The mathematical form of measurement and the argument for Proposition I in Newton’s Principia

Newton characterizes the reasoning of Principia Mathematica as geometrical. He emulates classical geometry by displaying, in diagrams, the objects of his reasoning and comparisons between them. Examination of Newton??s unpublished texts (and the views of his mentor, Isaac Barrow) shows that Newton conceives geometry as the science of measurement. On this view, all measurement ultimately involves the literal juxtaposition??the putting-together in space??of the item to be measured with a measure, whose dimensions serve as the standard of reference, so that all quantity (which is what measurement makes known) is ultimately related to spatial extension. I use this conception of Newton??s project to explain the organization and proofs of the first theorems of mechanics to appear in the Principia (beginning in Sect. 2 of Book I). The placementof Kepler??s rule of areas as the first proposition, and the manner in which Newton proves it, appear natural on the supposition that Newton seeks a measure, in the sense of a moveable spatial quantity, of time. I argue that Newton proceeds in this way so that his reasoning can have the ostensive certainty of geometry.     

History of Geometry and the Development of the Form of its Language

The aim of this paper is to introduce Wittgenstein’s concept of the form of a language into geometry and to show how it can be used to achieve a better understanding of the development of geometry, from Desargues, Lobachevsky and Beltrami to Cayley, Klein and Poincaré. Thus this essay can be seen as an attempt to rehabilitate the Picture Theory of Meaning, from the Tractatus. Its basic idea is to use Picture Theory to understand the pictures of geometry. I will try to show, that the historical evolution of geometry can be interpreted as the development of the form of its language. This confrontation of the Picture Theory with history of geometry sheds new light also on the ideas of Wittgenstein. This revised version was published online in June 2006 with corrections to the Cover Date.     

Reconsidering Reid's Geometry of Visibles

In his Inquiry , Reid claims, against Berkeley, that there is a science of the perspectival shapes of objects ('visible figures'): they are geometrically equivalent to shapes projected onto the surfaces of spheres. This claim should be understood as asserting that for every theorem regarding visible figures there is a corresponding theorem regarding spherical projections; the proof of the theorem regarding spherical projections can be used to construct a proof of the theorem regarding visible figures, and vice versa. I reconstruct Reid's argument for this claim, and expose its mathematical underpinnings: it is successful, and depends on no empirical assumptions to which he was not entitled about the workings of the human eye. I also argue that, although Reid may or may not have been aware of it, the geometry of spherical projections is not the only geometry of visible figure.     

Kant on geometry and spatial intuition

I use recent work on Kant and diagrammatic reasoning to develop a reconsideration of central aspects of Kant??s philosophy of geometry and its relation to spatial intuition. In particular, I reconsider in this light the relations between geometrical concepts and their schemata, and the relationship between pure and empirical intuition. I argue that diagrammatic interpretations of Kant??s theory of geometrical intuition can, at best, capture only part of what Kant??s conception involves and that, for example, they cannot explain why Kant takes geometrical constructions in the style of Euclid to provide us with an a priori framework for physical space. I attempt, along the way, to shed new light on the relationship between Kant??s theory of space and the debate between Newton and Leibniz to which he was reacting, and also on the role of geometry and spatial intuition in the transcendental deduction of the categories.     

NEWTON AND HAMILTON: IN DEFENSE OF TRUTH IN ALGEBRA

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

I—Time, Topology and Physical Geometry

The standard mathematical account of the sub-metrical geometry of a space employs topology, whose foundational concept is the open set. This proves to be an unhappy choice for discrete spaces, and offers no insight into the physical origin of geometrical structure. I outline an alternative, the Theory of Linear Structures, whose foundational concept is the line. Application to Relativistic space-time reveals that the whole geometry of space-time derives from temporal structure. In this sense, instead of spatializing time, Relativity temporalizes space.     

Free Variation and the Intuition of Geometric Essences: Some Reflections on Phenomenology and Modern Geometry

Edmund Husserl has argued that we can intuit essences and, moreover, that it is possible to formulate a method for intuiting essences. Husserl calls this method ‘ideation’. In this paper I bring a fresh perspective to bear on these claims by illustrating them in connection with some examples from modern pure geometry. I follow Husserl in describing geometric essences as invariants through different types of free variations and I then link this to the mapping out of geometric invariants in modern mathematics. This view leads naturally to different types of spatial ontologies and it can be used to shed light on Husserl's general claim that there are different ontologies in the eidetic sciences that can be systematically related to one another. The paper is rounded out with a consideration of the role of ideation in the origins of modern geometry, and with a brief discussion of the use of ideation outside of pure geometry. What would be the study that would draw the soul away from the world of becoming to the world of being?… Geometry and arithmetic would be among the studies we are seeking … a philosopher must learn them because he must arise out of the region of generation and lay hold on essence or he can never become a true reckoner … they facilitate the conversion of the soul itself from the world of generation to essence and truth… they are knowledge of that which always is and not of something which at some time comes into being and passes away.     

Kantian Themes in Contemporary Philosophy: Graham Bird

Michael Friedman criticises some recent accounts of Kant which 'detach' his transcendental principles from the sciences, and do so in order to evade naturalism. I argue that Friedman's rejection of that 'detachment' is ambiguous. In its strong form, which I claim Kant rejects, the principles of Euclidean geometry and Newtonian physics are represented as transcendental principles. In its weak form, which I believe Kant accepts, it treats those latter principles as higher order conditions of the possibility of both science and ordinary experience. I argue also that the appeal to naturalism is unhelpful because that doctrine is seriously unclear, and because the accounts Friedman criticises are open to objections independent of any appeal to naturalism.     

Frege’s philosophy of geometry

Synthese - In this paper, I critically discuss Frege’s philosophy of geometry with special emphasis on his position in The Foundations of Arithmetic of 1884. In Sect. 2, I argue that...     

The independence of the parallel postulate and development of rigorous consistency proofs

I trace the development of arguments for the consistency of non-Euclidean geometries and for the independence of the parallel postulate, showing how the arguments become more rigorous as a formal conception of geometry is introduced. I analyse the kinds of arguments offered by Jules Hoüel in 1860–1870 for the unprovability of the parallel postulate and for the existence of non-Euclidean geometries, especially his reaction to the publication of Beltrami's seminal papers, showing that Beltrami was much more concerned with the existence of non-Euclidean objects than he was with the formal consistency of non-Euclidean geometries. The final step towards rigorous consistency proofs is taken in the 1880s by Henri Poincaré. It is the formal conception of geometry, stripping the geometric primitive terms of their usual meanings, that allows the introduction of a modern fully rigorous consistency proof.     

Husserl on Geometry and Spatial Representation

Husserl left many unpublished drafts explaining (or trying to) his views on spatial representation and geometry, such as, particularly, those collected in the second part of Studien zur Arithmetik und Geometrie (Hua XXI), but no completely articulate work on the subject. In this paper, I put forward an interpretation of what those views might have been. Husserl, I claim, distinguished among different conceptions of space, the space of perception (constituted from sensorial data by intentionally motivated psychic functions), that of physical geometry (or idealized perceptual space), the space of the mathematical science of physical nature (in which science, not only raw perception has a word) and the abstract spaces of mathematics (free creations of the mathematical mind), each of them with its peculiar geometrical structure. Perceptual space is proto-Euclidean and the space of physical geometry Euclidean, but mathematical physics, Husserl allowed, may find it convenient to represent physical space with a non-Euclidean structure. Mathematical spaces, on their turn, can be endowed, he thinks, with any geometry mathematicians may find interesting. Many other related questions are addressed here, in particular those concerning the a priori or a posteriori character of the many geometric features of perceptual space (bearing in mind that there are at least two different notions of a priori in Husserl, which we may call the conceptual and the transcendental a priori). I conclude with an overview of Weyl’s ideas on the matter, since his philosophical conceptions are often traceable back to his former master, Husserl.     

Mathematical forms and forms of mathematics: leaving the shores of extensional mathematics

In this paper, I introduce the idea that some important parts of contemporary pure mathematics are moving away from what I call the extensional point of view. More specifically, these fields are based on criteria of identity that are not extensional. After presenting a few cases, I concentrate on homotopy theory where the situation is particularly clear. Moreover, homotopy types are arguably fundamental entities of geometry, thus of a large portion of mathematics, and potentially to all mathematics, at least according to some speculative research programs.     

Reconsidering Relativistic Causality

I discuss the idea of relativistic causality, i.e., the requirement that causal processes or signals can propagate only within the light‐cone. After briefly locating this requirement in the philosophy of causation, my main aim is to draw philosophers’ attention to the fact that it is subtle, indeed problematic, in relativistic quantum physics: there are scenarios in which it seems to fail.

I set aside two such scenarios, which are familiar to philosophers of physics: the pilot‐wave approach, and the Newton–Wigner representation. I instead stress two unfamiliar scenarios: the Drummond–Hathrell and Scharnhorst effects. These effects also illustrate a general moral in the philosophy of geometry: that the mathematical structures, especially the metric tensor, that represent geometry get their geometric significance by dint of detailed physical arguments.     

The perceptual organization of dot lattices

Bravais (1850/1949) demonstrated that there are five types of periodic dot patterns (or lattices): oblique, rectangular, centered rectangular, square, and hexagonal. Gestalt psychologists studied grouping by proximity in rectangular and square dot patterns. In the first part of the present paper, I (1) describe the geometry of the five types of lattices, and (2) explain why, for the study of perception, centered rectangular lattices must be divided into two classes (centered rectangular andrhombic). I also show how all lattices can be located in a two-dimensional space. In the second part of the paper, I show how the geometry of these lattices determines their grouping and their multistability. I introduce the notion ofdegree of instability and explain how to order lattices from most stable to least stable (hexagonal). In the third part of the paper, I explore the effect of replacing the dots in a lattice with less symmetric motifs, thus creating wallpaper patterns. When a dot pattern is turned into a wallpaper pattern, its perceptual organization can be altered radically, overcoming grouping by proximity. I conclude the paper with an introduction to the implications of motif selection and placement for the perception of the ensuing patterns.     

Evolutionary internalized regularities

Roger Shepard's proposals and supporting experiments concerning evolutionary internalized regularities have been very influential in the study of vision and in other areas of psychology and cognitive science. This paper examines issues concerning the need, nature, explanatory role, and justification for postulating such internalized constraints. In particular, I seek further clarification from Shepard on how best to understand his claim that principles of kinematic geometry underlie phenomena of motion perception. My primary focus is on the ecological validity of Shepard's kinematic constraint in the context of ordinary motion perception. First, I explore the analogy Shepard draws between internalized circadian rhythms and the supposed internalization of kinematic geometry. Next, questions are raised about how to interpret and justify applying results from his own and others' experimental studies of apparent motion to more everyday cases of motion perception in richer environments. Finally, some difficulties with Shepard's account of the evolutionary development of his kinematic constraint are considered.     

An Elementary System of Axioms for Euclidean Geometry Based on Symmetry Principles

In this article I develop an elementary system of axioms for Euclidean geometry. On one hand, the system is based on the symmetry principles which express our a priori ignorant approach to space: all places are the same to us (the homogeneity of space), all directions are the same to us (the isotropy of space) and all units of length we use to create geometric figures are the same to us (the scale invariance of space). On the other hand, through the process of algebraic simplification, this system of axioms directly provides the Weyl’s system of axioms for Euclidean geometry. The system of axioms, together with its a priori interpretation, offers new views to philosophy and pedagogy of mathematics: (1) it supports the thesis that Euclidean geometry is a priori, (2) it supports the thesis that in modern mathematics the Weyl’s system of axioms is dominant to the Euclid’s system because it reflects the a priori underlying symmetries, (3) it gives a new and promising approach to learn geometry which, through the Weyl’s system of axioms, leads from the essential geometric symmetry principles of the mathematical nature directly to modern mathematics.     

The Place Of Geometry: Heidegger's Mathematical Excursus On Aristotle

'The Place of Geometry' discusses the excursus on mathematics from Heidegger's 1924–25 lecture course on Platonic dialogues, which has been published as Volume 19 of the Gesamtausgabe as Plato's Sophist , as a starting point for an examination of geometry in Euclid, Aristotle and Descartes. One of the crucial points Heidegger makes is that in Aristotle there is a fundamental difference between arithmetic and geometry, because the mode of their connection is different. The units of geometry are positioned, the units of arithmetic unpositioned. Following Heidegger's claim that the Greeks had no word for space, and David Lachterman's assertion that there is no term corresponding to or translatable as 'space' in Euclid's Elements , I examine when the term 'space' was introduced into Western thought. Descartes is central to understanding this shift, because his understanding of extension based in terms of mathematical co-ordinates is a radical break with Greek thought. Not only does this introduce this word 'space' but, by conceiving of geometrical lines and shapes in terms of numerical co-ordinates, which can be divided, it turns something that is positioned into unpositioned. Geometric problems can be reduced to equations, the length (i.e, quantity) of lines: a problem of number. The continuum of geometry is transformed into a form of arithmetic. Geometry loses position just as the Greek notion of 'place' is transformed into the modern notion of space.     

Boolean considerations on John Buridan's octagons of opposition

This paper studies John Buridan's octagons of opposition for the de re modal propositions and the propositions of unusual construction. Both Buridan himself and the secondary literature have emphasized the strong similarities between these two octagons (as well as a third one, for propositions with oblique terms). In this paper, I argue that the interconnection between both octagons is more subtle than has previously been thought: if we move beyond the Aristotelian relations, and also take Boolean considerations into account, then the strong analogy between Buridan's octagons starts to break down. These differences in Boolean structure can already be discerned within the octagons themselves; on a more abstract level, they lead to these two octagons having different degrees of Boolean complexity (i.e. Boolean closures of different sizes). These results are obtained by means of bitstring analysis, which is one of the key tools from contemporary logical geometry. Finally, I argue that this historical investigation is directly relevant for the theoretical framework of logical geometry, and discuss how it helps us to address certain open questions in this framework.     

Hume on space, geometry, and diagrammatic reasoning

Hume??s discussion of space, time, and mathematics at T 1.2 appeared to many earlier commentators as one of the weakest parts of his philosophy. From the point of view of pure mathematics, for example, Hume??s assumptions about the infinite may appear as crude misunderstandings of the continuum and infinite divisibility. I shall argue, on the contrary, that Hume??s views on this topic are deeply connected with his radically empiricist reliance on phenomenologically given sensory images. He insightfully shows that, working within this epistemological model, we cannot attain complete certainty about the continuum but only at most about discrete quantity. Geometry, in contrast to arithmetic, cannot be a fully exact science. A number of more recent commentators have offered sympathetic interpretations of Hume??s discussion aiming to correct the older tendency to dismiss this part of the Treatise as weak and confused. Most of these commentators interpret Hume as anticipating the contemporary idea of a finite or discrete geometry. They view Hume??s conception that space is composed of simple indivisible minima as a forerunner of the conception that space is a discretely (rather than continuously) ordered set. This approach, in my view, is helpful as far as it goes, but there are several important features of Hume??s discussion that are not sufficiently appreciated. I go beyond these recent commentators by emphasizing three of Hume??s most original contributions. First, Hume??s epistemological model invokes the ??confounding?? of indivisible minima to explain the appearance of spatial continuity. Second, Hume??s sharp contrast between the perfect exactitude of arithmetic and the irremediable inexactitude of geometry reverses the more familiar conception of the early modern tradition in pure mathematics, according to which geometry (the science of continuous quantity) has its own standard of equality that is independent from and more exact than any corresponding standard supplied by algebra and arithmetic (the sciences of discrete quantity). Third, Hume has a developed explanation of how geometry (traditional Euclidean geometry) is nonetheless possible as an axiomatic demonstrative science possessing considerably more exactitude and certainty that the ??loose judgements?? of the vulgar.