Ilyenkov on mathematical logic

4 When we speak about formal logical tradition, we have in mind, here and below, philosophical theoretical interpretation of thinking, and not at all the rules and the schemes that for a long time constituted the content and the apparatus of ‘formal logic’, and that without a doubt have an important, if limited, significance and application. The same goes for the contemporary ‘mathematical logic’. Taken in and of itself the apparatus of this logic does not have any direct relation to the topic of our investigation. This apparatus is especially created and adopted for the solution of a well defined and very strictly determined class of problems – problems connected with the ‘calculation of propositions’, with the purely formal procedure for the transformation of propositions, i.e. for changing one set of signs into another set of signs. Mathematical logic as a special branch of contemporary mathematics entirely consciously limits the sphere of its attention by the relation of signs to signs as part of some strictly determined sign systems. Philosophy does not and cannot have any issues with this logic.

Another matter is when special schemes and rules of activity related to signs-symbols are interpreted as universal, absolute and indisputable ‘laws of thinking in general’, as laws of logic of any kind of thinking, regardless of the ‘subject matter’ thinking takes upon. This is already philosophy, and bad philosophy at that, and as such it can be judged from the point of view of philosophical criteria and must be regarded as completely illegitimate and false.

For it is very clear that this attempt to present the rules for treating immutable (and within the limits of strict formalism they must be immutable) signs as the universal ‘rules of treatment’ applicable to the mutable phenomena of reality, to the ‘things’ that change in reality and in experiments cannot lead to anything except to falsehood. These ‘rules’ cannot be used in these realms. Therefore the honour and the glory of mathematical logic as such are in no danger here.

SOURCE: Ilyenkov, Evald. “Hegel and the Problem of the Subject Matter of Logic,” in Intelligent Materialism: Essays on Hegel and Dialectics, edited and translated by Evgeni V. Pavlov (Leiden; Boston: Brill, 2018), [pp. 6-25,] p. 10, footnote 4.

Yet philosophy has even long developed an image (i.e., model) for the structure of thought: logic. We refer here not to “mathematical logic,” as a system for constructing formal, non-contradicting series of terms, but instead to the science of forms and rules for the development of thought which has been called for some time now dialectics. . . A logic whose stages of development are marked not by the names of Boole, Cantor, Russell, or Frege, but by the names of Aristotle, Spinoza, Kant, Hegel, Marx, and Lenin.

SOURCE: Ilyenkov, Evald. On Idols and Ideals (Ob idolakh i idealakh, 1968), translated by Trevor Wilson, chapter 11.