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General Epistemology        Chapter III-2       

 

Chapter III-2 Imaginary numbers:
to build a house which exists
with bricks which do not exist

 

 

(Permalink)(Was chapter 20 in version 1)

 

Preliminaries, for those who believe that numbers do not exist

(Permalink) Some mathematicians consider that mathematical entities do not exist of their own, that they are only pure creations of the human mind. I allow myself to question this opinion.

Let us take the equation: x² = 4

This equation has two solutions: +2 and -2. The only two numbers which, squared, give 4.

If mathematics are a creation of the human mind, then there should be a mean to create other solutions. No game! In law, we can create laws in this way: we write a law, we do the ritual to enact it, and hop, everybody is expected to know our law, even the ones who never read it. This is because laws are conventions, creations of the human mind. Is it the same in mathematics? Hear well, I actually ask for a third number which would satisfy the above equation. If somebody succeeds, I am ready to recognize too that mathematics are a creation of the human mind, as much as the laws. But I am definitively sure that nobody will never succeed in doing so.

First reason is that mathematical facts exist independently of time, so that they cannot appear or change.

By the way, let us make clear that I don't accept any illusionist trick such as saying that 10 (two in binary) is a solution, or II (two in roman). This is just the same number, expressed with different notations. Figures and notations are conventions, yes, and there are several conventions for the same number. I really ask for a third number for the above problem, not another notation for the numbers 2 or -2.

Two first metaphysical principles

(Permalink) This fundamental impossibility to do what we want in mathematics can be summarized in a very simple way, in saying that, in mathematics, some things exist, others not. This is precisely how speak the mathematicians, who even use the symbol ∃ to say «there exists». So ∃ two numbers which are the solution of the equation x² = 4.

This goes in the same way as in the material world: when an object imposes us its presence and properties, we say that it exists. If not, we say that it does not exist. This definition of existence will be our first metaphysical principle.

So, similarly in mathematics we can say that the solutions +2 and -2 match the definition «to exist», and not a third. I think this way of thinking is healthy, and any further discussion would be pure quibble.

 

So we can say that at least some «abstract» objects exist, in their way. The only counter argument would be that this is not «the same degree of existence» than for material objects. But this is just the implicit materialistic prejudice, attached to traditional science, as seen in chapter III-1. And this is precisely what we must not do in metaphysics, when we pretend to create a system explaining the world.

 

So our second metaphysical principle here is that precisely, abstract facts have the same degree of existence than physical facts. The only difference is that they do not appear to our sensory organs.

If we had to do a hierarchy of strength of existence, we could say that the hardest steel can be shattered, and even a mountain can be erased. But the solutions to the previous equation will never disappear. So it is rather the abstract facts which are infinitely more solid than any material fact.

Reminder: definition of imaginary numbers

(Permalink) The purpose of this sub-chapter is to introduce them in a simplified way (but mathematically exact), to the non-mathematician reader, while apologising to the purists for imaged short cuts.

To multiply a positive or negative number by itself, that is called to square. The obtained number is called a square, and this square is always positive. In the opposite operation, we start from a given number which we suppose that it is a square. We then seek of which number it is the square. This is called to extract the square root. (There is only in maths that we see square roots, I never found any in my kitchen garden). If our number is positive, all is right, we always find a square root, and even two: a positive and a negative, which even give us the choice. But if our number is negative? That cannot work, since no number, neither positive nor negative, can give a negative square. There are no square roots of negative numbers.

One day, to the great displeasure of other mathematicians, an eccentric had the idea to imagine a number i which square would be -1, and an infinity of multiple imaginary numbers of this i, all square roots of negative numbers. Imaginary numbers, numbers which do not exist, since no squared number can give -1. And to give a name to something which does not exist is not enough to make it real! This strange idea could have kept unnoticed, but it happened that a crafty one noticed that these imaginary numbers were very practical to make calculations in many technical fields, in particular with alternating electrical currents. They were not at all essential, we could have used serious trigonometry functions which have the advantage of existing. But only one line of simple imaginary calculation replaces a page of complicated trigonometry, and the results are much more clearly expressed.

So, if we are doing that way, let us do it completely. Since i was imaginary, we could give it any properties we wanted. As it does not exist, it cannot defend itself! Mathematicians thus stated that ei = cos(1) + sin(1)i. Even if you do not understand anything with this formula, you easily notice that this choice is very peculiar, very arbitrary. Why ei wouldn't have been equal to zero, Pi or the age of my aunt? We could as well, but this precise choice had the advantage of considerably simplifying the resolution of a type of differential equation (of the second order) essential in calculation of all the phenomena of damped oscillations: suspension of car, washing machine, pendulums, loudspeakers, radio circuits, all the speed regulations, temperature regulations, robot arms, planes rudders, servo-motors in remote controls, etc...

In other words imaginary numbers and their arbitrary properties govern everything in the house, on the road, in the plane... It is reassuring, when we are in flight above eleven kilometres of vacuum, to think that our security relies on something imaginary. And yet think: it is the only part of the plane which will never break down.

(Of course, we should not confuse: the current which passes in a coil is noted by an imaginary number, but if you put your fingers on it you will feel it as much! It is only the behaviour of the systems which calls upon imaginary elements, the systems themselves are all made of material elements).

What is interesting to be aware here is that we have build a whole theory which exists and which gives very concrete results, starting from an object, i, a base which does not exist, but about which we do as if it was there. And worse, we without shame abused of this situation to imagine it with the properties which we liked. A real object inevitably imposes to us its properties, qualities or defects. But this imaginary object, on its side, was defenceless, since it does not exist. Then we defined it in the most useful way for our little business. This is a concrete application of our first metaphysical principle.

 

So the mathematicians really built a house which exists, the set of the imaginary numbers and all its applications, starting from a brick i which does not exist.

They got some nerve, don't you think so?

 

 

 

 

 

 

General Epistemology        Chapter III-2       

 

 

 

 

 

 

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