Hi Ben, Clark, List,

I'm working on an essay for the conference on Peirce and mathematics that 
Fernando has organized in Bogota, and the topic is those three questions at the 
start of "The Logic of Mathematics."  In order to provide a coherent 
interpretation of what Peirce is trying to do, my efforts are focused on 
writings from that same time period.  So, I'm drawing on the explanations of 
the relations between the parts of geometry in the last lecture in Reasoning 
and the Logic of Things and the definitions he provides of generation and 
intersection, uniformity and the like in his work on topology in the New 
Elements of Geometry and Elements of Mathematics.  If I am not mistaken, most 
of this of this is from the same basic timeframe (around 1896-1898).   

The discussion of the fundamental properties of space in the introduction to 
the latter work was rejected by the editor as being too "philosophical" in 
character.  It looks to me like Peirce is drawing directly from William 
Benjamin Smith's Introductory Modern Geometry of Point, Ray, and Circle.  
Peirce's copy of the text is available through Google Books online.  In the 
annotations in the introduction, Peirce fills in missing words, so we know he 
was reading this section.  It is interesting to compare Smith's account of the 
fundamental properties of space with Peirce's account in the New Elements.  
Here are some features that stand out when making the comparison.  Both are 
explaining how the mathematical conceptions of continuity, uniformity and the 
like are drawn from common experience by a process of abstraction.  In addition 
to refining the explanations of those two properties, Peirce's account lays 
emphasis on the perissad character of the mathematical space that is drawn from 
experience.  Both characterize the introduction of such things as a ray in 
terms of relations between the homoloids in the space.  When one set is taken 
to be dominant, we move from projective to metrical spaces.

The key idea for understanding the character of the hypotheses that lie at the 
bases of both number theory and topology is that Peirce starts with a set of 
precepts that tell us what to do in constructing a figuring and then putting 
the parts into relation with one another.  As the hypotheses are formulated, 
additional precepts are derived that tell us what we are and are not allowed to 
do next.  I wonder:  what lessons can we learn about the relationships that 
hold between math and phenomenology by reflecting on the character of these 
precepts?  In what sense does the analysis of common experience involve 
precepts that govern what we should and shouldn't do by way of making 
observations?

Here is a particularly interesting passage (from a different time period) that 
appears to bear on this kind of question:

We have, thus far, supposed that although the selection of instances is not 
exactly regular, yet the precept followed is such that every unit of the lot 
would eventually get drawn. But very often it is impracticable so to draw our 
instances, for the reason that a part of the lot to be sampled is absolutely 
inaccessible to our powers of observation. If we want to know whether it will 
be profitable to open a mine, we sample the ore; but in advance of our mining 
operations, we can obtain only what ore lies near the surface. Then, simple 
induction becomes worthless, and another method must be resorted to. Suppose we 
wish to make an induction regarding a series of events extending from the 
distant past to the distant future; only those events of the series which occur 
within the period of time over which available history extends can be taken as 
instances. Within this period we may find that the events of the class in 
question present some uniform character; yet how do we know but this uniformity 
was suddenly established a little while before the history commenced, or will 
suddenly break up a little while after it terminates? Now, whether the 
uniformity observed consists (1) in a mere resemblance between all the 
phenomena, or (2) in their consisting of a disorderly mixture of two kinds in a 
certain constant proportion, or (3) in the character of the events being a 
mathematical function of the time of occurrence--in any of these cases we can 
make use of an apagoge from the following probable deduction:... (CP, 2.730)

This provides a really nice example of what it is to observe something like a 
uniformity.  It also provides some sense of how an analysis of the phenomena 
might enable us to sort out--as competing hypotheses--the possibilities 
represented in 1-3.  What is more, the elements provide us with guidance (they 
support the development of the precepts) needed to imagine the kinds of 
experiments that could be run to sort through the competing explanations.  
Stepping back from the particularities of the examples considered in this 
passage, I think we get a nice articulation of how a phenomenological account 
of the categories might supply us with the tools necessary to analyze the 
observations necessary to support, via an abductive argument, a set of 
conclusions in the normative theory of logic about what fair sampling really 
requires under different kinds of conditions.

--Jeff


Jeff Downard
Associate Professor
Department of Philosophy
NAU
(o) 523-8354
________________________________________
From: Benjamin Udell [bud...@nyc.rr.com]
Sent: Thursday, October 29, 2015 1:10 PM
To: peirce-l@list.iupui.edu
Subject: Re: [PEIRCE-L] Peirce's categories

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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