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Chance, Necessity and Convention

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Essay · Found or Made, Afterword · September 2026 Seven essays on whether mathematics, physics and biology are found or made turn out to have been circling the oldest pair in natural philosophy: chance and necessity. They also turn out to need a third term the pair does not have — and the third term is where the disciplines differ.
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The Frozen Accident

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Essay · biology · September 2026 Physics trusts beauty; Crick warned that biology cannot. Yet the genetic code is one in a million, the eye has been invented forty times, and the stripes on a fish obey an equation written in 1952. How much of life is found, and how much is the tape happening to run that way.
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The Diagram That Did The Sum

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Essay · physics · September 2026 Feynman's diagrams are the one case in which a private picture became the calculus of a whole field. They are also the clearest warning about what a picture can make you believe, and — in the last forty years — the strangest evidence that the picture was hiding something simpler underneath.
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The Bus at Coutances

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Essay · Found or Made, Part V · September 2026 Ask working mathematicians and physicists what is in their heads while they work, and almost none say words or equations. They say pictures, shapes, a sense of something moving. The symbols come afterwards. What that tells us about mathematics depends on what happens to the pictures when you take them away — and there are people to ask.
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Where Number Comes From

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Essay · Found or Made, Part IV · September 2026 Every brain that has been tested carries an approximate sense of quantity — newborns, crows, honeybees. Mathematics is exact. The gap between the two is the best evidence we have for where mathematics lives, and closing it is an open scientific problem, not a philosophical one.
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What a Proof Is Now

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Essay · Found or Made, Part III · September 2026 For twenty-three centuries a proof was something a person could read and be convinced by. In 1976 that stopped being true, and the question of what mathematics is has quietly turned into a question about who — or what — is doing the knowing.
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The Limits of the Rules

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Essay · Found or Made, Part II · September 2026 The first essay ended with a division of labour: we choose the rules, we discover what follows. In September 1930 a twenty-four-year-old showed that what follows can outrun any rulebook, and in 1963 another mathematician showed that some questions do not follow at all. Whether that helps the Platonist or the formalist is still being argued.
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Invented, Then Unavoidable

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Essay · Found or Made, Part I · September 2026 Imaginary numbers were dismissed as useless for two centuries before they became load-bearing in physics. That history is the sharpest test we have of whether mathematics is found or made — and of why physicists trust beauty.
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Symplectic Geometry

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A 1993 licenciatura thesis from Instituto Superior Técnico, transcribed and annotated. It asks what survives of symplectic geometry when the form is allowed to degenerate — when some directions become invisible to it — and shows that what remains, once those directions are collapsed away, is again a phase space on which mechanics works.
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Linear average time automorphism algorithm for random graphs.

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A 1996 Master's paper from the Department of Physics at Pennsylvania State University, reproduced in full. It develops an algorithm that finds the automorphisms of a random graph in linear average time, and with them the Variety — Lee Smolin's measure of how well each point of a structure can be told apart from every other, offered as a quantitative handle on complexity in diffeomorphism-invariant theories such as gravity.
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Contexto histórico internacional: a história mundial do trail como pano de fundo

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O trail running português não se entende sem o pano de fundo mundial que importou: o fell running britânico do séc. XIX, o ultra norte-americano (Western States 1974, Leadville 1983), o UTMB (2003), o skyrunning, e a institucionalização (ITRA 2013, World Athletics 2015). Este é o capítulo-referência da série — e o relógio contra o qual Portugal se lê, chegando por volta de 2005–2008, uma geração depois dos acontecimentos fundadores.
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Provas XL: a escalada da distância no trail português

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Em duas décadas, o trail português passou de uma prova mais longa que mal chegava aos 100 km para um pelotão de ultras «XL»: as 100 milhas normalizadas, um teto contínuo nos 300 km (a ALUT, no Algarve) e o backyard, que já nem fixa a distância à partida. Este dossiê traça a timeline datada de «a mais longa de Portugal» e mede a escalada com o arquivo de classificações do autor — corrigindo, logo à cabeça, a ideia de que «a Freita foi a mais longa».
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As corridas de montanha (1995–2005): o elo que ligou a montanha ao trail

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Entre a cultura de montanha que já existia (Dossiê 61) e o trail moderno que chegaria depois, houve uma disciplina intermédia: as corridas de montanha competitivas dos anos 90 e 2000. Este dossiê é o seu panorama nacional — e mostra que ela não correu numa casa federativa única, mas em duas linhagens paralelas, a do atletismo (com campeão nacional já em 1996) e a do montanhismo (a FPME, de 2002). Foi essa dualidade que o trail herdou.
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