Wait aren’t we then computing them to their limit? Well their limit/end is defined by processing speed relative to advancement of time… Sow we are except, its not a static limit! But a Dynamic one!

I should make it clear that I do not have any objection to what I see as the real content of Turing’s claims but only to the manner in which they are expressed. Let me briefly show why

Consider Turing’s procedure giving an example of a “non computable “number. I say procedure here rather than machine, algorithm or programme because Turing does not give any algorithm but only a procedure from which an algorithm could be constructed. We are invited to consider an infinite class of Turing machines and then construct algorithms to determine the halting property of each (or the circular property in Turing’s original terminology). The digits of a real number are then to be found in some way from the results of the halting tests or from the output of the machines that satisfy the halting test. Now it may be that the first Turing machines on our list are easily dealt with and a number of digits of the “non computable number” are obtained. Turing’s halting theorem however tells us that this easy success cannot continue since no single algorithm of finite size can decide the halting property of all machines in an infinite class. However Turing’s theorem does not show that any particular machine is absolutely un-decidable*. It remains possible to construct (in principle at least) some new testing algorithm that will do the job in any particular case. In this way unlimited progress can continue. There is no known limit to the number of digits of the “non computable” number that can be determined if one is prepared to deal with an algorithm of ever increasing size and complexity.
It must be admitted that such a task is of insuperable difficulty but my point is that it is misleading and confusing (to me at least) to attribute the responsibility for this difficulty to the intrinsic awkwardness of some particular real number. The difficulty lies in the horrendous nature of the algorithm. Thus I am led to consider that the concepts of computable numbers and non computable numbers are not to be taken too literally ( as I rather naively have been doing) but are only convenient metaphors for the more accurately descriptive terms computable procedures and non computable procedures or, better still, finitely programmable procedures and non finitely programmable procedures.

Footnote
*If this statement were untrue then the number in question would not be non computable but merely undefined.

I misunderstood your original point somewhat. I see now that your program is generating all numbers, not just a particular one.

Your program does not disprove non-computable numbers because it does not follow the rules: the program should generate a single number – the number that is being “computed” – not all possible numbers.

For example, it wouldn’t be meaningful to say that your program computes PI. Sure, PI is going to be somewhere in the output, but to identify it, I first need to compute PI myself. So, your program is not helpful at all in computing PI.

In the discussion of computable numbers, a program must be generating a single number – the number that is being “computed”.

Thanks for your response. However the process I sketched is surely realisable by an algorithm of finite size.
An array of 10^n rows x n columns will hold all possible decimal numbers of n digits in length. A procedure expressable in code of a quite modest size can then make a new array of size 10^(n+1) x (n+1) by, for example

1 Copy each row ten times
2 Insert the digit 0 – 9 in sequence onto the end of each row of a block of ten. This creates the new array containing rows listing all numbers of lenght n+1 digits

This procedure can, in principle, be repeated indefintely to any arbitrarly large value of n. To infinitely in fact.
Why is this not a proof of the non existence of any non computable number?
Examples that I have seen of “non computable numbers” make reference to programmes that could not be given since they are of infinite size. I could accept that this could be described as a non computable programme but it still seems meaningless to me to say that it corresponds to some non computable number.

If you have a computable number, there is a fixed-size algorithm that can generate the number to an arbitrary precision. So, I can give you the algorithm, and no matter how many digits of precision you want, you can (in theory) use the algorithm to generate the digits.

If you have a non-computable number, there is NO fixed-size algorithm that can generate the number to an arbitrary precision. You can have an algorithm that generates the number to 1,000,000 digits. But if you want 1,000,001 digits of precision, you can’t use the algorithm.

So, computable numbers are not a purely theoretical/imaginary concept.

A proof of this statement from this point of view might run like this.
A program can be written that will print ( or store on disc) every possible decimal number of,say, 6 digits length. There are only 10^6 of them. The same program could run on ( for 10 times longer) and generate all possible numbers of seven digits and so on to 100 digits and so on to infinity…….. There are no non-computable numbers of length 100 digits or 1000 digits or any other number you care to specify.
Can anyone explain to me why this is not a proof of the meaninglessness of the concept of non-computable numbers without invoking the equally meaningless idea of actually reaching infinity?

That there exist numbers that go against the grain of our current number system should not be too large of a surprise to anyone. The flaw is to assume that our number system is complete. In fact, other than the theorems proven by Kurt Godel and George Cantor, non-computable numbers is further validation of this simple fact. Whether or not this validation comes through computers or on paper through rigorous algebraic manipulation, is merely a matter of human abstract contrivance.

Another way the existence of non-computable numbers can be demonstrated is with complex valued functions. If it can be shown there exists at least one complex valued function that cannot have a solution that is real, irrational, transcendental, or complex, then the solution of the complex valued function is said either to “not exist” or is transcendent to any kind of number in our current system of algebra. There is more evidence in favor of the latter of these two (i.e. incompleteness, trans-infinite sets, diagonalization, …) than in the former of these two. Just because one cannot find a meaningful solution to a mathematical equation within the confines of the current math system does not mean “a solution” does not exist; it simply means a way to describe the solution does not currently exist.

We may use the same argument with our current knowledge of physics. Physics continues to demonstrate operative existential physical reality not currently discernible to us, whether directly, or through any clever instrumentation of our own making.

It would be hasty to assume all mathematics is known and all physical law is known just because it does not fit within the paradigm we are used to; that would just be unscientific.

We would do well to remember that everything we know is only a sliver of a greater and transcendent, abstract and physical, reality.

I thoroughly enjoyed your comments, Mr. Ostrovsky.