Monthly Archives: July 2011

Does Pi Exist?

To almost everyone this would seem a ridiculous question. Even to most mathematicians the existence of Pi (defined here as the ratio of a circle’s circumference to it’s diameter) is taken for granted. But the question of the existence of the class of number to which Pi belongs is a key example of a fundamental question in the philosophical understanding of Mathematics, and indeed of a broader issue in metaphysics and epistemology.

To understand the meaning of Pi, let’s start with a quick review of the types of numbers we encounter in elementary mathematics. We start with the “natural numbers” – the numbers 1,2,3 and so on. The very fact that these are the numbers we first learn and first teach to our children in their infancy is ultimately the key to understanding the issue in front of us. In the natural numbers, there are no negative numbers, and no zero.  The extension of the number system to include negative numbers I’ve discussed previously.  The concept of zero I’ll defer yet again, though I’ll hint that I believe it represents balance. 

 Both natural numbers and integers are “countable” – meaning that they can be ordered and indicated in order.  Considered as sets, they both contain a “countable infinity” of elements. 

The next extension of the number system is to include all fractions – the division of natural numbers by natural numbers, and while we’re at it we throw in division of signed integers by signed integers other than zero.  This new number system we designate as the “rational numbers”.  The natural number or the signed integer “n” is represented in the rational number system as ratios n/1, 2n/2, 3n/3, etc.  (Note that I say the signed integers are “represented” – I maintain that the signed integer n is different than the rational number n/1, 2n/2, …).   The nature in which rational numbers exist is again the topic for another discussion, though I’ll note that their nature is related to the epistemological process of measurement, and the mathematical concept of continuity.  I’ll also note that the rational numbers can be so ordered as to be countable, and so are “countably infinite” in number.

The next step is a bit tougher to follow.  The “algebraic numbers” are defined as all numbers which are solutions to polynomial equations whose coefficients and exponents are natural numbers.  In a simpler form, it is this number system that sparked the first significant philosophical debate about the meaning of numbers – back in Greece in the time of Pythagoras.  What we now refer to as the “Pythagorean Theorem” states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the other sides, or a^2+b^2=c^2, where c is the hypotenuse.  In its simplest form, if we are computing the length of the diagonal of a square with unit length sides (a=b=1), then c*c=2.  That the value of c, the square root of 2, cannot be expressed as a fraction of natural numbers was a major discovery of Greek mathematics. (This fact was proven by members of the Pythagorean school, possibly by one Hippasus who may have been executed for revealing the proof because it had been held as a state secret).  A good deal of debate occurred through the centuries over the existence of these “irrational numbers”.  I have not spent enough time considering the nature of these numbers to offer an opinion as to what they represent, but I will venture that this number system will be found to have an acceptable basis in reality.

As with the rational numbers, the algebraic numbers can be ordered in such a manner as to be countable.  There are therefore a countable infinity of algebraic numbers.

Both the rational and algebraic numbers can be shown to be “dense” on the real number line.  Think of the real number line as an “infinitely long” straight line, as you did in high school.  If we take any interval of this line, there will be contained in that interval an infinite number of rational numbers, no matter how small an interval is chosen.  This is the density property of a set of numbers.  This observation, in combination with the fact that all of these number classes are “countably infinite” leads to some confusing conclusions.  In the sense that we can order and count the elements of these number classes, we can establish a “one to one” relationship between all rational numbers with the natural numbers.  “In a sense”, this implies there are “as many” rational numbers as natural numbers – despite the fact that between any two natural numbers there are an infinite number of rational numbers!  The resolution of this confusion lies in the use made here of the concept “infinite” – which ultimately is not a valid concept.  But that is another digression.

The last development in this trail of number classes is the discovery of transcendental numbers.  The historical development of this number class derived from the problem of “squaring the circle”, which was a phrase used to describe the problem of computing the area of a circle with a rational-valued diameter in terms of algebraic numbers.  It was shown in 1882 that this is impossible, because Pi is not the solution of any polynomial with natural number coefficients.  Pi then belongs to yet another class of number.  Rather than deal with the area problem, let us define Pi in the simplest possible manner.  Pi is the ratio of the circumference of a circle to its diameter.  Hence, if a circle has a diameter equal to a natural number of units (or a rational, or even algebraic number of units), its  circumference will be a transcendental number. 

Using this definition, we have for the first time a number tied to a perfect geometrical figure.  A circle is an ideal figure that can never be found in the concrete world of objects.  Any “circle” you encounter in the physical world will be imperfect at some level.  “Circle” is a idealization of these experienced real-world circles.  It can be thought of as the limiting perfection of all physical circles.  And so, the question of whether Pi “exists” will become the question of whether this idealization can be said to exist.  More generally, it is the question as to whether a conceptual idealized abstraction can be said to exist.  Such an abstraction can never be exhibited in the physical world.

The answer to this question lies in the fundamental philosophical principles that are applied.  The empiricist will require that to “exist”, a thing must be present in the physical world, and furthermore, must be experienced.  Such an approach will then conclude that ideal circles do not exist, and therefore Pi does not exist.  The rationalist/Platonist will be happy to place the idealizations in their own realm (ultimately a realm occupied by a God), and declare that they in fact exist in this realm, but can only be approximated in the denigrated physical realm.

Finally, the Objectivist will answer with clarity.  The concept circle – as with all other human concepts – exists as a relationship between the human mind and the physical world.  To exist, the concept must have physical representations from which the details specific to each (such as imperfections) have been abstracted away (omitted as measurements) to arrive at the concept in human consciousness.   The concept circle then exists as an epistemological linkage between human understanding of the physical world, and the real world itself.  And since the circle exists, it is in the same sense that the number Pi (and ultimately all transcendental numbers) exist.

The acceptance of transcendental numbers leads to additional confusions involving the invalid concept of infinity.  Unlike the other number classes, it can be proven that there is an uncountable infinity of transcendental numbers.  A one to one relationship between the transcendental numbers and the natural numbers cannot be established.  Indeed between any two rational numbers (or even algebraic numbers) there is an uncountable infinity of transcendental numbers.  (Intriguingly, a one-to-one relationship can be established between transcendental numbers and the set of all subsets of natural numbers, which in turn implies that the set of all subsets of natural numbers is uncountably infinite as well). 

The further discussion of how these “infinities” and the concept of continuity and “infinite divisibility” should be approached to allow a resolution of the apparent contradictions will need to wait for another time.  Interested parties should consult writings and lectures given by Pat Corvini for excellent discussions of these topics.

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