Ghosts of the departed quantities

A)    About missing things

1.      When something is lost, we can still feel its presence around us. It is a memory in our mind which we will keep alive (even if it becomes fainter with time) for as long as we live. But it is not only the things which used to be but now are gone, but also the things which have never existed, yet they seem to be around us, sometimes as essential as real things, influencing our lives and defining reality. Leaving aside God (which is one of such ghostly things and a matter of religion), we can mention the numbers and the forces, which are the basis of the positive sciences, the soul or the mind (and its physical counterpart the brain), which are the subject of psychology and neurology, and finally the human being, its origin and purpose, as the subject of philosophy. 

B)    The notion of infinitesimals

2.      According to a definition, infinitesimals are defined as things so small that there is no way to measure them. Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz. In common speech, ‘infinitesimal’ means ‘extremely small,’ but not zero. To give it a meaning, it usually must be compared to another infinitesimal object in the same context (as in a derivative). The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope, even though these entities were quantitatively small. [1]

C)    Berkeley’s argument

3.      Although infinitesimals and their discipline (infinitesimal calculus) introduced the notion of instantaneous change (or simply of change) in modern science, and offered a method to calculate that change, their origin and validity as ‘real objects’ remains obscure and ambiguous. For example, are all infinitesimals equal in size? When we keep on dividing a quantity, will we be left with something measurable in the end? Is ‘zero’ a miniscule ‘something’ or an absolute ‘nothing?’ If we divide infinity into an infinite number of smaller parts, are those parts infinitesimals or infinites themselves?  What is the fundamental difference between something infinitely ‘small’ and something infinitely ‘large?’ Is there any difference after all? Or is it the quality of things (and of the mind thinking about the things) what finally decides their true size?

4.      Here is a relative argument stated by Berkeley, with respect to infinitesimals:

“It must, indeed, be acknowledged, that [Newton] used fluxions, like the scaffold of a building, as things to be laid aside or got rid of, as soon as finite lines were found proportional to them. But then these finite exponents are found by the help of fluxions. Whatever therefore is got by such exponents and proportions is to be ascribed to fluxions: which must therefore be previously understood. And what are these fluxions? The velocities of evanescent increments? And what are these same evanescent increments? They are neither finite quantities nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?” [2]

D)    The paradox of the derivative

5.      The notion of a ‘fluxion’ has been substituted by that of a derivative. Instantaneous velocity is the derivative of the position with respect to time- at the limit where the time interval is infinitesimally small. Can however division by zero be defined? If not then the whole discipline of infinitesimal calculus is invalid, since it uses ratios of infinitesimal quantities. Thus Berkeley’s objection.

6.      Although one may ignore the division by zero which appears in the calculation of a derivative, so that one may take in the end the correct result (e.g. a measurement of instantaneous velocity), the mathematical problem is still there: if zero is ‘nothing’ then how can one divide something by nothing? Even if we can finally ignore the problem, there is a logical fallacy in the roots of the mathematical method. Is this because human logic is imperfect? Is it perhaps because ‘zero’ is not ‘nothing,’ but a real quantity (although very small)? Or is it just because motion (and change in general) is impossible?

E)     Zeno’s paradox of motion

7.      This is a set of philosophical problems which have been devised to support the doctrine that contrary to the evidence of one’s senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion.

8.      Three of those paradoxes are the following:

a)      Achilles and the tortoise: In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.

b)      Dichotomy paradox: That which is in locomotion must arrive at the half-way stage before it arrives at the goal.

c)      Arrow paradox: If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless. [3]

F)     The method of exhaustion

9.      Perhaps the most ingenious method which has ever been discovered to treat the problem of infinity in logic and mathematics is Archimedes’ method of exhaustion. The method of exhaustion is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. As the difference in area between the polygons and the containing shape becomes arbitrarily small, the area of the shape is systematically ‘exhausted’ (its correct value is being approached). [4]

10.  An application of this method is finding the sum of an infinite series. For example if we add half a distance to half of what’s left, and so on, we take the whole distance as result (1/2+1/4+1/8+…=1). Thus one might argue that this way we can solve Zeno’s dichotomy paradox. However the previous series which converges to unity is in fact infinite. Thus we need an infinite number of subdivisions to have unity as result.

G)    Making a quantum step

11.  A way out of this logical problem of infinite regress- an infinitesimal quantity which gets smaller and smaller, without ever vanishing (or ‘departing,’ as Berkeley would say), is the notion of the quantum in quantum mechanics. According to this discipline all action (thus motion) is quantized. A quantum (such as a photon) is a very small quantity, but it is not zero. Even empty space is not completely empty, but consists of quanta of minimal energy (vacuum energy). Therefore, according to quantum mechanics, we can divide space into very small quanta which are not zero, so that adding them up we can get as result a macroscopic distance. Again however the problem is that the quantum is not a well- defined quantity. A quantum can be arbitrarily small, so that, after a limit, as quanta get smaller and smaller, quantum mechanics breaks down. Thus the problem of infinity resurfaces.

H)    Motion is reconstructed by the mind

12.  In relation to the arrow paradox, when we watch an arrow flying what we truly perceive is not the arrow moving but the arrow changing its position at regular and fixed intervals. What we really measure is the time, while the arrow helps us to perceive the passage of time. Therefore what is really passing by is not the ‘arrow’ but the ‘time.’

13.  Thus we might say that time is the invisible thing which is materialized and perceived in the form of an arrow. This is the same as measuring our pulse: What do we really measure? Is it the blood pressure? Is it the heart rate? Is it the ticking of the clock? It is as if we were counting the vibrations of our soul materializing into different perceivable objects. Thus we might say that motion is an aspect which -either it exists in the outer world or not- is reconstructed by the mind.

14.  A critical aspect of infinity is that in fact it can be either very big or very small. At the subatomic level even a grain of sand may look like the universe, but at the macroscopic level the whole universe may be just an infinitesimal part of the multiverse. Therefore it depends on the angle and the scale we use to approach the problem of infinity. In fact in our mind either a universe or a grain of sand ‘equally fit.’ Although we assume that the universe must be much bigger than a grain of sand, when we think about it the difference between these two objects is not their size but their meaning. Furthermore, with respect to time or distance to be traveled, the time it takes for us to travel at the other end of the universe with our imagination is no longer or shorter than the time it takes us to blink our eyes. 

15. Is it finally the qualities of our own mind what define the quantities of the objects which we perceive as real and moving? Is there something lying in imagination (beyond reason) what makes reason credible?

I)       What if the ghosts of the departed quantities are the manifestations of the incredible things to come?

[1] [https://en.wikipedia.org/wiki/Infinitesimal]

[2] [https://en.wikipedia.org/wiki/The_Analyst]

[3] [https://en.wikipedia.org/wiki/Zeno%27s_paradoxes]

[4] [https://en.wikipedia.org/wiki/Method_of_exhaustion]

8/27/2018

Picture: [https://sites.psu.edu/math140/sample-page/the-derivatives/definition-of-derivative/]