Hello lists,
As Gary Fuhrman posted two weeks ago, I will be leading discussion on
Chapter 10 of NP. I am sorry for posting a week later than planned.
In what follows, I will treat each section of the chapter, partly to
summarize the important points up for discussion, and partly to remind
listers of the contents of the chapter. Afterwards, I will finish with some
issues and questions regarding the chapter. Next week, I will post on the
relationship between this chapter and the other chapters of the book.
10.0
"The truth, however, appears to be that all deductive reasoning, even
simple syllogism, involves an element of observation; namely, deduction
consists in constructing an icon or diagram the relations of whose parts
shall present a complete analogy with those of the parts of the object of
reasoning, of experimenting upon this image in the imagination, and of
observing the result so as to discover unnoticed and hidden relations among
the parts" ("On the Algebra of Logic", 1885, 3.363) (p.268 of NP)
All deduction makes use of diagrams. A diagram is defined as an icon which,
by analogy, represents relations between objects. This means that diagrams
do not necessarily have to be graphic, visual representations, but can
include a much larger variety of representations, including even algebraic
formulas.
Mathematics is the science that has to do with drawing necessary
conclusions regarding hypotheses about the forms of relations of objects.
As Frederik restates it, mathematics has to do with hypothetical abstract
objects. In order to access such objects, a two-step process involving
diagrams is required:
First, a given diagram is stripped of its accidental qualities, in a way
similar to how everyday ordinary objects are stripped of their qualities in
order to grasp natural kinds. More formally, it is the process of
prescission, in which a token's accidental qualities are abstracted away so
that all that is left is what is essential to the type of which the token
is an instance. The diagram token, with extraneous considerations removed,
reveals only the essential relations between the objects involved in the
diagram, and thus reveals the diagram type of which it is a diagram token.
Second, the diagram token may be experimented upon according to certain
types of transformations that preserve truth through logical steps. By
experimenting on the diagram token, information can be garnered about the
diagram type. In this way, by manipulating the forms of relations of
objects according to rule-governed transformations that preserve logical
validity, we can learn about hypothetical abstract objects--the subject
matter of mathematics.
According to Peirce's system of the sciences, every other science borrows
principles from mathematics. In considering the relation between
mathematical diagrams and applied diagrams, this means that applied
diagrams, whether having to do with a science or with everyday reasoning,
employ mathematics either explicitly or implicitly. Thus, all deductive
reasoning, whether scientific or everyday, involves mathematical
diagrammatic reasoning. The following discussion about theorematic
diagrammatic reasoning is not only of significance then for mathematics,
but for epistemology as well. Recalling the quote from Peirce given above,
Stjernfelt notes that "[t]he 'unnoticed and hidden' relations obtainable by
diagram observation, of course, are what are later taken to require
theorematic deduction, in addition to mere inference from definitions"
(p.268).
The section finishes with introducing the corollarial/theorematic
distinction. However, the next section details the distinction more
precisely.
10.1
In this section is covered the various definitions given by Peirce over
time about what theorematic reasoning is.
Peirce's five definitions of theorematic reasoning:
1. Theorematic reasoning is not reducible to inferences from conceptual
definitions, i.e. conceptual analysis, in the way that corollarial
reasoning is (though both require observation).
2. Theorematic reasoning involves the introduction of new elements to the
premises, whether new individuals or foreign ideas, abstractions or
non-abstractions.
3. Theorematic reasoning involves performing an action that manipulates the
diagram as part of diagram experimentation.
4. Theorematic reasoning requires complex, or specially constructed,
schemata, as opposed to simple schemata in corollarial reasoning; a matter
of difference in degree of complexity.
5. Theorematic reasoning requires a new point of view of the problem.
"To sum up Peirce's different descriptions of theorematic reasoning, we can
say they exceed the mere explication from the combination of definitions by
introducing something further, be it new elements (particular or general),
be it experiments by diagram manipulation, be it the substitution of
schemata for words, or be it the gestalt shift of seeing the whole problem
from another point of view." (p.280)
10.2
The theorematic/corollarial distinction applies regardless of logic system
used, though what shall count as theorematic and what shall count as
corollarial is relative to the system of logic, i.e. the axiom and rule
systems.
10.3
Three levels of theorematic diagram experiment:
1. The appropriate selection of new particular objects that are permitted
by the formal system
2. Experimenting with one or more basic object or rule definitions
3. Establishment of a system of different versions of the object or rule
definitions
"Thus, the three theorematic levels distinguished here -- the introduction
of a new object, and the two types of introducing a foreign idea, the
experiment with one or more of the basic object or rule definitions, and
the establishment of a system of different versions of those definitions,
seem to to [sic] give us a hypothesis of three different levels of
theorematic diagram experiment." (p.285)
10.4
Examples of diagram experiments taken from geography:
1. Introduction of the ruler (new object?)
2. The relationship between domesticated species, isotherms, and the
development of human civilization (first level, new objects)
3. The connection between Africa and South America through the idea of
continental drift (second level, foreign idea)
4. The Pangaea hypothesis (third level, new perspective)
10.5
This section concludes the chapter with discussion of the relation between
theorematic reasoning and hypostatic abstraction. Hypostatic abstraction is
"the procedure Peirce described as making a second-level substantive out of
a first-level predicate, thereby creating a new object of thought" (p.291).
It is implied to be involved in second-level diagram experiments, ones that
involve the introduction of foreign ideas; it is explicitly said to be
involved in third-level diagram experiments, which involve many foreign
ideas synthesized into a new perspective, and which are cases of complex
hypostatic abstraction.
Issues/Questions
1. Is prescission here the same as prescision in "On a New List of
Categories", making it identical with the idea of abstraction?
2. "The truth, however, appears to be that all deductive reasoning, even
simple syllogism, involves an element of observation; namely, deduction
consists in constructing an icon or diagram the relations of whose parts
shall present a complete analogy with those of the parts of the object of
reasoning, of experimenting upon this image in the imagination, and of
observing the result so as to discover unnoticed and hidden relations among
the parts" ("On the Algebra of Logic", 1885, 3.363) (p.268) Stjernfelt says
"The 'unnoticed and hidden' relations obtainable by diagram observation, of
course, are what are later taken to require theorematic deduction, in
addition to mere inference from definitions" (p.268). Is this really true?
Peirce says in the quote that all deduction is of this sort; but
theorematic deduction is not all deduction. Why can't corollarial
reasoning, which involves observation and experimentation, reveal unnoticed
and hidden relations? After all, on p.285-6, Frederik mentions the work of
police detective Jorn "Old Man" Holm and his computer program, which
Frederik describes as a "practical example of corollarial map reasoning"
(p.285). In this example, Holm uses the corollarial reasoning to reveal
information about the whereabouts of suspects. Doesn't the comparison of
the map reasoning with suspects' testimony end up revealing unnoticed and
hidden relations?
3. Frederik's geography examples raises an issue about the relationship
between theorematic diagrammatic reasoning and abductive inference. In the
example where Wegener supposes that Africa and South America were once one
continent, Frederik refers to this as an example of theorematic diagram
experimentation, because Wegener observed in a map that the West coast of
Africa and the East coast of South America seemed to fit together, and
Wegener introduced the idea of continents moving over time to explain the
apparent fit. Now it is easy to see that a diagram was involved. But, it
also seems clear that this is a case of abductive reasoning, not deduction,
because the idea of continental drift is an idea which explains the
apparent, surprising fit of the two continents. Similar remarks could be
made of the Pangaea example.
The issue here is: how do theorematic reasoning and abductive inference
relate to each other? How can we distinguish which cases are theorematic
deductions and which cases are abductive inferences? Frederik mentions the
relationship on p.276, but that discussion does not make the relationship
very clear: "But in the course of conducting the experiment, an abductive
phase appears when investigating which experimental procedure, among many,
to follow; *which* new elements or foreign ideas to introduce". Is
abduction somehow adventitious to theorematic reasoning, or is it in fact
its inclusion in diagrammatic reasoning that marks the difference between
corollarial and theorematic diagrammatic reasoning? If it is a necessary
component, this throws doubt on Frederik's claims elsewhere that such
reasoning can be a priori; abduction is always an answer to some experience
calling for explanation.
4. In connection with the previous question, consider the idea of
diagrammatic experimentation. How does this kind of experimentation relate
to inductive experimentation? What is the place of diagrammatic
experimentation in scientific method?
5. Peirce's existential graphical logic is not mentioned in the chapter. A
claim in the chapter is that 'pure' diagrams are mathematical diagrams that
inevitably inform applied diagrams. However, it's not clear whether the
existential graphs are supposed to be mathematical diagrams, and yet they
seem to exemplify 'pure' diagrams par excellence. How should we view
graphical logic in light of Frederik's analysis of mathematical
diagrammatic reasoning?
These are just some possible ideas to think about. If anyone would like to
respond, or has something else to bring up with respect to the chapter,
please feel free to contribute!
-- Franklin
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