Acompanhamento da construção de três moradias unifamiliares: a prespetiva do dono de obra

Trabalho final de Mestrado para a obtenção do grau de Mestre em Engenharia Civil

On the applicability of mathematics : philosophical and historical perspectives

The present thesis is a contribution to the debate on the applicability of mathematics; it examines the interplay between mathematics and the world, using historical case studies. The first part of the thesis consists of four small case studies. In chapter 1, I criticize "ante rem structuralism", proposed by Stewart Shapiro, by showing that his so-called "finite cardinal structures" are in conflict with mathematical practice. In chapter 2, I discuss Leonhard Euler's solution to the Königsberg bridges problem. I propose interpreting Euler's solution both as an explanation within mathematics and as a scientific explanation. I put the insights from the historical case to work against recent philosophical accounts of the Königsberg case. In chapter 3, I analyze the predator-prey model, proposed by Lotka and Volterra. I extract some interesting philosophical lessons from Volterra's original account of the model, such as: Volterra's remarks on mathematical methodology; the relation between mathematics and idealization in the construction of the model; some relevant details in the derivation of the Third Law, and; notions of intervention that are motivated by one of Volterra's main mathematical tools, phase spaces. In chapter 4, I discuss scientific and mathematical attempts to explain the structure of the bee's honeycomb. In the first part, I discuss a candidate explanation, based on the mathematical Honeycomb Conjecture, presented in Lyon and Colyvan (2008). I argue that this explanation is not scientifically adequate. In the second part, I discuss other mathematical, physical and biological studies that could contribute to an explanation of the bee's honeycomb. The upshot is that most of the relevant mathematics is not yet sufficiently understood, and there is also an ongoing debate as to the biological details of the construction of the bee's honeycomb. The second part of the thesis is a bigger case study from physics: the genesis of GR. Chapter 5 is a short introduction to the history, physics and mathematics that is relevant to the genesis of general relativity (GR). Chapter 6 discusses the historical question as to what Marcel Grossmann contributed to the genesis of GR. I will examine the so-called "Entwurf" paper, an important joint publication by Einstein and Grossmann, containing the first tensorial formulation of GR. By comparing Grossmann's part with the mathematical theories he used, we can gain a better understanding of what is involved in the first steps of assimilating a mathematical theory to a physical question. In chapter 7, I introduce, and discuss, a recent account of the applicability of mathematics to the world, the Inferential Conception (IC), proposed by Bueno and Colyvan (2011). I give a short exposition of the IC, offer some critical remarks on the account, discuss potential philosophical objections, and I propose some extensions of the IC. In chapter 8, I put the Inferential Conception (IC) to work in the historical case study: the genesis of GR. I analyze three historical episodes, using the conceptual apparatus provided by the IC. In episode one, I investigate how the starting point of the application process, the "assumed structure", is chosen. Then I analyze two small application cycles that led to revisions of the initial assumed structure. In episode two, I examine how the application of "new" mathematics - the application of the Absolute Differential Calculus (ADC) to gravitational theory - meshes with the IC. In episode three, I take a closer look at two of Einstein's failed attempts to find a suitable differential operator for the field equations, and apply the conceptual tools provided by the IC so as to better understand why he erroneously rejected both the Ricci tensor and the November tensor in the Zurich Notebook.

Biblia sacra, pars.

BIBLIA SACRA

Biblia Clementis VII papae.

Avec prologues et arguments. Genesis (1v), etc. — Esdras I-III (146v), etc. — Oratio Salomonis (232) ; Isaias (232v), etc. — Evang. Matthaei (342v), etc. — XIV Epist. Pauli (384) ; Actus Apost. (410v), etc. — Interpretationes nominum hebraicorum : « Aaz, apprehendens... — ... consiliatores eorum. » (433). F. 464 Fragment d'un traité de droit (sur la preuve) : « ... Dixit quoque idem Innocentius quodquam jus... »

Biblia sacra.

Avec prologues et arguments. Genesis (7v), etc. — Oratio Manasse (167) ; Esdras I-III (167v), etc. — Macchab. I-II, avec prologues de RABAN MAUR (353v, 354). — Evang. Matthaei (377), etc. — XIV Epist. Pauli (421v) ; Actus Apost. (448v) ; VII Epist. canon. (462) ; Apocalypsis (467v). — Interpretationes nominum hebraicorum : « Aaz, apprehendens... — ... consiliatores eorum. » (473v). — « Ordo librorum presentis voluminis. » (513v).

Biblia sacra, pars.

Avec prologues et arguments. Genesis (3) ; Reg. I-IV (13) ; Tobias (43) ; Job (45) ; Proverbia (51), etc. — Oratio Salomonis (70v) ; Isaias (71v), etc. — Daniel (99) ; Ezechiel (104v) ; XII Proph. min. (116). — Evang. Matthaei (130), etc. — XIV Epist. Pauli (156) ; Actus Apost. (172), etc. — Interpretationes nominum hebraicorum : « Aaz, apprehendens... — ... consiliatores eorum... secundum Jeronimum. » (188). F. 187 Fragment de liste des épîtres et évangiles, suivi d'hymnes ou proses à la Vierge : « Ave, virgo singularis, mundi splendor... », « Vas auri... », « Ave stella matutina ex qua... », « Oliva virens... », « Murenulis aureis rutilans... », « Mons altissimus... ».

Biblia sacra.

Avec prologues et arguments. Genesis (6), etc. — Oratio Manasse (170v) ; Esdras I-III (171v), etc. — Macchab. I-II, avec prologues de RABAN MAUR (360v). — Evang. Matthaei (382v), etc. — XIV Epist. Pauli (426) ; Actus Apost. (452v) etc. — Interpretationes nominum hebraicorum : « Aaz, apprehendens... — ... consiliatores eorum. » (477) ; « Abba, pater, Marci XIIII... — ... zizaniam. » (519). F. 1 et 2 Listes des livres de la Bible (XIIIe s.). F. 526 Liste des chefs d'Israël, de Moïse à Salomon, avec renvois à la Bible.

Biblia Ludovici XI regis Franciae.

Avec quelques prologues, arguments et « capitula ». Genesis, sans prologue (2), etc. — Reg. I-IV (82) ; Isaias (125v) ; Jeremias (140v) ; Jeremiae Lament. et Oratio (157v, 159v) ; Ezechiel (160) ; Daniel, I, 1-XIV, 41 (175v) ; XII Proph. min. (181) ; Job (193v), etc. — Ecclesiasticus (239v) ; Oratio Salomonis (252v) ; Paralip. I-II (253v) ; Oratio Manasse (273) ; Esdras I-II (273v) ; Esther (282) ; Tobias (286) ; Judith (289) ; Macchab. I-II (293v). — Evang. Matthaei (311v), Marci (321v), Lucae (328v-329, 338-343v, 336-337, 330), Johannis (331-335v, 344-346) ; Actus Apost. (347) ; VII Epist. canon. (358) ; Apocalypsis (364v) ; XIV Epist. Pauli (370v).

Biblia sacra.

Avec prologues et arguments. Genesis (3v), etc. — Oratio Manasse (158v) ; Esdras I-III (159), etc. — Macchab. I-II, avec prologues de RABAN MAUR (320). — Evang. Matthaei (340v), etc. — XIV Epist. Pauli (381v) ; Actus Apost. (406v), etc. — Interpretationes nominum hebraicorum : « Aaz, apprehendens... — ... consiliatores eorum. » (432). — « Liber ystorialis et florum totius Novi et Antiqui Testamenti. In Genesi sunt L capitula. In primo... — ... In XXVIII (Act. Apost.)... quidam crediderunt et quidam non. » (474). F. 430v et 431 En marge, recettes de médecine (XIVe-XVe s.). F. 431v Table des livres de la Bible (XIVe s.).

Biblia sacra.

Avec prologues et arguments. Genesis (4v), etc. — Oratio Manasse (198v) ; Esdras I-III (199), etc. — Macchab.I-II, avec prologues de RABAN MAUR (418), — Evang. Matthaei (443), etc. — XIV Epist. Pauli (492) ; Actus Apost. (522), etc. — Interpretationes nominum hebraicorum : « Aaz, apprehendens... — consiliatores eorum. » (550). F. 603v Vers mnémotechniques sur les livres de la Bible : « Primus adest Genesis... » (XVIe s.). F. 604 Table des livres de la Bible (XIVe-XVe s.).

Biblia sacra. I

Genesis (7), etc. — Ruth (203).

Biblia sacra.

Avec prologues et arguments. Épître et préface de S. JEROME, incompl. (1 et 2v). Genesis XX,8-L,25 (3), etc. — Esdras I-III (159v), etc. — Oratio Salomonis (252v) ; Isaias (253), etc. — Daniel II,38-XIV,42, avec prologue jusqu'à : « gravi volasse cor[pore] » (315v) ; XII Proph. min., jusqu'à Malachiam III,1 (322v) ; Macchab. I,I,46-II,XV,40 avec prologues de RABAN MAUR (340). — Evang. Matthaei (361), etc. — XIV Epist. Pauli (401v) ; Actus Apost. (427), etc. — Interpretationes nominum hebraicorum : « Az, in hebreo... — ... Gailoth exultas... ante devorationes. » (452).

Biblia sacra.

Avec quelques prologues, arguments et « capitula ». Genesis (2), etc. — Reg. I-IV (109v) ; Isaias (147v) ; Jeremias (165v) ; Jeremiae Lament. et Oratio (186v, 187v) ; Ezechiel (188) ; Daniel (205v) ; XII Proph. min. (212v) ; Job (226v), etc. — Ecclesiasticus (276v) ; Oratio Salomonis (291) ; Paralip. I-II (291v) ; Esdras I-II (315v) ; Esther (324) ; Tobias (328) ; Judith (331v) ; Macchab. I-II (336v). — Evang. Matthaei (359v), etc. — VII Epist. canon. (409v) ; Apocalypsis (415) ; V Epist. Pauli, jusqu'à Epist. ad Ephes. I,20 (421v).

Vetus Testamentum, pars.

Avec quelques prologues et « capitula ». Genesis (sans prologue), avec « capitula » incomplets du début (1) ; Exodus (22) ; Levit. (40) ; Numeri (52) ; Deuteron. (69v) ; Josue (86v) ; Judices (97v) ; Ruth (109v) ; Reg. I-III,XI,38 (112) ; Isaias (150) ; Jeremias (175) ; Jeremiae Lament. et Oratio (209, 212) ; Ezechiel V,3-XIII,6 (213).

Vetus Testamentum, pars.

Genesis I,29-L,25 (avec l'Épître de S. JEROME incomplète de la fin) (4) ; Exodus (27) ; Levit. (49) ; Numeri (63v) ; Deuteron. (85) ; Josue cum prol. (105, 106) ; Judices (121) ; Ruth (134v).