Modern Measure Theory has something of a glitch. It asserts, as a main result, something which is rather obviously logically problematic (I am feeling polite this New Year’s morning!) Let’s talk a little about this subject today.
Modern measure theory studies, for example, the interval [0,1] of so-called real numbers. There are quite a lot of different ways of trying to conjure these real numbers into existence, and I have discussed some of these at length in many of my YouTube videos and also here in this blog: Dedekind cuts, Cauchy sequences of rationals, continued fractions, infinite decimals, or just via some axiomatic wishful thinking. In this list, and in what follows, I will suppress my natural inclination to put all dubious concepts in quotes. So don’t believe for a second that I buy most of the notions I am now going to talk about.
Measure theory texts are remarkably casual about defining and constructing the real numbers. Let’s just assume that they are there, shall we? Once we have the real numbers, measure theory asserts that it is meaningful to consider various infinite subsets of them, and to assign numbers that measure the extent of these various subsets, or at least some of them. The numbers that are assigned are also typically real numbers. The starting point of all this is familiar and reasonable: that a rational interval [a,b], where a,b are rational numbers and a is less than or equal to b, ought to have measure (b-a).
So measure theory is an elaborate scheme that attempts to extend this simple primary school intuition to the rather more convoluted, and logically problematic, arena of real numbers and their subsets. And it wants to do this without addressing, or even acknowledging, any of the serious logical problems that people (like me) have been pointing out for quite a long time.
If you open a book on modern measure theory, you will find a long chain of definitions and theorems: so-called. But what you will not find, along with a thorough discussion of the logical problems, is a wide range of illustrative examples. This is a theory that floats freely above the unpleasant constraint of exhibiting concrete examples.
Your typical student is of course not happy with this situation: how can she verify independently that the ideas actually have some tangible meaning? Young people are obliged to accept the theories they learn as undergraduates on the terms they are given, and as usual appeals to authority play a big role. And when they turn to the internet, as they do these days, they often find the same assumptions and lack of interest in specific examples and concrete computations.
Here, to illustrate, is the Example section of the Wikipedia entry on Measure, which is what you get when you search for Measure Theory (from Wikipedia at https://en.wikipedia.org/wiki/Measure_(mathematics) ):
Some important measures are listed here.
- The counting measure is defined by μ(S) = number of elements in S.
- The Lebesgue measure on R is a complete translation-invariant measure on a σ-algebra containing the intervals in R such that μ([0, 1]) = 1; and every other measure with these properties extends Lebesgue measure.
- Circular angle measure is invariant under rotation, and hyperbolic angle measure is invariant under squeeze mapping.
- The Haar measure for a locally compact topological group is a generalization of the Lebesgue measure (and also of counting measure and circular angle measure) and has similar uniqueness properties.
- The Hausdorff measure is a generalization of the Lebesgue measure to sets with non-integer dimension, in particular, fractal sets.
- Every probability space gives rise to a measure which takes the value 1 on the whole space (and therefore takes all its values in the unit interval [0, 1]). Such a measure is called a probability measure. See probability axioms.
- The Dirac measure δa (cf. Dirac delta function) is given by δa(S) = χS(a), where χS is the characteristic function of S. The measure of a set is 1 if it contains the point a and 0 otherwise.
Other ‘named’ measures used in various theories include: Borel measure, Jordan measure, ergodic measure, Euler measure, Gaussian measure, Baire measure,Radon measure, Young measure, and strong measure zero.
In physics an example of a measure is spatial distribution of mass (see e.g., gravity potential), or another non-negative extensive property, conserved (seeconservation law for a list of these) or not. Negative values lead to signed measures, see “generalizations” below.
Liouville measure, known also as the natural volume form on a symplectic manifold, is useful in classical statistical and Hamiltonian mechanics.
(Back to the regular channel) Now one of the serious problems with theories which float independent of examples is that it becomes harder to tell if we have overstepped logical bounds. This is a problem with many theories based on real numbers.
Here is a key illustration: modern measure theory asserts that the real numbers with which it is preoccupied actually fall into two types: the computable ones, and the uncomputable ones. Computable ones include rational numbers, and all irrational numbers that (supposedly) arise as algebraic numbers (solutions of polynomial equations), definite integrals, infinite sums, infinite products, values of transcendental functions; and in fact any kind of computer program.
These include sqrt(2), ln 10, pi, e, sqrt(3+sqrt(5)), Euler’s constant gamma, values of the zeta function, gamma function, etc. etc. Every number that you will ever meet concretely in a mathematics course is a computable number. Any kind of decimal that is conjured up by some pattern, say 0.1101001000100001000001…, or even by some rule such as 0.a_1 a_2 a_3 … where a_i is 1 unless i is an odd perfect number, in which case a_i=2, is a computable number.
And what is then an uncomputable real number?? Hmm.. let’s just say this rather quickly and then move on to something more interesting, okay? Right: an uncomputable real number is just a real number that is not computable.
Uhh.. such as…? Sorry, but there are no known examples. It is impossible to write down any such uncomputable number in a concrete fashion. And what do these uncomputable numbers do for us? Well, the short answer is: nothing. They are not used in practical applications, and even theoretically, they don’t gain us anything. But they are there, my friends—oh yes, they are there — because the measure theory texts tell us they are!
And the measure theory texts tell us even more: that the uncomputable real numbers in fact swamp the computable ones measure-theoretically. In the interval [0,1], the computable numbers have measure zero, while the uncomputable numbers have measure one.
Yes, you heard correctly, this is a bona-fide theorem of modern measure theory: the computable numbers in [0,1] have measure zero, while the uncomputable numbers in [0,1] have measure one!
Oh, sure. So according to modern probability theory, which is based on measure theory, the probability of picking a random real number in [0,1] and getting a computable one is zero. Yet no measure theorist can give us even one example of a single uncomputable real number.
This is modern pure mathematics going beyond parody. Future generations are going to shake their heads in disbelief that we happily swallowed this kind of thing without even a trace of resistance, or at least disbelief.
But this is 2016, and the start of a New Year! I hope you will join me in an exciting venture to expose some of the many logical blemishes of modern pure mathematics, and to propose some much better alternatives — theories that actually make sense. Tell your friends, spread the word, and let’s not be afraid of thinking differently. Happy New Year.