# Foundations of Mathematics

https://en.wikipedia.org/wiki/Foundations\_of\_mathematics

* the foundations of mathematics
* discovered inconsistencies in
  * geometry (non-Euclidean geometries)
  * set theory (Russell's paradox)
* Hilbert's program
* axiomatization of mathematics

The foundations of mathematics is the study of the _philosophical, logical and algorithmic basis of mathematics_.

More broadly, it is an investigation of _what underlies the philosophical theories concerning the nature of mathematics_.

The Foundations of Mathematics (FOM) is a mathematical field whose origin is tied to the early XXth century's events, when mathematics was plagued with inconsistencies discovered in geometry and the recently established set theory.

of mathematics primarily concerned with the study of the fundamental mathematical concepts and the way they compose and form hierarchies of more complex structures and concepts, especially those important structures that form the _language of mathematics_ and are also called _metamathematical concepts_, with an eye to the philosophical aspects and the unity of mathematics.

(formulas, theories, and especially models that give meaning to formulas, definitions, proofs, algorithms)

The study of the _basic mathematical concepts_ (set, function, geometrical figure, number, etc.) and how they form hierarchies of more complex structures and concepts,

especially the fundamentally important structures that form the _language of mathematics_ (formulas, theories and their models giving a meaning to formulas, definitions, proofs, algorithms) also called _meta-mathematical concepts_, with an eye to the philosophical aspects and the unity of mathematics.

The search for foundations of mathematics is a central question of the philosophy of mathematics; the abstract nature of _mathematical objects_ presents special philosophical challenges.

The foundations of mathematics as a whole does not aim to contain the foundations of every mathematical topic.

Generally, the foundations of a field of study refers to a systematic analysis of its most fundamental concepts and primitives, its conceptual unity and its hierarchy of concepts (natural ordering), which may help connect it with the entirety of the human body of knowledge.

Mathematics' many developments towards higher abstractions in the 19th century brought new challenges and paradoxes, urging for a deeper and more systematic examination of the nature and criteria of mathematical truth, as well as a unification of the diverse branches of mathematics into a coherent whole.

The systematic search for the foundations of mathematics started at the end of the 19th century and formed a new mathematical discipline called _mathematical logic_, which later had strong links to theoretical computer science. It went through a series of crises with paradoxical results, until the discoveries stabilized during the 20th century as a large and coherent body of mathematical knowledge with several aspects or components (set theory, model theory, proof theory, etc.), whose detailed properties and possible variants are still an active research field. Its high level of technical sophistication inspired many philosophers to conjecture that it can serve as a model or pattern for the foundations of other sciences.

## Math development

Mathematics is the science concerned with abstract concepts such as number, structure, space, change. _Pure mathematics_ , one of the two main branches of math, deals with these abstract concepts for their own sake (`maths-pur-maths`); the other branch, _applied mathematics_, looks for applications of math to other science.

Mathematicians try to curb complexity of various systems found in the universe, or inspired by them, by focusing on a particular system, then searching for recurring patterns which inspire new _conjectures_, which might lead to the discovery of new knowledge.

_Mathematical proof_ is the process used by mathematicians to analyze a conjecture, possibly resolving its truth value through the use of _rigorous argumentation_. Sometimes the effort to resolve conjectures takes years, even centuries; or it happens that the established proof is only later found incorrect. But often the correct proof benefits from the previous attempts. Sometimes the proof is never obtained; many conjectures are still neither proven true nor false. It can also happened that a conjecture is not provable at all, but there are never hints whether that might be the case.

Mathematics has started from counting, calculation and measurement, and further developed as the systematic study of objects from the nature, but then the use of abstractions propelled it into the science of abstract, widening its interest to also include the idealized objects living only in our minds. When mathematical objects and structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature.

_Rigorous argumentation_ was championed by ancient Greeks, most notably by Euclid. His book "Elements", from \~300 BCE, featured theorems, of what will later be known as _Euclidean geometry_, that were all deduced from a small set of axioms. Euclid justifies each proposition by a demonstration in the form of chains of syllogisms, showcasing excellence in rigor that set the standard for the future.

> "Elements" is one of the all-time most influential works, having served as the geometry textbook from the time of publication until the early XX century.

Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery.

Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others, on _axiomatic systems_ in the late 19th century, it has become customary to view **mathematical method** as establishing truth by _rigorous deduction_ from appropriately chosen _axioms_ and _definitions_.

> _Mathematical method_, championed by the ancient Greeks, is used to discover new truths and logically organize existing body of knowledge through the use of _rigorous deduction_ that stems from appropriately chosen _axioms_ and _definitions_.

Mathematical method is employed to build theories, using _primitive concepts_ and _axioms_ as the building blocks, and _rigorous argumentation_ for deriving proof.

**Primitive notion** is a given concept, a concept not defined in terms of previously defined concepts, that is to be taken for granted. Frequently, the primitive notions are unavoidable, either because the theory has to start somewhere, lest digress into infinity of definitions, or because a term is not defined (or it is undefinable). Therefore, primitives, like axioms, do not require proof.

**Axioms** are statements taken to be true that serve as starting premises for deriving further arguments. In philosophy, an axiom is a statement that is so evident or well-established that it is accepted without question or controversy.

The initiative to establish the foundations of mathematics came about very late in the course of math development, at the end of the XIX century, when some aspects of math were found to suffer from paradoxes and inconsistencies.

In 1900, at the international conference in Paris, David Hilbert presented a collection of 23 important mathematical problems, later known as **Hilbert's problems**, pushing for their solutions by the end of the XX century. One of the issues was about establishing the foundations of mathematics i.e. coming up with the basic set of axioms from which the whole of math could follow. Hilbert wanted to ground all existing math theories to a finite and complete set of axioms, together with the proof of consistency of these axioms.

The _foundation of a mathematical field_ refers to analysis of the most fundamental concepts within a field, an investigation into mean to convert that field into a coherent body of knowledge and integrate it with the rest of the human knowledge.

The **foundation of mathematics** can be regarded as the study of the basic mathematical concepts, especially the fundamental ones that form the language of mathematics, such as formulas, theories, models, definitions, proofs, algorithms, etc.

Hilbert's challenge was in response to the **foundational crisis of mathematics** that emerged at the end of XIX century, as a consequence of discovering inconsistencies and paradoxes in prevailing mathematical theories.

The search for theory that will serve as the foundations of all math settled down during the XX century, putting forward a ZF set theory. Recently, the commotion started again as the _Category theory_ began to draw support as the new contender for the title. Still, to this day the winner is undecided (undecidability as leitmotif).

***

**Foundational paradoxes**

If the goal of _mathematical ontology_ is taken to be the _internal consistency of mathematics_, it is more important that mathematical objects be definable in some uniform way, e.g. as sets, regardless of actual practice, in order to lay bare the essence of its paradoxes.

This has been the viewpoint taken by _foundations of mathematics_, which has traditionally accorded the management of paradoxes a higher priority than the faithful reflection of the details of mathematical practice as a justification for defining mathematical objects to be sets.

Much of the tension created by this foundational identification of mathematical objects with sets can be relieved without unduly compromising the goals of foundations by allowing two kinds of objects into the _mathematical universe_, _sets_ and _relations_, without requiring that either be considered merely an instance of the other.

These form the basis of _model theory_ as the domain of discourse of predicate logic. From this viewpoint, mathematical objects are entities satisfying the axioms of a formal theory expressed in the language of predicate logic.

**Category theory**

A variant of this approach replaces relations with operations, the basis of universal algebra. In this variant the axioms often take the form of equations, or implications between equations.

A more abstract variant is category theory, which abstracts sets as objects and the operations thereon as morphisms between those objects. At this level of abstraction mathematical objects reduce to mere vertices of a graph whose edges as the morphisms abstract the ways in which those objects can transform and whose structure is encoded in the composition law for morphisms. Categories may arise as the models of some axiomatic theory and the homomorphisms between them (in which case they are usually concrete, meaning equipped with a faithful forgetful functor to the category Set or more generally to a suitable topos), or they may be constructed from other more primitive categories, or they may be studied as abstract objects in their own right without regard for their provenance.

***

## Informal description prevents infinite regression

When developing a new theory, the notion most central to the theory is gonna be introduced, immediately provoking the need for the definition.

The definition that follows will certainly introduce terms that, in turn, need their own definition.

But then the terms used in that definition require definitions as well; and so on, always further down the downward spiral of infinite regression.

***

note-tip: to avoid letting this inconvenient situation linger and poison your further proceedings with the feeling of incompleteness (not the Godel's kind), you might entertain several strategies: constraints, triviality, vanity, digression.

The most straight forward way out is to keep defining terms until you reach the low-level fabric of the universe or existence whichever comes first loose yourself. However, excuses are the norm - blame the insufficient spacetime needed for proper deposition; to do the definition justice; lest be wasted on trivial matters, so trivial in fact, the reader's vanity could do it. Leaving it as an exercise to the reader comes as close second. Else, a call upon intuitive understanding might suppress the demand for rigorous argumentation, provided it is paired with authoritative use of a carefully placed "indeed".

A solid line of reasoning should form where the higher concepts clearly follow ("yes sequitur") from the terms below; sound complete consistent arguments that hold even in reference to the borderline cases from the edges of their domain.

The relief, inspired by the literally device out of ancient Greek tragedies, was promptly put to practice - "indeed", the theorists were granted a spell to break out of the infinite regression while simultaneously denying the obligatory definition. Such primitivism!
