PODCAST TRANSCRIPT – Season 2 – Episode 51 – Zeno’s Paradoxes

Gabrielle Birchak/ November 24, 2021/ Uncategorized

Have you ever been in a moment of your life where you wished some­thing would change, but it doesn’t? Time pass­es by, and noth­ing changes. Some­times grad­u­ate school can feel that way. Some­times get­ting a job or work­ing towards a job pro­mo­tion can feel that way. Some­times an exper­i­ment can feel that way. Like when you’re watch­ing a pot of water and wait­ing for it to boil.

Yet when you look back on your life, you real­ize that it has changed. It just took what felt like infi­nite­ly small mea­sure­ments of time to make that change occur. Through hind­sight, when we remem­ber those moments that felt like for­ev­er, we can see how much of our lives changed. In sci­ence and math­e­mat­ics, this is known as the Quan­tum Zeno effect.

By Francesco Pirane­si (etser) (kopie naar) Tomas­so Piroli (teke­naar) — http://hdl.handle.net/11259/collection.71954, Pub­lic Domain, https://commons.wikimedia.org/w/index.php?curid=87600372

Zeno of Elea was one of four promi­nent philoso­phers who attend­ed the Eleat­ic School of Phi­los­o­phy. The oth­er three promi­nent philoso­phers includ­ed Xeno­phanes, Par­menides, and Melis­sos. Zeno was born around 490 BCE and was close to his friend and pos­si­ble lover, Par­menides. We know about Zeno because of the writ­ings of Pla­to. In his work titled Par­menides, Pla­to alludes that Par­menides and Zeno were lovers. He writes, “Par­menides being at that time quite an old man with grey hair and a hand­some and noble coun­te­nance, and cer­tain­ly not over 65 years of age; Zeno about 40 years old, tall and ele­gant, said to have been the favorite of Par­menides.”[1] So, need­less to say, Zeno was not just Parmenides’s arm can­dy. He was also his spe­cial someone.

But, as most women would agree with me, it is wrong to objec­ti­fy Zeno because there is more than meets the eye. Zeno was also very bril­liant and came to be known as a philoso­pher who mas­tered the skill of con­tra­dic­tion.[2] Aris­to­tle wrote about Zeno’s method­olo­gies and believed that Zeno nev­er real­ly searched for truth in his argu­men­ta­tion. Instead, he argued for argument’s sake. This form of argu­ment makes sense because his para­dox­es are like puz­zles that do not so much require a dialect between two indi­vid­u­als found­ed on truth but rather a cre­ative analy­sis based on possibilities.

How­ev­er, this infor­ma­tion doesn’t come direct­ly from Zeno’s writ­ing. It comes from the teach­ings of Socrates and the writ­ings of Pla­to; we do not know exact­ly how many para­dox­es Zeno had writ­ten. At this cur­rent time, we know of ten para­dox­es. How­ev­er, it is pos­si­ble that Zeno wrote up to forty paradoxes. 

One of Zeno’s most pop­u­lar para­dox­es is Achilles and the tor­toise. In this para­dox, Achilles is rac­ing a tor­toise. As the race begins, Achilles gives the tor­toise a head start of 100 meters. Then Achille’s runs and meets the tor­toise at the 100-meter mark. How­ev­er, the tor­toise is still ahead of him by a cer­tain dis­tance. In this case, let’s say the tor­toise is ahead of him by fifty meters. That means Achilles now must run 50 meters to catch up with the tor­toise. So, Achilles runs the 50 meters, but now the tor­toise is lead­ing by 25 meters. And so, Achilles runs the 25 meters, only to real­ize that the tor­toise is 12 meters ahead of him. Thus, Achilles is always behind the tor­toise, which leads to this para­dox that Achilles will nev­er be able to catch up to the tor­toise. This idea is impos­si­ble because we all know that Achilles would be able to catch up to the tor­toise. Regard­less, this puz­zle has been pon­dered and ques­tioned by math­e­mati­cians, physi­cists, and philoso­phers for thou­sands of years!

By Mar­tin Grand­jean — Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=39999637

The alter­nate to this para­dox is called the Dichoto­my Para­dox. In this para­dox, an object will nev­er move a cer­tain dis­tance unless it meets the halfway point first. How­ev­er, because there is no defined dis­tance, there is no halfway defined dis­tance. For the object to move to half the length, it would first have to move to the first half of the first half. In oth­er words, it would have to move a quar­ter of a dis­tance. But, to move to that first halfway mark, it would have to meet the halfway mark before that halfway mark. In oth­er words, the dis­tance between the begin­ning and the end­point becomes so small that it is impos­si­ble for the object to move in the first place. And the rea­son this is called a dichoto­my is because it requires the con­stant process of split­ting the dis­tance into two parts.[3]

The next para­dox is called the Para­dox of Place. This para­dox cen­ters around the idea of dis­place­ment and states that if every­thing in the uni­verse has a place, then that place has a place, and that place has a place.[4]

Then there is the Para­dox of the Grain of Mil­let, which states that if a sin­gle grain of mil­let doesn’t make a sound when it falls, but if 1,000 grains of mil­let make a sound, then 1,000 noth­ings become some­thing. How­ev­er, with tech­nol­o­gy today, we now know that even a sin­gle grain of mil­let makes a sound when it falls.

The Para­dox of the Sta­di­um pro­pos­es two rows of run­ners, with each row con­sist­ing of the same num­ber of peo­ple. In this case, we will say four peo­ple in a row. Each row of four peo­ple runs on the same lane on a straight track. So, in this para­dox, there are two lanes on a run­ning track, and on each lane are four peo­ple. The run­ners in lane one run 400 meters from point A to point B. The run­ners in lane two also run 400 meters. How­ev­er, they are going from point B to point A. In each row, they are mov­ing at the same veloc­i­ty. And the con­cept behind this para­dox is that as each group of run­ners approach the cen­ter, the amount of time it takes them to reach the cen­ter will be dou­bled to reach the oth­er end of the track. It’s a prac­ti­cal math­e­mat­i­cal problem.

Zeno also has three para­dox­es that look at plu­ral­i­ty.[5] These include the Argu­ment from Dense­ness, the Argu­ment from Finite Size, and the Argu­ment from Com­plete Divis­i­bil­i­ty. If you would like to read and hear more on these para­dox­es, please come on over the www.Patreon.com/MathScienceHistory and sign up for a tier that will allow you to access ad-free, ear­ly release content!

By Mar­tin Grand­jean — Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=39999639

Final­ly, we get to one of my favorite para­dox­es, the Arrow Para­dox. This para­dox argues that in order to have motion, an object must change its occu­pied posi­tion. Aris­to­tle describes Zeno’s para­dox stat­ing, “that it is impos­si­ble for a thing to be mov­ing dur­ing a peri­od of time, because it is impos­si­ble for it to be mov­ing at an indi­vis­i­ble instant. This assumes that a peri­od of time is made up of indi­vis­i­ble instants, which can­not be grant­ed.”[6] What Zeno is say­ing is that at a par­tic­u­lar place on a lin­ear tra­jec­to­ry for an arrow, at a spe­cif­ic moment that arrow is at rest. But only at that moment. After that moment, it’s in anoth­er loca­tion which is also at rest. In oth­er words, it is not trav­el­ing dur­ing a moment. And the arrow at its des­ig­nat­ed size, let’s say two meters long, occu­pies an equal space for that instant. As a result, the entire length of its motion con­sists of moments of an arrow at rest. Thus, Zeno argues that the arrow can­not be mov­ing. But because the arrow moves from point A to point B, time con­sists of a mul­ti­tude of instants. This anal­o­gy helps us to under­stand how to mea­sure instan­ta­neous velocity.

Math­e­mat­i­cal­ly, this con­cept is used in cal­cu­lus, and it is defined as the deriv­a­tive of the object’s aver­age veloc­i­ty. In physics, this con­cept helps us to under­stand a con­tin­u­ous func­tion of time and deter­mine a particle’s veloc­i­ty at any point while it is in motion.

This para­dox is foun­da­tion­al to under­stand­ing the Tur­ing Para­dox, also known as the Quan­tum Zeno effect. It is called the Tur­ing Para­dox because it is also sim­i­lar to a para­dox pro­posed by British sci­en­tist Alan Tur­ing in 1958. The idea behind this is that in quan­tum sys­tems, a sys­tem can­not change if you are watch­ing it.[7] [8] How­ev­er, instead of watch­ing it, physi­cists can rapid­ly mea­sure a sys­tem in its known ini­tial state. What is most excit­ing about this ancient con­cept is that it is used in sci­ence and has been ver­i­fied through experimentation.

The idea behind this is that by rapid­ly mea­sur­ing a quan­tum sys­tem, physi­cists can slow down its evo­lu­tion and freeze it com­plete­ly while they mea­sure it. They can keep the sys­tem in its ini­tial state for any cho­sen finite time on one con­di­tion: they must per­form a mul­ti­tude of rapid mea­sure­ments. Back to Zeno’s para­dox, like that arrow that does not move in that one instant, the quan­tum sys­tem can­not change since it is con­stant­ly being observed through rapid mea­sure­ments while it is in its ini­tial state, at time equals zero. By observ­ing and mea­sur­ing it in its ini­tial state, they can apply the process of prob­a­bil­i­ty by pro­ject­ing the system’s state at a lat­er time and apply it to the state that they want the sys­tem to evolve to. The math behind this is not too com­plex. If you are an under­grad­u­ate stu­dent study­ing cal­cu­lus or cal­cu­lus-based physics, you could prob­a­bly under­stand the math that goes into freez­ing a Quan­tum Zeno Effect. You can find the math here at a YouTube chan­nel called Pret­ty Much Physics. The video is titled “How to Freeze a Quan­tum State.” The video is one of the best math­e­mat­i­cal expla­na­tions I have seen on the Quan­tum Zeno Effect.

Some­times life feels like it is stand­ing still. And that is real­ly not a bad place to be. It gives you a chance to observe and think about every­thing in that instant. By tak­ing a moment to observe the world, lis­ten care­ful­ly to what we are hear­ing, think about what we smell, and con­sid­er how we feel, we slow down. It freezes us for just an instant. And after that instant, you’re going to keep mov­ing for­ward. And you won’t be in the same place you are now. You will be mov­ing towards goals, plans, dreams, loved ones, maybe towards work where you can enjoy the com­pa­ny of your favorite co-work­ers, or maybe towards home where you can enjoy a warm din­ner and a cud­dle with your pet. And even if you don’t have any of these things at this moment, I hope you know that like that quan­tum state that encom­pass­es a posi­tion in a three-dimen­sion­al envi­ron­ment, with momen­tum, angu­lar momen­tum, spin, time, and ener­gy, you are mov­ing and evolv­ing. It is excit­ing to think that as humans, we have some­thing very much in com­mon with an ancient para­dox, an arrow, a phi­los­o­phy, physics, and math­e­mat­ics. That is the pow­er of math, sci­ence, and history!

Until next time, carpe diem!

Gabrielle


[1] Thomas David­son, The Frag­ments of Par­menides (St. Louis: E.P. Gray, 1869), 1.

[2] John Palmer, “Zeno of Elea,” Ency­clo­pe­dia, Stan­ford Ency­clo­pe­dia of Phi­los­o­phy, 2021, https://plato.stanford.edu/entries/zeno-elea/.

[3] Aris­to­tle, Aris­to­tle:  The Physics Vol. I, trans. Frances Corn­ford and Philip Wick­steed (Cam­bridge, MA: Har­vard Uni­ver­si­ty Press, 1933), VI:IX  http://archive.org/details/aristotlephysics0000aris.

[4] Aris­to­tle, Aris­to­tle : The Physics Vol. II, trans. Frances Corn­ford and Philip Wicksteed(Cambridge, Mass. : Har­vard Uni­ver­si­ty Press, 1970), IV:I, http://archive.org/details/aristotlephysics0000aris.

[5] Nick Huggett, “Zeno’s Para­dox­es,” Stan­ford Ency­clo­pe­dia of Phi­los­o­phy. Last mod­i­fied, 2018, https://plato.stanford.edu/entries/paradox-zeno/.

[6] Aris­to­tle, Aris­to­tle The Physics Vol. II, trans. Frances Corn­ford and Philip Wick­steed, vol. 2, 1 vols. (Lon­don: William Heine­mann, 1934), 179, http://archive.org/details/in.ernet.dli.2015.183610.

[7] Y. S. Patil, S. Chakram, and M. Ven­galat­tore, “Mea­sure­ment-Induced Local­iza­tion of an Ultra­cold Lat­tice Gas,” Phys­i­cal Review Let­ters 115, no. 14 (Octo­ber 2, 2015): 140402, https://doi.org/10.1103/PhysRevLett.115.140402.

[8] Bill Steele, “‘Zeno Effect’ Ver­i­fied: Atoms Won’t Move While You Watch,” Cor­nell Chron­i­cle, Octo­ber 22, 2015, https://news.cornell.edu/stories/2015/10/zeno-effect-verified-atoms-wont-move-while-you-watch.

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