Frege's arithmetical-platonism is glossed as the first step in developing the thesis; however, it remains silent on the subject of structures in mathematics: the obvious examples being groups and rings, lattices and topologies. The structuralist objects to this silence, also questioning the sufficiency of Fregean platonism is answering a number of problems: e.g. Benacerraf's Twin Puzzles of Epistemic and Referential Access. The development of structuralism as a philosophical position, based on the slogan 'All mathematics is structural' collapses: there is no single coherent account which remains faithful to the tenets of structuralism and solves the puzzles of platonism. This prompts the adoption of a more modest structuralism, the aim of which is not to solve the problems facing arithmetical-platonism, but merely to give an account of the 'obviously structural areas of mathematics'. Modest strucmralism should complement an account of mathematical systems; here, Frege's platonism fulfils that role, which then constrains and shapes the development of this modest structuralism. Three alternatives are considered; a substitutional account, an account based on a modification of Dummett's theory of thin reference and a modified from of in re structuralism. This split level analysis of mathematics leads to an investigation of the robustness of the truth predicate over the two classes of mathematical statement. Focussing on the framework set out in Wright's Truth and Objectivity, a third type of statement is identified in the literature: Hilbert's formal statements. The following thesis arises: formal statements concern no special subject matter, and are merely minimally truth apt; the statements of structural mathematics form a subdiscourse - identified by the similarity of the logical grammar - displaying cognitive command. Thirdly, the statements of mathematics which concern systems form a subdiscourse which has both cognitive command and width of cosmological role. The extensions of mathematical concepts are such that best practice on the part of mathematicians either tracks or determines that extension - at least in simple cases. Examining the notions of response dependence leads to considerations of indefinite extensibility and intuitionism. The conclusion drawn is that discourse about structures and mathematical systems are response dependent but that this does not give rise to any revisionary arguments contra intuitionism.

Frege is an open-source project which brings the popular functional programming language Haskell to the Java ecosystem. JetBrains MPS is an open-source language workbench which allows users to design a new language and build an integrated development environment with a projectional (structured) editor for the created language. In this work we analyzed Frege grammar and created an IDE based on MPS that assists developers with writing code in the Frege language. Our environment includes a set of intuitive editors for editing Frege syntax, provides a simple type checking and implements code generators for the Frege language. Aim of the Frege IDE is its usability. Additionally, the thesis compares projectional editors with the more common plain-text IDEs, such as Eclipse, and evaluates whether they offer any advantage for editing purely functional programming languages.

My thesis is an exposition and defence of Aristotle's philosophy of mathematics. The first part of my thesis is an exposition of Aristotle's cryptic and challenging view on mathematics and is based on remarks scattered all over the corpus aristotelicum. The thesis' central focus is on Aristotle's view on numbers rather than on geometrical figures. In particular, number is understood as a countable plurality and is always a number of something. I show that as a consequence the related concept of counting is based on units. In the second part of my thesis, I verify Aristotle's view on number by applying it to his account of time. Time presents itself as a perfect test case for this project because Aristotle defines time as a kind of number but also considers it as a continuum. Since numbers and continuous things are mutually exclusive this observation seems to lead to an apparent contradiction. I show why a contradiction does not arise when we understand Aristotle properly. In the third part, I argue that the ontological status of mathematical objects, dubbed as materially [hulekos, ÍlekÀc] by Aristotle, can only be defended as an alternative to Platonism if mathematical objects exist potentially enmattered in physical objects. In the fourth part, I compare Aristotle's and Plato's views on how we obtain knowledge of mathematical objects. The fifth part is an extension of my comparison between Aristotle's and Plato's epistemological views to their respective ontological views regarding mathematics. In the last part of my thesis I bring Frege's view on numbers into play and engage with Plato, Aristotle and Frege equally while exploring their ontological commitments to mathematical objects. Specifically, I argue that Frege should not be mistaken for a historical Platonist and that we find surprisingly many similarities between Frege and Aristotle. After having acknowledged commonalities between Aristotle and Frege, I turn to the most significant differences in their views. Finally, I defend Aristotle's abstractionism in mathematics against Frege's counting block argument. This whole project sheds more light on Aristotle's view on mathematical objects and explains why it remains an attractive view in the philosophy of mathematics.

The early Husserl wants to know what logic is, or what we should call ‘logic.’ He poses the question in a way that knowingly encompasses both what the 19th century (after Kant but before Frege) and the 20th century (since Frege) call ‘logic.’ But that he asks the question, and with such scope, has yet to be widely recognized. In particular, Husserl scholars still lack an overview of how Husserl’s early, explicitly logical inquiries, driven more by this single question than any worry about doctrinal consistency, does at least two things at once: probe what will later be called ‘pure phenomenology’ or ‘transcendental logic,’ and delimit logic as a positive yet mathematical discipline. With the aim of providing the neglected overview of this project, this dissertation takes the measure of Husserl’s two-volume Logical Investigations (1900-01) in four studies. Chapter I argues that the first volume, the Prolegomena to Pure Logic (1900), intends at once to resolve a 19th-century conflict and to establish logic’s possibility as its own discipline, all by means of demonstrating the confusion of psychologism (the view that empirical psychology could set the terms for logic as a discipline). Chapter II then contends that most of the Prolegomena’s first chapter falls outside this intention, departing from the book’s Bolzano-inspired argumentative framework yet thereby anticipating Husserl’s later ‘transcendental logic.’ Chapter III presents Frege and Husserl as two images of indecision as to how it falls to logic to know truth’s laws. Chapter IV concludes by expounding Husserl’s conception of logic as noetics, the self-clarification of knowing, thus completing the picture of Husserl’s indecision, while also laying groundwork to track the development of his thinking after the Logical Investigations.

A logic that utilizes higher-order quantification --quantifying over concepts (or relations), not just over the first-order level of individuals-- can be interpreted standardly or nonstandardly depending on whether one takes an intensional or extensional view of concepts. I argue that this decision is connected to how one understands the mathematical notion of function. A function is often understood as a rule that, when given an argument from a set of objects called a "domain," returns a value from a set of objects called a "codomain." Because a concept can be thought of as a two-valued function (that indicates whether or not a given object falls under the concept), having an extensional interpretation of higher-order variables --the standard interpretation-- requires that one adopt an extensional notion of function. Viewed extensionally, however, a function is understood not as a rule but rather as a correlation associating every element in a domain with an element in a codomain. When the domain is finite, the two understandings of function are equivalent (since one can define a rule for any finite correlation), but with an infinite domain, the latter understanding admits arbitrary functions, or correlations not definable by a finitely specifiable rule. Rejection of the standard interpretation is often motivated by the same reasons used to resist the extensional understanding of function. Such resistance is overt in the pronouncements of Leopold Kronecker, but is also implicit in the work of Gottlob Frege, who used an intensional notion of function in his logic. Looking at the problem historically, I argue that the extensional notion of function has been basic to mathematics since ancient times. Moreover, I claim that Gottfried Wilhelm Leibniz's combination of mathematical and metaphysical ideas helped inaugurate an extensional and ultimately model-theoretical approach to mathematical concepts that led to some of the most important applications of mathematics to science (e.g. the use of non-Euclidean geometry in the theory of general relativity). In logic, Frege's use of an intensional notion of function led to contradiction, while Richard Dedekind and Georg Cantor applied the extensional notion of function to develop mathematically revolutionary theories of the transfinite. / 2025-10-15

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Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

[pt] A dissertação apresenta e discute as idéias desenvolvidas por Crispin Wright no livro Frege´s Conception of Numbers as Objects (1983), em particular, a sua tese de que a aritmética é analítica. Wright deposita toda sua força argumentativa (em relação à analiticidade da aritmética) na derivação dos axiomas da aritmética de segunda ordem de Dedekind-Peano a partir do Princípio de Hume. Assim, é nosso principal objetivo apresentar e discutir em que medida o Princípio de Hume é capaz de fornecer, segundo Wright, um relato da analiticidade da aritmética, assim como, as objeções a esse relato. / [en] The dissertation presents and discusses the ideas developed by Crispin Wright in his book Frege's Conception of Numbers as Objects (1983), in particular his thesis that arithmetic is analytic. Wright concentrates all his argumentative efforts (in relation to the analyticity of arithmetic) on the derivation of the axioms of Dedekind-Peano's second order arithmetic from Hume's Principle. Thus, it is our main goal to present and discuss how Hume's Principle provides, according to Wright, an explanation of the analytic character of arithmetic as well as some objections to this account.

[pt] A intuição básica da noção de verdade como correspondência é que se uma proposição (ou sentença) p é verdadeira, então existe um s tal que s é o fazedor-de-verdade (truthmaker) de p. Essa idéia tem um apelo especialmente forte no que diz respeito a proposições verdadeiras em virtude de fenômenos ou objetos empíricos. Por outro lado, se não há alternativa para a tese de Frege segundo a qual a referência de uma sentença é o seu valor de verdade, uma teoria da verdade como correspondência é inviável. O argumento da funda (the slingshot argument) pretende defender a tese de Frege e inviabilizar uma teoria da verdade como correspondência. Os meus objetivos aqui são (i) investigar o que levou Frege a concluir que a referência de uma sentença é seu valor de verdade e (ii) investigar se uma teoria de fazedores-de-verdade de verdades empíricas evita o argumento da funda. / [en] The basic idea of the notion of truth as correspondence is that if a proposition (or sentence) p is true, then there is an s such that s makes p true (i.e. s is a truthmaker of p). This idea has a strong intuitive appeal, especially with respect to propositions (or sentences) true in virtue of empirical phenomena. On the other hand, if there is no alternative to Frege's thesis according to which the reference of a sentence is its truth-value, a theory of truth as correspondence seems to be undermined from the start. The slingshot argument intends to defend Frege's thesis and to undermine theories of truth as correspondence. My aims here are (i) to investigate why Frege concluded that the reference of a sentence is its truth- value and (ii) to investigate whether or not a truthmaker theory of empirical truths can avoid the slingshot argument.

Ce travail s’engage dans la reconstitution d’une intelligibilité globale nouvelle pour la logique qui est née avec Frege afin de restituer l’une des conditions décisives pour la philosophie contemporaine, à savoir celle qui concerne son rapport aux pratiques et aux savoirs formels. Son hypothèse initiale affirme que le projet premier et constant de Frege a été celui d’une logique du contenu. Pourtant, il ne s’agit pas de réinvestir l’œuvre de Frege d’une cohérence nouvelle dans le but de rétablir une unité stable. Car l’intelligibilité procurée par cette reconstitution permet de localiser dans les formulations de Frege de véritables lacunes qui ne semblent pas avoir été identifiées comme telles jusqu’ici. Que la logique de Frege soit efficace malgré ces lacunes, voilà ce qu’il faut expliquer. La réponse que nous donnons à ces questions est que l’efficacité de la logique de Frege en tant que logique du contenu provient d’un certain rapport à l’Arithmétique, à savoir celui par lequel c’est la logique qui est construite d’après les principes de l’Arithmétique, avant qu’elle ne soit capable de la construire à son tour. La question se pose alors de caractériser avec précision à ce niveau constitutif, non « fondationnel », la nature du rapport entre une logique du contenu comme forme spécifique de la logique dans le cadre de sa mathématisation, et l’Arithmétique comme domaine mathématique particulier. De l’analyse minutieuse de la constitution du système logique frégéen, une idée se dégage qui constitue la thèse centrale de notre travail : les différents systèmes de la logique mathématisée ou formelle ne reposent sur les mathématiques que par l’intermédiaire d’une dimension d’exercice, de réflexion et d’élaboration de signes, où les circulations et les emprunts entre ces deux savoirs formels contemporains que sont les mathématiques et la logique se construisent et se justifient. C’est donc cette thèse qu’il s’agit de démontrer, par une étude détaillée des processus d’émergence des deux plus grands projets de formalisation de la logique du XIXe siècle : celui de Frege et celui de Boole et des Booléens. Dans cet espace qui mène des pratiques mathématiques aux systématisations logiques à travers les fonctionnements des signes, deux régimes généraux se dessinent : celui d’ « Abstraction symbolique » qui mène de l’Algèbre abstraite à la Logique propositionnelle booléenne ; et celui de l’ « Expressionnisme », qui mène de l’Arithmétique au Calcul logique des prédicats, associée aux travaux de Frege. Mais plus profondément, par l’effet d’une lecture symptomale au plus près des dynamiques internes à ces processus, le présent travail décèle un lien transversal entre le contenu logique d’une part et l’Arithmétique comme ensemble des déterminations du nombre de l’autre. En suivant ce lien, qui s’avère le responsable de l’introduction de la catégorie de sens dans le cadre de la logique mathématisée, une théorie de l’expression formelle se dessine, définissant les conditions pour le développement d’une logique du sens. / This work aims at providing a new general interpretation of the logic that was born with the work of Gottlob Frege, in order to make explicit one of the most decisive conditions of contemporary philosophy: the one that concerns the relation of philosophy to formal practices and knowledge. Its initial hypothesis states that Frege’s primary and most constant project was that of building a logic of content. However, the intelligibility thus gained does not intend to unearth a new underlying unity of Frege’s thought; it rather aims at localising the real gaps within Frege’s formulations that have not been identified as such until now. Still, those gaps do not require to be filled, for Frege’s logic is indeed effective despite this indeterminacy. Rather than the gaps, it is this ungrounded effectiveness that needs to be explained. Our answer to this question is that the effectiveness of Frege’s logic as a logic of content comes from a certain relationship with Arithmetic; in fact, Frege’s logic is constructed on the template of Arithmetic, before it becomes capable of constructing Arithmetic in turn. The task then arises to characterise precisely, at this constitutive and non-foundational level, the nature of the relation between a logic of content as a specific form of logic in the framework of its mathematization, and Arithmetic as a particular mathematical domain. From the meticulous study of the constitution of the Fregean system, an idea can be drawn that constitutes the central argument of this thesis: the various mathematical or formalised logical systems rest upon mathematics only through an intermediary dimension consisting in the practice, the reflection and the elaboration of signs, where the circulations between these two contemporary domains of formal knowledge (mathematics and logic) are constructed and justified. From this point of view, we then lay out a detailed study of the rise of the two most significant projects for formalizing logic in the nineteenth century: Frege’s and Boole’s (and the Booleans’). In the space leading from mathematical practices to logical systematisations through semiotic functioning, two general schemes or semiotic formal regimes can be drawn: “Symbolic Abstraction”, leading from abstract Algebra to Boolean propositional logic; and “Expressionism”, leading from Arithmetic to Predicate Calculus, associated to Frege’s work. More deeply, our research reveals a deep connexion between logical content and Arithmetic (understood as the theory of integers), which horizontally crosses the different semiotic regimes. Following the multiple dimensions of this nexus – which is responsible for the introduction of the category of sense in the framework of mathematized logic – a formal theory of expression can be drawn, which defines the conditions for the actual development of a logic of sense.