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.

5 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.


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