Mathematics in school and higher mathematics - what are differences between them?

In Poland we sometimes use a phrase: "Oh, it is higher mathematics..." We say that about something that we don't understand and won't even try to understand. On the other hand, when someone doesn't understand what we say or ignores it, we may say: "Am I speaking Chinese?"

Mathematics and Chinese language have something in common: in order to understand them, you have to learn terms and symbols they use. In school mathematics it isn't hard. Although formal definitions of some basic concepts would surprise high school students by their level of abstraction (like definition of a function, a surface, construction of natural and real numbers), these concepts are more or less understood intuitively.

When it comes to symbols, they are used in notation of mathematical objects, operations and relations. And again, in school it is easy: objects are numbers (we are familiar with x, y, z in equations), functions (noted as f(x), g(x)), sets A, B, C and so on. We have also Greek letters - α, β, γ, sometimes δ- to denote measure of an angle in a geometric figure. Basic symbols of operations are known for everyone: +, −, ∙, √ etc. We can have relations between numbers (=, <, >, ≤, ≥, ≠) or between lines on a plane (⊥ - purpendicular, || - paralell). There are some other things, but I'm not trying to list everything here. Nonetheless, we can say that concepts in school mathematics are intuitive and there are not so many symbols you have to know. Does this hold for higher mathematics?

In school mathematics there's no need to go into the details of formal definitions (or they are simple, like definitions of trigonometric functions) but this is not the case for higher mathematics. It is often implied by practical problems. For example, we all know the concept of the distance and how to measure it without a formal definition. But how to measure how much a given curve is "curved"? It isn't so obvious. Mathematicians wondered about it and they have found the way to do this. As a result, a new concept developed: a curvature. We know intuitively which curves are more and less "curved". But to say precisely, how mouch a given curve is "curved", or to say whether a curve is "curved" as much as another, we need a definition of curvature.

We can see that on more advanced level we cannot avoid formal definitions. But to write them, we need appropriate symbols. For example, let's consider another intuitive concept: a limit. Let's see what it is on a concrete example. Take infinite sequence of numbers: 1, -1/2, 1/3, -1/4, 1/5, -1/6... We denote these numbers: a₁=1, a₂=-1/2, a₃=1/3 and so on, so n-th entry is denoted by a_n. We can see that this sequence "approaches" zero. Mathematically speaking, its limit in the infinity equals 0 (when a sequence has a limit, we say that it is convergent). How do we know this? We know this because we can see, that we can approach 0 on any small distance we want. So the distance between the sequence's entries and zero is arbitrarily small from some point. And this is a mathematical description of this fact:

This is how we read it: for every number epsilon bigger than zero, there exists a positive integer n, such that for every m bigger or equal n, the distance between a_m and zero is less than epsilon. So, roughly speaking, no matter how cose we want to come to 0 (and we want to come not further than epsilon), from some point (from m-th entry) we will be as close as we wanted (closer than epsilon).

We can see some new symbols here: ∀ - "for every", ∃ - "there exists", or Greek letter ε, which is used to denote a variable which may take as small value as we want.

But the need for new symbols is one thing. Another one is generalising ideas, which is commonly seen in mathematics. It was described very well by a phisicist and Nobel prize winner, Richard Feynman:

As a result of such generalising, various mathematical structures arise. And so mathematicians see the set of integers as an example of a ring or the set of real numbers as an example of a field. A field is a "nice" ring. Having a field, we can construct another structure, a vector space. The set of points on a plane is an example of such space. But another example is the set of continuous functions on a given interval. It seems that these two are completely different things, but in a way we can treat continuous functions as vectors.

Somebody would ask: "Alright, but vectors have a length. If continuous functions may be treated like vectors, we should be able to define the "length" of a function. But how?"
The answer is: That's right. And we are. What is more, we can also say about the "angle" between two functions. Although we lose a geometric interpretation of an angle.

A natural question arises: what do we gain? This was explained by Feynman. One general theorem may have many interpretations and applications depending on whether we choose the vector space to be the set of points on a plane or the set of continuous functions on a given interval.

Introducing new and more general structures implies new terms and symbols. As a result, mathematical text looks like Chinese for non-mathematician. There are almost no numbers and sometimes it is hard to find any equations, yet these are things which come to our minds when we think about mathematics. And the text between mathematical symbols doesn't help a lot, because this is how a typical mathematical theorem sounds like:

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

Simply saying, they don't associate with anything we know. This particular theorem is called Poincare conjecture. The proof of it done by Russian mathematician Perelman is one of the greatest results in modern mathematics. The interesting fact is that Perelman didn't proof exactly what this theorem says but more general theorem - Thurston's geometrization theorem, and Poincare conjecture is just one of its consequences, a "positive side effect"...

But has mathematics always been so well defined and precise? No, it wasn't. But I will write about how it happened and who was responsible for it in the future.