Clark, List, You ask: I wonder how we deal with things like quasi-empirical methods in mathematics (started I think by Putnam who clearly was influenced by Peirce in his approach). Admittedly the empirical isn’t the phenomenological (or at least it’s a complex relationship). I’m here thinking of mathematics as practiced in the 20th century and less Peirce’s tendency to follow Comte in a fascination with taxonomy.
Peirce draws on the distinction between pure and applied mathematics. When it comes to geometry, for instance, only topology is pure mathematics. Both projective geometry and all systems of metrical geometry import notions that are not part of pure mathematics, such as the conception of a ray, or a rigid bar. When it comes to pure mathematics, he is just as concerned about getting straight about the the kinds of observations we can draw on as he is concerned about getting straight on this question for the purposes of a pure science of cenoscopic inquiry. He makes the following point: The first is mathematics, which does not undertake to ascertain any matter of fact whatever, but merely posits hypotheses, and traces out their consequences. It is observational, in so far as it makes constructions in the imagination according to abstract precepts, and then observes these imaginary objects, finding in them relations of parts not specified in the precept of construction. This is truly observation, yet certainly in a very peculiar sense; and no other kind of observation would at all answer the purpose of mathematics. CP 1.240 So, I wonder, what kind of observation is it when a person observes the relations between the parts of the imaginary (or diagrammed) objects and learns something about the system that was not evident from the hypotheses and abstract precepts that the reasoning took its start? --Jeff
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