[lit-ideas] three

  • From: Adriano Palma <Palma@xxxxxxxxxx>
  • To: "lit-ideas@xxxxxxxxxxxxx" <lit-ideas@xxxxxxxxxxxxx>
  • Date: Mon, 14 Sep 2015 08:21:51 +0000

Only an eminent imbecil like speranza can make remarks of this form.
Note, if in fact
As this speranza claims'
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'
THEN it would impossible even to have true propositions expressed by
'three is prime and four is not' hence there is nothing like three to observe
no matter how long you stare at the 3 apples.
Speranza is so taken by sucking up turds from the chamber pot of 'pears' that
he is losing the remains of his mind

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Subject: [lit-ideas] Re: Copy of The Philosophy of Logical Atomism, 1918

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,
donalmcevoyuk@xxxxxxxxxxx writes:
"[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 in

"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.""

McEvoy's objection:

"But such a conjecture cannot be established empirically i.e. by 'observation'.
We cannot ever observe [in an empiricist SENSE], for example, whether "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 'observe']."

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 view 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 true)."

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.


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 door.

"Come in," he said.

I decided to wait a while, in order to test the validity of his proposition.

"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*."



"Is this just a slip by Pears?"

Reading Mill?

"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'
(even 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, because, 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
those entities.

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, ALMOST.

It is worth noting, however, that not everyone is dismissive of Mill's view of

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
psychological necessity.

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 'three

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, _via_



Lehman, Hugh, Introduction to the Philosophy of Mathematics, Totowa, NJ:
Rowna and Littlefield, 1979.

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