The Problem of Induction
Let’s say you’ve only seen white geese throughout your whole life. With this in mind, someone approaches you and asks you to describe the next goose you will see. Thanks to your past experiences, you cannot help but mention that it will be white.
Essentially, to conclude that all objects contain a property based on the observation that this property is contained within a few objects is called induction.
This tends to be a successful at times. When asked if the sun will rise tomorrow, one can merely cite all the past times that the sun has risen. Or when attempting to apply a scientific law, we can be sure that we will succeed based on the success of all the past instances of success. In mathematics, induction is powerful enough to allow mathematicians to use it to define and prove properties of numbers and mathematical objects.
For instance, it’s not difficult to prove that the square of a positive integer is the sum of the odd postive integers of that size:
- Base Case: 1 = 1^2, 1 + 3 = 2^2, 1 + 3 + 5 = 3^2
- Inductive Step: Assume that this is true for k.
We want to show that this is true for (k+1),
ie. 1 + 3 + … + (2k-1) = k^2 is true,
so we add (2(k+1)-1) to both sides:
1 + 3 + … + (2k-1) + (2(k+1)-1) = k^2 + 2k + 1
which is equal to:
1 + 3 + … + (2k-1) + (2k+1) = (k+1)^2
Beautifully simple, right?
Induction has proven itself to be useful in this way. However, induction suffers from a logical inconsistency. See, induction functions so well in mathematics because the objects that are being studied have a rigid definition. No mathematician will present confusion at the notion of a positive integer.
Yet, consider the example given before. Is it correct to conclude that all geese are white based on having seen just a few? Definitely not, as “white” would then be part of the definition of “goose.” And certainly, instances of non-white geese have been observed. Similarly, it is not in the definition of “sun” that we will observe it rise everyday. In fact, it is possible that something may occur to impede the sun from rising. That we cannot check this to be true enlightens on the nature of this logical inconsistency. With mathematics, there was the luxury of checking any instance I wanted. But in reality, I cannot check every instance of the run rising to make sure that I am right in concluding that the sun will rise tomorrow. The same applies to any scientific law, as no one can check every instance of gravity to make sure that the laws governing gravity will always hold. To suppose that we do have knowledge would mean to conclude what we want to prove, which itself is a logical fallacy.
So, when working in mathematics, it’s perfectly acceptable to make assumptions about some object that is aapproached. In reality, doing such a thing can easily lead us astray.