Complex Numbers

Inside The Box

Warning: Math entry.

Well, actually a philosophy entry.

Philosophy of math.

Some Math Problems Seem Impossible. That Can Be a Good Thing.  
Struggling with math problems that can’t be solved helps us better understand the ones we can.

Considering that simple arithmetic seems impossible to a lot of people, that's saying a lot.

Construct a convex octagon with four right angles.

It probably says a lot about me as a teacher that I assign problems like this. I watch as students try to arrange the right angles consecutively. When that doesn’t work, some try alternating the right angles. Failing again, they insert them randomly into the polygon. They scribble, erase and argue. The sound of productive struggle is music to a teacher’s ears.

Oh, that kind of teacher.

Finally someone asks the question they’ve been tiptoeing around, the question I’ve been waiting for: “Wait, is this even possible?”

This question has the power to shift mindsets in math. Those thinking narrowly about specific conditions must now think broadly about how those conditions fit together. Those working inside the system must now take a step back and examine the system itself.

I've known people who pride themselves on "outside-the-box" thinking. Without exception, every one of those people were utterly lousy at inside-the-box thinking. Otherwise known as "thinking."

It is true that breakthroughs happen when people think beyond the stated parameters of a problem. However, it is at least as important to acknowledge the constraints of the parameters.

Consider, for example, the nine-dots puzzle, which was my -- and probably a lot of peoples' -- first encounter with lateral thinking. I have only a very limited ability to post images here, but a full description of it, with illustrations (including solution), is provided at this Wikipedia page.  

From that entry: "The puzzle proposed an intellectual challenge—to connect the dots by drawing four straight, continuous lines that pass through each of the nine dots, and never lifting the pencil from the paper. The conundrum is easily resolved, but only by drawing the lines outside the confines of the square area defined by the nine dots themselves."

The solution, as noted, relies on not seeing the outer boundary of the nine-dot square as a constraint; to draw the required lines beyond that imagined border. However, the constraints posed in the instructions are inviolable; that is, "connect," "four," "straight," "continuous," etc. One might very well wonder why such constraints exist, and what practical use such constraints might have. Violating these necessary parameters is what most people do when they smugly "think outside the box." It's pernicious.

Anyway, the article goes on to explain why the octagon problem posed in its first sentence is, in fact, impossible (assuming a flat Euclidean plane). There's a bit of math involved, but merely geometry. You can probably skim those parts and still get a good idea what the article's about.

I'm talking about this mainly because, at least at the time I plopped the article into my blog fodder stash, I'd been seeing an uptick in people going, "Nothing is impossible! Everything is possible if you try."

And that's utter codswallop.

As noted in the article:

To consider impossibility, we need to understand that just asserting that a thing exists doesn’t make it so.

A well-known example is: Consider an omnipotent being. Can such a being create a boulder so heavy that even the being itself cannot lift it?

If the answer is "Yes," because the being is, after all, omnipotent, then from the question itself, the being cannot lift the stone -- but that's impossible because the being can do anything (which is the plain definition of omnipotence).

On the other hand, if the answer is "No," then the being is not omnipotent after all.

Conclusion: Omnipotence is impossible.

And yet, we can construct the sentence: "An omnipotent being can create an object so heavy that even the being itself cannot lift it."

Many more things are possible in language than in practice. Is "everything" possible in language? I can't answer that, because I'd have to use language itself to do it.

Math is a kind of language, too -- one that is far more precise than plain English. You can say almost anything and make it look like math (for example, "2+2=143," but saying it doesn't make it true. Mathematics is based on rigorous proofs, including proofs of impossibility. Language isn't.

I do think language has its limitations; I have yet to encounter a satisfactory time-travel story, for example, and whenever we start talking about infinity, people can't grasp it. But that doesn't mean that either is impossible.

And yet, I will generalize from the article's assertions about the impossible: it's in thinking about the impossible, in any subject, that we can find solutions to problems that are merely difficult. I personally think that this is one of the main purposes of fiction writing.

For a long time, mathematicians thought that there were no numbers that, when squared, yield a negative number. But such numbers turn out to be useful, so they simply invented new ones: the so-called "imaginary" numbers that I ramble on about in this blog's intro that no one ever reads. So yes, we might find that some things thought to be impossible are actually trivial, in math or in philosophy or in other areas.

But other things will always remain imaginary, thus keeping fiction writers in business for the foreseeable future.