Re: [Marxism] One more on Fermat's Last Theorem, or, Give Peace a Chance

[Rod Holt <rholt@xxxxxxxxxxxxxx>]

Jurriaan Bendien wrote:

> I had actually logged off MM but now I have to respond to this... Thanks to
> Rod for comment. Rod wrote:
>
> The notion of infinity you carry around is an intuitive notion, sort of
> like the "world is flat." It is not part of mathematics. We do not use
> the symbol (an eight lying on its side) except in analysis, which, as
> you know, is very sloppy. We do have another idea which is equivalent to
> your *infinity* and has the advantage of being precise. If you want me
> to expand on this topic I will be glad to later.
>
> I don't believe the world is flat at all, I believe it is round (not a
> perfect sphere but slightly conical, i.e. slightly egg-shaped), just as
> Gerhardus Mercator did in 1569 in confronting the problem of expressing
> three-dimensional space on a two-dimensional plane (a three-space open set
> if you like). I am keen to hear your other idea "which is equivalent to your
> *infinity* and has the advantage of being precise".
>
> Second week of May:
> Dear Jurriaan:
> Please pardon this late reply.
>         I think you already know all this, but I will write it out. This is all from memory. Even though it 
> has been 40 years since I’ve read a copy of “The Journal of Symbolic logic,” I 
> don’t think I have lost track of the broad truths.
> The problem lies in a confusion between cardinal numbers and ordinal numbers. Nobody should 
> confuse these but people do. Just remember that a cardinal number is a property of a set, not 
> of its elements. When a set contains as an element its own cardinal number, then everything 
> goes to pot. In 1905, Bertrand Russell published this observation, which became known as 
> Russell’s Paradox. Cantor’s contradiction is of this sort too.
>         Another idea which has eluded countless generations of mathematicians is the distinction between 
> statements OF a theory and statements ABOUT a theory. The statement that the positive integers form a 
> countably infinite set is NOT a statement of number theory; it is a metastatement. The “countable 
> infinity” referred to is an OBJECT of transfinite arithmetic, or an ELEMENT of Transfinite 
> Arithmetic Theory (given the name “aleph null” by Cantor).
> Number Theory says nothing about any “infinity.” All it says is that you never “run 
> out” of positive integers—which means that there is NO LARGEST POSITIVE INTEGER. You or I can always 
> make a larger one.
>         As far as proof by contradiction, the Constructivists just don’t 
> understand. Proof by contradiction is a metaproof. The theorem so found is a 
> metatheorem. Is all mathematics metamathematics? In a very real sense, yes. Kurt 
> Goedel, Alonzo Church and Kleene however proved that you could make a metatheory 
> mirror the theory under discussion so closely that the following holds: If the theory 
> is consistent, then any proof of a metatheorem about the system is a recipe for the 
> corresponding proof in the theory itself. There are some restrictions on this general 
> result, which I shall omit.
>
> The first written discussion of *infinity* was in Euclid (ca. 450 BC) when he proved there 
> was an *infinite* number of primes. His argument was of the sort “anything you can do, 
> I can do better,” the first such argument in mathematics that I know of. The so-called 
> law of the excluded middle was also fundamental to the proof. [see the special note below.]
>         Special note. Goedel’s startling results published before and during WWII distressed some 
> meta-mathematicians, mostly Europeans around Heytung, who thought they could get around his Incompleteness Theorem 
> by excluding the excluded middle. Starting with the statement calculus, which underlies all the stronger systems, 
> they created a new set of axioms where the statement “Either A or Not-A” would be un-provable. This 
> resulted in a weakened meta-system where proof by contradiction was not allowed. A lot of papers and several books 
> were written about this weakened system, the authors were called “Intuitionists.”
>         Around 1948, Goedel published a very short paper (1 page) in which he 
> pointed out that there could be only one reason to weaken the statement 
> calculus, that being to exclude an internal contradiction. Proof by 
> contradiction goes as follows: if the negation of a proposition is shown to 
> contradict an axiom or theorem, then the proposition is a theorem. Disallowing 
> this kind of proof can justified only if it prevents a potentially fatal 
> contradiction from creeping into the system. Having said this, Goedel then 
> proved (without using the excluded middle) that any internal contradiction 
> derivable with the use of the excluded middle could also be derived without 
> using the excluded middle. He did this by constructing a model within the 
> statement calculus. His proof was so direct and devastating that the 
> Intuitionists disappeared from the mathematical world shortly thereafter.
>         —end of special note
> Euclid’s proof went as follows: There are an infinite number of primes. You 
> don’t agree? OK. You say, then, that there is just a finite number of primes. Now, 
> multiply them all together to get G (G standing for the gigantic product of ALL the primes.) 
> G is a finite number since it is the product of a finite number of finite numbers. G + 1 
> cannot be divided by any of the primes you used to create G. (I suggest doing a couple of 
> examples since I do not have access here to the variety of mathematical symbols I would need 
> to present an abstract proof of this.) Since none of the primes you used to create G divide G 
> + 1, then G is either a new prime, which violates your assertion that you had multiplied ALL 
> primes together to get G, or G + 1 is itself divisible by a prime you left out, which also 
> violates your assertion. I.e., your assertion is false, and its negation must be true.
>         You notice that the excluded middle is used: we assume that there is 
> either a finite list of primes, or an *infinite* list of primes, and there are 
> no other choices. And again, we assume that either your assertion that there is 
> a finite number of primes and that you have multiplied ALL of them together to 
> get G is either true or false, not anything else.
>         There is another proof which goes like this: take the first N primes. Multiply them 
> together and get G(N). Now we can show that G(N) + 1 is divisible by at least one new prime. 
> (We omit the proof) Add the largest of them (which we will call P(N)) to our list. Multiply 
> again to get a new G, call it G(N + 1). We can continue this process as long as we like. We 
> have created a list of products of primes G(1), G(2), G(3), … G(N), G(N + 1) … 
> etc. We now observe that for every positive integer N, there corresponds at least one new 
> prime P(N). Now we conclude that if there is an *infinite* number of positive integers, then 
> there must be an *infinite* number of primes. This is the notion that two *infinities* are 
> equal if they can be organized in a one-to-one correspondence with each other and one sets is 
> *infinite*.
>         The second proof is based on what is called “Mathematical Induction.” 
> This notion was also known to Euclid. Except in geometry, the Greeks preferred proof by 
> contradiction because is most often short and elegant.
> Please note that the statement “There are an infinite number of primes” is a 
> metastatement and NOT one of number theory itself. It is a statement ABOUT a portion of 
> number theory.
>         Georg Cantor did the next significant work. I don’t remember who proved that the 
> number of rational numbers was the same as the number of positive integers, but it was probably him. 
> He showed that between every two rational numbers you could put a real number and, most startling, 
> the extension of that idea: between ANY two real numbers you could place a rational! Yet, there were 
> “more” real numbers than rationals.
>         The whole scheme starts with the notion of a cardinal number as a property of 
> sets. (Cantor was familiar only with naive set theory wherein any proposition 
> determines a set.) Two sets have the same cardinal number if their elements can be set 
> in a one-to-one correspondence with each other. The members of a string quartet in the 
> world of chamber music can be put in a one to one correspondence with the set of 
> numbers 1, 2, 3, 4. The cardinal number of any string quartet is then the same as the 
> cardinal number of bridge players at a table and the cardinal number of any four 
> consecutive integers, etc. (Don’t get confused; the cardinal number of the 
> COLLECTION of all string quartets is probably in the thousands and certainly not four.)
>         This contribution of Cantor is rarely appreciated. It was a very valuable thought, but, since Goedel 
> hadn’t come along yet to create a set theory that made a distinction between “sets” and 
> “classes,” Cantor discovered a contradiction within his own theory of transfinite arithmetic that 
> demolished the whole thing. Even though he knew of this contradiction, he hid it and stayed up all night trying to 
> find what was wrong. He later went crazy. (Goedel discarded the notion that any predicate determined a set. His 
> formalization of an expurgated set theory was printed by Princeton University in a very rare mimeographed paper 
> edition with an orange cover around 1949-50.)
>         You would like to extend the collection of non-negative integers by adding *inf* as 
> an element of the collection of integers. You would like to have *inf* to correspond in some 
> sense to the notion “infinity.” This can be done but nobody wants to do it 
> because it destroys the interesting (and useful) number system we have all come to know and 
> love.
> For example: It is either an axiom or a theorem that there is a unique 
> successor for every integer. The uniqueness allows us to count backwards (every 
> non-negative integer not 0 has a unique predecessor). Consider your *inf*. What 
> is its predecessor?
> Another example: The integers support an equality relation (a = a; if a = b then b = a; if a 
> = b and b = c then a = c), and so we can say there is always a unique integer y such that a + 
> y = b for ANY integers a and b. This little theorem creates the negative integers but it gets 
> screwed up if *inf* has the intuitive property that *inf* + 1 =*inf*. In other words, if any 
> of the usual properties of “infinity” are given to *inf*, then the system falls 
> apart.
> There are many useful extensions of the integers. If the “added elements” form 
> a finite field, then the results are quite interesting.
> The rational numbers (fractions) are an extension of the integers where (b not 0) the ordered pair 
> (a,b) becomes the new element. Equality is re-defined as (a,b) = (c,d) if and only if (a x d) = (b x 
> c). Now, as we all know, rational numbers can always be expressed as decimals although the 
> expression may be of infinite length. (There’s that word again.) For example, by repeated 
> division we find that 1/3 = 0.3333333…. (Remark: We don’t divide an infinite number of 
> times. Instead we observe that the remainder remains constant after a while. Since the remainder is 
> constant, the partial quotient remains constant.)
> Now consider the expression 0.9999999999….
> Let y = 0.9999999999…
> then
> 10y = 9.999999999… Now subtract y to get
> 9y = 9.0000000000… Then divide by 9 and get
> y = 1.00000000…
> We have shown that
> 1.00000000… = 0.99999999… EXACTLY.
> “Exactly” in the sense that there is NO difference between the two expressions.
>         People don’t understand this. In particular, people who work with the calculus (engineers) all believe that the 
> integral of a function creeps up on the right answer, but never ever “really” gets there. The point is that there 
> are many ways to express the same number. In the example of the decimal 0.999999…, if you think there is a difference 
> between it and 1.000…, what is it? It must be a number, although very small. Call your difference epsilon. Now I will 
> find an N such that by examining the difference expression out to N decimal places, the difference must be less than your 
> epsilon. All I have to do is pick N to be larger than the reciprocal of your epsilon. It doesn’t matter how small you 
> think epsilon is, I can always show that the difference between 1.00000000 and 0.99999999… must be less than that.
>         I’ll stop here. I don’t think there is a whole lot to be said for an 
> arithmetical notion of infinity other than what I’ve said. Further considerations must be 
> metaphysical.
> Thanks for you patience,
>                 —Rod
>
>
>
>
>
>
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