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## Divine Proportions192

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

Firstly, the definitions, given in the Introduction:-

quadrance = (distance)2 = (x2 2 - x1 2) + (y2 2 - y1 2)

N.B.When one has an equation to solve, (say it is a quadratic), one expects two solutions and deals with them accordingly. If, however, in order to solve an equation, that has a square root sign within it, then one has to square both sides of the equation at some time and this doubles the number of solutions. These extra solutions are regarded as inadmissible, despite their potential interest and possible geometric interpretation. (See worked example later.)

Here is a point of view which suffices to reject this book on its own merit, whether or not there are any other objections, although many other readers will already know of many other disapprovals to mine.

Let's consider someone proposing new variables in some geometric enterprise. This happened in Plane Geometry (for instance), post Descartes, when some bright sparks came up with Polar Coordinates, Pedal Coordinates, Contrapedal Coordinates, Bipolar Coordinates, Parabolic Coordinates, Elliptic Coordinates, Tangential Polar Coordinates, Cesaro Intrinsic Coordinates, Whewell Intrinsic Coordinates and Euler Intrinsic Coordinates, etc.

There are three essential steps to any such proposal:

1. The defining of these coordinates — either in words, with a geometrical description, or in clear mathematical symbology.
2. The relationship of these new coordinates with some other planar coordinate system. This amounts to a mathematical statement of a coordinate transformation. (e.g. From Cartesian to Polar and/or Polar to Cartesian.) Once this is so done, then one can transform any previously-found equations to the new symbology, and hence arrive at a new taxonomy for plane curves, or a new way of stating the conditions for two lines to be parallel, perpendicular or concurrent, or for points to be collinear or not, etc.
3. The demonstration how this new system can be a better system for certain types of problems, perhaps with some limitations in special cases, but not denying their right to be subsumed into mathematical texts, curricula, etc.(e.g. Curve-sketching made easier for plane curves, which are expressed in the new coordinate system, if it is to be preferred in selected examples.) Other pre-existing coordinate systems have shortcuts to finding such things as asymptotes, cusps, asymptotic circles, poles, points of inflection, maxima and minima etc., so the reader would expect to see similar findings by Wildberger.

This third step, in my humble opinion, is where Norman comes undone, and then some!

viz.1) Wildberger cites many plane curves and their concomitant equations in his new coordinate system, in Appendix A, (pages 279-286), but his diagrams have been drawn using software that is dependent on standard polar equations, which are then converted by the software to Cartesian form for plotting. In no way is his "Rational Polar Equation" suitable for being implemented by the software employed. Certainly, any programmer worth his salt could devise a not-so-easy and/or complicated routine to transform Rational Polar Equations back to the regular form, but that is no pat-on-the-back for Wildberger, rather it shows the counter-intuitive and flawed reason for using that coordinate framework.

viz.2) Wildberger's five laws are merely standard trigonometrical identities disguised by his new symbology, showing no advantage over the original forms. See table in Appendix.

He cites a triangle problem in his first chapter on page 14. He then gives a so-called "Classical Solution" in 5 equation lines, using a trig. table via a calculator, for part of this method.

Then, in the next page, he gives his so-called "Rational Solution"
, which requires three diagrams and 8 or 9 equation lines, and this is a flawed solution, to which he seems oblivious, and does not own to it therefore.

Anyone with a modicum of mathematical sense, who tackles this triangle problem, knows the following:-

The usual properties of arithmetic with respect to commutativity, associativity and distributivity also apply equally to common algebra.

When one has an equation to solve (say it is a quadratic), one expects two solutions and deals with them accordingly. However, in order to solve an equation that has a square root sign within it, one has to square both sides of the equation at some time, and this doubles the number of solutions. These extra solutions are regarded as inadmissible, despite their potential interest and possible geometric interpretation.

Viz. The worked example for the rational method for the triangle on page 15 accepts the inadmissible solution as though it is acceptable, whereas the better solution method is the classical method used properly, without recourse to trig tables, and in only four equation lines.

PROBLEM

A triangle ABC has sides a = 5, b = 4 and c = 6.

A st. line from C to AB, (length d), cuts AB at D,

where angle BCD = 45 degrees. What is the length d = CD?

MY SOLUTION

cos B = 3/4 sin B = 7/4, BDC = 180 - (45 + B)

sin BDC = sin (45 + B) = sin 45.cos B + cos 45.sin B

sin(45 + B) = (3/4 + 7/4)/2 = (3 + 7)/(42)

d = 5 sin B/sin BDC = 57/4 x (42)/(3 + 7) x (3 - 7)/(3 - 7)

= 52(37 - 7)/2 = 3.313693059

So, in this first instance, Rational Geometry does NOT provide anything worthwhile, contrary to Norman's hype.

In chapter two, Norman introduces a dissertation on Fields, as though this is an important factor for understanding and using Rational Geometry, despite the fact that up to a student's age of 17, schools don't find it necessary to introduce into his/her brain any Field lessons together with geometry and trigonometry.

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.

c.f. Cuisenaire of yesteryear.

However, you don't have to read between the lines to see on page 21 that Wildberger excludes 'characteristic two fields.' Although I am not versed in Field Theory, I opine that such an exclusion does not apply to classical geometry and/or trigonometry, otherwise he would have said so. So, he is already implicitly confessing, to a failure of Rational Geometry in the global sense.

I have to confess that I look upon his sojourn into Field Theory as a diversion in the same sense that a prestidigitator (magician), in his field of legerdemain (sleight of hand), distracts the audience members, thereby lessening their attention on what's really going on.

Wildberger then goes into proportions using the a:b = c:d symbology, as though it has more merit than the usual a/b = c/d, like we have in the Sine Rule, say. Warum? Wherefore?

On page 9, he states, without proof, the equation for the spread between two lines. From standard trig, one can easily calculate the angle between two lines, and when one squares the sine of that angle one has his equation without recourse to rational geometry. Now if one subtracts this expression from 1, one obtains the square of the cosine of the angle between these two lines. Naturally if one starts with these two terms and adds them one can see why they sum to unity, which he states on page 27 as Fibonacci's Identity.

A rose by any other name is still a rose, I believe; Pythagarose?

Then Wildberger presents variants of this, all of which are obtained with simple college algebra and are further diversions. Then he waffles on about the possibility of a denominator being zero and its implications. WOW.

(See table in Appendix).

Then, we have linear equations and their solutions using determinants as though it is a revelation. WOW WOW!

At this point, why not reinvent the wheel?

Remember, this book is not aimed at secondary students; such a lower level of presentation is promised in an intended future publication. So, why does he tell us cognoscenti' so much that, obviously, we would know before picking up his book?

Is he just filling up the pages, due to lack of the Step 3 material, so we are drooling to obtain an implied revelation or other especially informative disclosure?

N.B. We mustn't hold our breath, so as to avoid cyanosis!

So now, on page 31, we have Polynomial Functions and Zeros. Wildberger examines an example in F19, but does not explain why on earth that has any significance in curve sketching. After all, we expect our graph to be plottable in a Cartesian Framework in the usual field of numbers, which we, and our computer plotting software, always use by default.

Page 32 teaches us how to solve quadratic equations by completing the square. This is so deep, that I hope the reader's gray matter can cope, especially since he/she is, presumably, at tertiary level!

Now to chapter 3 starting on page 35: Cartesian Coordinate geometry. On page 40, he makes a special reference to the conditions of perpendicularity of two lines. This is easily calculated since the product of their gradients must be -1. However, he stresses "that this is the single most important definition in all geometry, it colours the entire subject." Then he follows this up by naming this "blue geometry."

So mind-boggling WOW WOW WOW! He then promises that other colours will appear. I can hardly wait. I hope the new colours match the colour scheme in my study.

Summarily, there has been nothing from Step 3 to illustrate a finding in Rational Geometry, that gives it an edge, at least. He is just making statements, that are already well-known in geometry and trigonometry, and he is an associate professor in mathematics, who should be able to do a lot better than that. I opine that he doffed his professorial hat and replaced it with a dunce's hat in order to write such pretentious garbage.

One must address one's audience, or write to one's intended readership, at a consistently-appropriate level. In matters of a so-called "New Mathematics," he must demonstrate actual advantages, and not attempt to hoodwink us, as he did in the earlier problem on Pg.14 and its badly worked out, so-called "Classical Solution".

If one searches the web, there appears to be no academic interest in "Rational Geometry" by the diasporic mathematical fraternity.

Especially, I had hoped to find that his fellow mathematicians at UNSW would have had something worthwhile to say, and thereby prove me to be an innumerate imbecile for daring to criticise "Divine Proportions."

Alas and alack, niente, gar nichts, zilch. Woe is me. Es tut mit leid.

CONCLUSION

In its present format, a better title would be:-

"LE GRAND PURPORTISSIMENT"

This book, overall, is a misrepresentation of the facts. It purports to be what it is not. The promotional literature on the author's web site is descriptive, but more of the author's dream for a mathematical breakthrough than an actual innovation.

If finances were no concern, I would suggest a complete re-presentation of all his original findings under a new title, that states, in effect, that this is a new coordinate framework, that, from time to time, has occasional advantage over the Cartesian Coordinate system, comparable to the other planar frameworks, stated on the first page of this review.

So mote it be. Amen.

APPENDIX

 RATIONAL TRIGONOMETRY LAWS ANALOGOUS LAWS IN TRIGONOMETRY 1. Triple Quad Formula for collinearity of three points Triangular area degenerated to zero. 2. Pythagoras' Theorem for right triangles Pythagoras' Theorem 3. Spread Law for any triangle Sine Rule 4. Cross law for any triangle Cosine Rule 5. Triple Spread Formula for any triangle (Quadrea) 16 x (Area)2

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## Divine Proportions

• #### Karma whoring (Score:5, Informative)

on Friday August 18, 2006 @02:51PM (#15936825) Journal
Slashdotters vetted this before [slashdot.org]
• #### I'm not that Smart! (Score:5, Funny)

on Friday August 18, 2006 @02:56PM (#15936867) Homepage
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

Polemic [wikipedia.org]
Tertiary [wikipedia.org]
Concomitant [wiktionary.org]
• #### Re:I'm not that Smart! (Score:5, Funny)

<keith.russellNO@SPAMgmail.com> on Friday August 18, 2006 @03:07PM (#15936947) Journal

Does the Architect know his thesaurus is missing?

• #### Re:I'm not that Smart! (Score:5, Funny)

<valuation&gmail,com> on Friday August 18, 2006 @03:10PM (#15936972)
I, too, found this book to be shallow and pedantic.
• #### Re: (Score:3, Insightful)

Really, what do you expect when you pick up a "groundbreaking" mathematical theory book? These books are all the same, some PhD spouting reasons why his solution to some trivial problem is more elegant than someone else's. Anything that requires an actual BOOK to explain probably isn't worth it, because the traditional means to solve problems are often much simpler. I actually had these types of discussions (though related to set theory, not trig/geometry) with grad assistants and professors in college,
• #### I'm not that funny either... (Score:4, Funny)

on Friday August 18, 2006 @03:11PM (#15936975)

A rose by any other name is still a rose, I believe; Pythagarose?

There's also the recurring WOW WOW WOW's which I believe delightfully attempts to break the morose ambiance that prevails throughout the maelstrom of words that the author has deemed fit to call a critique of Wildberger's latest publication.

• #### Re: (Score:2)

Btw, it's "Tut mir leid". not "mit".
Some people just try too hard...
• #### Re:I'm not that Smart! (Score:4, Funny)

<ed@membled.com> on Friday August 18, 2006 @04:35PM (#15937450) Homepage
I like that review... it starts off reasonable, and gets increasingly Time Cube [timecube.com] as it goes on.
• #### Re: (Score:3, Insightful)

I can't decide who's more crazy here, the reviewer, the original author, or the commenters. I now have more respect for the standard Slashdot story quality now that I've seen what happens when there's a major screw-up.
• #### Too bad (Score:5, Interesting)

on Friday August 18, 2006 @02:58PM (#15936879) Homepage Journal
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 [slashdot.org]. (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 (Score:5, Insightful)

on Friday August 18, 2006 @03:12PM (#15936988) Homepage
I do not include creationism in this category because it is not new

It also isn't science.
• #### Re:Too bad (Score:4, Interesting)

<<howard.salis> <at> <gmail.com>> on Friday August 18, 2006 @05:20PM (#15937637) Journal
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.
• #### Re: (Score:2)

I talked mathematicians who've had professional interaction with Wolfram and the general consensus is "very, very smart and very, very arrogant." Apparently he feels that being the head of a successful company makes him a better mathematician than the professors who aren't.
• #### Re: (Score:2)

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.

You sound like he ran over your dog :) I find the basic idea of that book is very thought provoking. So what if doesn't prove or disprove a hypothesis? It's really interesting and maybe it could even *lead* to new science in the future. Conv

• #### Re: Too bad (Score:3, Interesting)

"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 math

• #### A New Kind of Science (was Re:Too bad) (Score:4, Insightful)

by Anonymous Coward on Friday August 18, 2006 @03:56PM (#15937257)
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'm still trying to figure out if this was meant to be tongue-in-cheek or not, given the context. A New Kind of Science is a self-published, non-peer-reviewed, 2000-page testament to the sort of hubris that can only afflict mathematical prodigies who lack meaningful human contact at the age when normal people experience social development.

Wolfram performs an over-analysis of a very narrow subset of cellular automata while claiming to have invented the field, that 'mainstream science' refuses to look at this incredible discovery, and that his 'new kind of science' based on recursion and cellular automata will change the world, although he has no idea how.

It reads like something written after reading Godel, Escher, Bach, smoking pot, and thinking, "I'm thinking about thinking. Now I'm thinking about thinking about thinking. Now I'm....whoa, I wonder what that looks like on graph paper?"

From the reviewer's not-so-clear description, it appears this book falls into a similar category.

• #### geesh (Score:5, Insightful)

on Friday August 18, 2006 @03:06PM (#15936937)
Lay off the thesaurus, you're gonna put your eye out. I'm not sure who that overwrought prose is supposed to impress, but it makes me take an instant dislike to the author.

"I have to confess that I look upon his sojourn into Field Theory as a diversion in the same sense that a prestidigitator (magician), in his field of legerdemain (sleight of hand), distracts the audience members, thereby lessening their attention on what's really going on."

yes, thanks for providing an explanation for your \$10 college words, otherwise we plebs might not have understood you.

Also, what's up with the German and French from out of nowhere? I'm all for using them when there is no easy english equivalent, but what the hell, "Alas and alack, niente, gar nichts, zilch. Woe is me. Es tut mit leid." Those are just extra words.

• #### Re:geesh (Score:5, Funny)

<.loverevolutionary. .at. .yahoo.com.> on Friday August 18, 2006 @03:26PM (#15937088) Journal
You need the Pretentious Geek/English translator. Here, let me help:

"I have to confess that I look upon his sojourn into Field Theory as a diversion in the same sense that a prestidigitator (magician), in his field of legerdemain (sleight of hand), distracts the audience members, thereby lessening their attention on what's really going on."

"I have to confess that I'm really smart. Smarter than you. In fact, you're pretty damn dumb. So dumb that I have to explain what prestidigitator and legerdemain mean. A prestidigitator does not mean someone who spanks the monkey, and legerdemain does not mean a type of beer. They mean you are dumb."

"Alas and alack, niente, gar nichts, zilch. Woe is me. Es tut mit leid."

"Not only am I very smart, I know more languages than you, proving I am a cultered man of the world. And implying that you are a redneck hick. So suck it, hick, I'm going to go prestidigitate my legerdemain."

Hope that helps get you started. If you want to learn more Pretentious Geek, please first stick a broomstick up your ass and tilt your nose upwards at a 45 degree angle, it helps the learning process.
• #### Re:geesh (Score:4, Informative)

on Friday August 18, 2006 @03:34PM (#15937146)
well, no, I speak German, sorta. And Alas and alack, niente, gar nichts, zilch. Woe is me. Es tut mit leid translates into, roughly, "such a shame, nothing, nothing, zero, Woe is me, I'm afraid not." He's not saying anything different in German than he's already said in English. It's stupid.

also, it's 'es tut mir leid, but I'm not picky.
• #### Re:geesh (Score:4, Insightful)

on Friday August 18, 2006 @03:59PM (#15937268) Homepage

You need the Pretentious Geek/English translator. Here, let me help:

Pretentious, yes, but not Geek. Geeks strive for well-defined, unambiguous terms, rational organization of subject matter, and language that accomplishes exactly as much as is necessary, and no more. Geek writing is efficient.

The OP's analysis is excellent, but frought with writing that goes beyond pretentious. It's just bad. Disorganized, rambling, semi-coherent and full of useless jumbles of letters that communicate nothing.

• #### Re: (Score:3, Funny)

The OP's analysis is excellent, but frought with writing that goes beyond pretentious. It's just bad. Disorganized, rambling, semi-coherent and full of useless jumbles of letters that communicate nothing.

So... somewhat above average for Slashdot then?

• #### Re: (Score:3, Funny)

Also, what's up with the German and French from out of nowhere? I'm all for using them when there is no easy english equivalent, but what the hell, "Alas and alack, niente, gar nichts, zilch. Woe is me. Es tut mit leid." Those are just extra words.

Easy to explain: Legerdemain (sleight of hand).

It amplifies the prestidigitator (magician) author's drama induced authority.

• #### Re:geesh (Score:5, Insightful)

<Satanicpuppy@gma[ ]com ['il.' in gap]> on Friday August 18, 2006 @03:29PM (#15937118) Journal
Nothing like one mathematician being snarky about another mathematician.

Frankly they both bored the shit out of me after about 5 seconds. Why is it that math is always rendered this way? I've met interesting and articulate mathematicians before, so I know they exist...Are they not allowed to write textbooks? Or at least write reviews about textbooks?

I was pushed into a near-hatred of math by hordes of pretentious math prodigys that had zero use for any student who didn't start off with what they felt was obvious knowledge. The text book talks down to you, the professor talks down to you, and god forbid you ask for a practical example!

I'm not a math genius, but I'm damn good at practical math. The only way I managed to pass calculus the first time was because I happened to be taking it at the same time as a physics course, and I could figure it out where I could see an application in physics. For calc II I shopped around, trying to find a decent book with dismal results. Ended up dropping the class, and shopping for a decent professor the next semester.

Math is cool, but goddamn, the way it's taught is awful and jackasses like this reviewer and the joker who wrote the book he's reviewing are a prime reason why.
• #### Re: (Score:3, Insightful)

Sounds like someone had a string of shitty math teachers. I feel the same way about linux.

The symptoms you describe exist in every field, from math to literary critisism to welding to surfing.
• #### Re: (Score:2)

Broadly true, though I actually learned to weld from reading a book about it, and the book kicked ass. People who weld aren't usually brimming with hubris, though literary criticism is not just brimming, but overflowing with it.

Math is just as bad, and the thing is (unlike literary criticism), it shouldn't be! If you're doing theoretical math, you shouldn't need to be walking around trying to convince people what a big brain you have...you're doing theoretical math. Now if you're doing lit crit, you gotta t
• #### Re: (Score:2)

I'd be happy to recommend a bunch of kickass math books. Pretty much anything on Springer-Verlag with a yellow cover will fit the bill. ;)
• #### Re: (Score:2)

I have done all those things, and it seems to me that math college professors have a higher then average shitty attitude.

• #### Re: (Score:2)

My geometry text book was written by a smartass. It hasthe strangest off the wall analogies in it.
• #### Re: (Score:2)

"Nothing like one mathematician being snarky about another mathematician."

Only one of the protagonists here appears to be a mathematician.

• #### Re: (Score:2)

I'm a physics grad student, and I've had similar situations to you. Things I've learned in math classes were just abstract intangible theory, and I didn't really visualize them well enough to understand. Until I had to USE those methods in physics. Pretty much all of calculus, differential equations, linear algebra, group theory, etc.
• #### Re: (Score:2)

I'm sure you're familiar with the quote, but for everyone's benefit:

"Physics is to math what sex is to masturbation." (Feynman)
• #### Re:geesh (Score:4, Insightful)

on Friday August 18, 2006 @05:46PM (#15937755)
Mathematician? Well, the reviewer doesn't have a single article on MathSciNet and a quick Google search turns up some submissions to online vanity "general science" journals that have no criteria for acceptance. I tend to think it is a troll. It's certainly not coherent, in either case.

As to your prior experiences, articles like these are part of the reason why mathematicians are distrustful of people that don't find a way to prove themselves. It's an easy field to claim that you've come up with a result, and sometimes it can be a very technical logical fallacy that defeats your efforts. I just wasted a half hour of my time looking up this guy's name for any signs of credibility and reading through the comments.

In experimental fields, even if someone isn't very good, at least they can be used as a technician or research goon. In math, if you're not bright enough to come up with results, you're a non-starter. I know an undergrad who spent four years struggling through basic undergrad classes with the goal of grad school, and then got to his senior year and none of them would take him. It would almost have been a service if someone had been more blunt earlier on.

Of course, I'm not really talking about the calculus sequence, linear algebra I, that kind of thing. Those are more for engineers and scientists. But there you have to bear in mind that to math majors it's the equivalent of Humanities_Course 101, and I dunno about you, but I've taken my share of shitty-ass 101 courses. It's usually because it's foisted off on the newest professor that can't get out of it, they in turn foist off a lot of the work on the TAs, and it's not interesting for anyone's research. It's not a great situation, but then again there are exceptions. I went to a small, teaching-focused school, and my math professors were very personable and great teachers. They loved student research because they got so few who were motivated. I spent some time at a research school, and had a lot more opportunities, but the professors were a lot less accessible and not as good at teaching. It's a trade-off and something worth thinking about before you settle on a school.
• #### Maths As A Science (Score:3, Insightful)

Math is cool, but goddamn, the way it's taught is awful and jackasses like this reviewer and the joker who wrote the book he's reviewing are a prime reason why.

Because there are two types of mathematics practiced in the world today. Mathematics that follows the scientific method, and mathematics that does not follow the scientific method. The latter is regarded as a more laudable endevour.

Mathematics that follows the scientific method is the kind most geeks are familiar with, and which most engineers and ph

• #### Re: (Score:3, Insightful)

Not to mention that the reviewer's submission is one of the worst-written pieces of trash I've ever seen. It's as if this guy thinks he's a great writer because he happens to know a few esoteric words, some foreign phrases, but his writing is absolutely atrocious. Ironic that he's criticizing another guy for a poorly-written book.

.

If you want to see how a REAL scientist writes, without sound pretentious, but yet writing clearly without unnecessary obfuscation, check out anything by Richard Feynmann for

on Friday August 18, 2006 @03:13PM (#15936997) Journal
...the content of this book here [blogspot.com]. The core idea is sound and it looks like it has application to computer graphics.
• #### My idiosyncratic take (Score:5, Insightful)

on Friday August 18, 2006 @03:15PM (#15937008)
I believe this is the most pretentiously-worded article blurb that has ever been seen on Slashdot.
• #### Re: (Score:2)

I feel compelled to quote Disraeli: The review was clearly written by "a sophistical rhetorician, inebriated with the exuberance of his own verbosity, and gifted with an egotistical imagination that can at all times command an interminable and inconsistent series of arguments to malign an opponent and to glorify himself. "

Except of course, the reviewer's prose is so baroque it is impossible to tell whether is arguments are actually inconsistent.
• #### Re:My idiosyncratic take (Score:4, Interesting)

on Friday August 18, 2006 @05:27PM (#15937666)
There's a term for that - intellectual masturbation.
.

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.
.

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.

• #### Re: (Score:2)

Were you around for the Jon Katz articles? Those were worse.
• #### WTF? (Score:2)

Has Archimedes Plutonium taken over Slashdot?
• #### Re: (Score:2)

> Has Archimedes Plutonium taken over Slashdot?

SINUSOIDAL FUNCTIONS on a CARTESIAN PLANE?

(No, wait, that was ROBERT Mc ELWAINE! :)

• #### El Sucko (Score:5, Insightful)

on Friday August 18, 2006 @03:19PM (#15937040) Homepage
This review freakin' sucks.

I have an M.A. in Mathematics. I've read some of the "Rational Trigonometry" online before, and yes, it is pretty oddball and has its weakness and can be criticized.

But this review is borederline psychotic. It is poorly written, full of ad hominem attacks, lots of made-up grammar and word usage, wierd random abbreviations... it's scatterbrained, repetitive, and unnecessarily hostile.

There is a critical review to be written about "Rational Trigonometry", but this isn't it. I may not like our current government, but I'm still not going to listen to some incoherent homeless guy raving about it on the street.
• #### Re: (Score:3, Funny)

I'm still not going to listen to some incoherent homeless guy raving about it on the street.

And you don't have to, there are plenty of them on /.
• #### Re: (Score:2)

Agreed. Wildberger's manner is off-putting and the idea of replacing trigonometry with his rational trigonometry in high schools seems eccentric but for starters a trigonometry valid in a general field is interesting and this review just stinks.
• #### Re: (Score:3, Insightful)

It is poorly written, full of ad hominem attacks, lots of made-up grammar and word usage, wierd random abbreviations... it's scatterbrained, repetitive, and unnecessarily hostile.

Not to mention imprecise. In two instances the reviewer says

in order to solve an equation that has a square root sign within it, one has to square both sides of the equation at some time, and this doubles the number of solutions.

which is not true in all cases. Two examples are

\sqrt(x) = x, which has two solutions before and after
• #### New obligatory quote... (Score:5, Funny)

on Friday August 18, 2006 @03:31PM (#15937129)
"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"

"Ya hurt yer what?"
• #### Sack the reviewer (Score:2)

It's tough enough to get people interested in geometry and trig than to bloody some poor prof's attempt at unifying disciplines. It's nice that his review must demonstrate his various vocabularies and distainful lack of surprises..... yet conveying information about the actual content of the book is betrayed while the reviewer stands up and barks like a dog for a pet. Look! I killed this helpless little thing. Aren't I a good boy? Gotta bone.

The review, for its content, is perhaps as useless as the book he'
• #### Philosophically/Ideologically driven blather (Score:5, Interesting)

on Friday August 18, 2006 @03:33PM (#15937137) Homepage Journal
The author (of the book) is, to my mind, tending dramatically toward the loopy side. Take, for instance, this piece he wrote [unsw.edu.au]. 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") [graveyardofthegods.com] 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.
• #### Re: (Score:2)

That article of Wildberger's was hilarious! What's more entertaining than:

Perhaps you believe that even though you cannot write down numbers bigger than w, you can still abstractly contemplate them! This is a metaphysical claim. What does a number bigger than w mean, if there is nothing that it counts, and it can't even be written down? Believing you can visualize' an all-seeing Leprechaun or an unstoppable mouse in your mind, by some melange of images, descriptive phrases and vague feelings, does not m

• #### Re: (Score:2)

I particularly liked some of his railing against category theory, mostly for how remarkably off-target or otherwise false his claims were. His complaint about natural numbers though, are quite something else again... where to begin? How exactly does one argue with a platonist finitist?
• #### It's Friday (Score:5, Funny)

on Friday August 18, 2006 @03:40PM (#15937175) Homepage
I'm usually too lazy to read the article but holy shit I'm not reading the review either.
• #### A very odd mathematician (Score:3, Informative)

on Friday August 18, 2006 @03:50PM (#15937221)
The author, Norman Wildberger, is one strange mathematician. I could hardly believe his rant against set theory [unsw.edu.au], which borderlines on crankish or at the very minimum appallingly uninformed. For example, he calls the ZF (Zermelo-Fraenkel) axioms a "sorry list of assertions" - "these statements are awash with difficulties. What is a property? What is a parameter? What is a function? What is a family of sets? Where is the explanation of what all the symbols mean, if indeed they have any meaning? How many further assumptions are hidden behind the syntax and logical conventions assumed by these postulates?" In fact, these axioms are very precisely defined, and rank among mankind's greatest achievements.

(For the uninformed, consult Wikipedia [wikipedia.org]. For a very precise breakdown of these axioms translated to primitve symbols - Wikipedia still includes some higher-level defined symbols that Wildberger objects to because he can't seem to understand them - see the metamath version [metamath.org]. In other words, there is nothing fuzzy or ambiguous about these axioms.)

His set theory rant created quite a furor on Usenet, here [google.com] and here [google.com].

• #### Re: (Score:2)

Unless you want to admit that math has no connection to reality then you have to abandon completed infinities. At any given time there are a finite number of energy quanta and Planck 4-volumes in our past light cone, and the two together only allow a finite number of permutations.

This number may grow with time and might not even be bounded, but at any given time, numbers larger than this number (and almost certainly most of those smaller) are physically meaningless. So go on playing with your infinite and t
• #### Re: (Score:3, Insightful)

Physical reality? Is that the one where zero and sqrt(-1) can have no interpretation in any sort of physical meaning or even in interim calculation, the one where Newton's laws are the end-all be-all, or the one where spacetime is completely Euclidean?

I try to keep abreast of the current absolutely correct, final theories of everything.
• #### Re: (Score:2)

See Geometric Algebra (Clifford algebra over the reals - though rationals work at any precision), particularly Hestene's introductory papers, for a natural interpretation of imaginary numbers as bivectors.

The question of whether 0 is a physically legitimate number hadn't occured to me. As a denominator it surely isn't legitimate, and may be illegitimate as a factor in some situations. Quantum effects in the vacuum seem to preclude zero energy.

Newton's laws and the curvature of space seem to have no bearing
• #### Re: (Score:2)

Except that most physics assumes that spacetime is a continuum.
• #### Re: (Score:2)

As an approximation that can be shown to be in conflict with experiment. The ultraviolet catastrophe was the result of assuming a continuous and infinite distribution of energies. The need for renomalization is also an artifact of assuming continuity in a theory with point particles. The energy needed to probe very short distances at some point reaches a level that either creates a black hole, thus precluding getting a result of the attempted measurement (or, if black holes do not exist, the amount of energ
• #### cranky indeed... (Score:2)

if it wasn't for set theory, computing would be practically non-existant. At best it would be clockwork...

This guy sounds as if he needs to do an introductory course in discrete mathematics.

And he gets published????

What a crazy fsck'd up world

• #### I assume you meant... (Score:2)

Es tut mit leid./blockquote

Es tut mir leid.
• #### Compensating for something? (Score:4, Informative)

on Friday August 18, 2006 @03:51PM (#15937227)
Using long words doesn't make you look any smarter in the same way driving a flashy car doesn't make your dick look any bigger.
• #### Re: (Score:2)

Now you tell me.
• #### Re: (Score:2, Funny)

But driving a very tiny car might make your dick look bigger.
• #### Re: (Score:2)

But having your car built half size would make your dick look HUGH!
• #### There are a lot of math crackpots out there... (Score:4, Interesting)

on Friday August 18, 2006 @03:52PM (#15937232) Journal
...99% of whose [wikipedia.org] writings [crank.net] 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.
• #### Didn't bother to read it (Score:5, Funny)

on Friday August 18, 2006 @03:54PM (#15937241)
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

#1 - Humble my ass

#2 - Such excessive sesquipedalianism is an immediate flag that the writer is writing not to inform or help. He's just masturbating his brain in public.

#3 - Humble my ass
• #### Re: (Score:2)

Some people like to use words that are not ordinary. I like using them myself because I actually feel my mind going numb when I continuously use the same two thousands words over and over and over. When I do it, it is not to impress anyone but rather to exercise my brain. Maybe you are being too unkind to this gentleman (pretentious snob?).

Let's not force everyone into monosyllabic commentary by attacking erudite prose. :P

strike
• #### OK, stepping way out on a limb here... (Score:3, Insightful)

on Friday August 18, 2006 @03:59PM (#15937267) Homepage Journal
Hoping said limb does not break...

A few up-front things:
IANAMathematician;
I appreciate the reviewer's efforts to thoroughly discuss the reviewer's point of view;
I don't mind acknowledging that I'm not as smart as the vast population of Slashdot, but I like math even though I'm not top-notch;
I love to learn stuff, and like to read Slashdot articles/comments that are out of my field, and way over my head;

With the above said...
I don't mind looking up unfamiliar terms that appear in an article or in a review (I like learning) - when the words are concerned with the subject matter at hand. I do mind when I read something that attempts to completely fill up my "new word of the day" calendar (for the next millennium). Why? Because I'm interested in understanding the subject and the review, not in how many new non-topic-related words and phrases that can be crammed into a paragraph.

Lastly, a good review, IMVHO, is one that does not chastise, scold, or belittle the matter of review.
• #### How about a more qualified reviewer? (Score:4, Insightful)

on Friday August 18, 2006 @04:18PM (#15937353)
From the review:
However, you don't have to read between the lines to see on page 21 that Wildberger excludes 'characteristic two fields.' Although I am not versed in Field Theory, I opine that such an exclusion does not apply to classical geometry and/or trigonometry, otherwise he would have said so. So, he is already implicitly confessing, to a failure of Rational Geometry in the global sense.
However, you do have to have the slightest clue what you are talking about if you are going to call the author on the "characteristic two" exclusion.

Wildberger may be a little "out there" (alright, he's completely nuts), but this point is not one you can fault him for. There are a LOT of results which exclude fields of characteristic two. It's not a big deal. In fact, it's commendable that Wildberger has explored the ramifications of his framework in any fields with non-zero characteristic, as the "normal" pedestrian conceptualizations of geometry don't apply.

It would have been nice if /. could have posted a review by somebody who is actually qualified to critique the book. And no, I am not such a person, but I know a couple people who are.

• #### Re: (Score:2)

A friend of mine caught wind of the book a few months ago, and because I was his mathematically inclined friend, he asked me what I thought. I haven't read the entire book, but I looked at the sample chapters that he had posted on his website. My partially informed opinion is:

It is an interesting idea which simplifies the calculations in some cases by working only with intermediate values that might otherwise have to be square rooted and re-squared otherwise. However, it is not easier than trig in any re
• #### Re: (Score:2)

I'm curious, how do you know about fields of characteristic 2 but not feel you are qualified to critique the book?
I'm working on a PhD in math. I "know enough" to read the math in the book, but I don't think I'm qualified to critique the pedagogical angle well. That's a political minefield in mathematics. I haven't yet chosen a perspective to call my own. (Not that I have to choose only one.)
• #### Review is an obvious troll (Score:2, Informative)

by Anonymous Coward

The obvious mistake in the distance formula and the interpretation of the "fields of characteristic 2" exception are intended to rile up people who *are* familiar with these things.
• #### Very nice link... (Score:2)

... damn you! What was the WORD?!?!?

Grrrr...

• #### "Mighty odd-sounding?" (Score:4, Informative)

on Friday August 18, 2006 @04:35PM (#15937452)
The Divine Proportion is one of the most well-known geometric properties. Here is a link to the wiki page for the uninformed: http://en.wikipedia.org/wiki/Golden_ratio [wikipedia.org]
• #### "what's really going on" (Score:2)

I have to confess that I look upon his sojourn into Field Theory as a diversion in the same sense that a prestidigitator (magician), in his field of legerdemain (sleight of hand), distracts the audience members, thereby lessening their attention on what's really going on.

Like using big words to disguise a blatant troll, perhaps?

• #### Jessica Alba (Score:2)

Don't know about you, but I find her proportions pretty divine...
• #### distance-squared? (Score:2)

(distance)^2 = (x2^2 - x1^2) + (y2^2 - y1^2) [super-script replaced with "^"]

This is a definition of distance-squared with which I was previously unfamiliar.
• #### But it works! (Score:2)

For certain values of x and y
• #### Proposal to Slashdot Editors (Score:2)

Please, for the love of god, only publish articles that you understand.
• #### Fields (warning, this post contains math) (Score:2)

There is so much wrong with the review that I don't know quite where to begin. However, the one thing that nobody seems to have to responded to is

However, you don't have to read between the lines to see on page 21 that Wildberger excludes 'characteristic two fields.' Although I am not versed in Field Theory, I opine that such an exclusion does not apply to classical geometry and/or trigonometry, otherwise he would have said so. So, he is already implicitly confessing, to a failure of Rational Geometry in t

• #### Was that meant to be English? (Score:2)

Frankly, if you are planning on reviewing a book, it would be of benefit to actually learn how to express yourself in a way that at least slightly resemble coherent language first...
• #### Well, since you welcome comments... (Score:2)

... this is the poorest excuse of a review I have ever had the displeasure of reading. What a piss-poor hack! I think my eyeballs went on strike with this one! My first language is not english, and not maths either, and I could pull off something better that this!

Seriously hope you don't write for a living... and if you do, kindly let me know where that is so I can avoid it like the plague!

Alas and alack, niente, gar nichts, zilch. Woe is me. Es tut mit leid.

I mean, WTF?!? Are you choking on a hairball or s

• #### Ramblings of a madman. (Score:3, Insightful)

on Friday August 18, 2006 @08:49PM (#15938426) Homepage
At least that's how this reads.

Sigh... I'm irritated by people who think that their large vocabularies make them good communicators.
• #### Does reviewer remember high school math? (Score:2)

Okay, I'm just a "mere" high school math teacher with a bachelor's in math. And I'm certain that I'm not as "genius" a mathematician as the reviewer was.

Still, I can't believe the reviewer took 4 lines to find the length of 'd' in his example. He points out how the author used 7 or 8 lines to do it. That's what makes this ironic to me.

Has the reviewer ever heard of two delightful little formulas known as the Law of Cosines and the Law of Sines? I got the same answer in just two lines, personally. Perhaps no
• #### Gotta ask the reviewer .... (Score:2, Interesting)

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.
• #### Reviewer has wrong definition of Spread! (Score:2)

The actual definition given by Wildberger is this:

Definition The spread s(L1,L2) between the lines L1 = < a1 , b1 , c1 > and L2 = < a2 , b2 , c2 > as the number:

s(L1,L2) = (a1*b2-a2*b1)^2/( (a1^2 + b1^2) * (a2^2 + b2^2) )

[In Wildberger's line notation, a line L = < a , b , c > satisfies the equation a*x + b*y + c = 0 for all {x,y} in F^2]

The reviewer is entitled to his opinion, but does not have the right to present false information as fact. Definitions are very important in mathemati

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