Fibonacci Numbers to the Rescue

No, really. This time it’s true. Just wait.

Our favourite mathematics populariser at the moment is Evelyn Lamb. Lamb’s YouTube videos are great because they don’t exist. Evelyn Lamb is a writer. (That is not Lamb in the photo above. We’ll get there.)

It is notoriously difficult to write good popular mathematics (whatever one might mean by “popular”). It is very easy to drown a mathematics story in equations and technical details. But, in trying to avoid that error, the temptation then is to cheat and to settle for half-truths, or to give up entirely and write maths-free fluff. And then there’s the writing, which must be engaging and crystal clear. There are very few people who know enough of mathematics and non-mathematicians and words, and who are willing to sweat sufficiently over the details, to make it all work.

Of course the all-time master of popular mathematics was Martin Gardner, who wrote the Mathematical Games column in Scientific American for approximately three hundred years. Gardner is responsible for inspiring more teenagers to become mathematicians than anyone else, by an order of magnitude. If you don’t know of Martin Gardner then stop reading and go buy this book. Now!

Evelyn Lamb is not Martin Gardner. No one is. But she is very good. Lamb writes the mathematics blog Roots of Unity for Scientific American, and her posts are often surprising, always interesting, and very well written.

That is all by way of introduction to a lovely post that Lamb wrote last week in honour of Julia Robinson, who would have turned 100 on December 8. That is Robinson in the photo above. Robinson’s is one of the great, and very sad, stories of 20th century mathematics.

Robinson worked on Diophantine equations, polynomial equations with integer coefficients and where we’re hunting for integer solutions. So, for example, the equation x2 + y2 = z2 is Diophantine with the integer solution (3,4,5), as well as many others. By contrast, the Diophantine equation x2 + y2 = 3 clearly has no integer solutions.

Robinson did groundbreaking work on Hilbert’s 10th problem, which asks if there exists an algorithm to determine whether a Diophantine equation has (integer) solutions. Robinson was unable to solve Hilbert’s problem. In 1970, however, building on the work of Robinson and her collaborators, the Russian mathematician Yuri Matiyasevich was able solve the problem in the negative: no such algorithm exists. And the magic key that allowed Matiyasevich to complete Robinson’s work was … wait for it … Fibonacci numbers.

Label the Fibonacci numbers as follows:

F1 =1, F2 = 1, F3 = 2, F4 = 3, F5 = 5, F6 = 8, …

It turns out that with this labelling the Fibonacci numbers have the following weird property:

If Fn2 divides Fm then Fn divides m.

You can check what this is saying with n = 3 and m = 6. (We haven’t been able to find a proof online to which to link.) How does that help solve Hilbert’s problem? Read Lamb’s post, and her more bio-ish article in Science News, and find out.

Feynman on Modernity

We plan to have more posts on VCAA’s ridiculous curriculum review. Unfortunately.

Now, however, we’ll take a semi-break with three related posts. The nonsensical nature of VCAA’s review stems largely from its cloaking of all discussion in a slavish devotion to “modernity”, from the self-fulfilling prediction of the inevitability of “technology”, and from the presumption that teachers will genuflect to black box authority. We’ll have a post on each of these corrupting influences.

Our first such post is on a quote by Richard Feynman. For another project, and as an antidote to VCAA poison, we’ve been reading The Character of Physical Law, Feynman’s brilliant public lectures on physical truth and its discovery. Videos of the lectures are easy to find, and the first lecture is embedded above. Feynman’s purpose in the lectures is to talk very generally about laws in physics, but in order to ground the discussion he devotes his first lecture to just one specific law. Feynman begins this lecture by discussing his possibly surprising choice:

Now I’ve chosen for my special example of physical law to tell you about the theory of gravitation, the phenomena of gravity. Why I chose gravity, I don’t know. Whatever I chose you would’ve asked the same question. Actually it was one of the first great laws to be discovered and it has an interesting history. You might say ‘Yes, but then it’s old hat. I would like to hear something about more modern science’. More recent perhaps, but not more modern. Modern science is exactly in the same tradition as the discoveries of the law of gravitation. It is only more recent discoveries that we would be talking about. And so I do not feel at all bad about telling you of the law of gravitation, because in describing its history and the methods, the character of its discovery and its quality, I am talking about modern science. Completely modern.

Newer does not mean more modern. Moreover, there can be compelling arguments for focussing upon the old rather than the new. Feynman was perfectly aware of those arguments, of course. Notwithstanding his humorous claim of ignorance, Feynman knew exactly why he chose the law of gravitation.

This could, but will not, lead us into a discussion of VCE physics. It suffices to point out the irony that the clumsy attempts to modernise this subject have shifted it towards the medieval. But the conflation of “recent” with “modern” is of course endemic in modern recent education. We shall just point out one specific effect of this disease on VCE mathematics.

Once upon a time, Victoria had a beautiful Year 12 subject called Applied Mathematics. One learned this subject from properly trained teachers and from a beautiful textbook, written by the legendary J. B. “Bernie” Fitzpatrick and the deserves-to-be-legendary Peter Galbraith. Perhaps we’ll devote some future posts on Applied and its Pure companion. It is enough to note that simply throwing out VCE’s Methods and Specialist in their entirety and replacing them with dusty old Pure and Applied would result in a vastly superior, and more modern, curriculum.

Here, we just want to mention one extended topic in that curriculum: dynamics. As it was once taught, dynamics was a deep and incredibly rich topic, a strong and natural reinforcement of calculus and trigonometry and vector algebra, and a stunning demonstration of their power. Such dynamics is “old”, however, and is thus a ready-made target for modernising zealots. And so, over the years this beautiful, coherent and cohering topic has been cut and carved and trivialised, so that in VCE’s Specialist all that remains are a few disconnected, meat-free bones.

But, whatever is bad the VCAA can strive to make worse. It is clear that, failing the unlikely event that the current curriculum structure is kept, VCAA’s review will result in dynamics disappearing from VCE mathematics entirely. Forever.

Welcome to the Dark Ages.

Tweel’s Mathematical Puzzle

Tweel is one of the all-time great science fiction characters, the hero of Stanley G. Weinbaum’s wonderful 1934 story, A Martian Odyssey. The story is set on Mars in the 21st century and begins with astronaut Dick Jarvis crashing his mini-rocket. Jarvis then happens upon the ostrich-like Tweel being attacked by a tentacled monster. Jarvis saves Tweel, they become friends and Tweel accompanies Jarvis on his long journey back to camp and safety, the two meeting all manner of exotic Martians along the way.

A Martian Odyssey is great fun, fantastically inventive pulp science fiction, but the weird, endearing and strangely intelligent Tweel raises the story to another level. Tweel and Jarvis attempt to communicate, and Tweel learns a few English words while Jarvis can make no sense of Tweel’s sounds, is simply unable to figure out how Tweel thinks. However, Jarvis gets an idea:

“After a while I gave up the language business, and tried mathematics. I scratched two plus two equals four on the ground, and demonstrated it with pebbles. Again Tweel caught the idea, and informed me that three plus three equals six.”

That gave them a minimal form of communication and Tweel turns out to be very resourceful with the little mathematics they share. Coming across a weird rock creature, Tweel describes the creature as

“No one-one-two. No two-two-four”. 

Later Tweel describes some crazy barrel creatures:

“One-one-two yes! Two-two-four no!” 

A Martian Odyssey works so well because Weinbaum simply describes the craziness that Jarvis encounters, with no attempt to explain it. Tweel is just sufficiently familar – a few words, a little arithmetic and a sense of loyalty – to make the craziness seem meaningful if still not comprehensible.

But now, here’s the puzzle. The communication between Jarvis and Tweel depends upon the universality of mathematics, that all intelligent creatures will understand and agree that 1 + 1 = 2 and 2 + 2 = 4, and so forth.

But why? Why is 1 + 1 = 2? Why is 2 + 2 = 4?

The answers are perhaps not so obvious. First, however, you should go read Weinbaum’s awesome story (and the sequel). Then ponder the puzzle.

Update

Thanks to those who have posted so far. Everyone is circling with the right ideas, but perhaps people are searching for something deeper than intended. Anyway, for this first update (to which people are free to object in the comments), here is our suggested, simplest answer to why 1 + 1 = 2:

“1 + 1 = 2” is true by definition. 

To take a step back, what does 2 mean? It depends slightly on how you think of the natural numbers being given, but there are really only (ahem) two, similar choices. If you accept that addition is around then 2/two is simply a new symbol/name that stands for 1 + 1.

Or, more fundamentally, we can follow Number 8 and go Peano-ish, in which case 2 is defined as S(1), as the “successor” of 1. But then we have to define addition, and the first(ish) step for that is to define n + 1 = S(n); that is, 1 + 1 is defined to be S(1), which we have decided to call 2. There’s a good discussion of it all here.

With 1 + 1 = 2 done (modulo objections), why now is 2 + 2 = 4?

Second Update

It’s probably close enough to round this one off. To clearly state why 2 + 2 = 4, we first have to clearly state what 2 and 4 and + are. So, as discussed above, 1 + 1 = 2 by definition (more or less). And, similarly, we define 3 = 2 + 1 and 4 = 3 + 1. So, the question of why 2 + 2 = 4 comes down to understanding why

2 + (1 + 1) =  (2 + 1) + 1

So, our question amounts to a simple instance of the associative law of addition. And, how do we know the associative law is true? Naively, we can accept that’s the way numbers work. Or, we can go Peano-ish again, and the above example of associativity becomes part of the definition of addition.

In summary, to know that 1 + 1 = 2 all we need is the notion of natural numbers, of counting. To know that 2 + 2 = 4, however, requires the notion of addition.