# What does data mean mathematically

## General information on mathematics

### Chapter I: Ubiquity of Mathematics

#### A numbers in everyday life

We are literally showered with numbers these days: starting with

simple physical dimensions (times, distances, weights, speeds, power, etc.)
above
Money quantities (prices, costs, profits, credits, taxes, wages, etc.)
up to the
more refined ratio information (interest, growth rates, profit ratios, statistical values, etc.).
Obviously, numbers serve here as data and guidelines and mathematics as an informative and normative medium.

But in this increasing arithmetization of everyday life there is probably also an overburdening of people, sometimes even an overreaching. Just as the cultural technique of reading and writing arouses the tendency to consider what is written to be more credible than what is spoken, so does arithmetic nourish the belief that numbers are more reliable information than verbal attributes. It is certainly useful to have accurate data, but only if you know what to do with it. (What does 1 percent growth mean? How to average? What is tax progression?)

Finally: Mathematics as a guarantee for correct and substantial information is finally overused when the (apparent) precision of a number is no longer appropriate and consequently meaningless. (What does 64 percent probability of rain mean? Can the teaching performance of institutes be measured?)

#### B data processing, computers, calculators

Computers have conquered leisure and work and save their users - at least in the second area mentioned - considerable effort and costs. Of course, there is a fair amount of math involved in this technology, indirectly in the hardware and more directly in the software, but of which the ordinary user does not see much. But even without a thorough understanding of the internal construction and functioning of computers, one is constantly confronted with mathematics when dealing with them. It is important here to understand graphics, tables, diagrams, algorithms, that is, one must acquire an understanding of complex mathematical processes and data collections, even if one (like most people) has not been specially trained for them.

Obviously, basic concepts of the mathematical language have grown into the everyday communication structure to an extent that many are not aware of. Especially at the lowest level of this technology, namely when using pocket calculators, it becomes clear that mathematical understanding (here in the form of feeling for numbers) is necessary to assess and control the results. The well-known cliff in school lessons in this regard suggests a precautionary insight, namely:

You first have to learn to calculate yourself before you delegate this activity.
This may be indicative of the way math is turned into utility in the first place.

#### C school

Mathematics is inserted vertically into the educational canon, as a continuous main subject from the lowest to the highest level. Mathematics lessons should first of all convey the knowledge and skills that are useful for coping with the confrontation with mathematics mentioned under A and B. This is done in primary and secondary level I by treating the elementary arithmetic, geometric and functional objects and their regularities.

In the secondary level II and the grammar school level, the insight into mathematics is then expanded and the understanding of their way of thinking deepened, namely with the help of the introduction to classical areas of mathematics such as linear algebra, analysis, and stochastics. At the same time, this creates a link with other school subjects (e.g. physics and chemistry) so that the important application aspect of mathematics can already be seen at school level.

Of course, this educational phase also has a preparatory function for further training (e.g. at technical colleges and universities), also and especially when the graduate is not currently studying mathematics or natural sciences. Because the understanding of mathematical facts is now integrated into the broadest areas of science presentation (see also Section D).

One can ask the question - as it happens here and there - whether the mathematics education in the scope of elementary and secondary school is not enough as preparation for life. But one can also turn the matter around and ask whether mathematics does not have to be taught more and more in detail because of its growing global importance.

#### D Mathematics as an auxiliary science

While mathematics has a direct impact on areas A, B, C and affects everyone, be they laypeople or specialist, its influence on the various sciences is more hidden, namely through the work of the respective experts, which is usually difficult to understand for the layperson. It has always had its (if you will) strongest effect in scientific interaction, and it is precisely the fact that this area is so little public that it makes it difficult for the public to understand the nature and effect of mathematics.

In particular, there are the following areas in which mathematics is applied:

• Natural sciences (and medicine),
• Technology (and craft),
• IT (and communication technology),
• Economy and finance,
• Psychology and sociology,
in three main ways:
Provision of theories from various mathematical disciplines for the scientific description and for the derivation of knowledge in a purely deductive (non-experimental) way Development and evaluation of procedures for solving specific problems Statistical data analysis

To 1.: This is what is meant by saying that sciences are (or should be) written in mathematical language. In physics and computer science, for example, the integral part of mathematics is so immense that the widespread term `` auxiliary science '' is actually too weak.

Mathematics as the language of science forms the basis for the dream of scientists to organize the variety of natural processes and to get them under control by taking all complicated individual phenomena mathematically from a few basic principles that have been copied from nature, i.e. solely by means of pure thought , derives and understands.

To 2 .: The mathematical disciplines called up in this way are primarily those of so-called applied mathematics, such as B. Numerics, Scientific Computing, Operations Research, Optimization, Discrete Mathematics, etc. Thanks to the enormous power of today's computers, these disciplines are able to carry out concrete calculations for a growing number of problems of increasing complexity. However, this also increases the mental effort that one has to make in order to bring mathematics into a form suitable for processing by the computer (see also Chapter II.D.3). In addition, the modeling of the application situation and, in many cases, the simulation of real processes as well as their visualization play an important role.

The disciplines mentioned have experienced an enormous upswing due to the increasing use of computers; some of them came about in the first place.

To 3.: It is perhaps not generally known that in any kind of scientific, technical or economic investigation the data accruing to back up the conclusions to be drawn have to be statistically analyzed and processed in a professional manner. The mathematical disciplines of probability calculation and statistics concerned with this provide precisely formulated procedures which include precise rules that should be adhered to in every data analysis. Here, too, the strong increase in computing capacities has given decisive impulses for the development of new, high-performance statistical methods.

What is often presented to the general public as statistically confirmed knowledge has not always come about according to the rules of statistical art and with the participation of statistics experts and sometimes does not withstand scientific scrutiny. Here, mathematical statistics has the important task of providing a reliable basis for the maintenance and general implementation of scientific standards.