## 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:

Inductive Reasoning

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.

January, 2009 at 2:32 am

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 timeI recall trying to make the same point to Eliezer, but you did a much better job of it.

January, 2009 at 5:49 pm

My knowledge of academic philosophy is limited, mostly because I hold it in contempt due to the knowledge I do possess about it.

But I’ve consistently been surprised that the points I made above do not seem to have been addressed. At all.

Perhaps it’s a consequence of my not searching deeply through existing writings, but not even the general textbooks / sources I’ve consulted even refer to deduction being a subset of induction.

It is possible that very few people have explicitly recognized that — at least, in academic philosophy. I suspect early training causes most ‘professional’ philosophers not to bother thinking about the topic. Once something is categorized, it takes time and effort to realize it’s actually something completely different.

January, 2009 at 6:47 am

I disagree, but not with everything. But my attempted reply grew too long. So it’s here: http://considerables.tumblr.com/post/69528282/deduction-is-certain-but-trivial

Hope thats ok.

January, 2009 at 3:37 pm

Ah, but there are two problems.

1) How do you determine that your evaluation was correct, that you didn’t screw up the process and get a result that actually doesn’t follow in the model?

2) How do you determine that the model actually reflects the reality you’re trying to understand?

January, 2009 at 4:28 pm

1) The unreliability of induction and the unreliability of deduction are different things. Induction is inherently unreliable because the observed part of the state of affairs underdetermines the whole. As for deduction, only implementations of it can be unreliable. Unreliable brains. Unreliable pencils and papers. Unreliable computers. Etc.

2) That’s outside the realm of deduction. I did say deduction is pretty trivial.

January, 2009 at 3:10 pm

I think you’ll find a different take on it in academic philosophy – CS Lewis demonstrates this view in ‘achilles and the tortoise’. Deduction is obviously true, and if you don’t take it for granted then there’s probably no way to convince you of anything.

January, 2009 at 3:55 pm

“1) As for deduction, only implementations of it can be unreliable. Unreliable brains. Unreliable pencils and papers. Unreliable computers. Etc.”

Why?

You too, thom blake. How precisely do you know that the rules of deduction will not change, and that they have not changed?

You don’t seem to grasp that we can take a thing for granted without believing it to be ultimately true.

If you accept that your brain is unreliable, how do you determine which of its determinations reflect the “real” deduction, and which are errors?