# Defining INTERPRETATION

Let’s say you are thinking of something–about some species which might eventually become qualified to be called a theory–a theoretical structure about that thing you are thinking:

* _{<—}=^{—>} *

(I wish WordPress can read my mind and format accordingly, but until then what we have above is a pair of dots with an arrow going from the dot on the left to the dot on the right and another arrow going from the dot on the right to the dot on the left.)

With an eye on sharing–on communicating–my thoughts with my fellow gypsies, I think of describing–think of bringing my thoughts into being–of endowing my finite thought a material existence using a systematic procedure–a reproducible method.

I start with naming. Given my messianic complex I label one of my two dots, say, the one on the left, ‘true.’

Reflecting on my thought I notice that if I start with truth and go (along the arrow going from the dot on the left to the dot on the right) to the dot on the right, and then going from the dot on the right along the arrow going from the dot on the right to the dot on the left, then I return to the ‘true’ I started out with; all of which pleases me so much so that I write it down (even if it means invoking a negation a couple of times):

not (not (true)) = true

Of course I could have penned a [slightly] different script i.e. a different opening line naming the dot on the right as ‘false’ and then rewrite the whole story as

not (not (false)) = false

What we have here is a slow-moving plot of a finite [dynamical] system with two descriptions:

Description 1.

Generator **G**: true

Labels **L**: true, not (true)

Relation **R**: not (not (true)) = true

Description 2.

Generator **G**: false

Labels **L**: false, not (false)

Relation **R**: not (not (false)) = false

What about the ‘defining interpretation’ that you advertised in the title? Please don’t tell me you have to go away now–to go study. I’m sorry to disappoint you again, but I need to study (seriously; for my own [selfish] good):

An important method of defining an interpretation of one presented structure into another is in terms of mapping generators and verifying relations.

A prerequisite for understanding the above method is an even more careful study of the Presentations of dynamical systems in Conceptual Mathematics; pp. 182 – 6, 250 – 3.

Much of the above is to protect myself from falling prey [yet again] to scandalous brain theories that the Society for Neuroscience shamelessly palms off on the unsuspecting public all the while refusing to acknowledge an important thesis of science, which is absolutely necessary for the advancement of science–a sincere means of making sense of the ‘blooming buzzing confusion,’ simply because it is scared of learning.

Sad, a sad state-of-science. Please don’t get me wrong: technology is going great even if it means engineering deans have to spend a lot of time scratching their heads–whom to call in which company (google, facebook, twitter… so many—so confusing)–to figure out what to “teach” their students.

Tailpiece: One–only one–sensible brain theory, according to someone claiming to have coined computational neuroscience, in the study of brain, is the one he heard from his preteen daughter at their dinner table: Brain, while sitting on top of you, steals all the food you eat. It sort of makes sense not only in terms of energy, but also in light of my feeling–wasted :-(

Before I go, let me [leave you, in the spirit of the joyous season, on a happy note] thank Professor F. William Lawvere for the best Christmas gift (Algebraic Foundations of Physics and Engineering)** **I ever got :-) Thank you Professor Lawvere!

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Interpreting theories

F William Lawvere wlawvere@buffalo.edu, 12/24/12

Dear Posina

Let me say another word about the thermostatics which may not seem categorical at first sight. It had two motivations: to actually learn the subject as described by my teacher Truesdell by putting it into more strictly algebraic form and to provide a significant example for my 2nd semester calculus class of the role of partial derivatives (augmenting the physical chemistry courses that biological students will need to study). Because there are several traditional equivalent formulations as well as inequivalent formulations that describe strict subclasses of materials, I realized that this is an excellent arena to illustrate the method of presentations of algebraic theories and of interpretations between such. I think it is fairly clearly outlined in a chapter of my thesis (TAC Reprints), but for some reason people in several walks of mathematical life have great difficulty in grasping it, although it is widely used in group theory and commutative algebra.

A system of generators and relations is an important way of calling into idealized existence a particular algebraic structure of a certain species or doctrine, while an important method of defining an interpretation of one presented structure into another is to define generators of the first as combinations of generators of the second, and then carefully verify that the relations of the second follow from the relations of the first. The syntax of presentations has often been called theory in certain contexts but I thought (50 years ago) that in many categories it would be more accurate to use that word for the theoretical structures themselves.

The difficulty people have seems to stem from the prevailing dogma that in logic there are only the two, syntax and semantics, whereas for groups or rings in actual mathematics they are used all the time as a third that is both presented (by word problems, groebner bases etc which may not be actually called syntax) and represented (by permutations, modules, etc which are not usually called models but they are).

In any case terms like syntactical category or categorical counterpart as found on the internet only add to the confusion. (This point was already well understood for first-order theories by Halmos in the 1950s, but many seem to have missed that.)

Of course there are theories in constant use which did not arise from a presentation, such as Cinfinity rings or groups of distance-preserving maps. More generally my algebraic structure of a given functor (its natural endos etc) is theory which every value of the functor uniquely lifts to have. This provides useful invariants (eg Cohomology operations) even though the domain of the functor may not be algebraic in character.

Best wishes to you and your family.

Bill Lawvere

—————

F William Lawvere wlawvere@buffalo.edu, 01/10/13

Dear Posina

Please forgive my misuse of the pronouns “first” and “second” in my too-brief description of presentations of homomorphisms / interpretations. To avoid confusion it is best to start with the situation where the second structure is not equipped with a presentation but the first one is. The generators are mapped to elements of the second structure (“defined as”) in such a way that the relations become true. In case the second structure is also equipped with presentation, truth MEANS “following from” its relations by means of the appropriate rules of inference (in the case of algebras, the rules of inference are essentially just substitution of equals for equals).

I hope this is a little more clear.

Bill Lawvere

————

F William Lawvere wlawvere@buffalo.edu, Jan 11, 2013

Dear Posina

Please forgive me for having inadvertently led you into this exercise. I mean the one of trying to explain something precisely using words not symbols. In theory it should be possible but in practice it may take many words and a patient reader. For example, my 1990 paper Categories of Space and of Quantity has content that mathematicians who have read it deem very appropriate for the intended philosophical audience. But there is little evidence of any impact on philosophers themselves. I had heroically tried the same exercise, it having been impressed on me that philosophers might have difficulty with symbols. (Probably the mathematical readers “cheated” by introducing their own diagrams and equations.)

Truth in a presented structure (even if it is the codomain of a map) has in itself nothing to do with some other structure, but of course its definition, as provable from the relations, is required for correctly interpreting the notion of an interpretation in case the presented structure is indeed to be the codomain.

Elements of a presented structure are often called words or polynomials or WFFs, generalizing particular classical cases.

When a discussion of an OR becomes nonsensically extended, often the correct answer is YES (BOTH). A symbolic treatment can be the underpinning of a description in words. (And the Russell-Church metaphysic is just as metaphysical (=one-sided) as the above exercise).

I often explain to students that the need for symbols stems from the fact that natural language does not have enough pronouns (even if the latter is interpreted liberally to include words like former and latter which may not usually be considered such).

Of course that theory is incomplete, but I wonder if our cogsci friends can account for even that much.

Best regards

Bill Lawvere