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Innovation and mathematics


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Innovation and mathematics,

or a long way to Kant and beyond

or the self-development of science and its function as an auxiliary science


There is a very rich repository of books presenting the historical process of the fine arts. The number of works dealing with the history of mathematics may in fact be a smaller number than in the history of art, but it is usually extremely thorough and voluminous. What is common to both areas, however, is that each is based on exclusive human activity.

The development of mathematics as such can also be incorporated into a general system, a typifiable developmental trajectory.

Representation is historically much older, typical of the early stages of human coexistence, while mathematics can already be identified in a much more complex, hierarchical period of the organization of human society. Today, mathematics in the ordinary sense is made up of two basic branches: geometry based on visual basis and algebra based on counting. It is no coincidence that the “scientificization” of mathematics lies in the case of visual-based geometry: namely, the axiom, definition, statement, proof, principles and methods of proof are fixed in antiquity (here one should think above all of his basic work Euclid Elements). . The further development of counting (e.g., the Romans and their numbers used to this day are only for counting) occurs later, already through the demand of the Middle Ages and partly through the self-development of science (algebra).

In addition to the exclusivity associated with human activity, it is also characteristic that mathematics is rooted in reality, even if we think of developments in the late 19th and early 20th centuries in particular, in some areas this rooting seems very distant and highly rethinking. On the other hand, rethinking the history of mathematics is useful because it is increasingly linked to name, place, and time because of its self-development, its connection to other sciences, and its impact on other sciences, making it important and possible to integrate and analyze is.

So I described it with that, yes

• self-development as one's own activities expand,

• develops through its connection to other sciences (mainly technical sciences, physics) and by solving problems arising from the connection, new impulses generate new ideas

• its impact and repercussions in response to and influencing the development, innovation activities of other sciences

A feature of self-development is that it widens the circle of thought, e.g. the appearance of natural numbers, the appearance of fractions due to more complex tasks, negative numbers, decimal fractions, multi-dimensional “numbers” beyond the scalar, etc.

The formation of one's own scope and self-development - even the foundations laid in antiquity (principles, definition, statement, proof, rules of proof, etc.) expand, their thought content is also transformed, but it retains its original core, the root.

Solving the various problems that arise in practice generally gives a strong impetus to the development of mathematics: in the 18th and 19th centuries, mainly the technical sciences were an inspiring factor, but social and economic problems broadened e.g. strong development of statistics.










By the end of the 18th century (with name, place, address), the development of mathematics had reached a level that aroused the interest of philosophers. Thus, Kant is credited with the finding that there is as much absolute certainty in every science as there is in mathematics, or in other words, he made the statement to Antoine Lavoasier that every science is worth as much as there is mathematics in it. (although many large fields of mathematics, such as probability, etc., were still under development). Both man and human aspirations could increasingly be characterized by mathematics, e.g. on a statistical basis.

A special feature of the self-development of mathematics is the leap forward, when a seemingly ideological theoretical problem still seems to be self-serving (see the beginning of graph theory). (the same in painting, when one begins to apply new tools or a new approach to representation that one or two years later takes over). In other words, the practical utilization of the results of self-development, their transposition into the practical, is not always immediate, but once a time comes when it turns out that this is exactly the theoretical tool that is needed to solve a practical problem. A medieval problem of how many numbers there are, including how many special character numbers, regular numbers, and random numbers that can be produced, these questions are part of our daily practice today.

In the development of a discipline, sometimes in solving a technical problem, it may arise whether there is a mathematical toolkit for the solution that is at least partially or even more fully developed somewhere already in advance or in a less obvious way.

And it is also one of the cornerstones of the development of innovation. How to assess the development of an innovation area, to characterize it, or to guess its future development, to have a mathematical tool system with which to make it scientifically sound - or at least to make it visible.

In the case of a science called mathematics, which is in fact already very wide-ranging, digitization has brought an extraordinary turning point. In reality, however, digitization was essentially the end of the 19th-20th century. able to use the basics of the early twentieth century, the advent of the computer was primarily a quantitative and somewhat qualitative turnaround in solving classical computational problems. The possibility and need to treat classical theoretically solved or only assumed principles with a computer solution has emerged, but this is still in its infancy (eg the production of the largest primes, proofs), but in reality these are only experiments in the theoretical essence of mathematics. to approximate


In addition to its own development, innovation (of any kind, ie of a technical, social or other nature) raises another specific question: how much is in development and the factors that affect it and how much, as the human factor is used to say. The development generated by society is even harder to grasp, as society is not simply a mechanical aggregation of individual people, it also means a different level, a different level at the family level, small community almost, settlement, country or country-community level and back world as a whole.

At the individual human level, the “human factor” as such consists of a complex interplay of a minimum of three basic things.

Take, for example:

A person who is at a high level, takes his profession seriously and is committed will suddenly be shot on the open street.

• Due to his profession, it is a natural expectation from him, and even from within, that he thinks he will forgive the perpetrator. (Thought process)

• This does not mean that if you hear a bigger bang or see a bigger flash, it will not vibrate, even if you are on counter therapy for years. (emotional reaction type 1)

• And then we didn’t talk about waking up in your dreams, which is the least influential level. Plus the elements and needs of society outlined above. (emotional reaction type 2)

• Physical problems and their expansion are not negligible factors either. Chinese health philosophy attaches great importance to this, not to mention that the Romans already knew that a healthy body is a healthy soul. (reactions type 3)

This is the individual level, to which is added a wider circle:

• the social environment of the period, including expectations, as well as the narrower and wider geographical environment

• There are unseen, possibly inherited urges, qualities that are realized (or not) in the given environment

Thus, even named in this way, these are four well-distinguishable levels, which, however, are closely related and in a complex system of relationships. Statistical characteristics can be used to outline the characteristics of each array, and even to model the relationship between them. In fact, the types of functioning of individual people are also approachable, and what is difficult to approach at the moment is the creativity of individual reasoning.

It is clear that the number 1 thinking reaction is primary and decisive in the innovation process. It is difficult to imagine that e.g. for someone, the motivation to “only invent something” leads to results.

Human thinking has relatively easy-to-grasp elements and typifiable features. The oldest such form of typification is the science of logic itself, which has developed since ancient times. But with the development of the commonly commonplace human cognition, human knowledge, and psychology as well as psychiatry, the mapping of types of thinking is advancing.

A simple but probably surprising example. Agatha Christie culminates in her books the observation that unusual, unexpected, or there and then surprising sentences are the cornerstone of crime solutions: “Why did she say this,” “Strange that she said this,” and similar remarks corner the head of the detective. Such “experiences” also exist in the life of an ordinary person, preserving for decades why he said this, how he said it, said it, and so on. (I could list these myself).

But in addition, other typical thought patterns can be presented that can be used in some way.

It may be strange, but this scheme of Agatha Christie is already known to a small extent by the Google search and storage system, for example. Many, many years ago, for example, someone searches for something that can be targeted by ads, even after a break of many years, it brings up - obviously for lack of - that search, even though it may not have been relevant for a long time, but maybe.


There are already mathematical structures capable of analyzing complex systems, but due to the problems I have just indicated, a kind of extension is needed in order to examine the merits.

Creativity and innovation


What is still difficult to grasp with our tools today is the individual innovation skill, individual creativity. Although we look at different areas of innovation (be it fine art, a little technical novelty, or a bigger turnaround), individual creativity is hard to predict. There are historically and geographically favorable, conducive conditions that help unfold, but forecasting is not easy. Why?

The innovation capacity of societies, including countries and parts of countries, can be better defined and grasped, and shifts between large regions of the world can also be registered.


It is precisely this factor that makes the description of the innovation processes, and even their prediction, and the selection along the main directions uncertain. In our time, the so-called. scientific research often involves a predictable stage of development, the large number of people working on the problem, the time and costs involved, as well as standardization and the mass scale of the problems to be solved.


• external needs, problems to be solved - can be quantified and modeled

• self-development - can be quantified and modeled to a small extent

• historically favorable period - can be quantified to a small extent along historical trajectories and economic cycles

• geographical factor - broadly predictable, both temporal and spatial propagation

• individual intuition, kerativity - the least