The problem of the continuum

Mathematics throughout its history has wrestled with a major schism: between the discrete and the continuous. In the earliest times this was the difference between arithmetic and geometry.

Arithmetic ultimately comes down to the natural numbers 1,2,3,4,5,6…. These are so fundamental and familiar that most ordinary folk don’t see much point in `defining’ them. But us pure mathematicians like to ponder such things, and it is fair to say that the issue is still open to further insights coming from programmers and computer science. One way or the other natural numbers are symbols that we write down to help us count; the number of apples in a bushel, children around a campfire, stars in the sky.

Geometry, on the other hand, ultimately comes down to points and lines. It is not so easy to say exactly what these are. In the 19th century mathematicians started to acknowledge that the bible of geometry—Euclid’s Elements—didn’t deal adequately with this issue. Things were okay as long as you just assumed you knew about points, lines, and the plane; and accepted various physically obvious properties they satisfied.

Fortunately Descartes and Fermat some 300 years earlier had constructed a framework—the Cartesian plane—which allowed geometry to become subservient to arithmetic: a point is an ordered pair [x,y] of numbers, and a line is an equation of the form ax+by=c. This was a wildly successful conceptual leap. It allowed algebraic techniques to bear on higher order curves, like conics or cubics, gave a straightforward and uniform treatment of many geometrical problems, and led to the development of the calculus.

But there was a heavy price to be paid for this arithmetization, which was mostly unacknowleged for centuries. The precise and logical arithmetical form of geometry which Descartes’ system gives us has curious aspects that diverge from our everyday physical experience. No longer do two circles which pass through each others centers meet. We cannot guarantee that a line passing through the center of a circle meets that circle.

This explains, I believe, why Euclid shied away from an arithmetization of his geometry: he knew that standard geometrical constructions yielded `irrational’ numbers whose arithmetic he did not understand. The Greeks’ numerical system was cumbersome compared to our Hindu-Arabic system, they had no good notation for algebra, and they considered geometry more fundamental. So ultimately Euclid choose to consider a `line’ as a primitive object which need not be defined, and carefully avoided using distances and angles as the main metrical measurements. For him, logical purity trumped practical considerations.

Modern geometry has steered away from the concern and esteem for rationality of the ancient Greeks. The trigonometry we ostensibly teach in high school texts is logically half-hearted and involves a hefty amount of cheating; at the research level we have resorted to simply walking away from this challenge. Euclid would be appalled at the sad state of affairs in modern geometry, and would find it inexplicable that the majority of educated people have almost no understanding of this beautiful subject!

The problems with irrationals have been around for two and a half thousand years, and are still with us, whether we like it or not, acknowledge it or not. Deep at the bottom of modern mathematics lies a gnarled and warted toad: the lack of a true understanding of the continuum. Many (but not all!) modern mathematicians will view this statement with skepticism. We like to believe we understand the continuum—in the context of `real numbers’—and have faith that the definitions involving Dedekind cuts, Cauchy sequences, or just axiomatic assumptions, deal adequately with the problems. Unfortunately, they do not.

In my opinion, the continuum is actually much, much more complicated than mathematicians think. Our current view of the continuum is analogous to the simple-minded model of the heavens that ancient, and not so ancient, peoples had: that we live surrounded by a large celestial sphere on which the stars are pinned, and on which the sun, moon and planets move. For better or worse, the true celestial story is vastly richer and indeed more interesting than this, and so it is with the continuum.

Modern mathematics has accepted a confusion which has spread its poisoned tentacles into almost every aspect of the subject. By accepting the logically dubious, we come to accept also that some parts of mathematics are just inherently vague and obscure—that logic has its limits, and beyond that is a kind of no-mans land of convenient but arbitrary assumptions. Mathematics loses its certainty, and descends into shades of grey. This shrugging away the bounds of careful reasoning at the research level also naturally affects the integrity of mathematics education.

The reader will want some initial evidence to support these statements. Look in any modern Calculus textbook in the introductory section which purports to establish, or review, the fundamental properties of `real numbers’. Almost all resort to waffling or unwarranted assumptions, with a few honest exceptions that admit to the lack of proper foundations. Then consider how the modern computer programming community deals with `real numbers’. What you find is that they don’t, because they can’t; the rigour of their machines interferes with wishful thinking. Instead, the programmers work with floating point representations or rational number computations, which are light years away from working with `real numbers’.

So let me put some of my cards on the table: I propose to shine a clear light on this whole issue, to expose the logical confusion that underpins modern analysis, and most importantly, to provide a way forward that allows one to envision a time when mathematics will be logical and true, without waffling and the ugliness that naturally follows it around.

In my MathFoundations YouTube series at I will tackle the detailed mathematical aspects of this campaign. In this blog I hope to provide some overall framing and discussion of both the mathematical and the sociological aspects of this unfortunate delusion—a delusion that has got its stranglehold on mathematics education, as much as research level mathematics, since the beginning of the twentieth century. We need to move into a bigger, brighter, more honest space.

18 thoughts on “The problem of the continuum

  1. neena

    Arithmetization of Geometry :
    What you have written is very true. After reading todays post, I realized that, when we use arithmetical numbers(rational numbers) , only the lines with linear equations would satisfy the usual concepts of geometry, like two non-parallel lines in a plane, would always meet in a point. But the same thing cannot be said for every other curve.. Even the simplest curve like a circle does not satisfy very obvious looking properties.
    For example, two mutually ‘trespassing circles’, ( I am forced to use the word trespassing , instead of intersecting, because in the very next line, I am going to say that these circles are NOT intersecting !)
    So, coming back to the above example,
    Two mutually ‘trespassing circles may not even intersect ! Because if their points of intersections are irrationals, they won’t be intersecting as far as the rational coordinates are concerned !
    i.e. whenever we take the co-ordinates over the Arithmetical Numbers(Rational Numbers) , every curve, even the two dimensional XY plane will have holes into it, created by the absentees of irrational numbers ! In fact, the microscopic look of the XY plane will be very much like a very very fine mash.
    The one who has attempted to visualize the Dirichlet’s function,
    f(x) = 1 , if x = rational
    = 0 , otherwise , which is discontinuous everywhere
    may understand , the picture of the Geometry when it is not defined over the set of real numbers.
    Going one step further, we can assure that, the total ‘size’ of all these holes , is much more than the existing points, because the cardinality of rational numbers is , aleph nought, whereas the cardinality of the set of irrational numbers is ‘c’, the cardinality of the continuum !
    Once again, I took the help of the continuum to support my arguments ! The continuum which is taken for granted as the concept understood fully by one and all !
    This Arithmetization of Geometry opened an entirely new avenue to think about , and I am sure
    It will be interesting to hear about continuum from you.
    I am looking forward to it

  2. Jai H

    “I propose to shine a clear light on this whole issue, to expose the logical confusion that underpins modern analysis, and most importantly, to provide a way forward that allows one to envision a time when mathematics will be logical and true, without waffling and the ugliness that naturally follows it around.”

    As someone who doesn’t automatically dismiss your proposal, and wishes you to attempt or guide an attempt to create a program, may I recommend that you focus less on ‘exposing’ logical confusion, but instead on ‘providing a way forward’ and searching for unique results which may provide evidence of not just the problems but the solutions?

    Convincing people that a problem they don’t believe is a problem is a problem is a very inefficient effort. One can get a sense of this simply by reading that sentence! What’s more, you have to in essence ‘use their language’ to have the discussion, further entrenching yourself in the very matter which you are trying pry out of peoples’ minds in the first place. On a less abstract level, I caution against the use of certain language such as, ‘exposing’ the misunderstanding of others. This is often associated with quackery, which is a label which I’m sure you would like to avoid. Such language could become a barrier to looking at your work without bias for many. Finally, effort is best spent generating concrete (in a nod to D. Knuth’s terminology) results which illuminate new understanding. New results, proofs, and formal systems will ultimately be the tools people use to dislodge old ideas from their own minds. Rarely is it simply the passionate urging of another.

    You have not solicited my advice or sought opinion, but as someone who wishes you to succeed, I hope you are not offended and simply receive my critique as encouragement (which it is). I simply offer a bit of feedback from the marketing side of the marketplace of ideas. Hopefully my comment is not seen as rude or out of place. I sincerely wish you luck, and look forward to seeing more of your lectures and results on sequences and growth.

    Separate from the unsolicited advice above:

    How do ridiculously large numbers (far too large to ever enumerate, but provably finite) such as Graham’s, or Skewes’ number fit into your logical scheme?

    1. Pedro Mascarós

      It is true that from a marketing point of view, dealing only with the problems without providing solutions, can we become , in the eyes of others, Cranks

    2. njwildberger: tangential thoughts Post author

      Hi Jai, Thanks for the interesting comments. I am very aware of the need for putting out alternatives when one is criticizing. The lack of alternatives to the status quo in modern mathematics has been largely missing, a key reason why those who might otherwise have inclination to point out logical errors and weaknesses don’t perhaps do so. I have a new way of thinking about geometry, based on Rational Trigonometry. The understanding that has flowed from this development is a key driver to my program: I can now see where algebraic arguments can replace analytic ones in geometry. As a result we have a completely new way of thinking about hyperbolic geometry, called Universal Hyperbolic Geometry, which is explained in three of my papers (they all start with the titles Universal Hyperbolic Geometry I,II,III) as well as my YouTube series UnivHypGeom (40 videos so far). Also in my MathFoundations series I am in fact laying out alternatives to the usual approaches to calculus. So increasingly I hope people will see that there are options.

      1. gcina

        To me it seems enough to point out the problems without even providing any solutions at all. The way physics and other sciences are done is not to provide answers even when it’s not possible, which is exactly the opposite in pure mathematics. The problem of the continuum has confused me for many years, I could not at all see how a line can have an infinite number of points. I tried to picture the abstract line of the mathematician and still I could not understand what are these points they were talking about.

        These mathematicians point to you that there exists a set of real numbers, and then it is up to you how to picture it (if you are supposed to picture it at all). I’m really convinced by Prof. Wildberger that the definitions are just rhetorical ones and nothing more. You just find yourself in a kind of circular reasoning, having to rely on your own imperfect experience most of the time. Even calculus and measure theory rely on the notion of the interval and it’s length yet they pretend as if there is a continuum where there is none. Every thing they do and say rely on countably infinite sets, nothing uncountable there.

        Prof. Wildberger I don’t see why you should provide solutions because what you have done is a solution on it’s own. You’re helping us to see that the difficulties are not caused by our poor imaginations but there is just nothing these people are defining in the continuum. They wish they could come up with nice definitions for everything but they can’t, whether they use axioms to hide the difficulties the fact will remain that there problems are still there. I don’t see any reason why they are forcing such impossible things to exist at all. I think their freedom in the world of axioms has reached the domain of religion indeed.

        Thanks, for telling these people that what they are doing can not be done. It is just wishful thinking all the way. It has been seen by great mathematicians before, that it is much more difficult than they can imagine. All they are doing is that they are confusing us who are interested in mathematics with their definitions.

      2. njwildberger: tangential thoughts Post author

        Thanks for a very nice comment! Actually to be honest I enjoy the challenge of trying to figure out how things really ought to go, assuming we are aiming for 100% honesty. There are a lot of interesting things to discover along the way.

  3. johnpetrie

    I’ve studied math at the graduate-school level, though not nearly enough to become a professional mathematician, but I wonder if some of your doubts put forth in this post are in line with one major doubt that I’ve had for years: that the rational numbers really have measure 0 in the real numbers. I’ve read analogies or layman’s arguments in support of the mainstream view that seem pretty convincing, but on the other hand, there are infinitely many rationals between any two irrationals as well as infinitely many irrationals between any two rationals, which makes me think (on an informal, layman’s level) that one is not more “abundant” than the other. I gather that I am probably conflating the concepts of “dense” and “measure” and that intuition isn’t a good substitute for rigor, but I just wondered what you thought about this.

    1. njwildberger: tangential thoughts Post author

      Actually I don’t believe in “irrational numbers” at all. They are a figure of the pure mathematical imagination. Please watch my Math Foundations YouTube series, user njwildberger, or read my paper “Set theory:Should you believe?”

  4. SirTruffleberry

    While I personally have no qualms with infinite sets, I do see both the practical and academic interest in finitism. That being said, I find some of your positions confusing and would appreciate clarification.

    Firstly you seem fond of the rationals, but there is no avoiding their infinitude. I suppose you could avoid collecting all of them into a set but this makes defining functions of the rationals awkward. Do we only consider finitely many rationals at a time in their domains? Are you proposing an alternative to the very notion of function? It is hard for me to tell.

    If we do only use finite subsets then we lose ring/field structure of the rationals and we cannot consider morphisms we may want to study. And besides, the arithmetic of finite, commutative rings/fields is already well-understood, no? This seems to restrict further research.

    As for more analytical concerns: I can answer a question like “What should be the (rational) dimensions of a square garden if I want it to be within epsilon of 2 square units?” But to solve this problem for arbitrary epsilon, I must acknowledge the density and therefore infinitude of the rationals.

    Now if you are indeed right that the foundations of math are dubious then any shortcomings of the rationals doesn’t detract from that. But it seems to me that to speak of things like the field-like structure of the rationals or their ability to approximate “real-world” measurements arbitrarily well, you are still implicitly using their infinitude. So this hand-tying appears to me to be all for naught.

  5. Peter Tobias

    You write: “I believe, why Euclid shied away from an arithmetization of his geometry: he knew that standard geometrical constructions yielded `irrational’ numbers whose arithmetic he did not understand. The Greeks’ numerical system was cumbersome compared to our Hindu-Arabic system, they had no good notation for algebra, and they considered geometry more fundamental.”
    That seem FOUR reasons to me: (1) the arithmetic of irrational numbers, the inferiority of the Greek system of expressing (2) numbers and (3) equations compared to 16. century Europe, and the assumed superiority/noblesse of geometry (4). While I agree with (2), (3), and (4), do you have some evidence for (1)? If there is no evidence, then (2), (3), and (4) seem sufficient to explain why the Greeks never did geometry on the number plane.

    1. njwildberger: tangential thoughts Post author

      Sure, the entire work of Euclid is a careful attempt to avoid getting to grips with “irrational numbers”. These days that term is used without quotes by most people, the way “transcendental beings” might be used in religious cults. Euclid avoids “length measurement” and we may surmise that it is at least partially because of the Pythagorean distaste for square roots as numbers. And he avoids “angle measurement” because he wants to steer clear of “pi”. Curiously these days “length” and “angle” are considered the fundamental metrical ingredients in Euclidean geometry. Euclid would have found this absurd.

  6. Peter Tobias

    You: And he avoids “angle measurement” because he wants to steer clear of “pi”.
    – Why would angle measurements lead to pi? Was the radian used as unit for angle measurement by the ancient Greeks? I would have expected them using 90 degrees for a right angle. (To come up with 180 degrees for the sum of angles in a triangle.)

  7. njwildberger: tangential thoughts Post author

    You are right that the Babylonian angle measurement at first avoids “pi”. However once you start wanting to express lengths of sides of right triangles in terms of the angles at a vertex, irrationalities manifest themselves. For example a right triangle has an angle of 23 degree. What are then the side lengths of this triangle if the height is say 1, or if the hypotenuse is say 1. Another such question: what is the area of a segment of a unit circle formed by two radii making an angle of 10 degrees? Once you introduce angles, the usual difficulties with trigonometric ratios is unavoidable, and Euclid knew that these are generally not accessible with exact arithmetic — witness the approximate nature of almost all trig tables (with the exception of Plimpton 322 and Viete’s tables 3000 years later).

  8. iosephusgranicae

    I have been more and more suspecting that Newton and Leibniz had lead mathematics to a wrong way, who introduced an unnecessary concept, the infinitesimal, and made all mathematicians “waste” their paper and braincells around it. If there existed a world without calculus, that would also probably be perfect.

    Computer simulations of physical events use discrete arithmetics, only using finite decimals with addition, subtraction, multiplication and division, and there’s no performing of differential and integral inside it at all. The success of computer simulation might make people believe the world is really discrete rather than indefinitely divisible. So I wonder if there still be any points of calculating limits, differentials and integrals.

  9. Cargo

    I can imagine Kronecker, Brouwer and Wildberger marvelling all the shadows in the Plato’s cave, under the infinitely fruitful Cantor’s Paradise 😀


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