Mathematical truths

Mathematical truths seem to be absolute, for they seem to be universal and invariable. They seem to be objective, independent from place, culture, age, etc., and they also seem to be eternal. But mathematical truths are in the end subjective and relative.

First, mathematical truths are subjective simply because mathematics itself is humanly subjective. Mathematical systems are human constructions that don’t have any reality outside the human mind.
Mathematical spaces can be compared with a game of chess. Objectively, chess is nothing more than a squared board with pieces of different shapes and sizes. It is not the physical elements, but the mental space we create with those elements that makes the game. Algebra is for mathematics what the rules of chess is for the game. When we define the rules, we define the pieces, how they move and how they operate with each other. In both cases there is a construction of a mental space, where we define the elements, operations and functions between elements. Once the rules are defined and we are ready to play, chess, like mathematics, becomes independent from cultural, age, gender, etc. Chess and mathematics are humanly objective, but that doesn’t make them universally objective; they are just a mental spaces.

The subjectivity of mathematical truths is a sufficient condition for its relativity. In general, relative truths can be evaluated on their degree of logical consistency and consistency with experience. But since mathematics is a mental space, the validity of mathematical truths is reduced to its logical consistency (it is only in physics where it has to be complemented with consistency with experience).

So, unlike what Pythagoras, Leibnitz, Newton and many others believed, Nature is not an order written in a mathematical form, but mathematics is a human construction that we can use to explain Nature. And unlike the dominant belief during the Enlightenment that Nature is a rational order, Nature is non-rational, reason is a human faculty and we use reason to understand Nature often rationalising it. So logical consistency in mathematical truths doesn’t mean universal objectivity.

The relativity of mathematical truth not only is a necessary consequence of its subjectivity, but it has also some concrete manifestations like Gödel’s incompleteness theorems. The theorems states that mathematical systems (or at least those of any practical interest) include truths that cannot be proven within their system. Furthermore, any proof of their truthfulness would make the system inconsistent, and any attempt to prove those truths from outside the system would involve truths from another system that cannot be proven.
The consequence of Gödel’s incompleteness theorems is that, we might have a system, and truths within the system, that are logically consistent. But those truths are confined and relative to that system, and there is no way to prove them objectively. Mathematical truths then, are always relative.


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