# Science of Knowing (Cognitive Science)

The thesis is that the explicit adequate development of the science of knowing will require the use of the mathematical theory of categories. Even within mathematical experience, only that theory has approximated a *particular* model of the general, sufficient as a foundation for a *general* account of all particulars.

Category theory provides a guide to the complex, but very non-arbitrary constructions of concepts and their interactions which grow out of the study of any serious object of study.

F. William Lawvere (1994). Tools for the Advancement of Objective Logic: Closed Categories and Toposes, J. Macnamara & G. E. Reyes (Eds). The Logical Foundations of Cognition, Oxford University Press, 43-56.

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The Logical Foundations of Cognition

Posina Venkata Rayudu rvposina@yahoo.com, Date: 17 Dec 2001, Subject: categories: perception to knowledge, To: categories@mta.ca

Hi,

Recently Prof. Lawvere suggested a model for how experimental data lead to theory (see enclosed posting). In this context I would like to mention that there is similar problem in cognitive science. How do we acquire knowledge or how is perceptual experience transformed into conceptual knowledge? Moreover the discussion on generals & particulars readily lends itself to interpretation in cognitive terminology. We may compare particulars with sensation, abstract generals with categories (not the mathematical category) such as DOG, and concrete generals with prototypes. Stretching little further, presentation of abstract general can be compared with percept (of dog).

Of all the disciplines to which category theory is applied such as physics or computer science, I think the most natural domain of applications for category theory is cognitive science. Let me explain. Category theory captures mathematical practice. In the domain of mathematics, category theory provides a mathematical account of the process of transforming ignorance into knowledge. It is reasonable to treat the growth of mathematical knowledge as a particular case of knowing in general or cognition. Given that category theory models a particular case of cognition (mathematics), one strategy is to generalize the category theoretic description of the particular case of mathematical knowing to knowing in general or cognition. The ease with which we can implement and realize this research program is inversely proportional to the “distance” between mathematical practice and cognitive processes. The more cognition is similar to mathematics, the less are the changes or effort we need to make to the category theoretic model of mathematics for it to accommodate cognition in general. In view of the close resemblance between the cognitive process and the pursuit of mathematical knowledge (e.g. both describe real in terms of imaginary), category theoretic study of cognition is likely to be extremely fruitful.

One could motivate category theory in more concrete terms. For example, the problem of how perceptual experiences give rise to conceptual knowledge has a facet to which category theory has already provided solution. How do we go from figural or picturesque perception (geometry, topology) to propositional or symbolic thinking (algebra, logic)? Category theory explicated the connections between logic and topology and these insights can be brought to bear on comparable perception-thought transformations, and it possibly gives a cue to the role of thinking vis-à-vis perception.

I am sorry if I said something really stupid about category theory. I simply want to attract category theorists to cognitive science.

Thanking you,

Sincerely,

Posina Venkata Rayudu

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From: “F. William Lawvere” wlawvere@hotmail.com, To: categories@mta.ca, Subject: categories: Re: Sketches and Platonic Ideas, Date: 05 Dec 2001

In my 1972 Perugia Notes I had made an attempt to characterize the relation between these sorts of mathematical considerations and philosophy by saying that while platonism is wrong on the relation between Thinking and Being, something analogous is correct WITHIN the realm of Thinking. The relevant dialectic there is between abstract general and concrete general. Not concrete particular (“concrete” here does not mean “real”). There is another crucial dialectic making particulars (neither abstract nor concrete) give rise to an abstract general; since experiments do not mechanically give rise to theory, it is harder to give a purely mathematical outline of how that dialectic works, though it certainly does work. A mathematical model of it can be based on the hypothesis that a given set of particulars is somehow itself a category (or graph), i.e., that the appropriate ways of comparing the particulars are given but that their essence is not. Then their “natural structure” (analogous to cohomology operations) is an abstract general and the corresponding concrete general receives a Fourier-Gelfand-Dirac functor from the original particulars. That functor is usually not full because the real particulars are infinitely deep and the natural structure is computed with respect to some limited doctrine; the doctrine can be varied, or “screwed up or down” as James Clerk Maxwell put it, in order to see various phenomena.

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Posina Venkata Rayudu

http://www.mta.ca/~cat-dist/archive/2001/01-12

Memory Evolutive Systems