Franklin, lists,
I agree with Jon, thanks for your excellent starting post.
You wrote,
[....] 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?
There's a distinction that some make between complexity and mere
complication. Corollarial reasonings may accumulate mere complications
until the result becomes hard to see, although it involves little if any
complexity in, more or less, the sense of depth or nontriviality.
I don't know whether there's a theorematic approach to Jørn Holm's
diagrammatization that would show its result in a nontrivial aspect, and
anyway its diagrammatic, pictorial presentation already leaves one in no
doubt that a pattern is revealed. A good example involving alternate
proofs that seem corollarial and theorematic is the Monty Hall problem,
a popular puzzle based in probability theory. I remember reading an
essentially corollarial proof of the answer, and seeing a round diagram
that showed how alternatives lead inevitably to the conclusion in the
diagram's center. The answer to the Monty Hall problem remains, however,
notoriously counter-intuitive to people; the essentially corollarial but
multi-step proof - in words, even with the round diagram - often leaves
people with nagging vague doubts. They get that it must be true but they
feel that they don't fully get the problem, they keep re-examining the
problem, wondering whether it was well disambiguated, etc. (it describes
an actual standard scenario on a popular TV game show). But the
problem's answer has also a proof that deserves to be called theorematic
(even if it is not very much so) since it involves varying the
conditions of the problem, adding things not contemplated in the thesis,
going a little deeper into the mathematical possibilities. One increases
the number of doors in the scenario from 3 to 10. With 10 doors, the
basically the same solution makes obvious sense, then one reduces the
number doors from 10 to 9 to 8, etc. down to 3, and sees that the basic
solution does not change at all; people get satisfied (for whatever
that's worth). It has become hard to avoid running into that proof if
one searches the Internet for "Monty Hall problem". I also vaguely
remember a geometric problem involving the fitting of circles, shown to
me by a roommate during college; he was dissatisfied with a particular
usual proof, he wanted a proof that gave more satisfactory
understanding, and it turned out to be more imaginative and, as I'd call
it now, theorematic.
Nontriviality or depth of a result should not be confused with mere
complication and lengthiness of a proof; take the Pythagorean theorem,
which is considered both deep and not very hard to prove. The
nontriviality or depth of a theorem consists not in the difficult
complication of proving it but in its favorability as a bridge to
further nontrivial lessons or, to put it less recursively, its
favorability for use as a basis for further proofs almost as if it were
another postulate even though it is entailed by the postulates and
axioms already granted. It's a place where one can come to rest for a
while and set up camp; if I were to coin a word dedicated to expressing
it I'd say 'basatility'. Likewise the nontriviality or depth (apart from
mere complication as distinguished from complexity) of a proof of a
theorem is properly its favorability as a basis for further lessons.
(I'm not sure that there is much difference between 'depth' and 'power'
of a theorem or a proof.) The nontrivial or deep is more or less
_/difficult/_ (which is a usual connotation especially of 'nontrivial')
since, of course, it requires some corresponding depth or or
nontriviality of understanding and perspective.
(One should distinguish such depth, complexity, etc., of theorems and
proofs also from the logical complexity that a fact or datum, as a
relation or complexus of relations, possesses; I mean such 'complexity'
as quantified and characterized by valence, transitivity or
intransitivity, etc. This is likewise as one distinguishes the novelty
or new aspect of a deductive conclusion from Shannonesque quantity of
information or 'newsiness'.)
Best, Ben
On 4/19/2015 6:06 AM, Franklin Ransom wrote:
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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