bellasemplicita

Regaining a lost love for beauty and simplicity.

Tag: mathematics

A Few Updates

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Since I have been so long absent, I hope it is alright if I share a few of the things happening in my life.

– I am enjoying my summer – immensely!  I am trying to really soak in the beauty of the smells of rain and dirt, the brilliance of green, the warmth of the sun, the smell of roses.  I run through a small forest preserve nearly every morning and I love that I am able to follow the change of the seasons in this place.  At the moment there are myriads of birds and in the morning their singing is so calming.

– I am attending Wheaton College in the fall.  I am so excited about this!  My good friend and I will be rooming together so we are busy planning our room and discussing the new excitements of college life in general.  I am entering as a Mathematics major (and no, I do not want to teach) and an Art History minor.

– Now for me, this is really big news: I am running Cross Country in college!!  This is huge because about eight months ago I absolutely hated to run.  However, after forcing myself to ‘do it’ nearly every morning, I realized I had fallen in love.  I am excited to begin training with a team and I hope I will be able to greatly improve my speed.

– Yes, there are many other important things happening in my life – perhaps I will share them at a later time.  But for now, I will end with an extract from a poem by William Blake.

‘And we are put on this earth a little space

that we may learn to bear the beams of love.’

Solution

Here is the solution I promised for “A Problem” – Grandfather’s Breakfast.  I am sorry that it is so late.  I have not posted lately as I have been getting ready for this weekend.  O was gone Saturday through Tuesday at Summit.  (Maybe a post about it later.)  As I am not the best at explaining things myself, I will just give the book’s solution.

To restate the problem:

Grandfather is a very hard-boiled customer.  In fact, his eggs must be boiled for exactly 15 minutes, no more, no less.  One day he asks you to prepare breakfast for him, and the only timepieces in the house are two hourglasses.  The larger hourglass takes 11 minutes for all the sand to descend; the smaller, 7 minutes.  What do you do?  (Grandfather grows impatient!)

Solution:

You must begin by turning over both hourglasses.  Turning over only one can only return you to the initial situation!  Now there are two simple solutions:

(a).  After turning over both hourglasses, put the egg into boiling water right after the smaller one (7 minutes) empties.  Let the larger one run out (4 more minutes) and turn it over (11 more minutes).  After it runs out again, take the egg out of the water.  The egg will have been boiled for exactly 15 minutes (4 + 11), but this solution has one flaw: the whole procedure takes 22 minutes, and your grandfather is both impatient and a retired mathematician.  You need an optimal solution, so you boldly

(b). put the egg in the water at the instant you turn both hourglasses over.  Now, after seven minutes the small hourglass is inverted, and there are 4 minutes left in the large one.  When the large one runs out 4 minutes later, there are 3 minutes left in the top of the small one, but more important, there are 4 minutes worth of sand in the bottom!  With a sigh of relief, you invert the small hourglass at this instant, wait for it to run out, and present your grandfather with his breakfast.  This solution takes exactly 15 minutes, and you can do no better.

“Thinking About the Infinite”

A Mathematical Concept — and a Glimpse of God

Another article that I read in the SPU Response and thought interesting.

The word “infinite” comes from Latin in (meaning “not”) and finis (meaning “end”). Scripture does not actually use the word “infinite,” but refers to God as being great beyond our comprehension, eternal, or without end in time. So the word “infinite” used in reference to God means that some aspect of God’s character is without limit.

I come to this topic as a mathematician, not as a philosopher or a theologian. From my perspective, mathematical “sets” — specifically infinite sets — can provide a small insight into the nature of God.

When relating the language of infinity to God, we are using an analogy. The mathematical concept of infinity is very focused, and even though we sometimes don’t understand mathematical infinity very well, we know much more about it than we do about God.

Stated simply, a set is a collection of objects. These objects might be words, such as the number words {one, two, three}. Two sets have the same number of elements if they can be put into one-to-one correspondence. This means they can be paired up with each other so that nothing is missed, and nothing is counted (or paired up) more than once. All sets have the same number of elements as each other have the same name: the last number word we reach when counting the elements. This is called the “cardinal number” of the set. For example, if I count the number of words in the set {one, two, three}, the last number I say, three, is the cardinal number of that set. Any set that can be counted in this way, where you end at some number, is called a finite set.

What if I have a set for which no such stopping point exists, say the set of all counting or natural numbers, ℕ? If we try to count all these numbers, we never finish. This set is not finite, so we say it is infinite.

This is an indirect method of defining a concept — by defining what it is not. So to say a set is infinite is to say that it is “not finite.” Indeed, some of our descriptions of God also speak about what God is not — immortal (not mortal), invisible (not visible), unchangeable (not changeable), and others.

Mathematics also has a direct definition for infinite sets. Think about the even numbers, {two, four, six, eight …}. This set can be put into one-to-one correspondence with the natural numbers, by pairing any even number with half itself. We’re essentially counting the even numbers and finding that there are the same number of them as there are natural numbers.

So we have two sets, evens and natural numbers that are the same size, but the evens are a smaller set than the natural numbers because the set of evens fits inside the naturals with some numbers left over. (This is called a proper subset.) This property gives us a direct definition of an infinite set: a set that can be put into one-to-one correspondence with a proper subset of itself.

Looking at infinite sets in this direct way allows us to see some of the ways infinite sets are different from finite sets. Consider the simple act of subtracting. If you subtract with finite sets (think “take away”), you always get a smaller set. However, with the infinite set ℕ, taking away the odd numbers leaves the even numbers, another infinite set. Or, you can take the infinite set of all numbers greater than one from the infinite set ℕ and get a finite set, {one}. In other words, infinity minus infinity could be zero; it could be one; it could be infinity.

Infinite sets simply don’t follow the same rules as finite sets. Subtracting a finite set from an infinite set leaves a set the same size as before, infinitely large. Any finite set is negligible — it might as well be nothing — in comparison to the unimaginable size of an infinite set.

By analogy, as God allows us to make choices, giving up some power, God remains all-powerful. In mathematics, we are very careful when discussing the infinite, because we know that infinite sets don’t behave like finite sets. Our mathematical intuition built on our experience with the finite does not apply to the infinite.

Sometimes we Christians need to recognize with similar humility how much we don’t understand God’s ways. In our thinking we must allow God to be God, to be beyond our understanding, much as mathematicians allow the infinite sets of mathematics to be different from the finite.

Robbin O’Leary is a professor of mathematics at SPU.

A Problem

Recently someone allowed me to borrow a book entitled “The Chicken From Minsk”.  It contains many different traditional Russian math and logic problems, some of which are very irritating.  If you solve the problem below post a comment on the explanation.  For those of you who don’t feel like solving it, I will post the solution next week.  Have fun!

Grandfather’s Breakfast

Grandfather is a very hard-boiled customer.  In fact, his eggs must be boiled for exactly 15 minutes, no more, no less.  One day he asks you to prepare breakfast for him, and the only timepieces in the house are two hourglasses.  The larger hourglass takes 11 minutes for all the sand to descend; the smaller, 7 minutes.  What do you do?  (Grandfather grows impatient!)

Hint:  Think carefully about the first step.  The logic is unique!