Are Real Numbers "Real"?

As mentioned in my previous post, the scientific method is just about all the philosophy an engineer needs to know. In this post, I describe how this attitude can be applied to all the mathematics that engineers use.

The scope of the scientific method is defined by the assumption that the sole test of knowledge is experiment. That is, that which is outside the scope of experimental confirmation or falsification is not knowledge. Thus, to count mathematics as knowledge means that its axioms, operations, and results must be experimentally confirmed.

But there are inherent limits to the precision of experimental measurements in the real world. No quantity or phenomenon can be measured with absolute precision and no process can be performed without at least an infinitesimal possibility of error. For example, there is no experiment that can exactly measure any irrational, rational, or natural number. And since these numbers claim to be exact, this means they do not exist in reality. There is no zero because absolute zero cannot be observed. Just the infinitesimal. There is no infinity for the same reason. Just the very huge.

Does this affect the mathematical techniques useful to engineers -- such as the calculus? Are these techniques to be abandoned or viewed with suspicion? Not at all. Abraham Robinson was able to incorporate infinitesimals and huge numbers into what is called nonstandard analysis. Thus, the techniques remain the same, just the nature of the proofs change.

Interestingly, there is a number that is totally incommensurate with any attempt at direct measurement or observation yet inferred by the observed laws of Nature. It is the square root of minus one. In this sense it is the only number that is purely "imaginary."

No comments:

Post a Comment