| > In some sense, the existence of irrational numbers with no pattern in their digits is an illusory artifact of the number system. 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
As is often true in mathematics, this is easier to explain with a picture: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.