To answer this question precisely, we need to go back to the prehistory of this discipline. Algebra and geometry emerged for practical purposes. It's an unambiguous statement of how much I possess, how much I can acquire something for, and how many monetary resources I have to pay for the work and services of others. The key word here is "unambiguous" – defining the interpretation of a given quantity, perceived and understood by others exactly in the same way.
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People inherited from their animal ancestors a fuzzy counting system. Built on several synonyms like "a few," "many," "a lot," this metalanguage could approximate quantities but did not aid in barter or trade transactions. Because our primitive metalanguage contained three precise concepts - "one," "two," "three" - people began to apply multiplication of these concepts, laying the foundations of algebra and the measurement of flat spaces. Mathematics became a reflection of human nature and the process of perceiving stimuli from the environment. It effectively aided in conducting trade, winning battles, and waging wars.
I present this process in a simplified manner, referring to a feature discovered in antiquity. I call it the relativity of description. Each phenomenon can be described using degrees of simplicity. Mathematics provided us with the formalization of various degrees of simplified descriptions of phenomena or things. The development of writing allowed for the representation of quantities in a form that could be read in the same way by Egyptians, Phoenicians, and other peoples. Here we witness the evolution from basic primitive representations - the number of strokes as a synonym for the quantity of something - to the symbolic representation of a given quantity in the form of a abstract symbol.
Mathematics is thus a symbolic representation of the reality surrounding us and continues to serve this function to this day. It is a symbolic language of description, defining the manner of expressing quantities and their mutual relationships, simultaneously defining the level of simplificatio
The beginning of abstract forms of mathematics was the discovery of zero. The concept of zero does not fit into the human conceptual apparatus; it is purely abstract. We can easily visualize this by comparing two sentences describing the same quantity:
There ARE NO cars on this road.
There ARE ZERO cars on this road.
Our minds are geared towards registering what exists. But what if what exists is absent?
The spread of zero is owed to Arabs and Hindus. In Europe, it was not until the 10th century that Pope Sylvester officially legalized zero.
The development of abstraction has revealed another characteristic of mathematics to us: the organization of data - the ability to create multiple equivalent structures to describe a phenomenon. By distinguishing selected elements of a given structure and the relationships binding them, we obtain the means to describe the phenomenon. The simplest example is the concept of a function, where we analyze two selected quantities and their relationship to each other, disregarding other quantities and forces involved in the process.
I want to refer here to the principle of linguistic relativity, also known as the Sapir-Whorf hypothesis, and relate it to the symbolic language of mathematics. Although the hypothesis pertains to natural languages, its assertions have obvious implications in mathematics.
A stronger form of the hypothesis, known as linguistic determinism, asserts that language completely determines human thought and behavior - linguistic structures dictate an individual's perception of their "self" and embed them in certain cognitive schemas.
More precisely: the manner of thinking is a function of the language used.
While the phenomenon of linguistic relativity is often questioned in the context of natural languages (and rightfully so), when applied to the synthetic language of mathematics, it holds deeper significance. Based on existing linguistic structures, new abstract structures are created through negation, multiplication, combination, or the highlighting of selected elements. The language of mathematics determines the way data is analyzed, allowing for the creation of abstract entities. It is true that a mathematicians thinks "differently." Because their mode of thinking is determined by the very structure of the language.
This defines mathematics as the only scientific discipline that generates new cognitive values, creating nonexistent entities based on the synthetic language it employs.
It quickly became apparent that these nonexistent entities were increasingly suitable for practical application. The world turned out to be more complex than any conjectures of ancient philosophers, and "abstract" mathematics provided tools for their formal description. Set theory, topology, and quantum mechanics are primarily abstract languages for describing these phenomena. Their existence indirectly gave rise to these concepts (e.g., the concept of spacetime in GTR). Contemporary physics and increasingly practical technology are built upon them.
More than the queen of sciences, mathematics is thus the foundation of every possible field of knowledge. It is the soil in which scientific theories and new disciplines grow. However, not everything in mathematics is straightforward - contrary to appearances, it is closely related to our biology and the way we register and process stimuli from the environment.
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The basis of mathematics lies in definitions, theorems, and proofs. At this most fundamental level, doubts and ambiguity arise. Mathematicians disregard our biology by using substitute synonyms. They are called primitive concepts or intuition and assume that they are inherently undefinable.. Let's take a closer look:
A definition is a comprehensive description of a given object or phenomenon, capturing its key features without which the object or phenomenon does not exist. It is not a detailed description but rather the highest level of simplification, reducing the object or phenomenon to its fundamental characteristics.
Definition differs from axiomatics. Axiomatics consists of a set of statements that provide a description of a given phenomenon or object, provided that they are true. Often seen as a mathematical cry of despair – faced with the impossibility of constructing a precise definition, mathematics resorts to axiomatics.
To this day, there are no precise definitions of the most fundamental mathematical concepts. Statements are increasingly made that definitions are unnecessary and that a description based on axioms alone is more than satisfactory. The significant difference is often overlooked.
A precisely defined essence/traits and common properties/ of an object or phenomenon provided by a correctly constructed definition is always unambiguous. It delineates the scope of applicability of a given concept. This is not provided by axiomatics.
Negating the axiomatic approach, however, is an incorrect approach. Definition and axiomatic systems complement each other. They are two different facets of analyzing an object or phenomenon. It is worth mentioning that many valuable mathematical proofs are based precisely on axioms. Both forms are important and necessary.
So what is mathematics then? Primarily, it is a way of looking at the world, a means of understanding it, and a great adventure of humanity. It is a continuous journey into the unknown, exploring new worlds. It presents constant new challenges.
I argue that there is no such thing as pure mathematics or abstract mathematics. What is abstract today will be a formalized description of our world tomorrow. We do not know if our world is the only possible one. In order to survive, we must understand it, gain power over it. Mathematics is precisely what gives us that chance.
