Update: After I published this, Uncle Tom in the comment section points out that the struggle of Phidias with irrational numbers is a metaphor for our own struggles to understand God and his creation. Like Phidias in the fable, true wisdom and understanding of Faith, Science and Philosophy begins with humility.
As Uncle Tom put it:
…It also struck me as a perfect metaphor for faith in God. Much of faith is “irrational” –creation, the Holy Trinity, the physical presence of God in the transubstantiated bread and wine…
…Your story brought a smile to my face as I imagined God’s amusement at my frantic efforts to clearly understand Him. I’m sure it’s the same gentle laughter and head-shaking that He has for the wise pronouncement of learned scientists who smirk when believers attribute any natural phenomena to the work of our Creator…
Read the comment section for the rest…
Introduction: Most people are comfortable with the concept and manipulation of irrational numbers. After all, as we learned in junior high, irrational numbers are just numbers who’s decimal expansion “just goes on forever” to the right of the decimal point. Our junior high math teacher also told us that irrational numbers can’t be expressed as a fraction with integers at the top and bottom of the fraction.
We plug in irrational numbers in our calculators and move on with our lives with hardly any thought to the truly mysterious nature of these numbers. The truth is irrational numbers are crucial to the concept of continuity in physics and mathematics and without continuity calculus wouldn’t work. Without calculus we would not be able to describe motion and change. We would not be able to describe the motions of the planets or the flight of a rocket or just about any real world motion. Through calculus, irrational numbers have a deep connection to our dynamic universe.
The Greeks however, were totally perplexed by irrational numbers. Not surprisingly, they also struggled to come up with a mathematical definition of continuity and infinity. It was one of the reasons why the ancient Greeks never discovered Calculus although Archimedes came close to integral calculus. See: http://www.ancientgreece.com/s/People/Archimedes/
Why is this so? For the ancient Greeks, “number” and Geometry or Geometric measurements were the same thing. A number represented a length or area tied to geometric shapes and concepts like “line” , “square” “triangle” or circle”. What we call fractions were not numbers by themselves but rather a comparison of two numbers expressed in terms of each other. For example what we call 1/3 would be to the Greeks an expression that means “a length that is equivalent to 3 times of some another length”.
What shocked the Greeks was the discovery that they could construct through geometry two separate lines who’s lengths could not be expressed in terms of each other. Specifically, they found that the length of a diagonal of a square can not be expressed in terms of one of its sides!
For a geometric culture that valued precision, exactness and truth in its mathematics it’s discovery was a shock. The Greek term for irrational numbers was alogos or “the unsayable”
Legend has it that Hippassus of Metapontum, the man who discovered irrational numbers, was thrown overboard at sea by the Pythagorean cult who who believed that only rational numbers could exist.
The Greeks loved stories of arrogant humans full of hubris getting their comeuppance from the gods. So… for a little fun and in the spirit of the ancient Greeks, my daughter Lydia and I created our own Greek fable on the discovery of irrational numbers.
Before we visit Phidias and Tekton, you maybe asking : Why the focus on Mathematics in this blog? There has been a long standing debate among philosophers, mathematicians and scientists on the nature of mathematical objects. Are they discovered or invented? Do mathematical objects exist outside the human mind? If so, where do they exist? On the other hand, assuming that mathematics is a purely a human invention doesn’t explain their universal nature or how intertwined they are with the physical laws of the universe? Mathematics is at the same time both omnipresent and transcendent in our universe. It also seems to be true that Mathematics is alternatively discovered and invented. There is real mystery here …
Tekton: A young centaur and Phidias’s apprentice
Athena: Greek Goddess of Wisdom
Long ago, when the world was young, there lived in ancient Greece a brilliant mathematician named Phidias and his young centaur apprentice Tekton. Now from all around the world kings and philosophers would seek out Phidias and Tekton as solvers of mathematical puzzles. Mathematicians from the schools of Greece and Alexandria would marvel over Phidias’s elegant geometric proofs on the nature of squares, triangles and circles. The accolades and rewards poured in making Phidias and Tekton wealthy and perhaps just a bit a bit proud.
Now the Greek gods and goddesses who were watching all of this from on high on Mt Olympus were an easily bored and meddlesome bunch. Nothing was more irresistible and entertaining to them then playing tricks on puny, boastful humans who needed to be taken down a notch or two. One day, Athena, the goddess of Wisdom, Prudence and Mathematics was especially bored. She decided to visit the Earth and see Phidias and Tekton for herself and perhaps give them a needed lesson in humility (and amuse herself in the process). So disguised as an old woman, Athena paid a visit to Phidias’s workshop. “So Phidias”, croaked the disguised Athena, “Is it true you are the greatest mathematician in the entire world?”
“Of course it’s true, old woman,” boasted Phidias, “Why not only the whole world, but with a straight edge and compass, I could measure the heavens better than the gods of Olympus!”
“So Phidias, you can measure the heavens better than the gods? For your babbling insolence I am going to turn you into a babbling brook!”
Phidias stood frozen in terror but Tekton, finding his courage, pleaded for Phidias’s life: “Gracious Mistress, all Wise Athena, have pity for my master, who has treated me with kindness.”
Of course, this was just the opening the easily bored Athena was hoping for. “Very well”, said Athena, “I will spare you Phidias if you can tell me the answer to a simple geometric riddle.”
“I’ll do anything, merciful Athena!”
“So Phidias, the geometer who can measure the heavens, tell me what is the proportion of the length one side of a square to the diagonal of the same square. Here you can even use my simple square. “
Athena handed Phidias and Tekton a square with a diagonal that looked like this:
At this Phidias smiled. This will be easy! Tekton, who was wiser to the god’s mischief then Phidias, wasn’t so sure. “Master… I don’t know if…” “Have no fear my apprentice!” interrupted Phidias, and turning to Athena, “Oh Wise and Gracious One, it will done as you command.”
Phidias and Tekton carefully took apart the square Athena gave them and stood one of the sides and the square’s diagonal side by side. “Look Master! The length of the diagonal looks to be about one and a half times the length of the side of the square.”
“You’re right Tekton! All we have to do is cut the diagonal into three equal parts and show Athena that the side’s length is equal two of those parts.”
Phidias and Tekton carefully cut the diagonal into three equal parts and lined them up two of the parts along one of the square’s sides.
It wasn’t even close, the two diagonal parts exceeded the length of the square’s side. Hmm… “perhaps this will be harder then I thought”, Phidias mused.
The ever helpful Tekton suggested that they then cut the diagonal into 14 equal pieces and to line up 11 of them with the side of the square. With a fresh square provided by Athena, Phidias cut the diagonal into 14 equal parts. Tekton carefully lined them up the parts along the length of the square’s side. Seven, eight, nine…. it was close! Ten… eleven! Phidias and Tekton carefully peered at the eleven diagonal pieces to see if they exactly matched the length of the side of the square. If it did, they could tell Athena that “the length of a square’s diagonal to it’s side is the same as 14 is to 11.”
But alas, it still didn’t match up! The eleven pieces were oh so slightly longer then the side. But, they must be close! They just need to cut the diagonal into a larger number of equal pieces and line them up along the side of the square.
At Tekton’s suggestion, Phidias then cut the diagonal into 28 pieces of equal length and lined up 23 of those pieces along the length of the square’s side to see if it matched. If it did, then they would be able to tell Athena “the length of a square’s diagonal to it’s side is the same as 28 is to 23.”
Again, the 23 pieces were just a tiny, tiny bit longer then the side of the square, it was closer then before but it’s still longer. A wave of panic and dread came over Phidias and Tekton, “What kind of sorcery is this Master!”
Of course, Athena, the goddess of Mathematics, understanding exactly why Phidias and Tekton were struggling to meet her challenge, found their frantic cutting and measuring immensely amusing.
And so it went on through out the day. Phidias and Tekton would repeatedly cut the diagonal into a larger and larger number of pieces of equal length but try as they may, they never could find the exact number of pieces that would exactly match the length of the side of the square. At each attempt they would get closer and closer but it would never match up exactly.
Phidias, who was actually was a pretty good mathematician, soon realized that the length of the diagonal was actually a new kind of number. A number that cannot be expressed in terms of other numbers. Defeated, he finally admitted to Athena that it could not be done. Tekton was right, no one can go against the gods and win. Wincing and bracing himself, he fully expected to be turned into a babbling brook for his insolence.
When nothing happened he cautiously opened his eyes to see Athena smiling at him.
“Oh Phidias, you are so amusing!” At this Athena burst out laughing. “It is true, Phidias, you have discovered a new number, the Alogos”.
Again, finding his courage, Tekton boldly asked, “does this mean we will not be turned into a brook?”
“Yes Tekton, you and Phidias are safe, for now… but you will be my personal mathematicians, to be called on to serve me in a moments notice.” At that Athena vanished.
“What will you do now Master with the discovery of the Alogos? You could be the most famous mathematician in all of Greece!”
Phidias thought for moment. The Alogos is an important discovery but there was something about this new number that made him uneasy.
“Yes, Tekton, it is a remarkable discovery, but gifts from the gods are are only to be accepted with great trepidation.. Let’s visit my friend Hippasus in Metapontum and discuss the Alogos with him”
PostScript: This post took longer and was more challenging then I anticipated. What do you all think? I do have some ideas I am working for my next posts and I don’t plan and taking nearly as much time. So lease check back soon!
Some future topics: Science Fiction, Catholic Faith an forgiveness… all in one post! Augustine and military service and being a Christian soldier