But we're surely not talking about a finite number of programs.
I'm not even sure what the Kolmogorov complexity of a "complex irrational number" means. If you need the sequence of digits and you cannot use an algorithm to produce the digits, then the complexity is infinite?
The "countable" part means we can put the set into one-to-one correspondence with the natural numbers. The correspondence starts with 0 mapping to the empty program, then 1-256 mapping to programs of a single byte, then 257-65793 mapping to the programs of two bytes, and so on. This mapping will hit each program exactly once, and it will hit every program because every individual program has a finite length.
Different types of infinities are not intuitive so don't feel bad if the concepts are confusing. These issues troubled lots of very smart mathematicians for decades. The existence of irrational numbers was hugely troubling to Pythagoreans. The existence of uncountable infinities discovered by Cantor was shocking [1].
[1]: http://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_the...
But then, is any mathematical abstraction real? I guess it's all beside the point.
It's tough to Google for "Taylor series number system" as you just get pages about the guitars. ;-)
But irrational numbers remain irrational in any integer number base. Therefore no, they aren't illusory at all, nor are they an artifact of the "number system".
> ... such numbers lack any description or means of being separated as individuals.
Also false. Many irrational numbers are easy to identify unambiguously. For example, the square roots of numbers that are not perfect squares are irrational, but they can be easily identified by squaring them.
While we're on the topic, the terms in the running sum of odd numbers are all perfect squares:
0 + 1 = 1
1 + 3 + 4
4 + 5 = 9
9 + 7 = 16
16 + 9 = 25
http://arachnoid.com/example/#Math_Example
> But then, is any mathematical abstraction real?
Certainly. Consider general relativity. It's a mathematical abstraction, and it's also real.
Absolutely. Cantor proved that an infinite number of others can't be identified at all, because they outnumber all possible descriptions. Some numbers can only be described by an infinitely long list of digits, one that can't be produced by a Turing machine and contains an infinite amount of irreducible information. The Kolmogorov complexity is infinite.
Newton's equations describe a mathematical model of the universe that agrees well with measurements taken under familiar conditions. Einstein's equations describe a different mathematical model, one that has good agreement with experiment over a much wider range of conditions than Newton's. But general relativity has well-known problems. Its equations give nonsense solutions under some circumstances, e.g. singularities. Physical theories are models that predict the outcomes of experiments. They're reductionist out of necessity. It's unknown and probably unknowable as to whether a perfect model is possible, but there are likely things that can't be reduced.
I'm tempted to say that that definition places those examples in a unique set, thus at least unambiguously identifying the set to which they belong.
Turing machines might not capture all numbers that can be described, but, interestingly, descriptions and Turing machines have the same cardinality.
I actually think the reality of mathematical abstractions is hugely important because of... computer programs! In a real way, programs are the embodiment of mathematics. I want my programs to work so I need the underlying math to work as well.
That's why I'm a constructivist. I reject the law of the excluded middle because proofs that use it don't translate into real programs; they translate into programs that ask an omnipotent oracle to decide which branch to take. Constructive proofs translate into working programs.
It also ties into philosophy. I am a skeptic, so when someone tells me either A or not A must be true even if we can never know which one, I ask for proof or justification of that fact. The justifications that I get are remarkably similar to logical "proofs" that god exists, and just as fallacious. This isn't to say that there couldn't be an ultimate truth about A or a god, just that it is not logically necessary.
Philosophically, my worldview is like that of science: the way to know something is by making observations and formulating and testing hypotheses. What we can observe is limited, and hypotheses are only models that are tested by successive approximations. I'm not sure I understand what you mean that that statement isn't required to have an ultimate truth. Presumably a statement like that has an answer, but the fact that something is knowable in principle doesn't mean that there's any way to get the answer. For instance, the Hubble telescope can see far away galaxies that we can never get an up close look at. The question of whether there's life somewhere else in the universe must have an answer, but we can't know what's in those galaxies; even with a better telescope, we'd be seeing what they looked like a billion years ago. Many things will never be known.
You mean "computable".