Geometrical problem solving in early modern mathematics and practical reasoning * Philosophy of mathematics between history of mathematics and philosophy Sébastien Maronne Institut de Mathématiques de Toulouse & SPHERE
8th French Philosophy of Mathematics Workshop Marseille, 3-5 November 2016
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
Mathematical tasks
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction
When confronted with a geometrical problem to solve, early modern mathematicians had to build strategies in order to provide a legitimate construction of the solution. In his article “Philosophical Challenges from History of Mathematics” (2004), Henk Bos stresses that the legitimacy issue under discussion deals with the procedures which exhibit mathematical objects and not with the ontology of these objects.
Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
Practical reasoning and early modern mathematics
Geometrical problem solving and practical reasoning Sébastien Maronne
This leads Bos to bring into focus the mathematical tasks rather than the concepts which result from the performance of these tasks and to conclude by the following question : Where, in present-day or earlier philosophy of mathematics, can I find explorations of a view of mathematics as the performance of self-imposed tasks ? In my talk, I will try to address this question by considering jointly a part of the abundant literature on philosophy of action which studies the logic of practical reasoning and early modern mathematics sources.
Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
Bos and the interpretation of exactness I use the term interpretation of exactness for such actions as formulating rules for proper procedure in mathematics. In the case of theorems the exactness concerned truth and proof ; in the case of problems it concerned tasks and task performance The interpretation of exactness shows us mathematicians confronting questions which can only be answered through extra-mathematical arguments. One cannot, for instance, argue from mathematical principles that ruler and compass (straight lines and circles) provide the proper means of geometrical construction. The foundational issue was in the choice of these means of constructions. Bos, “Philosophical challenges from history of mathematics”
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
Bos’ requisites I
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For many historical periods in which mathematical activity can be traced, the image of mathematics as centred on objects and propositions whose existence and truth are safeguarded by logical proofs, is not applicable. The historical perception of mathematics is through the actions of mathematicians as well as through their mathematical results ; as the latter are often difficult to recapture, it is essential to develop the understanding of the former. The actions of mathematicians are to be understood as performing self-imposed tasks according to self-created criteria for quality control.
Bos, “Philosophical challenges from history of mathematics”
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
Knowledge-how and knowledge-that in philosophy and mathematics The debate on the nature of knowledge-how and its relation with knowledge-that is very intense nowadays in analytic epistemology. It seems however that no attention has been given in this literature to the phenomenon Rota sought to bring forward : the frontiers between knowledge-how and knowledge-that in mathematics are not fixed once and for all. What is regarded, at a certain time, as a disposition or a skill (for instance, the capacity of solving a certain kind of enumeration problems) can become, thanks to a theoretical breakthrough, a full-fledged knowledge-that (the theory of Mobius inversion). This neglect can be easily explained : philosophers involved in the epistemological discussion consider mathematical knowledge as the paradigm of theoretical knowledge. Gandon, “Rota’s Philosophy in Mathematical Context” (2016)
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
My entry points in philosophy of action
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction
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Vincent Descombes, Le raisonnement de l’ours et autres essais de philosophie pratique, Paris, Seuil, 2007
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Bruno Gnassounou, Philosophie de l’action. Action, raison et délibération, Paris, Vrin, Coll. textes clés, 2007
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Jean-Michel Salanskis, Modèles et pensées de l’action, Paris, L’Harmattan, 2000
and, of course, Elizabeth Anscombe, Intention, Harvard University Press, 1957.
Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
Understanding philosophical texts through practice
Geometrical problem solving and practical reasoning
Lié d’une manière ou d’une autre à l’événement de l’enseignement oral, donc s’adressant avant tout à un groupe qui écoute le maître ou discute avec lui, l’écrit philosophique antique demande donc pour être compris, non seulement qu’on en analyse la structure, mais qu’on le situe dans la praxis vivante dont il émane et dans laquelle il se réinsère. On peut donc dire que tout ce que les modernes considèrent, de leur point de vue, comme des défauts de composition, comme des incohérences ou même des contradictions, tout cela provient, en premier lieu, des contraintes propres à l’enseignement oral.
Sébastien Maronne
Hadot, “La philosophie antique : une éthique ou une pratique”
Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
The literary aspects of mathematics L’emploi de la première personne dans la Géométrie diffère de celui des traités d’algèbre classiques. Naturellement, lorsqu’il explique les procédures de son analyse, le géomètre se propose bien en modèle. Mais alors qu’ailleurs l’auteur montre plutôt à l’apprenti les procédés techniques qu’il doit appliquer, sans toujours en dévoiler le mystère, Descartes affecte surtout de chercher à faire comprendre sa pensée et ses opérations pour que d’autres puissent ensuite en user selon leur génie. Lorsqu’il propose sa manière de construire un problème, il souligne en même temps que le lecteur peut en trouver d’autres par lui-même, et il en donne des exemples. Descotes, “Aspects littéraires de la Géométrie de Descartes”
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
Aristotle and practical reasoning We deliberate not about ends, but about things that are conducive to ends. For a doctor does not deliberate about whether to cure, nor an orator whether to persuade, nor a politician whether to produce good order ; nor does anyone else deliberate about his end. Rather they establish an end and then go on to think about how and by what means it is to be achieved. If it appears that there are several means available, they consider by which it will be achieved in the easiest and most noble way ; while if it can be attained by only one means, they consider how this will bring it about, and by what further means this means is itself to be brought about, until they arrive at the first cause, the last thing to be found. Aristotle, Nicomachean Ethics, Book III, Chap. 3
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
Deliberation and mathematical inquiry
Geometrical problem solving and practical reasoning Sébastien Maronne
For the person who deliberates seems to inquire and analyse in the way described as though he were dealing with a geometrical figure (it seems that not all inquiry is deliberation – mathematics, for example – but that all deliberation is inquiry), and the last step in the analysis seems to be the first that comes to be. If people meet with an impossibility, they give up : take, for example, the case where they need money, but there is none available. But if it seems possible they will try to do it. Aristotle, Nicomachean Ethics, Book III, Chap. 3
Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
An example of practical reasoning taken by Kenny from Aristotle’Metaphysics I I I I I
The man is to be healed. Iff his humours are balanced, he will be healed If he is heated, his humours will be balanced If he is rubbed, he will be heated So I’ll rub him.
Here the balancing of the humours is the necessary means to health ; rubbing and heating are means to this ; and rubbing is not impossible, but is in the doctor’s power ; so he begins his treatment by this, which was the last thing to occur in his practical reasoning. if formalised [it] would look like a bit of deduction based on the propositional calculus (’S ; Iff R, then S ; If Q then R ; If P then Q ; So P’). Kenny, “Practical inference” (1966)
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
Theorems and problems Classical Greek mathematics customarily divided the propositions of mathematics in two kinds : theorems and problems. Theorems were assertions which had to be proved. The result, therefore, was a mathematical truth. Problems were different, they implied a task which had to be performed according to certain rules. The formal execution of the task was called the construction. The construction was to be completed by the proof that the result was correct and characteristically the construction and proof together were closed by the statement QEF , Quod erat faciendum (This was to be done). The end result of a solved problem, then, was not primarily a proven truth but a properly performed task. Bos, “Philosophical challenges from history of mathematics”
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
The distinction between theorems and problems
Geometrical problem solving and practical reasoning Sébastien Maronne
Selon Proclus les problèmes ont pour but de procurer, de rendre manifeste, de construire ce qui en un certain sens n’existe pas, tandis que les théorèmes se proposent de constater, de connaître et de démontrer qu’une propriété appartient ou non à un objet. Il est permis de se demander si une telle distinction donne effectivement un critère de décision. En de nombreux cas, il est facile de formuler un problème comme un théorème, et inversement. Caveing, “Introduction générale”, Euclide, Les Eléments
Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
Two sides of the same practice ? En effet, le problème demande de construire un objet A remplissant des conditions B au moyen d’une construction C à inventer ; le théorème énonce que l’objet A, résultat de la construction C , possède la propriété B (la démonstration de ce théorème corrélé est d’ailleurs celle qu’on trouve dans le texte, comme preuve que la solution donnée au problème est fondée et pertinente). On comprend dès lors l’existence chez les Anciens de deux positions “réductionnistes” extrêmes et opposées. Proclus nous en avertit en effet que certains soutenaient que toutes les propositions étaient des théorèmes, en tant que propositions d’une science théorétique portant sur des objets éternels [Speusippe, Platon]. En revanche, la réduction contraire était préconisée par les mathématiciens de l’Ecole de Ménechme, qui soutenaient que tout est problème. Caveing, “Introduction générale”, Euclide, Les Eléments
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
Euclid’s Data
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
The defeasibility of practical inference : Geach
Geometrical problem solving and practical reasoning Sébastien Maronne
Indicative reasoning from a set of premises, if valid, could of course not be invalidated because there is a premise “missing” from the set. But a piece of practical reasoning from a set of premises can be invalidated thus : your opponent produces a fiat you have to accept, and the addition of this to the fiats you have already accepted yields a combination with which your conclusion is inconsistent. Geach, “Dr Kenny on practical inference”
Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
An example of Descombes : “L’ours et l’amateur des jardins”
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction
L’ours est chargé d’écarter les mouches pendant que le vieillard dort. Lorsqu’il n’arrive pas à chasser la mouche qui se place sur le nez du dormeur, il ne renonce pas : « Je t’attraperai bien, dit-il ; et voici comme. » Aussitôt fait que dit : le fidèle émoucheur vous empoigne un pavé, le lance avec raideur, Casse la tête à l’homme en écrasant la mouche ; Et, non moins bon archer que mauvais raisonneur, Raide mort étendu sur la place il le couche.
Descombes, Le raisonnement de l’ours
Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
Un mauvais raisonneur Notre ours est un mauvais raisonneur, comme dit fort bien La Fontaine. Il n’est pas seulement un “extrémiste”, un “passionné” qui manquerait de “modération”. Il raisonne mal, parce qu’il raisonne comme un monomaniaque. Il se comporte comme un agent attaché à un but unique, visant obstinément un objectif posé de façon inconditionnelle ou inaccessible à toute révision au cours de la réflexion sur les moyens d’atteindre le but. La prémisse manquante est évidemment que le vieillard doit continuer à dormir (et donc à vivre). Le but à atteindre (chasser la mouche) faisait lui même partie d’un but plus général (assurer le confort et le bien-être de l’Amateur des jardins). Descombes, Le raisonnement de l’ours
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
Cartesian compasses and the construction of mean proportionals
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
Descartes, La Géométrie, Livre III
A fault in geometry
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action
Descartes, La Géométrie, Livre III
Conclusion
Practical prescription versus technical prescription La décision qui sert de conclusion à la délibération pratique est une sorte de prescription : c’est un jugement toutes choses bien considérées sur ce qu’il faut faire ici et maintenant. Ce jugement est immédiatement pratique : une fois formé, il doit logiquement déboucher sur l’action. Or le point important est que, en cela, il s’oppose essentiellement aux prescriptions dues à l’art (à l’expertise de l’expert), qui sont fondées sur des considérations appartenant à un domaine particulier et qui, comme telles, ne peuvent déboucher immédiatement sur une action. Gnassounou, Philosophie de l’action
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
Aristotle’s notion of phronesis Par exemple, il y a des prescriptions qui sont obtenues à partir d’un point de vue uniquement médical : il faut lui donner tel médicament (une “prescription médicale”). Mais cette prescription n’a une valeur pratique (devant déboucher sur l’action) que conditionnelle. La décision de mettre en œuvre [cette prescription médicale] n’est justement pas un verdict médical. C’est un verdict non pas technique (relevant d’un art), mais proprement pratique (relevant de la meilleure chose à faire pour un homme). C’est cette capacité intellectuelle à délivrer de tels verdicts qu’Aristote appelle “prudence”. Gnassounou, Philosophie de l’action
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
When and how do we decide to stop the resolution of a problem ?
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction
Enfin, retournant à l’une des trois premières équations, et au lieu d’y ou de z mettant les quantités qui leur sont égales, et les carrés de ces quantités pour y 2 et z 2 , on trouve une équation où il n’y a que x et x 2 inconnus ; de façon que le problème est plan, et il n’est plus besoin de passer outre. Car le reste ne sert point pour cultiver ou récréer l’esprit, mais seulement pour exercer la patience de quelque calculateur laborieux. Descartes to Elisabeth, November 1643
Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
Descartes’ provisional moral code
Geometrical problem solving and practical reasoning Sébastien Maronne
Afin que ie ne demeurasse point irrésolu en mes actions, pendant que la raison m’obligerait de l’être en mes jugements, je me formai une morale par provision. La première [règle] était d’obéir aux lois & aux coutumes de mon pays. Ma seconde maxime était d’être le plus ferme & le plus résolu en mes actions que je pourrais. Descartes, Discours de la méthode, III
Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
Applying the first rule of provisional moral
Geometrical problem solving and practical reasoning Sébastien Maronne
Si vous avez le premier [dessein d’écrire pour les doctes], il ne me semble pas qu’il soit nécessaire d’y employer aucun nouveau terme : car les doctes, étant déjà accoutumés à ceux d’Apollonius, ne les changeront pas aisément pour d’autres, quoique meilleurs, et ainsi les vôtres ne serviraient qu’à leur rendre vos démonstrations plus difficiles, et à les détourner de les lire. Descartes to Desargues, 19 June 1639
Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
Teaching mathematical formalism Les notions de familles liées, libres et génératrices, notions de base de l’algèbre linéaire, se définissent au moyen de formules respectivement du type ∃x P(x), ∀x P(x) ⇒ Q(x) et ∀x∃y G (x, y ). D’où la tentation de passer par l’outil symbolique pour enseigner ces notions. D’expliciter la forme logique des définitions. Pour l’élève, la mise à plat de la structure logique, qui pourtant constitue “objectivement” a beaucoup d’égards la difficulté de la notion, n’est pas un moyen de sa maîtrise. Par conséquent, la seule voie d’apprentissage est d’enseigner l’objet lui-même et, sans doute, la structure logique en lui du même mouvement, mais informellement : “Si tu peux trouver un système de coefficients non tous nuls qui annule la combinaison linéaire, les vecteurs de ta famille forment un système lié” Salanskis, Modèles et pensées de l’action
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
The transfer of a mathematical action À un premier niveau, il semble que la faute procède d’une confusion des registres théoriques et pratiques. La mise à plat logique serait un moyen de décrire l’incompréhension du destinataire plutôt qu’une façon de lever celle-ci, c’est-à-dire d’agir (destructivement) sur elle. Le dégagement de la forme logique, notamment de la structure quantificationnelle, n’est pas du tout quelque chose de déconnecté de la pratique mathématique visée par l’enseignement. Seulement l’idée est que la communication directe de cette forme logique à quelqu’un qui n’est pas dans le bain du langage du calcul des prédicats n’a aucune chance de transmettre la pratique mathématique en question, alors que la langue naturelle, avec sa façon ingénue de parler d’objets et de faits, sans réflexion des formes de discours à cette fin employées, le peut. Salanskis, Modèles et pensées de l’action
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
Descartes’ method of normals
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
Descartes, La Géométrie, Livre III
Descartes’ method of normals : practical register
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action
Descartes, La Géométrie, Livre III
Conclusion
Descartes’ method of normals : theoretical register
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
Descartes, La Géométrie, Livre III P(x) = (x − α)2 Q(x)
The transfer of a mathematical action Conclusion
The practical virtue of omissions
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics
Descartes, La Géométrie, Livre III
Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
Conclusion : from philosophy to history of mathematics
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction
Following the method that Hadot used to study Ancient Greek philosophy, I have tried to show that : I
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Practical reasoning is a relevant category to describe and analyse mathematical practice ; Using this category helps us to better understand (historical) mathematical texts by interpreting what could appear as incoherences on the level of theoretical reasoning as (essential) ingredients of practical reasoning which play a key role in the transfer of mathematical action.
Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
Philosophy of mathematics between history of mathematics and philosophy
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction Related perspectives
Reciprocally, I hope that the use of such mathematical examples can be helpful to elaborate some issues relative to philosophy of action en général, for instance the contrast between practical and theoretical reasoning. Finally, I have tried to give a piece of philosophy of mathematics by confronting the problems of history of mathematics with those of general philosophy.
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action Conclusion
Geometrical problem solving and practical reasoning Sébastien Maronne Introduction Related perspectives
Theoretical and practical reasoning in philosophy and mathematics Practical reasoning Theorems and problems The defeasibility of practical inference Practical versus technical prescription
The transfer of a mathematical action
THANK YOU
Conclusion
