# Mathematics: General

Mathematical elements:

* mathematical primitive
* mathematical axiom
* mathematical definition
* mathematical conjecture
* mathematical proof
* mathematical theorem
* mathematical lemma
* mathematical theory
* Mathematical method
  * Primitives
  * Axioms
  * Inference rules
  * Rigorous argumentation
* Although there is written evidence of mathematical activity in Egypt as early as 3000 BCE, many scholars place the birth of mathematics in ancient Greece around the 6th century BCE, when *deductive proof* was first introduced. Aristotle credited *Thales of Miletus* with recognizing the importance of not just what we know but how we know it, and finding grounds for knowledge in the deductive method. Around 300 BCE, *Euclid* codified a deductive approach to geometry in his treatise, the *"Elements"*. Through the centuries, Euclid's axiomatic approach was held as a paradigm of *rigorous argumentation*, not just in mathematics, but in philosophy and other sciences as well. The "Elements" takes the second place of the most printed book of all times. Until the XIX century it was used as a proper textbook for geometry classes across Europe. (at around that time, the other, the so-called non-Euclidean, geometries were established, spawning out of the problematic Euclid's 5th axioms about parallel lines).

Math\*=-related

* invented or discovered
* abstraction, generalization
* analysis, synthesis
* searching for patterns
* abstracting the details
* aggregating chunks of knowledge
* eventually giving rise to a mathematical field
* integration into the entire body of human knowledge
* Mathematics is about finding patterns and lifting them into abstraction in its ever going search for truth.
* *Mathematical fields and disciplines* are formed through the work of researches that, nowadays, publish their findings in dedicated academic journals.
* *Mathematical knowledge* is expressed beginning with *conjectures*, which are then explored in an effort to prove them. The worst kind of conjecture is the one that can neither be proven nor disproven (Goldblach's) because mathematicians hate the middle. Some conjectures are known as hypothesis (Riehmann's) to mix things up. Those conjectures shown to be true go on to become theorems. Conjectures that are proven become theorems. A set of theorems constitutes a theory. A small theorem used as a steppingstone when proving a larger theorem is called a lemma (from old norse 'Lemmi').
* *Abstraction* is the process of extracting the essence from a concept by severing its dependences on the real world, then generalizing it softly until it bleeds utility.
* Platonic view of math: mathematical objects have existence independent of the human mind. (Whose mind? Yours? Mine? "Collective mind"?)
* The goal of science is to understand *what is*, by describing and understanding the universe. Math, on the other hand, seeks to understand *what must be*.

***

* Mathematicians investigate mathematical concepts, try to formulate conjectures and establish their truth by rigorous deduction from a set of axioms and previously defined *theorems*, all within a particular formal system.
* A *formal science* is a branch of knowledge concerned with the properties of formal systems based on a set of axioms (definitions and rules of inference). Unlike other science, formal science is not concerned with the validity of theories based on observations in the physical world.
* *Logical reasoning* guides us in understanding and constructing *proofs*; it helpes us reason about formal mathematical objects (numbers, lists, trees, formulas, etc., even theorems themselves).
* *Definitions* define the objects of discussion. For instance, a straight line (versus a curved one) might be defined as "a line which lies evenly with the points on itself", as in Euclid.
* *Axioms* describe what we can do with the things we've defined. For example, the axiom of symmetry says that "If A=B, then B=A". In this example, you could see the axiom as something you can do ("you can switch the sides of an equation") or as defining what it means that two things are equal.
* On top of this foundation, mathematics is built with *logic*. Given the definitions and axioms, certain conclusions follow as inescapable consequences. These conclusions we call *theorems*.
* Mathematics is not static, and the axioms and definitions we use are not discovered (as in they were here all along waiting to be revealed) but invented (as in pulled out of ass).
* As we seek deeper understanding, we often come to a point where we realize our earlier understanding was incomplete or even incorrect, and we seek to fix the foundations. This has occurred over and over again in the history of mathematics.
* Mathematics is a quest for understanding what must be, but the very concepts we try to understand are not set in stone. The objects of mathematics are defined by people, and as we understand them better, the definitions and axioms we base our understanding on change.
* Math is a quest of understanding what must be. The basis for this quest are the axioms and definitions. Definitions define the objects of discussion. Axioms describe what we assume those objects can do.
* Another way that definitions are invented is when mathematicians want to generalize an idea to a more general situation. Another way to say this is that mathematicians are trying to somehow identify the intrinsics of an idea. This is a major theme of modern mathematics. "What does it mean to be a shape?", "What does it mean to multiply things?". These two questions lead to the complete reformulations of many branches of mathematics.

Until Bernhard Riemann, a shape was always visualized in the plane or in space (or perhaps a higher dimensional ℝⁿ). Pondering the question of *what makes a shape a shape*, Riemann concluded that the essential property of shapes is that, at any point on the shape, one can travel in a certain number of directions. The usual visualization of shapes in space was a crutch that distracted us from the intrinsic properties of that shape. These "many-fold quantities", as Riemann called them, now called *manifolds*, have become the basis for geometry.

By generalizing multiplication, group theory in abstract algebra emerged. A group is a collection of things you can multiply. It's immaterial whether these things are matrices, functions, numbers, shapes, symmetries, etc.; as long as they follow the group axioms, you will know how to multiply them.

These kinds of generalization often seem to further complicate something that is already complex. But abstracting away the details and focusing on the core ideas turns out to be very valuable. First, sometimes it makes it easier to prove results you care about. Second, by unifying very disparate ideas (such as matrix multiplication and rotations and normal multiplication), if you can prove a theorem about groups in general, than it applies to all of these very different situations.

## The system of logic

* Mathematical proofs are not eternal, existing independently of any other concept, shape or form. A proof is a logical consequence of a specific theory that was developed in a certain formal system of logic. The theory and the proofs derived are valid only with regards to some logic system.

For example, the *classical logic* is a formal logic system that has two truth values (`true` and `false`) and it holds that a proposition is either true or false, not both and not neither.

This and other behaviours are enforced by the axioms, one of which, called *the law of excluded middle*, states that either a proposition is true, or that its negation is true. Another axioms, called *the principle of double negation*, states that if a statement is true, then it is not the case that the statement is not true.

On the other hand, the *intuitionistic logic*, which is also a mainstream system of logic, doesn't accept neither of these two axioms that are fundamental in the classic logic.

## Sand castle

* Modern mathematics has the set-theoretic foundation (concretely ZF set theory), meaning the whole of mathematics could be derived from the set theory. In fact, it would be better to say that the mainstream mathematics is based on set theory because majority of mathematicians today agree with that. However, there are other groups of mathematicians that have different views.
* Recently, the *category theory* has gained enough momentum to present itself as the alternative candidate for the foundations of mathematics. With the introduction of sets and soon after the related paradoxes, the type theory has started development, which makes another alternative for the foundations of math. The foundations of mathematics based on functions (lambda calculus and combinatory SK calculus) have been, and still are, investigated for this role as well.
* The foundation of mathematics itself also became the mathematical discipline to encompass the effort of searching for mathematical foundations. However, it came about late in the history of mathematics, when many mathematical disciplines were already formed.

Namely, in the XIX century, after many surprises that questioned stability of mathematics, German mathematician David Hilbert has set forward the motion for *axiomatization of mathematics*.

The goal was to put the entirety of math on a solid cornerstone upon which the rest of mathematics could be build by devising a strong theory with a carefully chosen set of *axioms*.

Any *mathematical theory* consists of a set of *mathematical axioms*, given without the proof. Besides the axioms, *mathematical primitives* are similarly privileged, require no proof as well.

* A theory may start its development with a mathemtical primitive as its most central concept, with axioms describing its behavior (primitive as the central entity whose behavior is described by the axioms). Alternatively, a primitive itself may be introduced into a theory as one of the axioms.

The fact that primitives and axioms require no proof has to do with infinite regression (also nicknamed vicious circle). Any theory has to start somewhere, it introduces some concept, some term that wasn't previously defined. If it were to define that term, then all the terms used in a definition would aslo require a definition of their own. And so on and on, ad nauseam. This is the vicious circle of regression that would prevent any theory to even begin.

Therefore, it is allowed to "make a cut" and arbitrarily choose the starting concept for a theory. That concept is usually the most fundamental concept of the theory, either introduced as a primitive or explicitly given as an axiom. Then, a minimal set of axioms (*compactness*) is chosen in such a way that it prohibits deriving contradictory statements within this system (*consistency*). Also the axioms should allow deriving all the true statements within the system (*completeness*). No axiom should be chosen if it can be derived from other axioms.

One can choose to construct the new set theory, adding an axiom as a fix for every encountered problem.

## The higher ground

Even if the foundation of mathematics is not unanimously agreed upon, there's no time to waste waiting for the consensus, so mathematicians proceed exploring their fields of interest anyway, forming new theories and deriving new proofs from them. This means that the proofs depend on the theory and if you accept the foundation of some theory, you can be certain that the proof will be correct within that theory. However, if the future shows that the foundation of that theory was wrong, there are always alternative to get on board with.

You can even work with respect to two opposing theories and then produce your work in both versions, which would be the case if you were to work within the systems of both classical and intuitionistic logic.

Whether one mathematician chooses to construct natural numbers in set-theoretic terms (zero as the empty set, `{}`, the successor as `S(n) = n U {n}`), and another using lambda calculus (zero as the lack of function application, `λsz.z`, the successor as `λnfx.f(nfx)`), the other mathematicians will happily welcome them both at the next level of abstraction, where natural numbers are concerned as the standalone mathematical objects anyway, abstracted away by removing them from their foundational background.

So, just get to the higher abstraction level somehow and see everyone in agreement once again, at least initially.

***

* Mathematics tries to recognize patterns, and use them to formulate *conjectures*, which are resolved by means of *mathematical proof*. Those conjectures found to be true become *theorems*.
* When a *mathematical structure* turns out to be a good model for some real world phenomena, then *mathematical reasoning* can provide further insight or predictions about the nature of that phenomena.
* Math was developed from counting, calculating, measuring and the systematic study of shapes and motions of physical objects through the use of *abstraction* and *logic*,
* *Practical mathematics* has been a human activity from as far back as written records exist.
* The research required to solve *mathematical problems* can take years or even centuries of effort.
* The *rigorous method* of math had first appeared in ancient Greeks: it was established by Euclid in his "Elements".
* Since the pioneering work of Giuseppe Peano, David Hilbert and many others on *axiomatic systems* in the late 19th century, it has become customary to view **mathematical research as establishing truth by rigorous deduction based on the initial set of axioms and definitions**.
* Mathematics was in slow development until the Renaissance, which was a period when innovations and new scientific discoveries led to an increase in math discoveries, a trend that continued to the present day.
* *Applied mathematics* has led to entirely new mathematical disciplines, such as *statistics* and *game theory*.
* *Pure mathematics* ("math-pour-math-ism") is math for the sake of math, a pure exploration of math unburdened with practical application. However, an application is often discovered later.
* Many areas of math began as inquiry into practical problems, but then the underlying rules and concepts were identified and extracted - abstracted into *abstract structures*. For example, geometry originated from practical calculation of distances and areas; algebra started from methods for solving problems in arithmetic.
* **Definitions of mathematics**. Math has no generally accepted definition. Different schools of thought, particularly in philosophy, have put forth radically different definitions, all controversial.
* **Philosophy of mathematics** aims to provide an account of the nature and methodology of mathematics and to understand the place of math in people's lives.

## Refs

<https://en.wikipedia.org/wiki/Definitions\\_of\\_mathematics> <https://en.wikipedia.org/wiki/Philosophy\\_of\\_mathematics> <https://www.youtube.com/playlist?list=PLFJr3pJl27pIp1EsDD2rYaTI7GxoXqrLs> <https://infinityplusonemath.wordpress.com/2017/06/17/what-is-math/> <https://infinityplusonemath.wordpress.com/2017/06/21/where-do-axioms-come-from/>

Astronomy-and-trigonometry <https://www.britannica.com/science/geometry/Astronomy-and-trigonometry>

mathematics <https://www.britannica.com/science/mathematics>
