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← Mathematics: stupid and clever questions for people who understand

Jos Gibbons's Avatar Jump to comment 18 by Jos Gibbons

How can you possibly have "orders" of infinity? Infinity is not an actual number -- it's a concept that means that there is no end to numbers. So how can you multiply infinity times itself in the first place, and how could the result be anything other than infinity in the second place?

Further to the previous excellent discussions of infinite sizes of sets, or transfinite cardinals as they’re technically known, there are two other kinds of infinity in mathematics.

One of these is to do with set theory; in particular, sets whose members, members’ members etc. are also all sets. (It’s all built from the empty set originally. We say it’s set theory without urelements.) Those sets which are transitive (their members are subsets of them) and well-ordered by membership (so any non-empty subset of them has a member which has none of the other members of the subset as members, and given any two distinct members of the original set one is a member of the other but not vice versa) are called ordinals. The class of all ordinals is itself transitive and well-ordered by membership, which proves it’s not a set (this “Burali-Forti paradox” was the first paradox in naïve set theory, i.e. the first proof not all classes are sets). The “first” ordinal with a property at least one ordinal has is thus always well-defined, provided the ordering relation with which we define “first” is membership. For example, the first transfinite ordinal, i.e. ordinal with infinitely many members, is ω and all subsequent ordinals are also transfinite. Not all transfinite ordinals differ in cardinality, e.g. ω+1 and ω have the same cardinalities. In situations like these, while bijections between the sets exist, order-preserving bijections don’t; ordinals are used to define order types, which are more specific than mere cardinalities.

If the axiom of choice is assumed, all sets have the same cardinality as at least one ordinal, so all sets have a well-ordering (because a bijection between the set and the ordinal can be used to define one), and all well-orderings have the same order type as exactly one ordinal’s ordering relation under membership, although ordering relations other than well-orderings don’t correspond to any ordinal’s ordering relation under membership. If the negation of the axiom of choice is assumed, some sets don’t have the same cardinality as any ordinal, and these sets cannot be well-ordered.

The other has nothing to do with either set-theoretic notion of infinity, but is probably easier to understand. Real numbers are extended to include ∞, or +∞ & -∞, so as to get either the affinely or projected extended real number system. We define the arithmetic they obey to preserve as many of the old rules as possible, e.g. ∞×∞= ∞ and so it’s a root of x² = x as you suggested. This is also true of transfinite cardinals. One key difference between this kind of infinity and a transfinite cardinal is that the former satisfies 2 to the x = x and the latter satisfies 2 to the x > x (Cantor’s theorem). Extended reals have associated limit-theoretic rules, e.g. regarding what the phrase "limit as x tends to @#8734; f(x) = a" means when (i) a is real and (ii) a is infinite.

How is the rotation somehow smeared across the other 3 components, to end up with 4 components that are each some weird hybrid of part axis, part rotation? Does anyone know an intuitive way to visualize this?

I do know one way to make sense of quaternions’ characterisation of rotations, but I have to first enlarge the theory from 3 dimensions of space to 1 of time and 3 of space as per special relativity. Did you know quaternions are especially useful in special relativity because of the way Lorentz transformations work? A rotation of the spatial axes alone is expressible with a Lorentz transformation, so my explanation will begin by associating quaternions with Lorentz transformations, then looking at rotations specifically.

The four-vector (a, b, c, d) is mapped to the quaternion a+bi+cj+dk. Lorentz transformations are linear maps on the set of four-vectors. Linear maps on real or complex numbers are quite boring; you just multiply by a constant. Quaternions are a less trivial case, however, because the breakdown of commutativity means the most economic irreducible way to state all linear maps on quaternions is to say what are the 1,i,j,k components of the values it maps 1,i,j,k to. Indeed, thinking about what arbitrary linear maps on the set of four-vectors do in matrix terms makes this 16D phenomenon, and this way to parameterise it, intuitive. Another way to do it is by writing the map as a linear combination of the terms x, ix, jx, kx, xi, xj, xk, ixi, ixj, ixk, jxi, jxj, jxk, kxi, kxj, kxk. If you want an exercise, you can work out how to interconvert these two approaches.

Of course, only some linear maps are Lorentz transformations, so they correspond to only some linear maps on quaternions. This is where I get to enjoy those Greek-symbol letters various people have given me on this thread! If η is the familiar metric from special relativity, a Lorentz transformation is a matrix Λ iff ΛTηΛ = η (that T is meant to be a superscript; does anyone know how to get arbitrary superscripts rather than 1, 2 or 3?), T denoting taking a transpose. Another way to define them is that, if A=ΛA’, B=ΛB’ then ATηB = A’TηB’. That means we can write it in terms of how the map affects quaternion multiplication. Bear in mind that (a+bi+cj+dk)(e+fi+gj+hk)=(ae-bf-cg-dh)+(ch-dg)i+(df-bh)j+(bg-cf)k. In other words, if u, v are the 3D vectors (b,c,d), (f,g,h), and we treat quaternions as paravectors http://en.wikipedia.org/wiki/Paravector (scalar-vector sums), we can write (a+u)(b+v) = (ab - u.v) + u×v. That formula is even prettier when you realise its scalar part is the Minkowski inner product of the associated four-vectors.

Rotations are the Lorentz transformations in which time is unaffected and space rotates on its own, and so the matrix has first row and first column (1,0,0,0). All these quaternion calculations are heavy work and hard to summarise on here, but if you go through them all as an exercise hopefully the way quaternions correlate with rotations will “make more sense” afterwards. Let me know if it does.

Tue, 08 May 2012 06:56:51 UTC | #940494