Consider the set of those real numbers defined by the property that no specific example of one of the numbers in the set can be cited, even in principle. (And which therefore I refer to as the "dark matter" of the real number line, inasmuch as such a number could never be pointed to by any process, and is therefore "invisible" to mathematics.) Do such numbers exist? Where? How many are there?

We start by considering how one would point to a specific rational number. This poses no conceptual difficulty. You simply specify 2 integers the ratio of which is the number, or you specify the number in decimal format, which is either terminal or which enters a state of periodic repetition after a certain point.
The indication of a specific irrational number, however, is another matter. By definition,
irrationals cannot be expressed as a ratio of integers. They can be expressed decimally,
but the laws of division preclude the digits from occurring in a periodic pattern.
Of course, the digits could occur in a non-periodic pattern, in which case the
unambiguous indication of a specific number is possible by specifying the decimal pattern, but this process will typically be less convenient than the specification of a rational number. Unfortunately, most of the
irrationals we deal with do not have digits that occur in any recognizable pattern.
From the standpoint of decimal format, that leaves us without any general way of indicating
which irrational number we mean. However, we are not without resources: in such
cases, the normal thing to to is to give an expression which has the value of the
desired number, or to give a commonly recognized symbol for it. Thus, pi or e can be unambiguously indicated by mathematical expressions, without resorting to decimal evaluation.

The roots of difficult equations are more difficult to deal with. They may or may not be
expressable in terms of any finite combination of common mathematical operations and
quantities. The cases in which they are not so expressable, that we typically encounter, tend to involve
more complex ideas such as nonterminal summations, or integrations.
How do we indicate a numerical solution of a problem like this? The decimal evaluation of it is possible in
principle, as by solving the problem by numerical iteration, but gives us "the" answer only if all of the digits can be had.
Inasmuch as it is not possible to write out all the digits of an irrational number, nor
to indicate them at all using character-string methods if they do not occur accurding to some recognizable pattern (as is usually the case), the only way to indicate the answer is to give an algorithm for its generation.
Such an algorithm would appear of necessity to be, in general, possibly very complex,
the degree of complexity likely being related to its number class, such as the statement that the number is "algebraic" or
"transcedental". A complicating issue, of course, is whether an algorithm we come up
with for describing a given number is the only algorithm possible. We know of some cases for which
it clearly isn't, and some of the more difficult irrational numbers we come in contact
with can sometimes be expressed by a suprizingly simple algorithm. E.g.,
= 4 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 .......
(on the basis of which we might be inclined to conclude that is quite simple to deal with, until we observe that this series, considered as a problem in fraction addition, is perverse in that there is an extreme lack of commonness in the denominators, involving as they do every possible prime number other than 2).

In view of these difficulties associated with expressing the exact solution of some peculiar problem such as may not lend itself to normal algebraic solution; an interesting question is, in what language might we, in general, express an
arbitrary irrational number -- what kind of algorithms for doing so are possible, how we would find an algorithm in a given case, and how one might decide which of several possible algorithms was the
"best" algorithm to express that specific number.

The problem then is this: Given an arbitrary irrational number, find an expression (and preferably the best expression)
whereby it may be unambiguously identified for subsequent use.

To proceed, we must first obtain an arbitrary irrational number. "Arbitrary" means that it
cannot be confined to a preconcieved set, such as the set of all square roots of
integers, or of all roots of trigonometric equations. Nor of anything else, for that matter.

How can we select an arbitrary irrational number so that we can work out an
expression whereby it may be specified? Well, if we were talking about integers, we
could select an arbitrary integer (at least within specified limits) by selecting it
at random. Indeed, it is hard to think of any other way of getting an arbitrary integer.
So we will select our irrational number at random, and then try to work out an expression whereby it may be identified for use in a problem.

How does one select an irrational number randomly? To select it truly randomly, all
possible irrational numbers must be eligible. Inasmuch as there is no limit to the possible size of these, and because it is hard to say what it means to choose, "at random", a point on a line
of infinite length, we can perform a mapping whereby a function maps, 1 to
1, each element of a finite segment of the number line into the whole number line, such
as tan(x), which maps the interval -/2 through /2 into the entire number line, and
then choose a point randomly from the finite interval to select one in the infinite
interval. Of course, this results in a process where the probability density is not
uniform, and extreme numbers have a very low probability of being selected.
A fact which, in any case, will prove irrelevant to the present discussion. Let us
suppose that we proceed with such a scheme. First we need a process for selecting randomly the
initial number in the finite interval. If a number is selected at random from a
finite interval, then the probability of it falling within any of n equal divisions of
that interval is 1/n . The probability of it falling within n equal subdivisions of
that interval is 1/n² . The probability
of it falling then in a yet finer n-segment division of that is 1/n³ .
Etc. Each of these levels represents rolling an n-sided die to determine the position
in the subsequently finer division.
As to the mapping from the finite interval to the infinite, at some point the division
of intervals will become fine enough that this mapping is effectively linear with
respect to all dividing processes beyond a certain point. Because that is so, beyond a
certain level of division, the mapped process is effectively identical in character to
the unmapped, and therefore, as far as studying this process in the present application is concerned, we need not
concern ourselves at all with exactly how the selection process starts, and therefore, for purposes of our present discussion, may dispense with the mapping altogether. We may replace it by saying that by "some
means" a finite interval of the number line in which to start is selected, and that we
select our arbitrary number by sequentially dividing that interval into n equal segments,
selecting one of them at random, and within it repeating the process again, ad infinitum.
This is equivalent to selecting our number by a process of choosing each digit beyond
some point by a random toss of a 10-sided die. That will give us an "arbitrary" number
-- actually we are selecting a real (not necessarily irrational) number; however the odds
strongly favor irrationality.

Now that we have described how to choose a (presumably irrational) number at random, how
do we communicate exactly which number this is, once we have selected it? Well, we cannot actually write the
digits, even in principle, because there is no end to them, and because (by definition
of random toss) there is no identifiable pattern to them whereby we could determine, in principle, on the basis of the digits found up to a certain point, what the following digits would be. How, then, can we write an
expression to compute them? A random generation like this yields a number about whose
precise character we have no information, nor can we access the number directly by
reason of the nonterminality of its digits. Nevertheless, the number exists, because the process of
generation narrows the uncertainty in value arbitrarily close to zero, showing that the number
must exist and must be unique. So the number exists, yet we have no way of describing it.

Leading to a remarkable conclusion:
The set of real numbers contains elements that cannot be described, i.e. identified, by any process whatever.

Of course that does not mean that we cannot study such numbers collectively as a class,
such as is here being done, but we cannot ever pull one out and give it as an explicit example, nor feed it as an
operand into any process. We have no way of "seeing" a specific such number. We might
say, then, that the real number line is filled with "dark matter".

How much such "matter" is in the number line? Well, because the process of choosing a
real number at random can begin in any finite segment of the line, it is clear that such
numbers can be found virtually everywhere in it -- choose any point, and there exists a dark number arbitrarily close to it. The stuff is very prevalent.

One conceptual problem, however, remains not fully addressed, and that is the assumption
that we can, in principle, do successive subdivisions of a line segment using truly
random processes. This problem has to do with the fact that, unlike most lotteries, this one requires a sequence of random events that is truly nonterminal. That runs into some
problems with that theorem, related to the laws of division, which states that the
output of a finite-state machine must, at some point, either stop or become periodic.
What is the nature of the "machine" that does the die tossing? If we envision it as some
kind of a physical machine, one runs into the problem that the number of possible
quantum states in the universe may be finite. If that is so, then everything in the
universe must eventually repeat or stop altogether, ruining the process. However, I
suspect that this argument is ultimately of no relevance, because it confuses physics
with mathematics. Mathematics does not ask what physical processes are possible, but what
symbolic manipulations are conceptually possible. I know of no proof that a nonterminal sequence of random events is inherently untenable in a mathematical sense.

If this is so, then the finding that the real number line contains a plethora of numbers that have not been and can never be, even in principle, identified, stands.

Originally Posted by Atomic-S

Consider the set of those real numbers defined by the property that no specific example of one of the numbers in the set can be cited, even in principle. (And which therefore I refer to as the "dark matter" of the real number line, inasmuch as such a number could never be pointed to by any process, and is therefore "invisible" to mathematics.) Do such numbers exist? Where? How many are there?

I don't think you've shown that such "dark matter" exists. All you've shown is that it is impossible to describe every real number--I don't think you've shown that any one of those real numbers is impossible to describe. There is a difference!

The reason that it is impossible to describe every real number is much simpler. We know that the set of real numbers is uncountably infinite, but the set of (human-readable) descriptions is finite or, at most, countably infinite. Therefore, we could never describe more than a countable number of reals.

The indication of a specific irrational number, however, is another matter. By definition,
irrationals cannot be expressed as a ratio of integers. They can be expressed decimally,
but the laws of division preclude the digits from occurring in a periodic pattern.
Of course, the digits could occur in a non-periodic pattern, in which case the
unambiguous indication of a specific number is possible by specifying the decimal pattern, but this process will typically be less convenient than the specification of a rational number.

Louiville's number (0.11000100000000000000000100...) which has a one in each n! position, but zeroes elsewhere, is transcendental, an admittedly simple case, but not inconvenient.

How can we select an arbitrary irrational number so that we can work out an
expression whereby it may be specified?

::snip::

Now that we have described how to choose a (presumably irrational) number at random, how
do we communicate exactly which number this is, once we have selected it? Well, we cannot actually write the
digits, even in principle, because there is no end to them, and because (by definition
of random toss) there is no identifiable pattern to them whereby we could determine, in principle, on the basis of the digits found up to a certain point, what the following digits would be. How, then, can we write an
expression to compute them? A random generation like this yields a number about whose
precise character we have no information, nor can we access the number directly by
reason of the nonterminality of its digits. Nevertheless, the number exists, because the process of
generation narrows the uncertainty in value arbitrarily close to zero, showing that the number
must exist and must be unique. So the number exists, yet we have no way of describing it.

No, you cannot say this. For the same reason that you cannot write down all the digits (because of "nonterminality"). This process never specifies a unique number.

Besides, as you've noted, this process could converge on a rational number--which, in your terms, is easy to describe.

How much such "matter" is in the number line? Well, because the process of choosing a
real number at random can begin in any finite segment of the line, it is clear that such
numbers can be found virtually everywhere in it -- choose any point, and there exists a dark number arbitrarily close to it. The stuff is very prevalent.

Choose any point, and there is a rational number arbitrarily close to it.

If this is so, then the finding that the real number line contains a plethora of numbers that have not been and can never be, even in principle, identified, stands.

As I've said, you've not proven that there is any number which cannot be identified. All you have shown, is that we haven't described all of them individually.

By the way, it is easy to show that the set of all rational numbers that we can describe is smaller than the set of irrational transendental numbers we can describe.

The reason that it is impossible to describe every real number is much simpler. We know that the set of real numbers is uncountably infinite, but the set of (human-readable) descriptions is finite or, at most, countably infinite. Therefore, we could never describe more than a countable number of reals.

True, but beside the point. The point is not whether we can or cannot actually write a book contining the description of every real number, but whether there might be some real numbers that we would not be able to describe even one of.

Louiville's number (0.11000100000000000000000100...) which has a one in each n! position, but zeroes elsewhere, is transcendental, an admittedly simple case, but not inconvenient.

An interesting case. I stand corrrected.

No, you cannot say this. For the same reason that you cannot write down all the digits (because of "nonterminality"). This process never specifies a unique number.

You are correct that the process never specifies a unique number, in the sense that the process never can be completed in practice. What the process is, mathematically speaking, is a series of fractions whose denominators are powers of ten. We cannot form the sum if we do not know what the numerators are; however, if the numerators fall between fixed limits, which they do (0 through 9), then it is easy to prove that such a series is convergent no matter what the numerators are. Inasmuch as the series is convergent no matter what the numerators are, we are assured of the existence of its sum, even if we do not know what the sum is. We must not confuse finding the value of a quantity with proving its existence. A quantity may well exist even if we are unable to compute its value.

Originally Posted by Atomic-S

You are correct that the process never specifies a unique number, in the sense that the process never can be completed in practice. What the process is, mathematically speaking, is a series of fractions whose denominators are powers of ten. We cannot form the sum if we do not know what the numerators are; however, if the numerators fall between fixed limits, which they do (0 through 9), then it is easy to prove that such a series is convergent no matter what the numerators are. Inasmuch as the series is convergent no matter what the numerators are, we are assured of the existence of its sum, even if we do not know what the sum is. We must not confuse finding the value of a quantity with proving its existence. A quantity may well exist even if we are unable to compute its value.

That's not the biggest objection to that process, though. Even worse is that you have made no provision that it not be rational. Or worser, that it might be an easily described irrational, or even a transcendental (see my comment that there is at least as many easily described transcendentals as there are rationals (which you've taken to be easily described).

Still, as I said, there is an easier way!

That's not the biggest objection to that process, though. Even worse is that you have made no provision that it not be rational. Or worser, that it might be an easily described irrational, or even a transcendental

OK, there is the possibility that a real number formed by selecting digits at random could happen to turn out to be a definable irrational number, or a rational number, or even an integer. I had argued that because of the random process, we can not know definitely what the number is. But as you point out, it might accidently happen to be something that is describable, in which case it is not a "dark" number. But must it be describable? Does not the possibility exist that it is not? My first thought would be that it is much more likely to be indescribable than describable, based simply on chance (it is more probable that moveable type spilled on the floor will be unintelligible than that it will compose a sentence). Unfortunately, this analogy breaks down in that the number of possible descriptive algorithms is also infinite, and one might therefore make the argument that there exists a correspondence that can be set up between all algorithms and all real numbers, thus eliminating the possibility that any numbers are "dark". However, it would seem hard to prove that proposition. If it is so, one would start looking for a methodology whereby this correspondence could be made. How indeed does one find, for any real number whatever, the corresponding algorithm whereby it may be described?

Originally Posted by Atomic-S

OK, there is the possibility that a real number formed by selecting digits at random could happen to turn out to be a definable irrational number, or a rational number, or even an integer. I had argued that because of the random process, we can not know definitely what the number is. But as you point out, it might accidently happen to be something that is describable, in which case it is not a "dark" number. But must it be describable? Does not the possibility exist that it is not?

Well, of course! It's a possibility, if such "dark matter" exists, but that is exactly what you are trying to prove!

I think your "dark matter" numbers seem to be what we call "computable numbers", no? Computable numbers wiki

My first thought would be that it is much more likely to be indescribable than describable, based simply on chance (it is more probable that moveable type spilled on the floor will be unintelligible than that it will compose a sentence). Unfortunately, this analogy breaks down in that the number of possible descriptive algorithms is also infinite, and one might therefore make the argument that there exists a correspondence that can be set up between all algorithms and all real numbers, thus eliminating the possibility that any numbers are "dark". However, it would seem hard to prove that proposition. If it is so, one would start looking for a methodology whereby this correspondence could be made. How indeed does one find, for any real number whatever, the corresponding algorithm whereby it may be described?

I think your "dark matter" numbers seem to be what we call "computable numbers", no? Computable numbers wiki

No, the computable numbers would correspond to the light numbers, or at least a subset of them. The dark ones are what is left over.
Concerning which, please check out the following reference. This reference seems to be saying basically what I have been saying. It refers to non-definable numbers that strongly resemble what I have called "dark matter".

Definable real number - Wikipedia, the free encyclopedia

Originally Posted by Atomic-S

No, the computable numbers would correspond to the light numbers, or at least a subset of them. The dark ones are what is left over.

I stand corrected! I meant "noncomputable numbers"