Xeno’s Paradoxes and the Concept of Infinity

Without the audacity of any kind to overlook his contribution to mathematical and philosophical ways of thinking of modern ages, if Xeno lived today, his parents would most probably have had a YouTube channel making videos about how hard it is to raise a child like him, for he is against every straightforward and intuitive concept.

Xeno (or Zeno, or Xenon. ZHNΩN in Greek) was born in Elea in 5th century BC, located in today’s South-West Italy. He was a pre-Socratic, Ancient Greek philosopher, known to be a Monoist, refusing the existence of concepts like space, time and motion. He is famous for his paradoxes that challenge, at least at his time, the existence of these concepts.

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Bertrand Russel quoted that Xeno’s arguments, somehow have set the foundation of all theories built since his time till today, about space, time and infinity.?

His paradoxes are not intuitive at all but still it would take two millennia to bring mathematically sound proofs for them being wrong. Two of his many paradoxes were the Dichotomy Paradox and Arrow Paradox.

Dichotomy Paradox

Xeno states in this paradox that to reach your target destination while traveling from one point to the other, first you need to cover half of the total distance. Then, you have to cover half of the remaining distance, then again half of the remaining and this goes on forever. As forever is infinite, the sum of all these ‘half ways’ would be infinite and you could never reach your destination.

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Regardless of the fact that the numbers that he was adding up was getting smaller each time, he was claiming that adding infinite number of something would lead to infinity, thus never reach an end. Little did he know was the concept of divergent and convergent series. His mistake was to misinterpret the concept of infinity and his lack of understanding of how in different ways infinity could exist.

Mathematically we can write this problem as a series that stretches to infinity, like below:

1/2 + 1/4 + 1/8 + 1/16 + …

Xeno said the result of this summation would be infinity. He was wrong.?

Let us name this sum as S:

S = 1/2 + 1/4 + 1/8 + 1/16 + …

Let us multiply each side by 2:

2S = 1 + 1/2 + 1/4 + 1/8 + 1/16 + …

As seen, the part coming after the first figure (1) is the same with S. So, we can write:

2S = 1 + S?

2S - S = 1

And this makes S = 1.

Graphical illustration of this proof is even more mesmerizing. Below is a unit square, each side with 1 unit length.

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Sum of all these areas that get smaller each time add up to 1. This is a convergent series, which converges to 1. This means, contrary to his thinking, you can actually reach your destination and cover the whole distance.

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His Achilles and Tortoise Paradox is a different version of the same paradox. According to this, in a running race between these two, that the tortoise is allowed to start from the middle of the track due to his obvious disadvantage, and Achilles is twice as fast as his opponent, Achilles can never catch up and overcome tortoise, let alone reaching the finish line at all. He was telling that when Achilles reaches the previous position of the tortoise, tortoise would move a bit and Achilles can never reaches his opponent, yet the distance between them would get smaller each time. Solution was the same with his previous paradox.

Arrow Paradox?

In this paradox Xeno was stating that at one single instant, a thrown arrow could only exist at one single location and would be motionless. It was just an instant that he was talking about, so no motion could have been possible. But in the very next instant, the arrow should have changed its position to its next location, and again and again, from one instant to the other. So, he stated, there is no such a thing as motion. In previous paradox Xeno was dividing distance into infinity. Now he is doing the same thing to time. He is dividing time to infinity. He did not know the concepts of limit and derivatives.

Shutter speed of cameras is the time that the shutter remains open, so the camera sensor could get the light reflected from the object in, and create the image. It is expressed as a fraction of a second, like 1 / 500, 1 /1000, depending on your preference. 1 / 500 means the shutter remains open for 500th of a second. Xeno here was talking about a shutter speed of 1 over infinity. Mathematically, dividing something to infinity results 0 (zero), and time is no exception. If you divide time to infinity, you would get zero. Zero time. This means there is nothing there. No arrow, no target, no motion, no time, no space. So, if we are going to refuse the existence of time and motion, we need to refuse also the existence of arrow, bow, ourselves and everything else.

The mistake here, I believe is again about the interpretation of infinity. Infinity is a philosophical concept, rather than mathematical. There are many areas it is used in mathematics but it needs to be handled with care. I see it as the revolving door between the immense green fields of philosophy and mathematics, which you can easily pass from one to the other.

Before romanticizing the concept any further, I would like to give a little bit more glance on the mathematical solution of this paradox. It is about limit and derivatives, Calculus to be more clear.

This could be a very simple visualization of what Xeno was trying to achieve; reaching a point that there is no time and, thus no motion. But calculus found a heavenly way of showing how to do it. Change in f(x), when you change x by Δx is the first derivative of x, also displayed as df / dx, or dy / dx. When Δx approaches to zero (lim Δx → 0), you get the rate of change at the exact point of f(x). There is motion at this point. In the context of arrow specifically, there is also acceleration and deceleration, which is the second derivative of location, denoted as d^2f / dx^2, or d^2y / dx^2.

Talking about Calculus, founders of it is a controversial topic. Mathematicians are divided between Archimedes, Isaac Newton and Gottfried Leibniz. I will not dare to make any statement about my own opinion here, but rather express my gratitude to those who have contributed to this lovely topic, which takes my revenge from my son now at university, for all his tantrums he has thrown to us when he was small. Well, no, he did not have any tantrums, he was great. Some sleepless nights maybe. I am just enjoying seeing him struggling a little bit while studying it. Struggles make us stronger. As a side note, Newton was 23 years old when he invented Calculus in 1666.

Understanding the Concept of Infinity

Infinity is a curious thing. To me it is more of a philosophical concept, than mathematical. It is not a number, yet you can do calculations with it. Well, you should still be prepared not to get ordinary results when inserting it to your calculations. This was the very reason that I preferred to start this article by Xeno and his paradoxes to demonstrate how careful we should be while thinking about infinity and how misinterpretations could divert us from reason.

Infinity concept is also used in physics (spatial and temporal) and metaphysical contexts (God, etc.) but here I will focus on its mathematical interpretations. Its famous symbol, ∞, was first used by English mathematician John Wallis in 1655.

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The idea of infinity goes back to Ancient Greek philosophers. They used to name this as Apeiron, ‘a’ meaning ‘without’ and ‘peirar’ meaning ‘end’ or ‘limit’ or ‘bound’. This concept was used mainly in their cosmological theories, which goes back to 6th century BC. Mathematicians of that time had the tendency to think that any number could be expressed as a ratio of integers, until they have come across numbers otherwise, which would be named as irrational numbers centuries after them. Their first encounter with irrational numbers was after being able to calculate the hypotenuse of a unit square (thanks to Pythagoras) as √2. Its digits were going to infinity and it could not be written as a ratio of any two integers.

Cardinality

Set Theory could be a good starting point to have a better grasp on the topic. Cardinality means the number of members of a set. Natural numbers, N, start from 0 (sometimes from 1, do not ask why), include all positive integers and goes to infinity.

N = {0, 1, 2, 3, 4 …}

Integers, Z, include all positive and negative whole numbers.

Z = {… -4, -3, -2, -1, 0, 1, 2, 3, 4 …}

Natural numbers are clearly a sub-set of Integers and they are both have the same cardinality; infinite. But are these cardinalities same or different? Are these infinites same or different? If we could strip from the question how two infinities could be different, intuitively, we may think that the infinite cardinality of Z should be bigger than that of N. Not really.

In a Bijection, or a Bijective Function (sometimes called Isomorphisim, more of a term to describe two objects having the same shapes), the elements of the domain set sould correspond one-to-one to the elements of the output set.

Stating that cardinality of N and Z are different can be translated to mathematics as these two sets are not bijective. We mean that we cannot match the elements of domain set, N in this example, one-to-one with the elements of the output set, Z here. Let us prove otherwise. To do this we will benefit from the method developed by Georg Cantor (Georg Ferdinand Ludwig Philipp Cantor), who was a German-Russian 19th century mathematician. He is accepted to have an important role in the creation of Set Theory. He was suffering from manic-depression, which would later turn to paranoia.

Let us start by matching 0 in N with 0 in Z. Then lets us match the 1’s. Then let us match the 2 in N with -1 in Z. After that we can match the 3 in N with 2 in Z. Then 4 with -2, then 5 with 3, and so on. Do you see the pattern here? By this way we can match all elements in N one-to-one with that of Z.

How about rational numbers, Q? These are numbers that could be expressed as a ratio of two integers. Is the cardinality of Q bigger than that of N and Z? Answer is no, they are equal. Think of rational numbers as pairs of two integers that are divided to each other. You should be able to match each rational number to a pair of integers. Best way to understand this is to think it as a chart below. You can assign every integer to a number pair and these two sets would be bijective.

Cantor also proved that the cardinality of Real Numbers, R, the set of numbers that includes rational numbers plus irrational numbers, is bigger than the cardinality of rational numbers. Putting aside his slightly complex proof of this, this could be understood by thinking that there are no pairs of numbers to represent irrational numbers. This is called the Cardinality of the Continuum. Numbers line of rational numbers is not a continuum, there are gaps at irrational numbers, such as π, or e, or √2. These gaps are filled in real numbers and the lines becomes continuous, thus the name.

To differentiate the cardinalities of different infinite sets, Cantor used the first letter of Hebrew alphabet, Aleph (?). Cardinality of Q is referred as ?0, or Aleph Zero. Cardinality of R, on the other hand, is denoted as ‘??’. So, we can clearly state that:

?0 < ??

The next question was if there was any number between the cardinalities ?0 and ?1, but for the sake of simplicity, I prefer to leave it here and stay away from the relationship between ?0 , ?1 and other cardinalities in this reading. But I definitely recommend to explore this area to those who share the excitement of the topic with me. One sneak peek could be below equation, that demonstrates that the relationship between cardinal numbers is a function of powers of 2. ?1, for those who are interested, is the set theory symbol for the smallest infinite set larger than ?0.

By the way, Paul Cohen, 20th century American mathematician answered above question (if there was a number between two cardinalities) as ‘both yes and no’, as if the topic itself was not confusing enough.

Negative and Positive Infinity

We have stated that any number divided by infinity results 0. What would be the result if we divide a number with negative infinity? The same, 0. Does this mean that negative infinity and positive infinity are equal to each other? This is tricky. Below is the graph of 1 / x.

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Different interpretations of the concept result in different answers to the question. Both Yes and No can be given as an answer. Remember, we are revolving in the door between mathematics and philosophy. Results of both 1 over positive infinity and 1 over negative infinity being zero may make us think that they should be equal. But the function x^2 = 4 has also two solutions, 2 and -2. And clearly, they are not equal. Of course, one is a quadratic function and the other is not, but still.

Think of a straight line on Cartesian Plane, with one end at (0, 0) and the other at (5, 0). Keeping the end at (0, 0) stable, change the position of the other end (lift it up) and increase the slope of the line to 1, i.e. 45-degrees with x-axis. Keeping this motion, the slope will continue to increase and when the line become totally vertical, the slope will reach positive infinity.

Have a second line, again one end at (0, 0), but this time the other end at (-5, 0). Do the same thing, keep the end at (0, 0) stable and lift the other end up 45-degrees so that the slope will be -1. If you keep increasing the angle, the slope will start to have higher negative values. When you make it vertical, the slope will reach negative infinity this time. But two lines will have the same slope physically.

Two lines, both being 90-degrees vertical will have one single slope, and that will be infinite, not negative, not positive, only infinite. So, are negative infinity and positive infinity the same? This is why I preferred to make this resemblance for infinity as the revolving doors between mathematics and philosophy. Your definition to it could change from which side of the door you are looking at.

I believe studies on infinity concept contributed the discussion of if mathematics is a discovery or invention. It has many unnatural and counter-intuitive aspects that could support the invention argument. Then as a true supporter of the idea that mathematics is the language of the universe, it is the background of everything and explains how things are what they are, I am experiencing cognitive dissonance to drag away from the idea that it being a discovery. Questions truly worth thinking about, if you ask me. This concept kept mathematicians, philosopher, physicists and many more people busy thinking about it. It is obvious that this will continue till, well, infinity maybe.

Etymological post-script

First alphabets were created by Sumerians around 3000 BC in today’s Middle-East. Since then, many civilizations used several symbols as letters, some similar to each other, some different. Letters of the alphabets were the symbols derived from the items of those people, ranked in the order of their importance. Aleph, the first letter of Semitic alphabets was the symbol for ox. Second letter was Beth, and meant house. In Arabic, Aleph and Beth became Elif and Be. And yes, you are right in Greek they became Alpha and Beta. And still yes, in Latin they became A and B. I have always been fascinated by the viscosity of culture in the dimensions of time and location.

Cem Sürmen, CFA

14.07.2024, Budapest