Edwina, Jeff BD, and Jon AS, I have been trying to send this message to Peirce-L, but for some reason, it won't go through. In my previous versions, I had copied some excerpts from your notes. To see if they had caused some problems, I deleted those excerpts in this version.
In Peirce's classification of the sciences (see the attached CSPsciences.jpg) pure mathematics has no input about what exists, and It can't classify what exists. It can only reason about possibilities. But any physical classification could apply the mathematical reasoning methods to actuality. In his classification of modality, Peirce discussed the three "universes": Possibility, Actuality, and Necessity. Pure mathematics does necessary reasoning about possibilities, and applied mathematics adapts that reasoning to actualities. In CSPsciences.jpg, philosophy gets input from phenomenology, and the empirical sciences get input from organized experience. The dependence of philosophy and empirical science on mathematics enables them to adopt any methods of mathematical reasoning. With sensory input, they can apply that reasoning to what is actual. Peirce also said that all necessary reasoning is diagrammatic. Pure mathematics "experiments" with diagrams about possibilities, and draws necessary conclusions about anything that corresponds to (resembles) those diagrams. Applied mathematics (including informal common sense) can do the same kind of reasoning with diagrams. For everyday common sense, reasoning may involve approximate "experiments" on mental models. The highly disciplined sciences may begin with common sense for their initial insights (heuristics), but they can use the same methods of formal reasoning as mathematics. There is a continuum from mental models and mathematical diagrams to physical experiments based on those diagrams. John
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