In article <44b60791$0$22363$afc38c87@news.optusnet.com.au>, "Peter Webb" <webbfamily-diespamdie@optusnet.com.au> wrote:
> Because his "paper"s that I have seen either display a profound lack of > knowledge of set theory, or are deliberately false/misleading. The guy is a > professor of mathematics. I want to know which it is. I know you must know > him professionally, which is presumably why you are concerned. I would hope > that he can defend his ideas and papers for himself. > > So lets go. These are neccessarily out of context, but I don't think this > misrepresents his position: > > For each of these I would expect the statement is true, it is wrong out of > Norman's ingorance (eg crank material), or it is wrong and he knows it is > wrong (deliberately misleading): > > -------------------------- > To get you used to the modern magic of Cauchy sequences, here is one I just > made up: > > [2/3, 2/3, 2/3 ...] > Anyone want to guess what the limit is? Oh, you want some more information > first? The initial billion terms are all 2/3. Now would you like to guess? > No, you want more information. All right, the billion and first term is 2/3 > Now would you like to guess? No, you want more information. Fine, the next > trillion terms are all 2/3 You are getting tired of asking for more > information, so you want me to tell you the pattern once and for all? Ha Ha! > Modern mathematics doesn't require it! There doesn't need to be a pattern, > and in this case, there isn't, because I say so. You are getting tired of > this game, so you guess Good effort, but sadly you are wrong. The actual > answer is -17. That's right, after the first trillion and billion and one > terms, the entries start doing really wild and crazy things, which I don't > need to describe to you, and then `eventually' they start heading towards > but how they do so and at what rate is not known by anyone. Isn't modern > religion fun? > --------------------------- > > Compare and contrast: > > To get you used to the modern magic of decimal notation, here is one I just > made up: > 0.666... > > Anyone want to guess what the limit is? Oh, you want some more information > first? The initial billion digits are all 6. Now would you like to guess? > No, you want more information. All right, the billion and first digit is 6. > Now would you like to guess? No, you want more information. Fine, the next > trillion terms are all 6. You are getting tired of asking for more > information, so you want me to tell you the pattern once and for all? Ha Ha! > Modern mathematics doesn't require it! There doesn't need to be a pattern, > and in this case, there isn't, because I say so. You are getting tired of > this game, so you guess Good effort, but sadly you are wrong. The actual > answer is 2/3 - 10^billion. That's right, after the first trillion and > billion and one terms, the entries start doing really wild and crazy things, > which I don't need to describe to you. Isn't modern religion fun? > > These seem exactly the same argument to me. True, crank or deliberately > misleading?
My guess is that Norm would be just as uncomfortable with decimal expansions that are not generated by a rule as he is with (other) Cauchy sequences that are not generated by a rule, although there is the difference that when you know the first trillion decimals you know rather more than when you know the first trillion terms in a Cauchy sequence. I may be missing the point of your true-crank-misleading trichotomy here. If Wildberger is rejecting sequences not given by rules, he isn't the first serious mathematician to do so, to do so honestly and while in full possession of his faculties.
> How about: > > --------------------------- > Now that you are comfortable with the definition of real numbers, perhaps > you would like to know how to do arithmetic with them? How to add them, and > multiply them? And perhaps you might want to check that once you have > defined these operations, they obey the properties you would like, such as > associativity etc. Well, all I can say is---good luck. If you write this all > down coherently, you will certainly be the first to have done so. > -------------------------- > > True, crank or deliberately misleading?
This bothered me, too, as I have vague memories of having seen associativity and the rest proved from the Cauchy sequence definition in some course I attended decades ago. I don't know what to make of this paragraph. I hope he elaborates on it sometime.
> On Cauchy sequences: > > -------------------- > On top of the manifold ugliness and complexity of the situation, you will be > continually dogged by the difficulty that in all these sequences there does > not have to be a pattern---they are allowed to be completely `arbitrary'. > That means you are unable to say when two given real numbers are the same, > or when a particular arithmetical statement involving real numbers is > correct. > ---------------------- > > True, crank or deliberately misleading?
Is he saying anything that differs from what Bishop had to say about examples where you couldn't tell whether a real number was zero or not because the definition of the real number depended on knowing whether the decimal expansion of pi does or doesn't contain 17 consecutive zeros? Was Bishop a crank? or deliberately misleading?
> ----------------------- > A set of rational numbers is essentially a sequence of zeros and ones, and > such a sequence is specified properly when you have a finite function or > computer program which generates it. Otherwise `it' is not accessible in a > finite universe. > ------------------------- > > (My mind boggles at what he thinks he means here). > > True, crank or deliberately misleading?
Let's break it into pieces. "A set of rational numbers is essentially a sequence of zeros and ones." You could quibble about the word, "essentially," but I suppose it's true that anything you can do with a set of rational numbers, you can do with a sequence of zeros and ones. "A sequence of zeros and ones is specified properly [only] when you have a finite function or computer program which generates it." I'd class this as "opinion," an option you've omitted from your trichotomy, and an opinion that is unpopular but not unknown in the mathematical research community. "Otherwise, the sequence is not accessible in a finite universe." I think he makes it clear that what he means is that if the sequence is not generated by a rule the only way you could know enough about the sequence to answer all quesitons about it is to have it all written out, which you can't do in a finite universe.
> -------------------------- > Even the `computable real numbers' are quite misunderstood. Most > mathematicians reading this paper suffer from the impression that the > `computable real numbers' are countable, and that they are not complete. As > I mention in my recent book, this is quite wrong. Think clearly about the > subject for a few days, and you will see that the computable real numbers > are not countable , and are complete. > ------------------------- > > Oh, my God. He has written a book on it. I hope its not part of the UNSW > pure mathemtics syllabus. > > True, crank or deliberately misleading?
This one threw me for a loop, too. I know about the trigonometry book, although I haven't seen it. I don't know whether that book contains his argument that the computable numbers are complete and not countable. It would seem to be an odd thing to include in a book that purports to do high school trig.
Anyway, I haven't taken a few days to think clearly about the subject, and even if I did, I don't know that I'd see what he says about the computable reals.
I'd still reject your trichotomy, and label this paragraph, "unclear."
> He then goes on: > ------------------------- > Think for a few more days, and you will be able to see how to make these > statements without any reference to `infinite sets', and that this suffices > for Cantor's proof that not all irrational numbers are algebraic. > ------------------------- > > Sorry, I am having a lot of trouble understanding how I can think about > "all" irrational numbers without thinking about an infinite set. I have even > more trouble comparing the cardinality of "all computable numbers" and "all > algebriac numbers" without using set theoretic constructions. > > True, crank or deliberately misleading?
You can give a rule to construct an irrational number which you can then prove to be unequal to any algebraic number someone else hands you. You can do this without thinking about the set of all irrational numbers, just as you can prove the square root of two is irrational without thinking about the set of all rational numbers (heck, Euclid pulled that one off).
You can (Norm says) work with functions by knowing what kind of thing is an input, what kind of thing is an output, and what the rule is, without ever thinking of the set of all inputs. If you can show that any rule that takes an integer as an input and gives a computable number as an output must omit some computable number, then you have a proof that computable numbers are not countable, and you've done it without using set-theoretic constructions.
Don't ask me how to do it (but isn't all this stuff in Bishop?). Again, I'd reject your alternatives, and say, "needs elaboration."
> ------------------------------------ > In my studies of Lie theory, hypergroups and geometry, there has never been > a point at which I have pondered---should I assume this postulate about the > mathematical world, or that postulate? > --------------------------------- > > Hmm. Never seen a Group theorem that assumes a Group is commutative?
Well, now I think you're being misleading. I think it's clear that he means he has never had to ponder which model of set theory he is working in.
> ----------------------------- > but the nature of the mathematical world that I investigate appears to me to > be absolutely fixed. Either G2 has an eleven dimensional irreducible > representation or it doesn't (in fact it doesn't). > ----------------------------- > > The guy is a group theory expert. Presumably he has heard of the Whitehead > problem. http://en.wikipedia.org/wiki/Whitehead_problem > > True, crank or deliberately misleading?
Way beyond my area of expertise. I won't go there.
To sum up my reaction to your examples, I'd say Norm has a lot of explaining to do if he wants to gain acceptance for his ideas. But I'm not comfortable labelling him a crank, or accusing him of deceit.
-- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
