Dear List Members,
I am interested in determining the kinds of creativity involved in
theorematical reasoning. For this inquiry, I am adopting Margaret Boden's
typology, which identifies three distinct creative processes:
combinatorial, exploratory, and transformational. This framework has led me
to consider the nature of the constraints imposed upon the transformations
allowed within the diagram of the premises during a theorematical
deduction, particularly as expressed in what Campos terms "experimental
hypothesis-making." In this regard, I would greatly appreciate your
insights on two specific questions:
Is Campos accurate in equating "logical possibility" with "possible
within the mathematical system" as a constraint on the formulation of
experimental hypothesis-making in the following paragraph?
Is it reasonable to assume that the views expressed by Campos in this
paragraph represent the standard interpretation of this matter within
Peircean scholarship?
Here is the citation from Campos (2010):
"The logical possibility of the experimental diagram, within the assumed
mathematical framework, is a necessary condition that constrains what
experimental signs the imagination may submit for observation. As Peirce
put it generally in 1906, “that which is displayed before the mind’s
gaze—the Form of the Icon, which is also its object—must be logically
possible” (Peirce 1906b:C4.531). Logical possibility is in part a matter of
mathematical diagrams being subject to the normative rules of good
reasoning that Peirce’s logical critic prescribes, including for example,
for mathematical deduction, the law of the excluded middle and the law of
noncontradiction. But it is also a matter of what is possible to create
within a general hypothetical framework. As we have seen, for example, in
Euclidean geometry postulates affirm possibilities while axioms deny them.
Along with definitions, they are in fact the more general hypotheses that
frame a realm of possibility for mathematical investigation." (Campos,
2010, The Imagination and Hypothesis-Making in Mathematics: A Peircean
Account, p. 337)
I very much appreciate your invaluable comments. Thank you for your time
and consideration.
Best regards,
Matías Saracho
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