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[[Archivo:Russell, Whitehead - Principia Mathematica to 56.jpg|miniaturadeimagen|Principia Mathemática, obra de Bertrand Russell y Alfred North Whitehead,  representantes de la perspectiva filosófica del Logicismo.]]
Los '''fundamentos de las matemáticas''' son el estudio de conceptos matemáticos básicos como [[Número natural|números]], [[figuras geométricas]], [[conjuntos]], [[Función matemática|funciones]], etc. y cómo forman jerarquías de estructuras y conceptos más complejos, especialmente las estructuras fundamentalmente importantes que forman el lenguaje de las matemáticas: [[Fórmula matemática|fórmulas]], teorías y sus [[Teoría de modelos|modelos]], dando un significado a las fórmulas, definiciones, [[Prueba matemática|pruebas]], [[algoritmo]]s, etc. también llamados conceptos [[Metamatemática|metamatemáticos]], con atención a los aspectos filosóficos y a favorecer la unidad de la matemática. La búsqueda por los fundamentos de la matemática es una pregunta central de la [[filosofía de las matemáticas]]; la naturaleza [[Abstracción (filosofía)|abstracta]] de los objetos matemáticos presenta desafíos filosóficos especiales.

Los fundamentos de las matemáticas como un todo no apuntan a contener los fundamentos de cada tópico matemático. Generalmente, los ''fundamentos'' de un campo de estudio, se refieren a un análisis más o menos sistemático de sus conceptos más básicos, su unidad conceptual y su ordenamiento natural o jerarquía de conceptos, los cuales podrían ayudar a conectarlos con el resto del conocimiento humano. El desarrollo, surgimiento y aclaración de los fundamentos puede aparecer tarde en la historia de un campo, y podría no ser visto por algunos como su parte más interesante.

Las matemáticas siempre jugaron un rol especial en el pensamiento científico, sirviendo desde tiempos antiguos como modelo de verdad y rigor para la inquisición racional, dando herramientas o incluso fundamentos para otras ciencias (especialmente la [[física]]). Pero las matemáticas ya hacía abstracciones muy elevadas en el {{siglo|XIX||s}}, que trajeron paradojas y nuevos desafíos, exigiendo un examen más profundo y sistemático de la naturaleza y del criterio de la verdad matemática, así como también una unificación de las diversas ramas de la matemática en un todo coherente.

La búsqueda sistemática de los fundamentos de las matemáticas empezó al fin del {{siglo|XIX||s}}, y formó una disciplina matemática nueva llamada [[lógica matemática]], con fuertes vínculos con la ciencia de la computación teórica. Fue mediante una serie de crisis con resultados paradójicos, que los descubrimientos se estabilizaron durante el {{Siglo|XX||s}} con un amplio y coherente cuerpo de conocimiento matemático con muchísimos aspectos o componentes ([[teoría de conjuntos]], [[teoría de modelos]], [[teoría de pruebas]]...), cuyas detalladas propiedades y posibles variantes aún están en campo de investigación. Su alto nivel de sofisticación técnica inspiró a muchos filósofos a conjeturar que podrían servir como modelo para los fundamentos de otras ciencias.

== Crisis de los fundamentos ==
{{Referencias|t=20220421192343}}
La crisis fundacional de la matemática (llamada originalmente en [[Idioma alemán|alemán]]: ''Grundlagenkrise der Mathematik'') fue un término acuñado a principios del {{Siglo|XX||s}} para referirse a la situación teórica que llevó a una investigación sistemática y profunda de los fundamentos, que acabó inaugurando una nueva rama de la matemática.

Numerosas [[Filosofía de la matemática|escuelas filosóficas matemáticas]] incurrieron en dificultades una tras otra, a medida que la asunción de que los fundamentos de la matemática podían ser justificados de manera consistentes dentro de la propia matemática fue puesta en duda por el descubrimiento de varias paradojas (entre ellas la célebre [[paradoja de Russell]]).

El término "''[[paradoja]]''" no debe ser confundido con el término ''contradicción''. Una [[contradicción]] dentro de una teoría formal es una demostración formal de la existencia de un absurdo como resultado de un conjunto de asunciones inapropiadas (tales como 2 + 2 = 5), un conjunto de [[axioma]]s o teoría que da lugar a una contradicción se clasifica de [[Consistencia (lógica)|inconsistente]] y debe ser rechazada como teoría útil (ya que en ella cualquier proposición acabaría siendo demostrable). Sin embargo, una paradoja puede referirse o bien a un resultado contraintuitivo pero verdadero, o a un argumento informal que lleva a una contradicción, así que una teoría candidata donde se atente la formalización de un argumento debe inhabilitar al menos uno de sus pasos; en este caso el problema es encontrar una teoría satisfactoria sin contradicciones. Ambos significados pueden aplicar si la versión formalizada del argumento forma la prueba de una verdad sorprendente. Por ejemplo, la paradoja de Russell puede ser expresada como "no hay un conjunto que contenga a todos los conjuntos" (exceptuando algunas teorías axiomáticas marginales).

Algunas escuelas de pensamiento al buscar acercarse al enfoque correcto a los fundamentos de las matemáticas se oponían ferozmente entre sí. La escuela liderante era la escuela de enfoque [[formalista]], de la cual, [[David Hilbert]] era el proponente principal, culminando con lo que se conoce como [[Programa de Hilbert]], quien pensaba en fundamentar la matemática en una pequeña base de un sistema lógico sondeado en términos del [[finitismo]] [[metamatemático]]. El oponente principal era la escuela del [[intuicionismo]], liderada por [[L. E. J. Brouwer]], quien resueltamente descartó el formalismo como un juego fútil con símbolos (van Dalen, 2008). La disputa fue encarnizada. En 1920 Hilbert triunfó en sacar a Brouwer, a quien él consideraba una amenaza a la matemática, eliminándolo del cuadro editorial del ''[[Mathematische Annalen]]'', la revista líder en matemáticas en aquella época.

== Perspectivas filosóficas ==
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{{Ap|Filosofía de la matemática}}

Resumen de las tres filosofías matemáticas principales:

* [[Platonismo matemático|Platonismo]]: platonistas, como Kurt Gödel (1906–1978), sostienen que los números son abstractos, objetos necesariamente existentes, independientes de la mente humana.
* [[Formalismo matemático]]: formalistas, como David Hilbert (1862–1943), sostienen que la matemática no es ni más ni menos que un lenguaje matemático. Son simplemente una serie de juegos.
* [[Intuicionismo]]: intuicionistas, como L. E. J. Brouwer (1882–1966), sostienen que la matemática es una creación de la mente humana. Los números, como personajes de cuentos de hadas, son simplemente entidades mentales, que no existirían sin que nunca hubiera algunas mentes humanas que pensaran en ellos.

A principios del {{siglo|XX||s}}, tres escuelas de filosofía de las matemáticas tenían visiones contrapuestas sobre los fundamentos matemáticos: el [[Formalismo matemático|formalismo]], el [[Intuicionismo]] y el [[logicismo]].

=== Formalismo ===

{{AP|Formalismo (matemática)}}

La postura de los formalistas, tal como fue enunciada por [[David Hilbert]] (1862–1943), es que la matemática es sólo un lenguaje formal y una serie de juegos. De hecho, Hilbert usó el término "juego de fórmulas" en su respuesta de 1927 al criticismo de [[L. E. J. Brouwer]]:

: "And to what has the formula game thus made possible been successful? This formula game enables us to express the entire thought-content of the science of mathematics in a uniform manner and develop it in such a way that, at the same time, the interconnections between the individual propositions and facts become clear... The formula game that Brouwer so deprecates has, besides its mathematical value, an important general philosophical significance. For this formula game is carried out according to certain definite rules, in which the ''technique of our thinking'' is expressed. These rules form a closed system that can be discovered and definitively stated."<ref name="ReferenceA">Hilbert 1927 ''The Foundations of Mathematics'' in van Heijenoort 1967:475</ref>

Por tanto, Hilbert insistió en que la matemática no es un juego "arbitrario" con reglas "arbitrarias", sino más bien un juego que debe coincidir con nuestro pensamiento, que son el punto de partida de nuestra exposición oral y escrita.<ref name="ReferenceA" />

: "We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise".<ref>p. 14 in Hilbert, D. (1919–20), Natur und Mathematisches Erkennen: Vorlesungen, gehalten 1919–1920 in Göttingen. Nach der Ausarbeitung von Paul Bernays (Edited and with an English introduction by David E. Rowe), Basel, Birkhauser (1992).</ref>

La filosofía inicial del formalismo, tal como es ejemplificada por David Hilbert, es una respuesta a las paradojas de la [[teoría axiomática de conjuntos]], que se basa en la [[lógica formal]]. Prácticamente todos los teoremas matemáticos hoy en día se pueden formular como teoremas de la teoría de conjuntos. La verdad de un enunciado matemático, en esta teoría está representada por el hecho de que una declaración se puede derivar de los axiomas de la teoría de conjuntos utilizando las reglas de la lógica formal.<blockquote>El uso del formalismo por sí solo no explica varias cuestiones: ¿Por qué debemos utilizar estos axiomas y no otros, por qué debemos emplear unas reglas lógicas y no otras, por qué proposiciones matemáticas "verdaderas" (p. ej. las leyes de la [[aritmética]]) parecen ser verdad? y así sucesivamente. [[Hermann Weyl]] hará estas mismas preguntas a Hilbert:</blockquote><blockquote>"What "truth" or objectivity can be ascribed to this theoretic construction of the world, which presses far beyond the given, is a profound philosophical problem. It is closely connected with the further question: what impels us to take as a basis precisely the particular axiom system developed by Hilbert? Consistency is indeed a necessary but not a sufficient condition. For the time being we probably cannot answer this question...."<ref>Weyl 1927 ''Comments on Hilbert's second lecture on the foundations of mathematics'' in van Heijenoort 1967:484. Although Weyl the intuitionist believed that "Hilbert's view" would ultimately prevail, this would come with a significant loss to philosophy: "''I see in this a decisive defeat of the philosophical attitude of pure phenomenology'', which thus proves to be insufficient for the understanding of creative science even in the area of cognition that is most primal and most readily open to evidence – mathematics" (ibid).</ref></blockquote>En algunos casos, estas preguntas pueden ser contestadas satisfactoriamente a través del estudio de las teorías formales, en disciplinas como las matemáticas inversas y la teoría de la complejidad computacional. Como ha señalado por Weyl, los sistemas lógicos formales también corren el riesgo de inconsistencia; en la aritmética de [[Peano|G. Peano]], esto sin duda ya se ha salvado con varias pruebas de consistencia, pero hay debate sobre si son o no son suficientemente finitistas para que tengan sentido. El segundo [[teorema de incompletitud de Gödel]] establece que los sistemas lógicos de la aritmética no pueden contener una prueba válida de su propia consistencia. Lo que Hilbert quería hacer era probar que un sistema lógico <math>S</math> fuese consistente, basado en principios <math>P</math> que fueran sólo una pequeña parte de <math>S</math>. Pero Gödel demostró que los principios <math>P</math> ni siquiera podrían demostrar su propia coherencia, ¡por no hablar de la de <math>S</math>!

=== Intuicionismo ===

{{Extracto|Intuicionismo}}

=== Logicismo ===

{{Extracto|Logicismo}}

=== Constructivismo ===

{{Extracto|Constructivismo (matemáticas)}}

=== Platonismo ===

{{Extracto|Platonismo matemático}}

=== Argumento de indispensabilidad ===
{{Referencias|t=20220421192444}}
{{Extracto|Argumento de indispensabilidad}}

=== Realismo ''rudimentario'' ===

Pocos matemáticos suelen estar preocupados en su trabajo diario sobre el [[logicismo]], el [[Formalismo matemático|formalismo]] o cualquier otra posición filosófica. En cambio, su principal preocupación es que la empresa matemática en su conjunto siga siendo siempre productiva. Por lo general, esto se asegura al permanecer con una mente abierta, práctica y ocupada; potencialmente amenazada con volverse excesivamente ideológica, fanáticamente reduccionista o perezosa. Este punto de vista fue expresado por el Premio Nobel de Física [[Richard Feynman]]

: ''People say to me, “Are you looking for the ultimate laws of physics?” No, I’m not… If it turns out there is a simple ultimate law which explains everything, so be it — that would be very nice to discover. If it turns out it’s like an onion with millions of layers… then that’s the way it is. But either way there’s Nature and she’s going to come out the way She is. So therefore when we go to investigate we shouldn’t predecide what it is we’re looking for only to find out more about it. Now you ask: “Why do you try to find out more about it?” If you began your investigation to get an answer to some deep philosophical question, you may be wrong. It may be that you can’t get an answer to that particular question just by finding out more about the character of Nature. But that’s not my interest in science; my interest in science is to simply find out about the world and the more I find out the better it is, I like to find out…''<ref name="fey1">Richard Feynman, ''The Pleasure of Finding Things Out'' p. 23</ref>
: ''Philosophers, incidentally, say a great deal about what is absolutely necessary for science, and it is always, so far as one can see, rather naive, and probably wrong<ref name="fey2">Richard Feynman, [http://evolvingthoughts.net/2011/09/more-feynman-on-philosophers/ ''Lectures on Physics'', volume I, chapter 2.] {{Wayback|url=http://evolvingthoughts.net/2011/09/more-feynman-on-philosophers/ |date=20130705175401 }}</ref>''

y también por [[Steven Weinberg]]<ref name="weinberg">Steven Weinberg, chapter ''[http://libcom.org/library/unexpected-uselessness-philosophy Against Philosophy]'' in ''Dreams of a final theory''</ref>

: ''The insights of philosophers have occasionally benefited physicists, but generally in a negative fashion—by protecting them from the preconceptions of other philosophers.(...) without some guidance from our preconceptions one could do nothing at all. It is just that philosophical principles have not generally provided us with the right preconceptions.''
: ''Physicists do of course carry around with them a working philosophy. For most of us, it is a rough-and-ready realism, a belief in the objective reality of the ingredients of our scientific theories. But this has been learned through the experience of scientific research and rarely from the teachings of philosophers. (...) we should not expect [the philosophy of science] to provide today's scientists with any useful guidance about how to go about their work or about what they are likely to find. (...)''
: ''After a few years' infatuation with philosophy as an undergraduate I became disenchanted. The insights of the philosophers I studied seemed murky and inconsequential compared with the dazzling successes of physics and mathematics. From time to time since then I have tried to read current work on the philosophy of science. Some of it I found to be written in a jargon so impenetrable that I can only think that it aimed at impressing those who confound obscurity with profundity. (...) But only rarely did it seem to me to have anything to do with the work of science as I knew it. (...)''
: ''I am not alone in this; I know of no one who has participated actively in the advance of physics in the postwar period whose research has been significantly helped by the work of philosophers. I raised in the previous chapter the problem of what Wigner calls the "unreasonable effectiveness" of mathematics; here I want to take up another equally puzzling phenomenon, the unreasonable ineffectiveness of philosophy.''
: ''Even where philosophical doctrines have in the past been useful to scientists, they have generally lingered on too long, becoming of more harm than ever they were of use.''

Él creía que cualquier indecidibilidad en matemáticas, como la [[hipótesis del continuo]], podría potencialmente ser resuelta a pesar del [[Teorema de incompletitud de Gödel|teorema de incompletitud]], mediante la búsqueda de nuevos axiomas adecuados para añadir a la [[teoría de conjuntos]].
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==== Consecuencias filosóficas para el teorema de la completitud ====

The [[Completeness theorem]] establishes an equivalence in first-order logic, between the formal provability of a formula, and its truth in all possible models. Precisely, for any consistent first-order theory it gives an "explicit construction" of a model described by the theory; and this model will be countable if the language of the theory is countable. However this "explicit construction" is not algorithmic. It is based on an iterative process of completion of the theory, where each step of the iteration consists in adding a formula to the axioms if it keeps the theory consistent; but this consistency question is only semi-decidable (an algorithm is available to find any contradiction but if there is none this consistency fact can remain unprovable).

This can be seen as a giving a sort of justification to the Platonist view that the objects of our mathematical theories are real. More precisely, it shows that the mere assumption of the existence of the set of natural numbers as a totality (an actual infinity) suffices to imply the existence of a model (a world of objects) of any consistent theory. However several difficulties remain:

* For any consistent theory this usually does not give just one world of objects, but an infinity of possible worlds that the theory might equally describe, with a possible diversity of truths between them.
* In the case of set theory, none of the models obtained by this construction resemble the intended model, as they are countable while set theory intends to describe uncountable infinities. Similar remarks can be made in many other cases. For example, with theories that include arithmetic, such constructions generally give models that include non-standard numbers, unless the construction method was specifically designed to avoid them.
* As it gives models to all consistent theories without distinction, it gives no reason to accept or reject any axiom as long as the theory remains consistent, but regards all consistent axiomatic theories as referring to equally existing worlds. It gives no indication on which axiomatic system should be preferred as a foundation of mathematics.
* As claims of consistency are usually unprovable, they remain a matter of belief or non-rigorous kinds of justifications. Hence the existence of models as given by the completeness theorem needs in fact 2 philosophical assumptions: the actual infinity of natural numbers and the consistency of the theory.

Another consequence of the completeness theorem is that it justifies the conception of infinitesimals as actual infinitely small nonzero quantities, based on the existence of non-standard models as equally legitimate to standard ones. This idea was formalized by [[Abraham Robinson]] into the theory of [[nonstandard analysis]]. However this theory did not look so simple and did not have much success.

=== Más paradojas ===

{{See also|List of undecidable problems}}

1920: [[Thoralf Skolem]] corrected Löwenheim's proof of what is now called the downward [[Löwenheim-Skolem theorem]], leading to [[Skolem's paradox]] discussed in 1922 (the existence of countable models of ZF, making infinite cardinalities a relative property.

1922: Proof by [[Abraham Fraenkel]] that the [[axiom of choice]] cannot be proved from the axioms of Zermelo's set theory with urelements.

1927: [[Werner Heisenberg]] published the [[Uncertainty principle]] of quantum mechanics

1931: Publication of [[Gödel's incompleteness theorems]], showing that essential aspects of Hilbert's program could not be attained. It showed how to construct, for any sufficiently powerful and consistent recursively axiomatizable system – such as necessary to axiomatize the elementary theory of [[arithmetic]] on the (infinite) set of natural numbers – a statement that formally expresses its own unprovability, which he then proved equivalent to the claim of consistency of the theory; so that (assuming the consistency as true), the system is not powerful enough for proving its own consistency, let alone that a simpler system could do the job. It thus became clear that the notion of mathematical truth can not be completely determined and reduced to a purely [[formal system]] as envisaged in Hilbert's program. This dealt a final blow to the heart of Hilbert's program, the hope that consistency could be established by finitistic means (it was never made clear exactly what axioms were the "finitistic" ones, but whatever axiomatic system was being referred to, it was a 'weaker' system than the system whose consistency it was supposed to prove).

1935: Publication of the [[EPR paradox|article by Albert Einstein, Boris Podolsky and Nathan Rosen]] arguing that quantum mechanics was incomplete, as its formalism was non-local, which the authors assumed to not possibly reflect some true underlying mechanism that remained to be discovered.

1936: [[Alfred Tarski]] proved his [[Tarski's undefinability theorem|truth undefinability theorem]].

1936: [[Alan Turing]] proved that a general algorithm to solve the [[halting problem]] for all possible program-input pairs cannot exist.

1938: Gödel proved the [[Constructible universe|consistency of the axiom of choice and of the Generalized Continuum-Hypothesis]].

1936 - 1937: [[Alonzo Church]] and [[Alan Turing]], respectively, published independent papers showing that a general solution to the [[Entscheidungsproblem]] is impossible: the universal validity of statements in first-order logic is not decidable (it is only [[semi-decidable]] as given by the [[completeness theorem]]).

1955: [[Pyotr Novikov]] showed that there exists a finitely presented group G such that the word problem for G is undecidable.

1963: [[Paul Cohen (mathematician)|Paul Cohen]] showed that the Continuum Hypothesis is unprovable from ZFC. Cohen's proof developed the method of forcing, which is now an important tool for establishing independence results in set theory.

1964: [[John Stewart Bell]] published his [[Bell's inequalities|inequalities]] showing that the predictions of quantum mechanics in the EPR thought experiment are significantly different from the predictions of a particular class of hidden variable theories (the local hidden variable theories).
Inspired by the fundamental randomness in physics, [[Gregory Chaitin]] starts publishing results on Algorithmic Information theory (measuring incompleteness and randomness in mathematics)

1966: Paul Cohen showed that the axiom of choice is unprovable in ZF even without urelements.

1970: [[Hilbert's tenth problem]] is proven unsolvable: there is no recursive solution to decide whether a [[Diophantine equation]] (multivariable polynomial equation) has a solution in integers.

1971: [[Suslin's problem]] is proven to be independent from ZFC.
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== Resolución parcial de la crisis ==
{{Referencias}}

A partir de 1935, el grupo Bourbaki de matemáticos franceses empezaron a publicar una serie de libros para formalizar muchas áreas de matemáticas basados en los nuevos fundamentos de la teoría de conjuntos.

== Evolución histórica ==
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=== Matemática en la Antigua Grecia ===

Aunque que el uso práctico de la matemática fue desarrollada ya en civilizaciones de la edad de bronce, el interés específico por sus aspectos fundacionales y teóricos parece remontarse a la [[matemática helénica]]. Los primeros filósofos griegos discutieron ampliamente sobre qué rama de la matemática era más antigua, si la aritmética o la geometría. [[Zenón de Elea]] (490 a. C - ca. 430 a.{{esd}}C.) formuló cuatro aporías que aparentan mostrar que el cambio es imposible, que en esencia no fueron convenientemente aclaradas hasta el desarrollo de matemática moderna.

La escuela pitagórica de matemática insistía originalmente en que solo existían los números naturales y racionales. El descubrimiento de la [[Número irracional|irracionalidad]] de √2, la proporción de la diagonal de un cuadrado con su lado (data del {{siglo|V||s}} a.{{esd}}C.), fue un golpe filosófico a dicha escuela que solo aceptaron de mala gana. La discrepancia entre racionales y reales fue finalmente resuelta por [[Eudoxo de Cnido]], un estudiante de Platón, quien redujo la comparación de las proporciones de los irracionales a comparaciones de múltiples proporciones racionales, además de anticipar la definición de número real de [[Richard Dedekind]].

En su obra ''[[Segundos analíticos]]'', [[Aristóteles]] (384 a.{{esd}}C. - 322 a.{{esd}}C.) asentó el [[método axiomático]], para organizar lógicamente un campo del conocimiento en términos de conceptos primitivos, axiomas, postulados, definiciones, y teoremas, tomando una mayoría de sus ejemplos de la aritmética y la geometría.
Este  método llegó a la cumbre  con  Elementos de Euclides (300 a.{{esd}}C.), un proyecto monumental de la geometría estructural  con rigurosidad alta :cada proposición es justificable por una demostración mediante chains of syllogisms (though they do not always conform strictly to Aristotelian templates).
Aristotle's syllogistic logic, together with the Axiomatic Method exemplified by Euclid's Elements, are universally recognized as towering scientific achievements of ancient Greece.
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=== Platonismo como una filosofía tradicional de la matemática ===

Starting from the end of the 19th century, a Platonist view of mathematics became common among practicing mathematicians.

The objects of mathematics are abstract and remote from everyday perceptual experience: geometrical figures are conceived as idealities to be distinguished from effective drawings and shapes of objects, and numbers are not confused with the counting of concrete objects. Their existence and nature present special philosophical challenges: How do mathematical objects differ from their concrete representation? Are they located in their representation, or in our minds, or somewhere else? How can we know them?

The ancient Greek philosophers took such questions very seriously. Indeed, many of their general philosophical discussions were carried on with extensive reference to geometry and arithmetic. [[Plato]] (424/423 BC – 348/347 BC) insisted that mathematical objects, like other platonic ''Ideas'' (forms or essences), must be perfectly abstract and have a separate, non-material kind of existence, in a world of mathematical objects independent of humans. He believed that the truths about these objects also exists independently of the human mind, but is ''discovered'' by humans. In the ''[[Meno]]'' Plato’s teacher Socrates asserts that it is possible to come to know this truth by a process akin to memory retrieval.

Above the gateway to Plato's academy appeared a famous inscription: "Let no one who is ignorant of geometry enter here".

In this way Plato indicated his high opinion of geometry. He regarded geometry as ``the first essential in the training of philosophers", because of its abstract character.

This philosophy of ''[[Philosophy of mathematics#Platonism|Platonist mathematical realism]]'', is shared by many mathematicians. It can be argued that Platonism somehow comes as a necessary assumption underlying any mathematical work.<ref name="Podnieks">Karlis Podnieks, [http://www.ltn.lv/~podnieks/gt1.html#BM1_1 Platonism, intuition and the nature of mathematics: 1. Platonism - the Philosophy of Working Mathematicians]</ref>

In this view, the laws of nature and the laws of mathematics have a similar status, and the [[The Unreasonable Effectiveness of Mathematics in the Natural Sciences|effectiveness]] ceases to be unreasonable. Not our axioms, but the very real world of mathematical objects forms the foundation.

Aristotle dissected and rejected this view in his Metaphysics. These questions provide much fuel for philosophical analysis and debate.

=== Edad media y Renacimiento ===

For over 2,000 years, Euclid’s Elements stood as a perfectly solid foundation for mathematics, as its methodology of rational exploration guided mathematicians, philosophers, and scientists well into the 19th century.

The Middle Ages saw a dispute over the ontological status of the universals (platonic Ideas): [[Philosophical realism|Realism]] asserted their existence independently of perception; [[conceptualism]] asserted their existence within the mind only; [[nominalism]], denied either, only seeing universals as names of collections of individual objects (following older speculations that they are words, "''logos''").

[[René Descartes]] published [[La Géométrie]] (1637) aimed to reduce geometry to algebra by means of coordinate systems, giving algebra a more foundational role (while the Greeks embedded arithmetic into geometry by identifying whole numbers with evenly spaced points on a line). It became famous after 1649 and paved the way to infinitesimal calculus.

[[Isaac Newton]] (1642 – 1727) in England and [[Gottfried Wilhelm Leibniz|Leibniz]] (1646 – 1716) in Germany independently developed the [[infinitesimal calculus]] based on heuristic methods greatly efficient, but direly lacking rigorous justifications. Leibniz even went on to explicitly describe infinitesimals as actual infinitely small numbers (close to zero). Leibniz also worked on formal logic but most of his writings on it remained unpublished until 1903.

The Christian philosopher [[George Berkeley]] (1685–1753), in his campaign against the religious implications of Newtonian mechanics, wrote a pamphlet on the lack of rational justifications of infinitesimal calculus:<ref name="berkeley">''[[The Analyst]], A Discourse Addressed to an Infidel Mathematician''</ref> “They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?”

Then mathematics developed very rapidly and successfully in physical applications, but with little attention to logical foundations.

=== Siglo XIX ===

In the [[History of mathematics#19th century|19th century]], mathematics became increasingly abstract. Concerns about logical gaps and inconsistencies in different fields led to the development of axiomatic systems.
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==== Análisis sobre los reales ====

[[Cauchy]] (1789-1857) inició el proyecto de demostrar los teoremas de [[cálculo infinitesimal]] sobre una base rigurosa, rechazando el principio de [[generalidad del álgebra]] usado por diversos matemáticos durante el {{siglo|XVIII||s}}. En su ''Cours d'Analyse'' ('Curso de análisis) de 1821, Cauchy definió las cantidades infinitesimales como sucesiones decrecientes que convergen a 0, que pueden ser usadas para definir la continuidad. Aunque no formalizó ninguna noción de convergencia.

La definición moderna del [[criterio (ε, δ)]] y la noción de [[función continua]] fueron desarrollada por primera vez por [[Bolzano]] en 1817, pero durante un tiempo fue relativametne poco conocida. Estas nociones dan un fundamente riguroso al cálculo infinitesimal basado en el conjunto de los números reales, y resuelven claramente tanto las [[paradojas de Zenón]] como los argumentos de Berkeley.
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Mathematicians such as [[Karl Weierstrass]] (1815 – 1897) discovered pathological functions such as [[Weierstrass function|continuous, nowhere-differentiable functions]]. Previous conceptions of a function as a rule for computation, or a smooth graph, were no longer adequate. Weierstrass began to advocate the [[arithmetization of analysis]], to axiomatize analysis using properties of the natural numbers.
In 1858, [[Dedekind]] proposed a definition of the real numbers as [[Dedekind cuts|cuts]] of rational numbers. This reduction of real numbers and continuous functions in terms of rational numbers and thus of natural numbers, was later integrated by [[Georg Cantor|Cantor]] in his set theory, and axiomatized in terms of second order arithmetic by Hilbert and Bernays.

==== Teoría de grupos ====

For the first time, the limits of mathematics were explored. [[Niels Henrik Abel]] (1802 – 1829), a Norwegian, and [[Évariste Galois]], (1811 – 1832) a Frenchman, investigated into the solutions of various polynomial equations, and proved that there is no general algebraic solution to equations of degree greater than four ([[Abel–Ruffini theorem]]). With these concepts, [[Pierre Wantzel]] (1837) proved that straightedge and compass alone cannot [[trisect an arbitrary angle]] nor [[Doubling the cube|double a cube]], nor to construct a square equal in area to a given circle. Mathematicians had vainly attempted to solve all of these problems since the time of the ancient Greeks.

Abel and Galois's works opened the way for the developments of [[group theory]] (which will be used to study [[symmetry]] in physics and other fields), and [[abstract algebra]]. Concepts of [[vector spaces]] emerged from the conception of [[Barycentric coordinates (mathematics)|barycentric coordinates]] by [[August Ferdinand Möbius|Möbius]] in 1827, to the modern definition of vector spaces and linear maps by Peano in 1888. Geometry was no more limited to 3 dimensions.
These concepts do not generalize numbers but combine notions of functions and sets which are not yet formalized, breaking away from familiar mathematical objects.

==== Geometrías no-euclidianas ====

After many failed attempts to derive the [[parallel postulate]] from other axioms, the study of the still hypothetical [[hyperbolic geometry]] by [[Johann Heinrich Lambert]] (1728 – 1777) led him to introduce the [[hyperbolic functions]] and compute the area of a [[hyperbolic triangle]] (where the sum of angles is less than 180°). Then the Russian mathematician [[Nikolai Lobachevsky]] (1792–1856) established in 1826 (and published in 1829) the coherence of this geometry (thus the independence of the [[parallel postulate]]), in parallel with the Hungarian mathematician [[János Bolyai]] (1802–60) in 1832, and with [[Gauss]].
Later in the 19th century, the German mathematician [[Bernhard Riemann]] developed [[Elliptic geometry]], another [[non-Euclidean geometry]] where no parallel can be found and the sum of angles in a triangle is more than 180°. It was proved consistent by defining point to mean a pair of antipodal points on a fixed sphere and line to mean a [[great circle]] on the sphere. At that time, the main method for proving the consistency of a set of axioms was to provide a [[Model (mathematical logic)|model]] for it.

==== Geometría proyectiva ====

One of the traps in a deductive system is [[circular reasoning]], a problem that seemed to befall [[projective geometry]] until it was resolved by [[Karl von Staudt]]. As explained by Laptev & Rosenfeld (1996):

: In the mid-nineteenth century there was an acrimonious controversy between the proponents of synthetic and analytic methods in projective geometry, the two sides accusing each other of mixing projective and metric concepts. Indeed the basic concept that is applied in the synthetic presentation of projective geometry, the [[cross-ratio]] of four points of a line, was introduced through consideration of the lengths of intervals.

The purely geometric approach of von Staudt was based on the [[complete quadrilateral]] to express the relation of [[projective harmonic conjugate]]s. Then he created a means of expressing the familiar numeric properties with his [[Karl von Staudt#Algebra of throws|Algebra of Throws]]. English language versions of this process of deducing the properties of a [[Field (mathematics)|field]] can be found in either the book by [[Oswald Veblen]] and John Young, ''Projective Geometry'' (1938), or more recently in [[John Stillwell]]'s ''Four Pillars of Geometry'' (2005). Stillwell writes on page 120

: ...projective geometry is ''simpler'' than algebra in a certain sense, because we use only five geometric axioms to derive the nine field axioms.

The algebra of throws is commonly seen as a feature of cross-ratios since students ordinarily rely upon [[number]]s without worry about their basis. However, cross-ratio calculations use [[Metric (mathematics)|metric]] features of geometry, features not admitted by purists. For instance, in 1961 [[Coxeter]] wrote ''Introduction to Geometry'' without mention of cross-ratio.

==== Algebra Booleana y lógica ====

Attempts of formal treatment of mathematics had started with Leibniz and [[Johann Heinrich Lambert|Lambert]] (1728 – 1777), and continued with works by algebraists such as [[George Peacock]] (1791 – 1858).
Systematic mathematical treatments of logic came with the British mathematician [[George Boole]] (1847) who devised an algebra that soon evolved into what is now called [[Boolean algebra]], in which the only numbers were 0 and 1 and logical combinations (conjunction, disjunction, implication and negation) are operations similar to the addition and multiplication of integers.
Also [[Augustus De Morgan|De Morgan]] publishes his [[De Morgan's laws|laws]] (1847). Logic becomes a branch of mathematics. Boolean algebra is the starting point of mathematical logic and has important applications in [[computer science]].

[[Charles Sanders Peirce]] built upon the work of Boole to develop a logical system for relations and quantifiers, which he published in several papers from 1870 to 1885.

The German mathematician [[Gottlob Frege]] (1848–1925) presented an independent development of logic with quantifiers in his [[Begriffsschrift]] (formula language) published in 1879, a work generally considered as marking a turning point in the history of logic. He exposed deficiencies in Aristotle's Logic, and pointed out the 3 expected properties of a mathematical theory

# [[Consistency]]: impossibility to prove contradictory statements
# [[Completeness]]: any statement is either provable or refutable (i.e. its negation is provable).
# [[Decidability]]: there is a decision procedure to test any statement in the theory.

He then showed in ''Grundgesetze der Arithmetik (Basic Laws of Arithmetic)'' how arithmetic could be formalised in his new logic.

Frege's work was popularized by [[Bertrand Russell]] near the turn of the century. But Frege's two-dimensional notation had no success. Popular notations were (x) for universal and (∃x) for existential quantifiers, coming from [[Giuseppe Peano]] and [[William Ernest Johnson]] until the ∀ symbol was introduced by [[Gentzen]] in 1935 and became canonical in the 1960s.

From 1890 to 1905, [[Ernst Schröder]] published ''Vorlesungen über die Algebra der Logik'' in three volumes. This work summarized and extended the work of Boole, De Morgan, and Peirce, and was a comprehensive reference to symbolic logic as it was understood at the end of the 19th century.

==== Aritmetica de Peano ====

The formalization of [[Peano axioms|arithmetic]] (the theory of natural numbers) as an axiomatic theory, started with Peirce in 1881, and continued with Richard Dedekind and [[Giuseppe Peano]] in 1888. This was still a [[Second-order logic|second-order]] axiomatization (expressing induction in terms of arbitrary subsets, thus with an implicit use of [[set theory]]) as concerns for expressing theories in [[first-order logic]] were not yet understood. In Dedekind's work, this approach appears as completely characterizing natural numbers and providing recursive definitions of addition and multiplication from the successor function and mathematical induction.
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== Véase también ==

* [[Filosofía de las matemáticas]]
* [[Teoría de categorías]]
* [[Teoría de tipos]]
* [[Teoría de conjuntos]]
* [[Lógica matemática]]
* [[Teoría de la demostración]]
* [[Teoría de modelos]]
* [[Epistemología]]
* [[Elementos de Euclides]]
* [[Principia Mathematica]]
* [[Teoremas de incompletitud de Gödel]]

== Referencias ==

{{Listaref}}

== Bibliografía ==

* [[Avigad, Jeremy]] (2003) ''Number theory and elementary arithmetic'', Philosophia Mathematica Vol. 11, pp.&nbsp;257–284
* [[Howard Eves|Eves, Howard]] (1990), ''Foundations and Fundamental Concepts of Mathematics Third Edition'', Dover Publications, INC, Mineola NY, ISBN 0-486-69609-X (pbk.) cf §9.5 Philosophies of Mathematics pp.&nbsp;266–271. Eves lists the three with short descriptions prefaced by a brief introduction.
* [[Nicholas D. Goodman|Goodman, N.D.]] (1979), "Mathematics as an Objective Science", in Tymoczko (ed., 1986).
* [[Wilbur Dyre Hart|Hart, W.D.]] (ed., 1996), ''The Philosophy of Mathematics'', [[Oxford University Press]], Oxford, UK.
* [[Reuben Hersh|Hersh, R.]] (1979), "Some Proposals for Reviving the Philosophy of Mathematics", in (Tymoczko 1986).
* [[David Hilbert|Hilbert, D.]] (1922), "Neubegründung der Mathematik. Erste Mitteilung", ''Hamburger Mathematische Seminarabhandlungen'' 1, 157–177. Translated, "The New Grounding of Mathematics. First Report", in (Mancosu 1998).
* Katz, Robert (1964), ''Axiomatic Analysis'', D. C. Heath and Company.
* {{Cita libro |apellidos=[[Kleene]] |nombre=Stephen C. |título=Introduction to Meta-Mathematics |año=1991 |editorial=North-Holland Pub. Co |isbn=0-7204-2103-9 |ubicación=Amsterdam NY |edición=Tenth impression 1991 |año-original=1952}} In Chapter III ''A Critique of Mathematic Reasoning, §11. The paradoxes'', Kleene discusses [[Intuitionism]] and [[Formalism (mathematics)|Formalism]] in depth. Throughout the rest of the book he treats, and compares, both Formalist (classical) and Intuitionist logics with an emphasis on the former. Extraordinary writing by an extraordinary mathematician.
* Laptev, B.L. & B.A. Rozenfel'd (1996) ''Mathematics of the 19th Century: Geometry'', page 40, [[Springer Science+Business Media|Birkhäuser]] ISBN 3-7643-5048-2.
* Mancosu, P. (ed., 1998), ''From Hilbert to Brouwer. The Debate on the Foundations of Mathematics in the 1920s'', Oxford University Press, Oxford, UK.
* [[Hilary Putnam|Putnam, Hilary]] (1967), "Mathematics Without Foundations", ''Journal of Philosophy'' 64/1, 5–22. Reprinted, pp.&nbsp;168–184 in W.D. Hart (ed., 1996).
* Putnam, Hilary (1975), "What is Mathematical Truth?", in Tymoczko (ed., 1986).
* {{Cita publicación |título=The prime number theorem is PRA-provable |apellidos=Sudac, Olivier |publicación=Theoretical Computer Science |volumen=257 |número=1–2 |páginas=185–239 |doi=10.1016/S0304-3975(00)00116-X |año=Apr 2001}}
* [[Troelstra, A. S.]] (no date but later than 1990), "A History of Constructivism in the 20th Century", https://web.archive.org/web/20060209210015/http://staff.science.uva.nl/~anne/hhhist.pdf, A detailed survey for specialists: §1 Introduction, §2 Finitism & §2.2 Actualism, §3 Predicativism and Semi-Intuitionism, §4 Brouwerian Intuitionism, §5 Intuitionistic Logic and Arithmetic, §6 Intuitionistic Analysis and Stronger Theories, §7 Constructive Recursive Mathematics, §8 Bishop's Constructivism, §9 Concluding Remarks. Approximately 80 references.
* [[Thomas Tymoczko|Tymoczko, T.]] (1986), "Challenging Foundations", in Tymoczko (ed., 1986).
* [[Thomas Tymoczko|Tymoczko, T.]] (ed., 1986), ''New Directions in the Philosophy of Mathematics'', 1986. Revised edition, 1998.
* van Dalen D. (2008), "Brouwer, Luitzen Egbertus Jan (1881–1966)", in Biografisch Woordenboek van Nederland. URL:http://www.inghist.nl/Onderzoek/Projecten/BWN/lemmata/bwn2/brouwerle [13-03-2008]
* [[Hermann Weyl|Weyl, H.]] (1921), "Über die neue Grundlagenkrise der Mathematik", ''Mathematische Zeitschrift'' 10, 39–79. Translated, "On the New Foundational Crisis of Mathematics", in (Mancosu 1998).
* [[Raymond Louis Wilder|Wilder, Raymond L.]] (1952), ''Introduction to the Foundations of Mathematics'', John Wiley and Sons, New York, NY.

== Enlaces externos ==

* [http://www.math.psu.edu/simpson/papers/philmath/ Logic and Mathematics] {{Wayback|url=http://www.math.psu.edu/simpson/papers/philmath/ |date=20051025125944 }}
* {{Enlace roto|1=[http://www.math.osu.edu/~friedman.8/pdf/FOM12pt%5B1%5D.5.31.00.pdf Foundations of Mathematics: past, present, and future] |2=http://www.math.osu.edu/~friedman.8/pdf/FOM12pt%5B1%5D.5.31.00.pdf |bot=InternetArchiveBot }}, May 31, 2000, 8 pages.
* [https://web.archive.org/web/20130727184333/http://www.cs.auckland.ac.nz/~chaitin/lowell.html A Century of Controversy over the Foundations of Mathematics] by Gregory Chaitin.
* [http://settheory.net Set theory and foundations of mathematics]
* [http://www.cs.nyu.edu/mailman/listinfo/fom/ Foundations of Mathematics mailing list]

* Sir Roger Penrose & William Lane Craig • [https://divulciencia.blogspot.com/2023/05/roger-penrose-frente-william-craig.html?m=1| "The Universe: How did it get here and why are we part of It?"]. The Big Conversation. 2016.
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