This article is an attempt to present Gödel's discussion on concepts, from 1944 to the late 1970s, in particular relation to the thought of Frege and Russell. The discussion takes its point of departure from Gödel's claim in notes on Bernay's review of ?Russell's mathematical logic?. It then retraces the historical background of the notion of intension which both Russell and Gödel use, and offers some grounds for claiming that Gödel consistently considered logic as a free-type theory of concepts, called (...) intensions, considered as the denotations of predicate names. (shrink)

Leibnizian-Newtonian calculus was a theory that dealt with geometrical objects; the figure continued to play one of the fundamental roles it had played in Greek geometry: it susbstituted a part of reasoning. During the eighteenth century a process of de-geometrization of calculus took place, which consisted in the rejection of the use of diagrams and in considering calculus as an 'intellectual' system where deduction was merely linguistic and mediated. This was achieved by interpreting variables as universal quantities and introducing the (...) notion of function, which replaced the study of curves. However, the emancipation of calculus from its basis in geometry was not comprehensive. In fact, the geometrical properties of curves were attributed de facto to functions and thus eighteenth-century calculus continued implicitly to use principles borrowed from geometry. There was therefore no transition to a purely syntactical theory based on axiomatically introduced terms, a shift which only took place subsequently in modern times. (shrink)

The purpose of this article is to explore the extent to which mathematics is subject to open texture and the extent to which mathematics resists open texture. The resistance is tied to the importance of proof and, in particular, rigor, in mathematics.

Medieval Arabic algebra is a good example of an artificial language.Yet despite its abstract, formal structure, its utility was restricted to problem solving. Geometry was the branch of mathematics used for expressing theories. While algebra was an art concerned with finding specific unknown numbers, geometry dealtwith generalmagnitudes.Algebra did possess the generosity needed to raise it to a more theoretical level—in the ninth century Abū Kāmil reinterpreted the algebraic unknown “thing” to prove a general result. But mathematicians had no motive to (...) rework their theories in algebraic form. Because it offered no advantage over geometry, algebra remained a practical art in both the Islamic world and in Europe until the scientific uphevals of the 17th–18th centuries. (shrink)

The representation of mathematical objects in terms of (more) basic ones is part and parcel of (the foundations of) mathematics. In the usual foundations of mathematics, namely, ZFC set theory, all mathematical objects are represented by sets, while ordinary, namely, non–set theoretic, mathematics is represented in the more parsimonious language of second-order arithmetic. This paper deals with the latter representation for the rather basic case of continuous functions on the reals and Baire space. We show that the logical strength of (...) basic theorems named after Tietze, Heine, and Weierstrass changes significantly upon the replacement of “second-order representations” by “third-order functions.” We discuss the implications and connections to the reverse mathematics program and its foundational claims regarding predicativist mathematics and Hilbert’s program for the foundations of mathematics. Finally, we identify the problem caused by representations of continuous functions and formulate a criterion to avoid problematic codings within the bigger picture of representations. (shrink)

Este estudio aborda el conocimiento y el trabajo matemático del profesor durante la enseñanza de las representaciones de la función a través del uso en conjunto de dos modelos teóricos. Se analiza una sesión para el 1er año de enseñanza media (14-15 años) dada por un profesor de matemática con la categoría de experto. Los resultados dan cuenta de la relación entre el trabajo matemático que se propone en el aula y los conocimientos que permiten esta organización, aportando elementos a (...) la comprensión de la práctica docente. (shrink)

One of the distinguishing features of mathematics is the exceptional level of consensus among mathematicians. However, an analysis of what mathematicians agree on, how they achieve this agreement, and the relevant historical conditions is lacking. This paper is a programmatic intervention providing a preliminary analysis and outlining a research program in this direction.First, I review the process of ‘negotiation’ that yields agreement about the validity of proofs. This process most often does generate consensus, however, it may give rise to another (...) kind of disagreement: whether the original and new proof are effectively the same, leading to potential disagreement about the validity of the original proof.Second, I historicize the phenomenon of consensus. I show that in earlier European mathematics and other mathematical cultures, consensus about the validity of arguments was substantially weaker or conceived differently than it is today. This means that contemporary consensus about the validity of mathematical proofs should be explained by historical changes in mathematical practice.Finally, I conjecture what brought about the contemporary form of mathematical consensus. Since a sharp rise in consensus occurs around the turn of the 20th century, it makes sense to explain this consensus by the concurrent formalization of mathematics. However, this explanation has a major flaw: it explains a really existing phenomenon (consensus) by something that hardly ever happens (writing proofs in formal languages). I will therefore explain the ways in which formalization does enter mathematical practice so as to account for contemporary forms of mathematical consensus. (shrink)

We reconstruct essential features of Lagrange’s theory of analytical functions by exhibiting its structure and basic assumptions, as well as its main shortcomings. We explain Lagrange’s notions of function and algebraic quantity, and we concentrate on power-series expansions, on the algorithm for derivative functions, and the remainder theorem—especially on the role this theorem has in solving geometric and mechanical problems. We thus aim to provide a better understanding of Enlightenment mathematics and to show that the foundations of mathematics did not, (...) for Lagrange, concern the solidity of its ultimate bases, but rather purity of method—the generality and internal organization of the discipline. (shrink)

Le mathématicien au travail sait faire un geste que l'on appelle la« pulsation mathématique», qui s'exprime en tennes de bougé créatif nécessaire dans les diagrammes de pensée et d'interprétation des écrits mathématiques. Dans cette perspective Je statut d'objet est définitivement en révision, sous condition du jeu des relations. Le but ici est de construire aujourd'hui cette pulsation à partir de ce que Bachelard proposait hier comme épistémologie, aussi bien de la mathématique que de la science dite physique mathématique. Les liens (...) entre la pensée bachelardienne et les propositions plus récentes de Gilles Châtelet, Charles Alunni, ou René Guitart, sont mis en relief ainsi que ceux d'autres auteurs comme Jacques Lacan, Arthur Koestler, Alfred N. Whitehead, Charles S. Peirce. Nous proposons que le travail mathématique soit pensable comme du mouvement dans l'espace des diagrammes, du bougé pulsatif en icelui, et que cette manière de voir soit éminemment compatible avec la vue bacheJardienne suivant laquelle la forme intellectuelle précède l'objet empirique, lequel provient in fine, comme d'une concrétion de la pulsation de la pensée. Nous terminons donc sur un schème catégoricien de la pulsation. (shrink)

Frege's docent's dissertation Rechnungsmethoden, die sich auf eine Erweiterung des Grössenbegriffes gründen(1874) contains indications of a bold attempt to extend arithmetic. According to it, arithmetic means the science of magnitude, and magnitude must be understood structurally without intuitive support. The main thing is insight into the formal structure of the operation of ?addition?. It turns out that a general ?magnitude domain? coincides with a (commutative) group. This is an interesting connection with simultaneous developments in abstract algebra. As his main application, (...) Frege studies iterations of functions. He does not yet pose the question of existence proofs. Measurement of magnitudes is also connected to numbers, but the discussion is here ambiguous in a way which calls for the systematic account of numbers in Grundgesetze. (shrink)

Dedekind's mathematical work is integral to the transformation of mathematics in the nineteenth century and crucial for the emergence of structuralist mathematics in the twentieth century. We investigate the essential components of what Emmy Noether called, his ‘axiomatic standpoint’: abstract concepts, models, and mappings.

In his writings about hypergeometric functions Gauss succeeded in moving beyond the restricted domain of eighteenth-century functions by changing several basic notions of analysis. He rejected formal methodology and the traditional notions of functions, complex numbers, infinite numbers, integration, and the sum of a series. Indeed, he thought that analysis derived from a few, intuitively given notions by means of other well-defined concepts which were reducible to intuitive ones. Gauss considered functions to be relations between continuous variable quantities while he (...) regarded integration and summation as appropriate operations with limits. He also regarded infinite and infinitesimal numbers as a façon de parler and used inequalities in order to prove the existence of certain limits. He took complex numbers to have the same legitimacy as real quantities. However, Gauss’s continuum was linked to a revised form of the eighteenth-century notion of continuous quantity: it was not reducible to a set of numbers but was immediately given. (shrink)

In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. We survey implicit and explicit uses ofDirichlet characters in presentations of Dirichlet’s proof in the nineteenth and early twentieth centuries, with an eye toward understanding some of the pragmatic pressures that shaped the evolution of modern mathematical method.