Tag Archives: Mathematics

The Alexandrian–nay, Gaussian–Solution

Carl Friedrich Gauss

A year ago, I wrote about “the Alexandrian solution” to the Gordian Knot. I saw this as a metaphor for all instances in which genius lies in espying the simplicity hiding in a complex situation.

It just occurred to me that Carl Friedrich Gauss was, at the age of 10, just such an Alexander the Great. (Alexander was young, too, of course. In espying simplicity, it seems to help to be young — ie, intellectually daring, unspoiled by the complexity of life, et cetera.)

In about 1787, the young Carl Friedrich sat in class when the teacher told the kids to find the sum of the numbers 1 through 100. In other words:

1 + 2 + 3 … + 100 = ?

Think of this as the Gordian Knot. The teacher assumed that the kids would be busy for a long time, practicing their addition skills. Gauss reacted just as Alexander would have (I take poetic license):

This is too f***ing boring. There must be a simpler way.

Did Gauss get nervous as the other kids pulled ahead adding numbers, while he was still at 1, searching for simplicity? I don’t know. But he found it:

He realized that the numbers came in pairs:

1 + 100 = 101
2 + 99 = 101
3 + 98 = 101

(and so on until:)

50 + 51 = 101

So the sum of the numbers is simply (simply!)

50 x 101, or 5,050

You might, if you’re a regular on The Hannibal Blog, be guessing that I’m much less interested in sums of numbers than in, shall we say, Gordian Knots and Alexandrian Solutions in general — meaning in other, preferably surprising, walks of life.

If you can think of any instances in which daring simplicity blasted through mind-numbing complexity, drop me a line.

Great, if not greatest, thinker: Gödel

Gödel

Loyal readers of The Hannibal Blog are by now familiar with the wit of one Mr Crotchety who has already made cameos as a poet of Haikus, Senryus and Limericks. As soon as I began my series of posts in search of the world’s greatest thinker ever, Mr Crotchety began lobbying fiercely for Kurt Gödel as a candidate. Since we are now in the sub-series of posts on “honorable mentions”, I have invited Mr Crotchety himself to make the case for Gödel. Here it is, in Mr Crotchety’s words:

I am adding some additional criteria to the Great Thinker debate:

  • Do his/her great thoughts presently frame the basis of all other thoughts?
  • Do his/her great thoughts have anything to do with the meaning of life?
  • Did he/she go bonkers?

One of Gödel’s great thoughts is The Incompleteness Theorem. With respect to The Hannibal Blog‘s foremost criterion, the conclusion is simple (though not simply derived)… the First Incompleteness Theorem says that something can be true and unprovable. This is a very important conclusion for all of mathematics (hence, a great thought).

There is a conflict with the finite and infinite. No wonder Gödel went bonkers.* People who believe in the Bible and the Koran and the like must love this idea. Mathematicians must love this idea, too. Philosophers stay in business. Everyone is happy! Not only is it a great thought, but it inspires others to think great thoughts…

I’m tempted to go the route of some mystics and say that Gödel’s incompleteness theorem, like quantum mechanics**, is a paradox and really difficult to understand. Therefore it must have greater applicability (i.e., with respect to the meaning of life). I’m not prepared to go that direction, but it is good for the debate…

Notes and comment:

*Mr Crotchety refers to Gödel going “bonkers”. Apparently, he had an “obsessive fear of being poisoned” and “wouldn’t eat unless his wife, Adele, tasted his food for him.” In her absence, he refused to eat, “eventually starving himself to death.”

**Mr Crotchety likens the incompleteness theorem to quantum mechanics. Instinctively, this feels right. I am thinking of Werner Heisenberg and his famous Uncertainty Principle. It says that, in the context of observing sub-atomic particles such as electrons, it is impossible to observe with certainty both the position and the momentum of a particle. One suspects that what is true of the world at that little scale is also true of the world on our scale. So if Gödel reminds us that much of our “knowledge” will always remain “incomplete”, Heisenberg reminds us that much of our world is fundamentally “uncertain”. Simple and non-obvious: Great thoughts by great thinkers!

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