I'll buy that Newton. Might even have that information in one of these stacks.
You've done for Zim what I hesitated to do because I am usually much rougher than you. After all Zim does think what he is saying is valid and appears to mean well. What appalls me is that things have come to such a pass that folks who can write such garbage actually can acquire a degree, be it even a minor.
Of course folks like Steinbok are another kind of beastie with ego and who knows what else involved. I have not checked the rest of my messages so I don't yet know if he answered your questions.
Incidentally, I would have thought you would know it but Asimov, besides being the greatest writer of science fiction, was also a genuine scientist who occassionally wrote popular, but very good little bits about the real world and how it works. If I remember from early on, he once wrote an article that presages what DNA now proves beyond doubt, his article being about the related chemistrys that proves the common sources of all life on earth.
On 17 Aug 99, Newton Leibniz wrote re. The GRAND FINALE to this topic!:
> Once again Zim you have completely missed the point. Math art (as > defined below) is fine and is not dangerous. Paintings by Escher are > certainly fun and interesting to look at. Furthermore, NOBODY has > said so far that math art is dangerous in and by itself. Therefore I > do not see how you derive your conclusion. > > What we have said, and I repeat myself so I grow bored of this > conversation, is that your `mathematicsÂ is not well defined, doesnÂt > make any sense whatsoever, and is for the most part gibberish. Maybe > it is not obvious to you why I say your stuff is complete nonsense > (although I really hope is obvious to everyone else. ) So here is a > refutation of part of a paragraph from your page: > > "There may be a missing parameter in the equations of knowledge in > Mathematics and the other Sciences." > > -What `equations of knowledgeÂ are you referring to? I know of no > such thing in mathematics. It is exactly equivalent to me stating > `There are missing parameters in the equations of a hot dog.Â > > "Perhaps agreeing with Kurt Godel's Incompleteness Theorem, we can > never know everything about anything." > > - Perhaps agreeing? This is a completely proven mathematical theorem. > It is true based on the axioms of mathematics and therefore you are > forced to agree by logic. > > "Fleshing out this hypothesis I propose that the only axiomatic set(s) > of operands in Mathematics is: Everything, Anything, and Nothing." > > Do you even know what you said here? First off, there are two ways > to interpret this statement. This is due to very poor English and > lack of being able to express yourself clearly. First interpretation: > There are only three operands in mathematics: Everything, Anything, > and Nothing. Okay, if we accept this interpretation, you need to show > me how to build up to the multiplication operand (the one that allows > us to multiply numbers). Since by your thinking there is only three > operands, ALL other operands must be somehow defined from those three. > How do you use Everything, Anything, and Nothing to get 5 X 6 = 30?!! > You canÂt of course and so therefore you are writing gibberish. > > The second interpretation is that operands of mathematics fall into > one of three `sets.Â The `set of everythingÂ is not well defined as > a set (it is called a class and is outside the realm of mathematics), > the set of anything is just not well defined at all since there is no > way to test whether an element is in it or not, and the set nothing by > definition is the empty set. So really you are saying that each > operand is an element of either two mathematically undefined objects > or isnÂt an operand because it is contained in a set which is empty. > Again, your statements lead to complete gibberish. > > This goes on and on and on. I feel like a cat playing with a mouse > and I am growing bored. Go back to school and learn some more > mathematics. Before you can dance, you must first learn how to walk. > > Jack has given us an example of a much more interesting nature. This > example is better written and more clever than any of ZimÂs arguments. > It has the potential to sound convincing even though it is not based > on any real scientific evidence. Sure, the writer quotes Penrose and > other famous physicists and some of the science talked about has been > validated, but the writer is for the most part writing science > fiction. His actual claims are not based on any validated observable > phenomena. This is because we have not been able to accurately > measure conscious other than the binary "heÂs dead" and "your alive." > Until we can accurately measure conscious, science has absolutely > nothing to say about the matter. You might as well be reading Isaac > Asimov. While much of education theory does sound like ZimÂs > arguments, there are a few philosophers who are more clever in their > wording and are similar to JackÂs example. But their arguments fall > just as easily as ZimÂs arguments when scrutinized by a careful > reader. For example, I refer you to the article by a Nobel prize > winner in cognitive psychology where he refutes all of the main > tenants of constructivism: > > <a href="http://act.psy.cmu.edu/personal/ja/misapplied.html">http://act.psy.cmu.edu/personal/ja/misapplied.html
> > This brings me to the most important point of this discussion Â the > possible dangers of mathematical art. We need to distinguish between > two forms of mathematical `art.Â The first and most obvious are the > paintings that Escher has so kindly taken the time to draw. Certainly > there is a place for these pieces of artwork in elementary school as > long as the time studying them and others like them is spent during > the culture/art class period and is labeled as such. If the time > spent on math art infringes on true mathematical study then the > students will not learn real mathematics and this will have > devastating consequences on their future social class, job > opportunities, etc. In that context, it can be dangerous. > Furthermore, it is well known that elementary school teachers have > trouble teaching mathematics. Therefore you do not want to give them > any possible excuse to replace true mathematics with a > non-mathematical activity where they can claim `NoÂ I am doing math art > which is mathematicsÂ when in fact it is not. > > The second form of art deals with this `creativityÂ issue that Zim > keeps bringing up. Why not let the students be `creativeÂ and let them > make up their own mathematics? Sounds tempting and it reminds us all > of similar things done in elementary school where the teacher brings > out a bucket of paint and lets the students paint. Have you ever > looked at those paintings? They certainly are not artwork just like > ZimÂs mathematics is surely not mathematics. It took several years of > instruction and practice with a *trained* art instructor before I was > able to produce paintings and drawings that could be considered > artwork. It would take me several more years of intense study to > become a master. The point is that we do not need to become master > artist to live in todayÂs civilization, so buckets oÂ paint are fine > for elementary education. But we do require every child to become the > master of a certain amount of mathematics Â if they donÂt, it would be > as devastating as not knowing how to read. In this scenario asking > students to create their own mathematics would produce math which is > similar to their paintings and ZimÂs mathematics Â mostly gibberish > and definitely not true. The danger here is that the teacherÂs are > not well trained in mathematics and would not know the difference > between valid mathematics and invalid mathematics (much like some of > you might be fooled by ZimÂs arguments) and this opens up the very > real possibility that the students would not be learning the right > information, and in fact they are lead to believe in wrong > information. The `very real possibilityÂ means that I have already > seen such things occur. As students grow up, their poor training in > logic and mathematics opens the doors of fallacy allowing them to > accept the more clever example of JackÂs as valid and true. > > So we see that there are some dangers in what Zim and much of the > education community is proposing. I think it would be much better to > give the students a good training in the mathematics like the Japanese > or Chinese do. Their teachers are not worried about the students > doing `creative mathematicsÂ, they are trained teachers who expect > students to master a certain amount of very necessary mathematics. If > we can do that, then it will also prepare some of the more diligent > students to do true mathematical art To understand true art form in > mathematics, you will have to read a book called `A MathematicianÂs > ApologyÂ by G.H. Hardy. The only way to become a master of this art > form is through even more years of intense study and practice of > mathematics, much like a master artist. Only then can you truly be > `creativeÂ in mathematics. > > ********************************************************************** > ********************************************************************** > > Jack, this is tiring and I feel like I am talking to children (Zimmy > and Vicky). This is the case many times on normal news groups (which > is why I avoid them like the plague) but somehow I expected more from > a news group which is about such a serious topic. I propose we let > the little fish go and start after some of the bigger fish in the > pond. Surely there must be smart and intelligent educators reading > this forum whose arguments donÂt fall so easily and fast. If there > arenÂt and I am only speaking to you and the kids, then I am > completely wasting my time which would make me very angry. Give me a > few days to write up some questions that I would like answered by the > education community. Your job is to search the NCTM web site and > find names and email addresses of the people who run this > organization. We will pose these questions to themÂ no answers from > the kids will be taken seriously or commented upon. If we can get > them to answer, then maybe a good debate based on logic and scientific > evidence will ensue and I will be assured that I am not wasting my > time. If they donÂt answer, then that would be the biggest message of > all: > > "We here at NCTM do not care what you have to say! This news group is > only to give you the impression that we give a hoot. You can all go > blow it up your nose for all we care." >