Tag Archives: Infinity

Reflections on Zeno’s Paradox —— A Problem about Geometric Series, or Not?

Our knowledge of mathematics develops along with the long history of human civilization. Ancient Greece is usually considered as the cradle of western civilization and the birthplace of mathematics. Here I will discuss the famous Zeno’s Paradox, an intellectual legacy we inherited form those great thinkers in ancient Greece, whose philosophical thinking has been energetic and attractive since ancient times; Then I will have a brief introduction about the “solving” of the paradox using geometric series; In the end I will show that, in some sense, the paradox has not been fully unraveled, by reference to another problem proposed by contemporary scholars. I believe the charm of mathematics will be presented after these efforts.

The ancient Greek philosopher Zeno once created quite a few paradoxes to show his skepticism about some common phenomena. He thought plurality and change were not a universal truth, and in particular, motion was only our illusion. Among his paradoxes that survived today, most of them have equivalent math models. So I will pick up one of them, “Achilles and the Tortoise”, to represent his logic.

The problem is like this: Achilles, the most famous Achaean warrior in Homer’s Iliad, the “swift-footed” hero, is chasing a tortoise. Suppose the initial distance between them is 100-meters, and Achilles’ speed is 10 m/s while the tortoise’s speed is 1m/s. After the chasing begins, Achilles will spend 10 seconds to finish a first the 100-meters. Then he will be at the spot where the tortoise was, at 10 seconds ago; In this period (10 seconds), the tortoise also proceeds 10 meters. Then, to finish the second distance, 10 meters, Achilles spends 1 second, while in the same period, the tortoise proceeds 1 meter; Then it goes on, every time Achilles reaches the tortoise’s previous spot, he still needs to chase more because in that period the tortoise proceeds to another further spot. Hence, Zeno concludes, in this case, Achilles will never overrun the lucky tortoise, which is a very bizarre conclusion against our common sense.

This paradox raised in history of great interest. Many scholars tried to give an answer or explanation, including Aristotle, Archimedes, Thomas Aquinas, etc. The joint efforts of philosophers and mathematicians did not succeed immediately. Without the help of rigorous mathematical tools, their solutions cannot resist questioning from skepticism. To make it more clear, philosophical thinking alone could hardly solve this problem; even if it accomplished so, to convince others to believe this will be no less difficult. Immanuel Kant in his Critique of Pure Reason mentioned that rationality is not omnipotent. It has its own structure of a priori knowledge, and after itself combined with a posterior experience, it becomes useful knowledge, which guides our cognition. However, due to the nature of human’s longing for perfection, eternal, and universality (I would like to add “infinite” here), we are inclined to abuse our rationality and expands it to areas that it in fact does not apply. This is to say, human rationality arises from very specific experience, and is applicable there; but due to our preference, we create some concepts (like “perfection”, “eternal” and “universality” I mentioned above), which is non-existent in real life and also beyond rationality’s realm. But we are so confident and accustomed to our rationality that we apply it to those concepts generated by ourselves, without noticing it is not applicable there. After the abuse, confusion subsequently follows.

I really admire Kant’s genius in his noticing that a critique of human reasoning itself is very much needed. And I would use his theory to help form my personal understanding about this problem. But I will leave it here and deal with it later, after the introduction of the rigorous mathematical proving with respect to this problem.

Thanks to the invention of calculus and the epsilon-delta language, we now have the rigorous mathematical tool to deal with problems about infinity. A brief solution is to use geometric series. With respect to the “Achilles and Tortoise” problem we mentioned above, the time that Achilles needed to catch up with the tortoise can be represented as:


This means Achilles could overrun the tortoise after approximately 11.11 seconds. Thus, the sum of a series with infinite terms, are quite possibly finite, which may be beyond our predecessors’ understanding. But, does this problem stops here? Some modern scholars believes not. Why, because we are not sure what is Zeno’s true meaning. This is to say, the result of the formula may not answer Zeno’s question. Let me here give an example, which is called Thomson’s Lamp: suppose there is such a lamp with a toggle switch. After you start the game, it’s switched one after 1 minute, then switched off after half minute, then on after fourth minute, then off after eighth minute, and so goes on. The sum all the time we spend in the game is 2 minutes, according to the same method about sum of geometric series above. Then, one question follows: After exactly two minutes, is the lamp on or off?

This time we find it’s also very difficult to answer this variation of Zeno’s paradox, even if we know geometric series. And because of this, I believe to use geometric series could give a result, but could not solve the problem about the process, which may be Zeno’s real point. And Kant’s argument gives me guidance in understanding this paradox. Also, there is a scholar making this point more explicitly: according to Pat Corvini, this paradox arises from “a subtle but fatal switch between the physical and abstract”. When we expand our mathematical abstractions to the physical world, even it’s applicable almost everywhere, with respect to some concepts, it’s quite unimaginable and confusing. This time, we may still need mathematics as well as philosophy, to finally solve this paradox.


Binmore, K. G. & Voorhoeve, A. (2003). Defending Transitivity against Zenois Paradox, Philosophy and Public Affairs, Vol.31(3), pp.272-279

John, L. (2003). Key Contemporary Concepts from Abjection to Zeno’s Paradox, Ebrary, Inc.

Wikipedia, Zeno’s paradoxes, http://en.wikipedia.org/wiki/Zeno%27s_paradoxes

To Infinity and Beyond

Image: NASA/Paul E. Alers.

Image: NASA/Paul E. Alers.

In everyday life, infinity has multiple meanings. To most people, infinity means being bigger than any number, the entirety of everything, or something that has no ending. In mathematics, infinity is strictly defined, but for the purposes of this post we will use a more intuitive definition: bigger than any finite number or an unnamed number that is arbitrarily large.

Much like the number zero, infinity is a concept that had been controversial amongst ancient mathematicians and barely accepted by modern mathematicians before the twentieth century. The ancient Greeks are the first recorded society that used the concept of infinity, although they did not readily accept it. They believed (as did mathematicians and philosophers for two thousand years afterward) that there was a potential for something to be infinite but that it could not actually be infinite. Much like the set of integers could potentially be infinite by adding one to a get another larger number, but the set of all infinite integers did not exist.

It was not until the late 1800’s that mathematician Georg Cantor formalized and proved the concept of infinity. Not only did Cantor prove the existence of infinity but, much to everyone’s surprise, he also proved that there are different sizes of infinity.

Infinity. Image: Public domain, via Wikimedia Commons.

Infinity. Image: Public domain, via Wikimedia Commons.

How could something that is arbitrarily large be bigger than something else that is arbitrarily large? Take for example the set of all the even numbers, (2,4,6,…) and the set of all the natural numbers (1,2,3,…). The most obvious answer would be that the set of all even numbers is half as large as the set of all natural numbers, even though both of them are infinite. In fact, the correct answer is that the two sets of numbers are actually the same size. How is this so? Cantor showed this by assigning each element in each of the sets a number. The number 1 from the real’s and the number 2 from the evens was assigned the number 1, 2 and 4 were assigned the number 2 and so on. By doing this he showed that no matter how large either set becomes he can still assign them the same numbers, therefore proving that they are the same size. This is what is known as countable infinity.

So then how do we get infinities that are different sizes? Think of the integers on a number line, all the way from negative infinity (Yes, this is a real thing. It just means a negative number that is arbitrarily large) to positive infinity. It is quite large but as we talked about in the previous paragraph it is still countable. Now, take only the segment of line between one and two. In this segment you have the numbers 1.5, 1.2, 1.24, 1.247,… In fact you have so many numbers in this small segment that it would be impossible to list them all. To create a number that is different than any number you have chosen all I have to do is take the first digit of the first number and increase it by one, then take the second digit of the second number and increase that by one and so on and so forth. In doing this I have created a new number that you have not listed. We therefore have a set that is so large that it is uncountable.

The concept of infinity is very important in many areas of math. The field of calculus depends entirely on it and many other fields in mathematics use the concept in important theorems. In fact, the infinite arises much more in mathematics than does the finite.







My infinity is bigger than your infinity

When I was a child, I purposely found something to think about to help me fall asleep. Usually I picked cartoons or super powers, but sometimes things just came into my head, like it or not. What was the worst? Thinking about heaven. At first, heaven seems all right. There is a lot to do, gold everywhere (though no purpose for it), people are nice (it’s a prerequisite), you get to see most of your family, and there is plenty to eat (though no one is ever hungry). Anyway, I start thinking about FOREVER.

pic1cropAt first, it is just a sensation; a weird sensation like tingling and falling and nothingness. It is not a sensation that I can make sense of really because forever doesn’t really make sense, at least not to a 10 year old. I try to get away from forever but forever is a huge part of the definition of heaven. Then, the opening credits of the Twilight Zone, with the music, and starry sky, usually appear. Fade to myself standing, looking at heaven, in the dressing room mirrors of infinity. You know, when dressing rooms have those three mirrors that are angled just perfectly so the images are smaller and smaller replicas of one another, on and on, into infinity. This picture, and thoughts of the foreverness of heaven, kept me up at night as a child.

I am glad to say that forever no longer keeps me up at night. While I still find no comfort in the foreverness of heaven, the lack of a middle ground between forever and my time on earth is what usually keeps me up at night now. However; I still can’t stand it when mirrors are angled that way. It creeps me out, and I can’t help but wonder if there is an end, or if I can find a flaw from one image to the next. In my opinion, we are not meant to look into infinity like that, squarely.

When beginning to pursue mathematics, I thought math might clarify, or in some way define, forever (or as adults call it, infinity). On the contrary, Math has actually made it stranger. Theories in math have shown numerous types of infinity, and infinities within infinities, and sizes of infinities, and calculations of infinity. None of this brings me any comfort, except to say that we obviously don’t have this figured out yet because that is just not possible. Infinity is infinity, and it is very large, incalculable and non-denumerable, and there is only one kind; it is called forever. Heaven can only exist in one, all-encompassing infinity.


When reading A History of Mathematics, I read about Zeno’s paradox. That led to an internet search, and then to Numberphile. I watched the video, accepted the idea, and left it alone. The solution seemed reasonable enough. Later in the semester, I was required to do a research project. By some unknown scheme, we picked Georg Cantor, whom I had never heard of. If you haven’t either, he is the creator of set theory but also perhaps the mathematical or scientific father of infinity. You just can’t shake things off in life. They follow you.

My research for that project led me to question the mathematical view of infinity. Let me start by saying, I know very little of Math’s view of infinity. It seems to be an infinite topic. This is where I am in my understanding – so please comment, post, reply, educate me, and critique my understanding. Calculus one is a prerequisite for the course, and being a rule follower, I have that. So, I had experience computing limits to infinity. That is relatively easy. BUT, those are just numbers. They aren’t real things. Numbers aren’t real. So, of course I could compute the infinity of something that isn’t really real. What numbers represent is real; like Zeno’s paradox. Zeno’s paradox applies numbers to something real – something actually happening in the world (theoretically). In other words, when I take the limit of a sequence that goes to infinity, it has no relation to time or space. It is just numbers. But, if I were taking the limit of Zeno’s paradox to see how far Zeno actually travels, or to find the time it takes to travel, or to see if he can ever catch the turtle, I would have to do so in relation to time and space. When I do that, the exact opposite answer occurs. Zeno will never catch the turtle. That mathematics isn’t computing real infinity or perhaps all of infinity is perhaps echoed by the Numberphile narrator when he asks, “What I want to ask a physicist is, can you divide space and time infinitely many times?” Similarly, Kelly MacCarthur wonders in the Calculus 2 video used for online math courses, “Can I take infinitely many steps?”

However, if all of space and time existed at one instant, forever, then Math has it right. It could calculate the infinite because it occurs all at once. There is no sequence, event after event – in essence, no time or space really because it is all at once, everywhere. Yes, there are scientific theories, philosophies, and religions which believe this is the case. Of course, this idea is contrary to most people’s understanding of infinity. Whenever math instructors talk about infinity, they always say, “Infinity is only a concept. It is not a number.” Yes, it is only a concept but is it also something real? If it is only a concept then why are we computing something real that is a concept? Why would we bother to compute a concept? It seems like Math is walking a funny line here.

Math has worked something out though. I’m just not sure what it is. Math is summing an infinite process (as if infinity happened to end). Obviously, Math’s understanding of infinity has proven useful in mathematical calculations and many practical applications. To paraphrase others before Cantor, “It works. So, no need to define it. It works.” So, Math has worked something out about infinity but what has Math worked out, and is it really infinity?

pic3cropMathematicians always like to joke about engineers rounding numbers to 3 or 4 places because it doesn’t really matter to engineering after that, but is mathematics rounding off infinity or at least only capturing some aspect of infinity? After all, how can there be different types of infinity? My preferred illustration for the existence of multiple infinities is from Galileo. Galileo used a thoughtful but intuitive approach to understand infinity. He drew a circle. Then, he drew an infinite number of rays from the center of the circle. These rays filled up the space inside the circle. But then, he drew a larger circle around the smaller one and extended those rays to the larger circle. Though he drew as many rays as possible (an infinite number perhaps), the infinite number of rays did not fill up the larger circle; there were spaces between the rays. This led him to believe that first infinity was not large enough for the second circle; not even close. He would need another size of infinity to fill up the larger circle. [BAM! PHH! Did your mind just explode?] It is important to note that intuitively, his illustration makes sense. However, with today’s current understanding of infinity and better ability to calculate infinity, we now know that the infinity in the smaller circle leaves no space between the rays when extending to a larger circle. But, I liked his intuitive approach. Though intuition seems to be severely lacking when it comes to infinity.


Dangerous Knowledge: http://topdocumentaryfilms.com/dangerous-knowledge/
Georg Cantor His Mathematics and Philosophy of the Infinite by Joseph Warren Dauben
TML: The Infinities In Between (1 of 2): http://www.youtube.com/watch?v=WihXin5Oxq8
TML: The Infinities In Between (2 of 2): http://www.youtube.com/watch?v=KhgNiqI-bt0
Infinite Series: http://stream.utah.edu/m/dp/frame.php?f=f55f900bec01a3106121
Zeno’s Paradox – Numberphile: http://www.youtube.com/watch?v=u7Z9UnWOJNY

My new bumper sticker.