In an earlier post, I remarked that philosophy, including philosophy of mind, appears to be a syntactic enterprise, whereas I tend to think of the mind as primarily semantic. In this post, I want to suggest a way of thinking about the mind that better fits with the idea that it is primarily semantic.

I shall take it that the word “mind” names an abstraction, an idealization that attempts to account for what we refer to as our mental activity. And I shall take it that the brain is what implements that mental activity (among other things). So I will be presenting an idealized account. Most of mathematics is about idealizations so, as a mathematician, it is natural for me to think in idealized terms. I do not find identity theory (the brain is identical to the mind, or mental states are brain states) at all persuasive or useful, if only because mind is an idealization, while brain isn’t. We do, of course, have to consider what sort of brain activity is involved in implementing mental activity. But that consideration need not be a crude identification of the two.

## Properties

As we look around, we might notice that things seem to be green in some parts of our immediate environment. It seems that we have the ability to assess color properties. Likewise, as we look around, we seem to be able to assess distance and relative location of various things, suggesting that we have an ability to assess locational properties. When I touch things, they might seem warm, which I suppose that we could think of as “warmth” property.

One of the things that our mind allows us to do, is to assess such properties. I shall consider the ability to assess properties as an important aspect of what the mind does and what the brain must be able to implement. With the aid of science, we can also assess other properties such as temperature, density, etc. Since I am treating mind as an abstract idealization, it will be useful to assume that the mind, too, can potentially assess the other properties.

Let’s use *W* to represent the world, by which I mean the world of our immediate environment, the world that we perceive. As part of our idealization, we may think of *W* as a topological space. With that in mind, a property can be identified with a continuous function over the space *W*. When we look around and see the colors that are present, we can think of that as using functions such as *greenness* and *redness* to our world.

Following the notation from mathematics, we may use *C(W)* to denote the space of all continuous bounded functions over *W* and I shall use *P(W)* to denote those functions which we have the ability to evaluate. We may think of the *P* there as standing for **p**roperty functions or for **p**erceptual functions, since it is via our perceptual abilities that we evaluate these functions.

The space *C(W)* is actually an algebra. If *f* and *g* are continuous functions in *C(W)* then so are their sum *f + g* and their product *f*g* defined by

*(f+g)(x) = f(x) + g(x)*

(f*g)(x) = f(x)*g(x)

(using *** to denote multiplication). Likewise, we may take *P(W)* to be a sub-algebra of *C(W)*.

Scientific laws such as Newton’s *f = ma*, and Ohm’s law *V = IR* express some of the known algebraic relations between our property functions.

## Facts

A typical example of a scientific fact, might be

- The temperature is 25 degrees at latitude 45N and longitude 90W.

If we consider temperature, latitude and longitude to be properties, then that fact is of the form:

there exists *x* in *W*, such that

*f(x) = 25, g(x) = 45, h(x) = -90*

where *f, g, h* are the functions corresponding to temperature, latitude, longitude.

More generally, what we think of as a fact is something like a disjunction of statements of the kind given above. Our property functions are what we need in order to be able to have facts about the world.

## The world in itself

Kant made a distinction between the world that we experience (the phenomenal world), and the real world (the world in itself, or the noumenal world). This is illustrated in my example above of a fact. That the fact begins “there exists *x* such that …” illustrates that we do not have access to the world in itself. We assume that there is such a world, but we cannot identify any actual point *x* in that world. Our access to the world is filtered through the property functions that we use to access the world.

## Knowledge of the world

From the mathematics, we know that the algebraic structure of *C(W)* determines much of the structure of *W*. The classic treatment of this mathematics is in Gillman and Jerison, *Rings of Continuous Functions*, though it is also found in many textbooks on functional analysis. The google page for the book is here.

This suggests that the way to find knowledge of the world is to find property functions that we can use in our descriptions of the world, and to determine the algebraic relations between those property functions. And that does seem to be how much of science works.

## Implementation by the brain

My view is that the brain is massively involved in measurement activities. A measuring ability is what enables us to assess properties of parts of the world, so a measuring ability provides us with property functions. A mistake commonly made is to assume that perception is passive. Once we recognize that we are actively engaged in perceiving, we can recognize the importance of developing such measuring abilities (or property functions). This can roughly be summed up with my preferred slogan of Cognition is measurement.

An algebraic relation between functions is a special case of a functional relationship of the form *F(f,g,h,…) = 0*. Knowing such functional relations should be sufficient. It is not necessary that they be in an algebraic form. It seems likely that the brain represents functional relationships in the form of piecewise continuous functions that become wired into the neural network as a result of our learning from experience. The observed Hebbian learning should be about what is required to develop such functional relationships.