Category Archives: Mathematics

Happy Pi Day!

Happy Pi Day!!!

Yes math geeks, it’s March 14th otherwise known as 3.14, the approximation of the number Pi. The number Pi or “π” is the ratio of a circle’s circumference to its diameter. No matter how big or small  you make the circle that ratio will remain the same.  Pi has been known to the ancient Babylonians and Egyptians although it is through Greek mathematics that Pi has achieved cultural prominence.

Phidias thinks he can do ths

π  is an irrational number. Irrational numbers cannot be expressed as the ratio of two integers and their decimal expansions are infinite and non-repeating. In other words, the decimal expansion of π goes on forever. The first fifty digits of π are 3.14159265358979323846264338327950288419716939937510.

Using computers, mathematicians have calculated the first 10 trillion digits for π ! Now before you get excited about that accomplishment please stand up, reach to the sky and jump straight up. For that brief moment you are immeasurably closer to nearest galaxy 1 million light years away then we are to finding all of the digits of π.  Pi is so huge that no matter how far you go in expanding Pi you are just getting started.

Pi: Transcendental and Irrational

Pi has the extra distinction of not only being Irrational but Transcendental. There are two kinds of Irrational Numbers: plain old Irrational numbers like the square root of 2 and Transcendental Numbers like “e” and “π”.  Transcendental Numbers are not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. Here is an example of a simple polynomial equation: 0= x2 + 2x +5.  Because π is Transcendental, you could not calculate it exactly using addition, subtraction, division and multiplication using rational numbers in a finite number of steps. You could use any rational number(s) you want but you will never get there.

Though only a few classes of Transcendental numbers are known (in part because it can be extremely difficult to show that a given number is transcendental), Transcendental numbers are not rare. Indeed, almost all real and complex numbers are Transcendental, since the algebraic numbers are countable while the sets of real and complex numbers are both uncountable.

The most Beautiful Equation in Mathematics

Pi is mysteriously linked to other fundamental mathematical constants in Euler’s brilliant equation:

Euler's formula

Here linked together:

– The number 0, the additive identity.

– The number 1, the multiplicative identity.

– The number π

– The number e, the base of natural logarithms, which occurs widely in mathematical and scientific analysis (e = 2.718281828…).

– The number i, the imaginary unit of the complex numbers, a field of numbers that contains the roots of all polynomials (that are not constants), and whose study leads to deeper insights into many areas of algebra and calculus, such as integration in calculus.

 Is Pi normally distributed?

Another mystery of π.  Imagine a 10 sided die with the digits 0 through 9 on the sides and imagine using this die to determine the next digit of Pi.   Roll the die….. a “4”.  What was the odds of getting a 4? Why it’s 1/10 because there are 10 possible outcomes. What are the odds of getting two 4’s in a row? It’s 1/10 times 1/10 or 1/100. How about three in a row? 1/10 x 1/10 x 1/10 or 1/1000.  It’s possible but not very likely. Now imagine a hundred “0”s in row? Or a million? Or even a trillion… The odds of getting the same number on a trillion consecutive rolls of a die are extremely small.  However, If the digits of π are normally distributed like the would be if you rolled a ten sided die than any number sequence you can imagine would eventually show up in Pi if you calculated far enough.  Why? Because the occurrence of any event, not matter how small the odds as long as it isn’t zero, is 100% over infinity.  No one has proved that this is true for Pi or any irrational number but it does appear to be true.  The first 10 trillion digits of Pi look like they are normally distributed though.  Some examples  are the six “9”s at the Feynman Point (762nd digit) and  at the 193,034th place and there is a sequence of 12 zeroes starting at position 1755524129973.

Philosophical implications of Pi.

However, If it the digits of Pi are not normally distributed like we suspect then how are they generated?  Would that imply that there is a pattern?

On the other hand, saying that Pi is normal and the digits of Pi are like the random rolls of a ten sided die raises some problematic questions too.  Does it imply that the digits of Pi are determined  only as mathematicians and their computers calculate them just like the quantum states of sub-atomic particles are determined by our measuring them?  If that’s the case, could a highly advanced civilization  “force” the digits of Pi into patterns as they calculate them?  (Don’t laugh, it’s the premise of Carl Sagan’s novel “Contact”). If this is true it then the implications are stunning: There is a mysterious link between consciousness and the mathematical fabric of the universe.

Looking at it from the other side doesn’t let you off easy either.  If the entire infinite set of the digits of Pi are already predetermined then where are they stored? In what fashion does this information exist independent of human thought?

Hmm…Deep questions with even deeper implications for the meaning of human existence and the meaning of truth no matter what direction you go.  It’s probably a good thing that St Patrick’s Day is three days after Pi Day!

Post Script: For a fun way to look at the discovery of irrational numbers check out the Fable of Phidas and Tekton

https://augustinesalley.wordpress.com/2014/06/08/phidias-and-tekton-an-irrational-fable/

Phidias and Tekton: An irrational Fable

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! Picture2

 

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.

Picture1

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 …

Phidias: Ancient Greek builder and mathematician. Good natured but a little too sure of himself and boasts to the gods Phidas

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.  A bored Athena 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. Old Woman Athena “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!”

At these words Tekton exclaimed, “Master, you go too far, you know the gods love to bring down the prideful!” Tekton

Tekton had no sooner finished speaking when in a burst of light and thunder the old woman was gone and standing there in her place was the tall and graceful Athena with lightening in her eyes.  Athena surprises Phidias and Tekton

“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:

IRRAT1

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 thinks he can do ths

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.

Athena is amused

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”

New Adventures

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

 

The Limits of Human Knowledge

There is a fifth dimension beyond that which is known to man. It is a dimension as vast as space and as timeless as infinity. It is the middle ground between light and shadow, between science and superstition, and it lies between the pit of man’s fears and the summit of his knowledge. This is the dimension of imagination. It is an area which we call the Twilight Zone.

 Rod Serling, Writer (1924-1975)
Rod Serling, Writer (1924-1975)

Rod Serling’s introduction to his Twilight Zone series is so familiar to us that today we hardly give much thought to its poetic power or to the radical idea Serling invites us to contemplate: That there are real limitations to human knowledge.  So go head, get comfortable, grab that glass of wine or cup of coffee, forget what you know about the Twilight Zone and read Mr. Serling again.

Done?

Great!… If anyone of us would be  asked what was the most radically transformational idea to come out of the 20th Century most of us would say Einstein’s Theories of General Relativity and Special Relativity.  No debate right? But what about the second most transformative idea? Hmm… not so easy. Well here’s one to consider: Austrian  mathematician and philosopher Kurt Godel and his Incompleteness Theorems.

It’s the year 1900, the beginning of a century of promise and progress. In science and mathematics optimism and intellectual hubris was rampant. Conventional wisdom held that Science was almost complete and only some tidying up was needed.

On a hot August afternoon of that year in a conference in Paris, brilliant 38-year-old German mathematician David Hilbert proposed that Mathematics be put on a firm of foundation of a limited set of self-evident axioms (or facts) from which all mathematics could be derived from using simple rules of logic.  Sounded simple enough and intuitively, it made sense.  The project became known as the “Hilbert Program”.

David Hilbert  German Mathematician 1862-1943
David Hilbert
German Mathematician 1862-1943

For the next 30 years though, mathematicians and philosophers struggled to define any such list of axioms.  The difficulty seemed to be with coming up with enough axioms to cover all of mathematics without creating paradoxes and contradictions. It seemed you could have one or the other but you could not be both complete and paradox free at the same time. In 1931, Kurt Godel demonstrated why.

Kurt Godel with Albert Einstein
Kurt Godel with Albert Einstein

In 1931, Kurt Godel shattered the Hilbert Program with his Incompleteness Theorems.

The first Incompleteness Theorem states that an all-encompassing axiomatic system can never be found that is able to prove all mathematical truths, but no falsehoods.  The second Incompleteness Theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency.  If an axiomatic system can be proven to be consistent from within itself, then it is incomplete. In other words, there are mathematical truths that are not provable within the realm of mathematics itself.

 At that point, everyone threw down their pencils and went to their corners. David Hilbert himself was stunned into silence. Philosophically, Godel’s theorems imply that truth cannot be defined solely in terms of mathematical or scientific provability. So it is one thing to be provable, and a different thing to be true. Truth out runs provability.

 The implications are still being debated today. As mathematicians continue to specialize, any possible impact caused by the Incompleteness Theorem is assumed away as not impacting their limited areas of research. Some physicists, in their pursuit of the Theory of Everything that will completely explain the Universe, dismiss the Incompleteness Theorem by saying that it only affects arithmetic, not physics.

But is that a safe assumption? Today, do we live in an age of intellectual hubris like we did 114 years ago just before Einstein shattered everyone’s scientific complacency? Does the Kurt Godel’s Incompleteness Theorems imply limitations to human knowledge?  In other words, can we fully understand the universe while at the same time be within it?  Or will science soon develop a complete and consistent Theory of Everything without having to going outside of science itself?  Or is the “Theory of Everything” the Hilbert Program of our day, doomed for a shocking revelation?

Now, get up and go get yourself a stronger drink, relax and read Serling again and ask yourself: Just where and what are the limits of human knowledge?

Post Script: I love mathematics, and as a small child I was fascinated with its deeper truth and beauty but hated the mundane as taught by my elementary teachers.  While I struggled with simple multiplication tables I would surprise my teachers with my questions and insights. As my teachers explained to my parents about my mathematical aptitude: “Joe can climb mountains but he can’t cross a street.

Mathematics will continue to be part of this blog.  For example, did you know that late 19th Century Mathematician Georg Cantor proved that some infinite sets are bigger than other infinite sets? Or that as a consequence some sets of numbers are not only uncountable, but also unnameable?

Some good books on Mathematics that touch on topics discussed in this post:

    Godel’s Way, Exploits into an undecidable world ( 2012) by Chaitin, da Costa and Doria

Everything and More, A Complete History of Infinity (2003) by Wallace: A very enjoyable history of the mathematical concept of Infinity from the ancient Greeks to Georg Cantor

“It has forever been thus: So long as men write what they think, then all of the other freedoms – all of them – may remain intact.  And it is then that writing becomes a weapon of truth, an article of faith, an act of courage.” 
― Rod Serling