| I think you’ve just each used a different sense of the word ‘actual’. Sense 1) “‘Actual’ mathematics does not consist in notational embodiments” ‘Actual’ as in essence rather than appearance, territory rather than map. Implies platonic reality of an eternal ephemeral mathematics. Conceived of as beyond and beneath all the particulars of any given representation. Sense 2) “‘Actual’ mathematics consists in notational embodiments” ‘Actual’ as in embodied. Material. That ‘actual’ mathematics is the set of mathematics which has been ‘actually’ done. You describe the second sense as self evident but with mathematics in particular this is bound to cause trouble. The sense ‘actual as in essence’ is perfectly reasonable and has a particular affinity to a discussion of mathematics. I find it hard to even write the description of the second sense without tying some linguistic knot. If ‘actual’ mathematics is its embodiments, what is the thing being embodied? Either the notation embodies nothing (what does it mean to describe the characteristics of a thing which doesn’t exist?) or if it does refer to something then according to the designation of embodied mathematics as ‘material’ then it embodies something which exists immaterially. So either mathematics describes the characteristics of a non existing thing, or a thing which exists, but exists immaterially. If ‘actual’ mathematics is the materially existing embodied form of a separate immaterial thing, what is that thing? So that ‘Actual mathematics’ is the embodiment of a separate immaterial thing called ‘not actual mathematics’. I’m not trying to make a watertight argument here just trying to outline the course of thought that your assertions inflame. From another angle, what does it mean for something to ‘materially exist’ so that it can be considered ‘actual’. I think this concept so nebulous as to make impossible the patching up of all the holes a cursory investigation reveals. Personally I think the distinction between the two is the root of trouble here. In the sense that anything exists, it exists according to mathematics. The essence of material existence, to the extent that it can be specified, is a structure of formal relational principles. Such formal structures are the domain of mathematics. Of course ‘material existence’ is used and understood perfectly well in ordinary circumstances but when the topic at hand is “what counts as ‘actual’ mathematics?” it seems shaky ground to build a house. |