What is continuity? Or, better still, how do we conceive continuity? For example, if an arrow moves along a trajectory, we conceive its movement as continuous, that is, we would not allow the arrow to ever 'skip' any empty stretch of space. And that should be so from the very beginning. For an arrow to even start moving, it would have to pass from its initial position to a contiguous one smoothly, and then onwards. Unfortunately, this mental model raises a number of issues.
For one, if we were able to identify a contiguous position, then we would also be able to identify the location where both positions meet. Would that location be an intermediate position? Hardly so. Any intermediate position would be contiguous to the previous one and to the next one, and therefore the initial contiguity would be lost.
This conundrum was already pointed to, although in different terms, by Zeno of Elea in his renowned paradoxes, particularly the one about the 'impossibility' for an arrow to ever move. But also the one about Achilles and a tortoise, whereby the speedy Achilles could never get to reach a tortoise sluggishly pacing ahead of him.
As centuries went by, the development of mathematics --specifically, mathematical analysis-- brought about the notion of real number, which formally solved the problem of the discontinuity of rational numbers, i.e. any numbers which could be expressed as the quotient of two integers. The problem with real numbers, however, is that, by introducing infinity to define continuity, they exclude the notion of contiguity. Two real numbers can be selected as close to each other as we choose, but could never be contiguous to each other. There will always be an infinity of real numbers between them.
The mathematical definition of continuity has proved to be a most useful and indispensable tool in engineering, architecture, physics... and mathematics, among others, so it would be ludicrous to even think of dispensing with it. The problem seems to reside not in the formal rules of mathematics themselves, but in the realm of mental concepts and, hence, human language. We would very much like to think that, as Achilles sprints, he does not get lost in an dizzying maze of infinite infinities before being able to reach a crawling tortoise ahead of him.
For one, if we were able to identify a contiguous position, then we would also be able to identify the location where both positions meet. Would that location be an intermediate position? Hardly so. Any intermediate position would be contiguous to the previous one and to the next one, and therefore the initial contiguity would be lost.
This conundrum was already pointed to, although in different terms, by Zeno of Elea in his renowned paradoxes, particularly the one about the 'impossibility' for an arrow to ever move. But also the one about Achilles and a tortoise, whereby the speedy Achilles could never get to reach a tortoise sluggishly pacing ahead of him.
As centuries went by, the development of mathematics --specifically, mathematical analysis-- brought about the notion of real number, which formally solved the problem of the discontinuity of rational numbers, i.e. any numbers which could be expressed as the quotient of two integers. The problem with real numbers, however, is that, by introducing infinity to define continuity, they exclude the notion of contiguity. Two real numbers can be selected as close to each other as we choose, but could never be contiguous to each other. There will always be an infinity of real numbers between them.
The mathematical definition of continuity has proved to be a most useful and indispensable tool in engineering, architecture, physics... and mathematics, among others, so it would be ludicrous to even think of dispensing with it. The problem seems to reside not in the formal rules of mathematics themselves, but in the realm of mental concepts and, hence, human language. We would very much like to think that, as Achilles sprints, he does not get lost in an dizzying maze of infinite infinities before being able to reach a crawling tortoise ahead of him.
A solution to this quandary could perhaps be arrived at by formalizing the concept of acuity. Depending on the means we use to assign a number to a physical magnitude, the precision of such assignment will be variable, but unavoidable. Insofar as we could never exactly pinpoint a location in space, or determine a mass, a pressure value, or an atom's velocity, some degree of fuzzines is to be inexorably counted on, no matter what.
We could specify that degree of fuzziness, e.g., by expressing the value of a magnitude not as one single number, but two, the second one being the threshold of our accuracy. As an example, Achilles' position in his pursuit of the tortoise would be expressed not as 0.35, but (0.35, 3), 3 being the number of decimal places we are confident to have measured with precision. We could refer to it as the 'acuity' of the measurement.
According to this convention, then, one specific measurement could be expressed as any number of different pairs (n, q), where n would be a number, and q its acuity. Thus, if Achilles' position were determined as (0.35, 3), then we would know for sure that he is not at 0.36, but the positions 0.350001 and 0.349998 would be, as far as our determination is concerned, equally acceptable.
The above is not a quantitative definition. Any pair (n, q) could be used to describe Achilles' location at any instant (t, u), and what mathematicians call the equivalence class of all such pairs could be used as a definition of the concept of 'location'.
How would this new concept get over Zeno's paradoxes? Insofar as locations have been defined as fuzzy concepts, both Achilles and the tortoise would also occupy a fuzzy territory, and there would always be at least a pair (n, q) that would describe both Achilles' and the tortoise's location. By increasing the acuity, their positions could be discriminated, e.g. as resp. (n, q), (n', q), in such a way that, when Achilles's next step is greater than n' - n, we could always find an acuity that discriminates him as ahead of the tortoise.
Being a qualitative definition, the above does not seem to help much to the advancement of science--that is, until we enter the quantum realm. At a quantum level, we will find that there is a limit to the discriminating potential of a measurement, essentially determined by the Planck constant. Furthermore, the acuity of time measurements could also be expected to reach some limit, not because of the inherent nature of time, but rather because of the inherent nature of our own relation to reality, including our instruments' readings.
But the main lesson to be learnt from the above is semantic. Numbers are not really useful at describing mental processes. Instead, qualitative concepts such as 'range' and 'contiguity' are. Computers may provide a fashionable model to represent mental processes, but the human mind, I am afraid, does not generally compute. It categorizes, compares and connects. A hardware configuration that duly reflects those processes might therefore provide the key to understanding the human mind.
Fortunately, hardly anybody will ever read this post, much less take it seriously. For finding out how the mind works would --eventually will, but I will not be to blame-- open a door to the hell of total control of the population, which to this date is already worrysome enough. So, let this post remain insignificant, drifting forever in the vast ocean of the Internet. This is what I have found. It may be stupid, or useful, who knows. But I have to content myself with just having written it and getting it published. It is intimately frustrating, but morally reassuring.
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