# Is the overlap between Yoneda's lemma and Peirce's pragmatic maxim known?

In philosophy, particularly in ethics, the pragmatic maxim (WP) states that an object may be considered solely in terms of its effects on the surrounding context. In the special case of ethics, the maxim states that the ethical ramifications of an event are summarized by its physical effects upon the future.

In mathematics, particularly in category theory, Yoneda's lemma (WP, nLab) states that in a universe of transformations, an object is equivalent to all of the transformations which can be applied to it. In the special case of category theory, where everything is either an object or a one-way transformation between objects, Yoneda's lemma states that an object is equivalent to the collection of all transformations which start with that object.

Both of my summaries are not very deep, but a serious reader should hopefully notice the similarity between these two statements. I have found exactly one such reader. Can anybody find other philosophers who have noticed this overlap?

The goal would be to phrase the pragmatic maxim as yet another special case of Yoneda's lemma.

• Pragmatic maxim talks about the effect on "practical bearings", not "surrounding context", and at this level of vagueness pretty much anything is "similar" to great many things. Relational ontologies, where objects are placeholders for relations they enter, much predate both the maxim and the lemma, as does Leibniz's identity of indiscernibles or transformational geometry. So this reminds me another Peirce's quip: it is easy to be right, one has only to be sufficiently vague. Sep 12, 2022 at 13:35
• @Conifold: I thought it prudent to not do hard maths in my question. Consider a category on a differentiable manifold of spacetime; the objects of the category are events, and there exists an arrow between two events when the first event is in the past of the second event. (This category is a preorder.) Yoneda's lemma then states that we may exchange an event for all of the possible future events which have that event in their lightcone. This is one possible way to specialize the pragmatic maxim to category theory; there are others. Sep 12, 2022 at 16:48
• I think the revisionary force of the pragmatic maxim was not in the vague principle that objects/events can be characterized by or even reduced to their "effects" (of some sort). It was in overturning the classical view that meanings ultimately inhere in the "nature of things" rather than spring out from our practical dealings with them. The vague principle is compatible with the classical view, which manifests already in Plato, Aristotle and especially Leibniz, in some contexts, but Peirce's maxim is not. So its assimilation to Yoneda's lemma kind of misses the point, of both. Sep 13, 2022 at 4:04
• Perhaps see Leibniz's monadology: every monad mirrors the whole of the universe in that it expresses every other monad, but no monad has a window through which it could actually receive or supply causal influences.... Note here monad as a complete conceptual substance like an object in category has no efficient causal influence but only expresses all of its relations to other objects, and it's no accident that according to Mac Lane a monad is just a monoid in the category of endofunctors and is a singly typed object when viewed from outside but inside there're a lot of arrows dynamics... Sep 13, 2022 at 5:34
• Not Stone's duality, Stone's representation theorem for Boolean algebras (sets 'manifest' through their Boolean effects). Observations that we make upon objects is not what pragmatism is specifically about, you can relate that more to empiricism, and even then the relation is vague. Reasoning by loose association can be helpful, but typically analogies need to be more compelling to generate philosophical interest. Sep 17, 2022 at 0:52