In 1897 CP 4.218, Peirce makes his remarks that projective and metric
geometries are not pure geometry, but goes on to say that they are so if
the plane is defined so broadly as to make those geometries into
chapters in topics (topology).
in 1901 in "Truth (and Falsity and Error)"
http://www.gnusystems.ca/BaldwinPeirce.htm#Truth in the Baldwin
Dictionary, Peirce wrote:
CP 5.567. These characters equally apply to pure mathematics.
Projective geometry is not pure mathematics, unless it be recognized
that whatever is said of rays holds good of every family of curves
of which there is one and one only through any two points, and any
two of which have a point in common. But even then it is not pure
mathematics until for points we put any complete determinations of
any two-dimensional continuum. Nor will that be enough. A
proposition is not a statement of perfectly pure mathematics until
it is devoid of all definite meaning, and comes to this — that a
property of a certain icon is pointed out and is declared to belong
to anything like it, of which instances are given. The perfect truth
cannot be stated, except in the sense that it confesses its
imperfection. The pure mathematician deals exclusively with
hypotheses. Whether or not there is any corresponding real thing, he
does not care. His hypotheses are creatures of his own imagination;
but he discovers in them relations which surprise him sometimes. A
metaphysician may hold that this very forcing upon the
mathematician's acceptance of propositions for which he was not
prepared, proves, or even constitutes, a mode of being independent
of the mathematician's thought, and so a reality. But whether there
is any reality or not, the truth of the pure mathematical
proposition is constituted by the impossibility of ever finding a
case in which it fails. This, however, is only possible if we
confess the impossibility of precisely defining it.
[End quote]
I can't currently find the passage that I vaguely remember, where Peirce
describes geometry as partly empirical, or something like that.
Best, Ben
On 10/29/2015 4:10 PM, Benjamin Udell wrote:
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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