## Thursday, February 7, 2019

### Properties, relations and functions

Many philosophical discussions presuppose a picture of reality on which, fundamentally, there are objects which have properties and stand in relations. But if we look to how science describes the world, it might be more natural to bring (partial) functions in at the ground level.

Objects have attributes like mass, momentum, charge, DNA sequence, size and shape. These attributes associate values, like 3.4kg, 15~kg m/s north-east, 5C, TTCGAAAAG, 5m and sphericity, to the objects. The usual philosophical way of modeling such attributes is through the mechanism of determinables and determinates. Thus, an object may have the determinable property of having mass and its determinate having mass 3.4kg. We then have a metaphysical law that prohibits objects from having multiple same-level determinates of the same determinable.

A special challenge arises from the numerical or vector structure of many of the values of the attributes. I suppose what we would say is that the set of lowest-level determinates of a determinable “naturally” has the mathematical structure of a subset of a complete ordered field (i.e., of something isomorphic to the set of real numbers) or of a vector space over such a field, so that momenta can be added, masses can be multiplied, etc. There is a lot of duplication here, however: there is one addition operator on the space of lowest-level momentum determinates and another addition operator on the space of lowest-level position determinates in the Newtonian picture. Moreover, for science to work, we need to be able to combine the values of various attributes: we need to be able to divide products of masses by squares of distances to make sense of Newton’s laws of gravitation. But it doesn’t seem to make sense to divide mass properties, or their products, by distance properties, or their squares. The operations themselves would have to be modeled as higher level relations, so that momentum addition would be modeled as a ternary relation between momenta, and there would be parallel algebraic laws for momentum addition and position addition. All this can be done, one operation at a time, but it’s not very elegant.

Wouldn’t it be more elegant if instead we thought of the attributes as partial functions? Thus, mass would be a partial function from objects to the positive real numbers (using a natural unit system) and both Newtonian position and momentum will be partial functions from objects to Euclidean three-dimensional space. One doesn’t need separate operations for the addition of positions and of momenta any more. Moreover, one doesn’t need to model addition as a ternary relation but as a function of two arguments.

There is a second reason to admit functions as first-class citizens into our metaphysics, and this reason comes from intuition. Properties make intuitive sense. But I think there is something intuitively metaphysically puzzling about relations that are not merely to be analyzed into a property of a plurality (such as being arranged in a ball, or having a total mass of 5kg), but where the order of the relata matters. I think we can make sense of binary non-symmetric relations in terms of the analogy of agents and patients: x does something to y (e.g. causes it). But ternary relations that don’t reduce to a property of a plurality, but where order matters, seem puzzling. There are two main technical ways to solve this. One is to reduce such relations to properties of tuples, where tuples are special abstract objects formed from concrete objects. The other is Josh Rasmussen’s introduction of structured mereological wholes. Both are clever, but they do complicate the ontology.

But unary partial functions—i.e., unary attributes—are all we need to reduce both properties and relations of arbitrary finate arity. And unary attributes like mass and velocity make perfect intuitive sense.

First, properties can simply be reduced to partial functions to some set with only one object (say, the number “1” or the truth-value “true” or the empty partial function): the property is had by an object provided that the object is in the domain of the partial function.

Second, n-ary relations can be reduced to n-ary partial functions in exactly the same way: x1, ..., xn stand in the relation if and only if the n-tuple (x1, ..., xn) lies in the domain of the partial function.

Third, n-ary partial functions for finite n > 1 can be reduced to unary partial functions by currying. For instance, a binary partial function f can be modeled as a unary function g that assigns to each object x (or, better, each object x such that f(x, y) is defined for some y) a unary function g(x) such that (g(x))(y)=f(x, y) precisely whenever the latter is defined. Generalizing this lets one reduce n-ary partial functions to (n − 1)-ary ones, and so on down to unary ones.

There is, however, an important possible hitch. It could turn out that a property/relation ontology is more easily amenable to nominalist reduction than a function ontology. If so, then for those of us like me who are suspicious of Platonism, this could be a decisive consideration in favor of the more traditional approach.

Moreover, some people might be suspicious of the idea that purely mathematical objects, like numbers, are so intimately involved in the real world. After all, such involvement does bring up the Benacerraf problem. But maybe we should say: It solves it! What are the genuine real numbers? It's the values that charge and mass can take. And the genuine natural numbers are then the naturals amongst the genuine reals.

NobodyInportant said...

I know that this is realy off topic but your talk about properties reminded me of an issue that I have for a long time regarding necessary existence and physical reality.

Why could it not be the case that some fundamental layer of physical reality necessarily exists aka that it cannot fail to exist? Suppose that we are dealing with some form of "atomism". How do we know that some kind of physical particles or fields don't have necessary existence? Existence could be a PART of particles/fields or some other kind of fundamental physical reality. Then, what it means to be a particle/field ENTAILS, among other things, that this particle/field EXISTS.

In order to exist necessarily a thing would then have to be a kind of thing that has certain determinable physical properties (aka charge or mass or whatever...) and those properties then would be properties that necessarily have to be instantiated, because something HAS TO exist necessarily (since I believe that NOTHINGNESS as such is metaphysicaly incoherent). So, why not think that there is a set of determinable physical properties that out of ontological necessity cannot fail to be instantiated?

Even if hylomorphism is true, and anything physical is a composit of matter and form or an actualised potential of prime matter, or better to say, physical reality is INFORMED matter, then why not think that there are particles/fields or some other kind of physical reality that NECESSARILY cannot fail to EXIST aka that there is some ontological necessity that there has to be INFORMED matter in the form of particles/fields? Just because something is a COMPOSIT that doesn't mean that it can FAIL TO EXIST as it seems to me at least, I don't know. Because I don't see a reason why there cannot be certain composites that have to exist necessarily? Why can it not be that some kind of composition has to be necessarily instantiated?

Hence, are there any good reasons to prefer God as a necessary being aka a being who's essence and existence is identical and who cannot fail to exist, to some kind of physical reality? Why can it not be that some kind of physical reality necessary has to exist? This is my problem, I tried but I simply cannot solve this problem. How can we argue against the view that atheists present to us, that there is some kind of fundamental layer of physical reality, some physical thing with physical properties, that has to exist necessarily?

Again, I am sorry for being tedious with my questions, but I simply don't know whom should I ask and what to read because nobody deals with this question when it commes to arguments for a necessary being. Also, I hope that my question was clear, I tried to explain the problem as best as I could. I would realy appreciate your help since you are an expert. Again, thank you very much!

NobodyInportant said...

... a side note: when I said that existence could be a "part" of particles/fields, I ment by this that existence could be a "part" of the ESSENCE of particles/fields. What it means to be a particle/field (to exemplify properties that are essential to particles/fields/some other kind of fundamental physical reality) ENTAILS that the particles/fields/(fundamental physical reality) EXIST. Because, there could be a set of determinable physical properties that simply could not fail, in reality, to be instantianted out of ontological necessity. Hence, a particle/field would have to exist necessary in order for there to be anything in reality at all.......... I hope that now I am more clear. Again, I sincerely hope that I am not bothering you that much!

Alexander R Pruss said...

Given that particles go in and out of existence all the time, it doesn't seem to me to be very plausible that particles should exist necessary.

Fields are a better candidate. But there is still this thought. It seems unlikely that the precise laws of nature we have should be necessary. There seems to be too much of the arbitrary in the laws of nature: the number of dimensions, the constants, etc. But fields are tied to laws of nature: if the laws of nature were different, it seems we wouldn't have the same fields. So if the laws of nature are contingent, probably the fields are as well.

Walter Van den Acker said...

Alex

A necessary field can give rise to contingent laws, just as a necessary God can, supposedly, create different laws. If fact, the latter is much more problematic if God is timeless, simple and immutable.

Alexander R Pruss said...

Maybe, but I'm inclined to think that fields and particles are tied to laws. Consider a particular photon, call it Alice. Could *Alice* exist in a classical universe, where neither relativity nor quantum mechanics held sway? I have my doubts. And I have similar doubts about fields.

Walter Van den Acker said...

Alex

I am talking about a necessary field. There is no "classical" universe where neither relativity nor quantum mechanics hold sway in this scenario.
The only question that matters is whether this field is possible.