We can conceive of both kinds of structure; for example, we can count discontinuously (1, 2, 3,…) and measure continuously (never, or hardly ever, an exact counting number of centimeters or inches, or neat rational fractions thereof). But is the external world basically continuous or discrete?

Maybe it depends on which feature of the external world we are considering. Matter seems discontinuous down to molecules and atoms, as found by the early 18th century chemists with their laws of combination corresponding to small whole numbers. Never mind that prior to Lavoisier et al., solid and liquid matter (condensed phases) seemed quite continuous in naive perception. Sure, Democritus the Roman postulated atoms, but he really had no evidence, and the common folk of his time probably never heard of him.

So now we know about atoms (originally meaning “things that cannot be cut or further divided”), but of course we have smashed them (partly to our sorrow) to smaller particles. We have come to know that atoms consist of protons, neutrons, and electrons, and even that the first two consist of quarks and gluons. Did that give us pause? For various reasons, some that have to do with large scale cosmogenesis and some with the tiny-scale quantum world, we now postulate various versions of superstrings and M-branes, much tinier, by several orders of magnitude, than quarks and electrons. will this process of subdividing ever stop? Who knows?

Light, and electromagnetic radiation in general, was first characterized as waves, in that same 18th century dawn of modern science. Later (in the early 19th century), Albert Einstein found that light comes in tiny packets called photons or quanta. Soon after, electrons too were found to have complementary waves, making us wonder if matter and energy, electrons and photons, fermions and bosons, were really interconvertible, in some “supersyrometric” sense. So was radiation (and matter} continuous like waves or discontinuous like particles? The answer “both” given by Niels Bohr is not really satisfying.

What about numbers? That’s where we began. The counting numbers are discrete, so are negative numbers and zero (integers), so are rational numbers (the above plus fractions).

The problems come in with irrational numbers, like the square root of 2 or pi. Do the very numerous irrational numers between the fractions “touch” each other, and so make the number line continuous? No, because the “last” (though infinitely distant) numeral of an irrational number is an ordinary natural number.

George Lakoff and Rafael Nunez deal with this question in their book “Where Mathematics Comes From”. (Basic Books, 2000.) There are apparently two kinds of mathematics. The intuitive perceptions of geometry (point with no dimensions, line without width, surface without depth) belong to the continuous perception, but modern “discrete mathematics” tries to transform it to the discontinuous realm by defining certain axioms. According to the latter, which defines numbers as “sets”, real numbers are all there is on the real number line, but the line is a set of discrete points. According to the continuous mathematics, there is an infinite number of infinitesimal numbers between every pair of adjacent irrationals, which makes the number line continuous. Again, which version of mathematics corresponds to the external world? The answer “both” again is not satisfying.

There are also certain fractal curves that are said to “fill space”. Again, that is true according to one kind of mathematics, not the other.

What about space and time? In naive perception, again, they seem continuous, a framework for objects and events. However, when we get to extremely tiny dimensions, the so-called Planck dimensions (10 to the minus 45 for time and 10 to the minus 30 for length), space-time tears apart into a sort of a foam, physicists tell us.

And there is truth. In ordinary logic, a statement or proposition is either true or false, a simple binary choice. Yet along come statements like the Cretan liar, or simply “This sentence is false”, which, if true, are false, and if false, they are true. The structure is somewhat like that of an electric buzzer, which keeps on alternating rapidly between closing and opening the circuit. We have to admit that some statements can be half-true and half-false, like the Cretan liar paradox which figures so strongly in Goedel’s proof of either the incompleteness or the contradiction in the structure of mathematics. The logical structure which includes the Cretan liar paradox (Bertrand Russell would exclude it) is called “fuzzy logic”. (There might be other fractional “truths” besides one-half.) So is logic in the external world (in Penrose’s “Platonic sphere”) clear or fundamentally fuzzy? If we make logic (or truth) continuous, the Platonic sphere must be fuzzy.

Or is there no such thing as the Platonic sphere, and are numbers and logic and space and time and matter and energy only fictions in the mind-brain, as Lakoff argues? Kant too spoke of space and time as only mental categories. But the mental structures that the mind weaves in its metaphors must to some degree correspond to structures in the environment, or we and our ancestors would not have survived in that environment. It is another type of an anthropic principle.

So in the end, I am led to argue for a Fuzzy Platonic sphere, and continuity, though with many provisos.