[lit-ideas] Re: Euclid's Theorem
- From: Adriano Palma <Palma@xxxxxxxxxx>
- To: "lit-ideas@xxxxxxxxxxxxx" <lit-ideas@xxxxxxxxxxxxx>
- Date: Mon, 10 Jun 2013 07:39:25 +0000
speranza, did you take algebra in high school or is that prohibited?
yes, Virginia, there are infinitely many primes, and maybe, just maybe, there
infinitely many twins
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From: lit-ideas-bounce@xxxxxxxxxxxxx [mailto:lit-ideas-bounce@xxxxxxxxxxxxx] On
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Sent: 09 June 2013 10:24 PM
To: lit-ideas@xxxxxxxxxxxxx
Subject: [lit-ideas] Euclid's Theorem
Euclid said, somewhat out of the 'blue':
"There are infinitely many prime numbers".
He added, in Greek,
Ὅπερ ἔδει δεῖξαι. hoper edei deixai
Which Thomas Aquinas translates as:
"Quod erat demonstrandum"
Some demonstration! It got Popper into thinking that this proves the autonomy
of World 3 (or something).
----
What we should do is re-construct Euclid's proof (so-called), especially as it
depends on an 'intuitive grasp', as Popper calls it, of 'infinity' (or as it
doesn't, as the case may alternatively be).
Note that Euclid is, in Greek, using the epiphenomenal fuzzy quantifier, "many"
-- "There are INFINITELY _many_ prime numbers." (as implicating, "Don't ask he
how many").
So I will provide a closer commentary on part of the original post where
McEvoy introduced Euclid.
In a message dated 6/3/2013 12:41:49 P.M. UTC-02, donalmcevoyuk@xxxxxxxxxxx
writes:
The problem of understanding World 3 mathematical “objects”, such as
mathematical problems and mathematical arguments and theorems, is taken by
Popper as showing that it is “generally valid” that there is “direct grasp of
World 3 objects by World 2”."
where it seems a more specific reading would be: "a direct, INTUITIVE, grasp
of..." -- which seems to trade on INTUITIONISM.
---
McEvoy:
"In The Self and Its Brain, Popper seeks to illustrate this “direct”
relationship between World 3 and World 2 by discussion of a theorem of Euclid.
Popper comments (TSAIB p.548, Dialogue XI): “Although, of course, there are
some World 1 brain processes going on all the time while World 2 is awake, and
especially when it is busy in solving problems or in formulating problems, my
thesis is not only that World 2 can grasp World 3 objects, but that it can do
so directly; that is to say, although World 1 processes may be going on (in an
epiphenomenal manner) at the same time, they do not constitute a physical or
World 1 representation of those World 3 objects which we try to grasp. Let me
illustrate this by discussing Euclid’s theorem, that for every natural number,
however large, there exists a greater one which is a prime number; or, in
other words, that there are infinitely many primes."
cfr.
Grice,
"As far as I know, there are infinitely many stars."
Oddly, the use, here, of 'many' may be, metaphorically, a red herring, since
'many' is the pleonetetic quantifier (i.e. fuzzy) par excellence.
For we may start to wonder what Popper means "infinitely many" -- or Grice's
utterer (for it is not Grice, but his utterer, who utters, "As far as I know,
there are infinitely many stars").
Note that the English words
"every", "any", and "none"
can be qualified by certain adverbs, so that we may say, for instance,
"nearly every", "scarcely any", and "almost none".
Consider
1. Amost every man owns a car.
This is logically equivalent to
2. Few men do not own a car.
which in turn is equivalent to
3. Not MANY men do not own a car.
There is, indeed, a pretty close correspondence between two sets of words as
follows:
always -- ever -- often -- seldom -- sometimes -- never every -- any -- MANY
-- few -- some -- none
where the terms in the upper line interrelate in the same way as do those on
the lower, e.g.
4. I have a few books.
is equivalent to
5. I do not have MANY books. (Cfr. Popper, "I do not have infinitely many
books")
Similarly,
6. I seldom go to London.
is equivalent to
7. I do not often go to London.
Further quantifiers are discernible in English, unless the eyes deceives one.
As well as "few", "MANY" (and "infinitely many") and "nearly all", we have
"very few", "VERY (if not infinitely) MANY", "VERY (indeed infinitely) MANY"
and "very nearly all",
and yet more result from reiterated prefixing of "very".
In their representation in a formal syntax, all the foregoing expressions come
out as what Grice calls
(I,I)-quantifiers
- quantifiers which bind one variable in one formula.
A formal syntax, together with appropriate semantics, which gives an
appropriate treatment to all these is a significant generalisation of
classical quantificational methods on the pattern of ordinary logic.
The further generalisation to
(I, k)-quantifiers
- binding one variable in an ordered k-triple of formulae, gives a further
increase in power.
Thus, consider
8.
There are exactly as many Apostles as there are days of Christmas.
We have here a (1,2)-quantifier.
It seems significant that we can build up additional quantiifers in much the
same way as we can (1,1)-quantifiers.
For instance, from
"more than"
we can to to
"MANY more than" (indeed 'infintely many')
and
"very many more than".
We have such expressions as
"nearly as MANY as" (but hardly 'infinitely many') and "almost as few as",
and so on.
As to their truth-conditional semantics, one thing that is clear about the
truth conditions of
9. There are MANY As.
is that
10. It is not the case that there is only one A.
--
cfr. Grice, "As far as I know, there are infinitely many stars".
Popper, "Euclid saw that there were infinitely many primes."
It also seems that, in general, how MANY As (stars, numbers, primes) there need
to be for there to be "many As" depends on the size of the envisaged domain of
discourse.
E.g. in
11.
There are MANY communists in this constituency.
the domain of discourse would probably be the electorate of the constituency in
question. On the other hand,
"There are infinitely many comunists in this constituency" should, on occasion,
be treated as _hyperbolic_ (via conversational implicature).
This domain is smaller than the one envisaged in
12. There are MANY communists in England.
and consequently the number of communist there have to be for there to be
many communists in this constituency is smaller than the number there have
to be for there to be many communists in England.
This suggests the use of a numerical method in providing the appropriate
truth-conditional semantics, by selecting a number
"n"
which is
the LEAST number of things
there have to be with a certain property A (star, number, prime, comunist)
for there to be "many" things
with that property, an important constraint being, of course, that
n > 1.
Now, "n" varies with the domain of discourse, and its value relative to
numbers associated with other quantifiers should be correct.
Thus, the
quantifier
"a few"
is given a truth-conditional semantics in a way similar to those for
"many" (or "infinitely many" -- but never "infinitely few") in terms of
the LEAST number of things
that must have some property if there are to be "a few" things with that
property.
If such numbers are termed
THRESHOLD-NUMBERS,
the essential condition is that the threshold-number associated with "a
few"
should be smaller than that associated with "many".
This method can be used
also in the case of the quantifiers compounded with "very".
Thus, the
threshold-number associated with
"VERY many" (e.g. "very many stars, if not infinitely many")
will be
n + m,
with m positive, if n is the threshold-number for "many".
Of the other hand,
if "k" is the threshold for
"a few",
the threshold for "a very few" will be
k - l.
Repetitions of "very" can be coped with similarly, and, also, the
multiplicity of threshold-numbers is reduced by the possibility of defining
some quantifiers in terms of others.
Thus "nearly all" is "not many not".
Now consider
13.
There are many things which are both A & B.
This is one in which the quantifier is not "sortal", as Grice calls it, and
is
logically equivalent to
14.
There are many things which are both B & A.
In contrast,
15. Many As are Bs.
(e.g. many primes are infinite)
involves a queer "sortal quantifier", and is not equivalent to
16. Many Bs are As.
In (15), the quantifier's range is restricted to the set of As: thus the
set
of As becomes the domain of discourse whose size determines an appropriate
threshold number.
Consequently, since the set of As may NOT have even nearly the same
cardinal
number as the set of Bs, the threshold-number determined by one may be
different from that determined by the other, as in 17 vs. 18:
17.
Many specialists in Old Norse are university officers.
18.
Many university officers are specialists in Old Norse.
It seems that (17) is true and (18) false.
There are "more" university
officers than specialists in Old Norse, the threshold-number for "many
university officers" is correspondingly larger than that for "many
specialists in Old Norse".
Or consider the conditional,
19.
If many professional men own French motorcars, and
there are at least as many professional men as owners
of French motorcars, many owners of French motorcars
are professional men.
Now, take a segment of (19), viz.
20.
There are at least as many professional men as owners
of French motorcars.
How do we represent that formally?
Grice suggests this be done by the what he calls
(I,k)-quantifiers.
and involves the application of a method which enables sortal quantifiers
to
be replaced by more complex quantifiers which are *not* sortal.
Thus, it is
clear that the sortal
21. MOST As are Bs.
is logically equivalent to
22.
There are more things which are both A and B
than there are things which are A and not B.
which involves a non-sortal (1,2)-quantifier.
Similarly we may think that
23. Many As are B if not far from half the As are B.
In this case, we could give, as logically equivalent to (17)
24.
There are at least nearly as many specialists in Old Norse
who are university officers as there are specialists in
Old Norse who are not university officers.
where (24) is NOT, however sortal.
This transcription renders patent the lack of
equivalence between (17) and (18) above, for, as anyone can see, (18)
emerges as
25.
There are at lest nearly as many university officers
who are specialists in old Norse as there are university
officers who are not specialists in Old Norse.
Popper continues:
"Certainly, Euclid had impressed upon his memory (and thus presumably upon
his brain) some facts about prime numbers, especially facts about their
fundamental properties."
"But there can, I think, be little doubt about what must have happened."
"What Euclid did, and what went far beyond World 1 memory recordings in the
brain, was that he
VISUALISED [as in 'saw' -- this may be a feature of what Popper otherwise
calls his "Viennese German"]
the (potentially) infinite sequence of natural numbers – he saw them before
his mind, going on and on."
"And he saw that in the infinite sequence of natural numbers the prime
numbers get less and less frequent as we proceed."
"The distances between the prime numbers get, in general, wider and wider."
...
"Now, looking at this infinite sequence of numbers intuitively, which is
not a memory affair, Euclid ends up discovering that there was a problem:
the problem whether or not the prime numbers peter out in the end – whether
there is a greatest prime number and then no further ones – or whether the
prime numbers go on for ever."
-- and ever amen.
---
"And Euclid solved this problem" -- a tautological problem of arithmetics.
"Neither the formulation of the problem or the solution of the problem was
based on, or could be read off from, encoded World 3 material. They were
based directly on an [direct] intuitive grasp of the World 3 situation: of
the infinite sequence of natural numbers."
----
McEvoy comments:
"We might say the problem that Euclid discovered was inherent in the
INFINITE sequence of natural numbers, as primes are inherent in this sequence,
and so the question of whether the primes themselves constitute an infinite
sequence is inherent in the sequence of natural numbers."
This seems a bit stretched.
It is as if I were to say that when Grice utterered,
"As far as I know, there are infinitely many stars"
he was solving all problems of cosmology.
Surely there is nothing in a _phenomenon_ that is, per se, a problem. A
problem needs, first and foremost, a problematiser.
---
McEvoy quotes Popper:
"“Euclid’s proof operates with the following ideas: (1) A potentially
infinite sequence of natural numbers. (2) A finite sequence (of any length) of
prime numbers. (3) A possibly infinite sequence of prime numbers."
All negative ideas, according to Locke!
Note, incidentally, that for a time, the way Euclid arrived at that was
misunderstood.
Theorem [Euclid]: There are infinitely many prime numbers.
Proof:
We use the "Reductio Ad Absurdum" method.
Assume there are only finitely many prime numbers P1, P2, P3...Pn.
Now take the product of all these prime numbers and increase by 1, you get
A=P1xP2xP2…Pn+1.
Dividing A by P1 leads to remainder 1, the same for the other n prime
numbers, thus none of the prime numbers divides A.
Since A must have a unique factorization into prime numbers, the only
remaining possibility is that A itself prime, which is a contradiction to the
assumption that P1, P2, P3...Pn are all the prime numbers (note that A is
larger than all these).
That means that the opposite of the assumption must be true.
"Reductio ad absurdum, which Euclid loved so much,
is one of a mathematician's finest weapons.
It is a far finer gambit than any chess gambit: a chess player may offer
the sacrifice of a pawn or even a piece, but a mathematician offers the
game."
G.H. Hardy
Or not.
Cheers,
Speranza
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