For the pre-Socratic philosopher Heraclitus the world is moved by the struggle of opposite forces. (See https://en.wikipedia.org/wiki/Unity_of_opposites ) Much later, in the 19th century, Hegel, Schleiermacher, Grassmann, Marx and others developed a dialectical philosophy in which the unity of opposites plays a pro-eminent role.
William Lawvere, a mathematician who contributed significantly to the development of category theory in the 20th century, tried to identify the categorical structure hidden in these philosophical arguments. In the paper ‘Unity and Identity of Opposites in Calculus and Physics’ (accessible through https://ncatlab.org/nlab/show/Unity+and+Identity+of+Opposites+in+Calculus+and+Physics ) he claims that “cylinders” would fulfill this purpose.
Concretely, a cylinder in a category is the data of two parallel morphisms i,f : A->B and a morphism r : B->A with the condition r i = 1A = r f.
Later in the paper he gives, in the context of 2-categories, an example of such a cylinder in which the two parallel 1-morphisms respectively are left- and right-adjoint to the third 1-morphism. The generality of this construction and the connection to the objective to give a model for dialectical philosophy remains obscure to us. Indeed, he gives an example in physics (coexistence of two phases of a substance, liquid, and gas) but why considering in general left- and right-adjointness should be made clearer.
The purpose of this discussion is to make explicit what should be understood by “Opposites”, “Unity of opposites” and by “Identity of opposites”. How would you describe the link between a dialectics, a dialectical philosophy, and the categorical notion of adjunction? Do you have other examples in mind ?
As you know Tom Thomas Krantz I am not an expert on category theory. But I do have a view on the logic of dialectics. When we talk about thesis and antithesis, these are not propositions P and not-P because their synthesis would arguably be P or not-P, which is classically true. So an opposite is unlikely to be a negation or set theoretic complement. It is much more likely that thesis P and anti-thesis Q are inconsistent with each other, so that P and Q implies false, and that possible syntheses are consistent extensions of R and S that follow from P or Q, where P implies R and Q implies S and R and S is consistent. The opposite is then the (logically atomic?) propositions that lead to an inconsistency. In terms of categories, I assume that you need a Cartesian Closed category in order to have a logic.
Thank you, Andrew for your contribution. I see categories ass a sort of refined logic, where you distinguish several ways of getting from A to B. In so far logic is concerned you also would like some features like products (conjunction) co-products (disjunction) and cartesian closedness for implication. Well this is quite general and is certainly important if you speak of dialectics in logic.
To state it in a different way: Logic is about 0-categories and ordinary categories are 1-categories. Going further, and as far as one is interested in n-categories the notion of dialectics gets a new flavor, reminiscent of algebraic topology. To formulate it in an approximate way, one player would have an n-simplex to defend and the other(s ?) would try to detect the weakness.