Kant discovered a philosophical problem with mathematical proof. Despite being a priori , its methodology involves more than analytic truth. But what else is involved? This problem is widely taken to have been solved by Frege’s extension of logic beyond its restricted (and largely Aristotelian) form. Nevertheless, a successor problem remains: both traditional and contemporary (classical) mathematical proofs, although conforming to the norms of contemporary (classical) logic, never were, and still aren’t, executed by mathematicians in a way that transparently reveals (...) why these proofs—written in the vernacular to this very day—succeed in conforming to those norms. (shrink)

I argue that we find the articulation of a problem concerning bodily agency in the early works of the Merleau-Ponty which he explicates as analogous to what he explicitly calls the problem of perception. The problem of perception is the problem of seeing how we can have the object given in person through it perspectival appearances. The problem concerning bodily agency is the problem of seeing how our bodily movements can be the direct manifestation of a person’s intentions in the (...) world. In both cases what, according to Merleau-Ponty, obscures a recognition of the phenomenon in question is a conception of our bodily capacities, i.e. our sensibility and our motility, which reduces these to the workings of mechanisms that are blind to meaning. I argue that both the problem concerning perception and the problem concerning bodily agency can be properly called transcendental. The problem of perception is transcendental because it concerns the very intelligibility of appearances and judgements with empirical content. The problem of bodily agency is transcendental in the sense that it concerns the very intelligibility of our bodily capacity to carry out intentions and by implication the intelligibility of our intentions as such. (shrink)

This essay offers a novel interpretation of Michel Foucault’s original and often misunderstood conception of philosophy as a critical activity. While it is well known that Foucault’s critique undertakes to disclose contingent limits of thought that appear necessary in the present, the nature of the obstacle whose overcoming critique is meant to facilitate remains poorly understood. I argue that this obstacle, “the present limits of the necessary,” resides on the unconscious level of thought Foucault identified as the object of analysis (...) for an archaeology of knowledge. Therefore, Foucault’s conception of a critique can be grasped only against the background of the distinctive conception of thought that informs his archaeological analyses of discursive practices. According to that view, thinking is always shaped by some historically specific system of unconscious norms that define the contingent set of conceptual possibilities subjects are able to recognize in the present. Drawing on Foucault’s largely neglected remarks on the obvious and the habitual, I argue that these unconscious norms of thought are enacted habitually in a discursive practice, which endows them with an appearance of obviousness. In this way, I explain how something contingent appears in the guise of necessity and begins to function as part of the present limits of the necessary. Finally, I argue that the task of a critique to expand the scope of conceptual possibility by disclosing these unconscious limits of thought is motivated by Foucault’s commitment to the ideal of autonomy understood as subject’s self-determination. (shrink)

In his defense of innateness in New Essays on Human Understanding (1704), Leibniz attributes innateness to concepts and principles which do not originate from the senses rather than to the ideas that we are born with. He argues that the innate concepts and principles can be known in two ways: through reason or natural light (necessary truths), and through instincts (other innate truths and principles). In this paper I will show how theoretical and moral reasoning differ from each other in (...) Leibniz, and compare moral reasoning and instincts as sources of knowledge in his practical philosophy. As the practical instincts are closely related to pleasure and passions, which are by nature cognitive, my emphasis will be on the affective character of instinctive moral action and especially deliberation which leads to moral action. I will argue that inclinations arising from moral instinct, which lead us to pleasure while avoiding sorrow, can direct our moral action and sometimes anticipate reasoning when conclusions are not readily available. Acting by will, which is related to moral reasoning, and acting by instincts can lead us to the same moral knowledge independently, but they can also complement each other. To illustrate the two alternative ways to reach moral knowledge, I will discuss the case of happiness, which is the goal of all human moral action for Leibniz. (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)

Logic is usually thought to concern itself only with features that sentences and arguments possess in virtue of their logical structures or forms. The logical form of a sentence or argument is determined by its syntactic or semantic structure and by the placement of certain expressions called “logical constants.”[1] Thus, for example, the sentences Every boy loves some girl. and Some boy loves every girl. are thought to differ in logical form, even though they share a common syntactic and semantic (...) structure, because they differ in the placement of the logical constants “every” and “some”. By contrast, the sentences Every girl loves some boy. and Every boy loves some girl. are thought to have the same logical form, because “girl” and “boy” are not logical constants. Thus, in order to settle questions about logical form, and ultimately about which arguments are logically valid and which sentences logically true, we must distinguish the “logical constants” of a language from its nonlogical expressions. (shrink)

Abstract : In this paper, it is argued that Leibniz’s view that necessity is grounded in the availability of a demonstration is incorrect and furthermore, can be shown to be so by using Leibniz’s own examples of infinite analyses. First, I show that modern mathematical logic makes clear that Leibniz’s "infinite analysis" view of contingency is incorrect. It is then argued that Leibniz's own examples of incommensurable lines and convergent series undermine, rather than bolster his view by providing examples of (...) necessary mathematical truths that are not demonstrable. Finally, it is argued that a more modern view on convergent series would, in certain respects, help support some claims he makes about the necessity of mathematical truths, but would still not yield a viable theory of necessity due to remaining problems with other logical, mathematical, and modal claims. (shrink)

Vyrauja nuomonė, jog ankstyvoji Descartes’o filosofija angažuojasi metodologinėms problemoms, o tai laikytina naujosios epochos ženklu. Tačiau turint omenyje tokios problematikos išskirtinį aktualumą Aristoteliui bei faktą, kad Descartes’o metodologija aiškiai stokoja apibrėžtumo, jo santykis su ankstesniąja tradicija turėtų būti įvertintas naujai. Straipsnyje siekiama apginti požiūrį, kad „Proto vadovavimo taisyklėse“, savo pirmajame filosofiniame veikale, Descartes’as demonstruoja pastangas duoti tvirtą epistemologinį pagrindą teorinio mąstymo išlaisvinimui iš bet kokių formalių apribojimų. Pateikdamas naują mąstymo procedūrų traktuotę, Descartes’as numato perspektyvų įvairovės galimybę teoriniame reiškinių aiškinime ir (...) šiuo aspektu atsiskleidžia jo originalumas. (shrink)