Edward Brian Davies was born in 1944. He is a Fellow of the Royal Society and was Professor of Mathematics at King’s College London (1981–2010). Davies has written papers on spectral theory, non-self-adjoint operators, operator theory/functional analysis, elliptic partial differential operators, Schrodinger operators (in quantum theory) and so on. He was awarded a Gauss Lectureship by the German Mathematical Society in 2010.
This commentary is on the relevant parts of Davies’s book, Science in the Looking Glass. The following is not a book review.
A Short Introduction to Mathematical Empiricism
E. Brian Davies’s own position on mathematics (or at least on numbers) is generally called mathematical empiricism.
The following short introduction is a basic account of such a position as it relates specifically to what Davies has to say on the subject. Nonetheless, this isn’t to say that Davies would endorse everything in this account of empiricism and its relation to mathematics.
Mathematical empiricism has it (in very broad terms) that mathematics simply can’t be known a priori. (Something that is known a priori is something that can be known by “reason alone”.) As Davies will state later (if very broadly and in a slightly different way), the mathematical empiricist believes that “mathematical facts” are discovered by empirical research. Indeed this position can be traced back to the philosopher John Stuart Mill in the 19th century — and probably before him. Davies’s own position is close to Mill’s in that the latter believed that the empirical justification for mathematics (or at least for certain types of number) comes directly from empirical objects.
In more concrete terms, “quasi-empiricists” argue that when doing their research, mathematicians “test hypotheses”; as well as prove theorems.
Without going into great detail, the most obvious argument against mathematical empiricism is that this position must have in it that literally all the results (or theorems) of mathematics are as fallible as the results in the empirical sciences. Or, alternatively put, that mathematical results are always contingent and never necessary. Of course this position is — many would argue — hugely problematic and that’s obviously so (or so it would seem). However, this isn’t the place to go into detail on this specific issue. (See the short note at the end of this piece.)
E. Brian Davies’s Mathematical Empiricism
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 obviously the case for the mathematicians and philosophers whom call themselves Platonists). Nonetheless, most people are aware of the fact that mathematics is applied to the world and is an extremely useful — indeed necessary — tool for describing empirical reality. However, empiricists go one step further than this by arguing that mathematics itself is empirical in nature. Or they argue - at the least in E. Brian Davies’s case — that certain types of real number have an empirical status.
Quickly, it’s worth pointing out here that one can make a distinction between two things when it comes to empiricism and mathematics:
Having an empiricist (philosophical) position on mathematics itself.
Making one’s own philosophical empiricism more scientific by making it more mathematical.
Although these are two different positions, it can be said that both apply to Davies’s own position.
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”. Here there’s a reference to “counting”; which is a psychological (or cognitive) phenomenon. By inference, it also refers to what we count. And what we count are empirical objects (i.e., objects we can experience using our sense organs). That means that empirical objects need to be experienced (or observed) in the psychological (or cognitive) 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”.
Even if that means that we can count empirical objects easily enough with numbers, does that alone give small numbers “strong empirical support”? Perhaps we’re still talking about two completely different (or separate) things: small numbers and empirical objects. Simply because numbers can be used to count objects, does that (on its own) confer some kind of empirical reality on such numbers? We’re obviously justified in using numbers for counting objects; though that may just be a matter of usefulness. Again, do small numbers themselves have the empirical nature of objects (as it were) passed onto them simply by virtue of their being used in acts of counting?
Davies then mentions Peano’s axioms in this respect.
Surely small numbers existed before “assenting to Peano’s axioms”.
Davies believes that by accepting such axioms we then have the means to create, produce or construct small numbers. That is, we firstly take the axioms; and then we derive all the small numbers from them. However, before the creation of these axioms, and the subsequent generation of small numbers as theorems, didn’t the small numbers already exist? A realist (or Platonist) would say, “Yes”. A constructivist (of some kind) would say, “No”.
Davies appears to put a set-theoretic position on numbers in that he tells us that “‘counting’ numbers [do] exist in some sense”. (I say set-theoretic in the sense that the nature of each number is determined by its one-to-one correspondence — i.e., bijection - with other members of other sets.) 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”.
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 other equal-membered sets in a relation of one-to-one correspondence with the original set. In other words, how numbers are used can’t give them an empirical status. Something is used. However, does that use entirely determine the metaphysical nature of (small) numbers? (We use pens; though the use of a pen and the pen itself are two different things.)
In any case, what these “collections” have in common (according to Davies) is that we can “see” their equivalences. So do we also see the relevant numbers rather than count them? How do we count without using numbers? That is, even if there are equivalence classes, are numbers still surreptitiously used in the very definition of numbers?
Davies then goes on to argue a case for the empirical reality of small real numbers. There is a logical problem here, which he faces.
“If one is prepared to admit that 3 exists independently of human society.”
“Then by adding 1 to it one must believe that 4 exists independently…
“[Therefore] the number 1010100 must exist independently.”
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. However, according to Davies, counting to 1010100 may not be something “human society” can do. Yet 1010100 exists even though Davies believes that mathematics tells us that “It is not physically possible to continue repeatedly the argument in the manner stated until one reaches the number 1010100”.
Extremely Large 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”. Instead,
“they arise when one makes measurements and then infers approximate values for the numbers”.
The basic idea (again) is that there must be some kind of one-to-one correlation (or correspondence) between real numbers and empirical objects. If this isn’t forthcoming, then certain real numbers have a “questionable” (or “metaphysical”) status. From his position on small numbers, Davies also concludes that “huge numbers have only metaphysical status”.
I don’t really understand this.
Which position in metaphysics is Davies referring to? His use of the words “metaphysical status” makes it seem like some kind of synonym for “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. Perhaps the statement should be: “Huge numbers only have a … metaphysical status.” In that statement, the three dots should be filled with some kind of position (or “mode of being”) 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 status” (within this context) seems as if it is some kind of synonym for “lesser status”. That conclusion is backed up when Davies also uses the phrase “questionable status”. Thus a metaphysical status is also a questionable status. Nonetheless, I still can’t see how the words “metaphysical status” can be used in this way. Despite that, I’m happier with the latter locution (i.e., “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 Davies believes that there must be some kind of relation (or correspondence) between real numbers and empirical things (or objects), he also sees a problem with extremely small real numbers. Davies argues that physicists or philosophers may attempt to set up a relation between extremely small numbers and “lengths far smaller than the Planck length”. 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 (i.e., rather than with objects in the plural).
Yet Davies doesn’t believe that this approach works.
He argues that this is because Planck lengths “have no physical meaning anyway”. By inference, this also means that extremely small numbers don’t have any empirical support. In other words, they have either a “questionable status” or a “metaphysical status” (perhaps both).
Models, Real Numbers and the External World
Davies’s more general position is that
“real numbers were devised by us to help us to construct models of the external world”.
As stated earlier, does this mean that numbers gain their empirical status simply because they’re “used to help us construct models of the external world”? However, even though real numbers are used in this way, that still may not give them an empirical status. Can’t numbers be abstract objects and still have a role to play in constructing models of the external world? (There is the problem — among other problems - of our causal interaction with abstract non-spatiotemporal numbers or objects.)
In terms of a vague analogy. We use cutlery to eat our breakfast. Yet breakfast and cutlery are completely different things. Nevertheless, they’re both (as empirical objects) in the same ballpark. What about using a pen to write about an event in history? A pen is an empirical object. What about an historical event? We can say that the pen which writes about an historical event exists. Can we also say the same about the historical event itself? Yet there’s still a relation between what the pen does and a historical event even though they have two very different metaphysical natures.
Non-physicists 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 actually 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 an object (or bit of the external world)? Or do numbers actually (metaphorically?) belong to the external world just as much as they belong to the (mathematical) models we have of the external world? In other words, is the world already numerical (i.e., as in the Pythagorean position in which “all is number”)? Have we the philosophical right to say of the studied objects (or bits of the world) what we also say of the models of those studied objects (or bits of the world)?
E. Brian Davies puts the (or his) empiricist position on mathematics at its broadest by referring to philosophers and mathematicians whom he sees as being fellow empiricists. He cites John von Neumann, W.V.O. Quine, Alonzo Church and Hermann Weyl. These mathematicians and philosophers “accepted that mathematics should be regarded as semi-empirical science”. Of course saying that maths is “semi-” anything is open to many interpretations. Nonetheless, what Davies says about real numbers above (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”. What’s more, “the empirical aspect [is often] leading the way”. Indeed Davies says that this empiricist position is “increasingly common even among pure mathematicians”.
The bald empiricist position on mathematics — or on numbers — is very easy to attack. So just to take one example (which was aimed at J.S. Mill). The English philosopher A.J. Ayer (who was himself an empiricist… of sorts) stated (in his Language, Truth, and Logic) that if we took an empiricist position on mathematics (or specifically on arithmetic) itself, then the statement “2 + 2 = 4” would need to be seen as contingent and even uncertain. Indeed we’d also need to rely on observing two pairs coming together in order to formulate the statement in the first place.
This isn’t to say that empiricists don’t have counter-arguments to this. After all, once two pairs coming together (as it were) has been observed (either collectively or by an individual), then that doesn’t mean that all further uses of arithmetic require that we keep on matching empirical objects together in such a manner. So, sure, an abacus (or an empirical equivalent) may be psychologically and historically important for (most people) when it comes to learning arithmetic. However, that’s a fact about the human process of learning arithmetic: it’s not a fact about arithmetic itself.