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Isto é Matemática T02E01 A Parábola da Parábola - Parte 1

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Nicole d'Oresme

Nascimento

1 de janeiro de 1323
Fleury-sur-Orne ( Reino da França)

Morte

11 de julho de 1382
Lisieux ( Reino da França)

Cidadania

França

Alma mater

Ocupação

filósofo, matemático, economista, padre, astrônomo, musicólogo, professor, psicólogo, tradutor, físico, teólogo

Empregador

Universidade de Paris

Religião

Igreja Católica

Referências

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Nicole d'Oresme - Wikipédia, a enciclopédia livre

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Nicole d'Oresme Nascimento 1 de janeiro de 1323 Fleury-sur-Orne ( Reino da França) Morte 11 de julho de 1382 Lisieux ( Reino da França) Cidadania França Alma mater Ocupação filósofo, matemático, economista, padre, astrônomo, musicólogo, professor, psicólogo, tradutor, físico, teólogo Empregador Universidade de Paris Religião Igreja Católica

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Galileu Galilei foi um importante cientista, físico, matemático, astrônomo e filósofo italiano. Sua contribuição científica iniciou uma nova era na história da astronomia, foi o primeiro astrônomo a acessar novos conhecimentos com o uso do telescópio. Defendeu o conceito de que a Terra não era o centro do universo.

Galileu Galilei nasceu em Pisa, na Itália, no dia 15 de fevereiro de 1564, filho do casal Vincenzo Galilei e de Julia Ammannati. Seus pais notaram a grande inteligência e aptidões especiais de Galilei desde cedo. O garoto apresentava interesse em artes e realizava excelentes pinturas, demonstrando habilidade manual e criatividade para fabricar brinquedos e engenhocas. Tocava órgão e cítara com desenvoltura. Assim, Galilei destacou-se nos estudos na escola dominical de Vallombrosa e planejava ingressar no monastério, mas seu pai não concordou com a ideia e o inscreveu para estudar medicina na Universidade de Pisa. Dois anos após o ingresso, desistiu do curso e foi se dedicar ao estudo de matemática. A mudança não agradou o pai, e Galilei acabou abandonando a Universidade em 1585. Não concluiu nenhuma graduação, mas no mesmo ano foi para Florença e começou a dar aulas particulares para se sustentar. Destacou por suas pesquisas em geometria e continuou com seus estudos matemáticos.

Foi nessa época que inventou a balança hidrostática, mecanismo que seria publicado em tratado detalhado no ano de 1644. Em 1589, pelo reconhecimento de suas contribuições científicas e raciocínio brilhante, foi nomeado para integrar a cátedra de matemática na Universidade de Pisa. Não era bem acolhido pelos professores, pois tinha apenas 25 anos, formação acadêmica incompleta e desacreditava publicamente das teorias consagradas de Aristóteles. Em 1590 Galilei publicou um tratado sobre o movimento dos corpos. Em 1591 foi tirado do cargo de professor, após sucumbir às intrigas e disputas com os defensores de Aristóteles. Em 1592 foi indicado pelo senado de Veneza para lecionar matemática na Universidade de Pádua, cargo que manteria por 18 anos. Em 1609 construiu um telescópio baseado no inventado anteriormente por Hans Lippershey na Holanda. Galilei fez observações minuciosas do céu e descobertas incríveis: localizou as quatro maiores luas de Júpiter e as montanhas e crateras existentes na superfície da Lua. E quando detectou manchas presentes na superfície do Sol, a descoberta contribuiu para provar sua teoria de que a estrela girava sobre um eixo. Investigou Saturno e observou o que pareciam ser duas luas fixas, que eram as bordas do sistema de anéis de Saturno, mas o telescópio de Galilei não tinha precisão suficiente para determinar exatamente o que eram aqueles pontos.

Suas descobertas foram reunidas e publicadas em março de 1610 no livro "O Mensageiro das Estrelas". A obra foi aclamada e também gerou muita polêmica, pois Galilei defendia publicamente a teoria de Nicolau Copérnico de que o Sol era o centro de nosso Sistema Solar, e não a Terra. Naquela época, a Igreja Católica controlava totalmente a ciência e sustentava a visão oposta, de que o centro era a Terra.

Em 1616 Galilei foi acuado pelas autoridades da Inquisição e ameaçado de pena de morte, caso não renegasse publicamente as verdades científicas que tinha comprovado. Ficou expressamente proibido de ensinar e propagar ideias que fossem contrárias ao posicionamento da Igreja. Mesmo assim, em 1632 publicou o "Diálogo sobre os Dois Máximos Sistemas do Universo", ocasionando repúdio e intolerância total da Igreja. Impedido de seguir com suas pesquisas e teorias, o cientista recolheu-se em seu castelo localizado em Arcetri, vila próxima de Florença, onde se dedicou a prosseguir seus experimentos solitariamente.

Galileu Galilei faleceu no dia 8 de janeiro de 1642 em Arcetri, na Itália. Estava quase cego pela observação das manchas solares feita sem a proteção adequada durante décadas. Trezentos e cinquenta anos depois, através do Papa João Paulo II, em 31 de outubro de 1992 a Igreja Católica reconheceu formalmente a legitimidade das teorias de Galilei.

Texto originalmente publicado em https://www.infoescola.com/biografias/galileu-galilei/

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Galileu Galilei foi um importante cientista, físico, matemático, astrônomo e filósofo italiano. Sua contribuição científica iniciou uma nova era na história da astronomia, foi o primeiro astrônomo a acessar novos conhecimentos com o uso do telescópio. Defendeu o conceito de que a Terra não era o centro do universo.

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Em 1727, poucas semanas antes de sua morte, um dos homens mais famosos da época está ocupado queimando caixas cheias de seus manuscritos. O que poderia haver escrito para que fosse necessário tal atitude? Isaac Newton foi o pai da ciência moderna. Ele descobriu o segredo da gravidade. Mas parece que maçãs caindo não estava no topo de suas prioridades.

Newton passou a vida estudando secretamente a arte milenar da alquimia e códigos da Bíblia.

Newton era capaz de ser absolutamente centrado em alguma coisa e obcecado. Ao mesmo tempo, era uma figura grandemente contraditória.

No ano de 1661 o jovem Isaac Newton, com 18 anos, deixa sua casa para cursar a Universidade de Cambridge, na Inglaterra. Em 1664, assisti a palestras do matemático Isaac Barrow e fica bem claro, que mesmo tão moço, é capaz de fazer pesquisas totalmente independentes e originais.

Em 1665 a epidemia de peste negra o força a mudar-se para norte, buscando proteção na casa de sua família. Nos dois anos que passa ali, ele inventa o cálculo, faz importantes descobertas em relação à luz e a fatídica maçã supostamente cai, fazendo-o desenvolver a teoria da gravidade.

De volta a Cambridge Newton descreve leis fundamentais que governam o universo. Porém manteve tudo em sigilo.

Era um ser muito reservado, com poucas relações sociais, convivia com baixa ou nenhuma empatia.

Isaac Barrow, seu professor, começa a divulgar alguns trabalhos realizados por Newton e aos poucos a comunidade científica começa a reconhecer o seu talento.

Dessa forma Newton ganha atenção da Royal Society, o mais importante grupo de desenvolvimento científico da época

Isaac Newton passa a maior parte do tempo trancado em seu quarto. Fala com poucas pessoas e apenas sai desse isolamento quando seu professor resolve divulgar sua nova criação:

Newton havia criado um novo telescópio. Muito menor do que os que haviam na época. Além disso, sua criação resolveu um problema que era conhecido como aberração cromática (cores que se formavam em volta do objeto observado). Essa evolução foi possível graças a utilização de espelhos, no lugar de lentes.

Logo Newton divulgou seu estudo revolucionário sobre a luz e a natureza das cores. Por essas obras ele tornou-se reconhecido.

Uma outra figura eminente, o mais respeitado cientista, curador na Royal Socialty, Robert Hooke, escreve uma crítica contundente aos estudos de Newton. Essa crítica abala Newton profundamente que resolve se isolar novamente.

Nesse momento, Newton se concentra nessa prática ilegal na época, a alquimia. Isaac Newton se torna um alquimista. Lia obras e fazia experimentos ocultos.

Dez anos depois Edmund Halley visita Newton, em 1684, afim de debater as teorias de Kepler em relação ao movimento dos planetas. Newton, realiza um texto de onze páginas apenas, considerado a base da física moderna.

A partir disso, Isaac Newton escreve o livro Principia Mathematica, um livro surpreendente, escrito em latim. Considerado uma das maiores obras científicas já publicadas.

Estudos indicam que muitas descobertas científicas de Newton, são frutos de seus trabalhos com alquimia. Por essas questões Newton tomava muito cuidado com essa parte sigilosa de sua vida, as pessoas ainda consideravam alquimia um conhecimento fraudulento.

Nos últimos 20 anos de sua vida, Newton passa a estudar profundamente os escritos bíblicos, obcecado por códigos secretos e professais da Bíblia. Passa diariamente horas dissecando textos religiosos. Sobre as escrituras, ele escreve mais do que ciência e alquimia combinadas.

A verdade é que se suspeita que o material queimado antes de sua morte é referente a esses estudos ocultos de alquimia e teologia.

Newton serviu no parlamento, se tornou presidente da Royal Society e foi condecorado Sir.

Hoje se acredita que Isaac Newton sofria de síndrome de Asperger, que é uma condição apresentada em diversos gênios, como Einstein e Darwin. Essa síndrome determina uma mutação no funcionamento do cérebro, criando comportamentos diferentes das pessoas normais.

Quando Newton morreu, em 1727, era um dos homens mais famosos do mundo. Hoje é festejado com um dos homens mais influentes da história.

Bibliografia:
NetGeo, Isaac Newton Pai da Ciência era Ocultista e Alquimista: https://www.youtube.com/watch?v=0r2_hOigXTQ
http://www.e-biografias.net/isaac_newton
Texto originalmente publicado em https://www.infoescola.com/biografias/isaac-newton/

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Texto sobre Isaac Newton, quem foi, principais contribuições para a matemática, física, astronomia, etc, entre outras informações.

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Johann Bernoulli

(1667-1748)

Johann, também conhecido como Jean I ou John I, um dos muitos matemáticos proeminentes na família Bernoulli, é considerado um dos importantes fundadores do cálculo (junto com Newton e Leibniz).

Décimo filho de Nicolaus Bernoulli e de Margaretha Bernoulli, apesar das objecções dos pais, junto com o irmão Jacob Bernoulli (1654-1705), doze anos mais velho, os dois teriam forte influência sobre os descendentes da família no gosto pelos estudos matemáticos. Tentou afirmar-se noutros campos científicos tais como, a química, a medicina e o estudo do movimento dos animais de acordo com a escola de Borelli, mas foi na matemática aplicada que ele desenvolveu diversos estudos, nomeadamente no cálculo das variações, também desenvolveu estudos sobre a refracção da luz, trajectórias ortogonais de famílias de curvas, quadraturas de áreas em séries e sobre a braquistócrona.

Em Junho de 1696, Johann Bernoulli apresentou o problema da braquistócrona, publicado na Acta Eruditorum.

Denomina-se braquistócrona a trajectória de uma partícula que, sujeita a um campo gravitacional constante, sem atrito e com velocidade inicial nula, se desloca entre dois pontos no menor intervalo de tempo. Note-se que a questão não é qual o percurso mais curto entre os dois pontos, cuja resposta nas condições dadas é, obviamente, a recta que os une, mas sim, qual trajectória é percorrida no menor tempo!

Propôs a resolução deste problema a outros matemáticos famosos. Johann Bernoulli foi o primeiro a resolver o problema, seguido do seu irmão Jacob Bernoulli, depois Leibniz,, l'Hôppital e Newton A solução é um arco de uma ciclóide invertida.

Em 1695 foi nomeado professor de Matemática na Universidade de Grõningen, onde permaneceu até 1705 e teve como discípulo o grande Euler.

Johann Bernoulli tornou-se conhecido em todo o mundo pelos seus importantes trabalhos nas áreas da Matemática, Física e Engenharia.

Em 1747 foi abandonado pela família e morreu completamente louco a 1 de Janeiro de 1748, com 81 anos de idade.

Fontes:

MacTutor History of Mathematics archive [On line]. St Andrews : University of St Andrews, 2006. [Consult. 31 Ag. 2009]. Available in WWW: <URL:http://www-history.mcs.st-andrews.ac.uk/index.html>.

Wikipedia [On line]. [s.l.] : Wikipedia Foundation, [s.d.], mod. 31 Ag. 2009. [Consult. 31 Ag. 2009]. Available in WWW: <URL: http://pt.wikipedia.org >.

Weisstein, Eric W. - Scienceworld. [On line]. Ithaca, NY: Eric Weisstein's World of Science, [s.d.]. [Consult. 31 Ag. 2009]. Available in WWW: <URL:http://scienceworld.wolfram.com/>.

Lisboa. Faculdade de Ciências da Universidade de Lisboa. Departamento de educação [On line]. Lisboa : FCUL, [s.d.]. [Consult. 31 Ag. 2009]. Available in : <URL:http://www.educ.fc.ul.pt/icm/icm2000/icm27/matematicos2.htm>.

Wachsmuth, Berg G. - MATHS - Mathematics and Computer Science. [On line]. [South Orange, NJ] : [B.G.W.], 1998. [Consult. 31 Ag. 2009]. Available in: <URL: http://www.mathcs.org>

Livros disponíveis na Biblioteca Matemática:

BERNOULLI, Jean - Johannis Bernoulli opera omnia. Lausannae & Genevae : sumptibus Marci-Michaelis Bousquet & sociorum, 1742. 4 vols.
Cota: RE. 05.01-04
BERNOULLI, Jean - Der Briefwechsel von Johann Bernoulli. Basel : Birkhauser, 1955. 531 p.
Cota: RH.09.13
SÁNCHEZ FERNÁNDEZ, Carlos ; VALDÉS CASTRO, Concepción - Los Bernoulli : geómetras y viajeros. 1ª ed.Tres Cantos : Madrid, 2001. 276 p. ISBN 84-95599-21-X

Cota: 01A05/SAN

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(1667-1748) Johann, também conhecido como Jean I ou John I, um dos muitos matemáticos proeminentes na família Bernoulli, é considerado um dos importantes fundadores do cálculo (junto com Newton e Leibniz).

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Leonhard Paul Euler foi sem dúvida um dos matemáticos mais notáveis de todos os tempos e suas contribuições para a física e a matemática ecoam até hoje. Nascido na Suíça em 15 de abril de 1707, Euler teve uma bem sucedida carreira acadêmica com contribuições e descobertas em geometria, análise matemática, teoria dos números, cálculo, lógica e foi o matemático pioneiro da Teoria dos Grafos.

Euler publicou mais do que qualquer matemático havia feito até então, seu trabalho compreende cerca de 80 volumes. Fascinado por notações matemáticas e fórmulas, Euler define a maioria das notações usadas hoje como f(x) que indica uma função f que depende de uma variável x, a constante e que também é chamada de número de Euler, a parte imaginária i de um número complexo, o símbolo de somatório ∑ e popularizou o uso da letra grega 𝜋. Uma de suas fórmulas mais famosas, a identidade de Euler, que compreende o uso de constantes muito comuns na matemática, ficou muito conhecida, sendo nomeada por muitos a identidade mais bela da matemática:

Dentre as inúmeras contribuições de Euler, destaca-se a solução do problema das pontes de Königsberg, que abriu caminho para a Teoria dos Grafos.

O problema enunciava: É possível cruzar cada uma das sete pontes de Königsberg uma única vez, retornando ao ponto de partida? Euler publicou a solução do problema em 1736 onde, contando as pontes ao redor de cada área provou que isto não era possível, percebendo que só seria possível atravessar o caminho inteiro passando uma única vez em cada ponte se houvesse exatamente zero ou dois pontos de onde saísse um número ímpar de caminhos. A resolução deste problema é considerado como o primeiro teorema da Teoria dos Grafos e um dos primeiros resultados topológicos da geometria.

Na física, Euler também contribuiu muito. Publicou trabalhos acerca da cinemática dos corpos, introduzindo também o teorema de Euler para os ângulos de rotação. Seu nome é citado frequentemente na mecânica, onde obteve avanços significativos no Cálculo Variacional, onde ele generaliza o princípio da mínima ação, que posteriormente tornaram-se as bases para a mecânica Lagrangiana (que é uma formulação da mecânica clássica fundamentada por Joseph-Louis Lagrange, que combina a conservação do momento linear com a conservação da energia). A Equação de Euler-Lagrange foi descoberta em meados de 1750 por Euler e Lagrange na tentativa de solucionar um problema da mecânica que consistia em encontrar uma curva no qual o tempo gasto por um objeto para deslizar sobre ela até certo ponto fosse o mínimo, independente do seu ponto de partida. Este problema também é conhecido como problema da Braquistócrona. Lagrange soluciona o problema e envia a solução para Euler que, em suas mãos, desenvolveu o método de Lagrange levando a formulação Lagrangiana da mecânica e a descoberta do cálculo das variações. Essa e outras contribuições também são devidas à Euler no campo da astronomia, ótica, mecânica dos fluidos e até na teoria musical.

Euler morre no dia 18 de setembro de 1783, aos 76 anos, devido a uma hemorragia cerebral. Instantes antes de sua morte, na mesma data, Euler discutia com seu colega Anders Lexell sobre a descoberta do planeta Urano e a sua órbita ao redor do sol, morrendo horas depois. Seu trabalho lhe garantiu o posto de um dos grandes matemáticos de todos os tempos, assim como Newton, Arquimedes e Gauss.

Referências bibliográficas:

BOYER, Carl B. História da Matemática. São Paulo: Blucher, 2012.

ROQUE, Tatiana. História da Matemática - Uma visão crítica, desfazendo mitos e lendas. São Paulo: Zahar, 2012.

ROONEY, Anne. A História da Matemática. São Paulo: M. Books do Brasil Editora, 2012.

Texto originalmente publicado em https://www.infoescola.com/biografias/leonhard-euler/

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Leonhard Euler - Biografia do matemático suíço

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Leonhard Paul Euler foi sem dúvida um dos matemáticos mais notáveis de todos os tempos e suas contribuições para a física e a matemática ecoam até hoje. Nascido na Suíça em 15 de abril de 1707, Euler teve uma bem sucedida carreira acadêmica com contribuições e descobertas em geometria, análise matemática, teoria dos números, cálculo, lógica e foi o matemático pioneiro da Teoria dos Grafos.

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Vida & Obra de...

Joseph-Louis Lagrange, ou Giuseppe Lodovico Lagrangia (1736 - 1813) foi um matemático e astrónomo francês nascido em Turim, Itália; destacou-se em todos os campos da análise, da teoria dos números e da mecânica analítica e celeste.

Apesar do pai de Lagrange ocupar uma posição importante ao serviço do rei da Sardenha (Tesoureiro do Escritório de Obras Públicas e Fortificações), a família não era rica, pois o pai tinha perdido grandes somas de dinheiro em investimentos financeiros malsucedidos.

O interesse de Lagrange pela matemática começou quando aos 17 anos leu uma cópia do trabalho de 1693, do matemático e astrónomo Edmond Halley, sobre o uso da álgebra na óptica.

Mais tarde, Lagrange afirmaria:
"If I had been rich, I probably would not have devoted myself to mathematics."
A partir desta altura, Lagrange dedicou-se à matemática, mas grande parte do tempo foi autodidata e não teve o benefício de estudar com matemáticos importantes. Por isso, quando em 1754, com apenas dezoito anos, publicou o seu primeiro trabalho, descobriu que os seus resultados tinham aparecido na correspondência entre Johann Bernoulli e Leibniz. Lagrange ficou muito aborrecido com esta descoberta, pois temia ser rotulado como um impostor que copiava os resultados dos outros. No entanto, este começo nada notável fez com que redobrasse os seus esforços para produzir resultados de verdadeiro mérito em matemática.

Com apenas uma ano de estudos, Lagrange já dava nas vistas e, aos 19 anos foi nomeado professor de matemática na Escola Real de Artilharia de Turim em 1755.

Em 1756 Lagrange enviou a Euler os resultados que tinha obtido na aplicação do cálculo das variações à mecânica. Euler impressionado tenta atrair Lagrange para uma posição na Prússia, para um cargo que seria consideravelmente mais prestigioso do que aquele que ocupava em Turim. No entanto, Lagrange não procurava a grandeza, ele só queria dedicar-se à matemática e, por isso, timidamente, mas educadamente recusou o cargo.

Lagrange foi proposto por Euler para a Academia de Berlim e foi eleito membro a 2 de setembro de 1756. No ano seguinte, Lagrange foi fundador de uma sociedade científica em Turim, que mais tarde se tornaria na Academia Real de Ciências de Turim.

Neste período Lagrange fez contribuições para a teoria das cordas vibrantes, estudou a integração de equações diferenciais e fez várias aplicações na mecânica dos fluidos (onde introduziu a função de Lagrange), estudou as órbitas de Júpiter e Saturno e com apenas 26 anos, publicou o seu famoso método das equações de Euler-Lagrange no recente campo matemático do cálculo variacional, aplicando as suas ideias no campo da mecânica clássica.

Por volta de 1966 d'Alembert, que tinha visitado a Academia de Berlim e era amigo de Frederico II da Prússia, providenciou para que lhe fosse oferecido um cargo na Academia de Berlim. Apesar de nenhuma melhora na sua posição em Turim, Lagrange recusou novamente a oferta escrevendo:
"It seems to me that Berlin would not be at all suitable for me while M Euler is there."
Em março de 1766, quando Euler regressou a São Petersburgo, d'Alembert escreveu novamente a Lagrange para encorajá-lo a aceitar um posto em Berlim e Lagrange finalmente aceitou. Lagrange sucedeu Euler como Diretor de Matemática na Academia de Berlim a 6 de novembro de 1766. Lagrange foi saudado calorosamente pela maioria dos membros da Academia e logo se tornou amigo íntimo de Lambert e Johann (III) Bernoulli.

Pouco depois de chegar a Berlim, Lagrange casou-se com uma prima, mas não teve filhos porque não os quis, chegou a desabafar isso mesmo a d'Alembert numa das suas cartas.

Lagrange trabalhou durante 20 anos em Berlim, produzindo um fluxo constante de artigos de alta qualidade e ganhou regularmente o prémio da Academia das Ciências de Paris. Refira-se que Lagrange dividiu o prémio de 1772 sobre o Problema dos três corpos com Euler.

Em Berlim Lagrange desenvolveu trabalhos em astronomia, estabilidade do sistema solar, mecânica, dinâmica, mecânica dos fluidos, probabilidades e fundamentos do cálculo. Também apresentou resultados em teoria dos números provando em 1770 o "Teorema dos quatro quadrados" que diz que qualquer número inteiro positivo é a soma de quatro quadrados. Em 1771, provou o teorema de John Wilson de que n é primo se e somente se (n − 1)! + 1 é divisível por n. Em 1770, também publicou o seu importante trabalho Réflexions sur la résolution algébrique des équations, uma investigação fundamental apresentando a razão pela qual as equações de grau até 4 poderiam ser resolvidas por radicais. O artigo é o primeiro a considerar as raízes de uma equação como quantidades abstratas em vez de ter valores numéricos. Estudou as permutações das raízes que pode ser considerado como um primeiro passo no desenvolvimento da teoria dos grupos continuada por Ruffini, Galois e Cauchy.

Durante os anos em Berlim, a saúde de Lagrange foi bastante precária em muitas ocasiões, mas a da sua esposa foi ainda pior, tendo morrido em 1783, deixando Lagrange muito deprimido. Três anos depois, Frederico II morreu e a posição de Lagrange em Berlim tornou-se menos feliz. Recebendo várias ofertas aceitou a de Paris pois não teria de ensinar dedicando-se totalmente à investigação científica. A 18 de maio de 1787 deixou Berlim para se tornar membro da Academia das Ciências em Paris, onde permaneceu pelo resto da sua carreira.

Lagrange sobreviveu à Revolução Francesa, enquanto outros não, e isso pode ter sido devido, em certa medida, à sua atitude que ele havia expressado muitos anos antes quando escreveu:
"I believe that, in general, one of the first principles of every wise man is to conform strictly to the laws of the country in which he is living, even when they are unreasonable."
Lagrange escreveu em Berlim o trabalho, Mécanique analytique, que foi publicado em 1788. Este trabalho resumiu todo o trabalho feito no campo da mecânica desde a época de Newton e destaca-se pelo uso da teoria das equações diferenciais. Com este trabalho, Lagrange transformou a mecânica num ramo da análise matemática. Lagrange escreveu no Prefácio:
"One will not find figures in this work. The methods that I expound require neither constructions, nor geometrical or mechanical arguments, but only algebraic operations, subject to a regular and uniform course."
Lagrange foi nomeado membro da comissão da Academia das Ciências para estandardizar pesos e medidas, em maio de 1790. Esta comissão trabalhou no sistema métrico e defendeu uma base decimal.

Lagrange casou-se pela segunda vez em 1792, a sua esposa era a filha de um dos seus colegas astrónomos da Academia das Ciências.

Em 1793, o Reinado do Terror começou e a Academia das Ciências, junto com as outras sociedades eruditas, foi suprimida a 8 de agosto. A comissão de pesos e medidas foi a única autorizada a continuar e Lagrange tornou-se o seu presidente. Lagrange foi um dos responsáveis para a escolha do sistema de unidades para a massa, o Quilograma.

Em setembro de 1793, foi implementada uma lei que ordenava a prisão de todos os estrangeiros nascidos em países inimigos e confiscava todas as suas propriedades. Lavoisier interveio em nome de Lagrange, que certamente estava abrangido por esta lei, e foi-lhe concedida uma exceção. Note-se que Lavoisier salvou muitos da prisão e da morte, mas acabou por morrer na guilhotina pelas mãos dos revolucionários.

Lagrange foi o primeiro professor de Análise na École Polytechnique, inaugurada em 1794. Em 1795 foi fundada a École Normale com o objetivo de formar professores e Lagrange lecionou cursos de matemática elementar nesta escola. Lagrange tinha uma cláusula de 'não ensino' no seu contrato, mas com a Revolução, Lagrange foi obrigado a ensinar. Fourier, que assistiu às suas aulas na École Normale em 1795 escreveu:
"His voice is very feeble, at least in that he does not become heated; he has a very pronounced Italian accent and pronounces the s like z ... The students, of whom the majority are incapable of appreciating him, give him little welcome, but the professors make amends for it."
Da mesma forma, Bugge, que participou nas suas aulas na École Polytechnique em 1799, escreveu:
"... whatever this great man says, deserves the highest degree of consideration, but he is too abstract for youth."
Lagrange publicou dois volumes com o material das suas aulas de Cálculo. Em 1797, publicou sobre a teoria das funções de variável real o trabalho Théorie des fonctions analytiques - embora não tenha dado atenção suficiente às questões de convergência e em 1800 o trabalho Leçons sur le calcul des fonctions.

Nem todos acharam a abordagem do Cálculo de Lagrange a melhor, no entanto, Gaspard de Prony escreveu em 1835:
"Lagrange's foundations of the calculus is assuredly a very interesting part of what one might call purely philosophical study: but when it is a case of making transcendental analysis an instrument of exploration for questions presented by astronomy, marine engineering, geodesy, and the different branches of science of the engineer, the consideration of the infinitely small leads to the aim in a manner which is more felicitous, more prompt, and more immediately adapted to the nature of the questions, and that is why the Leibnizian method has, in general, prevailed in French schools."
Napoleão nomeou Lagrange para a Legião de Honra e Conde do Império em 1808. A 3 de abril de 1813, foi condecorado com o Grand Croix da Ordre Impérial de la Réunion. Morreu uma semana depois, na cidade de Paris, tendo sido sepultado no famoso Panteon, onde ainda hoje se encontra o seu túmulo.

Fonte: https://mathshistory.st-andrews.ac.uk/Biographies/Lagrange/

Fotos retiradas de: https://www.wikiwand.com/en/Joseph-Louis_Lagrange e https://www.alamy.com/stock-photo-joseph-louis-lagrange-1736-1813-italian-mathematician-and-astronomer-142308812.html

Imagem da estátua de Joseph-Louis Lagrange em Turim, Itália retirada de: https://cdn.britannica.com/86/134186-050-64A8EE90/statue-Joseph-Louis-Lagrange-Italy-Turin.jpg

Por Adília Marinho

Publicado/editado: 25/01/2021

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Vida & Obra de... Joseph-Louis Lagrange, ou Giuseppe Lodovico Lagrangia (1736 - 1813) foi um matemático e astrónomo francês nascido em Turim, Itália; destacou-se em todos os campos da análise, da teoria dos números e da mecânica analítica e celeste.

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Euler legou à posteridade um número assombroso de trabalhos sobre as mais diversas áreas, da Engenharia à Mecânica, da Óptica à Astronomia, da Música à Matemática (curvas, séries, cálculo de variações, cálculo infinitesimal, Geometria, Álgebra).

Euler (1707-1783)

Produziu tanto durante a sua vida que durante quase 50 anos depois da sua morte, os seus artigos continuaram a ser publicadas na Academia de S. Petersburgo. A lista bibliográfica das suas obras, incluindo itens póstumos, contém 886 títulos. A sua pesquisa Matemática chegava a ser, em média, de 800 páginas por ano, durante toda a sua vida.

No tempo em que esteve em Berlim, Euler ganhou o hábito de escrever artigos e colocá-los numa pilha. Sempre que era necessário material para as publicações da Academia eram retirados artigos da mesma. Como a produção de Euler era superior às publicações, os artigos na base demoravam muito a ser publicados. Isso explica o facto de quando alguns artigos surgirem, extensões e melhorias dos mesmos já terem sido publicadas antes, com a assinatura de Euler.

Jamais algum matemático terá superado a produção deste homem. Como tal, iremos referir somente algumas das contribuições de Leonard Euler para a ciência.

Uma das suas maiores contribuições foi ao nível das notações:

"(...) numa exposição manuscrita dos seus resultados, escrita provavelmente em 1727 ou 1728, Euler usou a letra e mais de uma dúzia de vezes para representar a base do sistema de logaritmos naturais." (Boyer, 1974, p.326). Apesar da base e ser já conhecida há um século, desde a invenção dos logaritmos, não havia sido padronizada uma notação. E é numa carta dirigida a Goldbach (1690-1764), em 1731, que Euler mencionou o e como "(...) aquele número cujo logaritmo hiperbólico [é] igual a 1." (Smith citado em Boyer, 1974, p. 326).

Livro Mechanica (1736/1737) de Euler

Em 1736, quando publicou o seu livro Mechanica, onde a dinâmica de Newton (1642-1727) foi apresentada de forma analítica,foi impresso pela primeira vez o número e. A partir deste momento, a notação do número foi facilmente aceite e adoptada nos cálculos matemáticos, bem como a padronização da denominação de "exponencial".

Newton (1642-1727)

A Euler também se atribui o uso definitivo da letra grega p como notação para a razão da circunferência e para o diâmetro do círculo. Não foi o primeiro matemático a utilizá-la, pois há registo de uma outra ocorrência em 1706, mas foi o primeiro a reconhecer a sua importância e utilidade. Digamos que, "foi a adopção do símbolo p por Euler em 1737, e mais tarde em seus muitos e populares livros de texto, que o tornou largamente conhecido e usado." (Boyer, 1974, p. 326)

A introdução do símbolo i para Ö (-1) foi mais uma notação adoptada em 1777, quase no fim da sua vida. Mas, só ficou conhecida em 1794 quando publicada numa obra posterior à sua morte.

Wallis (1614-1705)

Inicialmente, o fundamento da utilização baseava-se em representar um número infinito, tal como Wallis (1616-1705) usara o ¥. Desta maneira, Euler apresentava

e x = lim(1 + x/i) i

onde, actualmente se escreve
e x = lim(1 + x/n) n.

Mas somente após a adopção, por parte de Gauss (1777 - 1856), do símbolo i no seu livro Disquisitiones Arithmeticae em 1801, é que se assegurou a sua utilização nas notações Matemáticas.

Gauss (1777-1856)

Após a "apresentação" dos símbolos, cuja introdução e adopção se devem a Euler,foi possível combinar os números e, p e i com o 0 e o 1 na mais célebre igualdade que contém os cinco números:

epi + 1 = 0

Esta, revela uma importante relação entre os mesmos.

A Euler também é associada a introdução das seguintes notações:

- A sexta constante mais importante da Matemática, a Constante de Euler, g;

- O logaritmo de x, ln x;

- O uso da letra å para a adição;

- f(x) para uma função de x.

Segundo Boyer (1974), as nossas notações são hoje assim mais por causa de Euler do que qualquer outro matemático. Fica no ar se a afirmação se deveu à persuasão de Euler ou se foi devido à facilidade com que encarou e anotou conteúdos matemáticos.

* * *

É evidente que o seu reconhecimento na Matemática moderna não se reduz à introdução denúmeros importantes e posterior aceitabilidade da notação. Há mais áreas da Matemática em que o seu contributo foi notório, que vão desde a Geometria à Análise, passando pela Álgebra e Trigonometria. Em cada uma delas é usada uma terminologia própria e muitas vezes vêm mencionados os símbolos referidos anteriormente.

Geometria

Na área da Geometria, numa primeira abordagem, o seu reconhecimento deve-se ao uso das letras minúsculas a, b, c para os lados de um triângulo e das maiúsculas correspondentes A, B, C para os ângulos opostos, bem como a aplicação das letras r, R e s para o raio dos círculos inscrito e circunscrito e o semiperímetro do triângulo, respectivamente. Há a destacar, também, a "(...) bela fórmula (...)" (Boyer, 1974, p. 326):

4rRs = abc

que relaciona os seis comprimentos, referidos anteriormente, embora hajam resultados semelhantes na Geometria da Antiguidade.

No campo da Geometria Sintética o seu contributo não foi muito, apesar de actualmente se considerar a recta que contém o circuncentro, o ortocentro e o baricentro de um triângulo, como a recta de Euler.

Em Geometria Analítica fez-se notar, a partir de 1728, por usar coordenadas no espaço, por dar equações gerais para três grandes classes de superfícies – cilindros, cones e superfícies de revolução - e por definir o arcomais curto (geodésica) entre dois pontos de uma superfície cónica.

Análise e Cálculo

Os contributos de Euler no ramo da análise e do cálculo fizeram-se notar com a introdução de uma nova abordagem à resolução de equações diferenciais.

"Euler foi, sem dúvida, o maior responsável pelos métodos de resolução usados hoje nos cursos introdutórios sobre equações diferenciais, e até muitos dos problemas específicos que aparecem em livros de texto de hoje remontam aos grandes tratados que Euler escreveu sobre o Cálculo – Institutiones Calculi Differentialis (1755) e Institutiones Calculi Integralis (1768 – 1770, 3 volumes)". (Boyer, 1974, p. 333)

Através destes quatro volumes, deu a conhecer o uso de factores integrantes, os métodos sistemáticos para resolver equações lineares de ordem superior com coeficientes constantes, bem como a distinção entre equações lineares homogéneas e não-homogéneas, e entre solução particular e solução geral. Foi, desde então, que se pode afirmar que o tratamento mais completo realizado na área do cálculo se deveu a Euler.

Probabilidades

O interesse de Euler pelo estudo das Probabilidades, impulsionado por Daniel (1700-1782) e Nicolaus Bernoulli, também se fez notar. Escreveu sobre expectativas de vida, o valor de uma anuidade, lotarias, entre outros aspectos da vida em sociedade e como seria previsível, contribuiu, também, com algumas notações.

* * *

Introduction in Analysin Infinitorum

Uma parte significativa do trabalho de Euler foi publicada na obra Introduction in Analysin Infinitorum. Nela, Euler revelou um grande domínio na Análise, que fez com que esta obra fosse considerada como a "(...) chave de abóbada da análise." (Boyer, 1974, p. 327). Publicada em dois volumes, em 1748, serviu para mostrar alguns dos desenvolvimentos na Matemática que se deram durante a segunda metade do século XVIII.

No primeiro volume desta obra, foi apresentado:

O tratamento analítico das funções trigonométricas, bem como a respectiva notação: sen x, cos x, tg x, cotg x.
A título de exemplo, a função seno já não era considerada um segmento de recta, mas sim um número ou uma razão: era possível apresentá-la como a ordenada de um ponto sobre o círculo unitário, ou através de um número definido pela série

z - z ³/3! + z ⁵/5! + ...

para um determinado valor de z.

As identidades de Euler

sen x = [e Ö(-1)x – e-Ö(-1)x] / 2Ö(-1)

cos x = [e Ö(-1)x + e-Ö(-1)x] / 2

e Ö(-1)x = cos x + Ö(-1) sen x

que constituem instrumentos familiares da Análise, obtêm-se através das séries infinitas para ex , sen x e cos x.

"As funções transcendentes elementares – trigonométricas, logarítmica, trigonométricas inversas e exponencial – eram escritas e pensadas praticamente na forma em que são tratadas hoje. As abreviações sin, cos, tang, cot, sec e cosec, que foram usadas por Euler (...), são mais próximas das formas actuais em inglês do que as abreviações correspondentes das línguas latinas. Além disso, Euler foi dos primeiros a tratar os logaritmos como expoentes, do modo hoje tão familiar."(Boyer, 1974, p. 327)

Leibniz (1646-1716)

Processos infinitos, em que se destacam as séries infinitas. Apesar de apresentar a generalização das ideias de Newton, Leibniz (1646-1716) e dos irmãos Bernoulli, Euler foi capaz de obter resultados que tinham "escapado" a outros matemáticos, mesmo sendo um pouco descuidado no cálculo de séries divergentes.
Um dos resultados é a Soma dos Recíprocos dos Quadrados Perfeitos, que desafiou os irmãos Bernoulli durante alguns anos.
O resultado sobre a soma dos recíprocos dos quadrados dos inteiros data de 1736, apesar de ter sido publicada, somente, 11 anos depois. O seu interesse por este género de séries sempre foi grande.
Foram publicadas, também, somas de recíprocos de outras potências dos inteiros, por exemplo, as somas dos recíprocos de potências pares desde n = 2 até n = 26.
Segundo Boyer (1974), as séries de recíprocos de potências ímpares são tão intratáveis que ainda não se sabe se a soma dos recíprocos dos cubos dos inteiros positivos é ou não um múltiplo racional de p ³, ao passo que Euler sabia que para a 26ª potência a soma dos recíprocos é

O arrojado desempenho que demonstrou no tratamento das séries contribuiu para que Euler relaciona-se a Análise com a Teoria dos Números. Sem se preocupar com os perigos que as séries alternadas revelam, obteve resultados como:
p = 1 + ½ + 1/3 + ¼ - 1/5 + 1/6 + 1/7 + 1/8 + 1/9 – 1/10 + ...
em que o sinal de cada um dos termos obedece à regra de: se o denominador for um primo da forma 4m-1, o sinal é mais, se o denominador é um número composto (produto de dois primos), usa-se o sinal dado pelo produto dos sinais das componentes;

ln ¥ = 1 + ½ + 1/3 + ¼ + ...;

1/ln ¥ = 0 = 1 – ½ -1/3 – 1/5 + 1/6 – 1/7 + 1/10 - ...
em que os recíprocos de primos têm sinal menos e os recíprocos de produtos de dois primos distintos têm sinal mais;

0 = ½ * 2/3 * 4/5 * 6/7 * 10/11 * 12/13 * 16/17 * 18/19 *...;

¥ = 2/1 * 3/2 * 5/4 * 7/6 * 11/10 * 13/12 * 17/16 * 19/18 * ...
em que o símbolo ¥ é denotado como o recíproco do número 0.

No segundo volume da obra Introductio in Analysin Infinitorum:

Euler deu uma teoria geral de curvas, baseada no conceito de função, tema central no primeiro volume;

A introdução das funções trigonométricas como parte integrante da geometria analítica, não foi esquecida, bem como algumas funções não tão comuns como: y = x x , y x= x y ey = (-1) x ;

O uso de coordenadas polares;

Classes completas de curvas, tanto algébricas como transcendentes;

Uma maneira de transformar as coordenadas polares em coordenadas rectangulares, que são dadas em forma trigonométrica estritamente moderna;

A representação paramétrica de curvas.

* * *

Euler e os outros Matemáticos

Foram vários os matemáticos que trocaram ideias com Euler. Vejamos dois deles, d'Alembert e Fermat:

Euler e d'Alembert

O trabalho entre Euler e d'Alembert (1717-1783) sempre convergiu no mesmo sentido. Os seus interesses eram quase os mesmos, apesar de ter havido alguma controvérsia entre eles sobre o problema das membranas vibrantes, em 1757, cuja solução da equação de Bassel Euler conseguiu obter, o que ocasionou um afastamento. Mas, com a teoria dos números houve um grande apoio por parte de d'Alembert a Euler.

A contribuição de Euler para a teoria dos logaritmos não se restringiu à definição de expoentes, como usamos hoje. Trabalhou, também, no conceito de logaritmo de números negativos.

Em cartas enviadas a Jean Bernoulli (1667-1748), em 1740, Euler usou expoentes imaginários ao escrever e xÖ(-1) – e-xÖ(-1) = 2 cos x.

Enquanto se mantinha ocupado a pesquisar Matemática em Berlim, d'Alembert pesquisava em Paris. Em 1747, Euler escreveu a este matemático francês explicando correctamente a questão dos logaritmos dos números negativos. Mas ao contrário do que seria de se esperar, a fórmula

e qi = cos q + i sen q,

válida para qualquer ângulo (em radianos), não foi compreendida por Jean Bernoulli nem por d'Alembert pois, para estes os logaritmos de números negativos eram reais, o que não é verdade já que se tratam de números imaginários puros.

d'Alembert (1717-1783)

Através da sua identidade – mais tarde conhecida como Igualdade de Euler – é possível observar que os logaritmos de números complexos, reais ou imaginários, também são números complexos.

Usando as identidades de Euler é também possível expressar quantidades como sen (1 + i) ou cos (i), na forma usual para números complexos. Desta maneira, vê-se que ao efectuar operações transcendentes elementares sobre os números complexos, os resultados são números complexos.

Assim sendo, Euler foi capaz de demonstrar que o sistema de números complexos é fechado sob as operações transcendentes elementares, enquanto d'Alembert sugerira que o sistema de números complexos era algebricamente fechado.

A definição das funções Beta e Gama, através dos integrais eulerianos, permitiu a aceitabilidade destas funções transcendentes, tornando-se parte integrante do cálculo avançado e da Matemática aplicada. Um século depois, a função beta foi generalizada por Tchebycheff (1821 – 1894), e a sua aplicabilidade foi fomentada, principalmente no cálculo de probabilidades.

Também foi possível a Euler mostrar que uma potência imaginária de um número imaginário pode ser um número real. Tal é dado a conhecer através de uma carta, datada de 1746,a Goldbach, em que escreveu um resultado notável:

ii = e -p/2

Euler e Fermat

Tanto Fermat (1601-1665) como Euler sentiram-se bastante interessados pela teoria dos números. Embora não haja qualquer livro sobre este assunto, Euler escreveu cartas e artigos sobre vários aspectos desta teoria. Entre elas encontramos as conjecturas apresentadas por Fermat,que foram derrubadas por Euler.

Duas dessas conjecturas foram:

A primeira foi derrubada em 1732 com o auxílio do seu domínio em computação, evidenciando que 2 2*5 + 1 = 4294967297 é factorizável em 6700417 * 641.

No entanto, no recurso a um contra-exemplo para deitar por terra a segunda conjectura, Euler também errou, apesar do erro só ter sido descoberto em 1966, dois séculos depois e com o auxílio de um computador.

Fermat (1601-1665)

Euler também realizou a demonstração de uma conjectura bastante conhecida, denominada como Pequeno Teorema de Fermat. ( Clique para ver a demonstração). Tal demonstração foi apresentada numa publicação em 1736, denominada Commentarii. Posteriormente, demonstrou uma afirmação mais geral do Pequeno Teorema de Fermat, que veio a chamar-se Função de Euler.

Mas, contrariando o que seria esperado, Euler não foi capaz de demonstrar o Último Teorema de Fermat, embora provasse a impossibilidade de soluções inteiras de x n + y n = z n para n = 3.

Em 1747, definiu mais vinte e sete números amigáveis, que se juntaram aos três já conhecidos por Fermat. Mais tarde aumentou o número para sessenta.

Euler também provou que todos os números perfeitos pares são da forma dada por Euclides (séc. III aC), 2 n-1 (2 n – 1), onde 2 n – 1 é primo. Se existe ou não um número ímpar perfeito foi uma questão levantada por Euler e Goldbach, através de correspondência, ainda hoje sem resposta.

"Euler não só contribuiu para a teoria dos números como também escreveu um popular texto de álgebra que apareceu em edições alemãs e russas em S. Petersburgo em 1770 – 1772, em francês em 1774, e em numerosas outras versões, inclusive edições americanas em inglês. As qualidades excepcionalmente didácticas da Álgebra de Euler são atribuídas ao facto de ter sido ditada pelo autor cego a pessoa relativamente desesperada." (Boyer, 1974, p. 337)

* * *

As contribuições de Euler abrangeram muitas outras áreas. O seu domínio na Matemática permitiu-lhe desenvolver grandes avanços no campo da Astronomia, em que se inclui a determinação da órbita de cometas e planetas, baseando-se em poucas observações, métodos de cálculo da paralaxe do Sol, a teoria da refracção, considerações sobre a natureza dos cometas, entre outros aspectos.

Podem-se ainda referir trabalhos que aliavam a Matemática à música (apesar de serem pouco conhecidos) e em cartografia.

Como nota final sobre a obra de Euler, é correcto citar Boyer (1974, p.326):

"Pode ser dito com justiça que Euler fez pela análise infinita de Newton e Leibniz [e por áreas importantes da Matemática] o que Euclides fizera pela geometria de Eudoxo e Teaetetus, ou o que Viète fizera pela álgebra de al-Khowarizmi e Cardano."

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Euler legou à posteridade um número assombroso de trabalhos sobre as mais diversas áreas, da Engenharia à Mecânica, da Óptica à Astronomia, da Música à Matemática (curvas, séries, cálculo de variações, cálculo infinitesimal, Geometria, Álgebra). Euler (1707-1783) Produziu tanto durante a sua vida que durante quase 50 anos depois da sua morte, os seus artigos continuaram a ser publicadas na Academia de S.

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Karl Theodor Wilhelm Weierstrass nasceu a 31 de outubro de 1815 na Alemanha.

Em 1829 Karl ingressou no Catholic Gymnasium e alcançou um nível de competência matemática muito além do que seria esperado. Weierstrass lia regularmente o Crelle Journal e ensinava matemática a um dos seus irmãos. O pai de Weierstrass desejava que ele estudasse finanças e, assim, após se formar em 1834, Karl entrou na Universidade de Bonn com um curso que incluía o estudo de direito, finanças e economia. O pai queria que o filho assumisse um cargo no serviço público prussiano, mas Weierstrass sofria com o conflito de obedecer aos desejos do pai ou estudar o que gostava - matemática.

Durante quatro anos Weierstrass não compareceu às aulas do curso nem às aulas de matemática e viveu uma vida boémia (praticava esgrima e bebia cerveja em demasia).

Biermann refere que:
"... the conflict between duty and inclination led to physical and mental strain. He tried, in vain, to overcome his problems by participating in carefree student life ..."
No entanto, Weierstrass estudou matemática por conta própria, lendo Mécanique céleste de Laplace e um trabalho de Jacobi sobre funções elípticas. Weierstrass aprendeu os métodos necessários na teoria da função elíptica, estudando transcrições de palestras de Cristof Gudermann. Numa carta a Lie escrita quase 50 anos depois, ele explicou como decidiu definitivamente estudar matemática, apesar dos desejos do pai:
"... when I became aware of [a letter from Abel to Legendre] in Crelle's Journal during my student years, [it] was of the utmost importance. The immediate derivation of the form of the representation of the function given by Abel ..., from the differential equation defining this function, was the first mathematical task I set myself; and its fortunate solution made me determined to devote myself wholly to mathematics; I made this decision in my seventh semester ..."
Acabou por desistir do curso de finanças e no dia 22 de maio de 1839, Weierstrass matriculou-se na Academia de Münster (onde Gudermann lecionava) para se tornar professor do ensino secundário.

Weierstrass assistiu às aulas de Cristof Gudermann sobre funções elípticas, algumas das primeiras palestras sobre este tópico a serem dadas, e Gudermann incentivou fortemente Weierstrass nos seus estudos matemáticos.

Em 1841 Weierstrass obteve o seu diploma de professor e começou uma carreira de 14 anos como professor de matemática no Pró-Gymnasium em Deutsche Krone (1842-1848) e no Collegium Hoseanum em Braunsberg (1848-56). Era professor de matemática, mas também foi solicitado para ensinar outros tópicos - física, botânica, geografia, história, alemão, caligrafia e até ginástica. Mais tarde na vida, Weierstrass descreveu a "tristeza e tédio intermináveis" desses anos, Biermann refere sobre esse tempo:
"... he had neither a colleague for mathematical discussions nor access to a mathematical library, and that the exchange of scientific letters was a luxury that he could not afford."
Durante esse tempo de isolamento de outros matemáticos Weierstrass trabalhou incessantemente em Análise.

Por volta de 1850 Weierstrass começou a sofrer de tonturas muito fortes. Estes ataques frequentes dificultaram o seu trabalho e pensa-se que estes problemas podem ter sido causados pelos conflitos mentais que sofreu enquanto estudante, juntamente com o stress de aplicar-se à matemática em cada minuto livre do seu tempo e realizar o exigente trabalho de ensino.

O trabalho de Weierstrass em teoria das funções foi impulsionado pelo desejo de completar o trabalho iniciado por Niels Abel e Carl Jacobi.

Em 1854 Weierstrass saiu da obscuridade quando publicou o artigo Zur Theorie der Abelschen Functionen sobre funções abelianas, no Crelle Journal.

A Universidade de Königsberg conferiu-lhe o título de doutor honorário a 31 de março de 1854.

Quando Weierstrass participou de uma conferência em Viena em setembro de 1856, foi-lhe oferecido um lugar em qualquer universidade austríaca à sua escolha. Antes que ele decidisse, a Universidade de Berlim adiantou-se. Como era este o emprego que ele desejava há muito tempo, aceitou rapidamente. No entanto, só conseguiu ocupar formalmente o lugar na Universidade de Berlim, passados alguns anos.

As aulas de Weierstrass atraíram alunos de todo o mundo. Um grande número de alunos beneficiou do ensino de Weierstrass. A abordagem de Weierstrass ainda domina o ensino de hoje da Análise Matemática.

Os tópicos das suas aulas incluíam: a aplicação das séries de Fourier e de integrais à física matemática (1856/57), uma introdução à teoria das funções analíticas (onde expôs os resultados obtidos em 1841, mas nunca publicados), a teoria de funções elípticas (o seu principal tópico de investigação), e aplicações a problemas em geometria e mecânica. Nas suas aulas de 1859/60, Weierstrass deu Introdução à Análise, onde abordou os fundamentos do assunto pela primeira vez. Em 1860/61 lecionou Cálculo Integral.

Em dezembro de 1861, a sua saúde sofreu um agravamento. Weierstrass esteve cerca de um ano para voltar a dar aulas. Nunca mais recuperou a sua saúde completamente. A partir dessa época, lecionou sentado enquanto um aluno escrevia no quadro o que ele dizia.

No ano letivo de 1863/64 começou a formular a sua teoria dos números reais. Nas suas palestras de 1863, provou que os números complexos são a única extensão algébrica comutativa dos números reais. Gauss tinha prometido uma prova em 1831, mas não a conseguiu demonstrar.

Em 1872, a sua ênfase no rigor levou-o a descobrir uma função que, embora contínua, não tinha derivada em nenhum ponto. Riemann tinha sugerido em 1861 que tal função poderia ser encontrada, mas o seu exemplo falhou por não ser diferenciável em todos os pontos.

Weiertrass em conjunto com os colegas, Kummer e Kronecker, deram a Berlim a reputação de universidade líder para estudar matemática.

Weierstrass influenciou positivamente muitos dos seus alunos, muitos dos quais se tornaram matemáticos criativos, citamos alguns: Bachmann, Bolza, Cantor, Engel, Frobenius, Gegenbauer, Hensel, Hölder, Hurwitz, Killing, Klein, Kneser, Königsberger, Lerch, Lie, Lüroth, Mertens, Minkowski, Mittag- Leffler, Netto, Schottky, Schwarz e Stolz. Uma aluna em particular, no entanto, merece menção especial: em 1870 Sofia Kovalevskaya veio para Berlim e Weierstrass ensinou-a em privado, uma vez que não lhe foi permitida a admissão na universidade.

Os padrões de rigor que Weierstrass estabeleceu, definindo por exemplo números irracionais como limites de séries convergentes, afetaram fortemente o futuro da matemática.

Em Biography in Encyclopaedia Britannica é referido que:
"Known as the father of modern analysis, Weierstrass devised tests for the convergence of series and contributed to the theory of periodic functions, functions of real variables, elliptic functions, Abelian functions, converging infinite products, and the calculus of variations. He also advanced the theory of bilinear and quadratic forms."
Segundo Biermann, Weierstrass publicou pouco porque o seu sentido crítico obrigava-o a basear qualquer análise numa base sólida e, por isso, começava uma nova abordagem e continuamente revia-a e expandia-a.

No entanto, supervisionou a publicação de uma grande quantidade de material não publicado das suas palestras. Os dois primeiros volumes apareceram em 1894 e 1895, sendo os únicos a aparecer antes da sua morte em 1897.

Os volumes restantes dos seus Gesammelte Abhandlungen ( Obras completas) apareceram lentamente: o volume 3 em 1903, o volume 4 em 1902, os volumes 5 e 6 em 1915 e o volume 7 em 1927. Os sete volumes foram reimpressos em 1967.

Mais trabalhos continuam a ser publicados hoje, particularmente versões das suas aulas retiradas das notas feitas por quem assistiu às palestras.

Todos os grandes matemáticos que contribuíram para o desenvolvimento do Cálculo tinham um conceito intuitivo de limite, mas foi somente com o trabalho do matemático alemão Karl Weierstrass que uma definição formal, completamente satisfatória, do limite de uma sucessão foi obtida. Entre tantas e importantes contribuições na matemática destacamos a definição delta-epsilon de limite e o seu teorema: "Teorema de Weierstrass" que são continuamente e amplamente usadas.

Weierstrass foi membro estrangeiro da Royal Society.

Durante os seus últimos três anos ficou confinado a uma cadeira de rodas, imóvel e dependente. Weierstrass morreu a 19 de fevereiro de 1897 de pneumonia.

Fonte: https://mathshistory.st-andrews.ac.uk/Biographies/Weierstrass/

Por Adília Marinho

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Karl Theodor Wilhelm Weierstrass nasceu a 31 de outubro de 1815 na Alemanha. Em 1829 Karl ingressou no Catholic Gymnasium e alcançou um nível de competência matemática muito além do que seria esperado. Weierstrass lia regularmente o Crelle Journal e ensinava matemática a um dos seus irmãos.

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Síntese biográfica de Nicolai Lobachevsky
Nikolai Ivanovich Lobachevsky (1792 -1856) foi um matemático de origem russa que nasceu a 1 de dezembro de 1792.

A sua terra natal é Níjni Novgorod, uma cidade da Rússia, situada nas margens do rios Volga e Oka. Naquele tempo, os russos diziam que Moscovo era o coração da Rússia; São Petersburgo, o cérebro e Níjni Novgorod, o Bolso.

Desde muito cedo, revelou a sua paixão pela matemática, embora numa fase inicial a sua intenção fosse estudar medicina. Iniciou um complexo curso que envolveu duas ciências, a matemática e física. Aos 14 anos estava preparado para ingressar na Universidade de Kazan, fundada há pouco tempo, iniciando os seus estudos superiores em 1807 com apenas 15 anos. Passou quarenta anos da sua vida ligado a essa universidade, primeiro como aluno, depois como professor ajudante (1824), professor ordinário (1826) e, com apenas 23 anos foi bibliotecário, conservador do museu e, por fim, reitor (1827), cargo que ocupou até 1846.

Quando um novo reitor foi nomeado para a universidade em 1847, Lobachevsky compreendeu que não poderia retornar ao cargo. Triste pela decisão dos seus superiores, que o afastaram do cargo que desempenhara com tanto carinho e abalado pela perda de seu filho, Lobachevsky nunca mais recuperou e veio a falecer, já completamente cego, pouco depois do aniversário 50 da Universidade de Kazan.

A sua fama surgiu após ter sido o primeiro matemático a publicar uma descrição de uma geometria não-euclidiana. A geometria não-euclidiana que Lobachevsky desenvolveu é referida como a geometria hiperbólica.

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O matemático Nikolai Lobachevsky (1792 -1856) por Carlos Marinho

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Síntese biográfica de Nicolai Lobachevsky Nikolai Ivanovich Lobachevsky (1792 -1856) foi um matemático de origem russa que nasceu a 1 de dezembro de 1792. A sua terra natal é Níjni Novgorod, uma cidade da Rússia, situada nas margens do rios Volga e Oka.

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Johann Peter Gustav Lejeune Dirichlet (1805 - 1859)

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Richard Dedekind was born in Brunswick (Braunschweig), a city in northern Germany, in 1831. Much of his education took place in Brunswick as well, where he first attended school and then, for two years, the local technical university. In 1850, he transferred to the University of Göttingen, a main center for scientific research in Europe. Carl Friedrich Gauss, one of the greatest mathematicians of all time, taught in Göttingen, and Dedekind became his last doctoral student. He wrote a dissertation under Gauss, finished in 1852. As was customary, he also wrote a second dissertation ( Habilitation), completed in 1854, shortly after that of his colleague and friend Bernhard Riemann. Dedekind stayed in Göttingen for four more years, as an unsalaried lecturer ( Privatdozent). During that time he was strongly influenced by P.G.L. Dirichlet, Gauss's successor in Göttingen, and by Riemann, then a rising star. (Dedekind did important editorial work for Gauss, Dirichlet, and Riemann.) In 1858, he moved to the Polytechnic in Zürich (later ETH Zürich), Switzerland, to take up his first salaried position. He returned to Brunswick in 1862, where he became professor at the local university and taught until his retirement in 1896. He published most of his major works in this later period. He also had interactions with important mathematicians elsewhere; thus he was in correspondence with Georg Cantor, collaborated with Heinrich Weber, and developed an intellectual rivalry with Leopold Kronecker. He stayed in his hometown until the end of his life, in 1916. (Cf. Landau 1917, ch. 1 of Dugac 1976, Scharlau 1981, Mehrten 1982, ch. 1 of Ferreirós 1999, Harborth et al. 2007, Sonar 2017, for biographical information.)

Dedekind's main foundational writings are: Stetigkeit und irrationale Zahlen (1872) and Was sind und was sollen die Zahlen? (1888a). Equally important, especially in the eyes of historians of mathematics, is his work in algebraic number theory. That work was first presented in an unusual manner: as supplements to Dirichlet's Vorlesungen über Zahlentheorie. The latter was based on Dedekind's notes from Dirichlet's lectures, edited further by him, and published in a series of editions. It is in his supplements to the second edition, from 1871, that Dedekind's famous theory of ideals was first presented. He modified and expanded it several times, with a fourth edition published in 1893 (Lejeune-Dirichlet 1893, Dedekind 1964) and an unfinished fifth edition later found in his Nachlass (Scheel 2020). An intermediate version of Dedekind's theory appeared separately in a French translation (Dedekind 1877). Further works by him include: a long and influential article on the theory of algebraic functions, written jointly with Heinrich Weber (Dedekind 1882, Dedekind & Weber 2012); and a variety of shorter pieces in algebra, number theory, complex analysis, probability theory, etc. All of these works were re-published, together with selections from his Nachlass, in Dedekind (1930-32). Finally, lecture notes from some of his own classes were made available later (Dedekind 1981, 1985), as were further selections from his Nachlass (Dedekind 1982, also Dugac 1976, Sinaceur 1990, Schlimm 2000) and from his correspondence (Noether & Cavaillès 1937, Lipschitz 1986, Meschkowski & Nilson 1991, Scheel 2014).

As this brief chronology indicates, Dedekind was a wide-ranging and creative mathematician, although he tended to publish slowly and deliberately. It also shows that he was part of a distinguished tradition in mathematics, extending from Gauss and Dirichlet through Riemann, Dedekind himself, Weber, and Cantor in the nineteenth century, on to David Hilbert, Ernst Zermelo, Emmy Noether, B.L. van der Waerden, Nicolas Bourbaki, and others in the twentieth and twenty-first centuries. With some partial exceptions, these mathematicians did not publish explicitly philosophical treatises. At the same time, all of them were very sensitive to foundational issues in mathematics understood in a broad sense, including the choice of basic concepts, the kinds of reasoning to be used, and the presuppositions build into them. Consequently, one can find philosophically pregnant remarks sprinkled through their works, as exemplified by Dedekind works.

Not much is known about other influences on Dedekind, especially philosophical ones. While a student in Göttingen, he did attend a lecture class by Hermann Lotze, called "Deutsche Philosophie seit Kant" ("German philosophy since Kant"), as is clear from notes he took that are preserved in his Nachlass. In his short biography of Riemann (Dedekind 1876a), there is a reference to the post-Kantian philosopher and educator J.F. Herbart, professor in Göttingen from 1833 to 1841, as an influence on Riemann; and in his correspondence, Dedekind mentions the German Idealist J.G. Fichte in passing (Scharlau 1981, Reck 2017). However, he does not aligns himself explicitly with either of them, nor with any other philosopher or philosophical school. Little is known about which philosophical texts might have shaped Dedekind's views, especially early on. A rare piece of information we have in this connection is that he became aware of Gottlob Frege's most philosophical work, Die Grundlagen der Arithmetik (published in 1884), only after having settled on his own basic ideas; similarly for Bernard Bolzano's Paradoxien des Unendlichen (Dedekind 1888a, preface to the second edition). Then again, several of Dedekind's mathematical role models had strong philosophical interests, especially Gauss, Riemann, and Cantor; and German intellectual life at the time was saturated with discussions of Kantian and German Idealist views, including debates about the role of intuition for mathematics, and there is evidence that Dedekind was familiar with at least some of them.

The issues addressed in Dedekind's Stetigkeit und irrationale Zahlen ( Continuity and Irrational Numbers) grow out of the "rigorization" and "arithmetization" of analysis (the mathematical theory) in the first half of the nineteenth century. Yet their roots go deeper, all the way down, or back, to the discovery of incommensurable magnitudes in Ancient Greek geometry, e.g., diagonals in unit squares that have length \(\sqrt{2}\) (Jahnke 2003, ch. 1). The Greeks' response to this startling discovery culminated in Eudoxos' theory of ratios and proportionality, presented in Chapter V of Euclid's Elements (Mueller 1981, ch. 3, Corry, ch. 3). This theory brought with it a sharp distinction between discrete multitudes (numbers) and continuous magnitudes, which led to the long dominant view of mathematics as the science of number, on the one hand, and of magnitude, on the other hand. Dedekind's first foundational work concerns, at bottom, the relationship between the two sides of this dichotomy.

An important part of the dichotomy, as traditionally understood, was that magnitudes and ratios of them were not thought of as numerical entities, with arithmetic operations defined on them, but in a more concrete way based on geometry (as lengths, areas, volumes, angles, etc. and as relations between them). More particularly, while Eudoxos' theory provides a contextual criterion for the equality of ratios, it does not include a definition of the ratios themselves, so that they are not conceived of as independent objects (Stein 1990, Cooke 2005). Such features do little harm with respect to basic applications of the theory; but they lead to inner-mathematical tensions, e.g., when solutions to various algebraic equations are considered (some of which could be represented numerically, others only geometrically). This tension came increasingly to the fore in the early modern period, especially after Descartes' integration of algebra and geometry. What was called for, then, was a unified treatment of discrete and continuous quantities.

More directly and explicitly, Dedekind's essay was tied to the arithmetization of analysis in the nineteenth century-pursued by Cauchy, Bolzano, Weierstrass, and others-which in turn was a reaction to tensions within the differential and integral calculus, introduced earlier by Newton, Leibniz, and their followers (Jahnke 2003, chs. 3-6). As is well known, the inventors of the calculus relied on appeals to "infinitesimal" quantities, typically backed up by geometric or even mechanical considerations, although this was seen as questionable from early on. The nineteenth-century "arithmetizers" found a way to avoid infinitesimals (in terms of the epsilon-delta characterization of limits familiar from current introductions to the calculus). Yet this again, or even more, led to the need for a systematic characterization of various quantities conceived of as numerical entities, now in the form of a unified treatment of rational and irrational numbers.

Dedekind faced this need directly, also from a pedagogical perspective, when he started teaching classes on the calculus at Zürich in 1858 (Dedekind 1872, preface). Consequently, the goal for him was not just to supply a unified and rigorous account of rational and irrational numbers; he also wanted to do so in a way that established the independence of analysis from mechanics and geometry, indeed from intuitive considerations more generally. This indicates an underlying philosophical motivation for Dedekind's work on the foundations of analysis, beyond the pedagogical needs at issue, and it is natural to see an implicit anti-Kantian thrust in it. Finally, the way in which to achieve all of these objectives was to relate arithmetic and analysis closely to each other, indeed to reduce the latter to the former.

While the general idea of reducing analysis to arithmetic, as opposed to geometry, was not new at the time-Dedekind shared it with and adopted it from his teachers Gauss and Dirichlet (Ferreirós 1999, ch. 4, Merzbach 2018)-the particular manner in which he proceeded was quite original. The crucial issue, or the linchpin, for him was the notion of continuity. To get clearer about that notion, he compared the system of rational numbers to the points on a geometric line. Once a point of origin, a unit length, and a direction have been picked for the latter, the two systems can be correlated systematically: each rational number corresponds, in a unique and order-preserving way, to a point on the line. But a further question arises: Does each point on the line correspond to a rational number? Crucially, this question can be reformulated in terms of Dedekind's idea of "cuts" defined directly on the rational numbers, so that any geometric intuition concerning continuity can be put aside. Namely, if we divide the whole system of rational numbers into two disjoint parts while preserving their order, is each such division determined by a rational number? The answer is no, since some correspond to irrational numbers (e.g., the cut consisting of \(\{x : x^2 \lt 2\}\) and \(\{x : x^2 \gt 2\}\) which corresponds to \(\sqrt{2}\)). In this explicit, purely arithmetic, and precise sense, the system of rational numbers is not continuous, i.e. not line-complete.

For our purposes several aspects of Dedekind's procedure, at the start and in its subsequent steps, are important (cf. Ferreirós 1999, ch. 4). To begin with, Dedekind starts by considering the system of rational numbers seen as a whole. Noteworthy here are two aspects: Not only does he accept this system as an "actual infinity", i.e. a complete infinite set that is treated as a mathematical object in itself; he also considers it "structurally", as an example of a linearly ordered set closed under addition and multiplication (an ordered field). In his next step-and proceeding further along set-theoretic and structuralist lines-Dedekind introduces the set of arbitrary cuts on his initial system, thus working essentially with the bigger and more complex infinity of all subsets of the rational numbers (the full power set). It is possible to show that the set of those cuts can in turn be endowed with a linear ordering and with operations of addition and multiplication, thus constituting a totally new "number system".

It is not the cuts themselves with which Dedekind wants to work in the end, however. Instead, for each cut-those corresponding to rational numbers, but especially those corresponding to irrational quantities-he "creates" a new object, a "real number" determined by the cut (cf. Dedekind 1876b, 1888b). Those objects, together with an order relation and arithmetic operations defined on them (in terms of the corresponding cuts), form the crucial system for him. Next, two properties of the new system are established: The rational numbers can be embedded into it, in a way that respects the order and the arithmetic operations (a corresponding field homomorphism exists); and the system is continuous, or line-complete, with respect to its order. What we get overall is the long missing, unified criterion of identity for rational and irrational numbers, both of which are now treated as elements in an encompassing number system (isomorphic to but distinct from the system of cuts). Finally Dedekind indicates how explicit, rigorous, and straightforward proofs of various facts about the real numbers can be given along such lines, including ones that had been accepted without proof so far. These include: basic rules of operation with square roots; and the theorem that every increasing bounded sequence of real numbers has a limit value (a result equivalent to, among others, the more well-known intermediate value theorem.)

Dedekind's published this account of the real numbers only in 1872, fourteen years after developing the basic ideas on which it relies. It was not the only account proposed at the time; indeed, various mathematicians addressed this issue, including: Weierstrass, Thomae, Méray, Heine, Hankel, Cantor, and, somewhat later, Frege (Dieudonné 1985, ch. 6, Boyer & Merzbach 1991, ch. 25, Jahnke 2003, ch. 10). Most familiar among their alternative approaches is probably Cantor's, also published in 1872. Instead of using "Dedekind cuts", Cantor works with (equivalence classes of) Cauchy sequences of rational numbers. The system of such (classes of) sequences can also be shown to have the desired properties, including continuity. Like Dedekind, Cantor starts with the infinite set of rational numbers; Cantor's construction again relies essentially on the full power set of the rational numbers, here in the form of arbitrary Cauchy sequences; and the two resulting systems are isomorphic, as is well known. In these set-theoretic respects the two treatments are thus equivalent. What sets apart Dedekind's treatment of the real numbers, from Cantor's and all the others, is the clarity he achieves with respect to the central notion of continuity. His treatment is also more elegantly and maturely structuralist, in a sense to be spelled out further below.

Providing an explicit and precise definition of the real numbers constitutes a major step towards completing the arithmetization of analysis. But further reflection on Dedekind's procedure (and similar ones) leads to a new question: What is involved in it if thought through fully, i.e., what does this account of the real numbers rely on ultimately? As noted, Dedekind starts with the system of rational numbers; he uses a set-theoretic procedure to construct, in a central step, the new system of cuts out of them; and finally, the real numbers are "created" on that basis. This suggests three sub-questions: First, how exactly are we to think about the rational numbers in this connection? Second, can anything further be said about the set-theoretic procedures that have been used? And third, how is the creation at issue to be understood?

In his published writings, Dedekind does not provide an explicit answer to our first sub-question. What suggests itself from a contemporary point of view is that he relied on the assumption that the rational numbers can be dealt with in terms of the natural numbers together with some set-theoretic techniques. And in fact, in Dedekind's Nachlass explicit sketches of two now familiar constructions can be found: that of the integers as (equivalence classes of) pairs of natural numbers; and that of the rational numbers as (equivalence classes of) pairs of integers (Sieg & Schlimm 2005, earlier Dugac 1976). It seems that these constructions were familiar enough at the time for Dedekind not to feel the need to publish his sketches. (There is also a direct parallel to the construction of the complex numbers as pairs of real numbers, known to Dedekind from W.R. Hamilton's works, and a more general connection to the use of residue classes in developing modular arithmetic. For the former cf. Ferreirós 1999, ch. 7, for the latter Dedekind 1857 and Dugac 1976.)

All the material needed for analysis, including both the rational and all the real numbers, can thus be constructed out of the natural numbers by set-theoretic means. At this point the question arises: Do we have to take the natural numbers themselves as given; or can anything further be said about those numbers, perhaps by reducing them to something even more fundamental? Many mathematicians in the nineteenth century were willing to assume the former. A well-known example is Leopold Kronecker, for whom the natural numbers are "given by God" while the rest of arithmetic and analysis is "made by mankind" (Ferreirós 1999, ch. 4). In contrast Dedekind, and independently Frege, pursued the latter option: they attempted to reduce arithmetic and the natural numbers to "logic". This is the main goal of Was sind und was sollen die Zahlen? ( The Nature and Meaning of Numbers, or more literally, What are the numbers and what are they for?). Another goal is to answer the related second sub-question above: whether more can be said about the set-theoretic procedures used. For Dedekind, again similar to Frege, those procedures are part of logic. Along the way, Dedekind also attempts to clarify his notion of creation. But then, what are the basic notions of logic?

Dedekind's answer to this last question is: Basic for logic are the notions of object ("Ding"), set (or system, "System"), and function (mapping, "Abbildung"). These notions are fundamental for human thought, as he emphasizes especially for the notion of function. They are "logical" in the sense of being applicable in all domains, indispensable for exact reasoning, and not reducible further. (This interpretation of Dedekind, and with it whether he should be understood as a logicist, is not uncontroversial. For the corresponding controversy, cf. Klev 2017, Reck 2019, and Ferreirós forthcoming, on the one hand, and Benis-Sinaceur 2008, 2015, 2017, on the other hand. We will come back to this issue in Section 3.) While not analyzable in terms of anything even more basic, the fundamental logical notions are nevertheless capable of being elucidated, thus of being understood better. Part of their elucidation consists in observing what can be done with them, including how arithmetic can be reconstructed in terms of them (also analysis etc.). For Dedekind that reconstruction starts with the consideration of infinite sets, like in the case of the real numbers, but now in a generalized, more explicit, and more systematic manner.

Dedekind does not just assume, or simply postulate, the existence of infinite sets; he tries to prove it. For that purpose, he considers "the totality of all things that can be objects of my thought" and argues that this "set" is infinite (Dedekind 1888a, section v). He also does not just presuppose the concept of infinity; he defines it (in terms of his three basic notions of logic, as well as the definable notions of subset, union, intersection, etc.). The definition is as follows: A set of objects is infinite-"Dedekind-infinite", as we now say-if it can be mapped one-to-one onto a proper subset of itself. (A set can then be defined to be finite if it is not infinite in this sense.) Moving a step closer to arithmetic, this leads to the notion of a "simple infinity" (or "inductive set"). A rigorous introduction of that notion involves Dedekind's innovative idea of a "chain". As one would say in contemporary terminology, a chain is the minimal closure of a set A in a set B containing A under a function f on B (where being "minimal" is captured of in terms of the general notion of intersection).

What it means to be simply infinite can now be captured in four conditions: Consider a set S and a subset N of S (possibly equal to S). N is called simply infinite if there exists a function f on S and an element 1 of N such that: (i) f maps N into itself; (ii) N is the chain (minimal closure) of \(\{1\}\) in S under \(\,f\); (iii) 1 is not in the image of N under \(f\); and (iv) \(f\) is one-to-one. While at first unfamiliar, it is not hard to see that these Dedekindian conditions are a notational variant of Peano's well-known axioms for the natural numbers. In particular, condition (ii) is a version of the axiom of mathematical induction. These axioms are thus properly called the Dedekind-Peano axioms. (Peano, who published his corresponding work in 1889, acknowledged Dedekind's priority; cf. Ferreirós 2005.) As is also not hard to see, any simple infinity will consist of a first element corresponding to 1, a second element corresponding to \(f(1)\), a third corresponding to \(f(f(1))\), and so on, just like any model of the Dedekind-Peano axioms.

Given these preparations, Dedekind introduces the natural numbers as follows: First, he proves that every infinite set contains a simply infinite subset. Then he establishes that any two simply infinite systems (any two models of the Dedekind-Peano axioms) are isomorphic (so that the axiom system is categorical). Third, he observes that, as a consequence, exactly the same arithmetic truths hold for all simple infinities; or closer to Dedekind's actual way of stating this point, any truth about one of them can be translated, via the isomorphism, into a truth about the other. (That is to say, all models of the Dedekind-Peano axioms are "logically equivalent", which means that the axiom system is "semantically complete"; cf. Awodey & Reck 2002). In those respects, each simply infinity is as good as any other.

As a further step, Dedekind appeals again to the notion of "creation". Starting with some simple infinity constructed initially-it doesn't matter which one we start with, given their isomorphism-he "creates" new objects corresponding to its elements, thereby introducing a distinguished simple infinity Dedekind calls "the natural numbers". As we saw, this last step is parallel to his introduction of "the real numbers" (a parallel confirmed in Dedekind 1888b); but some aspects are clearer in the present case. Namely, the newly created objects are characterized completely by all arithmetic truths, i.e., by those truths transferable, or invariant, in the sense explained above. In other words, their identity depends on relational or structural properties alone, unlike for the elements of other simple infinities where intrinsic, non-arithmetic, or "foreign" properties are built in. In passing, Dedekind also uses the name "abstraction" for the step from the initially constructed simple infinity to the newly introduced one. (If Dedekind is interpreted along such lines, his position is a form of non-eliminative structuralism; cf. Reck 2003; but see Sieg & Morris 2018 and Benis-Sinaceur 2008, 2015, 2017, respectively, for two alternative interpretations. We will again come back to this issue in Section 3.)

Considered more generally, what Dedekind has introduced is the natural numbers conceived of as finite "ordinal" numbers (or counting numbers: the first, the second, etc.). Later he adds an explanation of how their usual employment as finite "cardinal" numbers (answering to the question: how many?) can be recovered. This is done by using initial segments of the number series as tallies: for any set we can ask which such segment, if any, can be mapped one-to-one onto it, thus measuring its "cardinality". (A set turns out to be finite in the sense defined above if and only if there exists such an initial segment of the natural numbers series.) Dedekind rounds off his essay by showing how several basic and so far merely assumed facts in arithmetic can now be proved as well. Especially significant is his purely "logical" justification of proofs by mathematical induction and definitions by recursion (based on his theory of chains).

Set-theoretic assumptions and procedures already inform Dedekind's Stetigkeit und irrationale Zahlen, as we saw. In particular, the system of rational numbers is assumed to be an infinite set; the collection of arbitrary cuts of rational numbers is treated as another infinite set; and when supplied with an order relation and arithmetic operations on its elements, the latter gives rise to a new complete ordered field. Parallel moves can be found in the sketches, from Dedekind's Nachlass, on how to introduce the integers and the rational numbers. Once more we start with an infinite system, here that of all the natural numbers, and new number systems are constructed out of it set-theoretically (although the full power set is not needed in those cases). Finally, Dedekind uses set-theoretic techniques in his other mathematical work as well (e.g. in his treatment of modular arithmetic and his construction of ideals as infinite sets, to be discussed further below). It should be emphasized that the application of such techniques was novel and bold at the time. While a few mathematicians, such as Cantor, used them as well, many others, like Kronecker, rejected them. In fact, by working seriously with actual infinities Dedekind took a stance incompatible with that of his teacher Gauss, who had allowed for the infinite only as a "manner of speaking" (cf. Ferreirós 1999, ch. 7, also Edwards 1983).

What happens in Was sind und was sollen die Zahlen?, in the context of Dedekind's logicist reconstruction of the natural numbers, is that the adoption of set-theoretic techniques is raised to a new level of clarity and generality. Dedekind not only presents set-theoretic definitions of various mathematical notions, he also adds a systematic reflection on the means used thereby and he expands that use in certain respects. For those reasons, the essay constitutes an important step in the emergence of modern set theory. In addition, we already saw that Dedekind presents the notion of set, together with those of object and function, as basic. Here an object is anything for which it is determinate how to reason about it, including having appropriate criteria of identity (Tait 1996). Sets are objects about which we reason by considering their elements, and this is all that matters about them. In other words, sets are to be identified extensionally, as Dedekind is one of the first to emphasize. (Even as important a contributor to set theory as Bertrand Russell struggles with this point well into the twentieth century.) Dedekind is also among the first to consider, not just sets of numbers, but sets of other kinds of objects as well.

Functions are to be conceived of extensionally too, as ways of correlating the elements of sets. But unlike in later axiomatic set theory, Dedekind does not reduce functions to sets of tuples. (Not unreasonably, he takes the ability to map one thing onto another, or to represent one by the other, to be fundamental for human thought; Dedekind 1888a, preface, cf. Klev 2017). Another important aspect of Dedekind's views about functions is that, with respect to their intended range, he allows for arbitrary functional correlations between sets of numbers, indeed between sets of objects more generally. (For the gradual development of his views in this connection, see Sieg & Schlimm 2005.) He thus rejects previous, often implicit restrictions of the notion of function to, e.g., functions presented by familiar formulas, functions representable in intuition via their graphs, or functions decidable by a formal procedure. In other words, he works with a generalized notion of function. In this respect he adopts, and expands on, the position of another of his teachers: Dirichlet (Stein 1988, Ferreirós 1999, ch. 7). Dedekind's notion of set is general in the same sense.

Such generalized notions of set and function, together with the acceptance of the actual infinite, were soon attacked by finitistically and constructively oriented mathematicians like Kronecker. Dedekind defended his approach by pointing to its fruitfulness (Dedekind 1888a, first footnote, cf. Edwards 1983, Ferreirós 1999, ch. 7). But eventually he came to see one feature of it as problematic: his implicit acceptance of a general comprehension principle (another sense in which his notion of set is unrestricted). We already touched on a specific way in which this comes up in Dedekind's work. Namely, in Was sind und was sollen die Zahlen? he introduces "the totality of all things that can be objects of my thought" (his "universal set"), together with arbitrary subsets of that totality (a general Aussonderungsaxiom is used). But then, any collection of objects counts as a set.

As just noted, Dedekind goes beyond considering only sets of numbers in his 1888 essay. This is a significant extension of the notion of set, or of its application, an extension about which one might be concerned. An even more worrisome aspect consists of the particular way in which Dedekind's all-encompassing totality is introduced, namely by reference to "human thought", since this leads to the question of whether crude psychologistic features are involved (more on that topic in Section 3). But the most problematic feature-and the one Dedekind came to take seriously himself-is a third one: his set theory is subject to the set-theoretic antinomies, including Russell's antinomy. (If any collection of objects counts as a set, then also Russell's collection of all sets that do not contain themselves, and this leads quickly to a contradiction.) Dedekind seems to have found out about such problems from Cantor in the late 1890s (who informed him that the collection of all ordinal numbers is an "inconsistent totality", as is the collection of all "objects of thought"). This news shocked him initially, so much that he delayed republication of Was sind und was sollen die Zahlen?; he even expressed doubts about "whether human thinking is fully rational" (Dedekind 1930-32, Vol. 3, p. 449, Dedekind 1996a, p. 836). Later he regained belief in the "inner harmony" of his approach, but without providing a solution to these problems himself (Dedekind 1930-32, Vol. 3, p. 343, Preface to the third edition, published in 1911).

Russell's antinomy and related ones establish that Dedekind's original conception of set is untenable. Yet they do not invalidate his other contributions to set theory. His analysis of continuity, the use of Dedekind cuts in the introduction of the real numbers, the definition of being Dedekind-infinite, the formulation of the Dedekind-Peano axioms, the proof of their categoricity, the analysis of the natural numbers as finite ordinal numbers, the justification of mathematical induction and recursion, and most basically, the insistence on extensional, general notions of set and function, as well as the acceptance of the actual infinite-all of these contributions can be isolated from the set-theoretic antinomies. As such, they have been built into the very core of axiomatic set theory, model theory, recursion theory, and other parts of current logic.

And there are further contributions to set theory we owe to Dedekind. Some of these do not appear in his published writings, but in his correspondence. Especially significant is his exchange of letters, starting in 1872, with Cantor (Noether & Cavaillès 1937, Meschkowski & Nilson 1991). These letters contain a discussion of Cantor's and Dedekind's respective treatments of the real numbers. Beyond that, they amount to a joint exploration of the notions of set and infinity. Among Dedekind's specific contributions in this context are the following: He impresses Cantor with a proof that not only the set of rational numbers but also the set of all algebraic numbers is countable (Ferreirós 1999, ch. 6). This result led, at least in part, to Cantor's further study of infinite cardinalities and to his discovery, soon thereafter, that the set of all real numbers is not countable. Dedekind also provides a proof of the Cantor-Bernstein Theorem (that between any two sets which can be embedded one-to-one into each other there exists a bijection, so that they have the same cardinality). This is another basic result in the theory of transfinite cardinals (Ferreirós 1999, ch. 7).

Finally, in the development of set theory during the twentieth century it became clear that several of Dedekind's results and procedures can be generalized in important ways. Perhaps most significantly, Zermelo and von Neumann succeeded in extending his analysis of mathematical induction and recursion to the higher infinite, thus expanding on, and establishing more firmly, Cantor's theory of transfinite ordinals and cardinals. Looking back on all of these contributions, it is no wonder that Zermelo-who knew the relevant history well-considered the modern theory of sets as having been "created by Cantor and Dedekind" (quoted in Ferreirós 2016b).

So far we have focused on Dedekind's contributions in his overtly foundational writings. We reviewed his innovative approaches to the natural and real numbers; we also reconsidered his role in the rise of modern set theory. Along the way, some philosophical issues have come up. A more extended reflection on them seems called for, however, especially concerning Dedekind's "logicism" and "structuralism" (both topics that have led to some controversies in the secondary literature, as already indicated parenthetically).

Like for Frege, the other main logicist in the nineteenth century, "logic" is more encompassing for Dedekind than often assumed today (as comprising only first-order logic). Both thinkers take the notions of object, set, and function to be fundamental for human thought and, as such, to fall within the range of logic. Each of them then develops a version of set theory (a theory of "systems", "extensions", or "classes") seen as part of logic. Also for both, logic in this encompassing sense is independent of intuitive considerations and, specifically, of traditional Euclidean geometry (understood to be grounded in pure or empirical intuition). A main goal in reducing analysis and arithmetic to logic is, thus, to establish that those fields too are independent of intuition (cf. Demopoulos & Clark 2005).

The view that analysis is not dependent on geometry, since it falls within the realm of arithmetical and basic logical thought, was not entirely new at the time-Gauss and Dirichlet already held such a view, as mentioned above (cf. also Ferreirós 2007). What Frege's and Dedekind's new contributions consisted in were original, detailed reductions of, on the one hand, analysis to arithmetic and, on the other hand, arithmetic to logic. Moreover, each of them complemented these reductions with systematic elaborations of logic. As Dedekind's work was better known than Frege's at the time, probably because of his greater reputation as a mathematician, he was seen as the main representative of "logicism" by interested contemporaries, such as C.S. Pierce, Ernst Schröder, and David Hilbert (cf. Ferreirós 1999, ch. 7, Ferreirós 2009).

In addition to these general commonalities in Dedekind's and Frege's versions of logicism, the two thinkers also agreed on a methodological principle encapsulated in the following remark by Dedekind: In science, and especially in mathematics, "nothing capable of proof ought to be accepted without proof" (Dedekind 1888a, preface). This principle ought to be adhered to not so much because it increases certainty. Rather, it is often only by providing an explicit, detailed proof for a result that the assumptions on which it depends become evident and, thus, its range of applicability established. Both Frege and Dedekind had learned that lesson from the history of mathematics, especially developments in geometry, algebra, and the calculus (see Reck 2013, building on Wilson 1992, 2010, Tappenden 1995, 2006, Ferreirós 1999, and Detlefen 2011).

Besides where they are in agreement, it is instructive to consider some of the differences between Dedekind and Frege too. First and put in modern terminology, a major difference is that, while Frege's main contributions to logic concern syntactic, proof-theoretic aspects, Dedekind tends to focus on semantic, model-theoretic aspects. Thus, nothing like Frege's revolutionary analysis of deductive inference, by means of his "Begriffsschrift", can be found in Dedekind's works. Dedekind, in turn, is much more explicit and clear than Frege about issues such as categoricity, completeness, independence, etc. This allows, among others, to see him as a precursor of the "formal axiomatic" approach championed later by Hilbert and Bernays (Hallett 1994, 1995, Sieg & Schlimm 2005, 2014, Sieg 2010, 2014).

Compared to Frege, Dedekind also has much more to say about the infinite, not just by formulating a definition of that notion, but also by exploring the possibility of different infinite cardinalities with Cantor. And he shows more awareness of the challenge posed by Kroneckerian computational and constructivist strictures to logicism. The differences between Frege's and Dedekind's respective treatments of the natural and real numbers are noteworthy too. As we saw, Dedekind conceives of the natural numbers primarily as ordinal numbers; he also identifies them purely "structurally". Frege makes their application as cardinal numbers central; and he insists on building this application into the very nature of the natural numbers, thus endowing them with non-structural, "intrinsic" properties. The case of the real numbers illustrates this divergence further. Finally, Frege's and Dedekind's underlying conceptions of logic and language differ significantly. (The latter is one of the main reasons for why Dedekind is not interpreted as a logicist in Sinaceur 2008, 2015, 2017.)

Apart from Frege, it is illuminating to compare Dedekind's approach more with later set-theoretic ones. We noted in the last section that many of his innovations have been built into axiomatic set theory. Yet here too, several differences emerge if one looks more closely. To begin with, Dedekind does not start with an axiom of infinity as a fundamental principle; instead, he tries to prove the existence of infinite sets. This can be seen as another application of the methodological rule to prove everything "capable of proof", as well as a crucial aspect of his logicism. However, few set theorists today will want to go back to this aspect of Dedekind's approach. (Dedekind's argument in this connection is similar to an earlier one in Bolzano's posthumously published work; cf. Ferreirós 1999, ch. 7, Klev 2019.)

A second marked difference between Dedekind and current set theory has already come up too, but deserves further comment. This is his appeal to "abstraction", and "creation", in the last step of his introduction of both the natural and the real numbers ("Dedekind-abstraction", as it is called in Tait 1996). For the real numbers, the standard procedure in axiomatic set theory is to follow Dedekind up to this last step, but then to work with the Dedekind-cuts themselves as "the real numbers". While aware of this option (Dedekind 1876b, 1888b), Dedekind tells us to apply "abstraction" to the cuts insofar as they consist of complex sets, thus to "create" additional mathematical objects determined by but not identical with them. Likewise, in the case of the natural numbers it is standard today to construct a particular simple infinity, usually the set of finite von Neumann ordinals, and then to identify the natural numbers with them (thus: \(0 = \emptyset\), \(1 = \{0\}\), \(2 = \{0, 1\}\), etc.). Once again, the step involving "abstraction" and "creation" of Dedekind's procedure is avoided by such identification.

In current approaches it is sometimes added that any other set-theoretically constructed system that is isomorphic to the system of Dedekind cuts or to the system of finite von Neumann ordinals, respectively, would do as well, i.e. be usable mathematically as "the real numbers" or "the natural numbers" for most practical purposes. This means that contemporary set theory is, often implicitly and without further elaboration, supplemented by a "set-theoretic structuralist" view about the nature of mathematical objects (Reck & Price 2000). The resulting philosophical position is different from Dedekind's. (But see again Sieg & Morris 2018; in it Dedekind is interpreted as changing his position over time, i.e., as having adopted a position close to set-theoretic structuralism by 1888.) Nor does Dedekind's position coincide with several other forms of structuralism prominent in contemporary philosophy of mathematics, e.g., those defended by Geoffrey Hellman, Michael Resnik, and Stewart Shapiro.

Why do most mathematicians and philosophers not follow Dedekind more closely in this respect? Here the following difficulties play a role (cf. Boolos 1990, Müller-Stach 2017, among others): How exactly should his notions of "abstraction" and "creation" be understood, if they can be made sense of at all? And why did Dedekind insist on their use in the first place, since we seem to be able to do without them as axiomatic set theory shows? A partial answer to the latter question is: For Dedekind, the real numbers should not be identified with the corresponding cuts because those cuts have "wrong properties"; namely, as sets they contain elements, something that seems "foreign" to the real numbers themselves. Similarly, the natural numbers should not be ascribed set-theoretic or other "foreign" properties; they too should be conceived of "purely arithmetically" (see again Dedekind 1876b, 1888b). Can anything further be said in this connection?

By philosophical critics, Dedekind's procedure is often interpreted as follows: As his language of "abstraction" and "creation" suggests, he is appealing to psychological processes. And the resulting entities-"the real number" and "the natural numbers"-must then be psychological or mental entities (existing in people's subjective consciousness). If that was correct, his position would amount to a form of psychologism about mathematics that is deeply problematic, as Frege, Husserl, and others have taught us (cf. Dummett 1995). Partly to avoid such a damning conclusion, partly to attribute more philosophical depth to Dedekind, the following reply can be found in the literature: While he does not say so explicitly, Dedekind's psychologistic-sounding language indicates a commitment to Kantian assumptions, in particular to Kant's transcendental psychology (Kitcher 1986, McCarty 1995, Klev 2019, cf. also Benis-Sinaceur 2008, 2015, 2018, Floyd 2013). Then again, it is not clear that this takes care of the psychologism charge fully (often also directed against Kant), so that such a response would have to be spelled out further (cf. Yap 2017, Reck 2020, building on Cassirer 1907, 1910). In itself, it also leaves unexplained the precise form of Dedekind's "abstraction".

Recently another suggestion for how to get clearer about the nature of "Dedekind abstraction" and, at the same time, avoid crude forms of psychologism has become prominent. Namely, instead of taking such abstraction to amount to a psychological process, it should be understood as a logical procedure (cf. Tait 1996, earlier Cassirer 1910). Consider again the case of the natural numbers, where Dedekind is most explicit about this issue. What we are provided with by him is this: First, the language and logic to be used are specified, thus the kinds of assertions and arguments that can be made concerning the natural numbers; second, a particular simple infinity is constructed; third, this simple infinity is used to determine the truth values of all arithmetic sentences (by equating them with the truth values of corresponding sentences for the given simple infinity); and fourth, this determination is justified by showing that all simple infinities are isomorphic (so that, if a sentence holds for one of them, it holds for all).

The core of the procedure just described is the following: Something is true for the natural numbers exactly when the corresponding statement holds for all simple infinities (i.e., is a semantic consequence of the Dedekind-Peano axioms). What, then, are the natural numbers? They are those mathematical objects whose properties are determined by all arithmetic truths; we "abstract away" from everything else in terms of a logical procedure. Moreover, this procedure can be spelled out, precisely and consistently, in terms of "structural abstraction principles" (cf. Linnebo & Pettigrew 2014, Reck 2018 for details). Along such lines, all that matters about mathematical objects, indeed all that is built into their identity and nature, is what the corresponding mathematical truths determine. Consequently, it is by specifying those truths that we "create", or better fully characterize, them. The resulting position seems clearly not psychologistic, and it might be called "logical structuralism" (Reck 2003, Yap 2009a).

Interpreted as such, Dedekind's approach is related to that of the "American Postulate Theorists", E. Huntington, O. Veblen, etc. (cf. Awodey & Reck 2002). There is also a close connection to, and a clear influence on, the formal axiomatics developed by Hilbert and Bernays later (cf. Sieg 2010, 2014, also Hallett 1994, 1995, 2003). For all of these thinkers, what is crucial in, and sufficient for, mathematics is that the completeness and consistency of certain basic concepts, or of the corresponding systems of axioms, are established. In Dedekind's case, completeness is to be understood in a semantic sense, as based on categoricity; likewise, consistency is to be understood semantically, as satisfiability (Dedekind 1890); and both are tied to his distinctive combination of logicism and structuralism.

The syntactic study of such issues, especially of consistency, that became a core part of Hilbertian "proof theory" is not present in Dedekind's work. As already noted, the proof-theoretic side of logic is not pursued much by him. Nor is the "finitism" characteristic of Hilbert & Bernays' later work present in Dedekind (an aspect developed in response to the set-theoretic antinomies and to intuitionist challenges), especially if it is understood in a metaphysical sense. Such finitism might have been acceptable to Dedekind as a methodological stance, but in other respects his position is strongly infinitary. In fact, the finite is explained in terms of the infinite in his work: the notion of finitude by that of infinity, the natural numbers in terms of infinite sets, etc. (cf. Hilbert 1922).

The summary of Dedekind's position in this section highlighted logical issues (his basic logical notions and procedures, compared both to Frege and to axiomatic set theory) and metaphysical issues (Dedekindian abstraction and creation, the structural nature of mathematical objects). However, there is another dimension of his position-another sense in which Dedekind's approach represents a unique combination of logicism and structuralism-that deserves philosophical attention. That side involves primarily methodological and epistemological aspects. Some of those aspects already played a role in our discussions of his overtly foundational writings; but in order to shed more light on them, it helps to turn to his other mathematical work, starting with algebraic number theory.

While historians of mathematics have long emphasized the influence that Dedekind's contributions to algebraic number theory had on the development of twentieth-century mathematics, the philosophical significance of these contributions has only started to be probed recently (with some early exceptions: Dugac 1976, Sinaceur 1979, Edwards 1980, 1983, Stein 1988). We should begin here by reviewing two related aspects: the roots of Dedekind's number-theoretic investigations in the works of Gauss, Dirichlet, and Ernst Kummer; and the contrast between Dedekind's approach and that of Leopold Kronecker.

The starting point for these number theorists is the solution of algebraic equations, especially their solution in terms of integers. A famous example is Fermat's Last Theorem, which concerns the (non-)solubility of the equation \(x^n + y^n = z^n\) by integers, for different exponents \(n\). The approach to the problem developed by Gauss, clarified and refined by Dirichlet, and pushed further by Kummer involves considering extensions of the (field of) rational numbers, as well as of the (ring of) "integers" contained in such extensions. Thus, Gauss studied the "Gaussian integers" (\(a + bi\), where \(a\) and \(b\) are regular integers and \(i = \sqrt{-1}\)) within the field of all complex numbers; Kummer considered more complicated "cyclotomic integers" in corresponding cyclotomic number fields (Edwards 1977, Stillwell 2000, Gray 2018).

What became clear along such lines is that in some of these extensions the Fundamental Theorem of Arithmetic-asserting the unique factorization of all integers into powers of primes-fails. If it had been available generally, solutions to important problems would have been within reach, including Fermat's Last Theorem. The question became, then, whether a suitable alternative to the Fundamental Theorem could be found. Kummer's response, obtained in a close study of certain particular cases, was to introduce "ideal divisors" in terms of which unique factorization could be recovered. While this move led to striking progress, the precise nature of these new mathematical objects was left unclear, as were the basis for their introduction and the range of applicability of the technique.

Both Dedekind and Kronecker knew this earlier work well, especially Kummer's, and they tried to build on it. Kronecker's strategy was to examine in concrete detail, and by exploiting computational aspects, certain kinds of extensions. Crucial for him was to proceed finitistically and constructively, thus in a self-consciously restricted way. This led to his "divisor theory", an extension of Gauss' and Kummer's "theory of forms" (Edwards 1980, 1990). Dedekind, in contrast, approached the issue in a more encompassing and abstract way. He considered algebraic number fields in general, thereby introducing the notion of a "field" for the first time (Edwards 1983, Haubrich 1992, Stillwell 1996, 1999, Ullrich 2017, Gray 2018).

Dedekind also replaced Kummer's "ideal numbers" by his "ideals"-set-theoretically constructed objects intended to play the same role with respect to unique factorization. Dedekindian ideals are infinite subsets of the number fields in question, or of the ring of integers contained in them, which again gives his approach an infinitary character. (An ideal I in a ring R is a subset such that the sum and difference of any two elements of I and the product of any element of I with any element of R are also in I.) This led him to introduce other fruitful notions as well, such as that of a "module". A particular problem Dedekind struggled with for quite a while, in the sense of finding a fully satisfactory solution, was to specify a suitable version, not only of the notion of "integer", but also of "prime number" (Ferreirós 1999, ch. 3, Avigad 2006, Haffner 2014).

Both Kronecker, with his divisor theory, and Dedekind, with his ideal theory, were able to obtain important results. Each theory also had a strong influence on later developments-Dedekind's by shaping the approaches to modern algebra (field theory, ring theory, etc.) of Hilbert, Emmy Noether, B.L. van der Waerden, Emil Artin, Nicolas Bourbaki, and others (Alten et al. 2003, Corry 2004, McLarty 2006, Gray 2008, 2018); Kronecker's by influencing Hilbert's work as well, and by being revived later in, among others, André Weil's and Alexander Grothendieck's work (class field theory, algebraic geometry) (Weil 1950, Edwards 1990, Reed 1995, Corfield 2003). The contrast between their approaches provides a clear example-historically the first significant example-of the opposition between "classical" and "constructivist" conceptions of mathematics.

Not infrequently this opposition is debated in terms of which approach is "the right one", with the implication that only one of them but not the other is legitimate. (This started with Dedekind and Kronecker. We already noted that Dedekind explicitly rejected constructivist strictures on his work, although he did not rule out corresponding projects as illegitimate in themselves. Kronecker, with his strong opposition to the use of set-theoretic and infinitary techniques, went further in the other direction.) But another question is more basic: Can the contrast between the two approaches to algebraic number theory, or to mathematics more generally, be captured more sharply and insightfully; and in particular, what is its methodological and epistemological significance?

An initial, rough answer to that question is already contained in our discussion so far: Dedekind's approach is set-theoretic and infinitary, while Kronecker's is constructivist and finitary. However, that leaves us with a deeper issue: What exactly is it that a set-theoretic and infinitary methodology allows us to accomplish that Kronecker's doesn't, and vice versa? The specific strength of a Kroneckerian approach is often summed up thus: It provides us with computational, algorithmic information (Edwards 1980, 1990, Avigad 2008). But what is the characteristic virtue of a Dedekindian approach? Here an informative and articulate answer is harder to find, especially one that is philosophically satisfying.

What tends to get in the way is that the set-theoretic and infinitary methodology Dedekind championed was so successful, and shaped twentieth-century mathematics so much, that it is hard to reach the analytic distance needed. Often all one gets, then, are platitudes: that it is very "general" and "abstract". Or one is left with negative characterizations: that it fails to be finitistically and constructively acceptable. Beyond that, there is the suggestion that Dedekind's approach is "structuralist". But as this term is now used to characterize a methodology, rather than a metaphysical position, one wonders what exactly it involves. Before attempting a clarification and elaboration, let us briefly consider some of Dedekind's other mathematical works, since this will make certain crucial features stand out more.

For our purposes, three additional areas in which Dedekind applied his set-theoretic, infinitary, and structuralist approach are relevant: the theory of algebraic functions, group theory, and lattice theory.

Dedekind realized early on that several of the notions and techniques he had introduced in algebraic number theory could be transferred to the study of algebraic functions (algebraic function fields, in later terminology), especially the set-theoretic introduction of ideals and ways of calculating with them (cf. Haffner 2014, 2015, 2017 for emphasizing the latter). This realization came to fruition in "Theorie der algebraischen Funktionen einer Veränderlichen", a substantive article co-written with Heinrich Weber and published in 1882. The approach in it led, among others, to a better understanding of Riemann surfaces, including a purely algebraic proof of the celebrated Riemann-Roch Theorem (Geyer 1981, Kolmogorov & Yushkevich 2001, ch. 2, Stillwell 2012). As these were issues of wide interest among mathematicians, the success gave Dedekind's approach significant legitimacy and publicity (cf. Edwards 1983, Dieudonné 1985).

Second, Dedekind's contributions to algebraic number theory were connected, in a natural and fruitful way, with Evariste Galois' revolutionary group-theoretic approach to algebra. In fact, Dedekind's early and systematic study of Galois theory-he was the first to lecture on it at a German university, already during his time as a Privatdozent in Göttingen-led to a transformation of that theory itself. It turned from the study of substitutions in formulas and of functions invariant under such substitutions into the study of field extensions and corresponding automorphisms, which deep connections to algebraic number theory (Mehrtens 1979b, Stein 1988, Ferreirós & Reck 2020). Dedekind also introduced additional applications of Galois theory, e.g., in the study of modular equations and functions (Gray 2000, ch. 4, Alten et al. 2003, ch. 8).

As a third spin-off, Dedekind's number-theoretic investigations led to the introduction of the notion of a "lattice" (under the name " Dualgruppe "). He examined that notion further, also as a topic in itself, in two relatively late articles: "Über Zerlegungen von Zahlen durch ihre größten gemeinsamen Teiler" (1897) and "Über die von drei Moduln erzeugte Dualgruppe" (1900). While these articles did not have the same immediate and strong impact that several of his other publications had, they were later recognized as original, systematic contributions to lattice theory, especially to the study of modular lattices (Mehrtens 1979a, ch. 2, Burris 1983, Alten et al. 2003, ch. 10) and to an axiomatic approach to mathematics more generally (Schlimm 2000, 2011, 2017).

None of these contributions by Dedekind can be treated in any detail here (and various others have to be ignored completely), but some general observations about them can be made. While they again exemplify a set-theoretic and infinitary perspective, they also display the following related features: the focus on entire systems of objects and on general laws for them; the move away from particular formulas, or from particular symbolic representations, to more general characterizations of the underlying systems of objects, specifically in terms of relational and functional properties; the consideration of homomorphisms, automorphisms, isomorphisms, and features invariant under such mappings; and the investigation of novel, abstract concepts, introduced in connection with specific cases but then studied in themselves.

These features are, in fact, characteristic for Dedekind's works overall, including his studies in algebraic number theory and his foundational investigations. It is by taking them into account that calling his methodology, not just "general" and "abstract", but "structuralist" starts to acquire further content. It also helps us to recognized in which way Dedekind's approach played a crucial role in the emergence of "modern" mathematics, as shaped by a structuralist methodology (cf. Reck & Schiemer 2020). Looking back on the corresponding history of algebra, B.L. van der Waerden, himself a major contributor to this field, concluded: "It was Evariste Galois and Richard Dedekind who gave modern algebra its structure; that is where its weight-bearing skeleton comes from" (Dedekind 1964, foreword, quoted in Mehrtens 1979b; cf. also Gray 2008).

In earlier sections we considered Dedekind's overtly foundational works. This led to a discussion of the logicist and metaphysical structuralist views emerging in them. In the last few sections, the focus shifted to his other mathematical contributions and the methodological structuralism they embody. Now we are in a position to provide a more explicit and systematic elaboration of the latter. Beyond just calling Dedekind's approach set-theoretic, infinitary, and non-constructive, the methodology that informs it can be analyzed as consisting of three parts.

The first part is closely tied to Dedekind's employment of set-theoretic tools and techniques. He uses these to construct new mathematical objects (the natural and real numbers, ideals, modules, etc.) or whole classes of such objects (algebraic number fields, rings, lattices, etc.), usually in terms of infinite sets. Closely related but even more characteristic, both in foundational and other contexts, is the following: These sets are then endowed with general structural features (order relations, arithmetic and other operations, etc.); and the resulting systems are studied in terms of corresponding higher-level concepts (continuity for the real number system, simple infinity for the natural numbers, unique factorization in algebraic number fields and function fields, etc.). Once more, philosophically relevant is not just that this procedure is infinitary (his acceptance of actual infinities) and non-constructive (the added features are not necessarily grounded in algorithms). A whole variety of relational systems, including many new ones, can now be investigated; and this brings with it a significant extension of the subject matter of mathematics. Mathematics is no longer tied to what is given intuitively or even empirically (concrete numbers of things, geometric magnitudes, physical processes that are observable, etc.); it has become the study of relational systems much more generally (based on a new "logic of relations", as Russell and Cassirer will later put it).

The second characteristic part of Dedekind's methodology consists of persistently (and from early on, cf. Dedekind 1854) attempting to identify and clarify fundamental concepts, including the higher-level concepts just mentioned (continuity, infinity, generalized concepts of integer and of prime number, also the new concepts of ideal, module, lattice, etc.). In this context, it is crucial for him to find "the right definitions"; and this involves not just basic adequacy, but also desiderata such as: fruitfulness, generality, simplicity, and "purity", i.e., the elimination of "foreign" aspects. (Geometric notions are foreign to the natural and real numbers, the definition of prime number must be at just the right level of generality, etc.). Moreover, once Dedekind has identified them, he studies the most important ones also in themselves, i.e., in terms of a separate part of the theory. This gives his approach a distinctive "conceptual" character (cf. Stein 1988, Avigad 2006), and it points forward towards an axiomatic approach (Sieg & Schlimm 2014, Schlimm 2017, also Haffner 2014, 2015, 2017)

The third main part of Dedekind's approach connects and complements the first two. Not only does he study infinite systems of objects or whole classes of such systems; and not only does he attempt to identify basic concepts applicable to them, especially higher-level concepts. He also tends to do both, often in conjunction, by considering mappings on the systems studied, especially structure-preserving mappings (homomorphisms, isomorphisms, implicitly also category-theoretic functors, etc.). This implies that what is crucial about a mathematical phenomenon may not lie on the surface (concrete features of examples, particular symbolisms, etc.) but go deeper. And while the deeper features are often captured set-theoretically (Dedekind cuts, ideals, quotient structures, etc.), this points beyond set theory in the end, towards category theory, especially in Dedekind's later works (Corry 2004, McLarty 2006, Ferreirós 2017a, 2017b).

These three parts of Dedekind's approach may not seem extraordinary to a contemporary mathematician. But that is just testimony to how much modern mathematics has been shaped by his work. His approach was certainly seen as novel, even revolutionary, at the time. Some negative reactions to it, by finitist and constructivist thinkers, have already been mentioned. The extent to which Dedekind's approach diverged from what had been common stands out further if we remember two traditional, widely shared assumptions: that mathematics is the science of number and magnitude; and that it has essentially to do with calculating and other algorithmic procedures. Relative to such assumptions, Dedekind's approach to mathematics involves a radical transformation and liberation (Stein 1988, Tait 1996).

Another way to bring out the radical character of Dedekind's work is by returning to his treatment of the infinite. It starts with taking what was long seen as a paradoxical property (to be equinumerous with a proper subset) to be a defining characteristic of infinite sets. Dedekind then adds a systematic and wide-ranging analysis of the finite in terms of the infinite, again a rather bold idea. For conceptual and methodological innovations like these to become accepted, they not only had to open up novel realms for mathematics; the resulting approach also had to lead to progress in more traditional realms (as it did: the clarification of continuity and of mathematical induction and recursion, results in algebra, algebraic number theory, the theory of algebraic functions, etc.).

While Dedekind was a great innovator, he was, of course, not alone in moving large parts of mathematics in a set-theoretic, infinitary, and structuralist direction. There was a whole group of mathematicians who promoted a more "conceptual" approach at the time, including several of his teachers in Göttingen. Within that group, Dirichlet has sometimes been picked out as the leader, or the "poet's poet", among others by being a big influence on Dedekind (Stein 1988, see also Merzbach 2018). As Hermann Minkowski, a major figure in this tradition himself, put it later (on the occasion of Dirichlet's 100th birthday): he impressed on other mathematicians "to conquer the problems with a minimum amount of blind calculation, a maximum of clear-seeing thought" (quoted in Stein 1988).

Riemann was another influential figure in this context. He too had a strong influence on Dedekind, in two ways: by emphasizing (in his development of complex function theory) the importance of using simple, characteristic, and "intrinsic" concepts, in contrast to "extrinsic" properties tied to, say, particular symbolisms (Mehrtens 1979b, Laugwitz 1999, Tappenden 2005a, 2006, Ferreirós & Reck 2020); and by exploring new "conceptual possibilities", including the systematic study of Riemannian geometries (Stein 1988). A further example of such conceptual exploration, again very familiar to Dedekind, was the study of transfinite ordinal and cardinal numbers by his correspondent Cantor (Ferreirós 1999, chs. 2 and 6). And of course, later mathematicians influenced by Dirichlet, Riemann, Dedekind, and Cantor pushed their innovations further, including Ernst Zermelo, Emmy Noether, and Nicolas Bourbaki (Corry 2004, 2017, McLarty 2006, Kahle 2017).

Can the significance of Dedekind's methodology be brought out even further? Or can it be articulated more in different terms? One way to do the latter, already touched on above, is by highlighting the methodological values embodied in it: systematicity, generality, purity, etc. (Avigad 2006, also Haffner 2014, 2015, 2017). Another is by focusing on the kind of reasoning involved: "conceptual" or "structural" reasoning (Stein 1988, Ferreirós 1999), or in terminology often associated with Hilbert, "axiomatic" reasoning (Sieg & Schlimm 2005, 2017, Schlimm 2000, 2011, 2017). We might even want to talk about a novel "reasoning style" in this connection, where "style" is to be understood in an epistemological sense, not in an aesthetic, psychological, or sociological sense. Such a style brings with it, not just new theorems and proofs, but a distinctive kind of "understanding" of mathematical facts (Reck 2009, cf. also Tappenden 2005b, Müller-Stach 2017).

These attempts to reveal the full significance of Dedekind's approach are initial forays and in need of further elaboration. However, one aspect seems clear enough: The methodological and epistemological perspective that shapes Dedekind's mathematical works is not independent of the logical and metaphysically views emerging in his more foundational writings. This is no accident; if one adopts his methodological stance, it is hardly possible to hold on to narrowly computational, formalist, empiricist, or intuitionist views about mathematics. More specifically, a structuralist methodology along Dedekindian lines calls for a structuralist metaphysics, as two sides of the same coin. Dedekind seems to have been keenly aware of this fact, even if he did not elaborate it much in his writings.

We started out by considering Dedekind's contributions to the foundations of mathematics in his overtly foundational writings: the two booklets, Stetigkeit und irrationale Zahlen (1872) and Was sind und was sollen die Zahlen? (1888). We then added a sketch and preliminary analysis of the methodological innovations in his more mainstream mathematical writings, from his work in algebraic number theory to other areas. We also noted that the logical and metaphysical position in his foundational work is closely tied to his more general methodological and epistemological perspective. In this sense, any attempt to distinguish sharply between his "foundational" and his other "mathematical" writings is misguided; Dedekind made foundationally relevant contributions throughout his work. In fact, his case provides a good argument and illustration for a more general lesson. Namely, any strict distinction between "foundational" or "philosophical" questions about mathematics, on the one hand, and "inner-mathematical" or "practice-based" questions, on the other hand, is counterproductive in the end, especially if one does not want to impoverish both sides.

If we look at Dedekind's contributions from such a perspective, the sum total is impressive. He was not only one of the greatest mathematicians of the nineteenth century, but also one of its subtlest philosophers of mathematics. With his structuralist views about the nature of mathematical entities and about how to investigate them, he was far ahead of his time. He was even ahead of much current philosophy of mathematics, especially with his sensitivity to both sides. This is not to say that his position is without problems. Dedekind himself was troubled by the set-theoretic antinomies, especially initially. And the twentieth century produced additional surprises, such as Gödel's Incompleteness Theorems, that are difficult to accommodate for everyone. Moreover, the methodology of mathematics has developed further since his time, including attempts to reconcile and integrate "conceptual" and "computational" thinking, e.g., in computer-assisted proofs. Then again, is there a philosophical position available today that answers all important questions about mathematics in a satisfactory way? If not, updating and reviving a Dedekindian position may be a worthwhile project.

Works listed in this section are by Richard Dedekind, unless otherwise specified.

1854

"Über die Einführung neuer Funktionen in der Mathematik; Habilitationsvortrag"; in Dedekind (1930-32), Vol. 3, pp. 428-438.

1857

"Abriß einer Theorie der höheren Kongruenzen in bezug auf einen reellen Primzahl-Modulus"; reprinted in Dedekind (1930-32), Vol. 1, pp. 40-67.

1876a

"Bernhard Riemanns Lebenslauf", in Riemann (1876), pp. 539-558.

1876b

"Briefe an Lipschitz (1-2)", in Dedekind (1930-32), Vol. 3, pp. 468-479.

1882

"Theorie der algebraischen Funktionen einer Veränderlichen", with H. Weber; reprinted in Dedekind (1930-32), Vol. 1, pp. 238-350; English trans. Dedekind (2012).

1888b

"Brief an Weber", in Dedekind (1930-32), Vol. 3, pp. 488-490.

1890

"Letter to Keferstein", in Van Heijenoort (1967), pp. 98-103; German original, with editorial introduction, in Scheel et al. (2017), pp. 205-211.

1897

"Über Zerlegungen von Zahlen durch ihre größten gemeinsamen Teiler"; reprinted in Dedekind (1930-32), Vol. 2, pp. 103-147.

1900

"Über die von drei Moduln erzeugte Dualgruppe"; reprinted in Dedekind (1930-32), Vol. II, pp. 236-271.

1901b

"Continuity and Irrational Numbers", in (Dedekind 1901a), pp. 1-27; English trans. of Dedekind (1872).

1901c

"The Nature and Meaning of Numbers", in (Dedekind 1901a), pp. 29-115; English trans. of Dedekind (1888a).

1981

"Eine Vorlesung über Algebra", in Scharlau (1981a), pp. 59-100.

1996a

"Julius Wilhelm Richard Dedekind (1831-1916)", in Ewald (1996a), Vol. 2, pp. 753-837; English trans. of various texts by Dedekind.

2017

Was sind und was sollen die Zahlen? Stetigkeit und Irrationale Zahlen, S. Müller-Stach, ed., Berlin: Springer.

Alten, H.-W., et al. (eds.), 2003, 4000 Jahre Algebra, Berlin: Springer; especially Chapter 9, "Algebra an der Wende zum 20. Jahrhundert", by K.-H. Schloteet al., pp. 475-550.

Avigad, Jeremy, 2006, "Methodology and Metaphysics in the Development of Dedekind's Theory of Ideals", in The Architecture of Modern Mathematics, J. Ferreirós & J. Grey (eds.), Oxford: Oxford University Press, pp. 159-186.

Awodey, S. & Reck, E., 2002, "Categoricity and Completeness, Part I: Nineteenth-Century Axiomatics to Twentieth-Century Metalogic", History and Philosophy of Logic, 23: 1-30.

Bell, E.T., 1937, Men of Mathematics; reprinted, New York: Simon & Schuster, 1965; especially Chapter 27, "Arithmetic the Second: Kummer (1810-1893), Dedekind (1831-1916)", pp. 510-525.

Belna, Jean-Pierre, 1996, La Notion de nombre chez Dedekind, Cantor, Frege: Théories, conceptions, et philosophie, Paris: Vrin.

Benis-Sinaceur, Hourya, 2008, Richard Dedekind, la creation des nombres, Paris: Vrin

---, 2015, "Is Dedekind a Logicist? Why Does Such a Question Arise?", in Benis-Sinaceur, Panza & Sandu (2015), pp. 1-58.

---, 2017, "Dedekind's and Frege's Views on Logic", in Scheel et al. (2017), pp. 50-62.

Benis-Sinaceur, H., Panza, M. & Sandu, G. (eds.), 2015, Functions and Generality of Logic: Reflections on Dedekind's and Frege's Logicisms, Berlin: Springer.

Biermann, Kurt R., 1971, "Dedekind", in Dictionary of Scientific Biography, Vol. 4, C.C. Gillespie (ed.), pp. 1-5.

Boolos, George, 1990, "The Standard of Equality of Numbers", in Meaning and Method: Essays in Honor of Hilary Putnam, G. Boolos (ed.), pp. 261-277; reprinted inLogic, Logic, and Logic, Cambridge, MA: Harvard University Press, 1998, pp. 202-219.

Bordogna, Francesca, 1996, "Interpreting the Ideal: Embedding Ideal Numbers in the Mathematical Programs of Kummer, Dedekind, and Klein", Max Planck Institute for the History of Science, Berlin.

Boyer, C. & Merzbach, U., 1991, A History of Mathematics, second edition (revised), Wiley & Sons: New York; especially Chapter 25, "Analysis", pp. 553-574, and Chapter 26, "Algebra", pp. 575-598.

Burris, Stanley, 1983, "Boolean Constructions", in Universal Algebra and Lattice Theory, Proceedings, Puebla 1982 (Springer Lecture Notes: Volume 1004), Berlin: Springer, pp. 67-90.

Cassirer, Ernst, 1907, "Kant und die moderne Mathematik ", Kant Studien 12, pp. 1-49; reprinted inGesammelte Werke, Band 9, Hamburg: Meiner, 2001, pp. 37-82.

---, 1910, Substanzbegriff und Funktionsbegriff, Berlin: Bruno Cassierer; English trans.,Substance and Function: Chicago: Open Court, 1923.

Cooke, Roger, 2005, "Richard Dedekind, Stetigkeit und irrationale Zahlen (1872)", Chapter 43 of Landmark Writings in Western Mathematics, 1640-1940, I. Grattan-Guinness (ed.), Amsterdam: Elsevier, pp. 553-563.

Corfield, David, 2003, Towards a Philosophy of Real Mathematics, Cambridge University Press; especially Chapter 8, "Beyond the Methodology of Mathematical Research Programmes", pp. 175-203.

Corry, Leo, 2004, Modern Algebra and the Rise of Mathematical Structures, second edition (revised), Birkhäuser: Boston; especially Chapter 2: "Richard Dedekind: Numbers and Ideals", pp. 66-136.

---, 2015, A Brief History of Numbers, Oxford: Oxford University Press

---, 2017, "Dedekind and Noether. Steht alles schon bei Dedekind? Ideals and Factorization between Dedekind and Noether", in Scheel et al. (2017), pp. 134-159.

Demopoulos, W. & Clark, P., 2005, "The Logicism of Frege, Dedekind, and Russell", in The Oxford Handbook of Philosophy of Mathematics and Logic, S. Shapiro (ed.), Oxford: Oxford University Press, pp. 166-202.

Detlefsen, Michael, 2011, "Dedekind against Intuition: Rigor, Scope, and the Motives of his Logicism", in Logic and Knowledge, C. Cellucci et al. (eds.), Cambridge: Cambridge Scholars Publishing, pp. 205-217

Dieudonné, Jean (ed.), 1985, Geschichte der Mathematik 1700-1900, Berlin: Deutscher Verlag der Wissenschaften, 1985; especially Chapter 5, "Zahlentheorie", by W. & F. Ellison, pp. 171-358, and Chapter 6, "Grundlagen der Analysis", by P. Dugac, pp. 359-421; originally published in French, 1978.

Dugac, Pierre, 1976, Richard Dedekind et les fondements des mathématiques, Paris: Vrin; with selections from Dedekind'sNachlass.

Dummett, Michael, 1995, Frege: Philosophy of Mathematics, Cambridge, MA: Harvard University Press.

Edwards, Harold M., 1977, Fermat's Last Theorem. A Genetic Introduction to Algebraic Number Theory, New York: Springer.

---, 1980, "The Genesis of Ideal Theory", Archive for History of Exact Sciences, 23: 321-378.

---, 1983, "Dedekind's Invention of Ideals", Bulletin of the London Mathematical Society, 15: 8-17.

---, 1990, Divisor Theory, Boston: Birkhäuser.

Ewald, William B. (ed.), 1996a, From Kant to Hilbert. A Source Book in the Foundations of Mathematics, Vols. 1-2, Oxford: Oxford University Press.

---, 1996b, Editor's introduction to Dedekind (1996a), pp. 753-754.

Feferman, S. & Hellman, G., 1995, "Foundations of Predicative Arithmetic", in Philosophical Logic, 24: 1-17.

Ferreirós, José, 1993, "On the Relation between Georg Cantor and Richard Dedekind", Historia Mathematica, 20: 343-363.

---, 1995, "'What Fermented in Me for Years': Cantor's Discovery of Transfinite Numbers", Historia Mathematica, 22: 33-42.

---, 1996, "Traditional Logic and the Early History of Set Theory, 1854-1908", Archive for History of Exact Sciences, 50: 5-71.

---, 1999, Labyrinth of Thought: A History of Set Theory and its Role in Modern Mathematics, Boston: Birkhäuser; especially: Chapter 3, "Dedekind and the Set-theoretic Approach to Algebra", pp. 81-116; Chapter 4, "The Real Number System", pp. 117-144; and Chapter 7, "Sets and Maps as a Foundation for Mathematics", pp. 215-255; second edition (with added postscript), 2007.

---, 2005, "Richard Dedekind (1888) and Giuseppe Peano (1889), Booklets on the Foundations of Arithmetic", Chapter 47 of Landmark Writings in Western Mathematics, 1640-1940, I. Grattan-Guinness (ed.), Amsterdam: Elsevier, pp. 613-626.

---, 2007, "The Rise of Pure Mathematics, as Arithmetic in Gauss", in The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae, C. Goldstein, N. Schappacher & J. Schwermer (eds.), Berlin: Springer, pp. 235-268.

---, 2009, "Hilbert, Logicism, and Mathematical Existence", Synthese, 170: 33-70.

---, 2016, "The Early Development of Set Theory", Stanford Encyclopedia of Philosophy (Fall 2016 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/fall2016/entries/settheory-early/>.

---, 2017a, "Dedekind's Map-Theoretic Period", Philosophia Mathematica, 25(3): 318-340.

---, 2017b, "The Role of Abbildungen in Frege's Late Work", in Scheel et al. (2017), pp. 160-173.

---, forthcoming, "On Dedekind's Logicism", in Analytic Philosophy and the Foundations of Mathematics, A. Arana & C. Alvarez (eds.), London: Palgrave.

Ferreirós, J. & Reck, E., 2020, "Dedekind's Mathematical Structuralism: From Galois theory to Numbers, Sets, and Functions", in Reck & Schiemer (2020), pp. 59-87.

Floyd, Juliet, 2013, "The Varieties of Rigorous Experience", in The Oxford Handbook of the History of Analytic Philosophy, M. Beaney (ed.), Oxford: Oxford University Press, pp. 1003-1042.

Gardies, Jean-Louis, 1984, "Eudoxe et Dedekind", Revue d'histoire des sciences, 37: 111-125.

Geyer, Wulf-Dieter, 1981, "Die Theorie der algebraischen Funktionen einer Veränderlichen nach Dedekind und Weber", in Scharlau (1981a), pp. 109-133.

Gillies, D.A., 1982, Frege, Dedekind, and Peano on the Foundations of Arithmetic, Assen: Van Gorcum.

Grattan-Guinness, Ivo (ed.), 1980, From the Calculus to Set Theory 1630-1910. An Introductory History, Princeton: Princeton University Press; especially Chapter 6, "Developments in the foundations of mathematics, 1870-1920", by R. Bunn, pp. 220-255.

Gray, Jeremy, 1986, "The Nineteenth-Century Revolution in Mathematical Ontology", in Revolutions in Mathematics, D. Gillies (ed.), Oxford: Oxford University Press; second edition, 1992, pp. 226-248.

---, 2000, Linear Differential Equations and Group Theory, from Riemann to Poincaré, second edition, Boston: Birkhäuser; especially Chapter IV, "Modular Equations", pp. 101-140.

---, 2008, Plato's Ghost: The Modernist Transformation of Mathematics, Princeton: Princeton University Press; especially Chapter 3, "Mathematical Modernism Arrives", pp. 113-175.

---, 2018, A History of Abstract Algebra, Berlin: Springer; especially Chapters 17-19, on Dedekind's theories of ideals, pp. 195-216.

Haffner, Emmylou, 2014, "D'un point de vue rigoureux et parfaitement général: Pratique des mathématiques rigoureuses chez Richard Dedekind", Philosophia Scientiae, 18, 131-156.

---, 2015, "The Science of Number in Action in Richard Dedekind's Works: Between Mathematical Explorations and Foundational Investigations", Ph.D. Thesis, History and Philosophy of Mathematics, University of Paris/Diderot (Paris 7), HAL archives available online.

---, 2017, "Strategic Use(s) of Arithmetic in Richard Dedekind and Heinrich Weber's Theorie der algebraischen Funktionen einer Veränderlichen",Historia Mathematica, 44: 31-69.

Hallett, Michael, 1994, "Hilbert's Axiomatic Method and the Laws of Thought", in Mathematics and Mind, A. George (ed.), Oxford: Oxford University Press, pp. 158-198.

---, 1995, "Hilbert and Logic", in Québec Studies in the Philosophy of Science, 1: 135-187.

---, 2003, "Foundations of Mathematics", in The Cambridge History of Philosophy, 1870-1945, T. Baldwin (ed.), Cambridge: Cambridge University Press, pp. 128-154.

Harborth, H. et al. (eds.), 2007, Gedenkschrift für Richard Dedekind, Braunschweig: Appelhans Verlag.

Haubrich, Ralf, 1992, Zur Entstehung der algebraischen Zahlentheorie Richard Dedekinds, Ph.D. dissertation, University of Göttingen.

Hawking, Stephen (ed.), 2005a, God Created the Integers. The Mathematical Breakthroughs that Changed History, London: Running Press.

---, 2005b, Editor's introduction to "Richard Julius Wilhelm Dedekind (1831-1916). His Life and Work", in Hawking (2005a), pp. 901-905.

Hilbert, David, 1922, "Neubegründung der Mathematik: Erste Mitteilung", reprinted in Gesammelte Abhandlungen, Dritter Band, New York: Chelsea, pp. 157-177.

Jahnke, Hans Niels (ed.), 2003, A History of Analysis, American Mathematical Society; especially: Chapter 1, "Antiquity", by R. Thiele, pp. 1-40, and Chapter 10, "The End of the Science of Quantity: Foundations of Analysis, 1860-1910", by J. Lützen, pp. 291-324; originally published in German, 1999.

Jourdain, P.E.B., 1916, "Richard Dedekind (1833-1916)", The Monist, 26: 415-427; reprinted inModern Logic, 3 (1993): 207-214.

Kahle, Reinhard, 2017, "Von Dedekind zu Zermelo versus Peano zu Gödel", in Scheel et al. (2017), pp. 174-185.

Kitcher, Philip, 1986, "Frege, Dedekind, and the Philosophy of Mathematics", in Frege Synthesized, L. Haaparanta & J. Hintikka (eds.), Dordrecht: D. Reidel, pp. 299-343.

Klev, Ansten, 2011, "Dedekind and Hilbert on the Foundations of the Exact Sciences", The Review of Symbolic Logic, 4: 645-681.

---, 2017, "Dedekind's Logicism", Philosophia Mathematica, 25: 341-368.

---, 2018, "A Road Map to Dedekind's Theorem 66", HOPOS: The Journal of the International Society for the History of the Philosophy of Science, 8: 241-277.

Kolmogorov, A.N. & Yushkevich, A.P., 2001, Mathematics of the 19th Century. Vol. I: Mathematical Logic, Algebra, Number Theory, Probability Theory, second edition (revised), Birkhäuser: Boston; especially Chapter 2, Section 3, "The Theory of Algebraic Numbers and the Beginnings of Commutative Algebra", by I.G. Bashmakova & A.N. Rudakov, pp. 86-135; originally published in Russian, 1978.

Landau, Edmund, 1917, "Richard Dedekind", Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Geschäftliche Mitteilungen, 50-70.

Laugwitz, Detlef, 1999, Bernhard Riemann 1826-1866. Turning Points in the Conception of Mathematics, Boston: Birkhäuser; originally published in German, 1996.

Linnebo, Ø & Pettigrew, R., 2014, "Two Types of Abstraction for Structuralism", The Philosophical Quarterly, 64: 267-283.

McCarty, David, 1995, "The Mysteries of Richard Dedekind", in From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics, J. Hintikka (ed.), New York: Springer, pp. 53-96.

McLarty, Colin, 2006, "Emmy Noether's 'Set Theoretic' Topology: From Dedekind to the Rise of Functors", in The Architecture of Modern Mathematics, J. Ferreirós & J. Grey (eds.), Oxford: Oxford University Press, pp. 187-208.

Mehrtens, Herbert, 1979a, Entstehung der Verbandstheorie, Hildesheim: Gerstenberg; especially Chapter 2, "Die 'Dualgruppe' von Richard Dedekind", pp. 66-126.

---, 1979b, "Das Skelett der modernen Algebra. Zur Bildung mathematischer Begriffe bei Richard Dedekind", in Disciplinae Novae, C.J. Sciba (ed.), Göttingen: Vanderhoeck & Ruprecht, pp. 25-43.

---, 1982, "Richard Dedekind - Der Mensch und die Zahlen", Abhandlungen der Braunschweigischen Wissenschaftlichen Gesellschaft, 33: 19-33.

Merzbach, Utah, 2018, Dirichlet: A Mathematical Biography, Birkhäuser: Basel.

Mueller, Ian, 1981, Philosophy of Mathematics and Deductive Structure in Euclid's Elements, Cambridge, Massachusetts: MIT Press, especially Chapter 3, "Magnitudes in Proportion", pp. 118-151.

Müller-Stach, Stefan, 2017, "Richard Dedekind: Style and Influence", in Scheel et al. (2017), pp. 26-35.

Potter, Michael, 2000, Reason's Nearest Kin: Philosophies of Arithmetic from Kant to Carnap, Oxford: Oxford University Press; especially Chapter 3, "Dedekind", pp. 81-104.

Reck, Erich, 2003, "Dedekind's Structuralism: An Interpretation and Partial Defense", Synthese, 137: 369-419.

---, 2009, "Dedekind, Structural Reasoning, and Mathematical Understanding", in New Perspectives on Mathematical Practices, B. van Kerkhove (ed.), Singapore: World Scientific, pp. 150-173.

---, 2013, "Frege, Dedekind, and the Origins of Logicism", History and Philosophy of Logic, 34: 242-265.

---, 2017, "Dedekind as a Philosopher of Mathematics", in Scheel et al. (2017), pp. 36-49.

---, 2018, "On Reconstructing Dedekind Abstraction Logically", in Logic, Philosophy of Mathematics, and their History: Essays in Honor of W.W. Tait, E. Reck (ed.), London: College Publications, pp. 113-138.

---, 2019, "The Logic in Dedekind's Logicism", in Logic from Kant to Russell: Laying the Foundations for Analytic Philosophy, S. Lapointe (ed.), Routledge: London, pp. 171-188.

---, 2020, "Cassirer's Reception of Dedekind and the Structuralist Transformation of Mathematics", in Reck & Schiemer (2020), pp. 329-351.

Reck, E. & Price, M., 2000, "Structures and Structuralism in Contemporary Philosophy of Mathematics", Synthese, 125: 341-383.

Reck, E. & Schiemer, G. (eds.), 2020, The Prehistory of Mathematical Structuralism, Oxford: Oxford University Press.

Reed, David, 1995, Figures of Thought, Routledge: London; especially Chapter 4, "Number Theory in the Nineteenth Century", pp. 76-116.

Scharlau, Winfried (ed.), 1981a, Richard Dedekind, 1831/1981: Eine Würdigung zu seinen 150. Geburtstag, Braunschweig: Vieweg.

---, 1981b, "Erläuterungen zu Dedekinds Manuskript über Algebra", in Scharlau (1981a), pp. 101-108.

Scheel, K., Sonar, T. & Ullrich, P. (eds.), 2017, In Memoriam Richard Dedekind (1831-1916). Number Theory-Algebra-Set Theory-History-Philosophy, WTM Verlag: Münster.

Schlimm, Dirk, 2000, Richard Dedekind: Axiomatic Foundations of Mathematics, M.A. thesis, Carnegie Mellon University, Pittsburgh; with selections from Dedekind'sNachlass.

---, 2011, "On the Creative Role of Axiomatics: The Discovery of Lattices by Schroeder, Dedekind, Birkhoff, and Others", Synthese, 183, 47-68.

---, 2017, "On Dedekind's Axiomatic Approach to the Foundations of Mathematics", in Scheel et al. (eds.), pp, 75-82.

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# data[].attributes.attachment_link.title
Dedekind's Contributions to the Foundations of Mathematics

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Richard Dedekind was born in Brunswick (Braunschweig), a city in northern Germany, in 1831. Much of his education took place in Brunswick as well, where he first attended school and then, for two years, the local technical university. In 1850, he transferred to the University of Göttingen, a main center for scientific research in Europe.

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10ºAno Funções Vídeo 00: Funções - um pouco de história