> On Oct 29, 2015, at 1:20 PM, Jeffrey Brian Downard <jeffrey.down...@nau.edu> > wrote: > > 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.
(Trying to remember my math classes - it’s been too long) That’s really helpful though. Thank you. Where does Peirce talk about this? It’s not in anything I have handy. The places I find him discussing topology seem more related to his logical diagrams and logic of relations. Admittedly that got connections to mathematical topology as I remember it from my undergrad years. That is the issue is over set theory and how within sets relations take place. Which does seem quite tied to his general semiotics. Most of the geometry I did back in the day involved metric spaces and so not pure mathematics in Peirce’s sense. I’m just trying to get clear in my mind the dividing line. Is it fundamentally between set theory (and its relations) as opposed to use of set theory? I’m familiar with the quote you give later where math is about possibilities. We make premises and trace out implications. It’s imaginary in that sense. I just don’t see how that leads to a divide between topology and metric geometry. The latter seems mathematical in this sense. Forgive my ignorance here. Like I said it’s been more years than I care to admit. The days when Nirvana and Soundgarden were fresh and new. Thanks for getting at this though. It’s extremely helpful to me. > 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? That bit about observation seems key. Peirce makes similar points in many places. It’s that reason I think Peirce is closer to Quine than Kant in this. However I can imagine many things - some tied closer to the regular world than others. Geometry is the best example of this since circles, lines, rays and so forth seem precinded from regular phenomenal objects. I’d go so far as to say that’s true even of sets or continuity.
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