Monday, October 03, 2005

Mathematics and Mathematicians

Stories abound when it comes to mathematicians. One famous mathematician, Norbert Wiener, was your stereotypical absent - minded professor. Here are some anecdotes about him.

John Challifour told this story in April 2005 to his complex analysis class: Knowing of his absent-mindedness quite well, Norbert Wiener's wife knew it would be necessary to remind him that they were moving house the morning of the last day in their old place. She told him, "Norbert, remember that when you come home from work today, you must go to the new house, because this house won't be ours anymore." "Yes, dear," Professor Wiener condescended to her, in the way academics often do to their better halves. Nevertheless, just as his wife predicted, at the end of the day Norbert did go back to his old house, and he was confused to see that his supper was not ready, and indeed that there were no recognizable objects in the house. After a glance inside, he looked out again and saw a young girl swinging on the door of the fence that enclosed the house. "Little girl," Professor Wiener asked, "who lives in this house?" The girl responded, "It's OK, Daddy. Mommy told me you'd forget to come to the new house." (Here is the reference.)

There are no limits, apparently, to what a mathematician can forget. Another Wiener anecdote:

A student happened upon Wiener pacing up and down in the post office, apparently deep in thought. Struck with the opportunity of meeting the great mathematician, the student was yet hesitant to approach the man. The student thought that interrupting the man might cause him to lose some great result. But the student finally mustered the courage to go and talk with him. "Good morning, Professor Wiener." He stopped his pacing, struck his forehead and exclaimed, "That's it! Wiener!"

Morris Kline, an eminent mathematician and educator, wrote a book entitled Why the Professor Can't Teach. In that book, he makes the comment that mathematicians are, after all, only human, and as regards ego, a rather disagreeable cross-section of humanity.

And there are, of course, plenty of jokes on mathematicians. Mathematicians are known for coming up with elegant, if useless, solutions to problems. An example: a farmer asked an engineer, a physicist, and a mathematician to fence off a maximum area with a fixed length of fence. The engineer makes the fence into a circle, and claims he has found the maximum area. The physicist puts the fence in a straight line, and then says, "We can assume the fence goes off into infinity. Thus, I have fenced off half the earth." The mathematician just laughes at them. He builds a tiny fence around himself and says, "I declare myself to be on the outside."

Here's a mathematical curiosity. Where does this series of equations fail?

1 = 1 * 1 = sqrt(1) * sqrt(1) = sqrt(1 * 1) = sqrt(-1 * -1) = sqrt(-1) * sqrt(-1) = i * i = -1.

The answer is not exactly obvious, I warn you.

Here's another curiosity. Pick out the pattern here. (For those of you to whom I have shown this already, you're not allowed to let the cat out of the bag!)

8, 5, 4, 9, 1, 7, 6, 3, 2, 0.

It's a finite sequence, so your task is simply to explain why the numbers are in this order, and not in some other order. This answer is quite different from the previous mathematical curiosity.

Calculus is responsible for the modern technological age. Here's what historian Arnold Toynbee wrote about it:

[begin quote]
... at about the age of sixteen, I was offered a choice which, in retrospect, I can see that I was not mature enough, at the time, to make wisely. This choice was between starting on the calculus and, alternatively, giving up mathematics altogether and spending the time saved from it on reading Latin and Greek literature more widely. I chose to give up mathematics, and I have lived to regret this keenly after it has become too late to repair my mistake. The calculus, even a taste of it, would have given me an important and illuminating additional outlook on the Universe, whereas, by the time at which the choice was presented to me, I had already got far enough in Latin and Greek to have been able to go farther with them unaided. So the choice that I made was the wrong one, yet it was natural that I should choose as I did. I was not good at mathematics; I did not like the stuff; ...

... Looking back, I feel sure that I ought not to have been offered the choice; the rudiments, at least, of the calculus ought to have been compulsory for me. One ought, after all, to be initiated into the life of the world in which one is going to have to live. I was going to have to live in the Western World ...; and the calculus, like the full-rigged sailing ship, is one of the characteristic expressions of the modern Western genius.

I would hasten to add that Latin and Greek are of exceptionally high value. I think Toynbee would not disagree. He's only saying that calculus was of much more value than he originally thought. I agree with these sentiments, to the point of concluding that no one ought to be allowed to hold a bachelor's degree unless he has seen and at least partially understood the Fundamental Theorem of the Calculus in all its glory. This theorem is easily the most important theorem in all of mathematics. Without it, technology as we know it could not possibly exist.

I would make one more very important comment. God gives us mathematics. It is not somehow independent of Him. If Christ is not Lord of all, then 2 + 2 = 5, or whatever you want it to be. There would be complete chaos. God gave us mathematics to help us in the Dominion mandate, which was to multiply (interesting choice of words there!), fill the earth, and subdue it.

In Christ.

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At 10/03/2005 06:12:00 PM , Anonymous Anonymous said...

"If Christ is not Lord of all, then 2 + 2 = 5, or whatever you want it to be."

What an odd thing to say. How on earth do you propose to justify such a claim?

At 10/03/2005 06:23:00 PM , Blogger Adrian C. Keister said...

Nothing easier. Jesus Christ is Lord of all. Therefore, if we claim He is not, we have a direct contradiction. The result of a contradiction in logic is any result you want.

As an example of such reasoning: the weather is either good or bad. If the weather is bad, the plane will land in Chicago. If the weather is good, the plane will land in San Diego. The plane landed in neither Chicago nor San Diego. Therefore, the plane landed in Denver.

This is a valid argument! Of course, the premises are contradictory, and so from a contradiction you can get any result you want.

At 10/04/2005 08:40:00 AM , Blogger Kim C. said...

Or, to state it another way:
If truth is relative (as so many claim) then, again, 2+2 can be anything I believe it to be.
Who are you to impose your beliefs upon me? You may say that 2+2=4; how do you know? What is your god, your measure for truth and reality? Your own eyes, your logic, your 1st grade teacher? Or perhaps it's majority rule: most people believe that 2+2=4, therefore it's true?
We must ask ourselves, How do we know what we know? Thus we identify our god - or our God.

At 10/04/2005 08:41:00 AM , Blogger Kim C. said...

BTW, I've got the answer to your finite number sequence. It's not just mathematical, right? :)

At 10/04/2005 12:33:00 PM , Blogger Susan said...

Yay, I love mathematics! My favorite class in college was Linear Algebra :).

In answer to the mistake in 1 = 1 * 1 = sqrt(1) * sqrt(1) = sqrt(1 * 1) = sqrt(-1 * -1) = sqrt(-1) * sqrt(-1) = i * i = -1, the square root product property was incorrectly used in this part: sqrt(-1 * -1) = sqrt(-1) * sqrt(-1). The property sqrt(ab) = sqrt(a) * sqrt(b) only holds for a,b in R positive.

One can prove just about anything by incorrectly using mathematical properties. Have you ever heard the proof that 1 = 2?

I'm guessing the sequence has something to do with counting letters for respective words in a common phrase, but I really don't know.

At 10/04/2005 12:42:00 PM , Blogger Adrian C. Keister said...

Reply to Susan,

You've sort of let the cat out of the bag.

Actually, sqrt( a * b ) = sqrt(a) * sqrt(b) works if at most one of them is negative. The only situation in which it does not work is if both a and b are negative real. Interestingly enough, there are no issues if a and b are complex. The exact difficulty here, and this is admittedly technical, is that there is a branch cut discontinuity in the square root function on the negative real axis. So if a and b are both negative real, you're trying to cross that branch cut discontinuity, which is not allowed.

There are many "proofs" that 1 = 2, most of which involve dividing by zero somewhere. One interesting proof that did not use division by zero involved integration by parts. The original integral was definite, and so the final integral must be definite, but many students forget that the boundary term must have limits as well. So if you leave off those limits on the boundary term, you can prove that 1 = 2.

In Christ.

At 10/04/2005 01:08:00 PM , Blogger Susan said...

Oops, I thought we were supposed to post our answer on that one. Sorry :). I thought it was only on the second one that you did not want people you had already told to give the answer. My mistake.

Good point about the product property working if at most one of the numbers is negative, since otherwise we could not simplify sqrt(-5) to 5i.

Have you heard the proof that all positive integers are interesting (my personal favorite)?

Theorem: All positive integers are interesting.

Proof: Assume the contrary, namely that there is at least one positive noninteresting integer. Then by the well-ordering principle there must be a lowest positive noninteresting integer. But hey, that's pretty interesting! A contradiction.

At 10/04/2005 01:12:00 PM , Blogger Adrian C. Keister said...

No, it was ok to post answers to the first one. I merely meant that you had already worked out something of a solution, and I wasn't expecting one to be quite so forthcoming. So congrats to you.

Yes, I've heard the "interesting" joke before. In fact, there was an AMS issue that had a bunch of jokes in it, and it had that one. That doesn't make it any less funny, though. Heehee.

At 10/05/2005 01:56:00 PM , Blogger Esther said...

Fascinating post. Can't say I quite agree with your conclusion about someone understanding calculus before they receive a bachelor's degree!

I detest math with a passion! Always have...and...well, I guess I shouldn't say "always will" because one of these days I just might have to force myself to master it, maybe even like it. Otherwise, homeschooling my children will be interesting!

*waves to Susan* Should have known such a post would interest you!! ;)

At 10/06/2005 07:27:00 AM , Blogger Susan said...

So, I'm lying in bed early this morning and I realize that I incorrectly simplified sqrt(-5) to 5i in my last comment. I'm so ashamed. *hides face*

sqrt(-5)= i[sqrt(5)]
sqrt(-25)= 5i


*waves to Esther* Should have known such a post would not interest you! :) You can just send your children to me and I'll teach them math in exchange for vitamins :).

At 10/22/2005 10:28:00 PM , Blogger natalie said...

Those anecdotes about Norbert Wiener are hilarious! I enjoyed sharing them with my family tonight!


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