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... derivation and decision making. The same applies to the couplets: order / disorder, element / system, evolution / revolution and not alive / alive. The latter is at the heart of the applied ethical issue of abortion: when should a foetus begin to be considered a live thing? Life springs suddenly. It is not more of the same.
It is not a matter of quantity of matter. It is a qualitative issue, almost in the eye of the beholder. All these are problems that call for a non-continuum approach, for the discrete emergence of new phases (order, life, system). The epiphenomena l aspect (properties that characterize the whole that are nowhere to be found when the parts comprising the whole are studied) is accidental to the main issue. The main issue being the fact that the world behaves in a sudden, emergent, surprising, discrete manner. There is no continuum out there, except in some of our descriptions of nature and even this seems to be for the sake of convenience and aesthetics.
But renaming or redefining a problem can hardly be called a solution. We selected the continuum idealization to make our lives easier. But WHY does it achieve this effect? In which ways does it simplify our quest to know the world in order to control it and thus enhance our chances to survive?
There are two types of continuum: spatial and temporal. All the other notions of continuum are reducible to these two. Take a wooden stick. It is continuous (though finite the two, we said, are not mutually exclusive or mutually exhaustive). Yet, if I were to break it in two its continuity will have vanished.
Why? What in my action made continuity disappear and how can my action influence what seems to be an inherent, extensive property of the stick? We are forced to accept that continuity is a property of the system that is contingent and dependent on external actions. This is normal, most properties are like this (temperature and pressure, to mention two). But what made the log continuous BEFORE I broke it and discontinuous following my action and (so it would seem) because of it? It is the identical response to the outside world.
All the points in the (macroscopic) stick would have reacted identically to outside pressure, torsion, twisting, temperature, etc. It is this identical reaction that augments, defines and supports the mental category of continuum. Where it ends discontinuity begins. This is the boundary or threshold.
Breaking the wooden stick created new boundaries. Now, pressure applied to one part of the stick will not influence the other. The requirement of identical reaction will not be satisfied and the two (newly broken) parts of the stick are no longer part of the continuum. The existence of a boundary or threshold is intuitively assumed even for infinite systems, like the Universe. This plus the identical reaction principle are what give the impression of continuity.
The pre-broken wooden stick satisfied these two requirements: it had a boundary and all its points reacted simultaneously to the outside world. Yet, these are necessary but insufficient conditions. Discrete entities can have boundaries and react simultaneously (as a group) and still be highly discontinuous. Take a set of the first 10 integers. This set has a boundary and will react in the same way, simultaneously, to a mathematical action (say, to a multiplication by a constant).
But here arises the crucial difference: All the points in the Stick will retain their identity under any transformation and under any physical action. If burnt they will all turn into ash, to take a radical example. All the points in the stick will also retain their relationship to one another, the structure of the stick, the mutual arrangement of the points, the channels between them. The integers in the set will not. Each will produce a result and the results will be disparate and will form a set of discrete numbers which is absolutely distinct from the original set. The second generation set will have no resemblance whatsoever to the first generation set.
An example: heating the wooden stick will not influence our ability to instantly recognize it as a wooden stick and as THE wooden stick. If burnt, we will be able to say with assuredness that a wooden stick has been burnt (at least, that wood has been burnt). But a set of integers in itself does not contain the information needed to tell us whence it came, what was the set that preceded it. Here, additional knowledge will be required: the exact laws of transformation, the function which was used to derive this set.
The wooden stick conserves and preserves the information relating to itself the set of integers does not. We can generalize and say that a continuum preserves its information content under transformations while discrete entities or values behave idiosyncratically and, thus, do not. In the case of a continuum, no knowledge of the laws of transformation is needed in order to extract the information content of the continuum. The converse is true in the case of discrete entities or values. These conditions: the existence of a boundary or threshold, the preservation of local information and the uniform reaction to transformation or action are what made the continuum such a useful tool in scientific thought.
Paradoxically, the very theory that introduced non-continuous thinking to physics (quantum mechanics) is the one that is trying to reintroduce it now. The notion of fields is manifestly continuous (the field exists everywhere, simultaneously). Action at a distance (which implies a unity of the Universe and its continuity) was supposedly exorcised by quantum mechanics only to reappear in space-like interactions. Elaborate and implausible theoretical constructs are dreamt up in order to get rid of the contamination of continuity. But it is a primordial sin, not so easily atoned for. The measurement problem (see: The Decoherence of Measurement) is at the very heart of Quantum Mechanics: if the observer actively participates in the determination of the state of the observed system (which, admittedly, is only one possible interpretation) then we are all (observer and observed) members of one and the same continuum and it is discreteness which is imposed on the true, continuous, nature of the Universe.
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Research essay sample on Quantum Mechanics Main Issue
