Abstract
I bring to light a set-theoretic reason to think that there are more (identifiable) mental properties than (identifiable) shapes, sizes, masses, and other characteristically “physical” properties. I make use of a couple counting principles. One principle, backed by a Cantorian-style argument, is that pluralities outnumber particulars: that is, there is a distinct plurality of particulars for each particular, but not vice versa. The other is a principle by which we may coherently identify distinct mental properties in terms of arbitrary pluralities of physical properties. I motivate these principles and explain how they together imply that there are more mental properties than physical properties. I then argue that certain parody arguments fail for various instructive reasons. The purpose of my argument is to identify an unforeseen “counting” cost of a certain reductive materialist view of the mind.
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Notes
The term ‘physical property’ is a term of art. I have in mind the sort of properties that can be sensed from a “third-person” perspective, such as shape, size, motion, etcetera, or properties that can be analyzed wholly in terms of such properties. Physical properties include, or are analyzable in terms of, the sorts of properties physicists qua physicists study. By contrast, candidate examples of non-physical properties include properties of mathematical entities, such as being divisible by two or being congruent.
Perhaps I should be more modest: by ‘physical property,’ I intend to mean whatever it is that “type-identity” theorists mean when they use the term. I am assuming (for the sake of argument, at least) that “type-identity” theory is itself intelligible. I should add that we can give counting arguments with respect to restricted classes of “physical” properties, such as shape, mass, size, etcetera, as I will explain in a later section.
The argument does not require that there be such things as sets or classes. Readers are welcome to translate “class” talk into plural reference talk.
To be clear, when I say that the mental properties are distinct, I do not mean they are each distinct from a physical property. Rather, I mean they are distinct from each other, since they feature different pluralities of physical properties.
In particular, we’ll consider attempts to show that premise 1 overly generalizes. I will explain why those attempts each include a premise that is importantly different from premise 1.
What follows is Cantor’s proof of Cantor’s theorem applied to the class of physical properties. Suppose there are no more classes of physical properties than physical properties. Then for each class (or plurality) C of physical properties, there is a distinct physical property, which we may call ‘C’s partner’. Now consider that some properties may be in the classes they are partnered with; for example, {p1, p2} might be partnered with p1. Nothing rules that out. But not every class can be partnered with one of its own members. For if every class were partnered with one of its members, then every singleton class (or “singleton plurality”) would be partnered with its member, leaving no other properties for the other, non-singleton classes to be partnered with. So, we cannot suppose that every class is partnered with one of its members. It follows, then, that there are at least some classes that are partnered with a property that is not one of its members.
We have just seen that some properties cannot be members of the classes they are partnered with. Now let ‘C’ be the class of just those properties that are not members of the classes they are partnered with. The existence of C follows from the Axiom of Separation, if the class of physical properties is a set. And even if they are not containable in a set, we may use what Pruss & Rasmussen (forthcoming) call plural comprehension: for any formula \(\uppsi \), if there are some things that satisfy \(\uppsi \), then there are the things that satisfy \(\uppsi \) (where, in this case, \(\uppsi \) = ‘\(x\) is not a member of its partner’). Next, let \(\hbox {P}_\mathrm{C}\) be a property that is partnered with C. There is such a property as \(\hbox {P}_\mathrm{C}\) since every class is partnered with a distinct property, given our starting assumption. A contradiction is now two steps away. Step one: \(\hbox {P}_\mathrm{C}\) cannot be a member of C, because C, by definition, only contains properties that are not in the classes they are partnered with. Step two: \(\hbox {P}_\mathrm{C}\) must be a member of C, because if \(\hbox {P}_\mathrm{C}\) is not a member of C, then since C contains all those properties that are not in the classes they are partnered with, it follows that C contains \(\hbox {P}_\mathrm{C}\), which contradicts the antecedent. So, \(\hbox {P}_\mathrm{C}\) cannot be in C, and \(\hbox {P}_\mathrm{C}\) must be in C, which is a contradiction. To avoid the contradiction, we must deny the starting assumption—that there is the same number of classes (pluralities) of physical properties as physical properties. (The “plurality” version of this proof replaces talk of classes with talk of pluralities, and the pairing is understood as a plurality of pairs.)
The conclusion of the counting argument is compatible with Searle’s proposal (2004) that mental properties, though irreducibly subjective in nature, are “physical” by virtue of being causally reducible to basic physical properties even if they aren’t ontologically reducible to such properties. But it is not compatible with typical formulations of type identity theory, such as the theory that Polger (2004) defends.
That isn’t to say that the counting argument rules out all psycho-physical identity theories. Some identity theorists may be happy to suppose that certain psychological properties are reducible to physical properties without supposing that all mental properties are reducible. For example, one might think that each property of the form being a thought that P has two components: (i) a psychological component, and (ii) a content component, where the content component is not itself a psychological in nature. Then identity theorists who reduce psychological properties may grant that non-psychological components of mental properties are themselves non-physical.
Those identity theorists may still appreciate the counting argument because the argument supplies a new reason to think that thoughts have a non-physical content component—and thus that the physical facts of reality do not exhaust the actual mental facts. I am grateful to an anonymous referee for emphasizing those identity theories that are unchallenged by my argument.
Distinct shapes are constructed from lines and angles, and the number of lines and angles is the same as the number of points on a continuum, which is \(\aleph _{1}\) (assuming the Continuum Hypothesis). The cardinality of the set of integers, by contrast, is only \(\aleph _{0}\).
A further difficulty is that not all conjunctions of distinct pluralities are themselves distinct. For example, the conjunction of \(p_{1}\) with \(p_{1}\) & \(p_{2}\) is plausibly not distinct from the conjunction of \(p_{1}\) with \(p_{2}\).
For more on the problems with packaging pluralities, see Pruss & Rasmussen (forthcoming).
I’m expressing the argument in terms of “physical properties,” but we could equivalently give the argument in terms of “physical state types.”
We can make sense of the limitations in terms of repeats: for example, if \(P\) is a package of \(p1\) and \(p2\), it’s unclear how to construct a distinct, coherent physical package consisting of \(P\), \(p_{1}\), and \(p_{2}\). See note 10.
By contrast, packages formed by the “thinking about” operator don’t plausibly produce repeats: for example, thinking about p1 and p2 is plausibly distinct from thinking about p1 and p2 and thinking about p1 and p2. (In the next objection, we’ll consider what happens when the “thinking about” operator applies to all mental pluralities; then, we probably get repeats.)
I am grateful to two anonymous referees who independently brought up the issue of packaging physical properties in ways other than the way of conjunction.
Cf. Grim and Plantinga (1993).
Cf. (Rasmussen (2014), p. 167).
I owe this remark to [an anonymous referee].
Putnam (1967).
Kirk (2005).
Jackson (1986).
Plantinga (2006).
I owe thanks to many, more than I remember. In this case, I will simply thank my wife, Rachel, who bore the burden of reviewing all previous drafts and listening to me talk about nearly all the objections and comments I’ve received from others.
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Rasmussen, J. Building thoughts from dust: a Cantorian puzzle. Synthese 192, 393–404 (2015). https://doi.org/10.1007/s11229-014-0575-2
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DOI: https://doi.org/10.1007/s11229-014-0575-2