Where does divisibility end ?

From late in the nineteenth century to early in the twentieth century, there was a brief island of resolution in one of the oldest and least tractable philosophical questions: how divisible is the continuum ?

The triumphant success of Cantorean set theory, building on Gauss's limit techniques for avoiding infinitesimals, was the development of a rigorous way of allowing the continuum to be made up of indivisible yet distinct points yet still retain its continuity. The twentieth century undermined that, via Gödel, thanks to its extensive and inescapable dependence on the use of reductio ad absurdum, along with its more ambitious cousins, which makes the post-Gauss synthesis unacceptable to constructivists.

That might be a baffling mess of words, if you haven't studied mathematics, philosophy, the philosophy of mathematics and the history of all three: I hope, in what follows, that I can manage to clarify.

Space vs matter

Ancient greek philosophers, the Muslims who synthesised their ideas into a frame-work compatible with monotheism and the Europeans who imported this synthesis into a Christian context (all the while playing down the Muslims' contribution in favour of the authoritative greek source) all struggled with the question, in their natural philosophy (which, since the seventeenth century, has come to be known as science, albeit via some substantial re-working), of the divisibility of, on the one hand, space and, on the other hand, matter.

Euclid's geometry allows that one can sub-divide any interval of a line into arbitrarily many equal sub-divisions. Splitting evenly in two is easy; but any number of equal parts is possible.

To sub-divide a line segment into seventeen equal parts (or 200560490131 of them, if you prefer), first construct a line parallel to the given one, but offset from it (Euclidean geometry gives a procedure for doing this); then, on this new line, mark off some interval (try not to use one at all close to one seventeenth of the length of the segment to be sub-divided, or 1/200560490131 as the case may be); then, continuing along that line, mark off duplicates of that same chosen interval, until you have seventeen (or 200560490131) of them all contiguous. Now draw a line through the start-point of the first of these (looking along the two parallel lines from some chosen direction) and the start-point of the segment to be sub-divided; likewise, draw a line through the end-point of the last and the segment's end-point; extend these until they meet. (If they don't meet, the interval you chose to repeat along the constructed parallel line was too close, in length, to one seventeenth – or 1/200560490131 – of the original segment; construct a fresh parallel line and use some longer or shorter interval. Let this newly constructed line take the place of the one previously introduced.) From the point where they meet, construct a straight line to each of the end-points of copies of the interval you duplicated along the constructed parallel line. These new straight lines, suitably extended if necessary, meet the original line segment in the points that sub-divide it evenly into your chosen number of parts.

So geometry believes in the indefinite subdivisibility of space; but if you actually want to perform the above-described construction, with a large number of sub-divisions, you may have trouble – unless the initial segment is very long. If you really want to sub-divide a length into 200560490131 equal parts, by the method just given, you're going to be in trouble unless the line segment to be sub-divided is significantly more than ten metres long. The ancient greeks wouldn't have been so specific (I multiplied the given big number by the Bohr radius and rounded down) about the threshold length; but they understood that there were practical limits to what one could actually do. They didn't have much time for practicalities, though, so they didn't worry about this; in principle geometry understood the construction, even if our command over matter wasn't adequate to effect the construction. Modern physics has a much more concrete sense of the limits of what effects can concretely be achieved, but its underlying mathematics still believes in the arbitrary divisibility of the underlying continuum.

So much for the continuum – the space that matter inhabits – but what about matter itself ? As far back as our knowledge of the discussion goes, there have been some who suppose matter is as divisible as the continuum and others who believe that matter (as distinct from the space it inhabits) has irreducible minimal parts, that cannot be sub-divided. Refinements on this position have also existed as far back as we know what folk thought: there may be parts that are essentially indivisible, or there may be parts that are minimal (no subdivision of it is capable of separate existence) yet describable in terms of sub-parts. This may seem a strange subtlety, but it allowed Muslim philosophers a millennium ago to resolve philosophical difficulties with material indivisibility contrasting with continuity of potential spatial movement: and it is a real feature of modern physical theory, in which assorted particles (including proton and neutron) are made up of quarks, which are not capable of independent existence.

I could go on at length about the conceptual problems of discrete matter in a continuum universe – and I would be recapitulating arguments from one, two and probably more millennia ago – but the essential argument for the atomist position (that matter has indivisible parts, even though the positions at which those parts may appear form a continuum which is indefinitely divisible) is that the alternative demands that we believe in continuum matter; in which we are forced to wonder how some kinds of matter possess different properties than others. If matter has indivisible parts, the differences between the diverse kinds of matter can be accounted for by differences in their indivisible parts – wax differs from sugar because they are made of different molecules.

Continuity and division

Euclid happily allowed lengths to be sub-divided arbitrarily; it seems natural enough in geometry. Yet, even here, there are problems. Consider a circle, crossed at some point by a radial line through its centre; construct a line at right angles to this radial line, through that point. Coming towards the circle from either end of that line, we are outside the circle but getting closer to it, until we finally meet it, then move away from it – yet on the same side. Both the circle and the straight line are continuous (they have no angled kinks in them) yet they meet without crossing; when two lines do that, they are said to be (mutually) tangent, which is just the Latin for touching.

Now ask the question: how big is the circle's intersection with, respectively, the radial line we started with and the tangent we constructed at right angles to it ? Draw the diagram and you shall rapidly see that the tangent meets the circle in more than the radius does; yet, follow Euclid and you will be forced to accept that neither is more, neither is less. The point where the radial line cuts the circle is indeed on the tangent line. If we mark off, on the tangent, a segment of length equal to the circle's radius, with one end at the point where tangent, radius and circle meet; then subdivide this into any whole number of equal parts that we care to chose; the one of these that ends at the meeting-point has its start emphatically not on the circle. Euclid only allows us to discuss positions on the tangent that are distant enough from where it meets the radius that the segment thus delimited would, if copied a large enough whole number of times, accumulate a total length that exceeds the circle's radius; yet my intuition insists that the circle and its tangent meet in more than the single point where they meet the radius.

Formally, Euclid has dismissed from discussion the segments, along the tangent, that are not homogeneous or rationally commensurate with the radius of the circle; to make sense of an intuition that the tangent meets more of the circle than the radius does, it is necessary to invent a hierarchy of infinitesimal segments (that is, ones so small that, no matter how huge a number of times you repeat them, you won't accumulate a length as big as the radius of the circle; that we get a hierarchy of such says that some of these tiny lengths are, none the less, bigger than others in the same way as the radius is bigger than any of them). There are ways to formalise that in mathematics (look up synthetic differential geometry or the surreal numbers for two approaches, of which the latter better suits my sense of naturality), but the formalism adopted by the ancient greeks (and essentially preserved, albeit in greatly developed form, by the post-Gauss set-theoretic analysis) specifically limited discussion to homogeneous quantities – one quantity, A, is homogeneous with another, B, if there are some positive whole numbers p, q, r and s for which: p.A > q.B yet r.A < s.B.

A thousand years ago, subtle thinkers were struggling with these questions. (I'll just call them thinkers: to distinguish science from philosophy would be an anachronism; as would any attempt to distinguish these from theology. It has aptly been noted that the distinction between philosophy and theology is an artifact of mediæval European university life.) The cornicular angle a circle forms with its tangent is of a wholly different kind to the angle between any two straight lines: just as circles of larger radius have bigger intersection with their tangents than do those of smaller radius, the latter form wider cornicular angles than the former.

See also

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