: Kant's 'argument from geometry' is usually interpreted to be a regressive transcendental argument in support of the claim that we have a pure intuition of space. In this paper I defend an alternative interpretation of this argument according to which it is rather a progressive synthetic argument meant to identify and establish the essential role of pure spatial intuition in geometric cognition. In the course of reinterpreting the 'argument from geometry' (...) I reassess the arguments of the Aesthetic and illustrate the origin of Kant's view of the role of pure intuition in geometry. (shrink)

The purported fact that geometric theories formulated in terms of points and geometric theories formulated in terms of lines are “equally correct” is often invoked in arguments for conceptual relativity, in particular by Putnam and Goodman. We discuss a few notions of equivalence between first-order theories, and we then demonstrate a precise sense in which this purported fact is true. We argue, however, that this fact does not undermine metaphysical realism.

It is widely believed that some puzzling and provocative remarks that Frege makes in his late writings indicate he rejected independence arguments in geometry, particularly arguments for the independence of the parallels axiom. I show that this is mistaken: Frege distinguished two approaches to independence arguments and his puzzling remarks apply only to one of them. Not only did Frege not reject independence arguments across the board, but also he had an interesting positive proposal about the logical structure of (...) correct independence arguments, deriving from the geometrical principle of duality and the associated idea of substitution invariance. The discussion also serves as a useful focal point for independently interesting details of Frege's mathematical environment. This feeds into a currently active scholarly debate because Frege's supposed attitude to independence arguments has been taken to support a widely accepted thesis (proposed by Ricketts among others) concerning Frege's attitude toward metatheory in general. I show that this thesis gains no support from Frege's puzzling remarks about independence arguments. (shrink)

This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or (...) imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, the information thus displayed is not metrical and it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions. (shrink)

This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or (...) imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, the information thus displayed is not metrical and it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions. (shrink)

The aim of this paper is to argue that there existed relevant interactions between mechanics and geometry during the first half of the nineteenth century, following a path that goes from Gauss to Riemann through Jacobi. By presenting a rich historical context we hope to throw light on the philosophical change of epistemological categories applied by these authors to the fundamental principles of both disciplines. We intend to show that presentations of the changing status of the principles of (...) mechanics as a mere epiphenomenon of the emergence of non-Euclidean geometries are inaccurate, that the relations between the two disciplines were richer than what is usually considered in the literature. These claims will be based on historical and philosophical arguments, starting from the fact that disciplinary boundaries at the time were not rigid as we are used today. It is widely known that the main figures we target worked in different areas, which is a first piece of evidence for the plausibility of our main thesis. (shrink)

§1. Introduction. With Hrushovski's proof of the function field Mordell-Lang conjecture [16] the relevance of geometric stability theory to diophantine geometry first came to light. A gulf between logicians and number theorists allowed for contradictory reactions. It has been asserted that Hrushovski's proof was simply an algebraic argument masked in the language of model theory. Another camp held that this theorem was merely a clever one-off. Still others regarded the argument as magical and asked whether such sorcery (...) could unlock the secrets of a wide coterie of number theoretic problems.In the intervening years each of these prejudices has been revealed as false though such attitudes are still common. The methods pioneered in [16] have been extended and applied to a number of other problems. At their best, these methods have been integrated into the general methods for solving diophantine problems. Moreover, the newer work suggests limits to the application of model theory to diophantine geometry. For example, all such known applications are connected with commutative algebraic groups. This need not be an intrinsic restriction, but its removal requires serious advances in the model theory of fields. (shrink)

Euclidean geometry, as presented by Euclid, consists of straightedge-and-compass constructions and rigorous reasoning about the results of those constructions. We show that Euclidean geometry can be developed using only intuitionistic logic. This involves finding “uniform” constructions where normally a case distinction is used. For example, in finding a perpendicular to line L through point p, one usually uses two different constructions, “erecting” a perpendicular when p is on L, and “dropping” a perpendicular when p is not on L, (...) but in constructive geometry, it must be done without a case distinction. Classically, the models of Euclidean geometry are planes over Euclidean fields. We prove a similar theorem for constructive Euclidean geometry, by showing how to define addition and multiplication without a case distinction about the sign of the arguments. With intuitionistic logic, there are two possible definitions of Euclidean fields, which turn out to correspond to different versions of the parallel postulate.We consider three versions of Euclid’s parallel postulate. The two most important are Euclid’s own formulation in his Postulate 5, which says that under certain conditions two lines meet, and Playfair’s axiom, which says there cannot be two distinct parallels to line L through the same point p. These differ in that Euclid 5 makes an existence assertion, while Playfair’s axiom does not. The third variant, which we call the strong parallel postulate, isolates the existence assertion from the geometry: it amounts to Playfair’s axiom plus the principle that two distinct lines that are not parallel do intersect. The first main result of this paper is that Euclid 5 suffices to define coordinates, addition, multiplication, and square roots geometrically.We completely settle the questions about implications between the three versions of the parallel postulate. The strong parallel postulate easily implies Euclid 5, and Euclid 5 also implies the strong parallel postulate, as a corollary of coordinatization and definability of arithmetic. We show that Playfair does not imply Euclid 5, and we also give some other independence results. Our independence proofs are given without discussing the exact choice of the other axioms of geometry; all we need is that one can interpret the geometric axioms in Euclidean field theory. The independence proofs use Kripke models of Euclidean field theories based on carefully constructed rings of real-valued functions. “Field elements” in these models are real-valued functions. (shrink)

A peculiarity of early medieval geometrical texts was that alongside Euclid's Elements they transmitted remnants of the corpus of Roman land surveyors and metaphysical digressions extraneous to geometry proper. Rather than dismissing these additions as irrelevant, this essay attempts to elucidate the cultural grounds for the indiscriminate mixture of the three disciplines -- geometry, surveying, and metaphysics. Inquiry into the broader context of early medieval culture suggests that neither geometry nor surveying was treated as an independent discipline. (...) Texts on geometry and surveying were studied mainly because they were thought to provide good opportunities for enlarging the scope of meditation on spiritual subjects. Accordingly, the main concern of this essay is to show how Christian symbolism influenced both the composition and the study of geometrical texts. The essay is based on a wide range of sources: treatises on geometry and surveying, pictorial representations of Creation, and philosophical literature in which geometry was used in arguments about ontological doctrines. (shrink)

ArgumentHermann von Helmholtz’s distinction between “pure intuitive” and “physical” geometry must be counted as the most influential of his many contributions to the philosophy of science. In a series of papers from the 1860s and 70s, Helmholtz argued against Kant’s claim that our knowledge of Euclidean geometry was an a priori condition for empirical knowledge. He claimed that geometrical propositions could be meaningful only if they were taken to concern the behaviors of physical bodies used in measurement, (...) from which it followed that it was posterior to our acquaintance with this behavior. This paper argues that Helmholtz’s understanding of geometry was fundamentally shaped by his work in sense-physiology, above all on the continuum of colors. For in the course of that research, Helmholtz was forced to realize that the color-space had no inherent metrical structure. The latter was a product of axiomatic definitions of color-addition and the empirical results of such additions. Helmholtz’s development of these views is explained with detailed reference to the competing work of the mathematician Hermann Grassmann and that of the young James Clerk Maxwell. It is this separation between 1) essential properties of a continuum, 2) supplementary axioms concerning distance-measurement, and 3) the behaviors of the physical apparatus used to realize the axioms, which is definitive of Helmholtz’s arguments concerning geometry. (shrink)

L’emploi par Poincaré de la notion de convention au sujet des hypothèses géométriques signale un déplacement par rapport aux problématiques traditionnelles. La découverte des géométries non euclidiennes montre qu’il n’y a pas de cadre spatial unique ; plusieurs systèmes sont possibles. On affirme ainsi l’existence d’un aspect essentiel de la connaissance qui ne dérive pas des faits et ne relève ni de l’inné ni de l’intuition. L’introduction de la notion de convention, dont il s’agit de prendre la mesure, ouvre la (...) voie à une prise en compte des facteurs décisionnels en science. Nous nous proposons, dans cet article, d’analyser l’argumentation de Poincaré en reconstruisant sa position face aux doctrines de Kant et de Mill.By understanding geometrical hypotheses as conventions, Poincaré significantly alters the traditional philosophical discussion of space. The discovery of non-Euclidean geometries proves that there is not a single spatial framework ; several systems are possible. Thus, there is an essential feature of knowledge that derives neither from empirical facts nor from intuition or innate understanding. The introduction of the notion of convention, whose significance is at issue here, provides a way of understanding the element of free choice in science. This article aims at analyzing Poincaré’s reasoning, by contrasting his viewpoint with that of Kant and Mill. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:66. ' HUME ON SPACE AND GEOMETRY * : A REJOINDER TO FLEW ' S 'ONE RESERVATION '.? Flew' s reservation about my assertion that the Enquiry contains no significant revision of the Treatise conception of geometry as a body of necessary and synthetic knowledge, appears to involve two charges. Firstly, he alleges that I dismiss but offer no substantial argument against his own view that (...) the Enquiry restores pure geometry "to its place alongside the other two elements of the trinity ". Secondly, he thinks that I have committed a serious sin of omission in failing to take into consideration a certain passage in the Enquiry (page 31, section 27) which, in his opinion, constitutes "a quite decisive reason" for supposing that Hume abandoned the Treatise position, in that it contains an account of applied mathematics, a subject on which the Treatise is silent. To the charge of omission I plead guilty, and I shall presently seek to remedy this defect and to state why I do not share Flew' s view of the significance of the said passage. To the former charge I am not so ready to plead guilty, and I shall counter-charge that Flew has failed not only to do justice to my proposed reading of Enquiry page 25, section 20, but also to provide clear and convincing support for his own. One difficulty encountered in answering Flew lies in understanding whether or not his claim about the status of geometry as a pure science in the Enquiry is put forward independently of his interpretation of E31. As originally presented in Hume's Philosophy of Belief this appears to be the case, and if so, on what evidence does it rest? 2 In his book, Flew' s statement about Hume replacing pure geometry alongside arithmetic and algebra follows directly upon a paragraph in which he comments that Hume's references to geometrical propositions at E25 signify "an important advance from the position of the Treatise". For whereas the status of perfect precision and certainty was 67. there withheld from geometry (T71), Hume now speaks of the certainty and evidence of the propositions of Euclidean geometry, and declares such propositions to be discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe. (E25) Now, I argued at some length in my paper (pp. 21-24) that Hume is prepared to grant scientific status to geometry in the Enquiry because he has come to regard its certainty as that proper to a mathematics of space. He has abandoned the Treatise contrast between an ideal or theoretical standard of equality, viewed therein as necessary to geometry considered as a science proper, and the merely sensible standard of equality upon which geometry as practised depends. It should not be forgotten that Hume recognises two kinds of certainty where "Relations of Ideas" are concerned, intuitive and demonstrative. Since he fails to provide any detail of the place of each of these in mathematics, the door is left open for speculation along the lines followed in pp. 28-29 of my paper. For there is nothing in?25 which implies that the certainty of all geometrical propositions is of the same kind, or that the process of geometrical demonstration is of the same logical nature as demonstrative reasoning in arithmetic and algebra. In his comment on my position Flew makes no reference to my proposed reason why Hume classes geometry as a science in the Enquiry.· Yet, even if this reclassification does mark an advance of a kind, the fact remains that E25 does not furnish incontrovertible evidence for that advance which Flew thinks he finds there. Nor, therefore, is it conclusive against my assertion that Hume in both works regards the propositions of geometry as necessary and synthetic truths. Given that Flew does not wish to deny the inclusion of a synthetic element even in the Enquiry account of mathematics (see H. P. B. pp. 64-66), I find it difficult to see why he supposes that the logical status of the propositions of geometry is different in the Enquiry from that in the Treatise, since in the latter too they are held 68. to be truths which depend... (shrink)

In this thesis, I investigate the nature of geometric knowledge and its relationship to spatial intuition. My goal is to rehabilitate the Kantian view that Euclid's geometry is a mathematical practice, which is grounded in spatial intuition, yet, nevertheless, yields a type of a priori knowledge about the structure of visual space. I argue for this by showing that Euclid's geometry allows us to derive knowledge from idealized visual objects, i.e., idealized diagrams by means of non-formal logical (...) inferences. By developing such an account of Euclid's geometry, I complete the "standard view" that geometry is either a formal system or an empirical science, which was developed mainly by the logical positivists and which is currently accepted by many mathematicians and philosophers. My thesis is divided into three parts. I use Hans Reichenbach's arguments against Kant and Edmund Husserl's genetic approach to the concept of space as a means of arguing that the "standard view" has to be supplemented by a concept of a geometry whose propositions have genuine spatial content. I then develop a coherent interpretation of Euclid's method by investigating both the subject matter of Euclid's geometry and the nature of geometric inferences. In the final part of this thesis, I modify Husserl's phenomenological analysis of the constitution of visual space in order to define a concept of spatial intuition that allows me not only to explain how Euclid's practice is grounded in visual space, but also to account for the apriority of its results. (shrink)

There are two natural ways of thinking about negation: (i) as a form of complementation and (ii) as an operation of reversal, or inversion (to deny that p is to say that things are “the other way around”). A variety of techniques exist to model conception (i), from Euler and Venn diagrams to Boolean algebras. Conception (ii), by contrast, has not been given comparable attention. In this note we outline a twofold geometric proposal, where the inversion metaphor is understoood (...) as involving a rotation o a reflection, respectively. These two options are equivalent in classical two-valued logic but they differ significantly in many-valued logics. Here we show that they correspond to two basic sorts of negation operators—Post’s and Kleene’s—and we provide a simple group-theoretic argument demonstrating their generative power. (shrink)

John Corcoran and George Boger. Aristotelian logic and Euclidean geometry. Bulletin of Symbolic Logic. 20 (2014) 131. -/- By an Aristotelian logic we mean any system of direct and indirect deductions, chains of reasoning linking conclusions to premises—complete syllogisms, to use Aristotle’s phrase—1) intended to show that their conclusions follow logically from their respective premises and 2) resembling those in Aristotle’s Prior Analytics. Such systems presuppose existence of cases where it is not obvious that the conclusion follows (...) class='Hi'>from the premises: there must be something deductions can show. Corcoran calls a proposition that follows from given premises a hidden consequence of those premises if it is not obvious that the proposition follows from those premises. By a Euclidean geometry we mean an extended discourse beginning with basic premises—axioms, postulates, definitions—1) treating a universe of geometrical figures and 2) resembling Euclid’s Elements. There were Euclidean geometries before Euclid (fl. 300 BCE), even before Aristotle (384–322 BCE). Bochenski, Lukasiewicz, Patzig and others never new this or if they did they found it inconvenient to mention. Euclid shows no awareness of Aristotle. It is obvious today—as it should have been obvious in Euclid’s time, if anyone knew both—that Aristotle’s logic was insufficient for Euclid’s geometry: few if any geometrical theorems can be deduced from Euclid’s premises by means of Aristotle’s deductions. Aristotle’s writings don’t say whether his logic is sufficient for Euclidean geometry. But, there is not even one fully-presented example. However, Aristotle’s writings do make clear that he endorsed the goal of a sufficient system. Nevertheless, incredible as this is today, many logicians after Aristotle claimed that Aristotelian logics are sufficient for Euclidean geometries. This paper reviews and analyses such claims by Mill, Boole, De Morgan, Russell, Poincaré, and others. It also examines early contrary statements by Hintikka, Mueller, Smith, and others. Special attention is given to the argumentations pro or con and especially to their logical, epistemic, and ontological presuppositions. What methodology is necessary or sufficient to show that a given logic is adequate or inadequate to serve as the underlying logi of a given science. (shrink)

The critique of my protophysical approaches to operational foundation of geometry by Lucas Amiras (Journal for General Philosophy of Science Vol. 34 (2003)) concerns my first publication from 1976 but not the further 30 years of work. It does not offer any argument leading from the (erroneous) judgement “lacking success” to the conclusion “impossible”. And it is, in general, based on a philosophical defect: it ignores the principle of methodical order as leading for constructivist protophysics.

Weyl's tile argument purports to show that there are no natural distance functions in atomistic space that approximate Euclidean geometry. I advance a response to this argument that relies on a new account of distance in atomistic space, called "the mixed account," according to which local distances are primitive and other distances are derived from them. Under this account, atomistic space can approximate Euclidean space (and continuous space in general) very well. To motivate this account as (...) a genuine solution to Weyl's tile argument, I argue that this account is no less natural than the standard account of distance in continuous space. I also argue that the mixed account has distinctive advantages over Forrest's (1995) account in response to Weyl's tile argument, which can be considered as a restricted version of the mixed account. (shrink)

This paper examines Helmholtz's attempt to use empirical psychology to refute certain of Kant's epistemological positions. Particularly, Helmholtz believed that his work in the psychology of visual perception showed Kant's doctrine of the a priori character of spatial intuition to be in error. Some of Helmholtz's arguments are effective, but this effectiveness derives from his arguments to show the possibility of obtaining evidence that the structure of physical space is non-Euclidean, and these arguments do not depend on his theory (...) of vision. Helmholtz's general attempt to provide an empirical account of the "inferences" of perception is regarded as a failure. (shrink)

Proposition I.1 is, by far, the most popular example used to justify the thesis that many of Euclid’s geometric arguments are diagram-based. Many scholars have recently articulated this thesis in different ways and argued for it. My purpose is to reformulate it in a quite general way, by describing what I take to be the twofold role that diagrams play in Euclid’s plane geometry (EPG). Euclid’s arguments are object-dependent. They are about geometric objects. Hence, they cannot be diagram-based unless (...) diagrams are supposed to have an appropriate relation with these objects. I take this relation to be a quite peculiar sort of representation. Its peculiarity depends on the two following claims that I shall argue for: ( i ) The identity conditions of EPG objects are provided by the identity conditions of the diagrams that represent them; ( ii ) EPG objects inherit some properties and relations from these diagrams. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:62. 'HUME ON SPACE AND GEOMETRY': ONE RESERVATION In so far as Rosemary Newman disagrees with any2 thing said in my 'Infinite Divisibility in Hume's Treatise ' - which seems, happily, not to be so very far - I hasten to report that I am now persuaded. Thus my suggested reason for refusing to allow that an impression of blackness could give rise to the idea of extension (...) was not, after all, 3 Hume's. And no doubt his conviction that whatever ^s capable of being divided in_ infinitum, must consist of an infinite number of parts (T26) was in part sustained by "the atomistic type of phenomenalism advanced in the first Part 4 of Book I of the Treatise". For to any objection that this nowadays so plainly unacceptable conviction had been accepted as obviously correct by such formidable predecessors as Berkeley and Hobbes, it could be replied that they too had their commitments to their own brands of atomism. In her further comments upon "Hume's phenomenalist atomism" Newman notices, as "a very curious claim", his statement that when - like the true philosopher he was! - he locked at a table:- My senses convey to me only the impressions of colour 'd points, dispos 'd in a certain manner (T34). So I was sorry that she seems to have missed the bold priority claim which, referring to the same passage, I made on Hume's behalf in that article: "Anyone familiar with the theories and the paintings of Seurat might also mischievously characterize the Hume of this Section as 'the Father of Pointillisme' "! But at the end of her paper, although I think only at the end, Newman begins to go very wrong indeed. She says: "Turning to the Enquiry there is little to indicate any shift in Hume's opinion of geometry as a body of synthetic but necessary knowledge." After remarking that Hume does not repeat "any of his earlier arguments" against infinite divisibility "nor those in support of a system of mathematical 63. points", she notices that, in the paragraph characterizing propositions stating only the relations of ideas, geometry "is classed as a science alongside arithmetic and algebra (E25) ".7 Next she quotes the end of that paragraph, and at once comments: "Flew thinks that in the Enquiry Hume is to be seen as 'restoring pure geometry to its place alongside Q the other two elements in the trinity',..." Newman does not, so far as I can see, offer any reason for dismissing this suggestion - apart, that is, from asserting that the paragraph from which she has just quoted "can be interpreted so as to remove any apparent inconsistency with the claim that the impossibility of conceiving the negation of a geometrical axiom has the same ground in the Enquiry as in the Treatise, namely, the invariant character of sensible space." She just tells us that "Where geometry is concerned I follow Atkinson's opinion in finding no reason to suppose that 9 Hume abandoned the Treatise stand." This I find, to say the least, surprising since the very sentence of Hume's Philosophy of Belief from which she quoted my contrary opinion does provide what I still consider to be a quite decisive reason. Since she does not discuss either my reason or the Hume passage to which I was referring, I can here do nothing else or better than repeat myself; adding only and by the way that Professor Atkinson's paper 'Hume on Mathematics' was published a year before that book:Besides restoring pure geometry to its place alongside the other two elements of the trinity, the Inquiry also sketches an account of applied mathematics. This is something the earlier book did not provide at all. 'Every part of mixed mathematics proceeds on the supposition that certain laws are established by nature in her operations, and abstract reasonings are employed, either to assist experience in the discovery of these laws or to determine their influence in particular instances... Thus it is a law of motion, discovered by experience, that the moment or force of any body in motion is in 64. the compound ratio or proportion of its solid contents and its velocity, and, consequently... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:62. 'HUME ON SPACE AND GEOMETRY': ONE RESERVATION In so far as Rosemary Newman disagrees with any2 thing said in my 'Infinite Divisibility in Hume's Treatise ' - which seems, happily, not to be so very far - I hasten to report that I am now persuaded. Thus my suggested reason for refusing to allow that an impression of blackness could give rise to the idea of extension (...) was not, after all, 3 Hume's. And no doubt his conviction that whatever ^s capable of being divided in_ infinitum, must consist of an infinite number of parts (T26) was in part sustained by "the atomistic type of phenomenalism advanced in the first Part 4 of Book I of the Treatise". For to any objection that this nowadays so plainly unacceptable conviction had been accepted as obviously correct by such formidable predecessors as Berkeley and Hobbes, it could be replied that they too had their commitments to their own brands of atomism. In her further comments upon "Hume's phenomenalist atomism" Newman notices, as "a very curious claim", his statement that when - like the true philosopher he was! - he locked at a table:- My senses convey to me only the impressions of colour 'd points, dispos 'd in a certain manner (T34). So I was sorry that she seems to have missed the bold priority claim which, referring to the same passage, I made on Hume's behalf in that article: "Anyone familiar with the theories and the paintings of Seurat might also mischievously characterize the Hume of this Section as 'the Father of Pointillisme' "! But at the end of her paper, although I think only at the end, Newman begins to go very wrong indeed. She says: "Turning to the Enquiry there is little to indicate any shift in Hume's opinion of geometry as a body of synthetic but necessary knowledge." After remarking that Hume does not repeat "any of his earlier arguments" against infinite divisibility "nor those in support of a system of mathematical 63. points", she notices that, in the paragraph characterizing propositions stating only the relations of ideas, geometry "is classed as a science alongside arithmetic and algebra (E25) ".7 Next she quotes the end of that paragraph, and at once comments: "Flew thinks that in the Enquiry Hume is to be seen as 'restoring pure geometry to its place alongside Q the other two elements in the trinity',..." Newman does not, so far as I can see, offer any reason for dismissing this suggestion - apart, that is, from asserting that the paragraph from which she has just quoted "can be interpreted so as to remove any apparent inconsistency with the claim that the impossibility of conceiving the negation of a geometrical axiom has the same ground in the Enquiry as in the Treatise, namely, the invariant character of sensible space." She just tells us that "Where geometry is concerned I follow Atkinson's opinion in finding no reason to suppose that 9 Hume abandoned the Treatise stand." This I find, to say the least, surprising since the very sentence of Hume's Philosophy of Belief from which she quoted my contrary opinion does provide what I still consider to be a quite decisive reason. Since she does not discuss either my reason or the Hume passage to which I was referring, I can here do nothing else or better than repeat myself; adding only and by the way that Professor Atkinson's paper 'Hume on Mathematics' was published a year before that book:Besides restoring pure geometry to its place alongside the other two elements of the trinity, the Inquiry also sketches an account of applied mathematics. This is something the earlier book did not provide at all. 'Every part of mixed mathematics proceeds on the supposition that certain laws are established by nature in her operations, and abstract reasonings are employed, either to assist experience in the discovery of these laws or to determine their influence in particular instances... Thus it is a law of motion, discovered by experience, that the moment or force of any body in motion is in 64. the compound ratio or proportion of its solid contents and its velocity, and, consequently... (shrink)

We argue that current constructive approaches to the special theory of relativity do not derive the geometrical Minkowski structure from the dynamics but rather assume it. We further argue that in current physics there can be no dynamical derivation of primitive geometrical notions such as length. By this we believe we continue an argument initiated by Einstein.

When national borders in the modern sense first began to be established in early modern Europe, non-contiguous and perforated nations were a commonplace. According to the conception of the shapes of nations that is currently preferred, however, nations must conform to the topological model of circularity; their borders must guarantee contiguity and simple connectedness, and such borders must as far as possible conform to existing topographical features on the ground. The striving to conform to this model can be seen at (...) work today in Quebec and in Ireland, it underpins much of the rhetoric of the P.L.O., and was certainly to some degree involved as a motivating factor in much of the ethnic cleansing which took place in Bosnia in recent times. The question to be addressed in what follows is: to what extent could inter-group disputes be more peacefully resolved, and ethnic cleansing avoided, if political leaders, diplomats and others involved in the resolution of such disputes could be brought to accept weaker geometrical constraints on the shapes of nations? A number of associated questions then present themselves: What sorts of administrative and logistical problems have been encountered by existing non contiguous nations and by perforated nations, and by other nations deviating in different ways from the received geometrical ideal? To what degree is the desire for continuity and simple connectedness a rational desire, and to what degree does it rest on species of political rhetoric which might be countered by, for example, philosophical argument? These and a series of related questions will form the subject- matter of the present essay. (shrink)

One of the most intriguing features of mathematics is its applicability to empirical science. Every branch of science draws upon large and often diverse portions of mathematics, from the use of Hilbert spaces in quantum mechanics to the use of differential geometry in general relativity. It's not just the physical sciences that avail themselves of the services of mathematics either. Biology, for instance, makes extensive use of difference equations and statistics. The roles mathematics plays in these theories is (...) also varied. Not only does mathematics help with empirical predictions, it allows elegant and economical statement of many theories. Indeed, so important is the language of mathematics to science, that it is hard to imagine how theories such as quantum mechanics and general relativity could even be stated without employing a substantial amount of mathematics. (shrink)

In this study we consider Kant's numerous attempts at finding a secure path to metaphysical truth via mathematical arguments. In 1756, he thinks that in natural philosophy we cannot do without metaphysical speculation, although he admits that there is no way to explain natural phenomena without the aid of experience and geometry. Kant's ideas in this matter have been mainly shaped by the influence of Euler, who proposed that the evident principles of mechanics ought to be taken as starting (...) point for metaphysical considerations. In a similar vein, after 1762 Kant looks for a metaphysical theory of space and time by taking mathematical arguments and observations as starting point for a genuinely presuppositive argument. Along these lines, Kant's so-called "Argument from Geometry" becomes more intelligible, although this paper ends with some sceptical remarks, questioning the existence of evident truths independent of any previous accepted set of metaphysical presuppositions. (shrink)

What if gravity satisfied the Klein-Gordon equation? Both particle physics from the 1920s-30s and the 1890s Neumann-Seeliger modification of Newtonian gravity with exponential decay suggest considering a "graviton mass term" for gravity, which is _algebraic_ in the potential. Unlike Nordström's "massless" theory, massive scalar gravity is strictly special relativistic in the sense of being invariant under the Poincaré group but not the 15-parameter Bateman-Cunningham conformal group. It therefore exhibits the whole of Minkowski space-time structure, albeit only indirectly concerning volumes. (...) Massive scalar gravity is plausible in terms of relativistic field theory, while violating most interesting versions of Einstein's principles of general covariance, general relativity, equivalence, and Mach. Geometry is a poor guide to understanding massive scalar gravity: matter sees a conformally flat metric due to universal coupling, but gravity also sees the rest of the flat metric in the mass term. What is the 'true' geometry, one might wonder, in line with Poincaré's modal conventionality argument? Infinitely many theories exhibit this bimetric 'geometry,' all with the total stress-energy's trace as source; thus geometry does not explain the field equations. The irrelevance of the Ehlers-Pirani-Schild construction to a critique of conventionalism becomes evident when multi-geometry theories are contemplated. Much as Seeliger envisaged, the smooth massless limit indicates underdetermination of theories by data between massless and massive scalar gravities---indeed an unconceived alternative. At least one version easily could have been developed before General Relativity; it then would have motivated thinking of Einstein's equations along the lines of Einstein's newly re-appreciated "physical strategy" and particle physics and would have suggested a rivalry from massive spin 2 variants of General Relativity. The Putnam-Grünbaum debate on conventionality is revisited with an emphasis on the broad modal scope of conventionalist views. Massive scalar gravity thus contributes to a historically plausible rational reconstruction of much of 20th-21st century space-time philosophy in the light of particle physics. An appendix reconsiders the Malament-Weatherall-Manchak conformal restriction of conventionality and constructs the 'universal force' influencing the causal structure. Subsequent works will discuss how massive gravity could have provided a template for a more Kant-friendly space-time theory that would have blocked Moritz Schlick's supposed refutation of synthetic _a priori_ knowledge, and how Einstein's false analogy between the Neumann-Seeliger-Einstein modification of Newtonian gravity and the cosmological constant \Lambda generated lasting confusion that obscured massive gravity as a conceptual possibility. (shrink)

Weyl famously argued that if space were discrete, then Euclidean geometry could not hold even approximately. Since then, many philosophers have responded to this argument by advancing alternative accounts of discrete geometry that recover approximately Euclidean space. However, they have missed an importantly flawed assumption in Weyl’s argument: physical geometry is determined by fundamental spacetime structures independently from dynamical laws. In this paper, I aim to show its falsity through two rigorous examples: random walks (...) in statistical physics and quantum mechanics. (shrink)

It may be a myth that Plato wrote over the entrance to the Academy “Let no-one ignorant of geometry enter here.” But it is a well-chosen motto for his view in the Republic that mathematical training is especially productive of understanding in abstract realms, notably ethics. That view is sound and we should return to it. Ethical theory has been bedevilled by the idea that ethics is fundamentally about actions (right and wrong, rights, duties, virtues, dilemmas and so on). (...) That is an error like the one Plato mentions of thinking mathematics is about actions (of adding, constructing, extracting roots and so on). Mathematics is about eternal relations between universals, such as the ratio of the diagonal of a square to the side. Ethics too is about eternal verities, such as the equal worth of persons and just distributions. Mathematical and ethical verities do both constrain actions, such as the possibility of walking over the seven bridges of Königsberg once and once only or of justly discriminating between races. But they are not themselves about action. In principle, neither mathematical nor ethical verities are subject to historical forces or disagreement among tribes (though they can be better understood as time goes on). Plato is right: immersion in mathematics induces an understanding of the necessities underpinning reality, an understanding that is essential for distinguishing objective ethics from tribal custom. Equality, for example, is an abstract concept which is foundational for both mathematics and ethics. (shrink)

The argument from reason states that if naturalism is true, then our beliefs are caused by physical processes rather than being causally based in their reasons, so our beliefs are not knowledge—including the belief in naturalism itself. Recent critics of the argument from reason provide dual process replies to the argument from reason—our beliefs can have both a naturalistic cause/ explanation and be caused/explained by its reasons, thereby showing that naturalism can accommodate knowledge. In (...) this paper I consider three dual process replies and conclude that none of them are successful. (shrink)

The correct locus (or loci) of binding theory has been a matter of much discussion. Theories can be seen as varying along at least two dimensions. The rst is whether binding theory is con gurationally determined (that is, the theory exploits the geometry of a phrase marker, appealing to such purely structural notions as c-command and government) or whether the theory depends rather on examining the relations between items selected by a predicate (where by selection I am intending to (...) cover everything from semantic dependencies to syntactic subcategorization). The second is the level of grammar on which binding is de ned. Attempting to roughly equate levels across di erent theories, suggestions have included the semantics/lexical conceptual structure (Jackendo 1992), thematic structure (Jackendo 1972, Wilkins 1988), argument structure/D-structure/initial grammatical relations (Manning 1994, Belletti and Rizzi 1988, Perlmutter 1984), surface syntax/grammatical relations, logical form, linear order, pragmatics (Levinson 1991), and discourse (Iida 1992). The data is su ciently varied and complex that many theories end up as mixtures, variously employing a combination of elements along both dimensions (for instance, Chomsky (1986) relies purely on con gurational notions for the relationship between an anaphor and its antecedent, but uses concepts from selection in the de nition of the binding domain of an anaphor). LFG has always rejected a con gurational account of binding. For instancError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMape, Simpson (1991) argues that a con gurational theory of binding in Warlpiri cannot be maintained, among other reasons because nite clauses lack a VP.. (shrink)

Three lines of argument are employed to show that Glymour's position on the epistemology of geometry is probably not as strong theoretically as the position of the underdeterminists whom he attempts to refute. The first argument centers on Glymour's implicit use of a realist position on intertheoretic reference, similar to that employed by Boyd and other realists. Citations are made to various portions of Glymour's work, and the relationship between the imputed theory of reference and Glymour's position (...) spelled out. The second line of argument refutes Glymour's contention that his criteria for choosing among theories are strongly based and not a priori, and a third line of argument rephrases Glymour's original position in a way which more clearly shows its implicit base and manifests as well the vagueness from which the formulation of the position suffers. It is concluded that a thorough examination of Glymour's deoccamization yields conclusions similar to those of the underdeterminists. * This paper is the result of work begun in a seminar on Philosophy of Science held at Rutgers University in the late 1970's. Work was continued at the University of California, Santa Barbara under the generosity of Grant DOE (OSERS) G0083/03651. I would like to thank colleagues from both institutions, and I owe a special debt of gratitude to an anonymous reviewer for the journal. (shrink)

According to informal logic, the possibilities of deductive logic as a tool for analysing and evaluating ordinary arguments are very limited. While I agree with this claim in general, I question it in the case of the argument from cause to effect. In this paper I first show, on the basis of carefully chosen examples, that we usually react differently to falsification of the conclusion of the argument from cause to effect than we do to the (...) falsification of the conclusion of other defeasible arguments. I then identify general conditions and assumptions under which the causal argument from cause to effect can be reconstructed as deductive. Finally, I compare the causal argument with the expert argument, which is a typical example of a defeasible argument in the narrower sense. (shrink)

ABSTRACTThis paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on ‘algebraical geometry’ and ‘geometrical algebra’ in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but (...) also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the ‘abstract algebra’ and ‘abstract geometry’ of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to ‘algebraical geometry’ and ‘geometrical algebra’ of the second generation of ‘scientific’ symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s–1840s. (shrink)

Arguments from fairness as described in Aristotle’s _Rhetoric_ are usually taken to aim at mitigating the strictness of the law or, in terms of procedure, to favour the defendant. This paper considers a more inclusive interpretation, that is, that arguments from fairness can work both ways. In the example given in the _Rhetoric,_ arguments from fairness are directed at a restrictive interpretation of the text. That may not be necessary however. Likewise, fairness may speak for the claimant. (...) Two examples may support this conclusion: a judicial speech by Hyperides, and the doctrine of issues appearing in Hellenistic school rhetoric. (shrink)

Corcoran, John. 1974. Aristotelian Syllogisms: Valid arguments or true generalized conditionals?, Mind 83, 278–81. MR0532928 (58 #27178) This tightly-written and self-contained four-page paper must be studied and not just skimmed. It meticulously analyses quotations from Aristotle and Lukasiewicz to establish that Aristotle was using indirect deductions—as required by the natural-deduction interpretation—and not indirect proofs—as required by the axiomatic interpretation. Lukasiewicz was explicit and clear about the subtle fact that Aristotle’s practice could not be construed as correctly performed indirect proof. (...) Lukasiewicz evidence is presented fully; it is irrefutable. But, instead of considering the possibility that Aristotle’s discourses were not intended to express indirect proofs of universalized conditions presupposing axiomatic premises, Lukasiewicz came to the amazing conclusion that Aristotle did not understand indirect proof. This paper builds on the admirable Lukasiewicz scholarship to establish a conclusion diametrically opposed to the one Lukasiewicz asserted. This paper points out that if Aristotle had not been establishing an underlying logic but he was instead axiomatizing a theory of terms, Aristotelians could not claim for Aristotle the title Founder of Logic. People who take Euclid, Peano, and Zermelo to have first axiomatized geometry, arithmetic, and set theory, respectively, do not think that this qualifies them for the titles Founder of Geometry, Founder of Arithmetic, and Founder of Set Theory, respectively. Such people do think that this would qualify the three for the titles Founder of Axiomatic Geometry, Founder of Axiomatic Arithmetic, and Founder of Axiomatic Set Theory, respectively: titles that carry no honors in logic. Being the Founder of Axiomatic Term Theory would likewise carry no honors in the field of logic. CORRECTION: The paper inadvertently implied that Euclid was the first to axiomatize geometry. In order to understand Aristotle’s Analytics as a treatise on demonstration it helps to realize that axiomatized geometry was studied in Plato’s Academy when Aristotle was a student. Euclid was the author of the only ancient axiomatized geometry now available. (shrink)

This chapter focuses on one of the common fallacies in Western philosophy, argument from fallacy. Also known as argumentum ad logicam, argument to logic, fallacy fallacy, and fallacist's fallacy, the argument from fallacy occurs when one reasons that because the argument for some conclusion is fallacious, the conclusion of that argument is false. Truth and falsity are features of claims. Fallacies are errors in reasoning, not errors about truth or falsity. That is, if (...) someone has committed a fallacy, then he has made an error in reasoning; but it does not follow that he has made a factual error. As a deductive argument, fallacy of affirming the consequent is invalid ‐ one cannot draw/infer the conclusion from the premises ‐ and yet the conclusion is true. As such, it should be clear that poor reasoning (committing a fallacy) does not entail a false conclusion. (shrink)

This paper offers a refutation of P. Duhem's thesis that the falsifiability of an isolated empirical hypothesis H as an explanans is unavoidably inconclusive. Its central contentions are the following: 1. No general features of the logic of falsifiability can assure, for every isolated empirical hypothesis H and independently of the domain to which it pertains, that H can always be preserved as an explanans of any empirical findings O whatever by some modification of the auxiliary assumptions A in conjunction (...) with which H functions as an explanans. For Duhem cannot guarantee on any general logical grounds the deducibility of O from an explanans constituted by the conjunction of H and some revised non-trivial version R of A: the existence of the required set R of collateral assumptions must be demonstrated for each particular case. 2. The categorical form of the Duhemian thesis is not only a non-sequitur but actually false. This is shown by adducing the testing of physical geometry as a counterexample to Duhem in the form of a rebuttal to A. Einstein's geometrical articulation of Duhem's thesis. 3. The possibility of a quasi a priori choice of a physical geometry in the sense of Duhem must be clearly distinguished from the feasibility of a conventional adoption of such a geometry in the sense of H. Poincare. And the legitimacy of the latter cannot be invoked to save the Duhemian thesis from refutation by the foregoing considerations. (shrink)

Abstract Einstein intended the general theory of relativity to be a generalization of the relativity of motion and, therefore, a radical departure from previous spacetime theories. It has since become clear, however, that this intention was not fulfilled. I try to explain Einstein's misunderstanding on this point as a misunderstanding of the role that spacetime plays in physics. According to Einstein, earlier spacetime theories introduced spacetime as the unobservable cause of observable relative motions and, in particular, as the cause (...) of inertial effects of ?absolute? motion. I use a comparative analysis of Einstein and Newton to show that spacetime is not introduced as an explanation of observable effects, but rather is defined through those effects in arguments like Newton's ?water bucket? argument and Einstein's argument for special relativity. I then argue that to claim that a spacetime theory is true, or to claim that a spacetime structure is ?real?, is not to claim that a theoretical object explains the observable. Rather, it is to claim that the fundamental definitions that link spacetime structure to physical phenomena are empirically sound, i.e. that they can be successfully applied empirically. This leads to a new and clearer view of the empirical content of spacetime theories and of the meaning of ?realism? about spacetime. (shrink)

Hyder constructs two historical narratives. First, he gives an account of Helmholtz's relation to Kant, from the famous Raumproblem, which preoccupied philosophers, geometers, and scientists in the mid-19th century, to Helmholtz's arguments in his four papers on geometry from 1868 to 1878 that geometry is, in some sense, an empirical science (chapters 5 and 6). The second theme is the argument for the necessity of central forces to a determinate scientific description of physical reality, an (...) abiding concern of Helmholtz's, and one that, as Hyder shows, has Kantian roots. Helmholtz's commitment to the necessity of central forces was key to his responses to rival views on electromagnetism, and is a deep and often under-appreciated element of his epistemology of science. (shrink)

The historian of philosophy often encounters arguments that are enthymematic: they have conclusions that follow from their explicit premises only by the addition of "tacit" or "suppressed" premises. It is a standard practice of interpretation to supply these missing premises, even where the enthymeme is "real," that is, where there is no other context in which the philosopher in question asserts the missing premises. To do so is to follow a principle of charity: other things being equal, one interpretation (...) is better than another just to the extent that the one produces a better argument than the other. We show that this principle leads to paradoxical conclusions, including the following: there is no objectively correct interpretation of any real enthymeme found in the text of a major philosopher; an interpreter will not regard a real enthymeme of a major philosopher as adequately interpreted until he has found a way of reading it that makes it into a good argument; every classical philosopher is infallible and omniscient; major philosophers never disagree. These conclusions are preposterous, but there are indications that they are in fact being reached, as we show by means of a case study of recent scholarship on Plato's Third Man Argument. To avoid the overinterpretation and anachronism that result from the unrestrained use of the principle of charity, one must employ a counterbalancing principle of parsimony: to seek the simplest explanation for the text under discussion. We study the role of the principle of parsimony by means of a mathematical case study, involving the suppressed premises in Euclid's Elements. Here the principle of parsimony plays a larger role than it does in the interpretation of philosophical texts, leading to a sharper distinction between Euclid's geometry and Euclidean geometry than we find between Plato and Platonism. We conclude by comparing two models of interpretation, which we call prospective and retrospective. Although the prospective model of interpretation leads to Platonism rather than to Plato, we argue that it still has a place in Platonic scholarship. (shrink)

Uriah Kriegel has attempted to describe the varieties of consciousness, that is, the primitive elements that constitute the phenomenal realm. Perceptual, imaginative, algedonic, cognitive, entertai...

In a recent exchange with prominent skeptical theists, Paul Draper has argued that skeptical theism bears no relevance to Humean versions of the argument from suffering. His argument rests, however, on a particular way of construing epistemically rational updating procedures that is not adopted by all forms of skeptical theism. In particular, a perspectival variety of skeptical theism, I argue, is relevant to his Humean arguments. I then generalize this result and explain how any argument (...) class='Hi'>from evil employing probabilistic premises is similarly threatened. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 39.3 (2001) 448-449 [Access article in PDF] James Van Cleve. Problems from Kant. New York and Oxford: Oxford University Press, 1999. Pp. xii + 340. Cloth, $45.00. The author acknowledges his debt to the "great Kant books of the 1960s, Jonathan Bennett's Kant's Analytic, and P. F. Strawson's The Bounds of Sense."Their analytical spirit lives on in this book, but the analyses (...) are fresh, take surprising turns, and make unexpected connections to current concerns. In seventeen chapters Van Cleve deals with necessity and the a priori, space and geometry, incongruent counterparts, time, the ideality of matter, experience and objects, substance in the First Analogy, the Second Analogy, noumena, problems of the self, rational theology, and Kant and contemporary irrealism. To sustain the flow of argument, the discussion of certain details, often important, is postponed and taken up in fifteen appendices, and there are nearly seventy pages of careful notes.Chapter 10, "Noumena and Things in Themselves," (134 ff.) criticizes in detail the prevailing "one world" or "double aspect" view of objects, namely that Kant allows just one set of objects, but two ways of "considering" them. Things in themselves are the very same objects as appearances, but considered as they are independent of our experience of them, a view developed in Henry Allison's Kant's Transcendental Idealism. Against this Van Cleve maintains (and I concur) that "appearances are intentionalia, and that intentionalia are constructions out of states of perceivers" (150). The distinction is between two separate universes of discourse, "not between two ways of discoursing [End Page 448] about the same objects" (ibid.). There is no space to report the sharply argued detail that led him to hold the double aspect view to be unfathomable and incoherent. He takes Kant to be a phenomenalist but holding to a version far removed from the reductionist phenomenalism of Russell, Broad, Price, and others. Here the basic elements are not made out to be indubitable givens, but are, rather, sensations. The structures built on them are, according to Van Cleve, "virtual objects" in the sense in which Quine speaks about "virtual classes." To say that something is an element of a class {F:Fx} is only to say that it is F. So, likewise, to say that, e.g., a patch of red exists is short for saying that a certain representation occurs. Carnap's Kantian masterpiece, TheLogical Structure of the World (The Aufbau) could have been consulted as a comprehensive guide to the kind of virtuality here envisaged.It is Van Cleve's stated aim never to discuss Kantian arguments without making their premise and conclusion structure clear (viii). Quite a few of them he finds persuasive; in other cases the arguments are valid, but one or the other premise doubtful. Sometimes he sides with Kant against his critics, sometimes the opposite, and occasionally, as in the case of the Second Analogy, it remains unclear what Kant's argument was. He has shown in this and earlier publications that all reconstructions are flawed, so that either Kant had no good argument, or his argument has not yet been discovered.The title Problems from Kant must be taken seriously. The book rarely deals with textual problems, historical context, or translation difficulties. Rather, the focus is on perennial and still compelling problems. Themes from the First Critique are tracked to and connected with current concerns and sometimes recently ventilated issues are traced back to Kant, as in the last chapter, where the question is raised if Kant was an internal realist in the style of Putnam, or perhaps an anti-realist like Dummett. The contrast with Kant shows that like all complex positions, these are shopping baskets with several goods in them, some in Kant's inventory, others not.Logical notation is used sparingly, and only where it clarifies—no attempt here to shoehorn arguments into unneeded symbolism. Deft examples, Kant's weakness, are a strength of this book. To explain the logic of considering Van Cleve asks, "Consider me apart from my shoes, am I barefoot... (shrink)