Constructing a World
Imagine yourself a deity idling away your days. It occurs to you to do something with your life (and the peer pressure has been mounting), so you decide to construct a world. The activity is new to you and you have never taken lessons, so you have to start from first principles. What is a world? It’s a way things are—or the way stuff is. It is something with definite being, the opposite of nothing. So far so good: but it’s not very far. How do you make one of those? You ponder the question and come up with the idea of objects: you have to create some objects. Okay, you can do that, but it quickly dawns on you that that is not going to be enough: the objects have to add up to a way things are, but mere objects can never do that—they need to be a certain way themselves. What is such a way? Here you feel stuck because you have been assuming that a world is a totality of objects—what else could it be? Eventually the solution occurs to you: you need to give the objects properties. These are things that confer a nature on objects—that give them definite being. Ways things are consist of properties objects have. It may seem ontologically excessive but there is no alternative: a whole new level of being must be invented or else your world will be indistinguishable from nothing. So you prepare to set a load of properties next to your bunch of objects. But then a problem strikes you: how are you going to connect the two? You can’t just have the objects and the properties; the objects must have the properties—they must (what’s the word?) instantiate the properties. Now this is going to require some serious world-engineering expertise! You wonder whether to consult the head god (old Grey Whiskers) but decide this would be embarrassing, so you set about solving the problem yourself—you too are a god after all. A clue comes to you in the form of the concept of a fact: you need to make facts from objects and properties, so there has to be a kind of glue that joins them to make facts. A fact is a type of combination, but combinations need methods of combination—combinatorial adhesive. Your objects and properties need to be formed into unities not just placed blindly next to each other.
Functions! These are things that apply to an object to yield another object: they are rules for mapping one object to another. Let the objects act as input to these rules and let the properties be the rules that yield facts as output. Properties must be so constructed that it is in their nature to generate facts from objects, and the notion of function is just what is needed. Design the properties as functions; then you will get the unity you need. So now the ingredients are all in place to make a nice shiny new world: objects, properties-as-functions, and facts. It’s a good thing you had the idea of functions or else your world might never have got off the ground. Functions are very useful in the business of world construction—functional, you might say. Now you just have to decide what kinds of objects and properties to create, but the real brainwork has been accomplished—you have a workable blueprint for constructing a world. A world is now within your reach.
Some time later, with the world construction finished, you are feeling a bit bored so you decide it would be nice to create a way of describing the world you have constructed. You want to create—what to call it?—a language. But what is a language? It has to say things, surely: it has to say how things are in the world. The world divides up into facts, so the language needs to divide up too, and these items must say something. Let’s call them sentences, conceived as complete expressions that say something. But what is a sentence made of? You are the one constructing them, so this is a question you need to resolve. Clearly you are going to need sentence parts that correspond to objects—you call these names. But these are not enough because you also need sentence parts that correspond to properties—call these predicates. Then the sentences will correspond to facts, since objects combined with properties (via functions) yield facts. So now you have names and predicates strung together: but hold on, that is just a list—how do we get from a list to a unified complete sentence? Remembering your similar question about objects and properties, the solution is staring you in the face: functions! A predicate is a function from names to sentences: input the name; output the sentence. Predicates act as functions, creating unities from mere lists: the predicate isn’t just next to the name; it acts on the name. By acting on the name it forms a sentence that has an internal unity and says something—it describes a fact. So language is functional as the world is functional: function-argument structure runs through it, forming it, making it possible. Functions are not just central to mathematics; they are also central to worlds and languages. They are the binding glue.
One Sunday afternoon you hit on another plan: you decide to create some thinking creatures. You are going to need some creatures, clearly, and some thoughts for them to have. How to create a thought? Well, thoughts are about things, so they need parts to do that—parts that are about objects, evidently. The thoughts need to be about the world you so carefully constructed. Let’s call these parts individual concepts: they are a bit like names, but let’s not jump the gun, so we won’t just make up some more names for thoughts to use. These are psychological entities. But a thought can’t just be a concatenation of individual concepts; thoughts need general concepts too. So a thought is a combination of an individual concept and a general concept: but it is not just any old combination but one with a special sort of unity. You have been this way before so the problem is no great stretch: general concepts need to be functions from individual concepts to thoughts. The complete thought is the output of the function and the individual concept is the input to the function. The general concept is a function that acts on another concept as argument to give a thought as value. This enables the creatures you have created to have thoughts about the world you created. If you make these creatures in such a way that they can speak as well as think, then we can say that their sentences express their thoughts as well as describe reality. So world, language, and thought all line up: combinations of parts unified by functions. Note that the functions don’t cross boundaries: thought functions don’t take objects as arguments and give facts as values but individual concepts as arguments and complete thoughts as values. Nor do sentences step outside of the linguistic domain, keeping arguments and values within language. The way you set things up the arguments of the functions are confined to their proper domain—whether objects, names, or individual concepts. The trick has been to exploit the concept of a function to solve a problem in reality construction—facts, sentences, and thoughts.
It is noteworthy that each step of your reasoning was substantive: there was no obvious move from the bare concept of a way things are to the apparatus of objects, properties and functions; and similarly for language and thought. The world as we have it conforms to these categories, but they are not part of the very idea of a world—the idea of definite being, or of a way things are. You made a conceptual choice; you worked with a theory. You imposed these concepts on a pretty blank slate: maybe you could have dispensed with the idea of objects and worked with an ontology of stuff, or just object-less instantiation (“feature-placing”). Once you made these decisions you faced certain challenges, notably the problem of joining properties to objects to generate facts, which luckily you solved. Senior gods may have shaken their heads over your decisions, preferring some other way to make a world (what about starting with undifferentiated substance and then carving it up?). The object-property scheme was bound to encounter the unity problem, which you solved by the skin of your teeth (or by dint of sheer brilliance, as you would prefer to put it). It might not have worked at all but for the concept of a function! What if there had been no mathematics to copy? You succeeded in constructing a world, but you might have failed at the early stages through lack of appropriate adhesive. Worlds don’t come easy.
 Frege’s insight in semantics (which I have not considered here) was that if you want to develop a workable theory of meaning you need to invoke the apparatus of function and argument. That is, if you were constructing a meaningful language, you should (must) build into it the function-argument structure. He thus took what he called concepts, the denotation of predicates, to be functions from objects to truth-values. This basic approach can be adopted without accepting Frege’s specific stipulation about concepts. I have transposed this theoretical approach to semantic theory into a recommendation about how to construct a meaningful language ab initio (as well as worlds and thoughts).