Let’s talk about Logic

In a recent blog entry, I moaned about the missing logical education in schools. But what is logic actually, and why is it important?

Let us take the viewpoint of the foundations of mathematics for a moment, and sketch the main ideas behind mathematical truth very briefly. We distinguish between axioms and theorems. The theorems are derived from the axioms by logical reasoning. We best understand correct logical reasoning if we look at models of our theory. A model of a theory is a system where the axioms are fulfilled. Now, logical reasoning is correct, if in any model where the axioms are true, the derived theorems are also true. If there is one model for the axioms where the theorem is not true, then the theorem cannot be derived by correct logical steps.

As an example, we take vector spaces. There are lots of them. First, the simple Euclidean plane and space. But then are also spaces of more dimensions, and even spaces of infinite dimensions. There are also vector spaces consisting of functions. In all these spaces, the axioms for vector spaces are true, and thus are all the theorems we derive for them. E.g., every finite set spanning the space contains a basis of the space, or the theorem that all bases have the same length.

Do we need that in school? You may argue that in school we have only models, and no theories. E.g., the kids learn about the numbers 1,2,3,… and how to compute with them. There is only one model of the natural numbers, or not? We have enough troubles learning how to handle them. So why bother with logical reasoning?

There are a number of arguments against that (besides the subtle fact that there are different models of arithmetic).

First, some theorems are not obvious. As an example, take the sum of the numbers 1 to N. Now we can just learn the formula for this sum and apply it, like drilled monkeys. But I cannot imagine one young kid that is not asking „Why?“.  Once they get older some are so frustrated by math that they just do not want to know. But if you see the formula applied for the first time, there must be wow effect.

Then, some things are wrong, albeit they look right. Examples are numerous in probability theory. To be able to tell false from right should be a desirable skill for everyone. In mathematics, you have means to achieve that goal. That is why the analysis of the sequence of equations in the previous blog post is a good exercise. The job is to fill in the logical steps correctly, and point to the one that does not hold.

Finally, finding a proof is a fun game. That is, unless you are trained to hate math. You may fail, and fail again. But if you succeed the first time you will feel great. It is like solving crosswords, or, more modern, succeeding all levels of a computer game. Why not offer these challenges to the kids instead of boring application of formulas?

I do not accept the argument that there is no time for that in schools. You can save so much time omitting boring things like hand computations. I also do not accept the argument that most kids are not clever enough. Reasoning can be on very low levels. Why is 56*98=98*56? Let the small ones think about such stuff and appreciate the demonstration of flipping over a rectangle. Why is the prime number factorization unique. Do teachers even ask this question? Sadly, they don’t.

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