# Set Theory and Psychology

Set-Theoretic Psychology

Set theory as a branch of mathematics is a comparatively recent invention, but the concepts it uses are surely planted deep in the mind. The idea of membership in a collection is primordial: things do not just exist separately; they exist as elements of something more inclusive. They make up totalities. Set theory formalizes these intuitive notions, bringing out hidden aspects of them. Perhaps the most important is the idea of *subset*: sets not only have members, they have other sets as subsets. This allows for recursion: we can form a set from a pair of elements and then form another set containing this set, and so on. It is easy to produce infinitely many sets this way, starting with finitely many basic elements (even one or zero). We also have notions like the unit set and the null set. A rich structure can be erected on a simple basis by iteration. Evidently set formation is a prodigious operation. But it all proceeds from intuitively accessible foundations—from elementary facts about the human conceptual system.

It is doubtful that other animals share this set-theoretic competence. No doubt they grasp the nature of collections and what it is to be part of a collection, but it is unlikely that they grasp the idea of recursive set formation, let alone the concept of the unit set or the null set. They don’t grasp the general theory of collections, just certain kinds of collection (pack, hive, herd, etc). Set theory seems distinctively human. Perhaps Neanderthals understood set theory, or were capable of understanding it, but today we alone on earth understand it. Other intelligences on other planets no doubt understand it, perhaps more easily than we do. In any case, I hazard that set theory is universal among humans: not explicitly, of course, but implicitly. We all grasp the basic notion of set formation and its recursive power (not so for calculus and topology, say). Why we do is a difficult question: How and why did this understanding evolve? Was there a specific mutation that led to it? Do we have dedicated genes for set theory? How modular is it? We can think of it as a cognitive schema, an organ of thought, with its own inner principles, existing alongside vision, language, theory of mind, and so on. It is a schema we can apply to the world, imposing a certain kind of order—the logic of collectives (it is close to logic conceptually).

Some of its applications are familiar, even famous. The use of set theory in the foundations of mathematics is the most obvious, but we also have the idea of the extension of a predicate, model theory, the ordered pair along with the generalized notion of a sequence, sets of possible worlds, and so on. But there are also less technical applications such as social cognition: we can think of groups of individuals, and groups of groups of individuals. There is membership in a club and clubs can be members of further social collectives (the league of all tennis clubs, say). Think of the concept of a family as a function from individuals to families: the arguments are individual people and the value is the family unit. This is an important function for us: it unites separate people into a larger unit—a set of family members (note the term). But this set can in turn be a member of a larger set—a tribe, a village, a province, a nation, or a whole world population. There are functions from sets to sets as larger social groupings are formed. Do chimps understand that there is a set of chimps in Africa and a bigger set of sets of chimps drawn from other places? We grasp this quite easily and it is just an application of the basic property of sets: set formation and set recursion, which delivers sets as subsets. This provides unlimited powers of combination, so that we can form the idea of arbitrary collections of people, and collections of collections, up to the set of all sets of people. Social cognition is thus set-theoretic in character: that is, social cognition exploits our general implicit competence in set theory. This makes it generative and unbounded—infinitely many groupings from finitely many basic elements (people).

We can also detect the workings of set theory in other areas of human cognition. Elsewhere I have outlined a general theory of human psychology emphasizing discrete segmented units combined according to recursive rules, including not just social cognition but also geometry, logic, and language.[1]Now I wish to add the principles of set theory to this general theory in the obvious way. In geometry we have geometrical figures that can be arranged to form patterns: concentric circles, recurring leaf shapes, snowflakes, bubbles, and so on. We apprehend these patterns as groupings, and they may have groups as their members (patterns within patterns). Architecture is set formation—from bricks to buildings. Elements are seen as members of a larger collective—as contributing to a totality not merely as isolated individuals. And we can also understand the constituent figures themselves as collectives—as made up of collections of lines, suitably arranged. A triangle is a collection of three straight lines, and so on for the other polygons. Further, lines themselves can be seen as sets of points. The concept of a set is extremely flexible, as well as topic-neutral. We can reduce arithmetic to pure set theory, but we can also conceptualize geometry set-theoretically. Again, we have a set of discrete elements, more or less fine-grained, and a recursive procedure that generates infinitely many geometrical forms. Similarly in logic: we collect propositions into the premises of an argument, forming a set of related propositions, and then we deduce a conclusion form this set. The premises are grouped together set-wise and the whole set leads to a certain conclusion. Just so a scientific theory is a set of propositions that leads to certain predictions. The web of belief is a precisely a collection of related beliefs—a belief set. Such a set can be a subset of a wider set—the beliefs of an entire community. Arguments are infinite in number, seeing that they consist of propositions that are infinite in number. There are only finitely many rules of inference, but these rules can operate on infinitely many propositions (themselves derived from finitely many elements): they can produce an infinite number of premise-conclusion sets. Conjunction elimination by itself can yield infinitely many arguments simply by adding conjuncts to the set of premises. So we can see logic (deductive and inductive) as constructed against a background of set-theoretic thinking, specifically the idea of an assembly of premises. If the premises were not seen as members of a logically significant set, we would not understand them as part of an argument (“From *this*set of premises you can derive *that*conclusion”). Set theory allows us to apprehend elements as elements *of*something—as members of a totality, as belonging together. The world is thus not the totality of objects but the totality of *totalities*of objects. That is the human way of conceiving things, thanks to the set theory embedded in our genes and shaping our minds.

What about language? An obvious first thought is that a sentence is a set of words: it is a unity made up of elements, with words constituting its membership; and speakers understand it *as*a set. But it is a special kind of set, for it has sets as its members, i.e. phrases, clauses, and sub-sentences. A noun phrase is a set of words, as is a verb phrase; they are word groupings. And spoken words are groupings of phonemes—sets of sounds. Thus we can break down a sentence into subsets at various levels of analysis: the sentence results from set formation beginning at the most elementary linguistic level. First a function takes phonemes into words; then a function takes words and forms them into phrases; finally a function takes the phrases into a sentence—all united by the subset relation. This set-theoretic structure readily generates an infinite array of sentences from a finite basis, simply from set recursion. Here we make contact with Chomsky’s work on *Merge*and the minimalist program.[2]As Chomsky explains the *Merge*operation, it is simply set formation: from two syntactic objects, *x*and *y*, we can form the set {*x*, *y*}, and then this new syntactic object can *Merge*with another syntactic object *z*to obtain {{*x*, *y*}, *z*}—and so on indefinitely. For example, from “reads” and “books” we can derive the set {“reads”, “books”}, and then this set can *Merge*with “John” to get {{“reads”, “books”}, “John”}. Intuitively, this is a representation of “John reads books”, where “John” is a noun phrase and “reads books” is a verb phrase. The nested sets provide groupings of words arranged hierarchically. What is interesting from the present perspective is that a mental process is construed in set-theoretic terms—operations on sets basically. The mental *structure*is set-theoretic. This means that what Chomsky calls the Basic Property of language is being explained in terms of set theory: we are deriving infinitely many sentences from a finite number of elements by means of an operation that iterates. In fact, the same schema that applies elsewhere also applies to the case of language: set theory in all its generative splendor. The language faculty operates by set-theoretic principles (at least in part), given that *Merge*is the fundamental principle. The elements are different from other elements—being words not people or geometrical figures or propositions—but the abstract schema is much the same. We are always thinking in terms of recursive set formation, forming sets and then making them members of other sets. Just as societies, geometrical arrays, and logical arguments are set-theoretic entities, as we conceive of them, so sentences are conceived as set-theoretic entities (and at several levels).

As Chomsky notes, the more minimal the machinery is, the easier is the evolution of that machinery. But it is also true that the more minimal the machinery, the more likely it is to be shared, because it won’t be as specific as richer machinery (such as transformational grammar). If *Merge*is the basic principle of linguistic construction, it is likely that it will be shared by other cognitive systems, being so general and abstract. The set-theoretic characterization of *Merge*thus invites the hypothesis that set theory might be operative in other areas too, which is precisely what we find. For example, we can apply *Merge*to several individuals to form a family, and then we can apply*Merge*to that family along with other families so as to produce a society. At some point the human mind acquired the abstract concept of a set, probably because of a chance mutation, and this concept, along with associated principles, began to shape human cognition in diverse ways. We became set-minded, generating sets all over the place. The different domains imposed their own content on the abstract principles, but the structure remained the same. A possible hypothesis is that social cognition came first, with set-theoretic thinking at an adaptive advantage, and only later did the cognitive machinery become co-opted by language; certainly we were an intensely social species long before we developed human language as it exists today (there might have been only simple signaling systems prior to that). We don’t know how individual words and concepts evolved, but grammar itself might have evolved from social forms—that is, social cognition might have been the precursor to grammatical cognition. If both are fundamentally set-theoretic, that is at least a possible evolutionary path. Once the elements exist it doesn’t take much to generate a rich structure built around sets of elements. Set theory has enough structure to generate arithmetic, so surely it has enough structure to generate language (or geometry, social reality, and logical deduction). The grammatical structure depicted by tree diagrams or by bracketing can equally be depicted by the nesting of sets. And when it comes to psychological processing, the notion of a set is entirely natural: we naturally think in terms of collections and membership. Phrases are thus conceived as collections that are subsets of larger collections—sentences, obviously, but also sequences of sentences such as conversations, arguments, articles, speeches, books, libraries, and world literature. The set-theoretic template provides a pleasing theoretical unification, as well paving the way for realistic evolutionary explanations. It’s set theory across the board. What is surprising is that set theory, as a branch of mathematics, was not developed until quite recently (early twentieth century), given its evident psychological ubiquity.[3]

Colin

[1]See my “Sketch for a General Theory of Human Psychology”. The present paper is designed to enrich the apparatus of this paper by adding the resources of set theory.

[2]See Robert C. Berwick and Noam Chomsky, *Why Only Us: Language and Evolution*(MIT Press, 2016). By the way, the word “merge” seems the wrong word to capture the intended notion, since the members of a set do not merge—they remain separate and distinct. The elements join and assemble, but there is no *merging*.

[3]It is an interesting fact that we have many synonyms for “set” in common usage, signaling the saliency of the notion: class, group, assortment, assemblage, batch, body, bunch, bundle, cluster, clan, company, collection, coterie, crew, gaggle, gang, lot, multitude, plurality, totality. Our language is set up for set theory.

Colin, quick comment on this point:

“Perhaps the most important is the idea of subset: sets not only have members, they have other sets as subsets. This allows for recursion: we can form a set from a pair of elements and then form another set containing this set, and so on.”

This isn’t quite right. It is not the subset relation which result in this outcome. Rather, it is pair formation, which is quite different,

x, y |-> {x,y}

Subset is, formally, the analogue of “less than” in a Boolean algebra or a set algebra (with union, intersection and complementation): “x is a subset of y” is short for “y = x union y”. This theory doesn’t implement recursion. There are finite Boolean algebras, such as the two-element algebra TOP, BOTTOM (or TRUE, FALSE). To use technical jargon, the theory of a Boolean algebra doesn’t interpret Q (Robinson arithmetic). And in order to implement recursion in some way, the theory needs to interpret Q (or a close relative, such as adjunctive set theory, AST, or the theory of concatenation, TC).

Theories that implement recursion in the way you’re interested are required to “represent recursive functions” (a more sophisticated condition is a theory’s being “sequential”). For example, Q represents all recursive functions. By some kind of coding, the other theories mentioned do so as well. So, in summary,, certain theories

– Robinson arithmetic Q,

– theory of concatenation TC,

– adjunctive set theory AST

do satisfy this condition. (Fancy theorems show how to “interpret” these theories in each other: these results require some advanced methods in mathematical logic which I can’t summarize.)

But theory of subset (basically Boolean algebra) does not satisfy this condition. So, the condition is not connected to the binary relation “x is subset of y”, but rather the far more complicated relation “x is an element of y”.

I haven’t read what Chomsky says about MERGE; but the formal properties of the MERGE operation will be connected to the concatenation operation “z is the concatenation of x and y”; and this does of course (as the theory TC shows) allow the implementation of recursion.

Jeff

OK, I’ll take your word for it.

There are different perspectives here. Is the basic notion of set theory elementhood and membership, is it subset, or is it more generally the concept of function (transformation of one set into another, of which both elementhood and subset are specific examples)?

Bill Lawvere proposed understanding (and doing) set theory in terms of category theory (an algebraic framework that has the notion of function or transformation of objects as a primitive). See for example:

http://www.tac.mta.ca/tac/reprints/articles/11/tr11.pdf

This approach has well known connections with the foundations of geometry (for examples, in terms of topos theory) as approached by the Fields Medalist, and one of the greatest mathematicians of the last century, Alexander Grothendieck.

Category theory has been used extensively in theoretical computer science; and quick google search on ‘category theory and psychology’ throws up a number of links that may be of interest.

Interesting to ask how much of set theory is universal and innate and how much is not.

Colin, a set is the sum of its parts, it is not something more.

This is not necessarily the case in geometry: a geometric object has parts, but there is also information in how these parts are glued together (e.g. a topological manifold); it may have intrinsic global properties such as the number of holes or its dimension; it may have quantities varying over it, for instance describing local curvature; the object may only make sense in relation to something else, for example a curve embedded in some surface (describing a trajectory), or a geometry varying over another object (e.g. the Mobius band varying over a circle, or a Hopf Fibration varying over a sphere – see https://wikivisually.com/wiki/Hopf_fibration); and a geometrical object may have a boundary (taking the boundary can be seen as an operation, which I believe has relevance to your post The Philosophy of Psychology).

Presumably many of the objects of interest in psychology have the property (structure?) that the collection, or ‘sum’, of their parts are not adequate enough to characterize the whole.

For example, consider compound concepts or words (and the controversies over whether other primates, e.g. Koko the gorilla, can also form them). A water fowl is more than a fowl that happens to be near water, and a ‘finger-bracelet’, or what we humans call a ring, is more than a bracelet that just happens to be on a finger.

Geometrical psychology has a lot of fine structure apparently.

On how much of set theory is innate: did Leucippus and Democritus use observation or introspection to develop the concept of atom?

[Note: I am unable to Reply to specific comments, but only post general comments.]