<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Mexico_-_Garden_Party_on_the_Terrace_of_a_Country_Home_-_Google_Art_Project.jpg/640px-Mexico_-_Garden_Party_on_the_Terrace_of_a_Country_Home_-_Google_Art_Project.jpg" class="right">
You are pleased to have been invited to a //salon// hosted by the Vicereina María Luisa Manrique de Lara y Gonzaga. The discussion for the evening kept coming back to the intriguing topic of paradoxes. For instance, the philosopher Epimenides said "All Cretans are liars," though he himself was Cretan. Is his statement true or false then?
[[next]]
{(set: $cards to (a:"A ♥","K ♥","Q ♥","J ♥","10 ♥","9 ♥","8 ♥","7 ♥","6 ♥","5 ♥","4 ♥","3 ♥","2 ♥","A ♦","K ♦","Q ♦","J ♦","10 ♦","9 ♦","8 ♦","7 ♦","6 ♦","5 ♦","4 ♦","3 ♦","2 ♦","A ♠","K ♠","Q ♠","J ♠","10 ♠","9 ♠","8 ♠","7 ♠","6 ♠","5 ♠","4 ♠","3 ♠","2 ♠","A ♣","K ♣","Q ♣","J ♣","10 ♣","9 ♣","8 ♣","7 ♣","6 ♣","5 ♣","4 ♣","3 ♣","2 ♣"))
(set: $deck to (shuffled: ...$cards))
(set: $place to 1, $log to (a:))
(set: $v1 to true, $v2 to true, $v3 to true, $v4 to true, $v5 to true, $v6 to true, $v7 to true, $v8 to true)
(set: $top to $deck's 1st, $second to $deck's 2nd, $third to $deck's 3rd, $fourth to $deck's 4th, $fifth to $deck's 5th, $sixth to $deck's 6th, $seventh to $deck's 7th, $eighth to $deck's 8th)}<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/6/66/Zeno_Achilles_Paradox.png/240px-Zeno_Achilles_Paradox.png" class="right">
Another paradox came from the Greek philosopher Zeno of Elea. Aristotle attributed to Zeno the following: "In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead."
This is often illustrated as a race between Achilles and the tortoise. Achilles allows the tortoise a head start of 100 meters, for example. By the time Achilles runs 100 meters, the tortoise has run a shorter distance, say 10 meters. Now Achilles needs some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles arrives somewhere the tortoise has been, he still has some distance to go before he can even reach the tortoise.
Zeno argues that it would take an infinite series of steps for Achilles to catch the tortoise, an impossibility.
[[next|one]]<img src="https://upload.wikimedia.org/wikipedia/commons/f/fe/Lady-Ranelagh-cropped.jpg" class="right">
The next day you received a letter from Katherine Jones Viscountess Ranelagh, writing from England. In it, she recommends a book by John Wallis called //Arithmetica Infinitorum//, published in 1656. "This most excellent book seeks to explain mathematical infinity and how to use the concept to solve problems, like the quadrature of a circle or parabola. He introduces a beautiful notation for unending infinity, the lemniscate."
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/c/c8/Infinite.svg/330px-Infinite.svg.png" class="left" width="200">
[[next|two]]<img src="https://upload.wikimedia.org/wikipedia/commons/8/87/Archimedes_circle_area_proof_-_circumscribed_polygons.png" class="right">
Jones continued "We have discussed here if it should be possible to find the area of a circle by thinking of regular polygons with many sides. With 100 sides, this polygon could scarcely be distinguished from a circle. The formula for a regular polygon is found by dividing it into triangles, each with area 1/2 times the base times the height. Summing all those together gives one half the perimeter times the common height. If we just extend the number of sides to infinity, then this matches the well-known formula of Archimedes for the area of the circle."
[[next|three]]<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Mel%C3%B3n_%C2%A1Loter%C3%ADa%21.jpg/328px-Mel%C3%B3n_%C2%A1Loter%C3%ADa%21.jpg" class="right">
"But we know that Zeno can't be correct. So must we accept infinite steps? And what about 1/∞? We know you have the lottery in Mexico. In a simple lottery, n tickets are sold and one is pulled as the winner. So the chance of anyone ticket winning is 1/n. And if we imagine an infinite number of tickets being sold, then the chance of any one ticket being the winner is 1/∞. Is that number zero? That would mean no ticket won but there would be some ticket that won. So it would be a number smaller than any we could think of but still not zero."
[[next|wrap]]<img src="https://upload.wikimedia.org/wikipedia/commons/3/30/Wallis_Arithmetica_Infinitorum.png" class="right" width="300">
Choose who you wish to talk to. Try not to draw too much attention to yourselves.
[[Calico merchant|Kerala]]
[[Holy Office Prefect|inquisition]]
[[Italian printer|Viviani]]
[[Jesuit philosopher|Kino]]
[[Mathematics professor|Gongora]]
<br> Cards drawn: $log<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Marguerite_G%C3%A9rard_-_A_Young_Woman_Just_Received_a_Letter_from_Her_Husband.jpg/400px-Marguerite_G%C3%A9rard_-_A_Young_Woman_Just_Received_a_Letter_from_Her_Husband.jpg" class="right">
"These are the questions our group here have about the concept of infinity: What is potential infinity versus actual infinity? Why did Aristotle argue against actual infinity? Why has infinity been a theological controversy? What paradoxes can we solve with a better understanding of infinity?"
"Looking forward to your reply, your servant, etc."
[[next|main]]{(if: $v1 is true)[(set: $log to $log + (a:$deck's ($place)), $place to $place + 1)]
(set: $v1 to false)}
<img src="https://upload.wikimedia.org/wikipedia/commons/b/b0/Tea_ceremony_sarasa.png" class="right" width="320">
The merchant of colorful textiles from India does have a moment to talk with you.
"I import textiles from Calicut, in southern India. In that region, astronomy and mathematics have been taken to new heights. You might know that the number system Europeans use was invented in India. It easily allows one to describe extremely large numbers (as needed in astronomy)."
And what about infinite numbers?
<img src="http://hyperphysics.phy-astr.gsu.edu/hbase/imgmth/triser.png" class="left" width="400">
"Yes, in Kerala specifically, the region Calicut is in, a school of mathematics has made powerful advances using series that are infinite in length. I'm not a mathematician, so I don't understand it completely, but it seems that we can add up an infinite number of terms and still get a finite number."
[[next|main]]
Cards drawn: $log{(if: $v2 is true)[(set: $log to $log + (a:$deck's ($place)), $place to $place + 1)]
(set: $v2 to false)}
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/8/87/Obispo_Manuel_Fern%C3%A1ndez_de_Santa_Cruz.jpg/296px-Obispo_Manuel_Fern%C3%A1ndez_de_Santa_Cruz.jpg" class="right">
<h4>Manuel Fernández de Santa Cruz</h4>
You try to stay calm as you ask to speak with the head of the Holy Office in Mexico City. The chief prefect does have a few moments to talk with you.
You thank his excellency for taking the time to speak with you. You ask about infinity and theology.
"The church has long taught that the powers of the Heavenly Father are infinite. We can perhaps find ourselves closer to God by considering infinity. Learned people will read the writings of the saints, like St. Augustine, and also of the great philosopher Aristotle."
"Aristotle pointed out that we can never actually take a process to infinity, we can only continue it indefinitely. A geometer doesn't need to draw a line with infinite length, merely that the line can be extended as far as we like."
<span class="left"><img src="https://upload.wikimedia.org/wikipedia/commons/1/15/Giordano_Bruno.jpg"><h5>Giordano Bruno</h5></span>
"Some have unwisely argued for an actual infinity. Even going so far as to suggest infinite worlds in the universe circling faraway stars. This speculation leads to heresy, inevitably. I'm sure you have heard of the unfortunate case of Giordano Bruno."
You don't want to take any more of the prefect's time and you thank him again.
[[next|main]]
Cards drawn: $log{(if: $v3 is true)[(set: $log to $log + (a:$deck's ($place)), $place to $place + 1)]
(set: $v3 to false)}
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Trattato_delle_resistenze_di_Vincenzo_Viviani_completato_da_Guido_Grandi_%28Firenze%2C_1718%29.jpg/355px-Trattato_delle_resistenze_di_Vincenzo_Viviani_completato_da_Guido_Grandi_%28Firenze%2C_1718%29.jpg" class="right">
<h4>Juan Pablos (Giovanni Paoli)</h4>
You enter the busy establishment. Since the Holy Office only gave licenses for six bookshops in Mexico City, they get a lot of customers. While you have been here before to buy books, today you are looking to talk to Juan Pablos. You know that back in Italy, when he was Giovanni Paoli, he was a colleague of Vincenzo Viviani, who in turn was one of Galileo's last students.
<span class="left"><img src="https://upload.wikimedia.org/wikipedia/commons/0/06/Vincenzo_Viviani.jpeg" width="300"><h5>Vincenzo Viviani</h5></span>
Pablos takes a moment and suggests a conversation in the back. "Yes, Viviani is continuing to try to publish Galileo's writings in Italy. The Church, while initially supporting Galileo, turned against him and so Viviani's efforts have been unsuccessful so far. He has been allowed to publish a defense of Galileo himself."
You say you have heard that Galileo wrote about infinity. "Yes, Galileo gave the example of two circles sharing the same center with one having twice the diameter of the first. The larger circle has twice the circumference of the smaller circle. Now each point on the smaller circle can be matched with a point on the larger circle sharing the same radius. So the number of points on the smaller circle equals the number of points on the larger circle."
You ponder this but Pablos has to get back to work.
[[next|main]]
Cards drawn: $log{(if: $v4 is true)[(set: $log to $log + (a:$deck's ($place)), $place to $place + 1)]
(set: $v4 to false)}
<img src="https://upload.wikimedia.org/wikipedia/commons/d/d3/Padre_Misionero_Francisco_Eusebio_Kino.jpg" class="right">
<h4>Father Eusebius Kino</h4>
Father Kino is preparing to leave on a missionary trip to the north but has a moment to talk with you.
"Ah, the infinite. I have closely observed the stars, planets, and occasional comet. Now with the invention of the telescope, we can obtain a much better understanding of these than the ancients could, like Ptolemy. But I don't think it's prudent to infer that the universe is infinite and there are infinite worlds. This speculation can lead to dangerous and wild arguments."
"In mathematics, my order, the Jesuit order, has trained many skilled practitioners. We carefully study astronomy. But we avoid using a completed or actual infinity and have completely ruled out the paradoxical infinitely small. How can something be not zero and yet then discarded as if it's nothing?"
[[next|main]]
Cards drawn: $log{(if: $v5 is true)[(set: $log to $log + (a:$deck's ($place)), $place to $place + 1)]
(set: $v5 to false)}
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Don_Carlos_de_Sig%C3%BCenza_y_G%C3%B3ngora.jpg/201px-Don_Carlos_de_Sig%C3%BCenza_y_G%C3%B3ngora.jpg" class="right">
<h4>Carlos de Sigüenza y Góngora</h4>
Gongora is just finishing a lecture on Archimedes and how he discovered formulas for the sphere, cone, and cylinder. He joins you after his students leave. "Dr. Wallis of England? Ah yes, a bold defense of infinity in his writings. But lately I have heard of a German philosopher named Leibniz. He declares 'I am so in favour of actual infinity.' and 'the rules of the finite are found to succeed in the infinite.'"
You ask to know more about actual infinity. "Well let's consider Euclid's famous proposition on the quantity of prime numbers. He didn't say that there's an infinite number of primes. He said 'There are more primes than found in any finite list of primes.' Meaning given any finite list of primes, we can prove that the list is incomplete."
Isn't that the same thing as saying there's an infinite number?
"Some philosophers, following Aristotle, say no, that it's more proper to say only that the list of primes is unbounded, we could //potentially// find as many primes as we wish. This makes mathematics more difficult however as we have to use difficult tools like the //method of exhaustion// for proofs like the area formula of a circle. Archimedes wrote a book where he used that to prove the area of a circle is half the circumference times the radius. But I suspect he had a method to first find such results. He made mention of it in his wonderful result on spheres, cylinders, and cones."
[[next|main]]
Cards drawn: $log{(if: $v6 is true)[(set: $log to $log + (a:$deck's ($place)), $place to $place + 1)]
(set: $v6 to false)}
<img src="" class="right">
<h4>Manuel Fernández de Santa Cruz</h4>
You try to stay calm as you ask to speak with the head of the Holy Office in Mexico City.
[[next|main]]
Cards drawn: $log{(if: $v7 is true)[(set: $log to $log + (a:$deck's ($place)), $place to $place + 1)]
(set: $v7 to false)}
<img src="" class="right">
<h4>Manuel Fernández de Santa Cruz</h4>
You try to stay calm as you ask to speak with the head of the Holy Office in Mexico City.
[[next|main]]
Cards drawn: $log{(if: $v8 is true)[(set: $log to $log + (a:$deck's ($place)), $place to $place + 1)]
(set: $v8 to false)}
<img src="" class="right">
<h4>Manuel Fernández de Santa Cruz</h4>
You try to stay calm as you ask to speak with the head of the Holy Office in Mexico City.
[[next|main]]
Cards drawn: $log