Divine Proportions

David Halprin writes with a review of a new (and mighty odd sounding) mathematics book: "In my humble opinion, we have an unjustified polemic in the world of mathematics, yet again. My background is tertiary level mathematics and concomitant research in specialised areas, so when a friend e-mailed me the link to this book, I was so excited after reading the author's hype, that I ordered a pre-publication copy. My expectations have not been met, unfortunately, hence my analysis precipitated this review." Read on for Halprin's idiosyncractic take on Norman John Wildberger's Divine Proportions: Rational Trigonometry to Universal Geometry. Divine Proportions - Rational Trigonometry to Universal Geometry author Norman John Wildberger pages 300 publisher Wild Egg Pty Ltd rating 2 reviewer David Halprin ISBN summary Wilberger presents an ultimately disappointing vision of a new descriptive system for geometry.

There are various ways to approach Norman's so-called "Rational Trigonometry" and/or "Universal Geometry." I have examined it from various perspectives and it does not live up to Norman's claims, whichever standpoint, that I have taken.

DEFINITIONS

Comments

Too bad

[andrewman327]

It is too bad that these new ideas are so poorly implemented and described. The ideas seem appealing at first glance, but they ultimately do not survive close scruntiny.

Sometimes it seems that the only really new ideas being tossed around (outside of lab research and the like) in science are from Wolfram in his book, A New Kind of Science. (I do not include creationism in this category because it is not new, so spare me the flames regardless of how you feel about it.) Scientists are great at empirically testing this and that theory but they often have problems altering their own perceptions on existing and accepted information.

I agree with the review that this form of geometry should never supplant the status quo:

Don't forget that his advocacy is to replace classical geometry and trigonometry, (especially lines and angles), at school level. He doesn't suggest retaining it and using his methods as a adjunct and/or complement, especially since some of those guys and gals will become architects, surveyors etc. etc. Were the academic institutions which set college and university curricula, to take Wildberger at his word, by eliminating regular trigonometry and geometry and replacing it with his concepts, it would be the downfall of current mathematical knowledge and standards for years to come. What's more, the damage would take years from which to recover; an almost irreparable predicament in education.

Re: Too bad

[Wolfbone]

"Sometimes it seems that the only really new ideas being tossed around (outside of lab research and the like) in science are from Wolfram in his book, A New Kind of Science."

Really new? No. Tossed around? Oh yes ;-)

" In ANKS Wolfram says that "the core of this book can be viewed as introducing a major gener- alization of mathematics" (p. 7). In this he is entirely mistaken, but there are at least two ways in which he has benefited mathematics: he has helped to popularize a relatively little-known mathematical area (CA theory), and he has unwittingly provided several highly instructive examples of the pitfalls of trying to dispense with mathematical rigor."

http://www.ams.org/notices/200302/fea-gray.pdf#sea rch=%22In%20ANKS%20Wolfram%20says%20that%22

Re:Too bad

[Salis]

Too bad Wolfram's book isn't science either.

Cellular automatons can "look" like some physical process, but that doesn't mean the two have any casual relationship whatsoever. I think Wolfram forgot that Correlation != Causation.

Or, more likely, he absolutely knows that the work is crap and so he publishes it in a book rather than submitting it to peer review in a respectable mathematical journal.

And, before I get a nasty reply, let me make this clear:

Science is about PROVING or DISPROVING a hypothesis. (Or, at least, making the attempt to do so.) Does Wolfram do this? Absolutely not. The title of his book makes sense, though. It is a new kind of science...the bad/wrong kind with zero consequences or illumination.

Philosophically/Ideologically driven blather

[Coryoth]

The author (of the book) is, to my mind, tending dramatically toward the loopy side. Take, for instance, this piece he wrote. It starts out as an interested discussion into some issues in the philosophy of mathematics, so skip down to the middle or closer to the end to read what has, by that point, devolved into an unmitigated rant from a finitist of the worst kind. Questioning the foundations of mathematics is not new, nor is questioning whether we wish to admit the concept of a "completed infinity" as compared to conceptions of "potential infinity", however even the Intuitionist school, hell even Brouwer himself (who was certainly not a man interested in compromise) would be rather appalled by the extremes here. Intuitionist mathematics has developed into a respectable field, with things like nonstandard analysis proving to provide interesting alternative constructions of real numbers and analysis. I can't see how Wilderberger's philosphy will lead anywhere.

Wilderberger's stance - that there is simply a finite "biggest number" and we shouldn't use or allow anything "bigger", and the resulting implications for irrational numbers - is just baffling. I'm guessing it is the extreme (and from what I can tell surprisingly uninformed) finitist philosophy that drives his Rational Geometry (he needs to somehow eliminate non-commensurable/irrational quantities from geometry lest they interfere with his fear of the infinite) - to him the superiority of Rational Geometry is presumably clear, in that it aligns with his extremist philosophy. The problem is that his philosophy seems, at best, half baked. He seems like a mathematician who took an interest in philosophy but couldn't be bothered seriously reading or considering any of the vast amounts of material on philosophy of mathematics. That is to say, he is, in many ways, little better than this lunatic ("Cubehead") who is hell bent of redefining mathematics to fit with the pronouncements of his idol, Gene Ray (creator of Time Cube), regardless of how shaky the grounding philosophy may be.

There are a lot of math crackpots out there...

[exp(pi*sqrt(163))]

...99% of whose writings would make a 5 year old's grasp of number theory seem advanced. People who have proved FLT (the easy way), that 0.999... recurring is less than 1, that there are countably many reals and so on. But the author of Divine Proportions is one of those unusual crackpots who's obsessed with an idea but hasn't allowed that to completely compromise their mathematics. These people don't deserve to be beaten down along with the others. I think that having no review of this book would have been better than this review.

Re:My idiosyncratic take

[Quantum Fizz]

There's a term for that - intellectual masturbation.
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But the irony is that despite the author's pretence, the review is horribly written and not clear at all. I'm a physics grad student, I've read my share of poorly-written texts and articles, but in even those instancs, at least, does the author convey his message in some understandable way.
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This review was atrocious, yet the author prides himself on his ability to use a thesaurus. It seems he wants so badly to be admired as a Renaissance man, yet he only comes out looking foolish.

Gotta ask the reviewer ....

[slightlyspacey]

The same question I once asked a mathematics professor after a 45 minute session on a single proof: "Someone actually pays you to do this?"

Didn't get a good grade, but the resulting stunned silence from the class was worth it.

Re:I'm not that Smart!

[linaron]

The reviewer is a moron. No only his euclidian distance formula is wrong (as in primary school wrong), but if he ever got past reading the first chapters of Wildberger's book, he should know that the whole idea is to get rid of cosines, sines and other trigonometric functions in algebra computations. Few people realize that the angle concept we use is axiomatic and ill-defined algebraically. The beauty of Wildberger's approach is to redefine a lot of algebraic formulas in the shape of quadrance and spread equivalent formulas that are much more precise, and easier to compute. The rest of the review was made on a mushroom-induced delirium
So the reviewer cheats by using cosines and sines in his example. Where did he get those values? From a table or calculator. Sines and cosines are computed by adding a (truncated) infinite series of numbers. You dont't have to do that using rational trigonometry, that's the wole point. It's like proposing going from New York to Boston without riding a car (by flying on a plane), and saying that it is ridiculous, because you like driving from New York to Boston just fine.
I remember an old southafrican (boer) joke.
"You need 2 legal workers at union wages and a bulldozer to dig a hole in a day? What, give me ten kaffirs and I'll make them dig it in a week, at half price".
Really