Suppose you observe someone tossing a coin and 'Heads' comes up every time in their first 99 tosses. What would you say about the 100th coin toss? One answer is that there have already been way too many 'Heads' and therefore it is about time to call 'Tails'. Another answer is that, as coin tosses are independent of one another, the next result is as likely to be 'Heads' as 'Tails'. A third answer is that the coin is evidently biased and the next result will be 'Heads'.
If there any readers who favour the first alternative they had better stop reading now as the rest of this post won't make much sense. The second and third alternatives characterise two different types of inference: deductive inference and inductive inference. It seems that they demand somewhat different abilities. In the situation I have just given, either might be deployed but sometimes one is overwhelmingly more effective than another.
I want to discuss the making of inferences (or coming to a judgment, or coming to a decision) where one does not have complete information. The goal is to try to make the right inference as often as possible.
In the sequel to this post I shall concentrate on deductive inference and how we humans seem to have an ingrained blind spot. Deductive inference is what, as a mathematician, I have been trained in but in this first post I want to argue that a good inductive reasoner may very well outperform a good deductive reasoner more often than not. My arguments begin with how we have managed to understand the natural world and the rate at which our understanding has advanced throughout history.
Of our present day sciences Mathematics is a huge outlier in that virtually nothing that it has discovered has been thrown away as invalid. Our Physics, Chemistry, Astronomy, Biology and Geology have existed (although not with these names) for just as long as Mathematics but almost everything we thought we knew about them 1000 years ago is now known to be entirely wrong. The reason for Mathematics being such an exception is that it proceeds by deductive inference (I oversimplify a little of course but, bear with me, it is clearly different to the other sciences). The other subjects had no chance to make the same progress while they were being pursued by deductive inference because there was an axiom that completely undermined our thinking. That false axiom was, of course, religion. Deductions based on false premises are correct only by great good luck so it is no surprise that advances in the other sciences had to await our discarding (or ignoring) the religion premise.
It was axiomatic that God created Man in His own image. How then could evolution even be contemplated? It was axiomatic that the Earth was the centre of the Universe. How then could astronomy develop? It was axiomatic that the Earth had been created for Man. How then could Geology say otherwise? But the religion axiom only applied to the natural world, not to the abstractions of Mathematics so it was not so hamstrung.
But, with the Renaissance and then the Enlightenment, the Church's grip on the minds of the people loosened to the extent that the religion axiom could be ignored (if not dismissed). Coupled with this occurred a revolution in thought: the rise of the experimental method. Experiments are not deductions. They are gatherers of information. A single experiment rarely gathers complete information about a natural phenomenon but it may suggest a hypothesis. Further experiments can then be conducted to test the hypothesis and sometimes a hypothesis survives all of these experiments and we can tentatively claim that we have discovered something. Of course I am aware that the connection between experiments, evidence, knowledge, falsifiability etc. is the subject of much philosophical debate but the simple picture I have presented is not very controversial. To put it another way our experiments allow us to infer knowledge about the natural world - and, quite obviously, this is inductive not deductive inference.
I will not deny that deductive inference has played a part in the successes of the non-mathematical sciences but I suggest that inductive inference through the experimental method has been the major player. So there's a very good justification for inductive over deductive inference: the triumph of most of science and its spin-offs for the way we live such comfortable lives. But it doesn't stop there!
As we progress through our lives we acquire more and more experience in dealing with situations that call for judgement. This experience can be very hard to describe. How can we describe the advice from a seasoned fisherman on where to cast our line? Does he have just a 'gut feel'? More likely he has many memories of similar conditions that prevailed where fish usually gathered in the spot he recommends to us. No guarantee - just the normal use of inductive inference even if carried out subconsciously.
Here is another example, personal for me and shared by many bridge players. How do I decide what to do at a particular point during the play of a bridge hand? As I age I get increasingly better at making the right choice (but I definitely know that my deductive abilities have waned over the years). I can do this because I have seen many similar instances in the past and can rely on inductive inference much of the time (and, in any case, there is often not enough time to go through a full deductive analysis).
In other words, personal experience is often grounded in inductive inference. We live and prosper by it.
Finally, if I still have not persuaded you of the merits of inductive inference, consider again the coin-tossing scenario I began with. Further suppose that you had been told that the coin was unbiassed. Would this make a difference to what you might think the 100th toss would be? I submit that it may very well not make any difference. Yes, you have been told something but all the evidence points to your having been told a lie. Just as we have released ourselves slowly and painfully from divine revelation and rejected a false axiom, so here we should trust the evidence. That coin is biassed - call 'Heads'!