Talk:Controversy over Cantor's theory

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Cantor Anti-Diagonal Argument — Clarifying Determinateness and Consistency in Knowledgeful Mathematical Discourse

Perhaps my unfinished manuscript "Cantor Anti-Diagonal Argument -- Clarifying Determinateness and Consistency in Knowledgeful Mathematical Discourse" would be useful now to those interested in understanding Cantor anti-diagonal argument. I was hoping to submit it to the Bulletin of Symbolic Logic this year. Unfortunately, since 1 January 2008, I have been suffering from recurring extremely blurred vision due to frequent “exploding optical nerves” brought on by my diabetes (I can’t afford laser eye surgery) and I had only about 20 productive days in the last 8 months. At this rate, it would take me a long while to finish my paper or may not be able to complete it if I go permanently blind soon. I just hope my endeavors to clarify mathematical infinity and modern logic would reach the next (if not the present) generations of mathematicians, philosophers, and logicians. [BenCawaling@Yahoo.com] BenCawaling (talk) 08:14, 4 September 2008 (UTC)

Computer Scientists Know Cantor Diagonal is Wrong

While most "main stream" mathematicians accept Cantor's Diagonal Proof, it's easily understood to be false by computer scientists.

How many real numbers are there? Simple. As many as there are computer programs capable of generating the nth digit of a real number when asked. The total number of finite computer programs is countable, and can be listed. Every real number is generated by an algorithm on the list.

So, why doesn't Cantor's Diagonal Argument disprove my simple list? My list of all algorithms contains invalid algorithms that will "hang" and never finish calculating the nth digit when asked. The Halting Problem shows that we cannot ever remove all the algorithms that hang from the list. Therefore, Cantor's algorithm hangs! QED.WaywardGeek (talk) 20:02, 2 June 2009 (UTC)

Interesting. I am not sold about the claim that all real numbers can be described. Certainly all finitely describable real numbers are countable since text is countable. Thus it is impossible to prove a real number is missing from such a list by finishing a description of it since any finitely describable real number would be in the list. Of course if one could prove the existence of any finitely indescribable real number, then such a number would not be in the list by definition.

QED.meadbert (talk) 20:02, 2 June 2009 (UTC)

WaywardGeek seems to be claiming that all real numbers are computable (not just describable), which is incorrect. But WaywardGeek is certainly right that the set of computable real numbers is countable, and that Cantor's diagonal argument does not apply to this set (because the diagonal number needn't be computable). --Zundark (talk) 21:10, 2 June 2009 (UTC)

You are absolutely correct. I would be classified a constructionist, I suppose.WaywardGeek (talk) 22:07, 2 June 2009 (UTC)

Cantor's theorem says "there is no surjection from the natural numbers to the reals". If you decline to recognize non-computable reals, then for coherency's sake you should also decline to recognize non-computable surjections. And now, guess what: You agree with the theorem.

I've indulged myself briefly here, going off-topic in response to an off-topic post, but please be aware that this is not what talk pages are intended for. Per WP:TALK, talk pages are meant to discuss improvements to the corresponding article, not to discuss their subject matter. If you want to argue about this, there's always Usenet, or a hundred other possibilities. --Trovatore (talk) 00:43, 3 June 2009 (UTC)

I think the article could be improved by pointing out that the only real numbers not included on the countable list of computable numbers are bazaar numbers who's value cannot ever be known (and thus never listed). These definable, but unknowable values do not impact our universe in knowable ways.WaywardGeek (talk) 14:29, 13 June 2009 (UTC)

Removal of Bishop quote

I removed this quote from the section about Hume:

"Classical mathematics concerns itself with operations that can be carried out by God… Mathematics belongs to man, not to God… When a man proves a positive integer to exist, he should show how to find it. If God has mathematics of his own that needs to be done, let him do it himself." (Errett Bishop (1967))

First of all, I'm looking at the 1967 book, and the second sentence occurs on page 2, but the first sentence doesn't seem be anywhere before it. Second, Bishop does not mention Cantor in this book on or before the page on which the quote occurs. According to the index, he mentions Cantor only on page 25:

"Theorem 1 is the famous theorem of Cantor, that the real numbers are uncountable. The proof is essentially Cantor's 'diagonal' proof. Both Cantor's theorem and his method of proof are of great importance."

Bishop praised both Cantor's theorem and Cantor's method of proof, in the only context where Bishop mentioned Cantor, which is entirely separate from the statement about God (where Bishop was saying positive things about Kant and Kronecker). Well, but what about Hume? Bishop's 1967 book has nothing to say about Hume, according to the index. Bishop's alleged role in this particular alleged controversy appears to have no documentation. Anyway, the first quote does not belong in the article as it stands. Maybe the one on page 25 does, but I doubt it; anyway it would seem to put Bishop on the other side of the controversy (i.e., supporting Cantor), if anywhere.66.245.43.17 (talk) 23:13, 15 October 2009 (UTC)

Time frame; synthesis

What is the time frame for this controversy? Cantor died in 1918. Kronocker died in 1891. Yet, the article quotes people in the late 20th century or even the beginning of the 21st century, as though they were all on two opposing basketball teams or political parties. Mathematicians probably can't be neatly categorized into "Cantorians" and "Anti-Cantorians". This idea of a controversy lasting more than a century seems to be a vague abstraction. The involvement of various mathematicians appears to be exaggerated by quotations taken out of context which may not in reality have been addressing the subject of this article. It's very interesting material, and certainly everything is related to everything else if you think about it hard enough, but there are problems with "original research" and "synthesis" in Wikipedia articles. The fact that this particular article is about "controversy" means it needs to be especially focused to avoid synthesis. I suggest that the editors of this article define its purpose and scope more precisely and narrowly. 66.245.43.17 (talk) 21:57, 16 October 2009 (UTC)