So what do I mean by this obscure phrase? It comes from a thought I had the other day. It started with pondering certain famous irrational numbers like square rout 2, the natural logarithm, or pi.
I think what these numbers stem from is the fact that they are derivable from the geometric properties of space itself as opposed to all rational numbers which are derived from human thought. That is to say, whereas neat and discrete quantities like the integers and the ratios between integers (which is what all rational numbers are) are idealizations that we tend to impose on the natural world to the full extent of precision we can muster, the irrational numbers are already given by nature, and they are packaged in the geometric properties of space itself. In fact, nature only gives us irrational numbers.
To prove my point, I propose the following thought experiment. Suppose you’re taking a measure of something. keep it simple - say the length of your arm. Suppose you measure it to be 1.5 meters. Now, how precise are you being? I mean, is it exactly 1.5m? Probably not. It might be more like 1.53m, or maybe 1.56m, or something thereabouts. But then again, maybe even greater precision is needed to get at exactly how long your arm is. So if we zoom in further, we may find that it actually measures 1.539m or 1.562m. Well, how precise is precise enough? We could take this to extremes. We could say that your arm actually measures 1.5392396293742893721398m. In fact, I’ll bet (and this is the key point) that you could keep zooming in ad infinitum and keep tacking on more digits to the decimal expansion without end.
But isn’t it possible to measure something out to be exactly 1.5m or any rational quantity? Well, supposing you did, how would you know it’s exactly 1.5m? You could say that you’ve measured it out to the greatest of your precision abilities, but this is one thing; it’s quite another to say that a measure is some quantity exactly in an absolute sense - that is, to infinite precision. To say the latter is essentially to say that no matter how much you zoom in, the reading of 1.5m, or whatever it was, will never change - that is, there will never be additional digits tacked onto the decimal portion of the number. Now, to say such a thing technically isn’t paradoxical, but it does sound absurd. I would imagine that if you kept zooming in forever, the appearance of extra digits to your decimal expansion is an inevitability. It may take an extremely long time, but it will happen. And if it happens once, it can happen again. That is, if, when you final zoom in enough to discern an extra digit that needs to be tacked onto your decimal expansion, you keep zooming in further, it will be another inevitability that you discern yet more extra digits. So because there is no end to how far one can zoom in (at least in principle), there is no final digit that marks the righthand end of the decimal expansion.
Now that’s measurement. Measurement is the way we glean quantitative values from nature herself. In other words, nature gives us her measurements; we don’t impose them. The only thing we impose are our idealized rational numbers, and we only take it for granted that certain measures in nature can be captured precisely by ration quantities. Put if you want absolute precision in any of your measures of nature, you’ll have to settle for irrational numbers. Pi, the natural logarithm, and the square root of 2 are irrational because they aren’t derived from our measurements. They are derived by the immutable properties of geometrical shapes and forms - that is, the properties of that part of nature we call space.