Early humans had to observe the world around them in order to begin to understand it, and, thereby, to survive in it. To go from observation to understanding requires categorisation, sticking labels on things, which becomes manifest, and takes a great leap forward with language. But, alongside it goes the need not only to categorise, but to count, which is why humans use as the basic means of such counting the fingers of their hands. Similarly, to count distances, they also use parts of the body – a foot, a stride, the length of a forearm, and so on. Only later do these things become merely names for those lengths, whilst an abstract, common unit of measurement takes their place.
The need to measure becomes vital to human survival. When settled agriculture begins, not only is it necessary to measure areas for cultivation, but measurement of time, of the seasons, and so on is required. None of it is possible without mathematics, but the mathematics itself could not develop without observation of the real world.
“... as in every department of thought, at a certain stage of development the laws abstracted from the real world, become divorced from the real world, and are set over against it as something independent, as laws coming from outside, to which the world has to conform. This is how things happened in society and in the state, and in this way, and not otherwise, pure mathematics is subsequently applied to the world, although it is borrowed from this same world and represents only one part of its forms of interconnection — and it is precisely only because of this that it can be applied at all.” (p 48)
One reason that mathematics develops concepts such as the point, which do not, and cannot exist in the real world, is precisely because it tries to model reality on the basis of a formal logic, of the syllogism, whereas reality is dialectical, i.e. it is inherently contradictory, because its essence is flux. The syllogism denies that something can simultaneously be something else. A cannot be -A. So, for example, a straight line cannot be a curved line. But, how then to explain a tangent? At the point of tangency, the angle of the straight line is equal to the angle of the curved line. In other words, at that point it is both curved and straight! How to resolve that? If the line is reduced to a point, which has no extension, i.e. is reduced to something that does not exist, and cannot exist.
Take another example, but, now, in relation to time. I have discussed these issues previously in relation to the arguments of supporters of the TSSI. If an object is in motion, say a ball that is dropped, this motion necessarily involves contradiction. It means that the object is in two places simultaneously, it is both at A and -A (not A). The syllogism cannot accept such a conclusion. But, the reality shows that, as well as starting at A, and ending at B, with no stops along the way, this very fact means that when I measure the position of the object, at any other moment in time, it must, similarly, start at one position and end at another. Each moment of time has a duration, a start and end, and, consequently, in this moment, the object is at two different positions. That is true no matter how short that duration might be. The only way around that is to theorise a zero in time, a point in time that has no duration. But, that, again, is something that does not, and cannot exist in the real world.
At the macro level, the arrow of time, points in one direction, and whilst this may not be true at the quantum level, it is still not true that what exists or can exist is a zero of time, but that there is uncertainty, with the arrow of time pointing in different directions, with different degrees of probability. Its only when the consequence of these backward and forward movements is viewed in their totality that we see the arrow of time moving in one direction. If you look at a mountain side, from a distance, it appears as a more or less straight upward line. But, look at it from a closer distance, and the straight upward line becomes a line with periodic changes in angle. Look even more closely, and you will see that, at some places on this line, it does not slope upwards at all, but downwards.
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