There are several approaches to the modern theory of “real numbers”. Unfortunately, none of them makes complete sense. One hundred years ago, there was vigorous discussion about the ambiguities with them and Cantor’s theory of “infinite sets”. As time went by, the debate subsided but the difficulties didn’t really go away.
A largely unquestioning uniformity has settled on the discipline, with most students now only dimly aware of the logical problems with “uncomputable numbers”, “non-measurable functions”, the “Axiom of choice”, “hierarchies of cardinals and ordinals”, and various anomalies and paradoxes that supposedly arise in topology, set theory and measure theory.
Some of the stumbling blocks have been described at length in my Math Foundations series of YouTube videos. In this blog we concentrate on the problems with real numbers and arithmetic with them.
The basic division in mathematics is between the discrete and the continuous. Discrete mathematics studies locally finite collections and patterns, and relies on counting, beginning with the natural numbers 1,2,3,⋯ and then extending to the integers, including 0, as well as -1,-2,-3,⋯, and to rational numbers, or fractions, of the form a/b with a and b integers, subject to the condition that b non-zero and with a particular notion of equality.
Continuous mathematics studies the “continuum” and functions on it, and relies on measurement, which these days involves also “irrational numbers” like √2,√5 and π that the ancients wrestled with, as well as more modern “irrational numbers” such as e and γ arising from integrals and infinite series. But what do these words and objects actually precisely mean and refer to? We should not presume that just because we use a common term or notion familiar from everyday life, that its mathematical meaning has been properly established.
Up to a hundred years ago, the notion of the “continuum” seemed intuitively straightforward, but difficult to pin down precisely. It appeared that we could rely on our intuition of space, following the philosopher I. Kant’s view that somehow humans had an innate understanding of three-dimensional geometry. However with the advent of modern physics, and in particular relativity and quantum mechanics, the true nature of the “continuum” grew increasingly murky: if time is relative and perhaps finite in extent, and space has an inherent graininess which renders it certainly not infinitely divisible, then what exactly are we modelling with our notion of the “infinite number line”?
While engineers and scientists work primarily with finite decimal numbers in an approximate sense, “real numbers” as infinite decimals are idealized objects which attempt to extend the explicit finite but approximate numbers of engineers into a domain where infinite processes can be ostensibly be exactly evaluated. To make this magic work, mathematicians invoke a notion of “equivalence classes of Cauchy sequences of rational numbers”, or as “Dedekind cuts”.
Each view has different difficulties, but always there is the crucial problem of discussing infinite objects without sufficient regard to how to specify them. I have discussed the serious logical difficulties at length around video 80-105 in the Math Foundations series.
For example the video Inconvenient truths with sqrt(2) has generated a lot of discussion. However not everyone approves of casting doubt on the orthodoxy: the video has more than 1000 likes, but also 316 dislikes. I doubt if I am saying anything in this video which is actually incorrect though — you can judge for yourselves.
Let’s return to the safe side of things. A finite sequence such as s = 1,5,9 may be described in quite different ways, for example as the “increasing sequence of possible last digits in an odd integer square”, or as the “sequence of numbers less than 10 which are congruent to 1 modulo 4”, or as the “sequence of digits occurring in the 246-th prime after removing repetitions”. But ultimately there is only one way to specify such a sequence s completely and unambiguously: by explicitly listing all its elements.
When we make the jump to “infinite sequences”, such as a sequence somehow implied by the expression “m = 3,5,7,⋯” the situation changes dramatically. It is never possible to explicitly list “all the elements” of an infinite sequence. Instead we are forced to rely on a rule generating the sequence to specify it. In this case perhaps: m is the list of all odd numbers starting with 3, or perhaps: m is the list of all odd primes. Without such a rule, a definition like “m = 3,5,7,⋯” is really rather meaningless.
We can say the words “infinite sequence”, but what are we actually explicitly talking about??
To a computer scientist, an “infinite sequence” is modelled by a computer program, churning out number after number perhaps onto a hard drive, or in former years onto a long tape. At any given point in time, there are only finite many outputs. As long as you keep supplying more memory, or tape, and electricity the process in principle never stops, but in practice will run short of resources and either grind forwards ever more and more slowly (the next output will take two years, hang on just a while!) or just come to a grinding halt when power or memory is inevitably exhausted.
So in this case the sequence is not to be identified by the “completed output tape”, which is a figment of our imagination, but rather by the computer program that generates it, which is concrete and completely specifiable. However here we come to an essential difficulty with “infinite processes”: the program that generates a given “infinite sequence”: is always far from unique. There is no escape from this inescapable fact, and it colours all meaningful aspects of dealing with “infinity”.
A finite set such as {2 4 6 8} can also be described in many ways, but ultimately it too is only properly specified by showing all its elements. In this case order is not important, so that for example the elements might be scattered over a page. Finite sets whose elements cannot be explicitly shown have not been specified, though we might agree that they have been described.
An example might be: let S be the set of all odd perfect numbers less than 10^{100000}. [A perfect number, like 6 and 28, is the sum of those of its divisors less than itself, i.e. 6=1+2+3 and 28=1+2+4+7+14.] Such a description of S does not deserve to be called a specification of the set, at least not with our current understanding of perfect numbers, which doesn’t even allow us to determine if S is empty or not.
With sets the dichotomy between finite and infinite is much more severe than for sequences, because we do not allow a steady exhibition of the elements through time. It is impossible to exhibit all of the elements of an “infinite set” at once, so the notion is an ideal one that more properly belongs to philosophy—it can only be approximated within mathematics.
The notion of a “completed infinite set” is strongly contrary to classical thinking; since we can’t actually collect together more than a finite number of elements as a completed totality, why pretend that we can? It is the same reason that “Leprechaun heaven” or “hierarchies of angels” are not generally recognized as proper scientific entities. Infinite sets, angels and Leprechauns may very well exist in our universe, but this is a philosophical or religious inquiry, not a mathematical or scientific one.
The idea of “infinity” as an unattainable ideal that can only be approached by an endless sequence of better and better finite approximations is both humble and ancient, and one I would strongly advocate to those wishing to understand mathematics more deeply. This is the position that Archimedes, Newton, Euler and Gauss would have taken, and it is a view that ought to be seriously reconsidered. I believe it is also closer to the view of modern giants such as H. Poincare and H. Weyl, both of whom were skeptical about our uses of “infinity”.
Why is any of this important? The “real numbers” are where Cantor’s “hierarchies of infinities” begins, and much of modern set theory rests, so this is an issue with widespread consequences, even within algebra and combinatorics. Secondly the “real numbers” are the arena where calculus and analysis is developed, so difficulties with their essential arithmetic foundation lead to weakness in the calculus curriculum, confusion with aspects of measure theory, functional analysis and other advanced subjects, and are obstacles in our attempts to understand physics.
In my opinion, it is possible, perhaps even likely, that we need to understand mathematics in the right way before we will be able to unlock the deeper secrets of the universe.
By reorganizing our subject to be more careful and logical, and by removing dubious axiomatic assumptions and unnecessary philosophizing about “real numbers” and “infinite sets” we make it easier for young people to learn, appreciate and contribute.
This also strengthens the relationship between mathematics and computing.
It is time to acknowledge the orthodoxy that silently frames our discipline. We need to learn from our colleagues in physics and computer science, and begin the slow, challenging but important and ultimately rewarding task of restructuring mathematics properly.
njwildberger: tangential thoughts 