The philosophy of mathematics can get complicated. Lakatos claim to fame
(infame to Popperians) is that he tried (and failed) to apply Popper's theory
of 'scientific progress' to mathematics, insulting, into the bargain, none
other than Fermat (whose surname he, on top, totally mispronounced!).
We are discussing Pears's idea of number [inter alia].
In a message dated 9/13/2015 4:06:47 P.M. Eastern Daylight Time,
"[Speranza] considers Pears a genius [...]"
whereas McEvoy considers Pears 'true to type'. The Oxford gives various
examples of this:
"true to form (or type):
"being or behaving as expected."
i.e. Pears is being as expected or worse behaving as expecting.
The Oxford examples are:
"True to type, they took it well."
"He, true to form, behaves like a cad and leaves her for the gambling
tables and his deserved fate."
"Ultimately, and true to form, the woman is portrayed as the weaker sex."
A final one:
"The whole centre is in need of regeneration and, as true to form, it is
the private investor that sets the standard."
Anyways [sic], McEvoy goes on:
"Pears writes (p.XI): "The two views may be combined without any
incoherence. They share the same conclusion, logical atomism, and they both
incorporate the assumption of a general correspondence between language and
reality. They differ only in their methods of establishing the conclusion.
According to one view, it is established empirically, like the conjecture that
every even number is the sum of two prime numbers, while the other view takes
it to be provable, as it is hoped that the arithmetical conjecture will be
proved one day. So Russell was not wrong when he allowed both views to be
represented in his treatment of logical atomism."
McEvoy provides an exegesis of this:
"There is a lead up and follow-on to this and both may be read to put it
"There remains the following puzzle."
"Pears speaks of a conjecture "like the conjecture that every even number
is the sum of two prime numbers", and says this example illustrates one of
the two delineated "methods of establishing the conclusion", and further
says "According to one view, it is established empirically.""
"But such a conjecture cannot be established empirically i.e. by
'observation'. We cannot ever observe [in an empiricist SENSE], for example,
"every even number is the sum of two prime numbers" [We cannot 'observe'
an even number or a prime number either, in an empiricist SENSE of
I think this may depend on how you use 'observe'. As Grice notes, first
comes IDIOSYNCRATIC meaning, and Pears is free to use 'observe' as he wishes.
In general, 'three' (a number) comes attached to a 'noun' as in "three
apples". And roughly, it may be argued that if you observe 'three apples', you
observe 'three'. Plato was different because he WANTED to _see_ the _idea_
of threeness (and that's where Platonism in Mathematics goes wrong).
McEvoy goes on:
"Even the empiricist view of such conjectures is not that they are
established "empirically" (by way of observation) but analytically: hence the
we may find in Hume and expounded in, say, Ayer's Hume-based version of
"Logical Positivism" Language, Truth and Logic - that the only true
propositions with sense are those true by virtue of the meaning of their terms
(analytically true) and those verifiable by sense experience (empirically
Pears, who only got into philosophy (via Lit. Hum.) because, as his
obituary read, a 'lucky accident' involving gass, may have been reading Mill
(recall Grice's adage, "Grice to the Mill"). For Mill's VERY empiricist theory
of number, 'three' HAS to be attached to 'pears', as in "three pears".
Russell knew this.
---- INTERLUDE ON MOORE'S THREE PEARS
As J. Miller reminds us in "Portrait from Memory", Bertrand Russell was
reminiscing on television a great deal in those days". Miller plays a tape:
Presenter: "This is the Bee Bee Cee Third Programme. We have in the studio
Bertrand Russell, who talks to us in the series, "Sense, Perception, &
Nonsense, Number Seven: Is this a *dagger* I see before me?".
Bertrand Russell: "One of the advantages of living in Great Court, Trinity,
I seem to recall, was the fact that one could pop across, at any time of
the day or night, into trap of the then young G. E. Moore, into a logical
falsehood, by means of a cunning semantic subterfuge. I recall one occasion
with particular vividness. I had popped across and have knocked upon his
"Come in," he said.
I decided to wait a while, in order to test the validity of his
"Come in," he said once again.
"Very well," I replied, "if that is in fact truly what you wish."
I opened the door accordingly, and went it. And there was Moore, seated by
the fire, with a basket upon his knees.
"Moore," I said, "Do you have any pears in that basket?".
"No," he replied, and smiled seraphically, as was his wont. I decided to
try a different logical tack.
"Moore," I said, "do you, then, have SOME pears in that basket?".
"No," he replied, leaving me in a logical position from which I had but
one way out.
"Moore," I said, "do you, then, have PEARS in that basket?".
"Yes," he replied. "Three". And, from that day forth, we remained the very
closest of friends. It helped that he offered the best pear to *me*."
---- END OF INTERLUDE.
"Is this just a slip by Pears?"
"A slip where he does not mean such claims or conjectures are "established
empirically" (as this is not even the view of the empiricist)?"
Unless the empiricist happens to be Mill. I understand that the only
mandatory reading in Logic at the time when Pears was pursuing his Lit.Hum at
Balliol was "System of Logic" and Mill takes the view that we 'observe'
"three" in collocations like "three pears".
"Does Pears instead mean by "established empirically" that they are
established, on one view, 'according to an empiricist theory of knowledge'
though this empiricist theory of knowledge holds that such claims are _not_
"established empirically")? But this would make Pears' expression the very
opposite of what he terms "tolerably clear". [Speranza] has spoken of
Pears' "genius" and perhaps will find it no trouble to clear this one up."
Well, I should quote from Mill -- after all "Grice to the Mill" is my
favourite adage of the day.
"Empiricism is a form of realism that denies that mathematics can be known
a priori at all. It says that we discover mathematical facts by empirical
research, just like facts in any of the other sciences. It is not one of the
classical three positions advocated in the early 20th century, but
primarily arose in the middle of the century. However, an important early
proponent of a view like this was Mill. Mill's view was widely criticized,
according to critics, it makes statements like
2 + 2 = 4
come out as uncertain, contingent truths, which we can only learn by
observing instances of two pairs coming together and forming a quartet."
I have two pears in this hand, and I have two other pears in this other
hand; therefore, I have four pears in total (in both my hands).
It is within this context that one must place the account of the necessity
of arithmetical truth that Mill develops in the System of Logic.
The truths of arithmetic had traditionally been taken to be necessary, or
as McEvoy says, 'analytic a priori' -- except of course for Kant for whom
7 + 5 = 12
If we add five pears to a basket containing seven pears, we get a basket
containing twelve pears.
is infamously synthetic a priori (since Popper draws a lot from Popper I
wouldn't be surprised if he thought Kant was talking sense here).
But alleged arithmetical truths clearly have more than verbal import.
They are therefore NOT necessary truths, given Mill's argument that the
only necessity is verbal necessity.
On Mill's metaphysics, therefore, they depend for their truth upon the
individuals (e.g. APPLES) and their attributes of the world as we experience
Mill's views on arithmetic are controversial for anyone but the Oxonian
philosopher (such as Pears was).
Mill's views were later vehemently disputed by the logician Gottlob Frege,
not without good grounds, some should confess.
Mill disagreed with those whom he called "conceptualists", who held that
arithmetical truths were truths about psychological states.
Mill also agreed with Kant (oddly) against Nominalists such as Hobbes that
the propositions of arithmetic are not true by definition.
They are, in Kantian terms, synthetic.
But that implies, for Mill, against Kant, that they are a posteriori,
inductive rather than a priori.
The only way that Mill could see one holding that they are both synthetic
and a priori, is to hold that they are truths about rationally intuited
forms not presented in ordinary experience.
This was the solution that Frege was later to adopt.
But Mill on empiricist grounds rejected this sort of Realism.
This makes Mill in more recent terminology a nominalist.
The problem is that arithmetic seems to have a necessity which is at once
more than verbal, as Mill correctly held, but also more than that which
attaches to the inductive truths of, say, physics or botany.
Mill's ontology of things and attributes is simply not sophisticated enough
to permit a solution to this problem.
Mill argues that a NUMBER (such as 'three' in "There are three pears in
the basket") is an attribute of an aggregate of units.
This brings him close to Frege's idea that the number of a given class is
the class of all classes equinumerous to that given class.
Granted, Mill does not clearly distinguish between the rather otiose
distinction between an aggregate and a class, nor the sum of two numbers from
the (Boolean) sum of two classes.
(Mill found Boole "boring to read").
Moreover, Mill takes measurement to be the empirical counting of units,
rather than a matter of relations among the members of an ordered dimension.
In both cases a more sophisticated account of relational form is necessary,
but this was developed only by later logicians.
Mill MAY THEN seem 'slightly confused' from the point of view of later
thinkers such as Frege or Russell -- BUT NOT PEARS, who's taken Grice to the
Certainly, the view of the later positivists that mathematical truths are a
matter of logical form would fit more comfortably with his empiricism,
It is worth noting, however, that not everyone is dismissive of Mill's view
Hugh Lehman for example, thought that Mill was, in his philosophy of
number, being a genius.
What Mill does argue about the necessity of arithmetic is that these
principles, while from the point of view of their truth are inductive
generalizations, are from the point of view of the thinker matters of
The appeal is to the principles of association.
The propositions of arithmetic record matters of fact that are very deep
and invariable in our experience -- and OBSERVATION -- as when we observe
Our repeated experience of these facts creates in the mind invariable
These inseparable connections create in the mind of the knower a sense of
the necessity of these propositions.
The necessity is there, as Whewell and others insist.
But the necessity is one of thought rather than one in the ontological
structure of things.
And Pears knows this.
If you've been to Australia (Pears was a traveller), you'll notice that
many Australian languages, for example, have just three number words, which
are glossed as 'one', 'two', and often 'three' -- but this is confusing,
since 'three pears' can also mean 'many pears'. Since there are only THREE
number words, the meaning of 'three' has to be grasped, as Grice would say,
Lehman, Hugh, Introduction to the Philosophy of Mathematics, Totowa, NJ:
Rowna and Littlefield, 1979.
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