Abstract

What is intelligence?  What makes reason reasonable?  What good is being sensible?  We begin to address these questions in the geometric (algebraic) terminology of figures (particulars), shapes (generals), and perspectives (doctrines / monads).  In doing so, we bring into figural salience for all to see science as ever-proper alignment of reason with experience.  The attendant sincere audit of the practice of science is in the ancient deep time-tested spirit of respecting / valuing comprehension / declarative understanding, as exemplified in teaching / learning, for the express purpose of which our thoughtful ancestors conceptualized education.

Introduction

In the following we introduce terms such as intelligence and reason using the methods of category theory, which explicates what has hitherto been implicit in our understanding of these familiar terms.  Category theory, to the extent objects populating our perceptual experience belong to one or another category, is the theory of our everyday experience.  Objects of a category partake in the essence that's characteristic of the category; as such, morphisms between objects of a category are respectful of the categorical essences, and hence are natural transformations.  Of course, objects can be functorially transported across categories.  However, it all follows the Dharma: Becoming consistent with Being.

What is intelligence?

Intelligence is what intelligence is 'good for'.  Stated differently, intelligence is that which wouldn't be but for human intelligence.  Science is what wouldn't be but for human intelligence.  Hence intelligence is defined in terms of science understood as ever-proper alignment of reason with experience.  Note that science is a connected system with research, education, and applications as its constituent components.

The good-for method can be understood in terms of the following parallel

Structure : Architecture :: Function : Good-for

which is intended to convey that the good-for method is a refinement of more familiar method of defining in terms of functions / uses.

To see the refinement that is our good-for method, let's consider a simple question: What is a pen?  A pen, if used to scratch one's itching back, going by the functional approach, whose flaws were pointed out by Stephen Jay Gould in the context of evolution (along with many scientists in their respective subjects of study), lends itself to be defined as a scratching tool.  Since there is no limiting the imaginative uses of pens, we find that pens can mean anything, everything, and nothing.  Now, adopting the advanced good-for method of definition, we have: pen is what pen is good for; or, equivalently, pen is that which wouldn't be but for the pen, which is writing.  The good-for method, a refinement of functional definitions, defines PEN in terms of WRITING (and excludes poking, scratching etc., as it should; note that uses other than writing to which a pen may be abused are functions that can be accomplished in the absence of a pen, and as such do not figure in the definition of PEN).

The scientific content that is our good-for method dictates conceptualizations and calculations, dismissing outright any and all hasty thoughtless mischaracterizations (e.g., rhetoric / verbal gymnastics).  As an illustration, consider the formula 2^|A| that is routinely used to calculate the number of subsets of a set A.  Where did the base 2 come from?  The number 2 in the formula is the size of the subset classifier / truth value set 2 = {false, true} that is good-for perfectly parameterizing every subset of any set.  More explicitly, there is a one-one correspondence between subsets of a set A and functions from the set A to the truth value set 2.  Thus, the number of subsets of a set A is given by |2|^|A| = 2^|A|).  This good-for method is used to define almost all mathematical objects (e.g., separator, which, in the case of sets, is a singleton set 1 = {*} that can completely parameterize any set and is adequate to test for the equality of any pair of parallel functions) and operations (e.g., exponentiation, which perfectly parameterizes functions from a domain set [as exponent] to a codomain set [as base]).

Mathematical objects / operations defined in terms of our good-for method, i.e., in terms of universal properties, which are unique with respect to the entire universe of discourse / category.   Definitions in terms of universal properties involve: ... for every... AND ...there exists exactly one....  Universal mapping property definitions include, in addition to the quantifiers: for every and there exists a unique, additional commutativity conditions that are required to be satisfied (e.g., sum of two sets A and B is a set A + B, along with inclusions iA: A --> A + B and iB: B --> A + B of the summands (A and B), such that for every set T (of the category of sets) and for every pair of functions from the summands to T, i.e., fA: A --> T and fB: B --> T, there exists exactly one function f: A + B --> T satisfying both the commutative conditions: f o iA = fA and f o iB = fB, here 'o' denotes composition of functions; Sets for Mathematics, pp. 26 - 30).  Given that the universal mapping definitions are "the best", statistical approaches, including gradient descent, can be used to abstract the definition of SUM for particulars (e.g., 1 + 1 = 2). 

Defining objects / operations in terms of universal mapping properties often assert that a process is invertible.  For example, the [higher] universal property definition of exponentiation asserts that a two-variable function can be transformed into a function-valued single-variable function and that this process is invertible.

Now that we have substantiated the good-for method we invoked earlier in defining intelligence, we are well-positioned to define intelligence as the basis of science; more specifically, with science as ever-proper alignment of reason with experience, intelligence is the basic mechanism aligning reason with experience.

What is reason?

A reason, simply put, is a proof of a proposition; proving this or that assertion involves substitutions, existential and universal quantifications, conjunction, disjunction, negation, implication, entailment, and then there are, of course, true and false.  Since our objective is more elementary---alignment of reason with experience---we get behind the reason, so to speak, to note that propositions are made up of concepts, and highlight the hitherto not adequately recognized primacy of concepts (e.g., the concept of SPACE is more basic than the truth or falsity of any statement about spatial objects, such as: parallel lines intersect).  Now this brings us to the elephant in the room: abstraction.  How do we abstract?  On a related note, how do we generalize?  When to abstract vs. generalize?  Why bother abstraction or generalization, especially given that, according to His Holiness (of the USA / pragmatism) William James: particulars make us wiser more so than generals.  We have the contemporary artificial intelligence (AI) with its large language models (LLMs) grabbing every particular, or so they think, not knowing particulars are unlimited: Maxwell's doctrine requires mental effort, which AI experts, popping up like mushrooms in a Mexican desert, might find alien; so here's the Fodor's dog, hopefully entertaining enough, while surreptitiously sneaking in a byte or two about the nature of particulars.  I don't know if all those behind corporate (RIP academia ;) AIs are evil-incarnate, but I know for sure its demonic daddy Bill Gates is going straight to hell in a wastebasket.  Insulating from the evil we are suspended in, let's stay focused on doing what little good we can do to make our lived experience meaningful, beginning with words and concepts.

Abstraction

Mathematical abstraction of general theories from a given family of particulars bears a profound resemblance to the commonplace abstraction of general concepts from everyday particular experiences.  Hence we focus on mathematical abstraction.

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