My favourite session at the recent Arizona Ontology Conference was on Daniel‘s paper Finite Quantities.

Daniel argues that there is suggestive evidence from science to the effect that certain fundamental quantities are quantized rather than continuous. That, for Daniel, is to say that not all properties of the form <em>having n units of X</em> are instantiated, for certain fundamental X such as mass, charge or perhaps distance. Rather, for these X, there is some minimal n such that the property <em>having n units of X</em> is instantiated, and for all other instantiated properties of this form, ‘n’ is replaced with some multiple of this minimum.

It is commendable to get clear about what the quantized hypothesis looks like, and Daniel gets quite a lot clearer about it than most other discussions I know of.

However, having clarified that it is not a claim about the necessity (nomic, metaphysical or otherwise) of this restriction on the instantiation of certain properties, or about the non-existence, unreality or other substandardness of such properties (assuming that properties can exist uninstantiated), the view does not seem so very surprising or controversial.

It strikes me as a much more modest and palatable claim than the claims that quantizers – including Daniel – often <em>sound</em> like they are making. It sounds considerably less shocking, for instance, than the claim that ‘there is no such thing as’ (say) 1/2 n units of mass, or that although I may express things like “1/2 n units of mass” in <em>language</em> there is ‘no quantity corresponding to these representations’ and that ‘these quantities are not physically real’ (p. 2).

Moreover, clarity as to the exact nature of the quantizer’s thesis seems to make some of Daniel’s argumentative moves puzzling.

One of Daniel’s main opponents in the paper is someone who says that every time (say) a mass of six units is instantiated, the thing which instantiates the property <em>having six units of mass</em> also instantiates <em>having three units of mass</em> (twice over) and <em>having two units of mass</em> (three times over).

But let’s be clear about two readings of ‘having three units of mass’. On the first, it means ‘having at least three units of mass’. On the second, it means ‘having exactly three units of mass (and no more)’.

Now no-one would deny that everything which instantiates <em>having six units of mass</em> also instantiates <em>having at least three units of mass</em>. That would be silly. The quantizer, in this (made up) case, must instead be looking at denying that the property of <em>having exactly three units of mass (and no more)</em> is instantiated by anything.

But once we are clear that this is what is meant, the <em>opponent</em>‘s position looks silly. Obviously something which instantiates <em>having six units of mass</em> does not instantiate <em>having exactly three units of mass (and no more)</em>.

On neither reading, then, does it seem as if a Daniel-style quantizer and the opponent he describes in his paper have a sensible dispute such that they might need to look at the science to resolve it.

“However, having clarified that it is not a claim about the necessity (nomic, metaphysical or otherwise) of this restriction on the instantiation of certain properties …”

Having looked briefly at the paper, this comment puzzled me. I thought the idea was that there may well be laws of nature that result in certain quantities being quantized. A good example is the argument, due to Dirac, showing that charge-quantization results from magnetic monopoles.

More generally, it would seem that, mathematically, there are many number systems, and there’s a basic (if boring) empirical question about which number system applies in which physical context. For instance, nobody has any difficulties with the statement that complex numbers do not describe certain physical quantities — for instance, there’s no such thing as “i bunny rabbits” or even an imaginary mass. (Yes, there’s the “Feynman prescription,” but …) Given that, I can’t see why anyone would be spooked by the thought that other quantities (such as, maybe, charge) are represented by integers rather than reals.

in fact, going back a bit further, the Greeks were spooked by the thought that lengths were represented by reals, rather than rationals. This is just the same problem in the other direction — you need different number systems to do different jobs, right?

Hi Dmitri,

It’s right that Daniel thinks there

mightbe nomic stuff going on, but the claim he is concerned to defend in the paper is non-nomic.Bunny rabbits are measured in integers because they are counted rather than measured (cf. Daniel’s discussion of the cats). The thought he’s interested in is that things which are measured rather than counted may be quantized.

I took Nolan to be a little less clear-cut than you did. While he’s primarily interested in the question of which physical properties are in fact instantiated, he writes: “Nevertheless, I suspect a lot of the interesting answers I discuss about which magnitudes of quantities there are will reflect nomically necessary facts about the quantities.” What I took from this is that he is motivated by the nomic stuff, but put it aside for the sake of argumentative clarity.

In any case, the more interesting substantive question is: why privilege the real numbers? There is, as far as I know, no disagreement about whether statements such as “the mass of this particle is 3i.” Nobody (though correct me if I’m wrong) thinks that statement makes sense. Complex numbers simply do not represent mass.

Given this, why should there be any additional hand-wringing about the thought that certain physical quantities might not be meaningfully measured real numbers? Is it just that we think we can imagine a charge of e/pi, whereas we can’t even begin to imagine a mass of 3i?

What worries me is that lurking in here is the thought that the real numbers are somehow special — they’re the “true” number system, whereas the others are somehow derived, secondary, or bogus. If, on the other hand, you think that we have a variety of number systems — nonnegative integers, real numbers, complex numbers, octonions, etc. — then the problem loses some of its intuitive force. We all accept that certain physical quantities (probability amplitudes) are measured in complex numbers while others (mass) aren’t. It then becomes relatively unsurprising that additional physical quantities (bunny rabbits, charge) are measured in integers, while others aren’t.

By the way, another quick comment. The Dirac argument shows that charge is quantized if magnetic monopoles exist. It does not show that charge is

necessarilyquantized, since there could be worlds in which there are no magnetic monopoles. However, if (for contingent reasons) there is one magnetic monopole, then charge is necessarily quantized.This strikes me as the sort of situation Nolan might have been trying to get at. Charge quantization (in our world) is contingent, but the nearest possible world in which charge is unquantized is also one that is different in other respects too — there’s no monopoles for instance. So there is nomic stuff going on, but it’s not the case that there is a law of nature that says “charge is necessarily quantized.” Instead, the law says “if X, then charge is necessarily quantized.”