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

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!

[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!

[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

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

[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:mod parent troll

[neonprimetime]

so cut me some slack, okay?

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

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

[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

[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

[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.

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.

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.

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

[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.

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

[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.

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?"

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.

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

[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.

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?

[rolyatknarf]

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

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.

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

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.

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.

Why don't you...

[Anonymous Coward]

Shut your pi hole!

"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

Re:All I could think about...

[Red Herring]

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

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.

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.