46. Zeno’s Paradoxes 2

Zeno of Elea's paradoxes of plurality spoke to infinite divisibility and the concept of the infinite within the finite.
The Greek philosopher Zeno of Elea showing his followers the doors of Truth and Falsehood. From a 16th century AD Fresco at the El Escorial, Madrid.

It all started when David Clayton, an internationally known artist, asked me a simple question: “How many colors are there?” Well, it turned out that that this quite simple question sent me on a trek at the end of which I ran into Zeno, of all people.

Of course, Zeno did not give me an easy answer to that question; rather he, as all good philosophers do, caused me to go much deeper into this question than I had ever anticipated, with some surprising results.

We all remember the 8th grade science experiment where we used a prism to break up white light into the colors of the rainbow. For most of us, it was our favorite, most interesting experiment to see simple white light explode into a colored rainbow with such a simple object as a prism.

We were taught that there were seven basic colors as represented by the acronym ROYGBIV, sometimes made into a person’s name, ROY G. BIV. This acronym stood for red, orange, yellow, green, blue, indigo, and violet. I was taught the same thing in optometry school years later.

Dispersion of light through a prism illustrates infinite divisibility.

My first inclination was to use this answer with David, but then I realized that as an artist, this answer would never satisfy him. He knew that the richness of the artist’s palette betrayed such a simplistic answer. In formulating my answer, I thought that I should first consider that part of the electromagnetic spectrum that contains the visible spectrum of light, which is roughly 400 to 700 nm (nanometers) in length. A nm is a billionth of a meter, but of course the boundaries change depending upon who you ask – I’ve seen ranges anywhere between 380 and 750 nm. Whoever said that science is a settled matter?

The Visible Spectrum of Light

Here is a chart of where the visible spectrum of light falls within the entire electromagnetic spectrum, ranging from harmful gamma rays to harmless AM radio waves.

The visible spectrum of light shows how we can find the infinite within the finite.

Considering the above spectrum, I first asked myself why there are only seven colors designated within this 300 nm range of visible light. Also, if the electromagnetic spectrum is continuous, where do each of the colors begin and end? In other words, how can there be such definite boundaries between the colors if the spectrum is continuous? I started suspecting that the ROYGBIV that we were all taught as represented by the spectrum below was an artificial construct that did not accurately represent reality.

The electromagnetic spectrum includes a finite range of visible light but within that range is an infinite number of colors.

As you can see, the question of how many colors there are is not an easy question to answer. I was in despair until I ran into Zeno, who brought clarity to this problem but also increased the mystery. I will pick this problem up again at the end, but first I want to set the table by discussing Zeno’s paradoxes of plurality.

Zeno’s Paradoxes of Plurality

Zeno invented the reductio ad absurdum argument. It is a very powerful argument that is often used today to destroy an opponent’s position. In Zeno’s case, he used this argument to defend his mentor Parmenides’ position that “all is one.” He illustrated that all is one by showing the utter absurdity of the opposite – that plurality is impossible. We call his arguments along this line the “paradoxes of plurality.”

In the last post, I discussed Zeno’s paradoxes of motion. I will discuss his other major set of paradoxes, those of plurality, in this post and show why it is impossible – according to Zeno – to have plurality, thus proving beyond a shadow of a doubt that everything is indeed one.

Zeno has four paradoxes of plurality, three of which are still discussed in modern times. The fourth one, the Paradox of Like and Unlike, is his weakest paradox and was easily refuted by Plato.1 Since it is the weakest argument, let’s dispatch that one first.

The Paradox of Like and Unlike

Zeno’s argument, as stated in Plato’s Parmenides dialogue, is as follows:

“If things are many, they must then be both like and unlike, but that is impossible, because unlike things cannot be like or like things unlike.2

Zeno relies on the Law of Noncontradiction, F cannot equal non-F. For example, if humans and dogs exist, then that would violate the Law of Noncontradiction since humans would be like dogs in that they both share existence or oneness, but also unlike dogs since they differ in kind and thus share in plurality. Since a human or a dog cannot be both like and unlike at the same time, then this proves that plurality cannot exist.

Plato’s solution is simple. Something can be like and unlike something else at the same time, but in different senses. Continuing with our example, humans are like dogs in many ways; both have eyes, ears, and noses, among other things. They are also unlike in that humans have the capacity for speech whereas dogs do not. Thus, the Law of Contradiction is not violated.

The Paradox of Limited and Unlimited

This is the second weakest of Zeno’s paradoxes of plurality. This is also called the Paradox of Denseness.3 This argument is summed up by the Neoplatonic commentator of Aristotle, Simplicius (c. 480-560 AD) as follows:

“‘If there are many, they must be as many as they are and neither more nor less than that. But if they are as many as they are, they would be limited. If there are many, things that are are unlimited. For there are always others between the things that are, and again others between those, and so the things that are are unlimited.’ (Simplicius(a) On Aristotle’s Physics, 140.29)”4

This argument simply states that if there is a plurality rather than a unity, then the number of objects in this plurality is “limited” since there will be a definite or fixed number of things.

For example, a plurality may consist of only two objects, A and B. But if two objects A and B are distinct objects, then there has to be something, entity C, in between them, a boundary so to speak, that allows A and B to maintain their distinctness. So now we have three things – A, B, and C. But if those things are to be distinct, then there has to be also a boundary, yet another entity, between A and C and B and C. Now we have five things, and so on, ad infinitum.

So, what was originally thought of as “limited” is in actuality “unlimited.” And therein lies the contradiction that nullifies the idea that plurality cannot exist since plurality cannot be at the same time limited and unlimited.

The fallacy of this argument lies in the assumption that “there are always others between the things that are.” He seems to be insinuating that there has to be a physical object between two other physical objects in order to maintain distinctness. We know this not to be true for we can have two objects in a vacuum with nothing between them except space. This proves that if plurality exists, it can be limited and does not have to be unlimited.

But this is not an open and shut case as it might seem. Enter Albert Einstein. According to his theory of relativity, space “moves, bends, snaps and has holes in it just like a tangible, material thing.”5 If this is the case then, even though space is not a physical object, it is nevertheless “something,” which would make Zeno’s argument not as easy to refute as first thought.

The Paradox of the Large and Small

There are two parts to this argument. I will cover the Paradox of the Small first.

The Paradox of the Small argument itself has two parts. The first part of it has been lost, but probably went something like this:

If there were a plurality, then the plurality would be composed of individual entities or parts. But if this were true, then the parts could have no size, for if they did have size, then the particular parts could be divided into more parts and so on. If distinct entities did exist, they would have to be so small that they could not be further divided into parts, and this would only be possible if they were nothing.

Once again we rely upon Simplicius to give us the second part of the argument for the Paradox of the Small:

“‘… if it should be added to something else that exists, it would not make it any bigger. For if it were of no size and was added, it cannot increase in size. And so it follows immediately that what is added is nothing. But if when it is subtracted, the other thing is no smaller, nor is it increased when it is added, clearly the thing being added or subtracted is nothing.’ (Simplicius(a) On Aristotle’s Physics, 139.9)”6

This second part really is a corollary to the first part. If the small were nothing, whether it was added or subtracted from something that existed, then it would make no difference in the size of that object.

The Paradox of the Large is as follows:

“‘But if it exists, each thing must have some size and thickness, and part of it must be apart from the rest. And the same reasoning holds concerning the part that is in front. For that too will have size and part of it will be in front. Now it is the same thing to say this once and to keep saying it forever. For no such part of it will be last, nor will there be one part not related to another. Therefore, if there are many things, they must be both small and large; so small as not to have size, but so large as to be unlimited.’ (Simplicius(a) On Aristotle’s Physics, 141.2)”7

If we weren’t confused already, Zeno (or Simplicius) seems utterly determined to confuse us even more. Really, this is just an argument in the opposite direction.

Say, for example, that a plurality existed of objects of definite size. Then each of those objects would have a front half and a back half extended into space. If that were the case, then the front half would also have a front half and a back half extended into space, and likewise the back half would have a front half and back half extended into space. Then each of those halves would have a front and back half extended into space ad infinitum. Thus, each object of definite size would be infinitely large.

It appears that in all of this, Zeno may be relying upon the concept of infinite divisibility that we will see more explicitly in the next and last paradox. In the Paradox of the Small, if we divide an object in half, and then each of those objects in half, we will end up with an infinite number of objects that are all the same infinitely small size. If we then lay all of those objects end to end, they will extend to infinity since the process of division into increasingly smaller fractional parts is endless. The absurdity of this demonstrates that plurality cannot exist.8

Conversely, in the Paradox of the Large, if we divide an object into two halves in order to extend each of those halves into space and then divide each of those halves into halves, and if we add all of those pieces together, we will get the following summation: (1/2 + 1/4 + 1/8 + 1/16 +….). Since we are talking about dividing the original object into a front half and back half, then this summation would go to infinity in both directions, thus making the object infinitely large.9

If Zeno’s Paradox of the Large and Small really is then a paradox of infinite divisibility, which I think it is, then we must turn to the most interesting of his paradoxes of plurality – that of the same name.

The Paradox of Infinite Divisibility

Finally we get to Zeno’s last and most important paradox, which is the Paradox of Infinite Divisibility. He uses this paradox not only in refuting plurality, but also in refuting motion as I discussed in the previous post. His refutation of both plurality and motion lies in the absurdity of infinite divisibility. Aristotle describes the argument as such:

“…whenever a body is by nature divisible through and through, whether by bisection, or generally by any method whatever, nothing impossible will have resulted if it has actually been divided…though perhaps nobody in fact could so divide it. What then will remain? A magnitude? No: this is impossible, since then there will be something not divided, whereas ex hypothesi the body was divisible through and through. But if it be admitted that neither a body nor a magnitude remain…the body will either consist of points (and its constituents will be without magnitude) or it will be absolutely nothing. If the latter, then it might both come-to-be out of nothing and exist as a composite of nothing; and thus presumably the whole body will be nothing but an appearance. But if it consists of points, it will not possess any magnitude.10

-Aristotle On Generation and Corruption, 316a19

Aristotle puts forth this argument, but oddly enough does not attribute it to Zeno. The reason we attribute the argument to Zeno is because Simplicius does so in his commentary on Aristotle, On Aristotle’s Physics, 139.24.

In this most important of his paradoxes of plurality, Zeno is describing a scenario where a body is continuously divided into increasingly smaller parts. The operative question is what is the final result? What do we end up with as the final product if there is infinite divisibility? According to Zeno, there are several possibilities.

The first possibility is that we end up with something. But that is impossible because if we end up with a body having magnitude, then by definition, that body can be further divided.

The second possibility is that the body is absolutely nothing. In this case, the body either arose from nothing or is composed of nothing, which is an absurd position.

The third possibility is that the body is composed of points without magnitude. A body composed of points without magnitude has zero size regardless of the number of points because the sum of a series of zeros is zero. Thus, the body is an illusion which makes this possibility absurd as well.

Infinite Divisibility Illustrated

At face value, it appears from the above arguments that Zeno was correct – plurality and motion are illusions. Think about having a line that is four meters long. We divide it into to equal sections, 1/2 x 4 meters = 2 meters. If we divide it in half again into quarters, we get 1-meter sections. We can divide it into smaller and smaller sections as the fraction gets smaller or the denominator of the fraction grows larger. For example, we can have a really small fraction such as 1/1025 which would divide the line into 4 x 10-25-meter sections. There is no limit to the smallness of the sections that the line could be divided into since the denominator can go to infinity.

As the denominator of the fraction increases toward infinity, then the entire function moves to zero, which seems to support Zeno’s claim of a body being composed of points of zero size in the third possibility mentioned above.

Solving 1 Divided by Infinity

According to the graph, as the denominator x moves to infinity (to the right), then the function of y moves toward zero (horizontal axis). This is a graph of the same function that I illustrated in the previous post on the Paradox of Motion.

The Infinite in the Finite

You can read of possible solutions to this paradox on the internet. The sites below, the Internet Encyclopedia of Philosophy and the Stanford Encyclopedia of Philosophy that I referenced, explain in detail some of these solutions which, in my mind, do not solve the paradox adequately. The Internet Encyclopedia of Philosophy states that if we divide a concrete object as such, then we will reach a physical limit in the size of the subatomic particles involved.

This is fair enough, but what about an abstract entity? Concerning this latter possibility, Internet Encyclopedia of Philosophy states:

“On the other hand, is Zeno dividing an abstract path or trajectory? Let’s assume he is, since this produces a more challenging paradox.”11

In other words, when we consider the abstract rather than the concrete, we are back to square one. Having said that, most of the solution comes down to the concept in modern mathematics that

“A geometric line segment is an uncountable infinity of points plus a distance function.”12

From my perspective, the above statement, as true as it may be, is more a description of what is than a solution of how it is possible that an infinite amount of points can be bound by finite limits. Regardless, it is important to note that what troubled Zeno has been troubling mathematicians ever since. Mathematics tells us that between each finite distance bounded by A and B lie an infinite number of points.

Our senses of reality tell us that the interval is finite, whereas mathematics tells us that infinity lies within those finite limits. Therein lies the paradox that infinity can fit inside the finite. This infinite divisibility bothered Zeno so much that he eventually denied reality all together, saying that plurality and motion are illusions and thus all is one.

Enter Georg Cantor

Georg Cantor (1845-1918) was born in Russia to a Russian mother and a father who had emigrated from Denmark.13 When he was 11, the family moved to Germany. It was evident from an early age that Georg had exceptional skills in mathematics. He eventually became a mathematician and made groundbreaking discoveries in the areas of set theory and transfinite numbers.14

Cantor picked up where Zeno had left off. Although many mathematicians in between had done significant work with the concept of the infinite, it was Cantor who really brought everything into focus. People have known about infinite sequences since the time of Euclid. For example, Euclid demonstrated that prime numbers could be continued to infinity.

Cantor took a different approach. Rather than looking at distance in regard to infinity, he did exactly what Zeno did and looked at the amount of numbers contained within line segments between two points of a specified distance.15 Cantor proved through rigorous mathematics what Zeno knew intuitively, that no matter how small a line segment, the amount of numbers contained within it is infinite.

A corollary of this theorem is that since there are different sizes of line segments, then there are different sizes of infinity. I would extend from there that since there are an infinite number of line segment distances, then there are an infinite number of sizes of infinity. Does your brain hurt yet?

Cantor demonstrated through rigorous calculations what Zeno was trying to tell us through his paradoxes of plurality. I include some of Cantor’s calculations below, not expecting most of us to understand them ( I certainly don’t), but to illustrate how far we have progressed since the time of the early Greek thinkers.

Cantor's calculations of different sizes of infinity.
Cantor's line segments illustrating infinity within the finite.

Infinity and the Color Spectrum

Let’s return to where we started, with the color spectrum. When trying to answer the question of how many colors there are, I asked myself the following question: What is the smallest unit in the visible light spectrum that can be considered a distinct color? As I chopped the spectrum up into smaller and smaller segments, I soon realized that the spectrum was a continuum and not a series of distinct points, which brought me to the conclusion that the amount of points within the visible spectrum is infinite.16 I saw firsthand that the infinite can exist within the finite.

I was in awe at this fact, probably in the same way Zeno and Cantor must have been when they each discovered this for the first time. I soon realized that the mystery did not end there – that was only the beginning. As I searched the internet for more information, I soon found an article by Ross Pomeroy of Real Clear Science entitled “How Infinity Can Fit Inside the Finite.”17 That is where I first met Zeno of Elea who opened up this whole mysterious world for me.

The Mystery of the Infinite Within the Finite

After his discovery, the aspect of infinity contained within the finite haunted Cantor for the rest of his life. He became increasingly depressed as he pondered this paradox and eventually went insane.18 Toward the end of his life, he was committed to a sanatorium where he died. Rather than going insane, Zeno took the approach of denying reality altogether, thus eliminating the paradox. What is it with this amazing discovery that caused Zeno to deny reality and Cantor to lose his mind?

It is difficult enough to wrap our minds around the concept of infinity, let alone how the infinite can fit within the finite. For those of us who remain in the secular matrix created by the Enlightenment, this paradox has no solution and would logically lead to insanity. The high priests and priestesses of modern secularism attempt to give us a rational explanation for a world full of mystery and wonder. They ask us to deny what we instinctively know to be true, that there is more than just the physical creation.

This has led to a whole host of mental and emotional illnesses that became prominent in the 20th century. That is why Existentialist writers such as Kafka, Camus, and Sartre rose to prominence in the 1900s; they wrote about the absurdities of an existence that appeared to have meaning but really didn’t.

The question becomes one of who really are the crazy ones – the ones who believe in the supernatural or those who don’t?

The Solution for the Infinite Within the Finite

The question is not how this is possible but why. Scientists and mathematicians cannot really say how the infinite can fit within the finite. They simply tell us that it is.

God left His mark on the universe when He created it. When we marvel at the beauty of creation, even in something as simple as a sunset or a nebula formation, we are seeing the handiwork of God. His fingerprints are all over creation.

It should be no surprise then that God, being infinite and eternal, has embedded the infinite within the finite universe. When we encounter a line segment of a specified length that contains an infinite continuum, it is because an eternal God has embedded an attribute of Himself within the DNA of the physical creation.

Like beauty, mysteries such as these are designed to act as windows to draw us to God through the creation. Paradoxes like the infinite in the finite are no reason to deny reality or to lose our sanity, but rather a cause for rejoicing in the one true God who made heaven and earth.

The Mystery of the Incarnation

The greatest expression of the infinite within the finite is the Incarnation, when God became a man in Jesus Christ. This is the ultimate example of God’s infinite divine nature in perfect union with His finite creation, the eternal and temporal in perfect harmony with the nature of both being preserved. St. Paul states the following in the epistle to the Colossians:

“For in Jesus all the fullness of Deity dwells in bodily form.”

-Colossians 2:9

When Jesus was a baby, the fullness of Deity dwelt within him. His small body, that had a specific length, width, and height, contained all of the fullness of the infinite, eternal God, the infinite within the finite in perfect harmony, with no corruption of either nature. Rather than being a point of consternation, the infinite within the finite should remind us of the Incarnation and point us to Christ where we can find peace. This does not solve the mystery; rather it glorifies the God who is incomprehensible.

As the book of Ecclesiastes states:

“He has made everything beautiful in its time. Also, he has put eternity into man’s heart, yet no one can fathom what God has done from beginning to end.”

-Ecclesiastes 3:11

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Deo Gratias

Footnotes and Endnotes

  1. Palmer, John, “Zeno of Elea”The Stanford Encyclopedia of Philosophy (Summer 2021 Edition), Edward N. Zalta (ed.)
  2. Plato, Parmenides, Translated by Mary Louise Gill and Paul Ryan with an introduction by Mary Louise Gill, p. 126, Hackett Publishing Company, Inc., Indianapolis, 1996
  3. Dowden, Bradley, “Zeno’s Paradoxes,” Internet Encyclopedia of Philosophy, California State University, Sacramento
  4. Palmer, John, “Zeno of Elea”The Stanford Encyclopedia of Philosophy, Section 2.1 The Argument from Denseness
  5. Webb, Richard, “Metaphysics special: Why is there something rather than nothing?” New Scientist, August 31, 2016.
  6. Palmer, John, “Zeno of Elea”, Section 2.2 The Argument from Finite Size, The Stanford Encyclopedia of Philosophy (Summer 2021 Edition), Edward N. Zalta (ed.)
  7. Ibid.
  8. Ibid.
  9. Ibid.
  10. Palmer, John, “Zeno of Elea”The Stanford Encyclopedia of Philosophy, Section 2.2 The Argument from Complete Divisibility
  11. Dowden, Bradley, “Zeno’s Paradoxes,” Internet Encyclopedia of Philosophy, Section b. iv. Infinite Divisibility
  12. Palmer, John, “Zeno of Elea”The Stanford Encyclopedia of Philosophy, Section 2.3, The Argument from Complete Divisibility.
  13. O’Conner, J.J. and Robertson, E.F., “Georg Ferdinand Ludwig Philipp Cantor,” MacTutor, Last Updated October 1998, https://mathshistory.st-andrews.ac.uk/Biographies/Cantor/
  14. Britannica, The Editors of Encyclopaedia. “Georg Cantor”. Encyclopedia Britannica, 27 Feb. 2021,https://www.britannica.com/biography/Georg-Ferdinand-Ludwig-Philipp-Cantor
  15. Creedon, Ryan, “The Mathematician: Georg Cantor,” Musings of a Math Nerd, October 4, 2012, https://sites.psu.edu/musingsofamathnerd/2012/10/04/40/
  16. To be specific, “color” is a subjective term. It is what our brain perceives based on the specific wavelengths of light that excite the photoreceptors in the retina. Mark Fairchild, Director of the Program of Color Science at the Rochester Institute of Technology and a leading expert in color science, has calculated that the number of color variations that the human race collectively can see is 18 x 1033. In his words, “That might not quite be infinity, but close enough since all those estimated numbers are probably on the low side.” For more information on this, please click the following link, http://markfairchild.org/WhyIsColor/files/ExamplePage.pdf
  17. Pomeroy, Ross, “How Infinity Can Fit Inside the Finite,” Real Clear Science, March 16, 2016
  18. Creedon, Ryan, “The Mathematician: Georg Cantor,” Musings of a Math Nerd

Bibliography

Aristotle, Physics, Translated by Robin Waterfield with introduction and notes by David Bostock, Oxford University Press, Oxford, 1996

Diogenes Laertius, Lives of the Eminent Philosophers, Translated by Pamela Mensch, Edited by James Miller, pp., Oxford University Press, Oxford. 2018

Grayling, A.C., The History of Philosophy, p. 36, Penguin Press, New York, 2019

Moore, A.W., The Infinite, Routledge; 3rd edition, October 29, 2018

Plato, Parmenides, Translated by Mary Louise Gill and Paul Ryan with an introduction by Mary Louise Gill, Hackett Publishing Company, Inc., Indianapolis, 1996

Pruss, Alexander R., Infinity, Causation, and Paradox, Oxford University Press (October 16, 2018)

Rucker, Rudy, Infinity and the Mind: The Science and Philosophy of the Infinite, Princeton University Press; Reprint edition (July 23, 2019)

Stewart, Ian, Infinity: A Very Short Introduction, Oxford University Press; Illustrated edition (July 23, 2017)

Sweeney, Leo, Divine Infinity in Greek and Medieval Thought, International Academic Publishers, September 1, 1992

Sweeney, Leo, Infinity in the Presocratics, A Bibliographical and Philosophical Study, The Catholic University of America, Martinus Nijhoff, The Hague, Netherlands, 1972

This volume was inspired by Etienne Gilson’s query, made in the early 1950s, as to why medieval authors spoke of God’s being as infinite, a statement found neither in Judeo-Christian Scriptures nor in Greek philosophy. Divine Infinity in Greek and Medieval Thought deals with Hellenic and Hellenistic philosophers such at the Presocratics, Plato, Aristotle, Plotinus, and Proclus, as well as with Eastern Church fathers such as Gregory of Nyssa and John Damascene. The book also draws on the works of Augustine and such medieval authors as Peter Lombard, Richard Fishacre, Bonaventure, and Aquinas. It concludes that infinity is predicated of God not only extrinsically but also intrinsically: His very being is infinite – a predication resting on an Aristotelian theory of act/potency or on a Platonic version of participation.

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