[fredilo]

Superb question! Interestingly, my math professor asked the exact same question after teaching Galois theory and stated that he does not have a good answer himself. Let me try to give sort of an answer. :)

We have fingers. These we can count. This is why we are interested in counting. This is gives us the natural numbers and why we are interested in them. What can we do with natural numbers? Well the basic axioms allow only one thing: Increment them.

Now, it is a natural question to ask what happens when we increment repeatedly. This leads to addition of natural numbers. The next question is to ask is whether we can undo addition. This leads to subtraction. Next, we ask whether all natural numbers can be subtracted. The answer is no. Can we extend the natural numbers such that this is possible? Yes, and in come the integers.

Now, that we have addition. We can ask whether we can repeat it. This leads to multiplication with a natural number. Next, we ask whether we can undo it and get division and rational numbers. We can also ask whether multiplication makes sense when both operands are non-natural.

Now, that we have multiplication, we can ask whether we can repeat it. This gives us the raising to the power of a natural number. Can we undo this? This gives radicals. Can we take the root of any rational number? No, and in come rational field extensions including the complex numbers.

A different train of thought asks what we can do with mixing multiplication and addition. An infinite number of these operations seems strange, so let's just ask what happens when we have finite number. It turns out, no matter how you combine multiplication and addition, you can always rearrange them to get a polynomial. Formulated differently: Every branch-free and loop-free finite program is a polynomial (when disregarding numeric stability). This view as a program is what motivates the study of polynomials.

Now, that we have polynomials, we can ask whether we can undo them. This motivates looking at roots of polynomials.

Now, we have radicals and roots of polynomials. Both motivated independently. It is natural to ask whether both trains of thought lead to the same mathematical object. Galois theory answers this and says no.

This is a somewhat surprising result, because up to now, no matter in which order we asked the questions: Can we repeat? Can we undo? How to enable undo by extension? We always ended up with the same mathematical object. Here this is not the case. This is why the result of Galois theory is so surprising to some.

Slightly off-topic but equally interesting is the question about what happens when we allow loops in our programs with multiplication and addition? i.e. we ask what happens when we mix an infinite number of addition and multiplication. Well, this is somewhat harder to formalize but a natural way to look at it is to say that we have some variable, in the programming sense, that we track in each loop iteration. The values that this variable takes forms a sequence. Now, the question is what will this variable end up being when we iterate very often. This leads to the concept of limit of a sequence.

Sidenote: You can look at the usual mathematical limit notation as a program. The limit sign is the while-condition of the loop and the part that describes the sequence is the body of the loop.

Now that we have limits and rational numbers, we can ask how to extend the rational numbers such that every rational sequence has a limit. This gives us the real numbers.

Now we can ask the question of undoing the limit operation. Here the question is what undoing here actually means. One way to look at it is whether you can find for every limit, i.e., every real number, a multiply-add-loop-program that describes the sequence whose limit was taken. The answer turns out to be no. There is a countable infinite number of programs but uncountably infinite many real numbers. There are way more real numbers than programs. In my opinion this is a way stranger result than that of Galois theory. It turns out, that nearly no real number can be described by a program, or even more generally any textual description. For this reason, in my opinion, real numbers are the strangest construct in all of mathematics.

I hope you found my rambling interesting. I just love to talk about this sort of stuff. :)

[dataflow]

Thanks so much, I feel like you're the only one who grasps the crux of my question!

> This is a somewhat surprising result, because up to now, no matter in which order we asked the questions: Can we repeat? Can we undo? How to enable undo by extension? We always ended up with the same mathematical object.

I think this is the bit I'm confused on - we have an operation that is a mixture of two operations, where previously we only looked at pure compositions of operations (let's call this "impure"). Why is it surprising that the inversion of an "impure" operation produces an "impure" value?

It's like saying, if I add x to itself a bunch, I always get a multiple of x. If I do the same thing with y, then I get a multiple of y. But if I add x and y to each other, I might get a prime number! Is that surprising? Mixing is a fundamentally new kind of operation; why would you expect its inversion to be familiar?

[fredilo]

What is surprising and what not is always very subjective.

Now that I think more about it, one could argue that everything you can do with inverting radicals can also be done by inverting polynomials. So You could look at radicals as the step after multiplication and at inverting polynomials as the step after radicals. With this may depiction that these are two competing extensions falls apart a bit.

My chain of argumentation was that one could expect that there is a single natural ever growing set of "numbers" starting with the natural numbers, then integers, then rational numbers, then real numbers, culminating in the real complex numbers and ever set is a superset of the previous one. This is the "natural" order in which they are taught in school and somewhat mirrors how they historically were discovered. In retrospect, this is obviously not true. Just look at the existence of rational complex numbers. However, when all you have are natural, integer and rational numbers, it seems like it could be true.

Let me try a different way of explaining why it is surprising to some.

In school, I learned that I can solve quadratic equations by combining the inverse operations of the basic operations that make up the quadratic polynom. This seems natural as it worked for solving the linear equations I had seen so far. Inverse of combination is combination of inverse. At some point the teacher showed the formula for degree three. Cubic radicals appeared. We were overwhelmed by it's size but the basic operations used matched what we expected. The teacher said that degree 4 is even drastically larger with degree 4 radicals and we definitely do not want to see that, which is true. Nothing was said about degree 5 but it felt like it was implied that the pattern continues and the main problem with degree 5 is that our brains are just not able to handle the amount of operations that make up the formula.

Fast forward to university. Now the professor proves in the Galois theory course that, no, it's not that you are too stupid to handle degree 5. It's just that degree 5 cannot be handled this way at all. I am still unsure about whether my teacher in school knew that degree 5 is impossible or just assumed that he too is just too stupid.

I guess this mathematicians must have felt something similar back then. You learn about linear equations. All is easy and works. You learn about quadratics. After mixing in quadratic radicals, all is well again. You try to grasp cubics, and yes, with a lot of work this too can be learned. You think about quartics and after lots and lots of time come to the conclusion that yes it is possible but impossible to master the formula. It feels like the pattern should continue and the reason you don't have a quintic formula with degree 5 radicals is not because it does not exist but because of it's sheer size and just stating it would fill a whole book. Turns out, there is no such book.

Suppose you are a renowned mathematician back then who has failed for years to find a quintic formula. Now this teenager named Évariste comes along and fails too but says that it's not because he's too stupid but because it's impossible. At first, this does sound like an excuse of a lazy student, doesn't it?

Let's say you are not surprised that roots of degree 5 polynoms cannot be computed using just addition, subtraction, multiplication, division, and radicals. Does it surprise you that degree 4 polynoms can? Why does this work for degree 2, 3 and 4 yet fails for 5 and higher? I can see that one can argue that there is no reason to assume that it always works. However, at least learning the fact that it starts failing at degree 5 should be non-intuitive.

[glompers]

Yep!