Thursday, January 31, 2019

Blue Crab/ Siri Azul (Torres, Brazil, Jan 2019) & the World Food Club (as they call it)







Torres, Rio Grande do Sul/Brazil, Jan 2019 (for more pictures see here);
Soup & Talk 2020: Vandana Shiva "Poison-free Food and Farming 2030"; 
Colin Tudge talk;
Colin Tudge interview;
A Message from the Future with Alexandria Ocasio-Cortez (The Intercept);
Greta Thunberg & other young activists (Dazed);

Preamble &/or how different things could & should be (which in no way becomes me to decide): 


"Sabia que degollar es una especialidad gaucha?"
El periodista miope (Vargas Llosa, La guerra del fin del mundo)

"The phrase, 'Be kind to Athos,' refers to Bloom's father's dog—and kindness to animals, who are so much like children, and can repay affection only with affection, is another of those quite ordinary and undistinguished aspects of human nature that Joyce underlines. Even the Citizen, like Homer's Cyclops, is good to Garryowen. The kindness of Bloom on June 16, 1904, begins with animals and ends with human beings. So he feeds his cat in the morning, then some sea gulls, and in the Circe episode a dog..."
Richard Ellmann

"— C'est que, dit le petit, le garde me mettrait en prison, s'il trouvait dans mes fagots du bois vivant... Et puis, quand j'ai voulu le faire, comme vous me l'aviez dit, j'entendais l'arbre qui se plaignait.
— C'est comme moi, dit la petite fille, quand j'emporte des poissons dans mon panier, je les entends qui chantent si tristement, que je le rejette dans l'eau... Alors on me bat chez nous!"
Gérard de Nerval (La Reine des Poissons)

"Os tiros ecoavam por todos os lados. Horrorizado, o garoto de quinze anos assistia a uma matança de macacos, que ia deixando um rastro de sangue e de animais agonizantes. Os adultos atiravam simplesmente para treinar a pontaria e pelo prazer de matar, abatendo tudo que passasse pela frente, mesmo que fossem fêmeas ainda carregando os filhotes nas costas. Na cabeça do adolescente, o convite para participar de uma caçada significava, até então, a possibilidade de matar um animal para comer, e não o imenso pesadelo em que aquilo se transformou... 'Naquele momento, alguma coisa aflorou dentro de mim definitivamente, e eu tive consciência do desvario humano de matar animais, e de que não compactuaria com aquela atitude. O estado de choque em que fiquei me fez procurar uma maneira harmônica de conviver com a natureza.'"
Denise Pires Vaz & Ney Matogrosso (Um cara meio estranho)
"Quando íamos de Jeep para o Guarujá, meu pai sempre dava uma paradinha nos vendedores de caranguejo no fim da Anchieta. Os bichos pendurados por um barbante balançavam as perninhas num frenesi alucinado, implorando por água, um horror. Comprava duas fileiras e, chegando em casa, lavava os cascos deles ainda vivos, mergulhava todos num caldeirão com água e fechava a tampa com um peso em cima. Dava para ouvir os bichinhos se debatendo no fogo do inferno, tadinhos. Quando o barulho cessava, imediatamente minha aflição em querer salvá-los dava lugar à gula em comê-los. Charles me ensinou a técnica de dissecar caranguejos: 'Isso é o pulmão, tem gente que come, eu não recomendo. Remova o intestino debaixo da água corrente, quebre a casca com cuidado para não sobrar uma lasquinha da cartilagem e machucar sua gengiva. As patolas e as perninhas a gente quebra, dá um chupão no buraquinho e a carne solta.'"
"Pois no exato dia da mudança de volta a São Paulo, depois de três meses em Ibiúna, vou carregar o Jeep, e quem está no capô tomando sol? Sim, Mouchie & Angel, as benditas cobras do maldito Alice Cooper. As duas me esperavam lá, tipo irmãs boazinhas do Jardim do Éden. 'As férias foram ótimas, mas queremos morar com você, mamãe.'"
Rita Lee

"Ces Indians répartis de part de d'autre de la frontière entre l'Équateur et le Pérou ne se distinguent guère des autres tribus de l'ensemble jivaro, auquel ils se rattachent par la langue et la culture, lorsqu'ils disent que la plupart des plantes et des animaux possèdent une âme (wakan) similaire à celle des humains, une faculté qui les range parmi les 'personnes' (aents) en ce qu'elle leur assure la conscience réflexive et l'intentionnalité, qu'elle les rend capables d'éprouver des émotions et leur permet d'échanger des messages avec leurs pairs comme avec les membres d'autres espèces..."
"La forme visible des animaux n'est en effet qu'un déguisement. Lorsqu'ils regagnent leurs demeures, c'est pour se dépouiller de leur apparence, revêtir parures de plumes et ornements cérémoniels, et redevenir de manière ostensible les 'gens' qu'ils n'avaient pas cessé d'être lorsqu'ils ondoyaient dans les rivières et fourrageaient dans la forêt."
"Si les animaux diffèrent des hommes, c'est donc uniquement par l'apparence, une simple illusion des sens puisque les enveloppes corporelles distinctives qu'ils arborent d'ordinaire ne sont que des déguisements destinés à tromper les Indiens. Lorsqu'ils visitent ces derniers en rêve, les animaux se révèlent tels qu'ils sont en réalité... il faut éviter le gâchis, tuer proprement et sans souffrances inutiles, traiter ave dignité les os et la dépouille, ne pas céder aux tartarinades ni même évoquer trop clairement le sort réservé aux proies."
"Comme le confiait le chamane Ivaluardjuk à Rasmussen, 'le plus grand péril de l'existence vient du fait que la nourriture des hommes est tout entière faites d'âmes.'"
Philippe Descola
"In einem solchen Zustande empfand ich einmal die Nähe einer Kuhheerde durch Wiederkehr milderer, menschenfreundlicherer Gedanken, noch bevor ich sie sah: Das hat Wärme in sich..."
Nietzsche

"In the past, most scientific research was carried out by amateurs... Charles Darwin, for example, never held any institutional post; he worked independently at his home in Kent, studying barnacles, writing, keeping pigeons, and doing experiments in the garden with his son Francis... There are now only a handful of independent scientists, the best known being James Lovelock, the leading proponent of the Gaia hypothesis... And although amateur naturalists and freelance inventors still exist, they have been marginalized."
Rupert Sheldrake
"I think having land and not ruining it is the most beautiful art that anybody could ever want to own."
"There should be supermarkets that sell things and supermarkets that buy things back, and until that equalizes, there'll be more waste than should be... People should be able to sell their old cans, their old chicken bones, their old shampoo bottles, their old magazines. We have to get more organized."
The Philosophy of Andy Warhol
"In der That, ich habe bis zu meinen reifsten Jahren immer nur schlecht gegessen, — moralisch ausgedrückt 'unpersönlich', 'selbstlos', 'altruistisch', zum Heil der Köche und andrer Mitchristen."
Nietzsche

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About the blue crabs in the pictures &/or how things really are (dejection, affliction, villainous affairs):


Ordinary people (in their subacid humour towards others) simply kill these animals as they come grabbed to their fishing nets. It doesn't look right to me. They kill them allot, during the Summer, while dragging their nets along the beach. Perhaps boats do even worse and we just don't see. 
"No Brasil, de modo geral, C. sapidus não constitui espécie-alvo de grandes pescarias, sendo basicamente explorada pela pesca artesanal no interior dos estuários, principalmente nas regiões sudeste e sul, bem como nas regiões costeiras como fauna acompanhante da pesca de arrasto de camarões. Geralmente os indivíduos capturados são devolvidos, já mortos, ao mar. Em alguns locais a espécie declinou consideravelmente, mas não existem dados suficientes para estimar o impacto das capturas na população de C. sapidus. Coleta de informações de captura e pesquisas voltadas a biologia pesqueira da espécie bem como o conhecimento do nível de degradação das áreas onde habita são necessárias para a alteração da condição de Dados Insuficientes (DD)," Avaliação do risco de extinção dos crustáceos no Brasil: 2010-2014 (ICMBIO);
"Nos Estados Unidos há só para o estudo e controle desta pescaria, uma série de comissões, leis, conferências, revistas científicas, livros, etc. Isto mostra a importância atribuída a esse recurso pesqueiro e a garantia da sua preservação," Siri azul do Atlântico americano: lixo ou tesouro do mar?

See also:
https://www.bluecrab.info
And also:
Environmental Issue (Brazil);

[***Obliquity out of the common track: one of the best quality things people living in cities such as Porto Alegre (capital of Rio Grande do Sul, South of Brazil) still have access to, besides alternative movie theatres, are the agroecological markets—relatively big ones, such as the "Feira Agroecológica do Bom Fim" (taking place Saturday mornings at the Av. José Bonifácio) and small ones, such as the "Feira Agroecológica da Travessa dos Lanceiros Negros" (Tuesday mornings). There you can buy a variety of organic fruits and vegetables direct from small producers for very affordable prices. The beach of Torres has an agroecological market as well, "Ecotorres" (Av. Gen. Osório, 158).]
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Other outrages against sense and decency:  


"Tereza Cristina não parece ser a ministra da Agricultura, mas do Envenenamento dos Consumidores. Nos primeiros 50 dias do novo governo, 54 novos agrotóxicos foram aprovados. O Ministério da Agricultura deu sinal verde para que novos fabricantes possam comercializá-los, e que novas combinações químicas entre eles sejam permitidas. Por causa de sua alta toxicidade, alguns desses produtos são proibidos no exterior. Mas isso parece ser secundário, desde que o agronegócio esteja bombando."
"Em outros países do mundo, procuram-se meios de plantar alimentos de forma mais saudável. Na Alemanha, até 2030, os orgânicos deverão ser plantados sobre 20% das terras agrícolas. Ao mesmo tempo, o mercado de alimentos orgânicos registra crescimento constante ano por ano – só em 2018, foram 5,5%." 
"Não é o caso do Brasil. A Fundação Oswaldo Cruz analisa 30 alimentos regularmente. Em algumas amostras, é possível encontrar até 15 agrotóxicos diferentes. E o que o legislador faz? Nada. No Congresso circula o chamado 'pacote do veneno', que almeja a aprovação de mais agrotóxicos."
"A catástrofe ambiental no Rio Grande do Sul passou quase despercebida pela opinião pública. A Associação dos Apicultores Gaúchos contabiliza a perda de 6 mil colmeias, inviabilizando a entrega de 150 toneladas de mel. Em 80% das análises das abelhas mortas, foi constatado algum tipo de agrotóxico presente. Nem a ministra da Agricultura, Tereza Cristina, e muito menos o ministro do Meio Ambiente, Ricardo Salles, se pronunciaram sobre o caso." 
"Com frequência, os inseticidas chamados de neonicotinoides são responsáveis pela morte das abelhas. Na Europa, os neonicotinoides são proibidos. No Brasil, não. A indústria agrícola é contra. Mais um produto licenciado recentemente é o Sulfoxaflor, igualmente conhecido por envenenar abelhas."
'At the beginning of this week the country was second only to the US with 1.88 million confirmed Covid-19 cases and 72,833 deaths. Its powerful agribusiness sector is allied with the country’s far-right president, Jair Bolsonaro, who has dismissed the pandemic as a “little flu”. The beef sector is worth $26bn (£20.7bn), according to the Brazilian Confederation of Agriculture and Livestock (CNA), while its chicken industry is worth another $8bn. Meat plants have stayed open during the pandemic, and staff work closely together, often in refrigerated areas. Other countries, including the US, Canada, Ireland and Germany, have also seen clusters around slaughterhouses...' 
'At a JBS plant in Dourados, in Mato Grosso do Sul state in the centre-west region, more than 4,000 employees were tested and nearly a quarter were positive, prosecutors said. The company suspended 1,600 workers on full pay but did not close the plant. As of 14 July, the town had 3,481 cases, a quarter of the state’s total.  The JBS plant in Dourados “was the initial focus for the outbreak”, said Andyane Tetila, an infectious diseases specialist in Dourados who works for the state health service. The JBS plant has 103 indigenous workers, many of whom live in  nearby reserves where more than 150 people were subsequently infected, said Indianara Machado, an indigenous nurse who works in the reserve.'
'Undercover footage at the Laboratory of Pharmacology and Toxicology (LPT) near Hamburg, published by Cruelty Free International and Soko Tierschutz, shows technicians with metal prongs grabbing macaque monkeys by the neck. The monkeys are restrained by braces during testing. The footage also shows primates being handled “violently” by technicians: in one incident a monkey has its head smacked against a door frame...'
'Some of the monkeys appeared to be kept alone in metal cages measuring less than a cubic metre and are seen spinning in circles, indicating high levels of distress. They were reportedly forced to stand for long periods. Dogs are pictured laying in what seemed to be their own blood and faeces, with one beagle in a cage appearing to be bleeding. Staff also appeared to mishandle cats.'
Barbaric tests on monkeys lead to calls for closure of German lab (The Guardian);
'Although recent reporting on deforestation has focused heavily on the Amazon, the fact is that the other ancient forests of South America – and elsewhere in the world – are disappearing too. These trees are probably falling to make way for a crop sometimes known as the “wonder bean” or even the “Cinderella bean”: soya. As the plane carries on, the huge soya fields that now patchwork this area unfold beneath us, small strips of forest still clinging on around their margins.'
'The discovery of a stable, cheap source of protein might have been a miracle for farmers – 75% of the world’s soya and maize is now fed to farm animals – but this monoculture is spreading over huge expanses of the Americas, and wiping out forests, wilderness and species as it goes. Soya is one of the four main culprits for deforestation (along with beef, wood and palm oil) and biodiversity loss as farmers clear land to grow this profitable oilseed.'
'... deforestation rate in Argentina is one of the highest in the world. According to Nasa’s Earth Observatory, 20% of the Gran Chaco’s forest, 55,000 square miles – an area larger than England – was lost between 1985 and 2016... The UK and EU don’t grow much soya and so the EU now imports about 15m tonnes of un-ground soya beans, and about 19m tonnes of crushed meal, while in 2017 the UK alone imported 2m tonnes of oilcake. The biggest importer by far however is China, the one-time home of soy. In 2017 China imported a breathtaking 96m tonnes of soya beans. For the exporters, America, Brazil and Argentina, soya is big business.'
- Rise of the 'wonder bean': from deforestation to your plate (The Guardian);
"In 2009, Walmart, Nike and other global companies vowed to stop buying beef and leather from Brazilian companies operating in the Amazon. They were responding to pressure from the environmental group Greenpeace, which had determined that cattle ranching there had become the largest driver of deforestation in the world, with an average of one acre of the Amazon cleared every eight seconds for grazing."
"... a decade later, the Amazon is in even graver danger, with 17% of its forests already gone and some scientists warning that losing as little as 3% more could begin turning it to savanna because the ecosystem will produce too little rainfall to sustain itself."
"Brazil produces more beef than any other country except the United States and exports more than anywhere else, sending 20% of its production to Hong Kong, China, the European Union and several smaller buyers.  The Brazilian company JBS, the world’s largest meatpacker with more than $50 billion in annual revenue, counts Walmart and Costco as major clients."
- Cows are killing the Amazon. Pledges from Walmart and Nike didn’t help save it (Los Angeles Times);
"Hotspots of antibiotic-resistant superbugs are springing up in farms around the world, the direct result of our overconsumption of meat, with potentially disastrous consequences for human health, a study has found."
"The scientists, reporting on their work in the peer review journal Science, said there was a “window of opportunity” to limit the rise of resistant bacteria “by encouraging a transition to sustainable animal farming practices” around the world, particularly in the countries highlighted."
"The rise of superbugs that are untreatable by even the strongest antibiotics is one of the greatest threats the world faces, according to the UK’s outgoing chief medical officer, Sally Davies, who warned of an “apocalyptic” threat and the end of modern medicine."
- "Superbugs hotspots emerging in farms across globe" (The Guardian);
"Esta era a mensagem que a procuradora da República em Altamira Thais Santi tentava passar aos jornalistas. Os incêndios são graves e devem ser denunciados e combatidos, mas é necessário compreender também que um rio está morrendo. Morrendo. “É ecocídio, e é genocídio”, ela afirma. A procuradora não exagera. Os fatos são eloquentes, investigados e mensurados pelos melhores cientistas da área do Brasil, e também por documentos oficiais. Na história recente da Amazônia, a grande causadora e reprodutora de violências na região do Médio Xingu, onde está a cidade de Altamira, foi e segue sendo a Usina Hidrelétrica de Belo Monte."
"... É também nesta região que, nos últimos anos, outra gigante, a mineradora canadense Belo Sun, pressiona a população local e assedia políticos de Belém para obter autorização para explorar aquela que seria a maior mina de ouro a céu aberto do Brasil – e também o sepultamento oficial da Volta Grande embaixo de toneladas de rejeitos tóxicos."
"Since President Jair Bolsonaro took office in January, Brazil has permitted sales of a record 290 pesticides, up 27% over the same period last year, and a bill in Congress would relax standards even further... About 40% of Brazil’s pesticides are “highly or extremely toxic,” according to Greenpeace, and 32% aren’t allowed in the European Union. Meanwhile, approvals are being expedited without the government hiring enough people to evaluate them, said Marina Lacorte, a coordinator at Greenpeace Brazil... The fertile nation is awash in chemicals. Brazil’s pesticide use increased 770% from 1990 to 2016, according to the Food and Agriculture Organization of the United Nations." 
"Desses, 16 são classificados pela Anvisa como extremamente ou altamente tóxicos e 11 estão associados ao desenvolvimento de doenças crônicas como câncer, malformação fetal, disfunções hormonais e reprodutivas. Entre os locais com contaminação múltipla estão as capitais São Paulo, Rio de Janeiro, Fortaleza, Manaus, Curitiba, Porto Alegre, Campo Grande, Cuiabá, Florianópolis e Palmas...."
"Os números revelam que a contaminação da água está aumentando a passos largos e constantes. Em 2014, 75% dos testes detectaram agrotóxicos. Subiu para 84% em 2015 e foi para 88% em 2016, chegando a 92% em 2017...."
"Do total de 27 pesticidas na água dos brasileiros, 21 estão proibidos na União Europeia devido aos riscos que oferecem à saúde e ao meio ambiente..."
"... somando todos os limites permitidos para cada um dos agrotóxicos monitorados, a mistura de substâncias na nossa água pode chegar a 1.353 microgramas por litro sem soar nenhum alarme. O valor equivale a 2.706 vezes o limite europeu..." 
Coquetel perigoso Levantamento aponta que 1 a cada 4 cidades brasileiras tem água contaminada por 27 tipos de agrotóxicos (Ana Aranha & Luana Rocha, UOL Notícias);
"“When I was 16, I used to bring a boat here with my uncle,” Ollivro said. “In those days, it was all about natural beauty and you didn’t see seaweed piled up. It’s a shame this place has come to be associated with death.”"
"For decades, potentially lethal green algae have amassed in shallow bays on Brittany’s beautiful north-western coast. Environmentalists say the blossoming of unusually large amounts of green algae are linked to nitrates in fertilisers and waste from the region’s intensive pig, poultry and dairy farming flowing into the river system and entering the sea. When the algae decompose, pockets of toxic gas (hyrogen sulfide) get trapped under its crust — potentially fatal to humans if they step on it."
"This summer, six Brittany beaches were closed because of a mass of dangerous seaweed. The bay of Saint-Brieuc was the focus, with bulldozers piling so much algae into dumper trucks on the beach that an inland treatment centre, where seaweed is dried out and disposed of, briefly closed due to an unbearable stench. The centre blamed the foul odour on the method used to collect the algae, which had mixed in mud and sand. Local residents complained the smell was so bad it woke them up at night."
"Jean-René Auffray, 50, was fit and preparing for a long-distance race when he set out on an afternoon jog from his home near the beach in Hillion. His dog returned alone and his wife and children went out to search for him. The area where he was found had already seen over 30 wild boar die in sludge five years before, with a likely link to rotting seaweed."
 - It can kill you in seconds': the deadly algae on Brittany's beaches (Angelique Chrisafis/The Guardian)

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the World Food Club:


"The world's food supply chain and hence the world at large are being seriously screwed up by the powers-that-be: governments, corporates, and the experts and intellectuals who advise and serve them (scientists, economists, lawyers, and professional bureaucrats known as MBAs)," Colin Tudge's The World Food Club.
"The aim of the WORLD FOOD CLUB is to replace the present food supply chain with something far better, controlled and answerable to people at large. The new food chain will be designed expressly to promote human wellbeing and cultural diversity, hugely to improve animal welfare, and to sustain and create landscapes that remain rich, diverse, and beautiful,Colin Tudge's The World Food Club.

Existing Initiatives:
- Soil Association;
- Slow Food International;
- Compassion in World Farming;
- Friends of the Earth;
- Fairtrade International;
- Forum for the Future;
- The Pari Center;
- Oxfordshire Community Land Trust;

"Pigs, sheep and goats all have "voices", distinguishing each individual from the herd, scientists have discovered in the past few years. This week, researchers at the University of Sydney added cows to the list... Taken together, the research suggests we need to think about barnyard animals less as a herd and more as a collection of individuals with their own characteristics (scientists in the field carefully avoid the word "personalities") and social bonds," "Here is the moos: Study finds cows' lows are unique way of saying 'hi'" (The Sidney Morning Herald)
"In the new report, released today, the European Food Safety Authority (EFSA) concluded that the welfare of rabbits is lower in conventional cages, compared to other systems. The key welfare issue for adult rabbits is that their movement is restricted. EFSA also concludes that organic systems are generally good," "EU Food Safety Agency Criticizes Rabbit Cages" (Compassion in World Farming)

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On the "Ecological Apocalipse" and how to avoid it: 


"For Isra, when it comes to discussing the environment, it’s crucial to talk about environmental racism – i.e., environmental and climate change issues that specifically affect black and brown communities. “There’s a pipeline being built in my state (Minnesota) and it’s going straight through indigenous treaty lands and sacred wild rice beds,” she explains. “This pipeline will destroy their water and their sacred wild rice and also just ruin their land. Some places are more privileged than others.” More than anything, we should make sure people “recognise their privilege in the movement”."
"As far as US Youth Climate Strike is concerned, Isra says, “we try and avoid things like sit-ins or a lot of marches, only because we deal with so many minors – we don’t want any of them to get arrested – especially with youth of colour. We try and focus on really safe forms of activism so everybody can participate.”"
"Everybody gets there at their own pace. It took me some time – it probably takes everybody some time. And whether or not you get media coverage or you get a great response it doesn’t mean what you’re doing is wrong. You should know what’s best for you and what you believe in, and nobody should be able to tell you any different.”"
"“It is a lot to handle, and it’s a full-time job,” says Isra, who is a high school student, juggling her school work along with her activism. “You have to make sure you balance time and that you get your priorities right.” Self-care is a must. “There are so many things that need to be tackled, but also you come first – make sure that you are OK enough to partake in that activism,”" "How to fight for climate action, according to Isra Hirsi" (Niellah Arboine/Dazed, 05/13/2019);
'The political class, as anyone who has followed its progress over the past three years can surely now see, is chaotic, unwilling and, in isolation, strategically incapable of addressing even short-term crises, let alone a vast existential predicament.'
'Even when broadcasters cover these issues, they carefully avoid any mention of power, talking about environmental collapse as if it is driven by mysterious, passive forces, and proposing microscopic fixes for vast structural problems. The BBC’s Blue Planet Live series exemplified this tendency.'
'Our system – characterised by perpetual economic growth on a planet that is not growing – will inevitably implode.'
'... for a peaceful mass movement to succeed, a maximum of 3.5% of the population needs to mobilise.'
'When a committed and vocal 3.5% unites behind the demand for a new system, the social avalanche that follows becomes irresistible,' "Only rebellion will prevent an ecological apocalypse" (George Monbiot, The Guardian, 04/15/2019);
'“We have shown that we are strong, we are determined,” she said. “I have never – I’ve been a police officer for 36 years – I have never known an operation, a single operation, in which over 700 people have been arrested.''Max Wedderburn, a 13-year-old from Milton Keynes, addressed crowds at the bridge and received cheers and applause. “We are getting bigger, we are getting stronger, we are gathering momentum. Together we can change the future,” he said.'
'Speaking afterwards, he said he became inspired to fight against climate change after learning about it from his mother. He explained: “My lifelong dream is to become a zoologist but I feel there is no point if half of the animals are all dead by the time I reach that goal.”'
'Around 20 activists, most of them aged under 17, had staged a brief demonstration on roads near Heathrow Airport on Friday morning and were arrested.
“One thing that is unusual about this demonstration is the willingness of those participating to be arrested and also their lack of resistance to the arrests,” said a spokesperson,' "Extinction Rebellion: More than 750 protesters arrested as climate change activists block London roads for sixth day" (Independent, 04/21/2019);

Tuesday, January 29, 2019

the most auspicious tetrahedron &/or two pairs of charming mathematicians of celestial extraction




Abel, Galois, Riemann, Clifford; 
Charles Fourier, Le reveur sublime: une vie une oeuvre (avec René Scherer et Simone Debout/France Culture, 2002); 
Jean-Pierre Serre & Alain Connes: Alexandre Grothendieck (Fondation Hugot du Collège de France/Youtube);

en exergue: accounts to reconcile, anecdotes to pick up, inscriptions to make out with the truth of the story orfracas (!):


"Ce qui m'a donné le plus de peine a été de toucher le fond au sujet de la conception de l'espace apportée par le cubisme: problème qui touche profondément non seulement à la métaphysique mais aux dernières conceptions de la physique: de Riemann, Einstein jusqu'à Jeans et Edington. J'ai fait là très attention et j'espère que le cubisme est bien montré en ce sens: qu'il fut une véritable révolution de la conception de l'espace dans la peinture, chose aussi importante sur le plan esthétique que le furent la Réforme et le Jansénisme sur le plan théologique."
André Masson (Lettre à Daniel-Henry Kahnweiler, 1939/Écrits, anthologie établie par Françoise Levaillant)
"Les notes musicales deviennent des nombre, et si votre esprit est doué de quelque aptitude mathématique, la mélodie, l'harmonie écoutée, tout en gardant son caractère voluptueux et sensuel, se transforme en une vaste opération arithmétique, où les nombres engendrent les nombres, et dont vous suivez les phases et la génération avec une facilité inexplicable et une agilité égale à celle de l'exécutant."
Baudelaire (Les Paradis artificiels)
"As the little world of abstract mathematicians is set a-quiver by some young Frenchman's deductions on the functions of imaginary values—worthless to applied science of the day—so is the smaller world of serious poets set a-quiver by some new subtlety of cadence. Why?"
Ezra Pound (The Wisdom of Poetry) 

"... Ceci montre déjà que notre étude relève de la théorie des permutations entre n éléments, théorie qui remonte à Lagrange et Galois, et qui a été poussée assez loin depuis."
André Weil (Sur l'étude algébrique de certains types de lois de mariage/Lévi-Strauss, Les Structures Élémentaires de la Parenté)

"Les travaux qui me valent la bienveillante attention de l’Académie royale datent d’il y a vingt-cinq ans, d’une époque où je faisait partie du milieu scientifique et où je partageais pour l’essentiel son esprit et ses valeurs. J’ai quitté ce milieu en 1970 et, sans renoncer pour autant à ma passion pour la recherche scientifique, je me suis éloigné intérieurement de plus en plus du milieu des scientifiques."
"Or, dans les deux décennies écoulées l’éthique du métier scientifique (tout au moins parmi des mathématiciens) s’est dégradée à un degré tel que le pillage pur et simple entre confrères (et surtout aux dépens de ceux qui ne sont pas en position de pouvoir se défendre) est devenu quasiment une règle générale, et qu’il est en tout cas toléré par tous, y compris dans les cas les plus flagrants et les plus iniques."
"Dans ces conditions, accepter d’entrer dans le jeu des prix et des récompenses serait aussi donner ma caution à un esprit et à une évolution, dans le monde scientifique, que je reconnais comme profondément malsains, et d’ailleurs condamnés à disparaître à brève échéance tant ils sont suicidaires spirituellement, et même intellectuellement et matériellement."
"C’est cette troisième raison qui est pour moi, et de loin, la plus sérieuse. Si j’en fais état, ce n’est nullement dans le but de critiquer les intentions de l’Académie royale dans l’administration des fonds qui lui sont confiés. Je ne doute pas qu’avant la fin du siècle des bouleversements entièrement imprévus vont transformer de fond en comble la notion même que nous avons de la « science », ses grands objectifs et l’esprit dans lequel s’accomplit le travail scientifique. Nul doute que l’Académie royale fera alors partie des institutions et des personnages qui auront un rôle utile à jouer dans un renouveau sans précédent, après une fin de civilisation également sans précédent…"
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main hall (out of the common track): 


Niels Henrik Abel
Évariste Galois
(pictures taken from the Internet)
What did they do?
Well, conjure the most remarkable consequences of an old ghost (Vieta's logistica speciosa): "solutions" for the general algebraic equations of degree higher than four.

- Bernhard Riemann
- Kingdon Clifford
(picture taken from the Internet)
What did they do?
Riemann simply solved the following riddle (and by doing so challenged classical and common sense understanding about space, which remains pretty much our understanding till nowadays): what is real space?
Clifford understood that.
"Riemann's allusions were ignored by the majority of contemporary mathematicians and physicists. His investigations were deemed too speculative and theoretical to bear any relevance to physical space, the space of experience. The only one who allied himself firmly to Riemann was the translator of his works into English, William Kingdon Clifford... already in 1870, Clifford saw in Riemann's conception of space the possibility for a fusion of geometry with physics... Clifford conceived matter and its motion as a manifestation of varying curvature [of space]... For Aristotle, space was an accident of substance; for Clifford... substance is an accident of space... These speculations aroused great opposition among academic philosophers..." (Max Jammer, Concepts of Space, Dover 1993);
"Riemann came to a deep philosophical conviction that a complete mathematical theory must be established, which would take the elementary laws governing points and transform them to the great generality of the plenum (by which he meant continuously-filled space)... the Riemann Integral calculus is defined as an infinite sum of integrals of step functions. Such infinite sums became the starting points for the study of infinite by Georg Cantor" (Amir D. Aczel, The Mystery of the Aleph, WSP 2000);
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from a History of Mathematics (wrote upon strong vellum): 


"... geometry, like counting, had an origin in primitive ritualistic practice. The earliest geometric results found in India constituted what were called the Sulvasutras, or 'rules of the cord.'These were simple relationships that apparently were applied in the construction of altars and temples. It is commonly thought that the geometric motivation of the 'rope-stretchers' in Egypt was more practical than that of their counterparts in India; but it has been suggested that both Indian and Egyptian geometry may derive from a common source—a protogeometry that is related to primitive rites in somewhat the same way in which science developed from mythology and philosophy from theology." 
''The Egyptians early had become interested in astronomy and had observed that the annual flooding of the Nile took place shortly after Sirius, the dogstar, rose in the east just before the sun. By noticing that these heliacal risings of Sirius, the harbinger of the flood, were separated by 365 days, the Egyptians established a good solar calendar made up of twelve months of thirty days each and five extra feast days..."
"Recognition by the Egyptians of interrelationships among geometric figures, on the other hand, has too often been overlooked, and yet it is here that they came closest in attitude to their successors, the Greeks."
"The fertile Nile Valley has been described as the world's largest oasis in the world's largest desert. Watered by one of the most gentlemanly of rivers and geographically shielded to a great extent from foreign invasion, it was a heaven for peace-loving people who pursued, to a large extent, a calm and unchallenged way of life."
"One is almost tempted to see in it [Babylonian ratio calculations] the genuine origin of geometry, but it is important to note that it was not so much the geometric context that interested the Babylonians as the numerical approximations that they used in mensuration. Geometry for them was not a mathematical discipline in our sense, but a sort of applied algebra or arithmetic in which numbers are attached to figures."
"It is indeed difficult to separate history and legend concerning the man [Pythagoras], for he meant so many things to the populace—the philosopher, the astronomer, the mathematician, the abhorrer of beans, the saint, the prophet, the performer of miracles, the magician, the charlatan."
"One of the tantalizing questions in Pythagorean geometry concerns the construction of a pentagram or star pentagon... In each case a diagonal point divides a diagonal into two unequal segments such that the ratio of the whole diagonal is to the larger segment as this segment is to the smaller segment. This subdivision of a diagonal is the well-known 'golden section' of a line segment, but this name was not used until a couple of thousand years later... The construction required is equivalent to the solution of a quadratic equation... Pythagoras could have learned from the Babylonians how to solve this equation algebraically. However, if a is a rational number, then there is no rational number x satisfying the equation. Did Pythagoras realized this?"
"... when the five diagonals of a regular pentagon are drawn, these diagonals form a smaller regular pentagon, and the diagonals of the second pentagon in turn form a third regular pentagon, which is still smaller. This process can be continued indefinitely, resulting in pentagons that are as small as desired and leading to the conclusion that the ration of a diagonal to a side in a regular pentagon is not rational. The irrationality of this ratio is, in fact, a consequence of the argument... in which the golden section was shown to repeat itself over and over again. Was it perhaps this property that led to the disclosure, possibly by Hippasus, of incommensurability?"
"The Pythagoreans had assumed that space and time can be thought of as consisting of points and instants; but space and time have also a property, more easily intuited than defined, known as 'continuity.'"
"... Plato in the Thaetetus says that his teacher, Theodorus of Cyrene—of whom Thaetetus also was a pupil—was the first to prove the irrationality of the square roots of the nonsquare integers from 3 to 17 inclusive."
"Plato looked upon the dodecahedron as composed of 360 scalene right triangles, for when the five diagonals and five medians are drawn in each of the pentagonal faces, each of the twelve faces will contain thirty right triangles. The association of the first four regular solids with the traditional four universal elements provided Plato in the Timaeus with a beautifully unified theory of matter according to which everything was constructed of ideal right triangles... He emphasized that the reasoning used in geometry does not refer to the visible figures that are drawn but to the absolute ideas that they represent..."
"Inasmuch as he hesitated to follow Platonic mathematicians into the abstractions and technicalities of the day, Aristotle made no lasting contribution to the subject. He is said to have written a biography of Pythagoras, although this is lost..."
"Among Euclid's lost works are also one on Surface Loci, another on Pseudaria (or fallacies), and a third on Porisms... The loss of Euclidean Porisms is particularly tantalizing, for it may have represented an ancient approximation to an analytic geometry."
"Book X of the Elements was, before the advent of early modern algebra, the most admired—and the most feared. It is concerned with a systematic classification of incommensurable line segments... Today we would be inclined to think of this as a book on irrational numbers... Book X contains 115 propositions—more than any other—most of which contain geometric equivalents of what we now know arithmetically as surds."
"No account of Ptolomy's work would be complete without mention of his Tetrabiblos (or Quadripartitum), for it shows us a side of ancient scholarship that we are prone to overlook..."
"The full title of [Boethius's] Expositio indicates, in fact, that it is an exposition of mathematical matters useful to an understanding of Plato. It explains, for example, that the tetractys consisting of the numbers 1, 2, 3, and 4 contains all the musical consonances inasmuch as it makes up the ratios 4:3, 3:2, 2:1, 3:1 and 4:1."
"In a sense it is not fair to criticize Diophantus for being satisfied with a single answer, for he was solving problems, not equations. In a sense the Arithmetica is not an algebra textbook, but a problem collection in the application of algebra."
"The idea of negative numbers seems not to have occasioned much difficulty for the Chinese since they were accustomed to calculating with two sets of rods—a red set for positive coefficients or numbers and a black set for negatives."
"Yang Hui's works included also results in the summation of series and the so-called Pascal triangle, things that were published and better known through the Precious Mirror of Chu Shih-chieh..." 
"The Mayas of Yucatan, in their representation of time intervals between dates in their calendar, used a place value numeration, generally with twenty as the primary base and with five as an auxiliary (corresponding to the Babylonian use of sixty and ten respectively)."
"It should be mentioned also that the Hindus, unlike Greeks, regarded irrational roots of numbers as numbers. This was of enormous help in algebra... Hindu algebra is especially noteworthy in its development of indeterminate analysis, to which Brahmagupta made several contributions... It is greatly to the credit of Brahmagupta that he gave all integral solutions of the linear Diophantine equation, whereas Diophantus himself had been satisfied to give one particular solution of an indeterminate equation."
"... ultimately the scheme of numeration making use of the Hindu numerals came to be called simply algorism or algorithm, a word that, originally derived from the name al-Khwarizmi, now means, more generally, any peculiar rule of procedure or operation—such as the Euclidean method for finding the greatest common divisor."
"Omar Khayyam (ca. 1050-1123), the 'tent-maker,' wrote an Algebra that went beyond that of al-Khwarizmi to include equations of third degree."
"The book in which Fibonacci described the new algorism is a celebrated classic, completed in 1202, but it bears a misleading title—Liber abaci (or Book of the Abacus). It is not on the abacus; it is a very through treatise on algebraic methods and problems in which the use of Hindu-Arabic numerals is strongly advocated."
"Perhaps through the Arabs Fibonacci had learned what we call 'Horner's method,' a device known before this time in China. This is the most accurate European approximation to an irrational root of an algebraic equation up to that time..."
"Even more imaginative than Oresme's notations was his suggestion that irrational proportions are possible."
"In 1545, the solution not only of the cubic but of the quartic as well became common knowledge through the publication of the Ars magna of Geronimo Cardano (1501-1576)."
"By birth illegitimate, and by habit an astrologer, gambler, and heretic, Cardan nevertheless was a respected professor at Bologna and Milan, and ultimately he was granted a pension by the pope."
"The solution of cubic and quartic equations [by Cardano] was perhaps the greatest contribution to algebra since the Babylonians, almost four millennia earlier, had learned how to complete the square for quadratic equations."
"Another immediate result of the solution of the cubic was the first significant glance at a new kind of number. Irrational numbers had been accepted by the time of Cardan, even though they were not soundly based, for they are readily approximated by rational numbers. Negative numbers afforded more difficulty because they are not readily approximated by positive numbers, but the notion of sense (or direction on a line) made them plausible. Cardan used them even while calling them numeri ficti..."
"Whenever the three roots of a cubic equation are real and different from zero, the Cardan-Tartaglia formula leads inevitably to square roots of negative numbers. The goal was known to be a real number, but it could not be reached without understanding something about imaginary numbers."
"Physics and astronomy had reached the point where there was increasing need for arguments concerning the infinitely large and small—the subject now known as analysis. Viète had been one of the first to use the word 'analysis' as a synonym for algebra, but he was one of the earliest analysts also in the more modern sense of one who studies infinite processes."
"Stevin, Kepler, and Galileo all had need for Archimedean methods, being practical men, but they wished to avoid the logical niceties of the method of exhaustion. It was largely the resulting modifications of the ancient infinitesimal methods that ultimately led to the calculus, and Stevin was one of the first to suggest changes."
"The infinitely small was of more immediate relevance to Galileo than the infinitely large, for he found it essential in his dynamics."
"Salviati then concludes that by bending the line segment into the shape of a circle, he has 'reduced to actuality that infinite number of parts into which you claimed, while it was straight, were contained in it only potentially,' for the circle is a polygon of an infinite number of sides... Galileo had intended to write a treatise on the infinite in mathematics, but it has not been found." 
"... the world remembers Cavalieri [disciple of Kepler] for one of the most influential books of the early modern period, the Geometria indivisibilibus continuorum, published in 1635."
"For a number of years Descartes traveled about in conjunction with varied military campaigns, first in Holland with Maurice Prince of Nassau, then with Duke Maximillian I of Bavaria, and later still with the French army at the siege of LaRochelle... The entire universe, he postulated, was made up of matter in ceaseless motion in vortices, and all phenomena were to be explained mechanically in terms of forces exerted by contiguous matter."
"The work of Descartes far too often is described simply as the application of algebra to geometry, whereas actually it could be characterized equally well as the translation of the algebraic operations into the language of geometry."
"... it is appropriate to follow Laplace in acclaiming Fermat as the discoverer of the differential calculus, as well as a codiscoverer of analytic geometry. Obviously Fermat was not in possession of the limit concept, but otherwise his method of maxima and minima parallels that used in calculus today..."
"The result was one of the most unsuccessful great books ever produced. Even the ponderous title was repulsive—Brouillon projet d'une atteinte aux événements des rencontres d'un cone avec un plan (Paris, 1639).... Moreover Girard Desargues used a bizarre new vocabulary full of terms borrowed from botany, a terminology that repelled scholars and practitioners alike."
"... in 1640, the young Pascal, then sixteen years old, published an Essay pour les coniques. This consisted of only a single printed page—but one of the most fruitful pages in history. It contained the proposition, described by the author as mysterium hexagrammicum, which has ever since been known as Pascal's theorem."
''On the night of November 23, 1654, from 10:30 to about 12:30, Pascal experienced a religious ecstasy which caused him to abandon science and mathematics for theology."
"Pietro Mengoli had rediscovered Oresme's conclusion, arrived at by a grouping of terms, that the ordinary harmonic series does not converge, a theorem usually attributed to Jacques Bernoulli in 1689; he also showed the convergence of the reciprocals of the triangular numbers, a result for which Huygens usually is given credit."
"Even proportion, the stronghold of ancient geometry, Wallis held to be an arithmetic concept. In this his attitude represented the tendency of mathematics for at least the following century, but it should be remarked that such a movement was without a solid foundation, since real numbers had not been defined."
"[In Arithmetica infinitorum, published in 1655] Wallis arithmetized the Geometria indivisibilibus of Cavalieri, as he had arithmetized the Conics of Apollonius... Wallis abandoned the geometric background after having associated the infinitely many indivisibles in the figures with numerical values. If, for example, one wishes to compare the squares of the indivisibles in the triangle with the squares of the indivisibles in the parallelogram, one takes the length of the first indivisible in the triangle as zero, the second as one, the third as two, and so on up to the last, of length n - 1, if there are n indivisibles."
"[Among Wallis] most important contributions in infinitesimal analysis [was his anticipation] of Euler on the gamma or factorial function."
" Thomas Hobbes (1588-1679) was foremost among those who criticized Wallis' arithmetization of geometry, objecting strenuously to 'the whole herd of them who apply their algebra to geometry' and referring to the Arithmetica infinitorum as 'a scab of symbols.'"
"... it is likely that it was in Italy, through Mengoli and Angeli, that James Gregory came to appreciate the power of infinite series expansions of functions and of infinite processes in general."
"Newton's first discoveries, dating from the early months of 1665, resulted from his ability to express functions in terms of infinite series—the very thing that Gregory was doing in Italy at about that time."
"... Newton's indirect approach... made clear to him that one could operate with infinite series in much the same way as with finite polynomial expressions... he had found that the analysis by infinite series had the same inner consistency and was subject to the same general laws as the algebra of finite quantities. Infinite series were no longer to be regarded as approximating devices only; they were alternative forms of the functions they represented... he wrote in De analysi per aequationes numero termonorum infinitas, composed in 1669: '... albeit we Mortals whose reasoning Powers are confined within narrow Limits, can neither express, nor so conceive all the Terms of these Equations as to know exactly from thence the Quantities we want...'"
"As was the case with Newton, infinite series played a large role in the early work of Leibniz."
"... it was on reading the letter of Amos Dettonville on Traité des sinus du quart de cercle that Leibniz reported that a light burst upon him. He then realized, in about 1673, that the determination of the tangent to a curve depended on the ration of the differences in the ordinates and abscissas, as these became infinitely small, and that quadratures depended on the sum of the ordinates or infinitely thin rectangles making up the area. Just as in the arithmetic and harmonic triangles the processes of summing and differencing are oppositely related, so also in geometry the quadrature and tangent problems, depending on sums and differences respectively, are inverses of each other. The connecting link seemed to be through the infinitesimal or 'characteristic' triangle, for where Pascal had used it to find the quadrature of sines, Barrow had applied it to the tangent problem."
"Leibniz by 1676 had arrived at the same conclusion that Newton had reached several years earlier, namely, that he was in possession of a method that was highly important because of its generality. Whether a function was rational or irrational, algebraic or transcendental (a word that Leibniz coined), his operations of finding sums and differences could always be applied."
"Whereas Descartes' geometry had once excluded all nonalgebraic curves, the calculus of Newton and Leibniz showed how essential is the role of these in their new analysis... Moreover, Leibniz seems to have appreciated, as did Newton, that the operations in the new analysis can be applied to infinite series as well as to finite algebraic expressions."
"The ambivalent status of complex numbers is well illustrated by the remark of Leibniz, who was also a prominent theologian, that imaginary numbers are a sort of amphibian, halfway between existence and nonexistence."
"The fourth and last part [of Jacques Bernoulli's Ars Conjectandi] contains the celebrated theorem that now bears the author's name, and on which Bernoulli and Leibniz had corresponded: the so-called 'Law of large numbers.'"
"Berkeley [in The Analyst] did not deny the utility of the techniques of fluxions nor the validity of the results obtained by using these; but he had been nettled on having a sick friend refuse spiritual consolation because Halley had convinced the friend of the untenable nature of Christian doctrine."
"Euler spent almost all of the last seventeen years of his life in total darkness."
"It may fairly be said that Euler did for the infinite analysis of Newton and Leibniz what Euclid had done for the geometry of Eudoxus and Theaetetus, or what Viète had done for the algebra of Al-Khwarizmi and Cardan. Euler took the differential calculus and the method of fluxions and made them part of a more general branch of mathematics which ever since ha been known as 'analysis'—the study of infinite processes... From 1748 [the time of publication of Euler's Introductio in analysin infinitorum] onward the idea of 'function' became fundamental in analysis. It had been adumbrated in the medieval latitude of forms, and it had been implicit in the analytic geometry of Fermat and Descartes, as well as in the calculus of Newton and Leibniz."
"The first volume of the Introductio is concerned from start to finish with infinite processes—infinite products and infinite continued fractions, as well as innumerable infinite series. In this respect the work is the natural generalization of the views of Newton, Leibniz, and the Bernoullis, all of whom were fond of infinite series."
"Opposing the views of Leibiniz and Euler, d'Alembert insisted that 'a quantity is something or nothing; if it is something, it has not yet vanished; if it is nothing, it has literary vanished... d'Alembert held that the differential notation is merely a convenient manner of speaking that depends for its justification on the language of limits. His Encyclopédie article on the differential referred to Newton's De quadratura curvarum, but d'Alembert interpreted Newton's phrase 'prime and ultimate ratio' as a limit rather than as a first or last ratio of two quantities just springing into being... D'Alembert denied the existence of the actually infinite, for he was thinking of geometric magnitudes rather than of the theory of aggregates proposed a century later."
"In his selection of the unifying principle, however, Carnot made a most deplorable choice. He concluded that 'the true metaphysical principles' are 'the principles of the compensation of errors.' Infinitesimals, he argued, are 'quantités inappréciables" which, like imaginary numbers, are introduced only to facilitate the computation and are eliminated in reaching the final result."
"The whole motive of [Lagrange's function theory] was not to try to make the calculus more utilitarian, but to make it more logically satisfying... In 1767 Lagrange published a memoir on the approximation of roots of polynomial equations by means of continued fractions; in another paper in 1770 he considered the solvability of equations in terms of permutations on their roots. It was the latter work that was to lead to the enormously successful theory of groups and to the proofs by Galois and Abel of the unsolvability, in the usual terms, of equations of degree greater than four." 
"The new branch of geometry that Gauss initiated in 1827 is known as differential geometry, and it belongs perhaps more to analysis than to the traditional field of geometry. Ever since the days of Newton and Leibniz, men had applied the calculus to the study of curves in two dimensions, and in a sense this work constituted a prototype of differential geometry. Euler and Monge had extended this to include an analytic study of surfaces; hence they sometimes are regarded as the fathers of differential geometry. Nevertheless, not until the appearance of the classical treatise of Gauss, Disquisitiones circa superficies curvas, was there a comprehensive volume devoted entirely to the subject."
"Roughly speaking, ordinary geometry is interested in the totality of a given diagram or figure, whereas differential geometry concentrates on the properties of a curve or a surface in the immediate neighborhood of a point on the curve or the surface. In this connection Gauss extended the work of Huygens and Clairaut on the curvature of a plane or gauche curve at a point by defining the curvature of a surface at a point... Gauss proceeds to show that the properties of a surface depend only on E, F, and G. This leads to many consequences. In particular, it becomes easy to say what properties of the surface remain invariant. It was in building on this work of Gauss that Bernhard Riemann and later geometers transformed the subject of differential geometry." 
"Oddly enough, however, no one before Wessel and Gauss took the obvious step of thinking of the real and imaginary parts of a complex number a + bi as rectangular coordinates of points in a plane."
"Functions no longer needed to be of the well-behaved form with which mathematicians had been familiar. Lejeune Dirichlet, for instance, in 1837, suggested a very broad definition of function: If a variable y is so related to a variable x that whenever a numerical value is assigned to x, there is a rule according to which a unique value of y is determined, then y is said to be a function of the independent variable x. This comes close to the modern view of a correspondence between two sets of numbers, but the concepts of 'set' and 'real number' had not at that time been established. To indicate the completely arbitrary nature of the rule fo correspondence, Dirichlet proposed a very 'badly behaved' function: When x is rational, let y = c, and when x is irrational, let y = d ≠ c."
"Rejecting the Taylor's theorem approach of Lagrange, Cauchy made the limit concept of d'Alembert fundamental, but he gave it an arithmetic character of greater precision. Dispensing with geometry and with infinitesimals or velocities, he gave a relatively clear-cut definition of limit... Where many earlier mathematicians had thought of an infinitesimal as a very small fixed number, Cauchy defined it clearly as a dependent variable..."
"The history of mathematics teems with cases of simultaneity and near simultaneity of discovery, some of which have already been noted. The work by Cauchy that we have just described is another case in point, for similar views were developed at about the same time by Bernhard Bolzano (1781-1848), a Czechoslovakian priest whose theological views were frowned upon by his church and whose mathematical work was most undeservedly overlooked by his lay and clerical contemporaries.... The similarity in their arithmetization of the calculus and of their definitions of limit, derivative, continuity, and convergence was only a coincidence. Bolzano in 1817 had published a book, Rein analytischer Beweis, devoted to a purely arithmetic proof of the location theorem in algebra, and this had required a nongeometric approach to the continuity of a curve or function. Going considerable further in his unorthodox ideas, he disclosed some important properties of infinite sets in a posthumous work of 1850, Paradoxien des Unendlichen."
"From Galileo's paradox on the one-to-one correspondence between integers and perfect squares, Bolzano went on to show that similar correspondences between the elements of an infinite set and a proper subset are commonplace. For example, a simple linear equation, such as y = 2x, establishes a one-to-one correspondence between the real numbers y in the interval from 0 to 2, for example, and the real numbers x in half this interval. That is, there are just as many real numbers between 0 and 1 as between 0 and 2, or just as many points in a line segment 1 inch long as in a line segment 2 inches long. Bolzano seems even to have recognized, by about 1840, that the infinity of real numbers is of a type different from the infinity of integers, being nondenumerable. In such speculations on infinite sets the Bohemian philosopher came closer to parts of modern mathematics than had his better-known contemporaries. Both Gauss and Cauchy seem to have had a kind of horror infiniti, insisting that there could be no such thing as a completed infinite in mathematics."
"The name of Cauchy appears today in connection with a number of theorems on infinite series, for, despite some efforts on the part of Gauss and Abel, it was largely through Cauchy that the mathematician's conscience was pricked concerning the need for vigilance with regard to convergence."
"Through a point Clying outside a line AB there can be drawn more than one line in the plane and not meeting AB. With this new postulate Lobachevsky deduced a harmonious geometric structure having no inherent logical contradictions. This was in every sense a valid geometry, but so contrary to common sense did it appear, even to Lobachevsky, that he called it 'imaginary geometry.'"
"Bolyai and Lobachevsky were far removed from Paris and Göttingen. Still, presence in Paris did not guarantee success to even the brightest young mathematical minds of the day. The most illustrious examples of men who felt frustrated by their failure to find the recognition they sought in paris are the Norwegian Niels Henrik Abel (1802-1829) and the Frenchman Évariste Galois (1812-1832)."
"Abel found Paris inhospitable... and wrote home to a friend: 'Every beginner has a great deal of difficulty in getting noticed here. I have just finished an extensive treatise on a certain class of transcendental functions... but Mr Cauchy scarcely deigns to glance at it'... Abel had returned to his native Norway; increasingly weakened by tuberculosis, he kept sending more material to Crelle. He died in 1829, scarcely aware of the interest his publications were creating."
"Young geniuses whose lives were cut short by death from dueling or consumption are part of the real and fictional literary tradition of the Romantic Age. Someone wishing to present a mathematical caricature of such lives could do no better than to create the characters of Abel and Galois."
"By the age of sixteen Galois knew what his teachers had failed to recognize—that he was a mathematical genius. He hoped, therefore, to enter the school that had nurtured so many celebrated mathematicians, the École Polytechnique, but his lack of systematic preparation resulted in his rejection. This disappointment was followed by others: A paper Galois wrote and presented to the Academy when he was seventeen was apparently lost by Cauchy..."
"His letter to Chevalier, published in September 1832, had contained an outline of the main results of the memoir that had been returned by the Academy. Here Galois had indicated what he considered to be the essential part of his theory. In particular, he stressed the difference between adjoining one or all of the roots of the resolvent, and related it to the decomposition of the group G of the equation. In modern terminology, he indicated that an extension of the given field is normal if and only if the corresponding subgroup is a normal subgroup of G."
"Inspired by Abel's proof of the unsolvability by radicals of the quintic equation, Galois discovered that an irreducible algebraic equation is solvable by radicals if and only if its group—that is, the symmetric group o its roots—is solvable... Lagrange had already shown that the order of a subgroup must be a factor of the order of the group; but Galois went deeper and found relations between the factorability of the group and an equation and the solvability of the equation. Moreover, to him we owe the use in 1830 of the word group in its technical sense in mathematics."
"Among Jakob Steiner's unpublished discoveries are those relating to the fruitful geometric transformation known as inversive geometry... Inasmuch as there is no outside point P' corresponding to P when P coincides with the center O, one has in a sense a paradox similar to that of Bolzano. The inside of every circle, no matter how small, contains, as it were, one more point than the portion of the plane outside the circle. In a exactly analogous manner one readily defines the inverse of a point in three-dimensional space with respect to a sphere."
"Another discoverer was A. F. Möbius (1790-1860), also a student of Gauss... He introduced his 'barycentric coordinates' by considering a given triangle ABC and defining the coordinates of a point P as the mass to be placed at A, B, and C so that P is eh center of gravity of these masses. Möbius classified transformations according to whether they were congruences (leaving corresponding figures equal), similarities (corresponding figures similar), affine (corresponding figures preserving parallel lines, or collineations (lines going into lines), and suggested the study of invariants under each family of transformations."
"The notations and patterns of reasoning of the four inventors of homogeneous coordinates [Plücker, Gergonne, Lamé & Möbius (?)] differed somewhat, but they all had one thing in common—they made use of three coordinates instead of two to locate a point in a plane."
"Plücker chowed that every curve (other than a straight line) can be regarded as having a dual origin: It is a locus generated by a moving point and enveloped by a moving line, the point moving continuously along the line while the line continues to rotate about the point."
"Cayley in 1843 had initiated the ordinary analytic geometry of n-dimensional space, using determinants as an essential tool."
"Non-Euclidean geometry continued for several decades to be a fringe aspect of mathematics until it was thoroughly integrated through the remarkably general views of G. F. B. Riemann (1826-1866). The son of a village pastor, Riemann was brought up in very modest circumstances, always remaining frail in body and shy in manner."
"[Riemann's Habilitationsschrift] urged a global view of geometry as a study of manifolds of any number of dimensions in any kind of space. His geometries are non-Euclidean in a far more general sense than is Lobachevskian geometry, where the question is simply how many parallels are possible through a point. Riemann saw that geometry should not even necessarily deal with points or lines or space in the ordinary sense, but with sets of ordered n-tuples that are combined according to certain rules."
"... the fundamental change in geometric thought that Riemann's Habilitationsschrift brought about was Riemann's suggestion of the general study of curved metric spaces, rather than of the special case equivalent to geometry on the sphere, that ultimately made the theory of general relativity possible."
"The unification of geometry that Riemann had achieved was especially relevant in the microscopic aspect of differential geometry, or geometry 'in the small.' Analytic geometry, or geometry 'in the large,' had not been much changed."
"The program that Felix Klein gave, which became known as the Erlanger Programm, described geometry as the study of those properties of figures that remain invariant under a particular group of transformations. Hence, any classification of groups of groups of transformations becomes a codification of geometries."
"Euclidean geometry in Klein's view, is only a special case of affine geometry. Affine geometry in its turn becomes only a special case of a still more general geometry—projective geometry."
"Analysis, the study of infinite processes, had been understood by Newton and Leibniz to be concerned with continuous magnitudes, such as lengths, areas, velocities, and accelerations, whereas the theory of numbers clearly has as its domain the discrete set of natural numbers."
"Gudermann had impressed upon the young Weierstrass what a useful tool the power series representation of a function was, and it was in this connection that Weierstrass produced his greatest work, following in the footsteps of Abel."
"... perhaps no one is more deserving to be known as the father of the critical movement in analysis than is Weierstrass."
"The importance of work such as that of Weierstrass is felt particularly in mathematical physics, in which solutions of differential equations are rarely found in any form other than as an infinite series."
"[The Frenchman H. C. R. Charles Méray, the four Germans Karl Weierstrass, H. E. Heine, Georg Cantor and J. W. R. Dedekind]" in a sense represented the climax in half a century of investigation into the nature of function and number that had begun in 1822 with Fourier's theory of heat and with an attempt made in that year by Martin Ohm to reduce all of analysis to arithmetic..."
"It was in seeking to liberalize Dirichlet's conditions for the convergence of a Fourier series that Riemann developed his definition of the Riemann integral; in this connection he showed that a function f(x) may be integrable in an interval without being representable by a Fourier series. It was the study of infinite trigonometric series that led also to the theory of sets of Cantor..."
"We have seen that the revolution in geometry took place when Gauss, Lobachevsky, and Bolyai freed themselves from preconceptions of space. In somewhat the same sense the thoroughgoing arithmetization of analysis became possible only when, as Hankel foresaw, mathematicians understood that the real numbers are to be viewed as 'intellectual structures' rather than as intuitively given magnitudes inherited from Euclid's geometry."
"Bolzano during the early 1830s had made an attempt to develop a theory of real numbers as limits of rational number sequences, but this had gone unnoticed and unpublished until 1962."
"Weierstrass sought to separate the calculus from geometry and to base it upon the concept of number alone. Like Méray, he also saw that to do this it was necessary to give a definition of irrational number that is independent of the limit concept, inasmuch as the latter had up to this point presupposed the former. To correct Cauchy's logical error, Weierstrass settled the question of the existence of a limit of a convergent sequence by making the sequence itself the number limit."
"... Now, Dedekind pointed out, the fundamental theorems on limits can be proved rigorously without recourse to geometry. It was geometry that had pointed the way to a suitable definition of continuity, but in the end it was excluded from the formal arithmetic definition of the concept. The Dedekind cut in the rational number system, or an equivalent construction of real number, now has replaced geometrical magnitude as the backbone of analysis."
"The life of Cantor was tragically different from that of his friend Dedekind. Cantor was born in St. Petersburg of parents who had migrated from Denmark, but most of his life was spent in Germany, the family having moved to Frankfurt when he was eleven. His parents were Christians of Jewish background—his father had been converted to Protestantism, his mother had been born a Catholic. The son Georg took a strong interest in the finespun arguments of medieval theologians concerning continuity and the infinite, and this militated against his pursuing a mundane career in engineering as suggested by his father. In his studies at Zurich, Göttingen, and Berlin the young man consequently concentrated on philosophy, physics, and mathematics—a program that seems to have fostered his unprecedented mathematical imagination."
"Cauchy and Weierstrass saw only paradox in attempts to identify an actual or 'completed' infinity in mathematics, believing that the infinitely large and small indicated nothing more than the potentiality of Aristotle—an incompleteness of the process in question. Cantor and Dedekind came to a contrary conclusion."
"The set of perfect squares or the set of triangular numbers has the same power as the set of all the positive integers, for the groups can be put into one-to-one correspondence. These sets seem to be much smaller than the set of all rational fractions, yet Cantor showed that the latter set also is countable or denumerable, that is, it, too, can be put into one-to-one correspondence with the positive integers, hence has the same power... The real numbers can be subdivided into two types in two different ways: (1) as rational and irrational or (2) as algebraic and transcendental. Cantor showed that even the class of algebraic numbers, which is far more general than that of rational numbers, nevertheless has the same power as that of the integers. Hence, it is the transcendental numbers that give to the real number system the 'density' that results in a higher power."
"More surprising still is the fact that dimensionality is not the arbiter of the power of a set. The power of the set of points in a unit line segment is just the same as that of the points in a unit area or in a unit volume—or, for that matter, all the three-dimensional space. (Dimensionality, however, retains some measure of authority in that any one-to-one mapping of points in a space of unlike dimensionality is necessarily a discontinuous mapping.) So paradoxical were some results in point-set theory that Cantor himself on one occasion in 1877 wrote to Dedekind, 'I see it, but I don't believe it'; and he asked his friend to check the proof."
"Dedekind and Cantor were among the most capable mathematicians, and certainly the most original, of their day; yet neither man secured a top-ranking professional position. Dedekind spent almost a lifetime teaching on the secondary level..."
"Kronecker is said to have asked Lindemann of what use was the proof that pi is not algebraic inasmuch as irrational numbers are nonexistent. Sometimes it is reported that his movement died of inanition. We shall see later that it can be said to have reappeared in a new form in the work of Poincaré and Brouwer."
"David Hilbert... described the new transfinite arithmetic as 'the most astonishing product of mathematical thought, one of most beautiful realizations of human activity in the domain of the purely intelligible.'"
"Liouville... constructed an extensive class of nonalgebraic real numbers. The particular class that he developed are known as Liouville numbers, the more comprehensive set of nonalgebraic real numbers being called transcendental numbers."
"Here, finally, was the answer to the classical problem of the quadrature of the circle. In order for the quadrature of the circle to be possible with Euclidean tools, the number pi would have to be the root of a algebraic equation with a root expressible in square roots. Since pi is not algebraic, the circle cannot be squared according to the classical rules. Emboldened by his success, Ferdinand Lindemann later published several purported proofs of Fermat's last theorem, but they were shown by others to be invalid."
"Boole carried the formalism to its conclusion. No longer was mathematics to be limited to questions of number and continuous magnitude. Here for the first time the view is clearly expressed that the essential characteristic of mathematics is not so much its content as its form. If any topic is presented in such a way that it consists of symbols and precise rules of operation upon these symbols, subject only to the requirement of inner consistency, this topic is part of mathematics."
"Boole's Investigation of the Laws of Thought of 1854 is a classic in the history of mathematics, for it amplified and clarified the ideas presented in 1847, establishing both formal logic and a new algebra, known as Boolean algebra, the algebra of sets, or the algebra of logic."
"De Morgan was a lover of conundrums and witticisms, many of which are collected in his well-known Budget of Paradoxes, a delightful satire on circle-squarers edited after his death by his widow."
"Just as Lobachevsky had created a new geometry consistent within itself, by abandoning the parallel postulate, so Hamilton created a new algebra, also consistent with itself, by discarding the commutative postulate for multiplication... Hamilton, quite naturally, always regarded the discovery of quaternions as his greatest achievement. In retrospect it is clear that it was not so much this particular type fo algebra that was significant, but rather the discovery of the tremendous freedom that mathematics enjoys to build algebras that need not satisfy the restrictions imposed by the so-called 'fundamental laws,' which up to that time, supported by the vague principle of permanence of form, had been invoked without exception."
"The concept of an n-dimensional vector space had received detailed treatment in Hermann Grassmann's Ausdehnungslehre, published in Germany in 1844. Grassmann also was led to his results by studying the geometric interpretation of negative quantities and the addition and multiplication of directed line segments in two and three dimensions. He emphasized the dimension concept and stressed the development of an abstract science of 'spaces' and 'subspaces' which would include the geometry of two and three dimensions as special cases."
"C. S. Peirce continued his father's work in this direction by showing that of all these algebras there are only three in which division is uniquely defined: ordinary real algebra, the algebra of complex numbers, and the algebra of quaternions."
"In 1870 William Kingdon Clifford wrote a paper 'On the Space-Theory of Matter' in which he showed himself to be a stauch British supporter of the non-Euclidean geometry of Lobachevsky and Riemann."
"The concept of field was implicit in work by Abel and Galois, but Dedekind in 1879 seems to have been the first one to give an explicit definition of a number field—a set of numbers that form an Abelian group with respect to addition and with respect to multiplication (except for the inverse of zero) and for which multiplication distributes over addition. Simple examples are the system of rational numbers, the real number system, and the complex number field."
"Not satisfied to leave the basic concepts of arithmetic, hence of algebra, in so vague a state, the German logician and mathematician F. L. G. Frege was led to his well-known definition of cardinal number. The basis his views came from the theory of sets of Boole and Cantor."
"Frege undertook to derive the concepts of arithmetic from those of formal logic, for he disagreed with the assertion of C. S. Peirce that mathematics and logic are clearly distinct."
"Peano's aim was similar to that of Frege, but it was at the same time more ambitious and yet more down to earth. He hoped in his Formulaire de mathématiques to develop a formalized language that should contain not only mathematical logic but all the most important branches of mathematics."
"Poincarés doctoral thesis had been o differential equations... which led to one of his most celebrated contributions to mathematics—the properties of automorphic functions."
"... as a date for the beginning of the subject [of topology] none is more appropriate than 1895, the year in which Poincaré published his Analysis situs."
"Poincaré referred to Cantor's Mengenlehre as a disease from which later generations would regard themselves as having recovered."
"Hilbert's Grundlagen opened with a motto taken from Kant: 'All human knowledge begins with intuitions, proceeds to concepts, and terminates in ideas,' but Hilbert's development of geometry established a decidedly anti-Kantian view of the subject."
"... the first of [Hilbert's] twenty-three problems... concerned the structure of the real number continuum. The question is made up of two related parts: (1) Is there a transfinite number between that of a denumerable set and the number of the continuum; and (2) can the numerical continuum be considered a well-ordered set? The second part asks whether the totality of all real numbers can be arranged in another manner so that every partial assemblage will have a first element. This is closely related with t he axiom of choice named for the German mathematician Ernst Zermelo..."
"Efforts to solve the second query of Hilbert led in 1931 to a surprising conclusion on the part of a young Austrian mathematician, Kurt Gödel. Gödel showed that within a rigidly logical system such as Russell and Whitehead had developed for arithmetic, propositions can be formulated that are undecidable or undemonstrable within the axioms of the system."
"In a sense Gödel's theorem, sometimes regarded as the most decisive result in mathematical logic, seems to dispose negatively of Hilbert's second query. In its implications the discovery by Gödel of undecidable propositions is as disturbing as was the disclosure by Hippasus of incommensurable magnitudes, for it appears to foredoom hope of mathematical certitude through use of the obvious methods."
"If any one book marks the emergence of point set topology as a separate discipline, it is Hausdorff's Grundzüge. It is interesting to note that although it was the arithmetization of analysis that began the train of thought that led from Cantor to Hausdorff, in the end the concept of number is thoroughly submerged under a far more general point of view. Moreover, although the word 'point' is used in the title, the new subject has a little to do with the points of ordinary geometry as with the numbers of common arithmetic. Topology has emerged in the twentieth century as a subject that unifies almost the whole of mathematics, somewhat as philosophy seeks to coordinate all knowledge."
"The first volume of Bourbaki's Éléments appeared in 1939, the thirty-first in 1965..." 

- Carl B. Boyer, A History of Mathematics: Revised by Uta C. Merzbach; 

See also:
- actual infinite falling (against Carlo Rovelli's pseudo-problem);
- the dogma of semantic uniformity & Python Gored Naturalism;
- the odd transformation of Der Herr Warum (Gödel & Resnais);
- the only three types of ingenuity (with Cantor & Dedekind);
- self-help books (with Rupert Sheldrake);
- what is REAL space? what is REAL number?
- Timothy Leary in the 1990s;
- 5G?! Get real...
- list of charming scientists/engineers;
- pick a soul (ass you wish);
- view from Berthe Trépat's apartment;
- list des déclencheurs musicaux;
- Dark Consciousness (with Yasuo Yuasa);
- The Doors of Perception (with Huxley);
- Structuralism, Poststructuralism (with Julia Kristeva);
- List des figures du chaos primordial (Deleuze);
- Brazilian Perspectivism (Viveiros de Castro vs. Haroldo de Campos);
- Piano Playing (with Kochevitsky);
- L'Affirmation de l'âne (review of Smolin/Unger's The Singular Universe);
And also:
- Dogen with Hagakure;
- L'articulation (Maurice Blanchot);
- L'intelligence des fleurs;
- Spooky Blue;
- Interactive while indifferent—Kinds & Phantasmagoria circa 1900;