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 http://www.youtube.com/course?list=EC5A714C94D40392AB&feature=plcp 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.