Franklin, lists,
Again, thanks for your opening post.
Some further comments. (I just noticed your reply to my previous message
as I added finishing touches to my message below, which addresses other
points than my first message did, so I figure that I'm not about to get
the discussion crossed up).
You wrote,
As Frederik restates it, mathematics has to do with hypothetical
abstract objects.
[End quote]
We should say, mathematics is about hypothetical abstract
not-conventionally-linguistic objects (compare with Quine's saying that
math deals with 'abstract nonlinguistic objects'). If _c/onceptual
definitions/_ are understood as definitions either in, or lending
themselves to, more-or-less conventional language, this makes more sense
of saying that corollarial inference is deduction merely from conceptual
definitions. Peirce says ("Truth (and Falsity and Error): /Logical/",
Baldwin Dictionary, 1902,
http://www.gnusystems.ca/BaldwinPeirce.htm#Truth CP 5.567) that
mathematical statements are never pure enough to live up to the name
'pure mathematics' because they keep using words like 'points' and
'lines' which need to be understood more generally than the words
themselves say. Elsewhere (CP 7.467) he says, "A concept is the living
influence upon us of a diagram or icon, with whose several parts are
connected in thought an equal number of feelings or ideas." Anyway, the
above seems pertinent to where you summarized Frederik:
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).
[....]
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.
[....] [End quote]
You wrote,
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.
[End quote]
Peirce indeed speaks in some places of the use of abductive inference in
mathematics. Now, generally, the choosing of a deductive procedure is
not itself a deductive act except for families of problems with
established procedures; but that is just what 'creative' mathematics
(Anellis's alternate term for 'pure' mathematical work) lacks; but this
is just to say that we don't know how to program a computer to do
creative mathematics. The conclusions are aprioristically true only
given the hypotheses, but the hypotheses themselves are not
aprioristically true nor asserted to be true except hypothetically, and
this hypotheticality is what allows such assurance of the conclusions,
although even the hypothesis is upended if it leads to such
contradictions as render the work futile (when a contradiction can be
safely 'cordoned off', then I guess it's like a birthmark of the
system). According to Peirce in 1904 in his drafts of an intellectual
autobiography
http://www.degruyter.com/view/books/9783050047331/9783050047331.35/9783050047331.35.xml
(Ketner 2009, among other places), the choosing of hypotheses for (pure)
mathematical exploration "is not a scientific act", and Peirce is
speaking of mathematics as itself a science:
This classification (which has been worked out in minute detail) is
to be regarded as simply Comtes classification, corrected. That is
to say, the endeavor has been so to arrange the scheme that each
science ought to make appeal, for its general principles,
exclusively to the sciences placed above it, while for instances and
special facts, it will find the sciences below it more serviceable.
Mathematics merely traces out the consequences of hypotheses without
caring whether they correspond to anything real or not. It is purely
deductive, and all necessary inference is mathematics, pure or
applied. Its hypotheses are suggested by any of the other sciences,
but its assumption of them is not a scientific act.
[End quote]
Yet can we confine mathematics to just the deductive part? Peirce
himself did not always do so. Consider his remarks quoted within a quote
from a Professor Fiske on page 7 in A Semicentennial History of the
American Mathematifcal Society 1888-1938:
"At a meeting of the Society in November 1894 in an eloquent oration
on the nature of mathematics, C.S. Peirce proclaimed that the
intellectual powers essential to the mathematician are
'Concentration, imagination, and generalization.' Then, after a
dramatic pause, he cried, 'Did I hear some one say demonstration?'
'Why, my friends,' he added, 'demonstration is but the pavement on
which the chariot of the mathematician rolls.'"
http://books.google.com/books?id=sOGifU-L_coC&pg=PA7&lpg=PA7&dq=%22Peirce%22+%22pavement%22+imagination
(Note that Peirce usually meant by (deductive) demonstration a
particular kind of deduction.) Anyway, Peirce discussed the importance
of concentration, imagination, generalization elsewhere too, as I
recall. It's hard to reconcile those different views. If we take
mathematics as all and only the deductions, then mathematics becomes
'but the pavement on which the chariot of the mathematician rolls'.
I'm not sure what conclusion to draw here. I might add that Peirce
believed that mathematics is not the science OF deductive conclusions,
but instead just the science which draws deductive conclusions.
Conclusions about the deductive relations among statements that
constitute a theory - when we consider _/theory/_ narrowly as a system
of logically related statements - might be considered a more
'theoretical' than 'hypothetical' science, and if this more purely
'theoretical' science were itself deductive, then it would seem to be a
kind of applied but quite general mathematics - mathematical logic. Here
one gets to the question of how to classify mathematical logic in terms
of Peirce's distinction between 'mathematics of logic' and philosophical
deductive logic, and, finding myself in over my head (as I so often am),
I've become rather unsure about it.
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, abst1ractions
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
-----------------------------
PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L
to this message. PEIRCE-L posts should go to [email protected] . To
UNSUBSCRIBE, send a message not to PEIRCE-L but to [email protected] with the
line "UNSubscribe PEIRCE-L" in the BODY of the message. More at
http://www.cspeirce.com/peirce-l/peirce-l.htm .
