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The problem of continuum versus discreteness seems to be related to the issue of infinity and finiteness. The number of points in a line served as the logical floodgate which led to the development of Set Theory by Cantor at the end of the 19 th century. It took almost another century to demonstrate the problematic nature of some of Cantor's thinking (Cohen completed Godel's work in 1963). But continuity can be finite and the connection is, most times, misleading rather than illuminating. Intuition tells us that the world is continuous and contiguous. This seems to be a state of things which is devoid of characteristics other than its very existence.
And yet, whenever we direct the microscope of scientific discipline at the world, we encounter quantized, segregated, distinct and discrete pictures. This atomization seems to be the natural state of things - why did evolution resort to the false perception of continuum? And how can a machine which is bound to be discrete by virtue of its 'naturalness' - the brain - perceive a continuum? The continuum is an external, mental category which is imposed by us on our observations and on the resulting data.
It serves as an idealized approximation of reality, a model which is asymptotic to the Universe 'as it is'. It gives rise to the concepts of quality, emergence, function, derivation, influence (force), interaction, fields, (quantum) measurement, processes and a host of other holistic ways of relating to our environment. The other pole, the quantized model of the world conveniently gives rise to the complementary set of concepts: quantity, causality, observation, (classic) measurement, language, events, quant's, units and so on. The private, macroscopic, low velocity instances of our physical descriptions of the universe (theories) tend to be continuous. Newtonian time is equated to a river. Space is a yarn.
Einstein was the last classicist (relativity just means that no classical observer has any preference over another in formulating the laws of physics and in performing measurements). His space-time is a four dimensional continuum. What commenced as a matter of mathematical convenience was transformed into a hallowed doctrine: homogeneity, isotropy, symmetry became enshrined as the cornerstones of an almost religious outlook ('God does not play dice'). These were assumed to be 'objective', 'observer independent' qualities of the Universe. There was supposed to be no preferred direction, no clustering of mass or of energy, no time, charge, or parity asymmetry in elementary particles.
The notion of continuum was somehow inter-related. A continuum does not have to be symmetric, homogenous or isotropic - and, yet, somehow, we will be surprised if it turns out not to be. As physical knowledge deepened, a distressful mood prevailed. The smooth curves of Einstein gave way to the radiating singularities of Hawking's black holes. These black holes might eventually violate conservation laws by permanently losing all the information stored in them (which pertained to the masses and energies that they assimilated).
Singularities imply a tear in the fabric of spacetime and the ubiquity of these creature completely annuls its continuous character. Modern superstring's and super membranes theories (like Witten's M-Theory) talk about dimensions which curl upon themselves and, thus become non discernible. Particles, singularities and curled up dimensions are close relatives and together seriously erode the tranquil continuity of yore. But the first serious crack in the classical (intuitive) weltanschauung was opened long ago with the invention of the quantum theoretical device by Max Planck. The energy levels of particles no longer lay along an unhindered continuum. A particle emitted energy in discrete units, called quanta.
Others developed a model of the atom, in which particles did not roam the entire inter-atomic space. Rather, they 'circled' the nucleus in paths which represented discrete energy levels. No two particles could occupy the same energy level simultaneously and the space between these levels (orbits) was not inhabitable (non existent, actually). The counter-continuum revolution spread into most fields of science.
Phase transitions were introduced to explain the behaviour of materials when parameters such as pressure and temperature are changed. All the materials behave the same in the critical level of phase transition. Yet, phase transitions are discrete, rather surprising, events of emergent order. There is no continuum which can accommodate phase transitions. The theory of dynamical systems (better known as 'Chaos Theory') has also violated long held notions of mathematical continuity. The sets of solutions of many mathematical theories were proven to be distributed among discrete values (called attractors).
Functions behave 'catastrophically' in that minute changes in the values of the parameters result in gigantic, divergent changes in where the system 'settles down' (finds a solution). In biology Gould and others have modified the theory of evolution to incorporate qualitative, non-gradual "jumps" from one step of the ladder to another. The Darwinian notion of continuous, smooth development with strewn remnants ("missing links") attesting to each incremental shift - has all but expired. Psychology, on the other hand, has always assumed that the difference between "normal" and deranged is a qualitative one and that the two do not lie along a continuous line. A psychological disorder is not a normal state exaggerated. The continuum way of seeing things is totally inapplicable philosophically and practically.
There is a continuum of intelligence quotients (I. Q. s) and, yet, the gifted person is not an enhanced version of the mentally retarded. There is a non-continuous difference between 70 IQ and 170 IQ.
They are utterly distinct and not reducible to one another. Another example: "many" and "few" are value judgements or cultural judgements of elements of a language used (and so are "big" and "small"). Though, theoretically, both are points on a continuous line - they are qualitatively disparate. We cannot deduce what is big by studying the small unless we have access to some rules of drive...
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