The Problem of Induction
There’s a good chance, O Moderately-Beloved Reader, that you’re already familiar with this topic. In case you aren’t (or you just want to establish precisely what we’re talking about), there are plenty of discussions in both philosophical literature and online sources, two of which are provided below:
The Problem of Induction
A rather common position among mathematicians is a sort of neo-Platonism, the idea that through the study of mathematics one can uncover profound and universal truths that defy the contingency and uncertainty of normal knowledge. Even Einstein is said to have remarked how curious it was that the universe could be understood by human reasoning, that the utility of mathematical analysis was a great mystery.
The very nature of the so-called “problem of induction” lies in the contrast that is supposed to exist between deductive and inductive reasoning. Deductive reasoning is held to be a certain guarantor of its conclusions; you might produce false results from false premises, but you can be certain that those results follow from them, as long as you perform the operations correctly. Your model might not match the reality you want it to, but there’s no question about the validity of the model itself.
This raises a deceptively-simple question: how do we know that what we call ‘deductive reasoning’ actually is certain? By what means did we reach the conclusion that deduction is reliable and universal?
Looking back at the history of mathematics and logic, we are forced to an equally simple conclusion: our ideas about deduction were generated inductively. We justify our intuition that certain results follow from certain premises by evaluating them over and over and noting that we get the same results every time. How do we know that this consistency is universal, both in space and in time? The same way we “know” that the sun will rise every morning – we don’t, in any rigorous sense. We take it for granted.
I have mentioned previously that mathematics is actually a science. Mathematicians perform experiments involving computation, either on external hardware or in their own neurology, and make predictions about the results of future experiments. They found one result when they evaluated the problem – will further evaluations produce the same result? Will other mathematicians reach that result when the experiment is repeated?
Deduction is actually a subset of induction. So the “problem” of induction is not how induction can be justified in the way that deduction is, but how logic itself can be justified. And that question not only presumes that an answer exists, but requires that one be implemented in order to be asked in the first place.