*)
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

**10****10****100**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

**10****10****100**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

**10****10****100****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

**10****10****100****” (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.)

**Context**

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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