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

Karma whoring

[UbuntuDupe]

Slashdotters vetted this before

The Proportions.

[Anonymous Coward]

She had a set of divine proportions that could cause cardiac arrest in a Yeti.

For all I know, Slashdot's populace may be affected too.

Re:The Proportions.

[PCM2]

She had a set of divine proportions that could cause cardiac arrest in a Yeti.

A Bulwer-Lytton candidate to be proud of!

I'm not that Smart!

[neonprimetime]

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
Tertiary
Concomitant

Re:I'm not that Smart!

[Keith Russell]

Does the Architect know his thesaurus is missing?

Re:I'm not that Smart!

[Rude Turnip]

I, too, found this book to be shallow and pedantic.

Re:I'm not that Smart!

[pVoid]

Btw, it's "Tut mir leid". not "mit".
Some people just try too hard...

Re:I'm not that Smart!

[Ed Avis]

I like that review... it starts off reasonable, and gets increasingly Time Cube as it goes on.

Re:I'm not that Smart!

[Anonymous Coward]

Hmm, yes, shallow and pedantic.

Re:I'm not that Smart!

[Anonymous Coward]

YOU ARE STUPID AND EVIL

Re:I'm not that Smart!

[bfields]

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.

Re:I'm not that Smart!

[innosent]

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, and even though my style of solution is always correct (I call it proof by construction), since it can be proven to be equivalent to other solution forms, I barely thought it was worth discussing to prove that the F I got on the assignment was actually an A, much less write a BOOK about it. I did get an A, but I had to prove that my methodology was correct [which was relatively trivial], and that it was essentially a set-oriented method combining several types of direct, indirect, and inductive proof methodologies into a simpler form (basically, that if you can show that the construction of two sets [i.e. formulae for all elements in the sets] are identical, then those sets are identical.)

As an extremely simple example, (A union B) intersect C = (A intersect C) union (B intersect C) because [skipping trival steps] (A union B) intersect C is the set of all x such that x is in C (commutative) and (definition of intersection) x is in either A or B (definition of union), and (A intersect C) union (B intersect C) is the set of all x such that x is in C (distributive) and x is in either A or B (definition of union). Since these two formulae are identical, the sets are identical. This sometimes leads to very long notations of sets, but essentially makes solving a lot of complex proofs as simple as flipping things around to take the same form and applying that to the problem. Indirect by contradiction is done when you can subtract one set from the other and show that something remains, and inductive is done by applying the defining formula (say elements n and n + 1) to the definition of the problem set to show that their inclusion in the problem set is equivalent (which is essentially the same as a traditional inductive proof, induction is not the strong point of this method, but it CAN be done).

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

Re:I'm not that Smart!

[tehcyder]

I remember an old southafrican (boer) joke

The fact that you are only (sort of) quoting racist shit does not absolve you from the charge of being a racist shit.

This is all

[jugglerjon]

Intuitively obvious. At least that's what my college professiors would say when lecturing.

Re:This is all

[__aaclcg7560]

Still greek to me, man. Got any hash for sale?

All I could think about...

[Anonymous Coward]

Mathematician 1+1 = 2
Scientist 1+1 is around 2
Accountant 1+1 = any thing your want

Re:All I could think about...

[Red Herring]

Engineer: 2+2=5, for large values of 2

Re:All I could think about...

[bunions]

Salesman: 2+2=4, but if we can close the deal right now, it can be 5. Ok, 6, but if you walk out that door, the deal is off.

Re:All I could think about...

[Anonymous Coward]

Programmer: 1 + 1 = 10

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

[eln]

I do not include creationism in this category because it is not new

It also isn't science.

Re:Too bad

[Anonymous Coward]

".. a rare blend of monster raving egomania and utter
batshit insanity"

Cosma Rohilla Shalizi on S.Wolfram, A new kind of science

http://cscs.umich.edu/~crshalizi/reviews/wolfram/

Re:Too bad

[kfg]

Scientists are great at empirically testing this and that theory but they often have problems altering their own perceptions on existing and accepted information.

That is the very reason that "scientists" is plural. The flip side of that, however is that scientists are always coming up with new ideas, because that's what they're supposed to do.

Sometimes they do this in the same manner that legislators come up with new laws because that's what they're supposed to do, because it has become a publish or perish world. It doesn't matter what you publish, just the quantity of publication; and how often you and your buddies cite each other to drive up your stats.

KFG

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.

Re:Too bad

[bunions]

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:Too bad

[markov_chain]

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. Conversely, there is plenty of real science out there that is pretty droll and uninteresting. So what if you can prove that the Boolean anti-binary least-square approach is NP-complete? I doubt you could get a credit card based on that.

Re:Too bad

[Threni]

> So what if doesn't prove or disprove a hypothesis?

If it doesn't then calling it a `new kind of science` is somewhat overblown, just like this weeks perpetual motion engine is hardly `a new kind of power generator`.

> It's really interesting and maybe it could even *lead* to new science in the future.

So at the very least the title is missing a question mark.

Re:Too bad

[Anonymous Coward]

We can take Wolfram seriously atleast in the realm of philosophy if not science. Philosophy does not demand proving or disproving as science does. Or it could be taken as pure math built up from axioms, but having no apparent significance

mod parent troll

[Anonymous Coward]

wikipedia is fine.

Re:mod parent troll

[phoenix321]

Nah, it's back up since five minutes and still not writable, so cut me some slack, okay? http://www.thewritingpot.com/wikistatus/

Re:mod parent troll

[neonprimetime]

so cut me some slack, okay?

Those who spend their day monitoring the status of wiki shall receive no slack.

Re:OT: Wikipedia down, what happened?

[andrewman327]

Wikipedia: Rational Geometry. I do not see any problems.

Postscript: I am ...

[Anonymous Coward]

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

PS. I am not a crackpot.

Re:OT: Wikipedia down, what happened?

[Phreakiture]

Wikipedia is down ATM with no explanation other than technical difficulties. All subdomains are affected, too.

Ten minutes have passed since you posted that, and I am seeing Wikipedia just fine.

geesh

[bunions]

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

[spun]

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

[tr0p]

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

[SatanicPuppy]

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:geesh

[bunions]

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

[Threni]

>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.

I couldn't have put it better myself.

-1 Scheisskopf (shit-head)

Re:geesh

[Anonymous Coward]

Es tut mit leid.

In his attempt to sound clever, he turns himself into a douchebag. It's Es tut mir leid. (ger.: I am sorry.)

I am sorry that a thesaur'ed review is accepted at all. It makes the author sounds like a junior high school student trying to bolster his report.

F, for effort.

Re:geesh

[bunions]

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:geesh

[Anonymous Coward]

"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."

Actually, I was insulted by the fact that he thought I'd not know what "prestidigitator" and "legerdemain" meant.

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.

I'm not 100% sure but I think those are examples of words which mean the same things. Which is what he's saying the book actually boils down to: a fancy way of saying something but which adds nothing to what was already known. Anyway, it's certainly a joke of the sort mathematicians seem to like.

Re:geesh

[Anonymous Coward]

The author of the review would do well to read Politics and the English Language by George Orwell.

Re:geesh

[friedo]

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:geesh

[H0p313ss]

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:geesh

[SatanicPuppy]

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 talk it up, or people might just think you're an asshole.

So while I'd expect to have to sift a lot of garbage while reading a book of literary criticism, I really resent having to do the same for a book of math. Math is simple, elegant, and objective, and it should be presented that way...preferably with a few concrete applications for people like me who like that crap.

Re:geesh

[Tyger]

My geometry text book was written by a smartass. It hasthe strangest off the wall analogies in it.

Re:geesh

[bunions]

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:geesh

[french_user]

My little brother who doesn't speak english understood the article. Unnecessary french you said ?

Re:geesh

[Wolfbone]

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

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

Re:geesh

[Quantum Fizz]

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 instance.

Re:geesh

[Quantum Fizz]

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:geesh

[Sage Gaspar]

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.

Re:geesh

[geekoid]

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

Re:geesh

[Anonymous Coward]

'Fraught', not 'frought'.

Re:geesh

[Anonymous Coward]

Actually, 'leger' means 'light' (as in not heavy), so it still works fairly well as a literal one-to-one translation.

Re:geesh

[failedtoinit]

AWW man. This dude just crashed my windows box with all these big words! (work laptop, and not by choice).

Re:geesh

[Matchstick]

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

[Anonymous Coward]

This is just filler to get around the retarded limitations of the pseudo-scientifically named "postercomment compression filter."

Here's the real comment:

"such a shame, nothing, nothing, zero, Woe is me, I'm afraid not."
                                                                                  ^= absolutely nothing ^= I'm sorry

Re:geesh

[Sage Gaspar]

I just wonder what kind of kinky stuff theoretical physicists are into then.

Re:geesh

[try_anything]

Why is it that math is always rendered this way?

*blinks in astonishment*

I have a whole shelf of math books here and read a few research papers in college. I also read school textbooks and popular math books when I was a kid. I have absolutely no idea what you're talking about. The review was simply bad, bad, bad writing of the type produced by someone under the influence of drugs, mania, or unrecognized stupidity. Math attracts more than its share of such crapola but far less than philosophy, religion, poetry, science fiction, and psychedelic rock. This stuff is never published in professionally edited (*ahem*, take a note, Slashdot editors) publications in any field.

I'm not that funny either...

[Chaffar]

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:I'm not that funny either...

[Anonymous Coward]

Amen !!

Re:OT: Wikipedia down, what happened?

[Eudial]

Ten minutes have passed since you posted that, and I am seeing Wikipedia just fine.

I experienced problems with wikipedia today as well, so I guess it was just bad timing posting it just before wikipedia was up again...

Some interesting comments about...

[exp(pi*sqrt(163))]

...the content of this book here. The core idea is sound and it looks like it has application to computer graphics.

Re:OT: Wikipedia down, what happened?

[phoenix321]

Yes, and I hang my head in shame.

It was down before I left my workplace today and wasn't up when I logged in at home again, more than two hours later. It's back up right now, write access is slow but possible. Let's see what Wikinews is saying about that...

My idiosyncratic take

[Junky191]

I believe this is the most pretentiously-worded article blurb that has ever been seen on Slashdot.

Re:My idiosyncratic take

[kfg]

I believe this is the most pretentiously-worded article blurb that has ever been seen on Slashdot.

Oooooooo, sounds like a challenge to me, excuse me, provocation (dare).

KFG

Re:My idiosyncratic take

[Angostura]

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

[Quantum Fizz]

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:My idiosyncratic take

[khallow]

Were you around for the Jon Katz articles? Those were worse.

Re:My idiosyncratic take

[Gli7ch]

Why are there dots between you paragraphs?

Re:My idiosyncratic take

[Quantum Fizz]

Why are there dots between you paragraphs?

Dunno if it's fixed, there used to be a bug in slashdot that the first paragraph separator didn;'t work (not sure if it's been fixed). so i inserted a period so it didn't matter. i added a sentence and forgot to take the first dot out.

btw, there should be a paragraph separator here, between the italicized quotes and my reply, if not then the bug is still there.

Re:My idiosyncratic take

[Gli7ch]

Why are there dots between you paragraphs?

Just wrap the quote in a parahraph tag?

Re:My idiosyncratic take

[Threni]

> Were you around for the Jon Katz articles? Those were worse.

They were more patronizing that pretentious. Whatever happened to him? I filtered him away, and noticed a year or so ago that the filter option itself had vanished.

Re:My idiosyncratic take

[bunions]

I'll pass along the same awesomely hilarious answer I got when I asked that question a few months ago:

Livin' in upstate NY, writin' books about dogs:

http://en.wikipedia.org/wiki/Jon_Katz

Search for him on Amazon. I don't know why, but it seems immensely fitting. If I was Jon Katz, I can imagine that being around nonhumans would probably be best for everyone.

WTF?

[bfields]

Has Archimedes Plutonium taken over Slashdot?

Re:WTF?

[Tackhead]

> Has Archimedes Plutonium taken over Slashdot?

SINUSOIDAL FUNCTIONS on a CARTESIAN PLANE?

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

El Sucko

[dcollins]

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:El Sucko

[EatHam]

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:El Sucko

[Wolfbone]

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:El Sucko

[Anonymous Coward]

Made up words?

"Wierd" is a new one to me.

Re:El Sucko

[Anonymous Coward]

It is poorly written, full of ad hominem attacks, lots of made-up grammar and word usage, wierd random abbreviations...

That and his *very first* formula is wrong, it should be (x1-x2)^2+(y1-y2)^2.

Re:El Sucko

[RackinFrackin]

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 squaring both sides, and

\sqrt(x)=10, which has one solution before and after.

Two words

[Anonymous Coward]

Pompous...windbag...

Missing Options!

[Anonymous Coward]

The most obvious Divine Proportions are: 36x24x36.

Oh wait: this is /.

In that case: Trigonometry is sexy!

Re:Missing Options!

[Impy the Impiuos Imp]

> The most obvious Divine Proportions are: 36x24x36.

That would be an odd-shaped palm. WTH are you talking about?

Re:Missing Options!

[mr_c0w]

> The most obvious Divine Proportions are: 36x24x36.

Only if she's 5' 3"!

Re:Missing Options!

[Eternauta3k]

Obviously, a 3d 35mm negative frame...

Save yourself some money by buying the book here!

[Anonymous Coward]

Save yourself some money by buying the book here: Divine Proportions. And if you use the "secret" A9.com discount, you can save an extra 1.57%!

Poor reading, or poor writing?

[dthmtluncrn]

When I first read the posting (in my mind anyways) the name of the book was purported to be: "In my humble opinion, we have an unjustified polemic in the world of mathematics, yet again. My background is tertiary level mathematics..." In my opinion this would have been a much more entertaining title. Along the lines of the full Dr. Strangelove title.

New obligatory quote...

[jpellino]

"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"

*blink*

"Ya hurt yer what?"

Sorry, but...

[Anonymous Coward]

I seem to have wandered onto a site for snobby, elitist mathematicians who write like pretentious twits, when I was really trying to get to a tech news site. Do you happen to know the way to slashdot?

Re:Sorry, but...

[Impy the Impiuos Imp]

Seriously, he could have shown the one example he counters with his own "way", to compare and contrast. Also, don't most solutions involving trig, where you solve in reverse for an angle, offer infinite solutions, all +/- 180 or 360 or something? If so, how does he select his "correct, one" solution?

Re:Sorry, but...

[Anonymous Coward]

He is not a mathematician.

Sack the reviewer

[postbigbang]

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's trying to describe-- it doesn't get beyond a sense of hopelessness. If hopelessness is the message then it should be stated, not a long sewn-together set of moans and oh-gawd-is-he-awful's.

There are some of us that get it. Others don't get it. He's obviously not the audience-- and his barbs at perceived accuracy shows how unbending Halprin is. Yes, math demands accuracy and rules, yet understanding trig, linear geometry, and other non-algebraic disciplines isn't a droll matter of lumping proofs after proof.

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.

Re:Philosophically/Ideologically driven blather

[nonlnear]

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 mean they exist.

Thank you for the entertainment. That article was better than a good BOFH episode.

Re:Philosophically/Ideologically driven blather

[Anonymous Coward]

You haven't read his book, have you?

Re:Philosophically/Ideologically driven blather

[Coryoth]

No I haven't. Is that particularly important? I was aware of Wilderberg before I had heard of the book, and thought he was a bit of a fruitcake then. After skimming sample chapters from the book I feel I can see how his particular brand of fruitcakery could come to inform the sort of view he is taking. Maybe the book is great. Perhaps the "Rational Geometry" is a wonderfully useful thing. That doesn't change the fact that, for quite separate reasons I think the author is a fruitcake.

Re:Philosophically/Ideologically driven blather

[Coryoth]

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?

Re:Philosophically/Ideologically driven blather

[Anonymous Coward]

No I haven't. Is that particularly important?

In a discussion about the book I would say that it is.

Re:Philosophically/Ideologically driven blather

[HalfFlat]

The argument for ultrafinitism is essentially sound. Why should we grant (mathematical) existence to entities which we can neither construct nor describe? Standard mathematics is interesting, powerful and highly applicable — but it might not even be consistent.

There really does seem to be a profound lack of interest in mathematical foundations. Most mathematicians don't care, and regard those who do as being at best a little odd. There are logicians working on ways of resolving the problem of potential inconsistency in mathematics that would allow such inconsistency to be contained, in that they would not make everything trivially true. The vast majority of mathematicians really couldn't care less.

From a finitist point of view, the advantage of working with the axiomatic set theory and dealing with all the inifinities and such is that very general statements can be proved which would remain true even in the finite domain. For example, any property that can be shown for all natural numbers in standard mathematics, is probably going to remain true and applicable to all natural numbers in a ultrafinitist mathematical system. On the other hand, there could well be properties that hold for numbers in a ultrafinitist system that are very much not held universally by what are normally regarded as natural numbers. (For example, that there is a natural number which is no less than every other natural number. Given that the number of possible numbers in a finite scheme is finite, it seems very possible that there are a very large number of properties that are enjoyed that do not hold for natural numbers generally.)

There is however a problem in formalising such extreme versions of finitism. As far as I know, no one has created an axiomatic system which well describes this sort of mathematical system. The larger problem seems to be that very few mathematicians feel that such an enquiry is at all worthwhile.

It's Friday

[papasui]

I'm usually too lazy to read the article but holy shit I'm not reading the review either.

A very odd mathematician

[ortholattice]

The author, Norman Wildberger, is one strange mathematician. I could hardly believe his rant against set theory, 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. 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. In other words, there is nothing fuzzy or ambiguous about these axioms.)

His set theory rant created quite a furor on Usenet, here and here.

Re:A very odd mathematician

[Savantissimo]

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 transfinite sets, trancendental numbers and so forth, so long as it's understood that from a universal point of view, such impossible fantasies can never be needed to describe physical reality.

Re:A very odd mathematician

[Sage Gaspar]

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:A very odd mathematician

[althai]

Just because an object does not exist in nature does not mean that theories about it cannot be useful for natural science. For example, you will never find imaginary quantities, and yet complex numbers play and important role in quantum mechanics, and make just about any equation involving periodic oscillations more manageable.

Re:A very odd mathematician

[colinrichardday]

Except that most physics assumes that spacetime is a continuum.

Re:A very odd mathematician

[Savantissimo]

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 either way on what I said. Are you suggesting that there is no physical reality, or that it cannot even be known well enough to determine whether it is continuous or discrete? Perhaps you can articulate some reason why infinities as sets reather than processes are required to accurately represent anything physical? Certainly Zermelo had to add as an axiom that infinite sets exist, and this implies that the remaining axioms and thus the basis of mathematics do not require that infinite sets exist.

Re:A very odd mathematician

[Savantissimo]

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 energy in the whole universe still sets a bound on how fine a scale anything can be probed).

Any physical or mathematical theory has to be in accordance with information theory, and thus must place finite limits on how much information is expressed in or or transmitted by any spatiotemporally limited entity, attribute, or behavior. Since any random real number contains infinite information, and any interval contains an infinity of random reals, any theory that truly depends on real numbers or continuity is inconsistent with the proven limits of information.

Re:A very odd mathematician

[colinrichardday]

Any physical or mathematical theory has to be in accordance with information theory

Why? Maybe it's the other way around.

Also, the energies of photons may be quantized, but what about the kinetic energies of objects? If I throw a ball, are its energies over time varying continuously? And when I said that spacetime is assumed to be a continuum, I meant a spacetime, not energy.

Re:A very odd mathematician

[Sage Gaspar]

Umm, perhaps you could articulate why they could never have any bearing on physical reality. Sure, go back to the current model, but the entire point of my post (whooooosh) is that there are times, historically, when we thought we had the universe nailed and some mathematical concept could never fit into it. Then we were proven wrong, and the good ol' useless, abstract mathematics in these areas suddenly became interesting to scientists.

As to its necessity in the basis of mathematics, there're plenty of axioms that aren't necessary. The law of the excluded middle, for one. In fact, no axiom is "necessary" by definition.

Eliminating infinite sets might well represent our current knowledge of physics better; if it's easier, or it fits your research, or it gives you a warm fuzzy feeling to throw out that axiom, by all means, go ahead and do it. Claiming that anything dependent on that axiom can never be applicable to reality is just short-sighted arrogance verging on stupidity. In either case, working in a certain axiomatic system is nowhere close to an admission that "math has no connection to reality."

I assume you meant...

[rolfwind]

Es tut mit leid./blockquote

Es tut mir leid.

Compensating for something?

[Nereus]

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:Compensating for something?

[xs650]

Now you tell me.

Re:Compensating for something?

[rolyatknarf]

But driving a very tiny car might make your dick look bigger.

Re:Compensating for something?

[Impy the Impiuos Imp]

Yeah!

I've known a few guys who thought they were pretty smart, but you've got being right down to an art. You think you're a genius-you drive me up the wall. You're a regular original, a know-it-all. Oh-oo-oh, you think you're special. Oh-oo-oh, you think you're something else...Okay, so you're a rocket scientist.

That don't impress me much. So you got the brain but have you got the touch? Now don't get me wrong, yeah I think you're alright, but that won't keep me warm in the middle of the night.

Re:Compensating for something?

[geekoid]

But having your car built half size would make your dick look HUGH!

His "Solution" isn't even math

[Anonymous Coward]

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

I don't know what this is supposed to be, but it's not math. The only parts that make sense are "BDC = 180 - (45 + B)" and the line following that. Sin B can't possibly be 7/4, which is greater than 1. And 4 * 2 = 8 not 42. The last three lines are complete jibberish.

Re:His "Solution" isn't even math

[svnt]

Giving the reviewer the benefit of the doubt and assuming he's working in metric or some other alien measurement system (joke!), all you have to do is look at his answer to determine how off-base he is. I'm assuming (since I can't see it) that the triangle ABC is named by its three vertices, A, B, and C.

If you have a triangle where the shortest side is 4, no line from any of the vertices to anywhere on the opposite line could possibly be less than 4. Anybody else see a different answer?

Re:His "Solution" isn't even math

[jtobin]

Actually, his solution is correct. His workings are complete bullshit though. That last line, for example, is hilarious. "52(37 - 7)/2 = 3.313693059". 780=3.313693059? This whole review has to be a joke, otherwise I want to go to college wherever this guy got his degree. Seriously, if they give a degree to a guy who believes cos B = 3/4 => sin B = 7/4, I'll have a PhD in no time! Here's some real maths, high school style, without any quadrance, and certainly without any lines like "4*2=42": As the author correctly stated, cos B = 3/4 => B = 41.4096 deg. Hence BDC = 180-45-41.4096=93.5904. Then, by the sine rule, (sin 93.5904)/5 = (sin 41.4096)/d => .199607447 = (sin 41.4096)/d => d = 3.31. Your statement "If you have a triangle where the shortest side is 4, no line from any of the vertices to anywhere on the opposite line could possibly be less than 4." is not quite correct. I could write a long dissection of this statement, but I don't think this is the right place :)

Re:His "Solution" isn't even math

[Anonymous Coward]

could write a long dissection of this statement, but I don't think this is the right place

If only the margins were bigger...

Re:His "Solution" isn't even math

[colinrichardday]

If you have a triangle where the shortest side is 4, no line from any of the vertices to anywhere on the opposite line could possibly be less than 4. Anybody else see a different answer?

You are incorrect. Take an equilateral triangle each of whose sides is 4. Then the altitudes are 2*sqrt(3). Let ABC be a triangle, and let AD be the altitude from A to BC. The AD is no longer than either AB or AC.

Re:His "Solution" isn't even math

[ScriptedReplay]

Actually, his solution is correct. His workings are complete bullshit though.

Not exactly. His workings are correct, too - what sucks is the formatting. Do it yourself and you'll see that specifically sqare roots are missing: cos B = 3/4 => sin B = sqrt(7)/4 and so on. However trivial the problem was, though, he wouldn't have gotten the right answer by bungling the math so badly as his submission formatting would suggest.

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.

Reviewer's equations aren't even right!

[Anonymous Coward]

distance**2 is not x2**2 - x1**2 + y2**2 - y1**2

It is (x1-x2)**2 + (y2-y1)**2

Re:Reviewer's equations aren't even right!

[Anonymous Coward]

Everyone with a tertiary level in mathematics can see that those two formulas are obviously equivalent.

Sign me up!

[Plaid Phantom]

If this sort of attack on each other's work is the basis for modern Mathematical debate, sign me up! This is hilarious!

Didn't bother to read it

[Versalis]

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:Didn't bother to read it

[strikethree]

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

awesome

[bunions]

great link, thanks.

A New Kind of Science (was Re:Too bad)

[Anonymous Coward]

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.

OK, stepping way out on a limb here...

[ursabear]

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.

Distance

[Anonymous Coward]

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

is he being polemic?

Wolfram

[pinkuff]

I slogged through the 1000+ pages of the tome three years ago - I remember all my friends were making fun of me because of the size of the book. While some of that stuff is interesting, anybody who claims he's invented a New Kind of Science is a raging lunatic. The way the book is filled with self promotion is a testimony to this, all the more sad because Stephen Wolfram's curriculum is impressive and he might (I'm not equipped to understand) be a genius

How about a more qualified reviewer?

[nonlnear]

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:How about a more qualified reviewer?

[Ibag]

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 real sense (assuming that trig is taught correctly, and not as some black box). Moreover, doing away with trig and replacing it with rational trig would be detrimental. Rational trig is at best a complement to normal trigonometry.

I should add that you'd be hard pressed to people both qualified to review the book for its mathematical content and interested in reading it. If he really included a section on field theory, it would require someone with the background of a decent undergraduate math curriculum, and the majority of the book would be so uninteresting and uninspired (just a reworking of how to solve trig problems using intermediate results to see that most problems actually are still solvable) that they would fall asleep. Most people at that level wouldn't want to review a standard trig textbook, and if you were at that stage, this would not be substantially different enough to change this fact.

I'm curious, how do you know about fields of characteristic 2 but not feel you are qualified to critique the book?

Re:How about a more qualified reviewer?

[nonlnear]

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

[Anonymous Coward]

This review is just an improved version of this classic adequacy troll: http://www.adequacy.org/public/stories/2001.10.14. 163749.94.html

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.

Re:Review is an obvious troll

[Anonymous Coward]

Thanks for the link, I had a good chuckle reading the conversation started by robotslave.

And now I too must know, WHAT was the WORD??

The answer to the question

[Anonymous Coward]

3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989

Why don't you...

[Anonymous Coward]

Shut your pi hole!

AMAZING!

[savage1r]

I was SO BORED by the intro to this article that I practically peed my pants when I realized that there was actually more and that by reading said article it would be possible to put me in a coma within seconds!

"Mighty odd-sounding?"

[complexmath]

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

"what's really going on"

[PCM2]

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?

meh

[krilli]

Maybe he just likes funny words. I like funny words, and funny letters. J is the funniest letter of the alphabet. Why the huff?

Seriously...

[Anonymous Coward]

The effects of this on constants which are congruent is marginal. In fact, the vast majority of differential mathematice relies soley on equations with multiple correct answers.

So I wonder why this was posted anyhow, it was vetted before, and it brings nothing new to Elaxalgesic Congruence Theory nor General Number Theory.

It's really just another impromptu disparaging of Dianetics.

/. is math notation for divide and multiply

[dbdweeb]

What's all this math? I thought /. was a reference to *nix stuff.

What a vocabulary too bad.....

[Chineseyes]

he misspelled specialize.

Re:What a vocabulary too bad.....

[xsonofagunx]

Actually, he's Australian, I believe, so he spelled it correctly, according to British English. Nice try though.

distances in what space exactly

[gsn]

From TFA:

DEFINITIONS

Firstly, the definitions, given in the Introduction:-

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

er... (x2-x1)^2 + (y2-y1)^2 or do cross terms no longer matter for calculating distances...
Seriously I stared at this for a while to make sure its what TFA says... because as we all know the physicists are the sloppy ones while the mathematicians are always rigorous...

Umm... the guy wrote a book

[modmans2ndcoming]

Lets see.... why would one think that a book has more voracity than a published article in a peer reviewed journal? One would think that if such a book was being published then the author would have at least had a paper on it in a journal somewhere.

Re:Umm... the guy wrote a book

[Anonymous Coward]

Well, books do tend to be written for a wider audience... Or did you mean veracity?

Re:OT: Wikipedia down, what happened?

[crabpeople]

I can confirm it was down earlier. Id give you the time, but i guess theres no timestamp in the firefox history page. Maybe ill submit it to them as a feature! it was aprox 2 hours ago or so...

Jessica Alba

[Eudial]

Don't know about you, but I find her proportions pretty divine...

Hash tables

[tepples]

Got any hash for sale?

That's not my department. You might want to go to the hash tables.

distance-squared?

[volpe]

(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.

Divine Proportions?

[Anonymous Coward]

I thought they were referring to Scarlett Johansson.

I won't make that mistake again.

Proposal to Slashdot Editors

[Sage Gaspar]

Please, for the love of god, only publish articles that you understand.

Re:Proposal to Slashdot Editors

[tehcyder]

Please, for the love of god, only publish articles that you understand.

Wouldn't that imply that the editors actually read them?

Fields (warning, this post contains math)

[Ibag]

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 the global sense.

I opine that if one does not know field theory, one should not comment on how it relates to classical geometry. For the sake of being more enlightening than the review, I will try to explain fields and how they relate to geometry (classical and otherwise). Before I begin, I want to note that classical geometry has no relation to fields of nonzero characteristic, so the complaint is completely off base.

A field is a mathematical object that contains the same kind of algebraic structure as the real numbers, the complex numbers, and the rational numbers. Namely, we have two commutative operations (+ and *), identity elements for each operation (0+k=k, 1*k=k), every number has a negative (k+(-k)=0), every nonzero number has a reciprocal (k*(1/k)=1), and multiplication distributes over addition (a*(b+c)=(a*b)+(a*c)).

For a field to be nontrivial, we must have that 1!=0. However, it is still possible to have that 1+1+1+...+1=0 in our field, for some number of 1's. The smallest number of 1's required is called the characteristic of the field. If there is no nonzero amount of 1's that work, the field is said to have characteristic 0. Most people only deal with fields of characteristic 0, but fields of characteristic 2 and useful for cryptography, and other fields pop up in random places.

Fields have a lot of structure, and give you just the right amount of generality for doing a lot of math. In particular, most linear algebra can be done over arbitrary fields, and a lot more math is done over fields with some additional restrictions (e.g algebraic closure: every polynomial factors).

While the Cartesian plane is generally though of as ordered pairs of real numbers, you can have a Cartesian plane over arbitrary fields. There is actually a good use of this (in my opinion):

(skip this paragraph if you don't like details) If you start with a line segment that you say is of length 1, then with straight edge alone, you can make all the rational numbers, it is straight forward to add two numbers or divide two numbers, and a compass allows you to multiply two numbers. However, it is easy to construct non-rational numbers: a right triangle with sides of length 1 has a hypotenuse of sqrt(2). If you look at the equations that you use when you intersect lines and circles, the length of any new line segment you can make comes from a finite number of additions, subtractions, multiplications, divisions, and square roots. We say that a field is constructable if it is the smallest field containing all lengths of line segments generated in the process of a ruler-compass construction. Basic field theory and the above observation shows that any constructable field is a vector space over the rational numbers of dimension 2^n for some n. Since cos(20) has a minimal polynomial of degree 3, and since we can construct equilateral triangles little more field theory tells us:

We cannot trisect a 60 degree angle, and therefore we cannot trisect arbitrary angles.

Greeks (and then non-Greeks) tried for a long time to prove or disprove that one could trisect an angle. It took field theory to give a proof.

There are more links between geometry and fields. For instance, transporting the dot product to arbitrary Cartesian planes, can can define cosines of angles by analogy (although not the angles themselves), and then try to do trigonometry. This approach sort of works if the field is nice, but fields of characteristic 2 are

Resolved for many centuries

[viking2000]

Eucledian geometry has been completed centuries ago. Isn't it all very simple?

Let's discuss how we add 2 and 2. Should we count fingers, or should we rather count coins?

Shouldn't /. discuss more recent issues?

David Halprins background is "tertiary level mathematics". What is that?

Was that meant to be English?

[vidarh]

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...

Does anybody else...

[Valley Redneck]

...feel as stupid as I do now? Og use calculator to addy big numbers.

what a terrible review...

[Anonymous Coward]

I'm surprised that this review does not end with numerous threatening references to the U.N. and the words "Flanders Sucks" repeated over and over again.

A review of the review

[Anonymous Coward]

A thesaurus turd.

cranky indeed...

[myowntrueself]

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

Maths As A Science

[ObsessiveMathsFreak]

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 physicists use. Under this type, basic properties are defined from the ground up, with examples, and theorems and proofs are given more concrete relations to basic numbers and geometry. In this reigieme, mathematics is, like the other sciences, an exploration, examination and classification of the universe, albiet in the case of mathematics a more abstract portion of the universe. Here mathematics is by default falsifable, as all our properties and theorems can be subjected to direct experiment by means of calculation of basic numbers and geometric measurements.

Mathematics that does not follow the scientific method is somewhat different. Instead of exploring the properties of basic numbers and geometry, proponents of this method instead propose structures that may or may not exist, defining them through axioms and other definitions. Examples are few and far between as the objects in question may or may not exist "in the real world", and even if they do exist, any concrete example would neccesarily restrict itself to only one minute subset of all possible manifestations of the object.

Here, mathematics is not falsifiable, as experiments to test the validity of properties are pointless, because the axioms restrict the objects we consider to only those with certain properties. Experiments to test the validity of theorems are also largely impossible or unfeasable, as most of the objects under consideration have never been constructed or explored, and indeed there is no guarantee that anyone can ever be able to construct them. In general, falsifiabilty is only really guaranteed when mathematics can be ultimately reduced to basic elements which we candirectly observe and manipulate, such as real numbers, finite sets, etc. Much of modern mathematics is not confined to this domain.

A lot of mathematicians would be in serious disagreement with me here. They would insist that their theorems are falisfiable, or even object that falsifiability is a nonsense concept in mathematics as everything is by definition true. I remain unconvinced of the validity of such world views, especially in the realm of science.

As someone who has read a lot of advanced mathematics, I can safely say that the standard of proof in modern mathematics is now very low. Most modern proofs essentially amount to proof by intimidation which most if not all readers must simply accept as an axiom. I recall recent stories about the "uncertainty" in many modern mathematical proofs. Apparently, the proofs were "unverifiable" by the academic referres assigned to validate them. To me, it sounded like the authors hadn't actually "proved" anything at all. But such is the state of modern mathematics.

I'd like to think that what I do is science. I really would. I endevour to make my proofs clear and above all repeatable, but I'm really just fighting the tide. Most advanced mathematics is a kind of pseudoscience. Undeservedly so, but that's the way it is.

Well, since you welcome comments...

[sh4na]

... 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 something?!? Jeez!

Very nice link...

[sh4na]

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

Grrrr...

Re:Very nice link...

[Anonymous Coward]

Yes?

Ramblings of a madman.

[shoolz]

At least that's how this reads.

Sigh... I'm irritated by people who think that their large vocabularies make them good communicators.

Very well written actually

[Frightening]

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.

Can someone please write a not-so-difficult and/or simple translation of the above paragraph? And why the crap did he have to mention the reasoning was counter-intuitive *if it was actually flawed*. Do you get extra points? What good would it do the reasing if it was intuitive if it was also flawed, genius?

Slashdot editors please think of the children. I'm going to bed.

Re:Very well written actually

[Frightening]

Correction:

What good would it do the reasoning to be intuitive if it were also flawed?

Sorry

Re:Very well written actually

[Anonymous Coward]

Man, you're just a faggot. Shut the fuck up and come back when you finally learn something about anything.

Re:Very well written actually

[Frightening]

If I get too annoyed with you, I will report you, get a treceroute to your ISP and make life very difficult for you. I don't know why you hate me so much(religion?) and I don't care, but you are in serious need of a life.

Does reviewer remember high school math?

[Londovir]

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 not very elegant in appearance, but certainly effective:

B = arccos( (b^2-a^2-c^2) / (-2*a*c) )

d = 5 * sin(B) / sin(180-45-B)

But then again -- I'm just a simple high school math teacher, so I could be completely wrong. :)
 
Londovir

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.

But it works!

[ishmaelflood]

For certain values of x and y

Reviewer has wrong definition of Spread!

[SaXisT4LiF]

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 mathematics. The fact that Wildberger's definition does not use the classical trigonometric concept of an angle is a key feature of rational geometry.

I think that Wildberger's biggest flaw in Divine Proportions was presenting the spread-equals-sine-squared equation in the "Indroduction". This put too much focus on the relationship between classical trigonometry and rational geometry, and not enough focus on the implications of rational geometry in higher dimensions.

Having said that, it was also the most enjoyable book I'd read in a long while. I despised classical trigonometry in high school. It felt so arbitrary and forced compared to the rest of mathematics. The scope of this book doesn't go that far beyond the tools provided by trigonometry, but it solves the same set of problems with a smaller set of assumptions. Simplicity is beautiful.

Personally, I read through this book with a multi-dimensional analog of spread dancing around in the back of my mind...
 
  For vectors v1 and v2 in an inner product space F^n, the spread can be defined as:

s(v1,v2) = 1 - ( <v1,v2>*<v2,v1> )/( <v1,v1>*<v2,v2> )

[the inner product, or "dot-product", is denoted here by < , >]

Being able to define a method for measuring relative orientation in an arbitrary number of dimensions is unnecessarily difficult in classical trigonometry. I think that Wildberger was on right on track with this book, and that readers can get as much out of it as they are willing to put in.

Law of Cosines

[The Grassy Knoll]

Did anyone else read that as "Law Of Cosiness"?

Aw, bless.

.

I think you missed the main points, buddy...

[bitbucketeer]

1. an "angle" is not rigorously defined, nor do you attempt to refute Wildberger by defining it. 2. rational trig allows you to solve problems exactly with algebraic techniques. besides, recommendations of looking up approximate answers in trig tables (or using a calculator) is a form of "legerdemain". I think it wholly appropriate to use rational trig when teaching high school geometry and to save traditional trig for calculus when the foundations for rigorously defining an "angle" have been laid.

"David" sounds more like "Sanjay"

[Anonymous Coward]

Or if not, maybe some odd attempt at a Turing test.

Law of Cosines isn't taught everywhere

[Cassini2]

The law of cosines is only taught in some jurisdictions. Everywhere else, they appear to use some tortured workaround based on right triangles. This sucks, the law of cosines makes for a really quick solution when used appropriately.

Incidentally, in some countries they don't teach the vector cross-product either. This really makes a mess of vector math when you try to read many North American textbooks. Some branches of physics make frequent use of the vector cross-product.

On my display, some math notation is missing

[Cassini2]

In screen, some of the math notation is missing. It might be confusing for someone trying to follow his proof.

Specifically:
On the first line: sin B = sqrt(7) / 4
On the third line: sin (45+B) = (3 + sqrt(7)) / (4 * sqrt(2))
On the fourth line: d = 5 sin B/sin BDC = 5 * sqrt(7)/4 * (4*sqrt(2))/(3 + sqrt(7))*(3-sqrt(7))/(3-sqrt(7))
The (3-sqrt(7))/(3-sqrt(7)) comes from an effort to complete the square.

On the fifth line: d = 5*sqrt(2)*(3 * sqrt(7) - 7)/2 = 3.313693059