The farmer fed the chicken every morning at the same. The chicken realised this and ran up to the farmer every morning to be fed. One morning the chicken ran up to the farmer who grabbed it and chopped off its head. This demonstrates the dangers of inductive reasoning. The old turkey was a little more sophisticated however. When asked by a younger turkey when Thanksgiving was, he replied that it was on the fourth Friday in November. The younger turkey was incensed to find out that it was the fourth Thursday in November. The older turkey said to him “Boy, the humans celebrate it on the Thursday, but if I wake up on Friday morning, then I give thanks”.
Induction is looking at the past in a particular way to predict the future. Specifically, induction looks at a series of events in the past to predict the future. The sun has risen like clockwork every day, whether or not you can see it, for as long as anyone can remember and for as long as we can determine from reports from the past. Will it rise tomorrow morning? I would put money on it because either it will, and I win, or it won’t and it won’t matter because we will almost certainly be dead. The argument comes down to “It has always happened in the past, so it will (or it is extremely like to) happen in the future.
The alternative method of reasoning is deductive reasoning. The deductive argument is that the rising of the sun is a consequence of the rotation of the earth. As the earth rotates, the sun appears to us on the earth’s surface to appear from beneath the horizon and travel across the sky. Actually, it is us who move, a good demonstration of relativity (but maybe I’ll go there another day). The argument goes stepwise from fact to fact and leads inevitably or logically to a conclusion.
The trouble with this approach is that, for all its logical stepwise approach it is built on two things, a theory and a set of past observations. A scientist has a theory or decides to check a theory, so he does an experiment, and the results of his experiment support or do not support the experiment. The scientist assumes that the theory is true and bases his predictions on this. Unfortunately there is an inductive element to this – if the theory is true for the experiment, there is no guarantee that it will be true for subsequent experiments, even given that ‘ceteris paribus’ (all things remain the same). Some other unconsidered cause could affect the result. The argument is deductive, proceeding in logical steps from the theory, but the practise is inductive – the data has always supported the theory in the past, so it will continue to support the theory in the future.
To be fair to the inductivists, todays’ inductivists tend to specify the results of their arguments in terms of probabilities: the probability of the sun rising tomorrow is close to 100%, given that it has always risen in the morning for as far back as we can see, but there is a minute but finite possibility that it won’t for known or unknown reasons.
Let’s consider the case of the sun rising each day and suppose that the fact that the earth rotates is not known. To make the argument more deductive we can postulate causes and so long as the cause fits the facts, we can tentatively label the cause as a hypothesis. Suppose we conjecture that some deity causes the sun to rise each morning. This hypothesis certainly fits the facts and predicts with accuracy that the sun will continue to rise each morning. Such a hypothesis would not be accepted today, of course, except by some individuals.
Is there any great difference between the theist and the scientist? The theist says “all things happen because of God”. The scientist says “all things happen because of the laws of nature”. They both explain things on the basis of their fundamental beliefs.
It is possible that people in the future may look at our theories of the sun rising and other things and consider them naive and consider our view of everything happening according to the laws of nature to be a quaint misunderstanding, in much the same way as many people would consider the “deity hypothesis” to be today.
In mathematics the situation is different. Induction is a much more formal process and is applied on top of an axiomatic system. Proved theorems are the results of the applying the axioms repeatedly to another proved theorem or the axioms themselves. Unproven assertions can be proved and turned into theorems or disproved and discarded (or possibly modified so that they can be proved). If something is proved in an axiomatic system, it is true for all time, and cannot be disproved in that system.
Specifically an inductive proof would go something like this: firstly the theorem would be proved for a generic case (eg if statement N is true, then statement N + 1 is true) and secondly it is proved for a specific case (eg statement 1 is true). Then all applicable statements are true because, if statement 1 is true, the generic case means that statement 2 is true, and so on for all cases. Because of the rigor of the argument and the undeniable conclusion of the argument, mathematical inductive proofs are of the same order of reliability as deductive proofs, that is, they are only wrong if there is an error in the logic.
Why the difference between scientific induction and mathematical induction? Well, I think that it is related to the fact that mathematics is axiomatic and therefore certain, whereas scientific induction is based on the laws of nature which are not and never will be, in my opinion, completely defined. If the basis of your argument is not certain, how can your conclusion be certain?