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Friday, May 1, 2009

Imagine a Perfect Cube

This post grew from a comment I was going to write on my post Pi Whats, but as the comment grew and grew I felt it deserved its own post. It's about measurement and precision, and how whether you think they matter in mathematics depends on your perspective.

Do you think a string can be pi inches long?

You need to ask yourself this question:

Do you think a string can be 1 inch long? I mean EXACTLY 1 inch long?

If your answer is yes, a string can be exactly 1 inch long, then you automatically think it can be exactly pi inches long. Get a sphere whose diameter is exactly one inch, wrap the string around its circumference one time and cut it the exact length of the circumference, and the string is exactly pi inches long. If you are going to try to argue with me on this particular conclusion, then you are wrong. Pi is not some ill-defined Mysterioso of a number. It's very precisely defined. It's the circumference of any circle divided by its diameter. It's the circumference (in u units) of a circle whose diameter is 1 u.

If your answer is no, a string can't be exactly 1 inch long, then I'm okay with you saying it can't be exactly pi inches long, either.

The larger philosophical concept is that mathematics deals with ideal, perfect, abstract constructs that can really only exist in our minds. I can imagine a line segment of unit length. Cutting a physical piece of string with scissors to be exactly 1 inch long is much more treacherous terrain.

Scientists understand this, that's why they invented significant figures, and their calculations incorporate expected percent error. Whenever you measure something, you are necessarily rounding your measurement to the nearest level of precision on your measuring device. If you use a metric ruler with millimeter markings, and you measure something and say it's 2.6 centimeters, that means the thing you measured was closest to the marking at 2.6. It could be anywhere from 2.55 to 2.65 centimeters long.

Therefore, if I am thinking like a scientist or engineer (possibly using math but not thinking like a mathematician) dealing with only objects I can hold in my hand, I don't think you can point at a piece of string and say "that string is exactly one inch long".

But in math we don't often discuss these problems of measurement. Because in math we deal in abstract perfection that can really only exist in our minds. We don't stop every time we give a measurement in a math problem and talk about the imprecision inherent in measuring something. In the physical realm there is no perfect square. No perfect right angle. No perfect circle. No length that is the square root of 2. But in my mind, the string can be exactly one inch long because I say so. These things can exist in our imaginations. In my mind, I can picture a square with sides of unit length, and draw in a diagonal, and I have created a length that is exactly the square root of two. In my mind, I can reason that somewhere between a string that is clearly less than one inch long and a string that is clearly greater than one inch long, there is a place I could theoretically cut that would be exactly one inch long. So as a mathematician, I say the string can be one inch long.

I'm reminded of this litmus test from Out of the Labyrinth: Setting Mathematics Free:

Close your eyes and imagine a perfect wooden cube. Now saw this cube three times, along each of its three axes.

Question: did you see sawdust?

If so, you're no mathematician.