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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