Professional mathematician. Doctorate in (a subarea of) mathematics from a German university.

Germany
Joined July 2018
»..diligite homines, interficite errores, sine superbia de veritate praesumite, sine saevitia pro veritate certate, orate pro eis quos redarguitis atque convincitis.« St. #Augustine(354–430), Contra Litteras Petiliani, Book 1, Chapter 29 #Christianity #ChristianCulture #Europe
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#PetermannPracticalHerbrand-3 [AppliedMathematics Logic] 'Relying on completeness proofs based on an algebraization technique presented in [Rasiowa-Sikorski (1964)], but taking a more optimistic view, we show how Herbrand disjunctions lead to a practical method for >
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>building-in theories into automatic theorem provers [Journal on Computer and Artificial Intelligence 11(2):105–142, 1992.]. Surveying newer results we discuss how to treat heterogeneous theories as well as practical implications of different normal form transformations. >
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Mathematics is about understanding proofs. (One may not delete either of the last two words.)
After 25 years of being told that theoretical physics isn’t mathematics because we prioritize understanding over proofs, I find the math-AI debate rather heartwarming. Suddenly, many mathematicians are explaining that mathematics is about understanding, not just proofs. Welcome!
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"The great task before us is to frame a covenant under which what has become optional—understanding—remains required: that nothing counts as mathematics until a human being has understood it—and can show the next person why." (Alexander Gamburd) xitter.cf/AGamburd18746/status/2…
Over three centuries ago our Enlightenment forefathers brought forth in Europe and carried to this continent a new way of knowing, and of telling what is known, conceived in distrust of authority, and dedicated to the proposition that truth is accessible to human reason. Mathematics, the Enlightenment’s firstborn, was its existence proof—knowledge that compels assent without appeal to authority—because the only way to make a theorem certain was to show it to a human mind. Now we are engaged in an existential struggle testing whether mathematics, or any discipline so conceived and so dedicated, can long endure. We are met in the wake of the OpenAI press release: 166 pages, 616,000 lines of Lean, produced in 88 hours by some 10,000 software agents, and read in full—at the time of this Siren Call—by no human being. A proof has historically been two things: a certificate that a theorem is true, and a demonstration that lets another person see why. OpenAI delivered the first without the second. But, in a larger sense, what matters is not what the Silicon Leviathan (OpenAI and its peer competitors) produces but - rather - what we accept. By accepting the certificate without the demonstration, we adopt the Hobbesian bargain: authority, not truth, makes the law. The age it augurs is a Middle Ages in reverse, in Panofsky’s phrase, with a sub-human superintelligence for its authority. The great task before us is to frame a covenant under which what has become optional—understanding—remains required: that nothing counts as mathematics until a human being has understood it—and can show the next person why. Whether we keep our sovereign inheritance will be decided at the moment of acceptance -- which is not yet past -- and will determine whether this Aufklärungsdämmerung spells the Enlightenment perishing from the earth, which nourishes all. arxiv.org/abs/2609.28591
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None of the following three is furthered by AI-math. 𝗣𝘂𝗿𝗽𝗼𝘀𝗲 of mathematicians: increase the world's SURETY. 𝗗𝗲𝘀𝗶𝗿𝗲 of mathematicians: increase their own UNDERSTANDING. 𝗖𝗼𝗻𝘁𝗿𝗶𝗯𝘂𝘁𝗶𝗼𝗻 of mathematicians: increase the world's STRUCTURE; see below.
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None of the following three is furthered by AI-math. 𝗣𝘂𝗿𝗽𝗼𝘀𝗲 of mathematicians: increase the world's SURETY. 𝗗𝗲𝘀𝗶𝗿𝗲 of mathematicians: increase their own UNDERSTANDING. 𝗖𝗼𝗻𝘁𝗿𝗶𝗯𝘂𝘁𝗶𝗼𝗻 of mathematicians: increase the world's STRUCTURE; see below.
The main contribution of mathematics to the world is the dialectical process of (1) adding mathematical structure(s) to the world (e.g., buildings made from right-angled pieces) (2) providing methods to PREDICT the behavior of said structures. There is back-and-forth (1)<->(2).
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Replying to @LGcommaI
"The great task before us is to frame a covenant under which what has become optional—understanding—remains required: that nothing counts as mathematics until a human being has understood it—and can show the next person why." (Alexander Gamburd) xitter.cf/AGamburd18746/status/2…
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"Ce qu’Euclide a fait avec le savoir de son temps, les mathématiciens n’ont jamais arrêté de le faire. Il a créé les conditions.. [Euclide] a rendu possible une compréhension structurée et universelle des choses, à l’usage de toutes les autres sciences." Simon Coste (2026)
"was"? Euclidean Geometry REMAINS beautiful. Exhibit A is Byrne's edition of The Elements, which is still being printed in new editions:
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The main contribution of mathematics to the world is the dialectical process of (1) adding mathematical structure(s) to the world (e.g., buildings made from right-angled pieces) (2) providing methods to PREDICT the behavior of said structures. There is back-and-forth (1)<->(2).
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Logic Geometry, Information retweeted
Les maths ont une place particulière dans l'imaginaire du grand public, entre détestation et fascination, et c'est pareil pour l'IA. Forcément, quand les deux se percutent et que ça fait la révolution scientifique la plus brutale jamais vue dans les sciences, ça génère beaucoup, mais alors beaucoup de fantasmes… J'ai écrit un essai dans Le Grand Continent, où je raconte ce que ça fait intimement d'être mathématicien en cette période. J'espère que mon petit récit enthousiaste parlera à des gens éloignés du monde de la recherche qui voudraient un aperçu de notre petit monde feutré. Les mathématiques du monde entier sont actuellement en plein déménagement, c'est une bonne occasion de rappeler les souvenirs, de sortir les vieux meubles cassés et de bien réfléchir au futur aménagement :) legrandcontinent.eu/fr/diman…
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#KreiselHilbertsProgramme-2 "Abstract Hilbert's plan for understanding the concept of infinity required the elimination of non-finitist machinery from proofs of finitist assertions. The failure of the original plan leads to a hierarchy of progressively less elementary, but >
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> Throughout open problems and alternative approaches are emphasized."
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> "Zusammenfassung Hilbert sah das Verstehen des Begriffs des Unendlichen in der Eliminierung nichtfiniter Methoden aus Beweisen finiter Sätze. Das Versagen dieses Unternehmens führt zu einer Hierarchie progressiv weniger elementarer, aber noch konstruktiver Methoden, >
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"Throughout, open problems and alternative approaches are emphasized." Georg Kreisel: Hilbert's Programme (1958)
#KreiselHilbertsProgramme-2 "Abstract Hilbert's plan for understanding the concept of infinity required the elimination of non-finitist machinery from proofs of finitist assertions. The failure of the original plan leads to a hierarchy of progressively less elementary, but >
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'Every good metaphor is itself a huge scientific contribution.' Mikhail Kapranov, in 2026 [v=eDkMHX7jBvQ; 26m17s]
Replying to @LGcommaI
> If we observe attentively, however, we realize that in the traditional exposition of the laws of logic certain fundamental arithmetic notions are already used [!], e.g., the notion of set and, to some extent, also that of number. Thus we find ourselves turning in a circle, >
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#ZachEpsilonCalculus-3 "Hilbert first presented his philosophical ideas based on the axiomatic method and consistency proofs in the years 1904 and 1905, following his exchange with Frege on the nature of axiomatic systems and the publication of Russell’s Paradox. >
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> In the text of Hilbert’s address to the International Congress of Mathematicians in Heidelberg, we read: ›Arithmetic is often considered to be part of Logic.. traditional fundamental logical notions are usually presupposed when [we think about the] foundation of arithmetic. >
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> If we observe attentively, however, we realize that in the traditional exposition of the laws of logic certain fundamental arithmetic notions are already used [!], e.g., the notion of set and, to some extent, also that of number. Thus we find ourselves turning in a circle, >
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#ZachEpsilonCalculus-1 [#BellottiVonNeumann1927 #HistoryOfMathematics #SlaterEpsilonCalculi #VonNeumann1927] Synthese 137(1/2), 211–259 (2003) The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert’s Program Richard Zach (University of Calgary, Canada) >
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> M.t. Die Praxis des Finitismus: Epsilon-Kalkül und Widerspruchsfreiheitsbeweise im Hilbertprogramm
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#SiegRavaglia-1 [BellottiVonNeumann1927 Logic SlaterEpsilonCalculi #VonNeumann1927 ZachEpsilonCalculus] Technical Report No. CMU-PHIL-154 (2004) David Hilbert and Paul Bernays, 𝘎𝘳𝘶𝘯𝘥𝘭𝘢𝘨𝘦𝘯 𝘥𝘦𝘳 𝘔𝘢𝘵𝘩𝘦𝘮𝘢𝘵𝘪𝘬 I and II: A Landmark Wilfried Sieg Mark Ravaglia
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#SiegRavaglia-1 [BellottiVonNeumann1927 Logic SlaterEpsilonCalculi #VonNeumann1927 ZachEpsilonCalculus] Technical Report No. CMU-PHIL-154 (2004) David Hilbert and Paul Bernays, 𝘎𝘳𝘶𝘯𝘥𝘭𝘢𝘨𝘦𝘯 𝘥𝘦𝘳 𝘔𝘢𝘵𝘩𝘦𝘮𝘢𝘵𝘪𝘬 I and II: A Landmark Wilfried Sieg Mark Ravaglia
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"Cardinalities feel like an artifact designed to hide some self-referential aspects." Leonid Levin, in 2026[!]
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This is from 2209.07497, a preprint with unusually-many versions:
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"Some of the uses to which Hilbert wanted to put his programme were ill-thought out [and] even its success would not have provided the separation of mathematics from philosophy which Hilbert wanted." Georg Kreisel [Dialectica 12, pp. 346–372 (1958)]
#KreiselObituaryForGoedel-7 "Without losing sight of the permanent interest of his work, Gödel repeatedly stressed—at least, during the second half of his life—how little novel mathematics was needed; only attention to some quite commonplace (philosophical) distinctions; >
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Relevant: xitter.cf/doomslide/status/21143… ( "there's urgent need to revisit the foundations of the subject from deliberately *non*-formalist pov.")
Why did I write such a long meandering essay? Well, at risk of sounding intelligible, there's urgent need to revisit the foundations of the subject from deliberately *non*-formalist pov. I feel this is difficult to do w/ purely formal arguments, so i went for show over tell.
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