[This is a post using Newman-style reasoning to argue for the existence of natural properties and relations.]

Consider a claim like:

(1) The mind-independent world is *complicated*.

One might

*deny* that there is a mind-independent world (

*Idealism*) or one might accept that there is, while insisting that it is "unknowable", while adding that what is

*known* is

*mentally* constituted (Kantian Transcendental Idealism). Here, in asserting the latter, one does merely mean

*representations* are mentally constituted, for this is a truism that no one denies. One means that what knowledge is

*about* is also mentally constituted (e.g, that physical objects

*are* representations; that space and time

*are* representations). Idealism is not the truism that our thoughts and representations are somehow in, or connected with, our minds; it is the much stronger metaphysical claim that everything (Idealism) or almost everything (Kant) is

*mind*-

*dependent*.

Assuming that we're not Idealists, what might this statement (1) mean? It might mean:

(2) The cardinality of the mind-independent things is quite large (e.g., $>10^{50}$).

If this is what (1) means, then the complexity of the world is

*solely* its cardinality. Therefore, a

*sound and complete description* of the mind-independent world consists in a statement of the form:

(3) The cardinality of the mind-independent world $= \kappa$,

where $\kappa$ is some cardinal number. It should strike anyone as surprising that the ultimate goal of physics, chemistry, biology, etc., is simply to identify this number $\kappa$. (Cf., the punchline of Douglas Adams's joke "

42".) So, I take it that this is

*not* what the statement (1) means.

So, perhaps (1) means,

(4) There are mind-independent properties and relations amongst the mind-independent things and *their* relations (e.g., scientific laws) are complicated.

Here "complexity" may mean something like the

*structural complexity* of the

*truth set* for a language containing predicates for these properties and relations. For example, the truth set for full arithmetic is more complicated than the truth set for arithmetic with just addition. For the latter is a recursive set, while the former is not recursive -- and in fact not even arithmetically definable. There are other ways of measuring complexity, notably Kolmogorov complexity, for finite strings, and various notions of computational complexity. Perhaps, if the world is finite, "complexity" might involve the Kolmogorov complexity of the simplest program that answers soundly all questions about the world.

However, independently of how one understands the concept of "complexity", one has to be careful. Suppose that by "property" or "relation" one means just

*any* set of things, or

*any* set of ordered pairs of things. These are properties in a very broad sense. It then follows, by Newman-style reasoning, that (4) is reducible to (3). For

*any* structure (or classification, if you like) $\mathcal{A}$ can be imposed on some collection $C$ of things so long as there are enough of them.

To illustrate: consider a finite set $X = \{1, \dots, n\}$ of numbers, and partition it any way you like. Let the partition be $(Y_i \mid i \in I)$, where $I$ is the index set. I.e., the sets $Y_i$ are non-empty and disjoint, and $X = \bigcup_i Y_i$. Now, suppose that we have a collection $C$ of $n$ things, or physical objects, or what have you. Then it is easy to

*define* a partition $(C_i \mid i \in I)$ of these things which is

*isomorphic* to $(Y_i \mid i \in I)$. For since $C$ and $X$ have the same cardinality, let $f : C \to X$ be a bijection (this function enumerates the elements of $C$). Then, for each $i \in I$, define $C_i$ by:

$c \in C_i$ iff $f(c) \in Y_i$.

By construction, this gives us an isomorphism. So, if we have a partition of $n$ natural numbers (the "mathematical model") and collection $C$ of physical things of size $n$, we can partition $C$

*isomorphically* to the original partition. If there are no

*independent* constraints built into $C$ itself beyond cardinality, we can impose any structure $\mathcal{A}$ we like onto $C$, modulo $C$ having cardinality at least as large as that of $\mathcal{A}$.

Consequently, if the reasonable sounding (4) is not to trivialize down to (3), the quantifier in "there are ... properties" must range over a special

*subset* of the set of all properties in the broader sense. In principle, this might be

*any* special subset. But, usually, what is intended is what metaphysicians call "natural properties". This is because what "selects" that subset as special is not the

*mind*, but

*Nature*. If one intends it to mean "there is a

*mind-dependent subset* of properties ...", then one is back to Idealism, this is almost certainly not what (1) is taken to mean by anyone.

So, if this reasoning is right, the most reasonable interpretation of "the mind-independent world is complicated" is:

(5) There are mind-independent *natural *properties and relations amongst the mind-independent things and *their* relations (e.g., scientific laws) are complicated.

And this is much more in keeping with scientific inquiry. However, note that (5) implies *the existence of mind-independent natural properties and relations*.

So, *if* there is a mind-independent world (Idealism is incorrect) and the mind-independent world *is* complicated, then either this *mind-independent* complexity consists merely in its *cardinality*, or it consists in the complexity of the laws and relations amongst *natural* properties and relations. In particular, if Idealism is incorrect but there are no natural properties or relations, then the complexity of the mind-independent world consists solely in its *cardinality*.

(I'm inclined to think that this latter position is, more or less, Kant's metaphysical view.)