Infinity. What is it, really? Perhaps you remember, as I do, the moment in your life when you realized one could (in principle) count forever, and there would always be integers there for one to count. Yet, how can our finite minds comprehend the truly infinite? Do infinte things even exist?
My own thoughts on this subject have gone full circle. I accepted the infinitude of the integers at a young enough age. Since I recall this as an act of acceptance, I infer that I must previously have assumed that only finite things exist. But accept I did, and proceeded, over the course of my mathematical education, to grow more and more steeped in the mysteries of the non-finite. One can measure infinities by bijection; so the even numbers are no more and no less infinite than all the integers, and than the positive numbers divisible by 1000. Yet the real numbers are more infinite—they are too many to be counted, for there is no 1-1 mapping between them and the counting numbers.
It went on from there: One can measure infinities more finely by order-preserving bijection. This leads to the ordinals, to transfinite induction, to the continuum hypothesis. The axiom of choice rears its controversial head. I was in my element, an initiate of eldritch powers delving (or, at least, learning how to delve) into eternal secrets beyond the (finite) universe. I accepted the Axiom for the power it gave, and looked down upon the doubters and detractors as only a teenager can.
But the study of computing sowed the seeds of doubt, and the practice of probability germinated them, and with the passage of years they have begun to sprout. Real numbers are awkward things to try to compute with. It may be a little trite to say that you can’t store all of the digits of a real number in the memory of the computer, but the limits on precision really are a consistent source of trouble in practice.
Probabilities over real numbers are awkward too. It’s easy enough to say “consider a real number drawn uniformly between 0 and 1”, but you can’t actually draw one: the probability of any given \(x\) under that distribution is 0 (whether \(x\) is between 0 and 1 or not!). Mathematically, one is forced to resort to the usual song and dance about probability densities: the limit as \(\eps\) goes to zero of the ratio of the probability in the interval to the size of the interval, yada, yada. Computationally, finite precision makes this even worse: the joke in probabilistic computing is that probability zero events happen alarmingly often. To say nothing of trying represent those probability densities, and of periodically being bitten by having forgotten whether one was referencing Lebesgue or counting measure.
So what is it like to sit in the shade of the tree of doubt? The epigraph from which I started this essay is the foliage between me and the burning sun of irresolution. If mathematics is to be a study of mental objects, one cannot simply ban infinities, because infinite mental objects are too easy to construct (they even form a useful programming technique). But they must have reproducible properties: so the place to draw the line is at objects that take an infinite amount of information to describe. This is the article of faith that defines finitary finiversalism: infinity only as the consequence of a finite description.
What, then, is admissible? The integers may be infinite, but they are probably the most reproducible mental object in all of mathematics—every kid learns the integers. The rationals are fine too, as each is given as a ratio of two finite integers—nothing non-reproducible about that. What of the reals? They are conventionally defined as the completion (in the order topology) of the rationals. That is, a real number is the limit of a sequence of rational numbers. How are we to reproduce such a sequence? We can’t just write it all down, because the whole point is that the sequences with irrational limits are infinite. But we can of course specify some sequences. For instance, “the \(k\)th term is the perimeter of a regular \(2^k\)-gon of diameter two.” The limit is a perfectly good irrational number, with very many very well-reproduced properties.
Which sequences of rationals are describable with a finite amount of information? Well, certainly any given Turing machine is fine: it can be encoded as a finite string in a finite alphabet. The only thing infinite about (some) Turing machines is their behavior; and the behavior can be reproduced by reproducing the machine.1 So any sequence of rationals that we can program a (finite) computer to print is reproducible, at least in principle. And the Church-Turing thesis tells us that’s it.
In fact, the Church-Turing thesis tells us that’s it for all of mathmatics, not just the reals. Computable objects are the only objects that can be described with a finite amount of information. This view reassuringly eliminates various controversies in mathematical foundations:
Among sets of computable objects, the well-ordering theorem is obvious, because we can always order objects lexicographically by (the description of) the smallest Turing machines that compute them.2
The conventionally controversial axiom of choice follows from well-ordering, but of necessity it’s now an axiom of finitely-specified choice.
The conventional alternative to choice, namely the axiom of determinacy, is moot. It concerns perfect play in (certain) infinite games, but since it’s not in general possible to compute the outcome of such a game even with a given pair of strategies, dicusssing existence of perfect strategies seems excessive.
What of the real numbers? There are only countably many Turing machines, so the reals can’t really exist in a universe of only computable objects.3 What we have instead are the computable numbers, which is the field of limits of computably-Cauchy computable sequences of rationals. The famous Cantor diagonalization proof that the classical reals are uncountable turns into a proof that there is no computable bijection between the computable numbers and the integers.
On the other hand, the unique characterization of the reals still holds, and the computable numbers are indeed not complete. Not only is it not possible in general to compute the limit of a computable sequence of rationals, there are even specific computable sequences whose limits are not computable numbers. (To get the limit one also needs to compute the Cauchy condition, i.e., bound how much farther the sequence may move.) Never mind that if one tries to make computable numbers totally ordered, one finds that equality is not computable.
So the analysis one gets out of computable numbers is different from real analysis. Which analysis is better? Computable analysis is more faithful to reproducible reality: you really can’t reliably compute the limits of infinite sequences, and must instead ever watch for accumulating errors. On the other hand, real analysis is simpler: even formulating the computable analogue of any nontrivial statement requires thinking about what is to be computed, in what generality, and given what inputs. Real analysis is also better developed, having predated attempts at computable analysis by centuries.
And so it is with infinity. It’s a simplification. Only behvaiors (of, e.g., computing processes) are infinite, but it’s often convenient to noun verbs and treat summaries of those behaviors as objects in their own right. But sometimes one needs information that was summarized away, and the predictive power of the summary fails.
And that’s the finitary finiversalist catechism:
What exists? Only what can be communicated about.
What can be communicated? Only finite descriptions.
What is infinite? Only behaviors of finite machines.
What are infinities? Only summaries of infinite behavior.
Do infinite or infinitesimal things exist? Only while they summarize behavior faithfully.
Notes
And its input, and the model of computation in which it’s interpreted, but those are finite too.↩
This ordering is not, in general, computable, because checking whether a given Turing machine is the smallest that emits a given (infinite) object is not computable. But while I don’t know offhand whether every computable set of computable objects can be computably well-ordered, this is a technical rather than philosophical question.↩
One can, of course, study sets as formal entities in their own right—“the unique Dedekind-complete totally-ordered field extension of the rationals” is itself a mental object with reproducible properties, even if nearly all of the limit points it consists of are not. But how useful of a mental object is it?↩