flexibeast.space - gemlog - 2021-05-03
Wikipedia's definition of ‘category theory’
Category theory (CT), in the mathematical sense, is not ‘just’ a subset of graph theory[a]. Categories do indeed form graphs with a particular structure, but this fact no more reduces category theory to graph theory than analysis[b] is ‘just’ set theory. In this sense, i find the opening sentence of the Wikipedia article for ‘category theory’ to be misleading:
Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category
even though it does go on to say, in the next sentence,
A category has two basic properties: the ability to compose the arrows associatively, and the existence of an identity arrow for each object.
The fact that i have, on a few occasions, encountered people thinking that they can form a ‘category’ of arbitrary non-mathematical things (e.g. gaming genres) has made me wonder whether this article might be a source of such thinking. Which is why i'd prefer that the phrase “labelled directed graph” be removed from the introduction, and instead used elsewhere in the article:
Category theory formalizes mathematical structures and its concepts in terms of a category: a collection of ‘morphisms’ (or ‘arrows’) and ‘objects’, such that morphisms can be composed associatively, and each object has an identity morphism.
i use ‘morphism’ in preference to ‘arrow’ due to the former seeming much more common nowadays than the latter. Also, the use of the word ‘formalizes’ is a small can of worms[c].
All that said, i'm leery of making edits to the opening of such a significant Wikipedia article; i'm simply not interested in putting my hand up for possible edit wars, especially since i'm not at all an expert in CT. And anyway, maybe there's something important about the current opening that i'm failing to consider.
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[a] Nor is it a ‘mathematisation’ of Kantian categories, though, yes, the name was inspired by the work of Kant (and Aristotle):
“Why were Kant's categories used in the mathematical category theory?”
[b] i.e. the mathematical area that includes calculus.
Wikipedia: ‘mathematical analysis’
[c] There are at least two current notions of something being ‘formal’, which appear when talking about ‘formal proof’. In one context, a ‘formal proof’ is the usual sufficiently-rigorous proof in a mixture of natural and mathematical language; it's counterposed to a ‘handwavy proof’ in which one simply says “Here's the overall idea of the proof, but various details need to be worked out.” In another context, a ‘formal proof’ is a machine-checkable proof, one that involves all the tedious technical details usually elided by the other type of ‘formal proof’. Mixing the two contexts could create sentences like “This formal proof needs to be formalised in a formal proof.”