Mathematics and the Universe

[Purple Rabbit]

Purple Rabbit, my dad has replied to you:

'Peace', 'slavery', 'war', 'socialism', 'torture', etc, are abstract concepts, relating specifically to human conditions. (They are in a word, relative.) Mathematical truths are Absolute, and therefore, universal - ie non relative. (Gödel's theorum demonstrates that mathematics cannot be reduced to a positivistic, formal system, because truth is absolute, proof merely relative).


Regards Michael

Mathematical and logical truths (even so called absolute ones) are absolute only in a limited sense. I'll give an example:

Suppose that from premisses A and B with logical deduction it follows that conclusion C is true, then the inference of C from A and B is logically absolute. Still C is true only under the assumption that A and B are both true. To establish the logical truth of A and B we need other syllogisms. The regress is stopped in mathematics at the level of axioms, statements that are assumed to be true because they are considered impossible to refute. Axioms are said to be evident statements, they have no proof, they are the bedrock of logic and mathematics. This means that mathematics is build on some human choice after all. Furthermore the applicability of A and B to our reality requires some way to establish correspondence between the logical realm and the physical realm. This again requires some assumptions to be made. Apart from deductive logic you need inductive methods, and inductive logic is not absolute (problem of induction).

The dethroning of Euclidean Geometry (EG) as absolute truth about our universe again suffices as a good example. Given Euclid's axioms, EG's logical conclusions follow and are said to be absolute. But EG turned out to be not applicable to our universe. This shows that logical absolutes not necessarily are absolutes in the physical realm. In other words the logical does not necessarily prescribe the physical reality. Because mathematics is always based an axiom's, asserted by man guided by the rules that seem to rule our physical universe (doesn't that make these axioms relative too?), mathematical absoluteness has no prescriptive meaning in our physical world.

So this means that, contrary to the assertion made in your reply, the proof (i.e. the deductional part) may be absolute, absolute truths about reality cannot be obtained by mathematics. Gödel's incompleteness theorems have little relevance for the above. They only cripple mathematical deduction further in the sense that undecibibility creeps in, in case you use first-order predicate calculus.

PS: Thank your father for his reply.