Jeff D., Clark, list,
I think it's important in this to get the quotes and dates. I recall
Peirce's views as changing, and partly it's his acceptance of changing
terminology. Earlier, he had regarded geometry as mathematically applied
science of space; later he accepted the idea that geometers were not
studying space as it is, but instead studying spaces as hypothetical
objects. Digging those quotes up is another little research project.
Best, Ben
On 10/29/2015 3:20 PM, Jeffrey Brian Downard wrote:
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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