Irrational numbers cannot be represented in the real physical world, I’ve been told. So my question is: if I have a one meter by one meter square of wood, which is a perfect square precisely to the atom, is its diagonal length not sqrt2?
In: Mathematics
This is a good question, and it’s one I’ve seen asked before without a satisfactory answer. I think that in order to answer it well, we need to talk about what it means for a number to “be represented in the real physical world”, and to do that we’ll have to take a step back.
Numbers in a math problem or textbook don’t have to relate to the real world. They simply “are”, in some ineffable sense. Mathematicians and philosophers argue over what it means for a number to “exist”, or whether that’s a meaningful thing to say at all.
But you’re talking about “the real physical world”. We do, frequently, use numbers to describe the physical world. When we do, it is usually in one of two ways. Most of the time, we are either counting something or measuring something.
Counting is simple; it’s one of the first things we learn as children. One apple; three sheep; nineteen dollars and ninety-nine cents. It makes a real, qualitative difference whether we have one sock or two socks, because most of us have two feet. There’s no need to decide how to measure how many socks or feet you have; it’s inherently natural to use the number “two” to describe them.
Counting doesn’t have to involve only positive, whole numbers. A bank balance is a count, but it can be negative. And you can combine counting numbers: one hundred cents make a dollar, so a cent is one hundredth of a dollar.
Measurement is different. Measurements are by nature approximate, not exact. When you say that an object is “one meter” long, you’re fundamentally limited by the precision of your measuring instrument. Even if the object is, in fact, *exactly* one meter in length, there is no measuring instrument that could verify it. And the meterstick is itself arbitrary: you could substitute a yardstick and get a different number without anything about the situation really changing.
What does it mean for a number to “be represented in the real physical world”? Well, if you have two shoes, then that is in some sense an exact representation of “two” in the real physical world. There’s no error or uncertainty there; there’s no possibility that you actually have 2.00000001 shoes. The same applies to negative integers and to rational numbers: we can construct simple real-world situations in which these numbers have an exact, unambiguous relationship to reality.
The same isn’t true for measurements. A stick that you’ve measured at one meter is not a perfect, absolute representation of the number “one”. All you can say is that you can’t distinguish the length of the stick from one meter with the equipment you have. The same is true for a stick whose length you measure to be indistinguishable from pi meters, or the square root of two meters.
In your example, you have “a one meter by one meter square of wood, which is a perfect square precisely to the atom”. This is like a math problem from a textbook. If you stipulate that the plank is *exactly* one meter by one meter, then its diagonal is exactly √2 meters. But there is no piece of wood in the real world that we can guarantee to be *exactly* one meter in the way that you can say that are wearing exactly two socks.
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