Friday, 23 September 2016

E. Brian Davies's Empiricist Account of Real Numbers

*) This commentary is on the relevant parts of E. Brian Davies's book, Science in the Looking Glass.  


At first glance it's difficult to see how mathematics generally, and numbers specifically, have anything to do with what philosophers call “the empirical”. This is also the case for mathematicians and philosophers whom class themselves as “realists” or “Platonists”. Nonetheless, everyone is aware of the fact that maths is applied to the world. Or, at the least, that maths is a useful tool for describing empirical reality.

Nonetheless, empiricists go one step further than this by arguing that mathematics (or, in Davies's case, a real number) is empirical in nature. Or, at the least, that certain types of real number have an empirical status.

At first I had to decide whether to class E. Brian Davies's position “empiricist mathematics” or “mathematical empiricism”. The former is a philosophical position regarding maths. The latter, on the other hand, is a position on empiricism itself. In other words, in order to make one's empiricism more scientific, it would make sense to make it mathematical. Empiricist maths, on the other hand, is a philosophical position one could take on mathematics itself. Although these are different positions, I can only say that both apply to Davies's account.

Small Real Numbers

E. Brian Davies puts his position at its most simple when he says that for a “'counting' number its truth is simply a matter of observation” (81). Here there seems to be a reference to the simple act of counting; which is a psychological phenomenon. By inference it must also refer to what we count. And what we count are empirical objects or other empirical phenomena. That means that empirical objects need to be observed in the psychological act of counting.

Prima facie, it's hard to know what Davies means when he writes that “[s]mall numbers have strong empirical support but huge numbers do not” (116). Even if it means that we can count empirical objects easily enough with numbers, does that, in and of itself, give small numbers “strong empirical support”? Perhaps we're still talking about two completely different/separate things: small numbers and empirical objects. Simply because numbers can be utilised to count objects, does that - on its own - confer some kind of empirical reality on them? We are justified in using numbers for counting; though that may just be a matter of practicality. Again, do small numbers themselves have the empirical nature of objects passed onto them simply by being used in acts of counting?

Did these small numbers exist before “assenting to Peano's axioms”? Davies make it seems as if accepting such axioms is a means to create or construct small numbers. That is, we take the axioms; from which we derive all the small numbers. However, before the creation of these axioms, and the subsequent generation of small numbers as theorems, did the small numbers already exist? A realist would say 'yes'. A constructivist, of some kind, would say 'no'.

Davies appears to put the set-theoretic or Fregean/Cantorean position on numbers in that he writes that that “'counting' numbers exist in some sense” (82). What sense? In the sense that “we can point to many different collections of (say) ten objects, and see that they have something in common” (82). I say Fregean/Cantorean in the sense that the nature of each number is determined by its one-to-one correspondence with other members of other sets.

Prima facie, I can't see how numbers suddenly spring into existence simply because we 'count' the members of one set and them put the members of equal-membered sets in a relation of one-to-one correspondence. How numbers are used can't give them an empirical status. Something is used, sure; though that use doesn't entirely determine its metaphysical nature. (We use pens; though that use of a pen and the pen itself are two different things.)

The other problem is how we 'count' without using numbers? Even if there are "equivalence classes", are numbers still surreptitiously used in the very definition of numbers?

In any case, what these “collections” have in common, according to Davies, is the number of members, which we “see” (rather than count?).

Davies goes on to argue a case for the empirical reality of small real numbers. There is a logical problem here, which he faces.

Davies offers a numerical version of the sorites paradox for vague objects or vague concepts. Let me put his position in argument-form. Thus:

 i) “If one is prepared to admit that 3 exists independently of human society.
ii) “then by adding 1 to it one must believe that 4 exists independently...”
         iii) “[Therefore] the number 1010100 must exist independently.” (82)

This would work better if Davies hadn't used the clause “exists independently of human society”. I say that because it's empirically possible, or psychologically possible, that there must be a finite limit to human counting-processes. Thus counting to 4 is no problem. But counting to 1010100 may not be something “human society” can do.

I mentioned the simpler and more effective argument earlier. Thus:

i) If 3 exists.
ii) Then by adding 1 to 3, 4 must exist.
iii) Therefore, by the repeated additions of 1 to the previously given number, the number 1010100 must also exist.

It may exist; though Davies thinks that mathematics tells us “it is not physically possible to continue repeatedly the argument in the manner stated until one reaches the number 1010100 ” (82).

Extremely Large and Extremely Small Real Numbers

Davies begins his case for what he calls the “metaphysical” or “questionable” nature of extremely large numbers by saying that they “never refer to counting procedures” (67). Instead, “they arise when one makes measurements and then infers approximate values for the numbers”.

The basic idea is that there must be some kind of one-to-one correlation between real numbers and empirical objects. If this isn't forthcoming, then certain real numbers have a “questionable” or “metaphysical” status. (Again, this is like the idea of a one-to-one correspondence between members of one set and the members of another set. This is – or was - a process used to determine the set-theoretic status of numbers.)

From his position on small numbers, Davies also concludes that “huge numbers have only metaphysical status” (116). I don't really understand this. Which position in metaphysics is Davies talking about? His use of the word “metaphysics” makes it sound like some kind of synonym for “lesser” (as in a “lesser status”). However, everything has some kind of metaphysical status, from coffee cups to atoms. Numbers do as well. So it makes no sense to say that “huge numbers have only metaphysical status” until you define what status that is within metaphysics. The phrase should be: “huge numbers only have a … metaphysical status”; with the three dots filled in with some kind of position within metaphysics.

Davies goes on to say similar things about “extremely small real numbers” which “have the same questionable status as extremely big ones”. I said earlier that the word “metaphysical” (within this context) sounded as if it were some kind of synonym for 'lesser'. That conclusion is backed up by Davies using the phrase “questionable status”. Thus a metaphysical status is also a “questionable status”. Nonetheless, I still can't see how the word “metaphysical” can be used in this way. Despite that, I'm happier with the latter locution (“extremely small real numbers have the same questionable status as extremely big ones”), than I am with the former (i.e., “huge numbers have only metaphysical status”).

Since there must be some kind of relation or correspondence between real numbers and empirical things, Davies also sees a problem with extremely small real numbers. It seems that physicists or philosophers may attempt to set up a relation between extremely small numbers and “lengths far smaller than the Planck length” (117). Thus the idea would be that Planck lengths divide up single empirical objects. Small numbers, therefore, correlate with individual empirical objects; whereas extremely small numbers correlate with the various Planck lengths of an object (rather than with objects in the plural).

Davies doesn't appear to think that this approach works. That is because Planck lengths “have no physical meaning anyway” (117). This means that extremely small numbers don't have any empirical support. They have a “questionable” or “metaphysical status”.

Models, Real Numbers and the External World

Davies's general position is that “real numbers were devised by us to help us to construct models of the external world” (131). As I said earlier, does this mean that numbers gain an empirical status simply because they're “used to help us construct models of the external world”? Perhaps, again, even though real numbers are used in this way, that still doesn't give them an empirical status. Can't numbers be abstract platonic objects and still have a role to play in constructing models of the external world? Why do such models and numbers have to be alike in any way? (Though there is the problem, amongst others, of our causal interaction with abstract numbers.)

In terms of a vague analogy. We use cutlery to eat our breakfasts. Yet breakfasts and cutlery are completely different things. Nevertheless, they're both, as empirical objects, in the same ball park. What about using a pen to write about an event in history? A pen is an empirical object; though what about an historical event? Can we say that the pen exists; though the historical event no longer exists? Nonetheless there is a relation between what the pen does and a historical event even though they have two very different metaphysical natures.

As non-physicists, we may also want to know how real numbers “help us to construct models of the external world”. Are the models literally made up of real numbers? If the answer is 'yes', then what does that mean? Do real numbers help us measure the external world via the use of models? That is, do the numbered relations of a model match the unnumbered relations of a object (or bit of the external world)? Would that mean that numbers belong to the external world as much as they belong to the models we have of the external world? Is the world, in other words, numerical? Thus, have we the philosophical right, as it were, to say of the studied objects (or bits of the external world) what we also say about the models of studied objects (or bits of the external world)? Platonists (realists) would say 'yes'. (Perhaps James Ladyman and Donald Ross, or ontic structural realists, would say 'yes' too.)


E. Brian Davies puts the empiricist position on mathematics at its broadest by referring to von Neumann, Quine, Church and Weyl. These mathematicians and philosophers “accepted that mathematics should be regarded as semi-empirical science” (115). Of course saying that maths is “semi-” anything is open to many interpretations. Nonetheless, what Davies says about real numbers, at least in part, clarifies this position.

Davies then brings the debate up to date when he tells us that contemporary mathematicians are “[c]ombining empirical methods with traditional proofs” (114). What's more, “the empirical aspect [is often] leading the way”. Indeed, Davies says, this position is “increasingly common even among pure mathematicians”.

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