Even Great Mathematicians Guess Wrong
How Hamilton spent ten years on an impossible search
William Hamilton, Sir William Rowan Hamilton, was one of the great mathematician-physicists of the 1800’s. He invented a key way to look at physical systems—now called Hamiltonian mechanics—that is used in most theories of modern physics. He also discovered the famous Cayley-Hamilton theorem, a matrix satisfies its own characteristic polynomial. Actually he presented a special case and Arthur Cayley stated the general case —another Matthew effect?
Today I want to talk about perhaps his most prized invention—quaternions—and how we can all guess wrong.
Hamilton searched for a way to define a useful algebra for reasoning about three-space. His intuition, apparently, was that complex numbers are a powerful tool for studying the two-dimensional plane: each point corresponds to a unique complex number . The beauty of this correspondence is that it allows you to add, subtract, and multiply points in the plane. This power, especially of multiplication, is immense. It simplifies many proofs, gives us new intuition about points, and in general is extremely powerful.
He hoped to do the same for points in three-dimensional space. That is, he wanted to be able to map the point to generalized complex numbers
so that you could again add, subtract, and multiply points. He reasoned by analogy that such an algebraic formulation of three-space would yield wonderful new insights, and would make thinking about difficult problems easier.
Hamilton spent over ten years of his life trying to figure out how to make this work. That is, he sought a way to define how to multiply
so that the right properties of algebra were preserved. Of course, one could define the product to be anything, but an arbitrary definition would be of little use. The definition of multiplication had to fit together with addition and subtraction; otherwise, the ability to multiply vectors would not be useful. It was not demanded that the algebra be commutative—Hamilton already understood that matrices are fundamental to Nature and their multiplication is not commutative.
All his attempts failed. No matter how he defined multiplication the resulting algebraic system was unusable. The reason was: he guessed wrong. It is in fact possible to prove that there is no such algebra. None.
The story is that Hamilton, while walking with his wife, suddenly realized that while he could not handle triples he could handle quads. And the notion of quaternion was born. He carved the equations into a bridge during this famous walk—on Monday 16 October 1843.
Prior to that day, during his search, his son used to ask:
Well, Papa, can you multiply triples? Which he answered each day before his great discovery: No, I can only add and subtract them.
Hamilton considered quaternions to be one of his great discoveries—he felt that the ability to have an algebra of four-space was absolutely fundamental. Let’s return to this in a moment, but first discuss how he could have saved over a decade of work.
Real numbers, complex numbers, and quaternions are closely related to algebraic identities. In particular, it is not hard to see that famous identities arise in a natural manner from the existence of real numbers, complex numbers, then quaternions, and finally octonions. A trivial example is this identity for the reals:
The more interesting identity
underlies an important one for complex numbers. Let be the norm of the complex number , OK technically the norm-squared. Then the above says that
This states that the norm is multiplicative. Note that this identity really can be guessed and checked, but a better method is to use complex numbers. The norm function can be defined as follows:
where is the conjugate of . Then
This is the way to really “see” why the identity
is true—don’t you agree?
Here is the four square identity that arises from the quaternions:
Even though quaternions are not commutative, a similar norm proof can be given for this more involved identity.
John Baez, in a delightful page which I have already quoted from above, relates that Hamilton’s friend John Graves took this “alchemy” one step further. Here is the eight square identity that arises from the octonion numbers:
But this is as far as it goes. There is a fun quote from the top of Baez’ page that I must include:
The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative.
Why Hamilton Failed
Let us use the notation to stand for the set of numbers that are sums of squares. Then the above show that
The key insight that would have saved Hamilton a decade of hard work is simple: if there were a way to multiply triples, that had reasonable properties, then it would follow that
But this is impossible. Note that and are in , but their product is not. To see this, assume that
Now if this is true then it must be true modulo . It is easy to check that the square of every integer modulo is either , , or . The punchline is that there is no way for , which is , to be obtained as a sum of three numbers chosen from .
To verify this, note we can assume that . We argue by looking at how many of are . If none are , then the maximum we can get is , which is not . If one is , then . But this is impossible, since the maximum this can be is , which again is not . If two are , then , which is clearly impossible. Finally, if three are , then this also is clearly impossible. A question: should I have included this paragraph—is the proof too simple?
The Quaternion Scandal
As I stated earlier Hamilton thought quaternions were the answer, the answer to almost all questions. He wrote extensively on them, pointing out the many advantages of basing physics on them. The world quickly divided into two groups: those who thought quaternions were the answer to all, and the those who thought they were unneeded, useless, and dominated by general linear algebra. Two quotes, which I do not have to label, should make this point:
The invention of the calculus of quaternions is a step towards the knowledge of quantities related to space which can only be compared, for its importance, with the invention of triple coordinates by Descartes. The ideas of this calculus, as distinguished from its operations and symbols, are fitted to be of the greatest use in all parts of science.
Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including James Maxwell.
These are both from an interesting article by Simon Altmann on quaternions: “Hamilton, Rodrigues, and the Quaternion Scandal.” There are two parts of the “scandal:” Who really invented quaternions? And how important are they? The first question raises another example of the Matthew effect. Olinde Rodrigues published the notion in 1840, while Hamilton discovered them in 1843. Rodrigues did not name them—he was not the famous mathematician that Hamilton was—and he never pushed quaternions like Hamilton did. Besides that, Rodrigues was a Socialist banker, who was certainly not the mathematician Hamilton was.
The other part of the scandal was: were quaternions really the answer to all? The battle was over using quaternions or using more general vector methods. Eventually vectors methods mainly won, although quaternions are used quite a bit today in modern computer graphics. They are quite good at representing orientations and rotations of objects in three dimensions—see this for more details. I wonder what the pure mathematician in Hamilton would have thought about that.
Hamilton guessed wrong: he thought for over ten years that there might be a way to define multiplication on triples that would make them into an algebra. He could have saved himself a great deal of work if he had thought: what happens if such an algebra does exists? He might have quickly realized that it would imply, in our notation, that
Since this is easily seen to be impossible, he would either have given up or tried four dimensions years earlier.
I wonder sometimes what complexity conjectures have we guessed wrong on, and are there short proofs that they are wrong? Noticing that
is not trivial. But once it is explicitly stated, it is not too hard to prove.