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Julius Hamilton (Nov 16 2024 at 17:26):The issue of choosing some concepts as “primitives”, and others as constructed from primitives by rules, is a subject I wish to study in its own right, rather than leaving as a background assumption in some theory.
A common presentation of logic and formal systems includes the concept of “terms”, as well as “function symbols”, and the notion of “application” of a function symbol to terms. For example, here is how one paper regarding a certain term-rewriting algorithm begins:
Implicitly, it seems like there are already a number of “mental concepts” being used here, to present some of this information.
We are told there is something called “terms”, and something called “function symbols”. This, alone, already seems to imply that:
Even those rudimentary concepts probably contain more structure than one might initially realize. This may become more evident when one tries to “formalize”, or at least “analyze”, those concepts in turn.
For example: a “type” of a “thing” is something that stands in relation to a given thing; is associated with it. But, is a “type of a thing” itself also a “thing”? Is “thing” the most general mental concept possible, which anything conceivable or expressible is one of? Or, in this scenario, is “thing” a special type, and there are other concepts which “stand above” “thing” - properties, qualities, conditions of/on things? One could say that “the notion of ‘type’” is itself a “thing”, just by virtue of me being able to talk about it. One could also say that “thing” is a type, given that everything that exists (apparently) is of some type - at minimum, the type of “thing”.
Given these issues, I have recently started trying to define fundamental mathematical concepts in terms of “mental concepts”, rather than thinking there is some “canonical formal system” which I can formalize everything in.
I believe one of the most essential mental concepts is “putting two things together”. I have actually considered calling this “the axiom of reification”, which says something like “if you have two things, you can put them together, and that itself counts as ‘a thing’”.
Is this a good way to “get off the ground”, to define “the application of a function symbol to a term”? We might conceive of “application” in numerous ways. For example, it could just be seen as string concatenation, where we literally take a symbol “” and put it in front of a string of symbols “”. In set theory, it seems to me that “putting two things together” would be forming the set pairing the two things. If we needed them to be ordered, we could form the ordered pair.
But a lot of the time, if we try to use pre-existing theoretical frameworks to formalize something so “primitive”, I think we might be putting the cart before the horse, and it will only introduce more problems, more things to be formalized.
I think the most barebones aspect of what we are calling “function application”, in this context, is just “putting two things together”. It does not matter “in what order they are packaged together” (as in, an ordered pair like ); so long as whoever is “perceiving” this theory can mentally differentiate which element has which role (this symbol is the function symbol, this is the first argument to the function, this the second one, etc.) (To be extra nuanced: even though the arguments to the function carry information about their “order” relative to the function, they way we “store” this information does not itself have to be “ordered”.)
Is this a good way to avoid a bottomless chain of formalization? That the “application of a function symbol to terms” is just a pure mental concept, prior to any mathematical theories of sets, pairing, unions, functions, products, types, etc., which just says “putting them together”?
Ryan Wisnesky (Nov 17 2024 at 03:40):it's turtles all the way down. you pick your object language, pick your meta language, and then start proving theorems or writing programs. anything else is philosophy,
Rich Hilliard (Nov 17 2024 at 16:26):Regarding "putting two things together" you might find MERGE in generative linguistics of interest. See: https://arxiv.org/abs/2305.18278 and related papers by same authors.
Julius Hamilton (Nov 26 2024 at 21:09):That paper looks really interesting. Although it's too advanced for me, I wanted to share this thing I am working on, which might turn out to be similar, once I understand it better. Thanks.
a-minimalist-framework-for-ontologies.pdf
Posina Venkata Rayudu (Nov 29 2024 at 17:37):Putting Together those that Fit Together
Going by ~your terminological order:
The idea of theories themselves as structures whose mutual interpretations would form a category was also evidently possible if one cared to carry it out, and indeed Hall, Halmos, Henkin, Tarski, and possibly others had already made significant moves in that direction, p.10.
A key discovery around 1962 (foreshadowed by the algebraic logic of the 1950s) was that a theory A is also a category! That is because the most basic operation of calculation and reasoning (which as it turned out uniquely determines the other operations) is substitution, and substitution correctly objectified is composition, p. 236.
Composition is not always defined, and in fact is defined precisely under geometric conditions
A little too pedantic and a somewhat unfamiliar variants of putting together, so to speak.
Then there is the trinity: Thing vs. Body vs. Object; here object is a geometric objectification of a concept, the structure and logic of which is non-trivial; if we consider propositions as morphisms of the category of concepts, then the foregoing needs to corrected: Quotient Mind, as kindly pointed by Professor Andree Ehresmann.
Not to be forgotten is the rational passage between concepts that got lost in the rational exuberance of inferring sentences from sentences.
Be that as it may, given that concepts / theories can be objectified as categories, one definitive statement that one could assert about the rational passage between concepts is: functorial (A2. Functor, pp. 177-178).
While I am at it, I might as well note that mathematical objects can be represented as structures, but they're not structures; confusing models of reality with reality is a mistake!, then there is the absolutely fascinating correspondence:
consciousness : mind :: particular : general
along with indispensable scientific advances involving Professor Bastiani and Ehresmann Sketches, Grothendieck's Descent, and Professor F. William Lawvere's Naturality of formation of concepts in general.
Of all the two things I'm dying to be put-together are epistemology and ontology
I might even go a step further and say that all that's really interesting: putting together in science
It ain't shocking that Newton beat me by quite a few centuries ;)
As Mathematicians have two Methods of doing things wch they call Composition & Resolution & in all difficulties have recourse to their method of resolution before they compound so in explaining the Phaemoena of nature the like methods are to be used & she that expects success must resolve before he compounds. For the explications of Phaenomena are Problems much harder then [sic] those in Mathematicks. The method of Resolution consists in trying experiments & considering all the Phaenomena of nature relating to the subject in hand & drawing conclusions from them & examining the truth of those conclusions by new experiments & new conclusions (if it may be) from those experiments & so proceeding alternately from experiments to conclusions & from conclusions to experiments untill you come to the general properties of things, [& by experiments & phaenomena have established the truth of those properties.] Then assuming those properties as Principles of Philosophy you may by them explain the causes of such Phaenomena as follow from them: wch is the method of Composition.
P.S. According to Proust, things don't move because of the statis of our conceptions of them things ;)