The Metaphysics of Philosophy, Knowledge and Morality Chapter 6A

Chapter 6A

Numbers of measurements

You might think that the formula 1+1=2 is evidently true. It's not. You cannot demonstrate that 1+1=2.

Take an object that represents 1. Take another object that also represents 1. Herein lies the problem: no two objects are the same, so there is no such thing as 1 object +1 object in practice. This being the case, you must believe a priori that 1+1=2.

Take a symbol that represents 1, e.g. draw an apple and say that's 1 apple. Now draw another apple and say that's the same symbolic apple representing 1. You actually know that no two symbols are technically the same, yet you regard them as though they were the same. You can regard them as the same for theoretical purposes, but practically they're not the same. It turns out that you can't demonstrate in practice that 1+1=2, but you'd very much like 1+1 to be 2, so you ignore the finicky practical details.

Why do we do that? So we can build houses, bridges and cars, etc. Accepting on a purely theoretical basis that 1+1=2 is very useful and though it cannot be demonstrated in practice, this a priori approximation certainly helps us produce very practical results.

Measurements are a similar problem. Let's say you want to measure the length of an object that you intuit to be about 1 millimeter. How do you do that? You take a measuring tape or a ruler, put it next to the object and check if it visually matches 1mm on your scale. If it matches, you've established that it's about 1mm long.

If we really wanted to measure the length of the object, we should use a more precise instrument like a micrometer or an electron microscope and then... Then what? Can we measure the length of the object to the last atom? We can't. Even if we could, we don't have scales that are precise to the last atom. So we'll never know the actual length of anything, we can only approximate. However, approximation is good enough for practical purposes, which is why we don't sit down next to our object and weep helplessly. In practice, we use rough-and-ready measurements to build makeshift structures that we call technology. Technology lies at the heart of our civilisation.

Our civilisation relies on an a priori use of numbers and measurements.