Difficulties with real numbers

This is an important topic on which I will have a lot to say. Today, let’s just gently introduce a big stumbling block for modern analysis.

There are several approaches to the current theory of “real numbers”. Unfortunately, none of them makes sense. One hundred years ago, there was vigorous discussion about the difficulties, ambiguities and even paradoxes. The topic was intimately linked with 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 mathematics community, 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.

A hundred years ago, the notion of the “continuum” appeared intuitively straightforward, but it was difficult to pin down precisely. The Greeks had struggled with irrational numbers, but the decimal number approach of Simon Stevin in the 16th century seemed reasonable, especially considering that in practice the further digits beyond the three dots in pi=3.14159263… are hardly of practical significance. Great mathematicians like Newton, Euler, Gauss, Lagrange and others were always interested in a combination of applied questions (physics and astronomy mostly) along with more theoretical questions. In the later part of the nineteenth century an increasing preoccupation with trying to pin down the fundamentals of analysis led to both more careful definitions but also to the realization that the default view of irrational numbers as infinite decimals was shaky.

However with the advent of relativity theory and quantum mechanics, the concept of the continuum again became murky: if time is relative and perhaps finite in extent, and space has an inherent graininess which is not infinitely divisible, then what exactly are we modelling with our notion of the `infinite number line’?
While engineers and scientists viewed real numbers primarily as “decimals which go on till we don’t care anymore”, nineteenth century mathematicians introduced the ideas of  “equivalence classes of Cauchy sequences of rational numbers”, or as “Dedekind cuts”, or sometimes as “continued fractions”. Each view has different difficulties, but always there is the crucial problem of discussing `infinite objects’ without sufficient regard to how to specify them.

The twentieth century saw an entirely new sleight-of-hand; the introduction of “axiomatics” removed the time-honoured obligation of defining mathematical objects before using them. This was a particularly unfortunate and wrong-headed turn of events that has done much to diminish the respect for rigour in modern mathematics.

A finite sequence such as S=1,1,2,5,14,42,132,429,1430 may be described in many different ways, 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 the sequence of Catalan  numbers C=1,1,2,5,14,42,132,429,1430… (sequence A000108 in Sloane’s Online Encyclopedia of Integer Sequences (OEIS) at http://oeis.org/ ) the situation changes dramatically. It is never possible to explicitly list “all” the elements of such a “sequence”; indeed it is not clear what the word “all” even means, and in fact it is not clear even what the term “infinite sequence” precisely means. (Do you think you have a good definition?)

But assuming for a moment that we have some idea of the terms involved: still we are obliged to admit that in the absence of a complete list of the elements, we can specify the Catalan sequence C essentially only by giving a rule which generates it. A quick look at Sloane’s entry for the Catalan numbers shows some obvious problems: which of the potentially many rules that generate the Catalan numbers are we going to use? How are we going to tell when one rule actually agrees with a seemingly quite different one? Is there some kind of theory of `rules’ that we can apply to give meaning to the generators of a sequence?

If we think in terms of computation, an infinite sequence can also be modelled by a computer program, churning out number after number onto a long tape (or these days your hard drive). At any given point in time, there are only finite many outputs. As long as you keep supplying more tape, electricity, and occasionally additional memory banks, the process continues. 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 the same  essential difficulty with infinite processes: the program that generates a given infinite sequence is never unique. There is no escape from this inescapable fact, and it colours all meaningful aspects of dealing with `infinity’. It seems that any proper theory of real numbers presupposes some kind of prior theory of algorithms; what they are, how to specify them, how to tell when two of them are the same.

Unfortunately there is no such theory.

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 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 `God’ or `the hereafter’ are not generally recognized as proper scientific entities. Both infinite sets, God and the hereafter 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.

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 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, we need to understand mathematics in the right way before we will be able to unlock the deepest secrets of the universe.

By reorganizing our subject to be more careful and logical, and by removing dubious `axiomatic assumptions’ and unnecessary philosophizing about `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 ultimately rewarding task of restructuring mathematics properly.

13 thoughts on “Difficulties with real numbers

  1. T.Tobler

    Should you not include natural numbers of sufficient magnitude to also be included in your set of “distasteful” numbers — let’s call these SFN = “superfinite natural” numbers? To be precise, X is in SFN if it was not generated in any explicit form in any finite base by any computing device (humans included), within the space time region centered at the middle of the planet earth at 100,000,000 BC, and extending G lightyears in all directions, where G is Graham’s number.

    Obviously, SFN is defined empirically rather than intrinsically, however, it should be reasonable to assume that N – SFN should be finite — It seems inconcievable to me that we would figure out how to build a computation device, the number of instructions / digits it can execute/output exceeding all bounds within a finite time period, or the means to create a number computation devices exceeding all bounds within some finite space. I believe it should be reasonable to assume we will never achieve such a exception.

    It should also be possible express (probably itself in SFN) a number as an arithmetic expression on G that, if solved, would gives an upper bound to the count of all numbers that are in N but not in SFN.

    Unless I misunderstand you then, the problems / choices you present with Q vs R are symmetric on ( N – SFN ) vs N. To me, there seems a clear correlation between this and your objection that we must “pretend” to be able to treat infinity as a mathematical object …if we “pretend” that we had more time than our realistic expectations on existence would allow for.

    Hence, as to proposing which number system we should study, it seems you would be more consistent by arguing that pure mathematics should focus only on the non-SFN numbers, those that could possibly be expressed by a computation device (feel free to change the limit G to some other suitable number for this purpose). This excludes N, Z, Q, R, and leaves only the finite groups for study… it seems a lot of very interesting mathematics is missed with this approach, and would be similar to missing some of the very interesting mathematics that can only be examined with ZFC.

    Or, perhaps, your objection is one more of the lack of care that mathematicians apply to proofs, where they assume ZFC unnecessarily? E.G, Euclid delayed quite a while before he started utilizing postulate 5 (If I recall, his first ~25 axioms were proven in natural geometry?), even though some of the earlier proofs are much easier, and more natural, with P5 compared to the machinations one must make in neutral geometry…? (When mathematicians actually looked at the problem of P5 assumptions, they discovered entirely alien, but “interesting” geometries… — you can apply the initial theorems of euclid in many of these primarily because he did not use P5 — is this your point — that we should be more careful in stating the assumptions in proofs, or trying to find alternate proofs in ZF/C?)

    1. M. Beale

      Any natural number of any magnitude can be represented simply by its digits, which are finite.

      So there are no superfinite natural numbers.

      I don’t think “distastefulness” is the issue here, but firm foundations. (Not sure if you were making a point with that word, or just were adding color.)

      1. Trent Tobler

        > Any natural number of any magnitude can be represented simply by its digits, which are finite.

        And how do you propose to express a number so large it is beyond any chance of us expressing it or looking at it or evaluating properties on it directly? NJ Wildberger discussed this to some level in several videos, the example being the finite tower of 10’s exponent + 21 , and asks about the prime factors of this number. The primes of this composite number are almost certainly so large that even if you wrote down the digits at the scale of the planc constant, there won’t be enough room in the visible universe to write it down.

        This is what I expressed about the idea of “distastefulness” quality here — do you allow these numbers to “exist” (under some definition of exist), where the only access to these are via an axiomatic approach? Or do you exclude them, and take a more constructionist approach?

        As I allude, the limits of the constructionist approach will almost certainly not be deterministic, but we should be able to specify some upper and lower limits to the range in which this discontinuity in “distasteful numbers” would begin to occur.

  2. Mark Botirius

    Dr. Wildberger,

    In the fall semester of 2011 I took an introductory course in philosophy at Owens Community College in Ohio. One day, my instructor was going around the room asking each student the question “what do you know?”. Like all good philosophy professors he was challenging us to think outside our comfort zone. I watched as he easily cast doubt an each reply, but I was certain my answer was going to be the only solid reply given. So when he asked me, “what do you know?”, I replied, “I know that “a” cannot be “a” and “non a” at the same time and in the same sense.” To which he answered, “What about light?, is it a particle or is it a wave? What about Godel’s Incompleteness Theorem, or the pardoxes of Zenos? Of course, I disagreed even though I had never heard of Godel or his theorem. And so he told me to write a paper defending my position, and it was then that I came face to face, for the first time, with the very issues you mentioned in this post.

    I learned a great deal while researching my paper, and I have my philosophy instructor to thank for that. I remember trying to get out of writing the paper halfway through. “How about I write a nice comparison and contrast paper between objectivism and utilitarianism?” I remember asking. “No.” he replied, “I like this topic, I think you can do this”. So I soldiered on.

    Since then I wondered about other mathematical “paradoxes” and I found your channel while surfing through utube videos of people trying to convince me that point nine repeating equals one. To me, your channel is the light of reason in a world trying to destroy reason itself, and I am relieved. I am truly horrified when mathematics, the pinnacle of proof and logic, is used to destroy the very thing that enables us to differentiate between what is true or false. You bring mathematics back onto solid ground, and I am grateful. I sincerely thank you for what you do.

    I posted my paper recently online at markbotirius.blogspot.com. I would be honored if you could find the time to read it. In it I discuss Godel’s theorem and Cantor’s diagonal argument as it applies to infinity. Thank you for your time.


    Mark Botirius

  3. Aaron Michaux

    Don’t buy it. You can use induction to show that two generators produce the same infinite series. There is a mathematical language for proving that an algorithm will produce a particular series. (Formal methods aren’t really that popular in computer programming.) Sure induction means that you have to deal with “infinity” — whatever that is — but here we enter what programmers like myself call “the gap of formality”, and in a very subtle way. You see, there will be always be agreed upon axiomatic assumptions that mathematicians use, but when we ask what is being *modeled* then we surely are asking about a relationship to an informal world. And there will always be a gap, because our minds just can’t cross every lower-case j when it comes to specifying a formal rule about the real world. But there is also a far deeper problem with what you are saying: where do these “rules of logic” come from anyway? Why does 1+1=2? I’d argue that they are related to brain processes that are present in almost everyone we talk to in everyday life. Different species — different cognitive experience — maybe different mathematical axioms if any. So perhaps 1+1=2 is an evolutionary accident, or perhaps there is some sort of mathematical realism going on, and the mind is modeling something veridical. However you slice it, one cannot escape that the “rules of logic” are constructed in the mind in a very real way — a human phenotype. The number 1 is an abstract idealization of sorts. Heard of the uncarved block? One what! There is just no such thing. In fact, there is no guarantee that the rules of logic are any more real than any other mental construction. There is no guarantee that our mental landscape should be self-consistent, and we know it is not, logically speaking, because of paradoxes. So when you encounter a paradox, one part of the mind will always complain when the other becomes ascendant. That’s life, and that’s my cognitive-science point of view on why I think the problems with real numbers are no big deal.

  4. Dan

    “some kind of prior theory of algorithms; what they are, how to specify them, how to tell when two of them are the same.

    Unfortunately there is no such theory.”

    It’s called computer science and while infinity may not exist, computer science definitely does.

  5. iosephusgranicae

    After reading your ideas toward real number, I recalled that once I read an article on the Internet
    where the author claim probability doesn’t exist, and I had this kind of idea since long time ago, that the concept “probability” may share the same difficulity with the concept real number.

    Probability involves a sequence of outcomes, and it’s claimed that the frequency of outcome will (almost surely) converge to a certain value called probability. The a difficult problem rises: how can we prove that the probability of an event exist, or how can we assign a probability to a given event?

    On one hand, we’re unable to toss infinite times of coins and look for the probability value. on the other hand, after any toss of coin, no matter the outcome frequency is, there is always two possibilities: the frequency is close enough to the probability, or you’re “lucky” enough to get a frequency much deviated from the probability. So nothing can make one assure that the probaility of getting a head or a tail is 1/2.

    And the modern definition given by Kolmogorov (which himself is an analyst) uses the measure concept, and says the probability of events can only be assumed apriori. The concept “convergence in probability” says nothing about the rate of convergence, and how long we can get a frequency close enough to the probability.

    I’m quite curious about your opinion on probability, and how can we define it reasonably.

  6. iosephusgranicae

    In spite of that, I believe the theory of real numbers itself, as a thought model, is self-consistent, although no-one has proven or unproven this statement. The only issue is that it may not satisfy the intuition of ongiong sequences (or any other objects), and it may not represent perfectly the physical intuition of a perfect continuum in the nature.

    Every statement in the real number theory, when considered as pure logical formulas involving universal and existential quantifiers, may be exactly true if we comprehend them in another way and another perspective, instead of thinking them as a statement about “collections of infinity amount of things” or “continuous changings/movements from one position to another”. But which way is the best is the problem, and maybe that way is beyond one’s usual comprehension.

  7. Cargo

    “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.”

    But haven’t your heard that the essence of mathematics lies in its freedom? Math is not only primitive finger-counting but also art that expands consciousness!

  8. iosephusgranicae

    Hello Professor Wildberger. Recently I came up with a question related to whether real numbers exist.

    If real nunbers don’t exist, then does that mean that something — like the ‘actual’ length of a wall, the ‘precise’ weight of an apple or the ‘exact’ volume of water inside a lake — are all meaningless concepts? If they don’t have meaning, then what other kinds of concepts can replace them (or replace the function of them)? I’m sincerely hoping for your answer if you have any idea on it.


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