Discrete and Continuous
Discrete and Continuous
Philosophy is awash in grand dichotomies—particular and general, mind and body, fact and value, finite and infinite, being and nothingness. Reality is held to divide into two large categories and the relations between them are mapped. But there is one dichotomy that is seldom discussed by philosophers, though it is generally recognized elsewhere: that between the discrete and the continuous. These concepts are not easy to define, though they are widely accepted at an intuitive level, no doubt because they pervade our everyday experience. The discrete consists of separate, distinct, self-contained objects that can be distinguished and counted: animals, mountains, tables, cells, atoms, words, concepts, numbers, propositions, gods. The continuous consists of undivided, unbroken, uninterrupted, seamless, smooth, homogeneous…what? Not objects or things–for then they would be discrete–but what we call mediums or manifolds or dimensions or magnitudes: stuff of some sort. Space and time are the paradigms, but we also regard other things as continuous: intensity of emotion, milk and honey, geometrical figures, colors, motion, fundamental matter. Of course, things that seem continuous have sometimes been discovered to be discrete, as with the atomic theory of matter or the quantum theory of energy; but we have a clear idea of what continuity might be even in these cases. We have a commonsense concept of the continuous that meshes with our ordinary perception of things, in which discrete objects are perceived to be internally continuous (possibly falsely). We thus feel ourselves to be surrounded by two kinds of being: discrete separated entities that can be counted, on the one hand, and smoothly varying continua that can only be measured, on the other. There are the discrete objects in space and time and the continuous mediums of space and time. The latter require their own mathematics, which nowadays involves the real numbers, infinitesimals, the concept of a limit, and calculus. We employ the modern notion of a dense array of points between any two of which there is always a third (this may be viewed as a way to discretize continuity). There is even a distinctive type of paradox associated with continuity (Zeno et al). So we accept a kind of ontological dualism: two kinds of being with different essential natures. Descartes used the concept of extension to unite space and matter, but that concept papers over the deep difference between the discrete and the continuous, both of which can be said to have extension—though we should note that not everything that is continuous is physical, e.g. emotional strength. The discrete-continuous distinction cuts across the mental-physical distinction, and brings its own brand of dualism.
Like other dualisms, this one invites philosophical scrutiny. How solid is the distinction? Might we not view each as a special case of the other? Is one derivable or emergent from the other? Are there illusions of continuity and discreteness? Is it possible to be a monist with respect to one or the other type of being? For instance, we are told that in the first moments after the big bang the temperature was so high that no particles could exist, so there were no discrete objects then—they came into being only when the universe cooled. Then wasn’t physical reality entirely continuous at that early point? If so, our current discrete universe emerged from a continuous universe, rather as we suppose that mind emerged from matter (which took more than mere cooling). Might not other universes stay at that initial high temperature and never evolve into discrete universes? On the other hand, it has been maintained that continuity is a mathematical fiction—everything real consists of discrete entities with no smooth transitions anywhere. Motion is really jerky and jumpy, space and time are particulate, and the mind is purely digital. Or we could just decide to eliminate entities that don’t meet our ontological expectations: there is no such thing as motion, space and time are unreal, and there are no emotions to vary continuously. We have the usual panoply of philosophical options to choose from: dualism, monism, reductionism, elimination, and invocations of God to get over ontological humps (e.g. the miracle of discrete entities springing from continuous stuff). Our experience suggests a dualism of the discrete and the continuous, but maybe reality is not so constituted; maybe in the noumenal world all is discrete (or all continuous). Continuity certainly presents problems of understanding, and it was only in the nineteenth century that mathematicians began to be comfortable with it (but at what cost—is a smooth line really reducible to a collection of points?). And why is the universe made this way to begin with? Why the ontological division? Wouldn’t it be simpler to make a universe that was just one way or the other? Why did God introduce continuity at all, given that his main purpose was to create discrete moral beings like us? What has continuity got to do with morality? We appear to live in a mixed world, but this doesn’t seem like a logical necessity—unless it really is once you get down to basics (maybe space and time couldn’t exist without their smooth structure). It is all quite puzzling—the mark of a good philosophical problem.
That was about the metaphysics of the discrete and continuous, but there is also the epistemology. Do we know about these things in the same way? Do we perceive continuity as we perceive discreteness? How do we get the concepts? There is a kind of primitive impression of continuity in vision that exists side by side with impressions of discreteness, but what exactly does this amount to? Is it just an absence of perceptible discreteness or is it a positive sense datum in itself? Is the child’s mind a continuous visual blur until sensations of discreteness supervene? What does it mean to say that a surface looks continuous—does it look as if all potential gaps have been filled? What if we look closer and see that the object is made up of lots of little discrete entities? Were we under an illusion? But is it even possible to see a discrete object without some parts of the visual field giving an impression of continuity? The gaps between objects look to be filled with continuous space and the objects themselves look like they are composed of continuous matter. And the cognitive mechanisms that process perception must recognize the discrete-continuous distinction: they deliver different kinds of mental representation to handle the sensory input. Is consciousness itself continuous or discrete or both? Is it quantized or infinitely divisible? Are the features of the brain that account for consciousness discrete properties of neurons or continuous features? Neural firings are discrete, but electrical charges can vary as continuous magnitudes—do both contribute to generating consciousness? Behaviorism in effect treated the mind as continuous, because behavior is just a type of motion, but how does that square with the discrete character of so much of the mind, particularly language and concepts? There is no such thing as applying half a concept, but the body can move half a meter. Your utterances must be either meaningful or not, but your voice can be louder or softer. How do we derive the discrete mental notions from concepts of continuous bodily motion? That is like trying to define atomic structure in terms of motions of matter—a sort of category mistake.
The natural position to take is that the world contains two sorts of ontological structure corresponding to two types of mathematics: discrete structure and continuous structure. The former can be dealt with using finite mathematics (or the mathematics of discrete infinity), while the latter requires the infinite mathematics of the continuum. Space and time have a continuous structure, while atoms and species have a discrete structure. This is just an irreducible fact. The two coexist and intermingle. Correspondingly, we have two sorts of phenomenology and mental representation geared to these objective structures—discrete cognition and continuous cognition. These might be conceived as distinct modules located somewhere in the brain. We know how to handle continuous magnitudes and we know how to handle discrete objects. When we see an object in motion we separate it from its surroundings as a distinct individual thing (using our discrete module) and we also track its movement through space as a continuous path with no gaps or interruptions (using our continuous module). We are capable of seeing the world in both ways simultaneously. The dualism is present but it is integrated, fused. It is rather like the perception of shape and color: different properties, different perceptual modules, but a unified perception. Just as there is a division of primary and secondary qualities despite perceptual unity, so there is a division of discrete and continuous properties despite perceptual unity. We see the same object as a discrete entity and as moving through a space without internal discreteness. Phenomenology thus recapitulates ontology. The distinction between discrete and continuous deserves a place in the pantheon of philosophical dualities.
 We could call it the problem of the granular and the gradual: are both equally real, and how do they meet up? The grainy and the graded, the chopped up and the smoothed out, the lumpish and the soupy: which form does reality prefer, and how does it combine them?
 The distinction is entrenched in mathematics along with other dualities (finite and infinite, odd and even, rational and irrational, prime or non-prime); time for philosophy to catch up.
The circle plays a special role in mathematics – connecting geometry and analysis. It is in a way the smallest object that non-trivially embodies both discrete and continuous aspects. It is a boundary, it can be used as a component, but it also gives rise to discrete frequencies (smooth functions from the circle to itself are caharacterised by the number of windings they take), iteration and feedback.
Is there an analogue to the circle in non-mathematical areas – I mean a canonical entity that combines discreteness and contntinuity in a nontrivial way?