There are some numbers that cannot be expressed as ratios of a pair of integers, or whole numbers. As the numbers that can be so expressed are called rational, those that cannot are, quite appropriately, called irrational. Each irrational number falls between two rational numbers, and can be approximated to any degree of precision desired, although it always leaves a residuum. Parenthetically, a problem of great historical portent to mathematics was whether irrational numbers were numbers at all.

It would be tempting to proceed by analogy and to suggest that some concepts could be expressed as ratios of other concepts and their contents, if these concepts had failed to exhaust their contents, thus implying that the relationship between the latter two could be exhausted by the former. Such concepts could be called rational, while those that cannot be so expressed could be called irrational. The only problem with such a suggestion is that all concepts are actually irrational, as residua inevitably spring, again and again, from the fact that every concept, a product of abstract thought, fails to exhaust the residuum resulting from all the preceding attempts at exhaustion of the relationship between some initial concept and its content, an integral part of concrete reality. The incommensurability between thought and reality, that is, between a concept and its content, is intrinsic to the problem from the very inception, and consequently cannot be removed by this, or any other abstract procedure. It remains a practical matter that may benefit from abstract thought only to the extent that it, the thought, abandons its misguided ambition to ultimately eradicate all residua, while persevering in its pursuit of the residuum.

Parenthetically again, a problem of some portent to philosophy is whether irrational concepts are concepts at all. It follows not only that they are, but that rational concepts are but a figment of our imagination. The so-called mathematical concepts, for example, that may appear to contradict this argument, are not concepts by virtue of the fact that they remove the content a priori, and thus circumvent the residuum by circumventing, or avoiding, the problem itself. They are not concepts because they do not even strive to be concepts sensu stricto. We are left with nothing but a misnomer. This misnomer is nevertheless revealing insofar as it unwittingly identifies rational concepts with abstract thought in its purest form, that is, the thought propelled by a paranoid avoidance of any contact with historical reality, which indeed threatens to spoil the thought’s conquests by rendering them palpably immaterial and thus irrelevant. And indifference is the ultimate price for calculated irrelevance.