A study of the making of George Peacock's highly influential, yet disturbingly split, 1830 account of algebra as an entanglement of two separate undertakings: arithmetical and symbolical or formal.

Non-Archimedean probability functions allow us to combine regularity with perfect additivity. We discuss the philosophical motivation for a particular choice of axioms for a non-Archimedean probability theory and answer some philosophical objections that have been raised against infinitesimal probabilities in general.

Esta enciclopédia abrange, de uma forma introdutória mas desejavelmente rigorosa, uma diversidade de conceitos, temas, problemas, argumentos e teorias localizados numa área relativamente recente de estudos, os quais tem sido habitual qualificar como «estudos lógico-filosóficos». De uma forma apropriadamente genérica, e apesar de o território teórico abrangido ser extenso e de contornos por vezes difusos, podemos dizer que na área se investiga um conjunto de questões fundamentais acerca da natureza da linguagem, da mente, da cognição e do raciocínio humanos, bem (...) como questões acerca das conexões destes com a realidade não mental e extralinguística. A razão daquela qualificação é a seguinte: por um lado, a investigação em questão é qualificada como filosófica em virtude do elevado grau de generalidade e abstracção das questões examinadas (entre outras coisas); por outro, a investigação é qualificada como lógica em virtude de ser uma investigação logicamente disciplinada, no sentido de nela se fazer um uso intenso de conceitos, técnicas e métodos provenientes da disciplina de lógica. O agregado de tópicos que constitui a área de estudos lógico-filosóficos é já visível, pelo menos em parte, no Tractatus Logico-Philosophicus de Ludwig Wittgenstein, uma obra publicada em 1921. E uma boa maneira de ter uma ideia sinóptica do território disciplinar abrangido por esta enciclopédia, ou pelo menos de uma porção substancial dele, é extrair do Tractatus uma lista dos tópicos mais salientes aí discutidos; a lista incluirá certamente tópicos do seguinte género, muitos dos quais se podem encontrar ao longo desta enciclopédia: factos e estados de coisas; objectos; representação; crenças e estados mentais; pensamentos; a proposição; nomes próprios; valores de verdade e bivalência; quantificação; funções de verdade; verdade lógica; identidade; tautologia; o raciocínio matemático; a natureza da inferência; o cepticismo e o solipsismo; a indução; as constantes lógicas; a negação; a forma lógica; as leis da ciência; o número. (shrink)

A version of nonstandard analysis, Internal Set Theory, has been used to provide a resolution of Zeno's paradoxes of motion. This resolution is inadequate because the application of Internal Set Theory to the paradoxes requires a model of the world that is not in accordance with either experience or intuition. A model of standard mathematics in which the ordinary real numbers are defined in terms of rational intervals does provide a formalism for understanding the paradoxes. This model suggests that in (...) discussing motion, only intervals, rather than instants, of time are meaningful. The approach presented here reconciles resolutions of the paradoxes based on considering a finite number of acts with those based on analysis of the full infinite set Zeno seems to require. The paper concludes with a brief discussion of the classical and quantum mechanics of performing an infinite number of acts in a finite time. (shrink)

Two alternative accounts of quantum spontaneous symmetry breaking (SSB) are compared and one of them, the decompositional account in the algebraic approach, is argued to be superior for understanding quantum SSB. Two exactly solvable models are given as applications of our account -- the Weiss-Heisenberg model for ferromagnetism and the BCS model for superconductivity. Finally, the decompositional account is shown to be more conducive to the causal explanation of quantum SSB.

The small, the tiny, and the infinitesimal have been the object of both fascination and vilification for millenia. One of the most vitriolic reviews in mathematics was that written by Errett Bishop about Keisler’s book Elementary Calculus: an Infinitesimal Approach. In this skit we investigate both the argument itself, and some of its roots in Bishop George Berkeley’s criticism of Leibnizian and Newtonian Calculus. We also explore some of the consequences to students for whom the infinitesimal approach is congenial. The (...) casual mathematical reader may be satisfied to read the text of the five act play, whereas the others may wish to delve into the 130 footnotes, some of which contain elucidation of the mathematics or comments on the history. (shrink)

We seek to elucidate the philosophical context in which one of the most important conceptual transformations of modern mathematics took place, namely the so-called revolution in rigor in infinitesimal calculus and mathematical analysis. Some of the protagonists of the said revolution were Cauchy, Cantor, Dedekind,and Weierstrass. The dominant current of philosophy in Germany at the time was neo-Kantianism. Among its various currents, the Marburg school (Cohen, Natorp, Cassirer, and others) was the one most interested in matters scientific and mathematical. Our (...) main thesis is that Marburg neo-Kantian philosophy formulated a sophisticated position towards the problems raised by the concepts of limits and infinitesimals. The Marburg school neither clung to the traditional approach of logically and metaphysically dubious infinitesimals, nor whiggishly subscribed to the new orthodoxy of the “great triumvirate” of Cantor, Dedekind, and Weierstrass that declared infinitesimals conceptus nongrati in mathematical discourse. Rather, following Cohen’s lead, the Marburg philosophers sought to clarify Leibniz’s principle of continuity, and to exploit it in making sense of infinitesimals and related concepts. (shrink)

In this essay we introduce the idea of an epistemic triangle, with factual, theoretical, and conceptual investigations at its vertices, and argue that whereas scientific progress requires a balance among the three types of investigations, psychology's epistemic triangle is stretched disproportionately in the direction of factual investigations. Expressed by a variety of different problems, this unbalance may be created by a main operative theme—the obsession of psychology with a narrow and mechanical view of the scientific method and a misguided aversion (...) to conceptual inquiries. Hence, to redress psychology's epistemic triangle, a broader and more realistic conception of method is needed and, in particular, conceptual investigations must be promoted. Using examples from different research domains, we describe the nature of conceptual investigations, relate them to theoretical investigations, and illustrate their purposes, forms, and limitations. (shrink)

We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy’s foundational work associated with the work of Boyer and Grabiner; and to Bishop’s constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research.

Recently, there have been several attempts to generalize the counterfactual theory of causal explanations to mathematical explanations. The central idea of these attempts is to use conditionals whose antecedents express a mathematical impossibility. Such countermathematical conditionals are plugged into the explanatory scheme of the counterfactual theory and—so is the hope—capture mathematical explanations. Here, I dash the hope that countermathematical explanations simply parallel counterfactual explanations. In particular, I show that explanations based on countermathematicals are susceptible to three problems counterfactual explanations do (...) not face. These problems seriously challenge the prospects for a counterfactual theory of explanation that is meant to cover mathematical explanations. (shrink)

1. Overview and organizing themes 2. Historical Review: Aristotle to Mill 3. Logic of method and critical responses 3.1 Logical constructionism and Operationalism 3.2. H-D as a logic of confirmation 3.3. Popper and falsificationism 3.4 Meta-methodology and the end of method 4. Statistical methods for hypothesis testing 5. Method in Practice 5.1 Creative and exploratory practices 5.2 Computer methods and the ‘third way’ of doing science 6. Discourse on scientific method 6.1 “The scientific method” in science education and as seen (...) by scientists 6.2 Privileged methods and ‘gold standards’ 6.3 Scientific method in the court room 6.4 Deviating practices 7. Conclusion Bibliography Academic Tools Other Internet Resources Related Entries . (shrink)

The importance of mathematics in the context of the scientific and technological development of humanity is determined by the possibility of creating mathematical models of the objects studied under the different branches of Science and Technology. The arithmetisation process that took place during the nineteenth century consisted of the quest to discover a new mathematical reality in which the validity of logic would stand as something essential and central. Nevertheless, in contrast to this process, the development of mathematical analysis within (...) a framework that largely involves intuition and geometry is a fact that cannot go unnoticed amongst the mathematics community, as we shall show in this paper through the research made by Bernhard Riemann on complex variables. (shrink)

For several decades now philosophers have discussed apparent examples of internally inconsistent scientific theories. However, there is still much controversy over how exactly we should conceive of scientific theories in the first place. Here I argue for a new approach, whereby all of the truly important questions about inconsistency in science can be asked and answered without disagreements about theories and theory-content getting in the way. Three examples commonly described as ‘internally inconsistent theories’ are analysed in the light of this (...) approach. In the process, the question ‘Is the theory inconsistent or not?’ is identified as a bad, or at least unimportant, question. (shrink)

Minimalists, following Horwich, claim that all that can be said about truth is comprised by all and only the nonparadoxical instances of (E) p is true iff p. It is, accordingly, standard in the literature on truth and paradox to ask how the minimalist will restrict (E) so as to rule out paradox-inducing sentences (alternatively: propositions). In this paper, we consider a prior question: On what grounds does the minimalist restrict (E) so as to rule out paradox-inducing sentences and, thereby, (...) avoid contradictions? We argue that there is no good reason for thinking that the minimalist can furnish such grounds. Accordingly, while we are tempted to conclude from this that the minimalist should acknowledge the contradictoriness of truth, instead, we end with a challenge: Provide grounds, compatible with minimalism, for banning the paradoxical instances of (E), or embrace dialetheism. (shrink)

– We offer a new motivation for imprecise probabilities. We argue that there are propositions to which precise probability cannot be assigned, but to which imprecise probability can be assigned. In such cases the alternative to imprecise probability is not precise probability, but no probability at all. And an imprecise probability is substantially better than no probability at all. Our argument is based on the mathematical phenomenon of non-measurable sets. Non-measurable propositions cannot receive precise probabilities, but there is a natural (...) way for them to receive imprecise probabilities. The mathematics of non-measurable sets is arcane, but its epistemological import is far-reaching; even apparently mundane propositions are liable to be affected by non-measurability. The phenomenon of non-measurability dramatically reshapes the dialectic between critics and proponents of imprecise credence. Non-measurability offers natural rejoinders to prominent critics of imprecise credence. Non-measurability even reverses some of the critics’ arguments—by the very lights that have been used to argue against imprecise credences, imprecise credences are better than precise credences. (shrink)

Although it is clear that Sir William Rowan Hamilton supported a Kantian account of algebra, I argue that there is an important sense in which Hamilton's philosophy of mathematics can be situated in the Newtonian tradition. Drawing from both Niccolo Guicciardini's (2009) and Stephen Gaukroger's (2010) readings of the Newton–Leibniz controversy over the calculus, I aim to show that the very epistemic ideals that underpin Newton's argument for the superiority of geometry over algebra also motivate Hamilton's philosophy of algebra. Namely, (...) Hamilton's defense of algebra, like Newton's defense of geometry, is driven by the claim that a mathematical science must have a proper object and thus a basis in truth. In particular, Hamilton aims to show that algebra is not a mere language, or tool, or a mere “art”; instead, he argues, algebra is a bona fide mathematical science, like geometry, because its methods also provide true and accurate insight into a genuine subject matter, namely, the pure form of temporal intuition. (shrink)

The conceptualisation of movement has always been problematical for Western thought, ever since Parmenides declared our incapacity to conceptualise the plurality of change because our self-identical thought can only know an identical being. Exploiting this peculiar feature and constraint on our thought, Zeno of Elea devised his famous paradoxes of movement in which he shows that the passage from a position to movement cannot be conceptualised. In this paper, I argue that this same constraint is at the root of our (...) incapacity to conceptualise the unseen movement at the micro-level and that the aporetic idea of super-position far from opening the gate on a deeper reality is a symptomatic word for this lack of understanding. (shrink)

We apply Benacerraf’s distinction between mathematical ontology and mathematical practice to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like (...) Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler’s own. Euler’s use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. Euler’s principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro’s assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler’s work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves. (shrink)

With the arrival of the nineteenth century, a process of change guided the treatment of three basic elements in the development of mathematics: rigour, the arithmetization and the clarification of the concept of function, categorised as the most important tool in the development of the mathematical analysis. In this paper we will show how several prominent mathematicians contributed greatly to the development of these basic elements that allowed the solid underpinning of mathematics and the consideration of mathematics as an axiomatic (...) way of thinking in which anyone can deduce valid conclusions from certain types of premises. This nineteenth century stage shares, possibly with the Heroic Age of Ancient Greece, the most revolutionary period in all history of mathematics. (shrink)

Many historians of the calculus deny significant continuity between infinitesimal calculus of the seventeenth century and twentieth century developments such as Robinson’s theory. Robinson’s hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic; thus many commentators are comfortable denying a historical continuity. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others to consider the history of the infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies, (...) Robinson regards Berkeley’s criticisms of the infinitesimal calculus as aptly demonstrating the inconsistency of reasoning with historical infinitesimal magnitudes. We argue that Robinson, among others, overestimates the force of Berkeley’s criticisms, by underestimating the mathematical and philosophical resources available to Leibniz. Leibniz’s infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Leibniz’s defense of infinitesimals is more firmly grounded than Berkeley’s criticism thereof. We show, moreover, that Leibniz’s system for differential calculus was free of logical fallacies. Our argument strengthens the conception of modern infinitesimals as a development of Leibniz’s strategy of relating inassignable to assignable quantities by means of his transcendental law of homogeneity. (shrink)

Because of the complex interdependence of physics and mathematics their relation is not free of tensions. The paper looks at how the tension has been perceived and articulated by some physicists, mathematicians and mathematical physicists. Some sources of the tension are identified and it is claimed that the tension is both natural and fruitful for both physics and mathematics. An attempt is made to explain why mathematical precision is typically not welcome in physics.

One of the most influential scientific treatises in Cauchy's era was J.-L. Lagrange's Mécanique Analytique, the second edition of which came out in 1811, when Cauchy was barely out of his teens. Lagrange opens his treatise with an unequivocal endorsement of infinitesimals. Referring to the system of infinitesimal calculus, Lagrange writes:Lorsqu'on a bien conçu l'esprit de ce système, et qu'on s'est convaincu de l'exactitude de ses résultats par la méthode géométrique des premières et dernières raisons, ou par la méthode analytique (...) des fonctions dérivées, on peut employer les infiniment petits comme un instrument sûr et commode pour abréger et simplifier les demonstrations.Lagrange's renewed enthusiasm for .. (shrink)

The ubiquitous assertion that the early calculus of Newton and Leibniz was an inconsistent theory is examined. Two different objects of a possible inconsistency claim are distinguished: (i) the calculus as an algorithm; (ii) proposed explanations of the moves made within the algorithm. In the first case the calculus can be interpreted as a theory in something like the logician’s sense, whereas in the second case it acts more like a scientific theory. I find no inconsistency in the first case, (...) and an inconsistency in the second case which can only be imputed to a small minority of the relevant community. (shrink)

I provide an interpretation of Wittgenstein's much criticized remarks on Gödel's First Incompleteness Theorem in the light of paraconsistent arithmetics: in taking Gödel's proof as a paradoxical derivation, Wittgenstein was right, given his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. I show that the models of paraconsistent arithmetics (obtained via the Meyer-Mortensen (...) collapsing filter on the standard model) match with many intuitions underlying Wittgensteins philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any meaningful mathematical question. (shrink)

We explore the potential of Simon Stevin’s numbers, obscured by shifting foundational biases and by 19th century developments in the arithmetisation of analysis.

In various arguments, Descartes relies on the principles that conceivability implies possibility and that inconceivability implies impossibility. Those principles are in tension with another Cartesian view about the source of modality, i.e. the doctrine of the free creation of eternal truths. In this paper, I develop a ‘two-modality’ interpretation of the doctrine of eternal truths which resolves the tension and I discuss how the resulting modal epistemology can still be relevant for the contemporary discussion.

Scientific pluralism is the view according to which there is a plurality of scientific domains and of scientific theories, and these theories are empirically adequate relative to their own respective domains. Scientific monism is the view according to which there is a single domain to which all scientific theories apply. How are these views impacted by the presence of inconsistent scientific theories? There are consistency-preservation strategies and inconsistency-toleration strategies. Among the former, two prominent strategies can be articulated: Compartmentalization and Information (...) restriction. Among the inconsistency-toleration strategies, we have: Paraconsistent compartmentalization and Dialetheism. In this paper, I critically assess the adequacy of each of these four views. (shrink)

We here attempt to address certain criticisms of the philosophical import of the so-called Brazilian approach to paraconsistency by providing some epistemic elucidations of the whole enterprise of the logics of formal inconsistency. In the course of this discussion, we substantiate the view that difficulties in reasoning under contradictions in both the Buddhist and the Aristotelian traditions can be accommodated within the precepts of the Brazilian school of paraconsistency.

Kant, in various parts of his treatment of causality, refers to determinism or the principle of sufficient reason as an inescapable principle. In fact, in the Second Analogy we find the elements to reconstruct a purely phenomenal determinism as a logical and tautological truth. I endeavour in this article to gather these elements into an organic theory of phenomenal causality and then show, in the third section, with a specific argument which I call the “paradox of phenomenal observation”, that this (...) phenomenal determinism is the only rational approach to causality because any logico-reductivistic approach, such as the Humean one, would destroy the temporal order and so the very possibility to talk of a causal relation. I also believe that, all things said, Kant did not achieve a much greater comprehension of the problem than Hume did, in his theory of causality, for he did not free a phenomenal approach from the impasse of reductivism as his reflections on “simultaneous causation” and “vanishing quantities” indeed show, and this I will argue in Sect. 4 of this article. (shrink)

Johann Bernoulli in 1710 affirmed that Newton had not proved that conic sections, having a focus in the force centre, were necessary orbits for a body accelerated by an inverse square force. He also criticized Newton's mathematical procedures applied to central forces in Principia mathematica, since, in his opinion, they lacked generality and could be used only if one knew the solution in advance. The development of eighteenth-century dynamics was mainly due to Continental mathematicians who followed Bernoulli's approach rather than (...) Newton's. The ways of thinking of the British Newtonians have, therefore, been somewhat forgotten. This paper is an attempt to assess what Bernoulli was criticizing and what were the immediate reactions of the Newtonians. In particular, I will concentrate on two papers by John Keill, submitted to the Royal Society in 1708 and 1714, in which the results on central forces achieved by the British were summarized and their methods defended. (shrink)

The widespread idea that infinitesimals were “eliminated” by the “great triumvirate” of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the continuum (...) with a single number system. Such anachronistic distortions characterize the received interpretation of Stevin, Leibniz, d’Alembert, Cauchy, and others. (shrink)

The notion of indivisibles and atoms arose in ancient Greece. The continuum—that is, the collection of points in a straight line segment, appeared to have paradoxical properties, arising from the ‘indivisibles’ that remain after a process of division has been carried out throughout the continuum. In the seventeenth century, Italian mathematicians were using new methods involving the notion of indivisibles, and the paradoxes of the continuum appeared in a new context. This cast doubt on the validity of the methods and (...) the reliability of mathematical knowledge which had been regarded as established by the axiomatic method in geometry expounded by Aristotle’s younger contemporary Euclid. The teaching of indivisibles was banned within the Society of Jesus, the Jesuits. In England, indivisibles were used by the mathematician John Wallis, and there was an acrimonious and extended feud between Wallis and the philosopher Thomas Hobbes over legitimate methods of argument in mathematics. Notions of the infinitesimal were used by Isaac Newton and Gottfried Leibniz, and were attacked by Bishop Berkeley for the vagueness of the concept and the illegitimate reasoning applied to it. This article discusses aspects of these events with reference to the book Infinitesimal by Amir Alexander and to other sources. Also discussed are wider issues arising from Alexander’s book including: the changes in cultural sensibility associated with the growth of new mathematical and scientific knowledge in the seventeenth century, the changes in language concomitant with these changes, what constitutes valid methods of enquiry in various contexts, and the question of authoritarianism in knowledge. More general aims of this article are to widen the immediate mathematical and historical contexts in Alexander’s book, to bridge a gap in conversations between mathematics and the humanities, and to relate mathematical ideas to wider human and contemporary issues. (shrink)