Critical Thinking Toolkit: Inductive Reasoning

Deductive reasoning is a very useful way to gain knowledge, but it is also very limited. Deduction is a type of reasoning that intellectually compels you to believe something if you accept some number of other things. But very few aspects of reality are like this. In most situations, there are a range of available options for explaining certain aspects of the way the world is. This is the case in science, history (especially ancient history), psychology, and so many other areas of our daily lives. It is therefore important to discuss methods of reasoning that help us deal with situations where a variety of options are available to us within the bounds of deductive reasoning.

In this post, I want to discuss one of the two most important methods of reasoning that are not deductive in nature. This method is called inductive reasoning or induction.

What is Induction?

Although most of us have a familiarity with induction in one particular context, I don’t want to define induction using a specific kind of induction, but by using the broadest form I possibly can. Inductive reasoning, in the most general sense, draws from specific examples some kind of general conclusion. (Notice that specific examples do not necessarily lead you to some conclusion, but nonetheless it is often quite reasonable to use specific examples to justify a conclusion).

I think probably the most helpful general example I can use for induction is the way that a toddler learns about the world around them. It is probably quite obvious to every one of us that when we are born, we don’t really know anything about the world – at least not consciously. Perhaps we have certain inherent instincts when we are born – an example of such an instance might be something like when I experience such-and-such a sensation, that means I need urinate. Beliefs about hunger and pain are probably quite similar. But consider a different belief – that liquids are wet. You certainly are not born with this belief. But over time, as you gain experience with more and more things that look liquid, you will discover that each of them feel wet. As a toddler, you learn that liquids feel wet.

Is learning that liquids feel wet deductive? Not at all. For a toddler, liquid is basically something visual – you would identify liquid in terms of what a thing looks like (namely, fills the space it occupies, sloshes around). For the same toddler, wet is a tactile experience – you feel wetness on your skin. Since the senses are distinct, you won’t be able to use anything deductive to connect them (at least not for a toddler – perhaps there are such arguments that I am unaware of). So if the toddler isn’t using some kind of deductive reasoning in their subconscious, what is happening here? The answer is that the toddler is learning about their world inductively. They have enough examples in their lives where such-and-such a visual experience is related to such-and-such a tactile experience that they come to conclude that the two are closely related to one another.

Could they be wrong? Well, yes. There are non-Newtonian liquids (so-called oobleck is an example) that sometimes become hard when you press on them – this doesn’t seem wet. And yet when left to themselves, non-Newtonian fluids certainly look liquid. So this would be a counterexample to the toddler’s experience. So deductively, “wet if and only if liquid” is not really true. But nonetheless, we all know that it is quite true that wetness and liquidity are quite related, even if not identical. This is the kind of knowledge that we can gain by induction. The toddler experiences enough examples of wetness and of liquids, and they come to infer – in an inductive way, not a deductive way – that wetness and liquidity are related.

This is the sort of learning that induction is all about. But… how do we actually use induction?

The Scientific Method

This is the example I was referring to vaguely earlier. All of modern science is at its core an inductive enterprise. The scientific method involves conducting experiments over and over again, trying to figure out how any slight variations in how you do your experiment affect the result of that experiment. When you do this, what you are really doing is trying to build up examples of physical experiments in an organized way. The organization and scientific control here is supposed to help you do induction faster than would otherwise be reasonable. It might take a baby a year to realize that liquids are wet – a scientist could arrive at a conclusion like that much, much faster.

There is such a thing as theoretical science, that uses theoretical principles to derive theoretical conclusions – but at the end of the day, these results are only meaningful when verified experimentally, which is where the inductive method comes in. If theoretical science turns out to be false, then it still has value as a piece of theoretical mathematics, but not as genuine physics that applies in the real world.

Daily Life

We often use induction in daily life as well. Muscle memory is sort of like induction – although it is not cognitive, muscle memory is acquired in a way that resembles in a lot of ways the way that we obtain knowledge of the world through induction. This is also probably similar to the way we learn language and skills like driving, cooking, a sport, or even child-rearing. Anything that you learn “by experience” usually has a lot to do with induction.

Conclusion

Induction is an important way that we come to learn information. Although deduction does give us more certainty than induction does, induction is more widely applicable than deduction. Both are useful and they have their contexts, but it is important to recognize both.

As a mathematician, I am all too familiar with the failures of induction. There is a reason that mathematicians use deduction and not induction. For the mathematician, Skewes’s number is a key example of why we don’t place our absolute trust induction. It would take a while to explain all of the details that go into Skewes’ number, so I will be brief. Prime numbers are an important kind of whole number, and in mathematics we often write \pi(x) to represents the amount of prime numbers between 0 and x. Mathematicians now have an easier to compute function, which we will call Li(x), that very closely approximates the much harder to compute \pi(x). For a long time, it was believed that Li(x) was always an underestimate for \pi(x). But the South African mathematician Stanley Skewes has shown that eventually this is no longer true. But all we know today is that the failure happens when x is less than 10^{10^{10^{964}}}. The number 10^{964} already has essentially no meaning in the physical world we live in – the number of subatomic particles in all of the world is so much smaller than 10^{964} that it is funny. The number 10^{10^{10^{964}}} is insanely larger than this. What is our point here? The point is that you can have millions of millions of millions of examples of a true claim in mathematics, only for that claim to turn out to be false later. This ought to give us pause about any absolute trust in induction. Induction gives us a huge degree of trust when done correctly – as something like science usually is – but it is still not infallible.

2 thoughts on “Critical Thinking Toolkit: Inductive Reasoning

  1. I’m not talking here about water molecules, I know the debate. What I mean is the following. When a large quantity of liquid molecules are together and your sense of feeling perceives them, it is quite distinct from the sensation of solid molecules or gaseous molecules. The particular sensation of liquidity is what I mean by “wet” here.

    And if you are referring to Skewes’ number, it is an upper bound, not a lower bound. So less than is appropriate. It is just such an insanely large upper bound that not even modern computers can make any use of it computationally.

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