[lit-ideas] Re: Griceian Numbers
- From: Jlsperanza@xxxxxxx
- To: lit-ideas@xxxxxxxxxxxxx
- Date: Sat, 16 Jun 2012 09:59:41 -0400 (EDT)
I haven't checked all sources, but there doesn't seem to be ONE (ouch) good
online account (summary type) of philosophy of number. The Stanford has a
good entry on FREGE on number, and this on Aristotle, etc. It may do to
relate this to Ayer, Witters, and Grice, say.
(We are discussing Ayer's moot point that Witters's use of 'number' in PI
has a bearing on the 'key tenet' about the show/say distinction, :) -- vide
D. McEvoy).
_http://plato.stanford.edu/entries/aristotle-mathematics/#10_
(http://plato.stanford.edu/entries/aristotle-mathematics/#10)
"Greek mathematicians tend to conceive of number (arithmos) as a plurality
of units."
This allows for a Griceian analysis (if not Grecian).
"There were a number of eggs in the basket".
I would think that '1' applies. For the Greeks, there had to be at least
_two_.
"Perhaps a better translation, without our deeply entrenched notions, would
be ‘count’."
"The Greek conception involves a few items."
"A number is constructed out of some countable entity, unit (monas)."
"Numbers are more like concatenations of units and are not sets."
"To draw a contrast with modern treatments of numbers, a Greek pair or a
"two" is neither a subset of a triple, nor a member of a triple."
"It is a part of "three"."
"If I say:
Ten cows are hungry.
I am NOT saying:
A set is hungry.
Or to point to another use of ‘set’, my 12 piece tea-set is in a cabinet,
not in an abstract universe.
So too, these ten units are a part of these twenty units:
"One (a unit) typically is not a number (but Aristotle is ambivalent on
this), since a number is a plurality of units." -- cfr. Grice, "I am one
genius." (versus Whitman, "I contain multitudes").
"At least in theoretical discussions of numbers, a fractional part is not a
number.
In other words, numbers are members of the series:
2, 3, …,
with 1 conceived as the ‘beginning’ (archê) of number or as the least
number."
"In early Greek mathematics (5th century), numbers were represented by
arrangements of pebbles."
--- this possibly inspired Witters to bring in red and yellow bricks.
"Later (at least by the 3rd century BCE) they were represented by evenly
divided lines." (Hardly a progress).
"For Aristotle and his contemporaries there are several fundamental
problems in understanding number."
"The precision problem of mathematicals is similar in the case of
geometrical entities and units (see Section 6)."
"Consider, for example, Plato's discussion of incompatible features of a
finger as presenting one or two things to sight."
"Aristotle deals with the problem in his discussion of measure (see Section
10.1)."
"The separability problem is the same as for geometry (see Section 6)."
"The plurality problem of mathematicals (Section 6) is similar in the case
of geometrical entities and units, with some differences."
"To count ‘perceptible’ units, or rather units from abstraction (cf.
Section 7.1), one needs some principle of individuating units, what one is
counting, whether cows or categories of predication."
--- This is a VERY important point emphasised by Ayer that McEvoy dismisses
rather abruptly. Just because didn't care ONE jot about empereia (and
dismissed verificationism, science, and the rest of it) it does not mean that
Ayer should not be given credit for bringing in the EMPIRICAL issues
involved in our use of 'number words'.
"Aristotle says that one can always find an appropriate classification (we
may assume that some classifications would be fairly convoluted, but that
this is at best an aesthetic and not a logical problem)."
But surely number belongs in LOGIC -- philosophy of number is part of
philosophy of logic.
"For units, one will use the same principle that allows one to individuate
triangles."
"This is why Aristotle can describe a point as unit-having-position."
"Arithmetic involves the study of entities qua indivisible."
"The unity problem of numbers bedevils philosophy of mathematics from
Plato to Husserl."
"What makes a collection of units a unity which we identify as a number?"
"It cannot be physical juxtaposition of units. Is it merely mental
stipulation?"
"Aristotle does not seem bothered by the overlap problem."
"What guarantees that when I add this 3 and this 5 that the correct result
is not 5, 6, or 7, namely that some units in this 3 are not also in this
5."
"Aristotle presents three Academic solutions to these problems."
"Units are comparable if they can be counted together (such as the ten
cows in the field)."
"the ten cows."
"the three stooges"
--- I once dwelled on this for the concatenation of the iota operator
(uniqueness 'the') with a number word seems to trigger the wrong implicature:
"The ten cows were eating."
"There are TEN cows in the field." -- the dissertation I cited, re "there
are" sentences -- quite a field).
Cfr. Russell,
"the king of France."
"the two kings of France."
---- and so on.
Note that 'one' Grice dealt with quite a lot:
"Smith is meeting a woman tonight". Implicature: "Not his wife" (WoW:
"Logic and Conversation").
Grice is considering "one" as "some" ("at least one").
Indeed, this is important:
Grice has
"at least ONE" (a Griceian number if ever there was one) in the opening
passage to "Logic and Conversation" focused in this dissertation I was
mentioning.
Grice:
"It is a commonplace of philosophical logic [for Husserl and Frege knew
what they were talking and philosophy of number belongs to
philosophy of logic] that there are, or appear to be [but this is false]
divergences in _meaning_ ['sense'?] between, on the one hand,
at least some of what I shall call the formal devices --
-,&,v,->,(x),(Ex), (ix) --
(when these are given a standard two-value interpretation -- and, on the
other [hand], what are taken to be their ANALOGUES or counterparts
in [English] -- such expressions as "not", "and", "or", "if", "all" [rather
than 'every'], "SOME" (or "AT LEAST **ONE**"), "the"". (WoW:22)
"At least one"
Pass me at least one red brick.
Equivalent to:
"Pass me some brick."
Cfr.
"That one some good party."
--- i.e. at least one good party.
---
The Stanford goes on:
Units are not comparable, if it is conceptually impossible to count them
together (a less intuitive notion)."
Incomparable Units:
"Form numbers are conceived as ordinals, with units conceived as being well
ordered."
"What makes this number 3 is not that it is a concatenation of three units,
but that its unit is the third unit in this series of units."
"Hence, it is simply false that there is a unity of the first three units
forming a number three."
"What makes an ordinary concatenation, e.g.,
a herd of cows,
ten cows is that they can be counted according to the series of
Form-numbers."
"The notion of incomparable numbers lacks the basic conception of numbers
as concatenations of units."
"Comparable/Incomparable Units."
"Form numbers are, of course, special."
"Each is a complete unity of units."
"For example, the Form of 3 is a unity of three units."
"Since it is a unity, it cannot be an accident that these three units form
this unity."
"They are comparable with each other in the sense that together they
comprise Three Itself and perhaps cannot be conceived separately."
--- e.g. the three stooges.
"Hence, they cannot be parts of any other Form number."
cfr. Quine's example:
The twelve apostles were intelligent.
(in logical form (ix12)Ix.
"We cannot take 2 units from the Three Itself and add them to 4 units in
the Six itself, to get a Form-number of the Seven itself."
"Myles Burnyeat once suggested an analogy with a sequence of playing cards
of one suit, say diamonds.[1]"
"Each card from ace (unit) to ten contains the appropriate diamonds (from 1
to 10) on each card, unified by their being on their particular card."
"Yet we don't count up two diamonds from the deuce and two from the trey,
but treat each card as a complete unity."
Comparable Units:
"Comparable units are intermediate or mathematical numbers (see Section
6)."
"There is a unlimited number of units (enough to do arithmetic), which are
arranged and so forth."
"Comparable numbers solve the plurality problem, but not the unity
problem."
"Aristotle reports that some Academics opted for a version of Incomparable
or Comparable/Incomparable Units to solve the unity problem and introduced
comparable units as the objects of mathematical theorems, e.g., given some
comparable units, they are even if they can be divided in half, into two
concatenations corresponding to (participating in) the same Form-number."
And so on.
Cheers,
Speranza
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