I was working in a nearby coffeeshop (an air-conditioned oasis) and the guy sitting next to me noticed my stack of Geometry books. We started talking, and he told me about a book called "Rational Geometry". I looked it up on the web, and the full title is DIVINE PROPORTIONS: Rational Trigonometry to Universal Geometry, by N J Wildberger.

This professor from Sydney has looked at the logical foundations of Euclidean Geometry and found them not only lacking, but fundamentally flawed. His claim is that the basic notions of distance and angle are problematic, and that the classical trigonometry relationships (can I get a SOHCAHTOA?) are cumbersome and unneeded.

He proceeds to recreate Geometry using concepts called Quadrance and Spread, which replace distance and angle. This allows him to calculate everything using only rational numbers (he even says that maybe the Pythagoreans were right - had they thought to consider the squares of numbers as more fundamental than the numbers themselves). This is, apparently, a more accurate way to compute, and does not require the use of extensive trig tables or modern calculators. He also says that this rational geometry is logically sound, not requiring the use of any undefined axioms, and therefore is easier for students to learn and make sense out of.

On the website above, you can view pdfs for the intro and first chapter. They are quite interesting. (There is also a posting for the last chapter, but the content is, let's just say, beyond my current scope of understanding.)

I wonder if anyone has had experience with this type of Geometry, especially with regards to trying to teach it to students? Is it the wave of the future, or is it a passing fad?

## Sunday, July 23, 2006

### Rational Geometry

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## 6 comments:

I don't know if you're aware of this, but a large thread was recently started on sci.math by Wildberger. He initially posted a brief message in which he expressed dissatisfaction with the current state of thought on mathematical foundations and included a link to a recent paper of his on set theory. It soon degenerated into a mess with plenty of anti-Cantoran crankery. There is also some discussion of the book in the thread.

See http://tinyurl.com/jzxmq

More discussion in http://science.slashdot.org/article.pl?sid=05/09/17/1313249

Wildberger is nothing more than another anti-Cantorian crank. It sickens me to think that he's poisoning the minds of students at the University of New South Wales.

This one quote (from his paper which can be linked from the link in the first comment):

"Mathematics does not require ‘Axioms’. The job of a pure mathematician is

not to build some elaborate castle in the sky, and to proclaim that it stands up

on the strength of some arbitrarily chosen assumptions. The job is to investigate

the mathematical reality of the world in which we live."

Oh, mathematics doesn't require axioms? Then there are no undefined terms. But the dictionary is finite. So what Wildberger is trying to say is that mathematics is based upon circular reasoning. Oops!

And if mathematics doesn't require axioms, then maybe Wildberger can tell me what he means by a number. After all, he uses natural numbers in his calculations in his rational trigonometry book...certainly, he can tell us what a natural number is.

And finally, we come to Wildberger's most idiotic assumption of all: that the job of mathematics (and presumably that of mathematicians) is "to investigate

the mathematical reality of the world in which we live."

However, mathematics is not limited to the real world. Many mathematical ideas are inspired by the real world, and some mathematical ideas have applications in the real world. However, much (I'd dare to say most) of mathematics has nothing to do with the real world.

Wildberger seems to be confusing mathematics with the sciences.

Asst. Prof - I was wondering the same thing myself, because clearly axioms are needed as a starting point for number theory itself. Based on reading the first chapter of the book, I thought Wildberger was opposed to the specific axioms used in traditional Geometry, but your quote shows that's not true.

What about the actual concepts of Spread and Quadrance, and the theorems he derives? I wonder if these are valid and if they can be useful for students, as long as we admit that he really is using certain axioms as a foundation. I should have realized there would be tons of discussion out there already.. I'll have to go read up on the links that the first two comments provided.

There are good reasons not to use quadrance and spread in daily life, they are not linear. The quadrance between city A and city B is 17, what is the round trip? Spread 1 is 24/25, spread 2 is 24/49, what is the total spread? These are common daily problems, and if the problem is in distance and angle you can do it in your head, but it is impossible without calculator in quadrance and spread. Maybe the point is to convert distance and angle into quadrance, do the calculation with the easy “rational trigonometry” and convert them back. However now you have to use sine and square root table, defeating the claim that they can be avoided.

I looked into it and read the downloadable chapters as a prelude to buying the book. I decided against the purchase.

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