Can philosophy integrate the irrational as mathematics can?
- Yes.
- No.
Can the irrational be dealt with in philosophy in the same way as in mathematics?
The irrational is that which cannot be grasped by reason, which is considered “superrational”, “subrational”, “unreasonable”, but not “counterrational”, “counterreasonable”, “anti-rational”, “anti-reasonable”.
N. Hartmann speaks of the “transintelligible” and means that which is beyond the reach of human understanding.
Friedrich Wilhelm J. Schelling calls the irrational “in things the incomprehensible basis of reality, that which cannot be dissolved into understanding with the greatest effort, but remains eternally at the bottom. Out of this incomprehensible, in the proper sense, understanding is born”. Schelling teaches that all rule-like, all form arises from the rule- and formless.
Irrational numbers.
If one is to be able to perform exponentiation or root extraction with any rational numbers (in the exponent), it is necessary to introduce new numbers: the irrational numbers. There are algebraically irrational and transcendentally irrational numbers.
The totality of all irrational numbers (algebraic and transcendental) and all rational numbers gives the set of real numbers: “|R”.