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

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

[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]

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

[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

[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

[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:Too bad

[eln]

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

It also isn't science.

My idiosyncratic take

[Junky191]

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

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

[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.

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.

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: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.

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.

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.

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).