Alexander Kuklev (akuklev) wrote,
Alexander Kuklev
akuklev

Mathematical introduction

What is mathematics good for?
Mathematics provides instruments that can be used for many practical concerns. Historically, first uses of mathematics were trade, land surveying and architecture. Now mathematics is used primarily in engineering, economics and all mature sciences. In physics and engineering essentially nothing can be done or understood without involving mathematics. Instruments provided by mathematics are notions, notations, facts and methods: either strict algorithms or general strategies involving some creativity at one or another point. Let's look at examples:

Trade is exchange of goods directly or involving intermediate carrier good called money. Either way, you need the notion of amount (incl. addition, multiplication and division) for doing reasonable trade. You need to understand facts like commutativity of multiplication (triple of a double is the same as double of the triple), which was an unobvious fact for ancient people and remains so for the majority of current first graders. And you can make great advantage by knowing decimal system which is a notation and pencil-and-paper addition/multiplication/division, which are methods. In ancient land surveying one had to divide a piece of agricultural land into a number of equally big pieces or to compare the existing fields in their size. There were no exact algorithms for doing it, but a general strategy (dissecting into appropriate trapezia and triangles) based on notions of shape and area and knowledge on areas of several common geometric figures. In modern social sciences one often uses mathematical instruments like principal component analysis and its generalizations for processing and analyzing data, theory of dynamic systems and differential equations to model some processes and various methods for testing hypotheses. The range of mathematical instruments used in natural sciences and engineering is too enormous to be described here.

Well, understanding a field always involves learning the appropriate notions, comprehensive net of interconnected facts, methods and convenient notations. That applies to any kind of science-based discipline, so what's specific about mathematics? Mathematics concerns distilled mental constructs (like numbers and spaces) appropriate for strict synthetic reasoning.

What's a “Mental Construct”?
Here is a quotation from Paul Lockhart's “Mathematician’s Lament”:
“I might imagine a triangle inside a rectangular box:
I wonder how much of the box the triangle takes up? Two-thirds maybe? The important thing to understand is that I’m not talking about this drawing of a triangle in a box. Nor am I talking about some metal triangle forming part of a girder system for a bridge. [..] For one thing, the question of how much of the box the triangle takes up doesn’t even make any sense for real, physical objects. Even the most carefully made physical triangle is still a hopelessly complicated collection of jiggling atoms; it changes its size from one minute to the next. [..] The mathematical question is about an imaginary triangle inside an imaginary box. The edges are perfect because I want them to be— that is the sort of object I prefer to think about. This is a major theme in mathematics: things are what you want them to be. [..] On the other hand, once you have made your choices (for example I might choose to make my triangle symmetrical, or not) then your new creations do what they do, whether you like it or not. This is the amazing thing about making imaginary patterns: they talk back! The triangle takes up a certain amount of its box, and I don’t have any control over what that amount is. There is a number out there, maybe it’s two-thirds, maybe it isn’t, but I don’t get to say what it is. I have to find out what it is.

[..] There’s no experiment I can do with test tubes and equipment and whatnot that will tell me the truth about a figment of my imagination. The only way to get at the truth about our imaginations is to use our imaginations, and that is hard work.
In the case of the triangle in its box, I do see something simple and pretty:
If I chop the rectangle into two pieces like this, I can see that each piece is cut diagonally in half by the sides of the triangle. So there is just as much space inside the triangle as outside. That means that the triangle must take up exactly half the box!

This is what a piece of mathematics looks and feels like. That little narrative is an example of the mathematician’s art: asking simple and elegant questions about our imaginary creations, and crafting satisfying and beautiful explanations. There is really nothing else quite like this realm of pure idea; it’s fascinating, it’s fun, and it’s free!

Now where did this idea of mine come from? How did I know to draw that line? How does a painter know where to put his brush? Inspiration, experience, trial and error, dumb luck.”

What's “Synthetic Reasoning”?
The game of making up imaginary objects and reasoning about them is only of esthetic value to the player, unless one gives an exact specification of the objects in a conveyable manner so that others can reason about them too and obtain compatible results. Fortunately, this is possible.

About 2300 years ago a mathematician calles Euclides codified the rules of planar geometry, i.e. found a set of basic rules (he called axioms) of planar geometry. There are only 6 of them necessary to derive everything belonging to planar geometry. Well, Euclides don't get in completely right, some other things are actually assumed to be too obvious rules to be codified. If one includes them, there are 20 (so called Hilbert axioms).

(We now know that there are other geometrical systems like hyperbolic and spherical geometry. They are neither worse, nor better than the euclidean one. They just have different sets of rules, comparing them is like comparing checkers and chess. The question what geometry describes the world we live in is a physical one and isn't related to mathematics.)

Later on, Aristoteles discovered that true judgements can be derived from other true judgements using few obvious rules. Furthermore, it seemed that all arguments (including also ones involving additional constructions, like the one demonstrated in the previous section) can be translated to sequential proofs where every each step is just application of one of these obvious and unquestionable rules. So, proofs can be written down in a way that every mathematician (or a machine, if suitable machine-readable symbolic language is used*) can check their correctness and nobody can deny them.

Furthermore, if one can convert proof of certain statement A under assumption of some axioms (like the Euclidean ones) into universal proof of the statement “Under assumption of classical logic and rules of Euclidean geometry A is true”. Such statement is universally true, no matter what one thinks about appropriateness of Euclidean geometry and applicability of classical logic.

Mathematics is the only discipline, where the facts are derived rather than observed or presumed. The exception is the part of mathematics concerning its own foundations, where some tend to to justify basic notions on philosophical reasons or practical observations (both perform well).

____
Machine-readable language for verifiable proofs, constructions and calculations is called Construction Calculus. To date there are several of them, the difference being convenience of use.

The Geometrical Idea
At first, people learn to point at objects and use their names: “mom”, “dad”, “chair”. Then they start to use adjectives: “green”, “sour”, “big”, “warm”. And then come the first abstractions: “color”, “taste”, “size”, “temperature”, “location”. That are the names of attributes, and they give rise to one of the most important notions of mathematics, the one which connects it to natural sciences: the concept of space.

In the picture on the right you can see a slice through the color space, i.e. the totality of all perceivable colors. Every attribute of objects or phenomenon of the world has an associated space of possible attribute values. Space of lengths, space of temperatures and the most familiar 3d-space of point locations.

From the side of natural sciences, a space is defined by precise description of measurement process. The word “color” can have multiple different meanings, after all one can think of colors as perceived by people and colors as perceived by cats; these are two different notions of color, they give rise to different color spaces. But once you agreed on exact measurement procedure, you specified exactly what is ment by “color” or “length” etc.
(Warning: Here the word “measurement” doesn't necessarily mean “quantitative measurement”, it's not about assigning numbers but about what we measure and how we distinguish same and different.)

What's interesting for mathematicians is what you can do with measured values after you got them. In other words, what operations are available on the space and which rules they obey to. On every space there is an operation of comparing if two values are the same, this operation can be inferred directly from measurement process. (Measurement process in non-discrete spaces also provides a topology.) But in the most cases there is something more: you can lay lines through objects and measure distances as in euclidean geometry, or you can add objects and multiply them by real factors (as in space of lengths). Studying intrinsic properties of different kinds of spaces makes up a major part of mathematics.

The Algebraic Idea
Here is a lengthy quotation from “Basic Notions of Algebra” by Igor Shafarevich:
“What is algebra? [..] One can attempt a description by [..] drawing attention to the process for which Hermann Weyl coined the unpronounceable word “coordinatisation”. An individual might find his way about the world relying exclusively on his sense organs, sight, feeling, on his experience of manipulating objects in the world outside and on the intuition resulting from this. However, there is another possible approach: by means of /quantitative/ measurements, subjective impressions can be transformed into objective marks, into numbers, which are capable of being preserved indefinitely, of being communicated to other individuals [..] and most importantly, which can be operated on to provide new information concerning the objects of the measurement.

The oldest example is the idea of counting (coordinatisation) and calculation (operation), which allows us to draw conclusions on the numbers of objects without handling them all an once. Attempts to “measure” or to “express as a number” a variety of objects gave rise to fractions and negative numbers in addition to whole numbers. The attempt to express the diagonal of a square of a side 1 as a number led to a famous crisis of the mathematics of early antiquity and to the construction of irrational numbers.

Measurement determines the points of a line by real numbers, and much more widely, express many physical quantities as numbers. To Galileo is due to the most extreme statement in his time of the idea of coordinatisation: “Measure everything that is measurable, and make measurable everything that is not yet so”. The success of this idea, starting from the time of Galileo, was brilliant. The creation of analytic geometry allowed us to represent points of the plane by pairs of numbers, and points of the space by triples, and by means of operations with numbers, led to the discovery of ever new geometric facts. However, the success of analytic geometry is mainly based on the fact that it reduces to numbers not only points, but also curves, surfaces and so on. For example, a curve in the plane is given by an equation F(x, y) = 0; in the case of a line, F is a linear polynomial and is determined by its 3 coefficients: the coefficients of x and y and the constant term. In the case of a conic section we have a curve of degree 2, determined by its 6 coefficients. If F is a polynomial of degree n than it is easy to see that it has (n+1)(n+2)/2 coefficients; the corresponding curve is determined by this coefficients in the same way that a point is given by its coordinates.

In order to express as numbers the roots of an equation, the complex numbers were introduced, and this takes a step into a completely new branch of mathematics, which includes elliptic functions and Riemann surfaces.

For a long time it might have seemed that the path indicated by Galileo consisted of measuring “everything” in terms of a known and undisputed collections of numbers, and that the problem consists just of creating more subtle methods of measurements, such as Cartesian coordinates or new physical instruments. Admittedly, form time to time the numbers considered as known (or simply called “numbers“) turned out to be inadequate: this led to a “crisis”, which had to be resolved by extending the notion of number, creating a new form of numbers, which themselves soon came to be considered as the unique possibility. In any case, as a rule, at any given moment the notion of number was considered to be completely clear, and the development moved only in the direction of extending it:
“1, 2 many” ↪ Natural Numbers ↪ Integers ↪ Rationals ↪ Reals ↪ Complex Numbers.

But matrixes, for example, form a completely independent world of “number-like objects”, which cannot be included in this chain. Simultaneously with them, quaternions were discovered, and then other hypercomplex systems (now called algebras). Infinitesimal transformations led to differential operators, for which the natural operation turns out to be something completely new, the Poisson bracket. Finite fields turned up in algebra, and p-adic numbers in number theory. Gradually, it became clear that the attempt to find a unified all-embracing concept of number is absolutely hopeless. [..]

The principle of coordinatisation can nevertheless be preserved, provided we admit that the set of “number-like objects” by means of which coordinatisation is achieved can be just as divise as the world of physical and mathematical objects they coordinatise. The objects which serve as coordinates should satisfy only certain conditions of a very general character.

They must be individually distinguishable. For example, whereas all points of a line have identical properties (the line is homogeneous), and a point can only be fixed by putting a finger on it, numbers are all individual: 3, 7/2, sqrt(2), π and so on. They should be sufficiently abstract to reflect properties common to a wide circle of phenomenons. Certain fundamental aspects of the situations under study should be reflected in operations that can be carried out on the objects being coordinatised: addition, multiplication, comparison of magnitudes, differentiation, forming Poisson brackets and so on.”

You'll see that there are plenty weird number-like objects (say, infinite dimensional generalizations of matrices) which are vividly used for coordinatisation in physics.

Algebraic objects can be used to give very concise description for spaces. Once we know, that an algebraic objects (group, ring, algebra etc) homogeneiously acts on a certains space, the complete structure is fices. Spaces of lengths/temperatures/etc can be unambigously characterized as ℝ+-scale (ordered 1d space). ℝ-spaces and ℂ-spaces (called real and complex vector spaces respectively) are particularly well behaved and often met throughout the whole mathematics. Spaces with euclidean geometry (as well as spherical, hyperbolic, projective, conformal and some others) can be characterized as homogeneous G-spaces, where G is the corresponding symmetry group. Such characterisations are much more convinient than characterisations by 20+ axioms.
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