My contribution to a book symposium on Graham’s commentary on Kant’s Critique of Pure Reason, sponsored by the North American and the UK Kant Societies, held in conjunction with the Central Division Meeting of the American Philosophical Association, Chicago, 20 February 2009. Comments also delivered by Adrian Moore, Gary Banham, Jill Buroker and Manfred Kuehn, with relplies by Graham Bird.

It is occasionally claimed that the important work of philosophers, physicists, and mathematicians in the nineteenth and in the early twentieth centuries made Kant’s critical philosophy of geometry look somewhat unattractive. Indeed, from the wider perspective of the discovery of non-Euclidean geometries, the replacement of Newtonian physics with Einstein’s theories of relativity, and the rise of quantificational logic, Kant’s philosophy seems “quaint at best and silly at worst”.1 While there is no doubt that Kant’s transcendental project involves his own conceptions (...) of Newtonian physics, Euclidean geometry and Aristotelian logic, the issue at stake is whether the replacement of these conceptions collapses Kant’s philosophy into an unfortunate embarrassment.2 Thus, in evaluating the debate over the contemporary relevance of Kant’s philosophical project one is faced with the following two questions: (1) Are there any contradictions between the scientific developments of our era and Kant’s philosophy? (2) What is left from the Kantian legacy in light of our modern conceptions of logic, geometry and physics? Within this broad context, this paper aims to evaluate the Kantian project vis à vis the discovery and application of non-Euclidean geometries. Many important philosophers have evaluated Kant’s philosophy of geometry throughout the last century,3 but opinions with regard to the impact of non-Euclidean geometries on it diverge. In the beginning of the century there was a consensus that the Euclidean character of space should be considered as a consequence of the Kantian project, i.e., of the metaphysical view of space and of the synthetic a priori character of geometry. The impact of non-Euclidean geometries was then thought as undermining the Kantian project since it implied, according to positivists such.. (shrink)

There seem to be some very good reasons for a philosopher of science to be a deductivist about scientific reasoning. Deductivism is apparently connected with a demand for clarity and definiteness in the reconstruction of scientists' reasonings. And some philosophers even think that deductivism is the way around the problem of induction. But the deductivist image is challenged by cases of actual scientific reasoning, in which hard-to-state and thus discursively ill-defined elements of thought nonetheless significantly condition what practitioners accept as (...) cogent argument. And arguably, these problem cases abound. For example, even geometry--for most of its history--was such a problem case, despite its exactness and rigor. It took a tremendous effort on the part of Hilbert and others, to make geometry fit the deductivist image. Looking to the empirical sciences, the problems seem worse. Even the most exact and rigorous of empirical sciences--mechanics--is still the kind of problem case which geometry once was. In order for the deductivist image to fit mechanics, Hilbert's sixth problem (for mechanics) would need to be solved. This is a difficult, and perhaps ultimately impossible task, in which the success so far achieved is very limited. I shall explore some consequences of this for realism as well as for deductivism. Through discussing links between non-monotonicity, skills, meaning, globality in cognition, models, scientific understanding, and the ideal of rational unification, I argue that deductivists can defend their image of scientific reasoning only by trivializing it, and that for the adequate illumination of science, insights from anti-deductivism are needed as much as those which come from deductivism. (shrink)

There are some puzzles about G¨ odel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, G¨ odel’s writings represent a smooth evolution, with just one rather small double-reversal, of his view of finitism. He used the term “finit” (in German) or “finitary” or “finitistic” primarily to (...) refer to Hilbert’s conception of finitary mathematics. On two occasions (only, as far as I know), the lecture notes for his lecture at Zilsel’s [G¨ odel, 1938a] and the lecture notes for a lecture at Yale [G¨ odel, *1941], he used it in a way that he knew—in the second case, explicitly—went beyond what Hilbert meant. Early in his career, he believed that finitism (in Hilbert’s sense) is openended, in the sense that no correct formal system can be known to formalize all finitist proofs and, in particular, all possible finitist proofs of consistency of first-order number theory, P A; but starting in the Dialectica paper.. (shrink)

This book brings together philosophers, mathematicians and logicians to penetrate important problems in the philosophy and foundations of mathematics. In philosophy, one has been concerned with the opposition between constructivism and classical mathematics and the different ontological and epistemological views that are reflected in this opposition. The dominant foundational framework for current mathematics is classical logic and set theory with the axiom of choice. This framework is, however, laden with philosophical difficulties. One important alternative foundational programme that is actively pursued (...) today is predicativistic constructivism based on Martin-Löf type theory. Associated philosophical foundations are meaning theories in the tradition of Wittgenstein, Dummett, Prawitz and Martin-Löf. What is the relation between proof-theoretical semantics in the tradition of Gentzen, Prawitz, and Martin-Löf and Wittgensteinian or other accounts of meaning-as-use? What can proof-theoretical analyses tell us about the scope and limits of constructive and predicative mathematics? (shrink)

I argue that Kant's refutation of Mendelssohn's proof of the immortality of the soul also refutes his own proof of the permanence of material substance. To evade this result, Kant would have had to rely on premises that can only be established empirically. This difficulty brings up deep and disturbing difficulties with Kant's theory of matter and body in his Metaphysical Foundations of Natural Science and suggests that his early Physical Monadology offered a better account, one he was constrained by (...) later commitments to abandon to the detriment of the coherence of his matter theory. (shrink)

The article attempts to reconsider the relationship between Leibniz’s and Kant’s philosophy of geometry on the one hand and the nineteenth century debate on the foundation of geometry on the other. The author argues that the examples used by Leibniz and Kant to explain the peculiarity of the geometrical way of thinking are actually special cases of what the Jewish-German mathematician Felix Hausdorff called “transformation principle”, the very same principle that thinkers such as Helmholtz or Poincaré applied in a more (...) general form in their celebrated philosophical writings about geometry. The first two parts of the article try to show that Leibniz’s and Kant’s philosophies of geometry, despite their differences, appear to be preoccupied with the common problem of the impossibility to grasp conceptually the intuitive difference between two figures (such as a figure and its scaled, displaced or mirrored copy). In the third part, it is argued that from the perspective of Hausdorff’s philosophical-geometrical reflections, this very same problem seems to find a more radical application in Helmholtz’s or Poincaré’s thought experiments on the impossibility of distinguishing distorted copies of our universe from the original one. I draw the conclusion that in Hausdorff’s philosophical work, which has received scholarly attention only recently, one can find not only an original attempt to frame these classical arguments from a set-theoretical point of view, but also the possibility of considering the history of philosophy of geometry from an uncommon perspective, where especially the significance of Kant’s infamous appeal to “intuition” can be judged by more appropriate standards. (shrink)

My goal in this paper is to develop our understanding of the role the imagination plays in Kant’s Critical account of geometry, and I do so by attending to how the imagination factors into the method of reasoning Kant assigns the geometer in the First Critique. Such an approach is not unto itself novel. Recent commentators, such as Friedman (1992) and Young (1992), have taken a careful look at the constructions of the productive imagination in pure intuition and highlighted the (...) importance of the imagination’s activity for securing the universality of geometry knowledge. Specifically, as their respective examinations bring to light, it is only with due attention to the imagination that we can make sense of how a .. (shrink)

J. H. Lambert proved important results of what we now think of as non-Euclidean geometries, and gave examples of surfaces satisfying their theorems. I use his philosophical views to explain why he did not think the certainty of Euclidean geometry was threatened by the development of what we regard as alternatives to it. Lambert holds that theories other than Euclid's fall prey to skeptical doubt. So despite their satisfiability, for him these theories are not equal to Euclid's in justification. Contrary (...) to recent interpretations, then, Lambert does not conceive of mathematical justification as semantic. According to Lambert, Euclid overcomes doubt by means of postulates. Euclid's theory thus owes its justification not to the existence of the surfaces that satisfy it, but to the postulates according to which these "models" are constructed. To understand Lambert's view of postulates and the doubt they answer, I examine his criticism of Christian Wolff's views. I argue that Lambert's view reflects insight into traditional mathematical practice and has value as a foil for contemporary, model-theoretic, views of justification. (shrink)

Friedrich Paneth’s conception of “chemical element” has functioned as the official definition adopted by the International Union of Pure and Applied Chemistry since 1923. Paneth maintains a distinction between empirical and “transcendental” concepts of element; furthermore, chemical science requires fluctuation between the two. The origin of the empirical-transcendental split is found in Immanuel Kant’s classic Critique of Pure Reason (1781/1787). The present paper examines Paneth’s foundational concept of element in light of Kant’s attempt, late in life, to revoke key distinctions (...) made in his Critique, including that of regulative and constitutive functions of reason. In a section of his Opus postumum devoted to the “Transition from the Metaphysical Foundations of Natural Science to Physics,” Kant bends his philosophical system to address the newly emerging sciences of matter of his time. Specifically, he tried, without success, to develop the transcendental ground for microscale motions of bodies encountered in physical, electrical and chemical processes. Paneth’s discussion of chemical element does not take the Opus postumum into account, which is why it begins with a rejection of Kant’s rejection (in his earlier writings) of chemistry’s status as science. I make the case that Paneth’s definition of element effectively maintains something very like Kant’s critical separation of regulative and constitutive principles, while a advancing the concept of chemical science. (shrink)

This paper analyses the actual meaning of a transcendental philosophy of biology, and does so by exploring and actualising the epistemological and metaphysical value of Kant's viewpoint on living systems. It finds inspiration in the Kantian idea of living systems intrinsically resisting objectification, but critically departs from Kant's philosophical solution in as far as it is based in a subjectivist dogmatism. It attempts to overcome this dogmatism, on the one hand by explicitly taking into account the conditions of possibility at (...) the side of the subject, and on the other hand by embedding both the living and the knowing system into an ontology of complexly organized dynamical systems. This paper fits into the transcendental perspective in acknowledging the need to analyse the conditions of knowability, prior to the contents of what is known. But it also contributes to an expansion and an actualisation of the issue of transcendentality itself by considering the conditions of possibility at the side of the object as intrinsically linked to the conditions of possibility at the side of the subject. (shrink)

Idealist Heresies in Philosophy of Science: Cassirer, Carnap, and Kuhn. As common wisdom has it, philosophy of science in the analytic tradition and idealist philosophy are incompatible. Usually, not much effort is spent for explaining what is to be understood by idealism. Rather, it is taken for granted that idealism is an obsolete and unscientific philosophical account. In this paper it is argued that this thesis needs some qualification. Taking Carnap and Kuhn as paradigmatic examples of positivist and postpositivist philosophies (...) of science it is shown that these accounts share important features with Cassirer's idealist philosophy of science developed in the first half of this century. As it turns out, often Cassirer is more modern than those classical philosophers of (post)posivitist philosophy of science. For instance, Quine's criticism against Carnap's empiricist philosophy of science launched in Two Dogmas of Empiricism is anticipated by Cassirer for several decades. (shrink)

Im ersten Teil des Anhangs zur Transzendentalen Dialektik entwickelt Kant einen Übergang zwischen dem logischen Gebrauch der Vernunft und den transzendentalen Prinzipien der Vernunft. Basierend auf methodologischen Überlegungen zur Chemie, zur physiologischen Anthropologie und zur Entwicklungsgeschichte astronomischer Grundannahmen versucht Kant, diese Prinzipien in ihrem transzendentalen Status zu rechtfertigen und ihre Kompatibilität mit anderen Teilen der Kritik der reinen Vernunft nachzuweisen. Ausgehend von diesen wissenschaftstheoretischen Beispielen wird für eine heuristisch- pragmatische Rechtfertigung der transzendentalen Prinzipien der Vernunft und eine ideelle Funktion der (...) durch diese Prinzipien gewonnenen Ideen argumentiert. Dabei lässt sich die enge Verbindung zwischen den transzendentalen Prinzipien des ersten Teils des Anhangs zur Transzendentalen Dialektik und Kants Konzept der rationalen Naturwissenschaft in den Metaphysischen Anfangsgründen der Naturwissenschaft nachweisen. (shrink)

This article aims to make further progress in revising the standard account of Wolff’s philosophy as a popularisation and systematisation of Leibniz’s doctrines. It focuses on the topic of the communication among substances and the metaphysics of simples and activity underlying it. It is argued that Wolff does not accept the pre-established harmony in its orthodox Leibnizian version. The article explains Wolff’s departure from Leibniz’s PEH as stemming from his rejection of Leibniz’s construal of the activity of every simple as (...) representational power and of the metaphysics of unity and activity in which that construal is rooted. (shrink)

The ‘triumph of the anti-phlogistians’ is a familiar story to the historians and philosophers of science who characterize the Chemical Revolution as a broad conceptual shift. The apparent “incommensurability” of the paradigms across the revolutionary divide has caused much anxiety. Chemists could identify phlogiston and oxygen, however, only with different sets of instrumental practices, theoretical schemes, and philosophical commitments. In addition, the substantive counterpart to phlogiston in the new chemistry was not oxygen, but caloric. By focusing on the changing visions (...) of chemical body across the revolutionary divide with a more sensitive probe into the historical actors’ material manipulations and linguistic usage, we can historicize their laboratory realities and philosophical agenda. An archeology of chemical bodies that configures the fragile stability of the material worlds chemists created in succession promises a philosophical horizon that would recognize our hybrid (natural–artificial) environment as an evolving investigative object of science. (shrink)

The project behind this and the following1 special issue of Angelaki first assumed concrete form in the shape of a three-day international conference, “Continental Philosophy and the Sciences,” hel...

This article argues that the limited influence of Ludwik Fleck's ideas on philosophy of science is due not only to their indirect dissemination by way of Thomas Kuhn, but also to an incommensurability between the standard conceptual framework of history and philosophy of science and Fleck's own more integratedly historico-social and praxis-oriented approach to understanding the evolution of scientific discovery. What Kuhn named "paradigm" offers a periphrastic rendering or oblique translation of Fleck's Denkstil/Denkkollektiv , a derivation that may also account (...) for the lability of the term "paradigm". This was due not to Kuhn's unwillingness to credit Fleck but rather to the cold war political circumstances surrounding the writing of The Structure of Scientific Revolutions . Following a discussion of Fleck's anatomical allusions, I include a brief discussion of Aristotle (on menstruation and darkened mirrors) and conclude with a reference to the productivity of error in Mach and Nietzsche. (shrink)

I survey the debate over rationalism and empiricism in science. This chapter contains sections titled: The New Experimental Science as a Challenge to Intuition, Geometry and Intuition, The Mathematical Tradition and Theoretical Science.

Este ensaio introdutório faz uma breve apresentação do tratado de óptica atribuído a Euclides de Alexandria, inserindo-o no contexto das teorias sobre a visão formuladas pelas doutrinas filosóficas antigas. Ressalta-se o antagonismo entre a análise geométrica da visão, empreendida por Euclides, e as considerações filosóficas acerca dos processos físicos subjacentes à sensação visual. Pretende-se mostrar que o objeto da óptica euclidiana é a percepção visual daquilo que Aristóteles denomina "sensível comum". This introductory essay provides an abridged presentation of the optical (...) treatise attributed to Euclid of Alexandria, placing it in the context of theories about vision formulated by the ancient philosophical doctrines. I emphasize the antagonism between the geometric analysis of vision, undertaken by Euclid, and the philosophical considerations about the physical processes underlying visual sensation. In addition, I aim to show that the object of Euclidean optics is the visual perception of what Aristotle calls "common sensible". (shrink)