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Computer and Mathematics

We humans process information and also have abstracted the rules by which this is done. The result is mathematics.

The computer processes information by applying these rules. It can even be used to find new rules, new paths, new proofs.

Computer science in this sense is a part of mathematics, now with many extensions which are specifically applicable to the computers of our times.

Since we have the computer now, we will not only let it calculate, but gradually use it for all kinds of formalized thought processes. Autonomous learning, abstracting and synthesizing (creativity), can also be taken over by computers, although still in its infancy.

The computer not only makes calculations unbelievably fast, but has the potential to take over most of the thought tasks, we humans are capable of.

Mathematics deals with information processing. But what is that?

Information

The communication between two humans, in which information is exchanged, can be broken down to an elementary process.

This process consists of

The selection is a repeated process. All selections make up a mapping. An element can be chosen exclusively only, and every element gets its turn.

Note

The smallest set one can still choose from has 2 elements. That is why the bit is the smallest unit of information.

The sender chooses concepts in his head, maps them to words, maps them to phonetic combinations or sign combination and sends them over a medium (air, paper) to the receiver. The receiver processes in the opposite direction.

Phonemes, letters and digits are there to create a multitude (words, numbers) to which concepts can be mapped (coding).

Note

With numbers one can choose everything one can choose with words. One way is to put all concept into a sequence and let the number choose the position. Words in our brain are used associatively, but that is possible with numbers, too.

Basically all dynamic systems function with the above elementary selection processes. There are always sets and choices.

Examples:

Since quantum mechanics we know that randomness is an inherent principle of nature, that not everything is predetermined, but that selection processes do really create new combinations, fleeting ones and staying ones.

Nature processes information by distributing states The universe can thus be compared with living systems like us. Both function with the same abstract principle of information processing.

Energy, Entropy

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Mathematics and Information

We have recognize the elementary process of information (transport) to be the selection which consists of

Let’s connect this better with mathematics?

Sets and Variables

The set we find in the set theory, the foundation of mathematics.

To choose elements from a set one can reserve a bit for every element and code a selection via 0 (not used) and 1 (used). One can do with less bits, if one first finds out, which elements exclude each other.

A set from which an element is chosen exclusively, is a variable. The element is the value.

Variable

Variablen bestehen aus Werten.

With variable we mean the real thing, not a placeholder for a number. Only via a mapping, e.g. by comparing to a unit, the value is linked to a number (coordinate).

Mathematics describes reality by finding variables and then dependencies between them.

Strukture

An important aspect of information processing is abstraction. By comparing one finds common patterns and these are used to build a smaller encoding (description) and to recognize later. In programming there is an important guideline: Don’t Repeat Yourself (DRY). That’s how mathematics does it as well. Patterns that repeat are described abstractly and concrete objects become examples of these structures (group, ring, field, …).

This makes information exchange (communication) more effective. Instead of repeatedly describe, e.g. the rules for \(+\) in \(\mathbb{Z}\) and \(V\) and so on one can say \((\mathbb{Z},+)\) and \((V,+)\) and so on are groups.

Mathematics as science builds a structure as a whole by successively adding new concepts and theorems. New works build on these concepts, choose what to work on, and thus extend the structure.

Struktur

This successive extension and sophistication of structure is a general developement of dynamic systems (evolution): biological evolution, economy, … (see above).

In order for complex and lasting structure to develop energy must be supplied in the right dosage. For the economy this is the money.

Algorithmics

To determine (choose) a value of a variable, mathematics uses the function. This is also the name in informatics, but with a slightly different meaning, Other names are subprogram, subroutine, procedure, …

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The value of one variable can depend on the values of more other variables. Function therefore have more parameters. They are called formal parameters in the definition of the function. A first selection process there is already when choosing the actual parameters to correspond to the formal ones when calling the function.

Functions in programming languages do not necessarily return the value of a variable, but it is a good design to name variables and the elementary dependencies via functions in order to clearly separate them from others.

Functions consist of calls to other functions

This algorithmics is hidden everywhere in mathematics behind numbers, expression and symbols (polynomials, \(\sqrt{}\), \(\lim\), \(\int\), …) in theorems and proofs. The whole of mathematics can be regarded as a huge program, but unfortunately still most of it in a language that the computer does not understand.

Representation

A big challenge is to convert the representation of mathematics into a language the computer can understand. There are many computer languages, some especially made for mathematics.

Many different representations increase the effort and reduce the applicability.

The same is true for human languages.

Numbers

When introducing the number systems algorithmic aspects are combined with the count (the natural number):

  • Count with + and -: Integers

  • Count with * and /: Rationals

Then we can regards

  • Numbers as elementary instruction (add 2, subtract 2, …)

  • analytic expressions as programs/functions

  • term simplification as program simplification, profiling

Equations

The equation is a function that returns the result of a comparison. Equivalent transormations of an equation is a kind of profiling, too.

Equations and inequalities are used to implicitly define sets.