It has been argued that Hume's denial of infinite divisibility entails the falsity of most of the familiar theorems of Euclidean geometry, including the Pythagorean theorem and the bisection theorem. I argue that Hume's thesis that there are indivisibles is not incompatible with the Pythagorean theorem and other central theorems of Euclidean geometry, but only with those theorems that deal with matters of minuteness. The key to understanding Hume's view of geometry is (...) the distinction he draws between a precise and an imprecise standard of equality in extension. Hume's project is different from the attempt made by Berkeley in some of his later writings to save Euclidean geometry. Unlike Berkeley, who interprets the theorems of Euclidean geometry as false albeit useful approximations of geometrical facts, Hume is able to save most of the central theorems as true. (shrink)

We use Herbrand’s theorem to give a new proof that Euclid’s parallel axiom is not derivable from the other axioms of first-order Euclidean geometry. Previous proofs involve constructing models of non-Euclidean geometry. This proof uses a very old and basic theorem of logic together with some simple properties of ruler-and-compass constructions to give a short, simple, and intuitively appealing proof.

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 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)

Independently of any eighteenth century work on the geometry of parallels, Thomas Reid discovered the non-euclidean "geometry of visibles" in 1764. Reid's construction uses an idealized eye, incapable of making distance discriminations, to specify operationally a two dimensional visible space and a set of objects, the visibles. Reid offers sample theorems for his doubly elliptical geometry and proposes a natural model, the surface of the sphere. His construction draws on eighteenth century theory of vision for (...) some of its technical features and is motivated by Reid's desire to defend realism against Berkeley's idealist treatment of visual space. (shrink)

In this paper we extend the NeutroAlgebra & AntiAlgebra to the geometric spaces, by founding the NeutroGeometry & AntiGeometry. While the Non-Euclidean Geometries resulted from the total negation of one specific axiom (Euclid’s Fifth Postulate), the AntiGeometry results from the total negation of any axiom or even of more axioms from any geometric axiomatic system (Euclid’s, Hilbert’s, etc.) and from any type of geometry such as (Euclidean, Projective, Finite, Affine, Differential, Algebraic, Complex, Discrete, Computational, Molecular, Convex, etc.) (...) Geometry, and the NeutroGeometry results from the partial negation of one or more axioms [and no total negation of no axiom] from any geometric axiomatic system and from any type of geometry. Generally, instead of a classical geometric Axiom, one may take any classical geometric Theorem from any axiomatic system and from any type of geometry, and transform it by NeutroSophication or AntiSophication into a NeutroTheorem or AntiTheorem respectively in order to construct a NeutroGeometry or AntiGeometry. Therefore, the NeutroGeometry and AntiGeometry are respectively alternatives and generalizations of the Non-Euclidean Geometries. In the second part, we recall the evolution from Paradoxism to Neutrosophy, then to NeutroAlgebra & AntiAlgebra, afterwards to NeutroGeometry & AntiGeometry, and in general to NeutroStructure & AntiStructure that naturally arise in any field of knowledge. At the end, we present applications of many NeutroStructures in our real world. (shrink)

For over half a century I have been interested in the role of intuitive spatial reasoning in mathematics. My Oxford DPhil Thesis (1962) was an attempt to defend Kant's philosophy of mathematics, especially his claim that mathematical proofs extend our knowledge (so the knowledge is "synthetic", not "analytic") and that the discoveries are not empirical, or contingent, but are in an important sense "a priori" (which does not imply "innate") and also necessarily true. -/- I had made my views clear (...) in courses on philosophy of science and mathematics when teaching at Sussex University (from 1964) which was why one of our former students, Mary Pardoe (then Mary Ensor) who had become a mathematics teacher informed me, while visiting the university, that she had found a new diagrammatic proof of the triangle sum theorem. I reported her proof in some papers and presentations on methods of representation and reasoning, e.g. here, but neither she nor I has encountered anyone else who knew about the proof. (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)

Most commentators agree that (part of what) Kant means by characterizing the propositions of geometry as synthetic is that they are not true merely in virtue of logic or meaning, and that this characterization has something to do with his views about the construction of geometrical concepts in intuition. Many commentators regard construction in intuition as an essential part of geometrical proofs on Kant’s view. On this reading, the propositions of geometry are synthetic because the geometrical theorems (...) cannot be proved in purely conceptual or logical terms. Other commentators see the main role of pure intuition and the figures constructed in pure intuition in that they provide a model for Euclidean geometry. On views of this kind, the propositions of geometry are synthetic because the geometrical axioms are substantive truths about one of our forms of intuition. On the interpretation proposed in this essay, what Kant means by claiming that the propositions of geometry are synthetic is not only that the Euclidean axioms and theorems cannot be reduced to tautologies or logical truths, but also that they apply to really possible objects. Construction in intuition plays no essential role in (what we now call) ‘pure’ geometry on Kant’s view. But the fact that the concepts of geometry can be constructed in intuition is of crucial importance in the context of Kant’s transcendental philosophy of geometry, because, among other things, it allows him to explain how Euclidean geometry is possible as an a priori synthetic science in the sense just indicated. (shrink)

In a brief but remarkable section of the Inquiry into the Human Mind, Thomas Reid argued that the visual field is governed by principles other than the familiar theorems of Euclid—theorems we would nowadays classify as Riemannian. On the strength of this section, he has been credited by Norman Daniels, R. B. Angell, and others with discovering non-Euclidean geometry over half a century before the mathematicians—sixty years before Lobachevsky and ninety years before Riemann. I believe that (...) Reid does indeed have a fair claim to have discovered a non-Euclidean geometry. However, the real basis of Reid’s geometry of visibles is subtle and easy to misidentify. My main aim in what follows is to make clear what the real basis is, separating it from several fallacious or irrelevant considerations on which Reid may seem to be relying. A secondary aim is to air the worry that Reid’s case for his geometry can succeed only at the cost of compromising the achievement for which Reid is best known—his direct realist theory of perception. (shrink)

In a brief but remarkable section of the Inquiry into the Human Mind, Thomas Reid argued that the visual field is governed by principles other than the familiar theorems of Euclid—theorems we would nowadays classify as Riemannian. On the strength of this section, he has been credited by Norman Daniels, R. B. Angell, and others with discovering non-Euclidean geometry over half a century before the mathematicians—sixty years before Lobachevsky and ninety years before Riemann. I believe that (...) Reid does indeed have a fair claim to have discovered a non-Euclidean geometry. However, the real basis of Reid’s geometry of visibles is subtle and easy to misidentify. My main aim in what follows is to make clear what the real basis is, separating it from several fallacious or irrelevant considerations on which Reid may seem to be relying. A secondary aim is to air the worry that Reid’s case for his geometry can succeed only at the cost of compromising the achievement for which Reid is best known—his direct realist theory of perception. (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)

The present first part about the eventual completeness of mathematics (called “Hilbert mathematics”) is concentrated on the Gödel incompleteness (1931) statement: if it is an axiom rather than a theorem inferable from the axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic, this would pioneer the pathway to Hilbert mathematics. One of the main arguments that it is an axiom consists in the direct contradiction of the axiom of induction in arithmetic and the axiom of infinity in set theory. (...) Thus, the pair of arithmetic and set are to be similar to Euclidean and non-Euclidean geometries distinguishably only by the Fifth postulate now, i.e. after replacing it and its negation correspondingly by the axiom of finiteness (induction) versus that of finiteness being idempotent negations to each other. Indeed, the axiom of choice, as far as it is equivalent to the well-ordering “theorem”, transforms any set in a well-ordering either necessarily finite according to the axiom of induction or also optionally infinite according to the axiom of infinity. So, the Gödel incompleteness statement relies on the logical contradiction of the axiom of induction and the axiom of infinity in the final analysis. Nonetheless, both can be considered as two idempotent versions of the same axiom (analogically to the Fifth postulate) and then unified after logicism and its inherent intensionality since the opposition of finiteness and infinity can be only extensional (i.e., relevant to the elements of any set rather than to the set by itself or its characteristic property being a proposition). So, the pathway for interpreting the Gödel incompleteness statement as an axiom and the originating from that assumption for “Hilbert mathematics” accepting its negation is pioneered. A much wider context relevant to realizing the Gödel incompleteness statement as a metamathematical axiom is consistently built step by step. The horizon of Hilbert mathematics is the proper subject in the third part of the paper, and a reinterpretation of Gödel’s papers (1930; 1931) as an apology of logicism as the only consistent foundations of mathematics is the topic of the next second part. (shrink)

I start by considering Mark Steiner’s startling claim that Hume takes geometry to be synthetic a priori, which engenders the Kantian challenge to explain how such knowledge is possible. I argue, in response, that Steiner misinterprets the (deceptive) relevant passage from Hume, and that Hume, as the received view has it, takes geometry to be analytic, although in a more expansive sense of the word than the modern one. I then note a new challenge geometry engenders for (...) Hume. Unlike Euclidean space, Humean space is finitely divisible, for which several Euclidean axioms and theorems do not hold. I argue, in response, that (crucially for his scientific project) Hume can account for our belief in the truth of Euclidean geometry on the basis of non-Euclidean ideas, although (innocuously for him) not all should be true. I conclude by arguing, on a less optimistic note, that Hume cannot point to a geometry that is true of our (discrete) ideas. (shrink)

I describe two approaches to modelling the universe, the one having its origin in topos theory and differential geometry, the other in set theory. The first is synthetic differential geometry. Traditionally, there have been two methods of deriving the theorems of geometry: the analytic and the synthetic. While the analytical method is based on the introduction of numerical coordinates, and so on the theory of real numbers, the idea behind the synthetic approach is to furnish the (...) subject of geometry with a purely geometric foundation in which the theorems are then deduced by purely logical means from an initial body of postulates. The most familiar examples of the synthetic geometry are classical Euclidean geometry and the synthetic projective geometry introduced by Desargues in the 17th century and revived and developed by Carnot, Poncelet, Steiner and others during the 19th century. The power of analytic geometry derives very largely from the fact that it permits the methods of the calculus, and, more generally, of mathematical analysis, to be introduced into geometry, leading in particular to differential geometry (a term, by the way, introduced in 1894 by the Italian geometer Luigi Bianchi). That being the case, the idea of a “synthetic” differential geometry seems elusive: how can differential geometry be placed on a “purely geometric” or “axiomatic” foundation when the apparatus of the calculus seems inextricably involved? To my knowledge there have been two attempts to develop a synthetic differential geometry. The first was initiated by Herbert Busemann in the 1940s, building on earlier work of Paul Finsler. Here the idea was to build a differential geometry that, in its author’s words, “requires no derivatives”: the basic objects in Busemann’s approach are not differentiable manifolds, but metric spaces of a certain type in which the notion of a geodesic can be defined in an intrinsic manner. I shall not have anything more to say about this approach. The second approach, that with which I shall be concerned here, was originally proposed in the 1960s by F.. (shrink)

We describe a first order axiom set which yields the classical first order Euclidean geometry of Tarski when used with classical logic, and yields an intuitionistic Euclidean geometry when used with intuitionistic logic. The first order language has a single six place atomic predicate and no function symbols. The intuitionistic system has a computational interpretation in recursive function theory, that is, a realizability interpretation analogous to those given by Kleene for intuitionistic arithmetic and analysis. This interpretation (...) shows the unprovability in the intuitionistic theory of certain “nonconstructive” theorems of the classical geometry. (shrink)

In the article, an argument is given that Euclidean geometry is a priori in the same way that numbers are a priori, the result of modelling, not the world, but our activities in the world.

In this paper we give probably an exhaustive analysis of the geometry of solids which was sketched by Tarski in his short paper [20, 21]. We show that in order to prove theorems stated in [20, 21] one must enrich Tarski's theory with a new postulate asserting that the universe of discourse of the geometry of solids coincides with arbitrary mereological sums of balls, i.e., with solids. We show that once having adopted such a solution Tarski's Postulate (...) 4 can be omitted, together with its versions 4' and 4". We also prove that the equivalence of postulates 4, 4' and 4" is not provable in any theory whose domain contains objects other than solids. Moreover, we show that the concentricity relation as defined by Tarski must be transitive in the largest class of structures satisfying Tarski's axioms. We build a model (in three-dimensional Euclidean space) of the theory of so called T*-structures and present the proof of the fact that this is the only (up to isomorphism) model of this theory. Moreover, we propose different categorical axiomatizations of the geometry of solids. In the final part of the paper we answer the question concerning the logical status (within the theory of T*-structures) of the definition of the concentricity relation given by Tarski. (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)

Some ancient Greek coins from the island state of Aegina depict peculiar geometric designs. Hitherto they have been interpreted as anticipations of some Euclidean propositions. But this paper proposes geometrical constructions which establish connections to pre-Euclidean treatments of incommensurability. The earlier Aeginetan coin design from about 500 bc onwards appears as an attempt not only to deal with incommensurability but also to conceal it. It might be related to Plato’s dialogue Timaeus. The newer design from 404 bc onwards (...) reveals incommensurability, namely in the context of ‘doubling the square’. It thereby covers the same topic but a different geometry as passages in Plato’s dialogue Meno (385 bc). This coin design incorporates important elements of ancient Greek geometrical analysis of the fifth century bc like the gnomon, Hippocrates’ squaring of the lunule (ca. 430 bc), and a geometrical version of monetary equivalence. Through this venue, the design’s conceptual lineage might be traced as far back as Heraclitus’ cosmology of about 500 bc. (shrink)

A new form of the Hyperbolic Pythagorean Theorem, which has a striking intuitive appeal and offers a strong contrast to its standard form, is presented. It expresses the square of the hyperbolic length of the hypotenuse of a hyperbolic right-angled triangle as the “Einstein sum” of the squares of the hyperbolic lengths of the other two sides, Fig. 1, thus completing the long path from Pythagoras to Einstein. Following the pioneering work of Varičak it is well known that relativistic velocities (...) are governed by hyperbolic geometry in the same way that prerelativistic velocities are governed by Euclidean geometry. Unlike prerelativistic velocity composition, given by the ordinary vector addition, the composition of relativistic velocities, given by the Einstein addition, is neither commutative nor associative due to the presence of Thomas precession. Following the discovery of the mathematical regularity that Thomas precession stores, it is now possible to extend Thomas precession by abstraction, (i) allowing hyperbolic geometry to be studied by means of analogies that it shares with Euclidean geometry; and, similarly (ii) allowing velocities and accelerations in relativistic mechanics to be studied by means of analogies that they share with velocities and accelerations in classical mechanics. The abstract Thomas precession, called the Thomas gyration, gives rise to gyrovector space theory in which the prefix gyro is used extensively in terms like gyrogroups and gyrovector spaces, gyroassociative and gyrocommutative laws, gyroautomorphisms, gyrotranslations, etc. We demonstrate the superiority of our gyrovector space formalism in capturing analogies by deriving the Hyperbolic Pythagorean Theorem in a form fully analogous to its Euclidean counterpart, thus contrasting it with the standard form in which the Hyperbolic Pythagorean Theorem is known in the literature. The hyperbolic metric, which supports the Hyperbolic Pythagorean Theorem, has a dual metric. We show that the dual metric does not support a Pythagorean theorem but, instead, it supports the π-Theorem according to which the sum of the three dual angles of a hyperbolic triangle is π. (shrink)

A reply to Risjord's defense of the view that there is no conflict between non-Euclidean geometry and Kant's philosophy of geometry because, while the form of intuition restricts which systems of concepts may be accepted as a geometry, it does not do so uniquely ("Stud Hist Phil Sci, 21", 1990). I argue that under these circumstances it is difficult to sustain the synthetic "a priori" status of geometrical propositions. Two broad ways of attempting to do so (...) are considered and criticized. (shrink)

Geometry defines entities that can be physically realized in space, and our knowledge of abstract geometry may therefore stem from our representations of the physical world. Here, we focus on Euclidean geometry, the geometry historically regarded as “natural”. We examine whether humans possess representations describing visual forms in the same way as Euclidean geometry – i.e., in terms of their shape and size. One hundred and twelve participants from the U.S. (age 3–34 years), (...) and 25 participants from the Amazon (age 5–67 years) were asked to locate geometric deviants in panels of 6 forms of variable orientation. Participants of all ages and from both cultures detected deviant forms defined in terms of shape or size, while only U.S. adults drew distinctions between mirror images (i.e. forms differing in “sense”). Moreover, irrelevant variations of sense did not disrupt the detection of a shape or size deviant, while irrelevant variations of shape or size did. At all ages and in both cultures, participants thus retained the same properties as Euclidean geometry in their analysis of visual forms, even in the absence of formal instruction in geometry. These findings show that representations of planar visual forms provide core intuitions on which humans’ knowledge in Euclidean geometry could possibly be grounded. (shrink)

Kant argued that Euclidean geometry is synthesized on the basis of an a priori intuition of space. This proposal inspired much behavioral research probing whether spatial navigation in humans and animals conforms to the predictions of Euclidean geometry. However, Euclidean geometry also includes concepts that transcend the perceptible, such as objects that are infinitely small or infinitely large, or statements of necessity and impossibility. We tested the hypothesis that certain aspects of nonperceptible Euclidian (...) class='Hi'>geometry map onto intuitions of space that are present in all humans, even in the absence of formal mathematical education. Our tests probed intuitions of points, lines, and surfaces in participants from an indigene group in the Amazon, the Mundurucu, as well as adults and age-matched children controls from the United States and France and younger US children without education in geometry. The responses of Mundurucu adults and children converged with that of mathematically educated adults and children and revealed an intuitive understanding of essential properties of Euclidean geometry. For instance, on a surface described to them as perfectly planar, the Mundurucu's estimations of the internal angles of triangles added up to ∼180 degrees, and when asked explicitly, they stated that there exists one single parallel line to any given line through a given point. These intuitions were also partially in place in the group of younger US participants. We conclude that, during childhood, humans develop geometrical intuitions that spontaneously accord with the principles of Euclidean geometry, even in the absence of training in mathematics. (shrink)

For the classical Geometry, in a geometrical space, all items (concepts, axioms, theorems, etc.) are totally (100%) true. But, in the real world, many items are not totally true. The NeutroGeometry is a geometrical space that has some items that are only partially true (and partially indeterminate, and partially false), and no item that is totally false. The AntiGeometry is a geometrical space that has some item that are totally (100%) false. While the Non-Euclidean Geometries [hyperbolic and (...) elliptic geometries] resulted from the total negation of only one specific axiom (Euclid’s Fifth Postulate), the AntiGeometry results from the total negation of any axiom [and in general: theorem, concept, idea etc.] and even of more axioms [theorem, concept, idea, etc.] and in general from any geometric axiomatic system (Euclid’s five postulates, Hilbert’s 20 axioms, etc.), and the NeutroAxiom results from the partial negation of any axiom (or concept, theorem, idea, etc.). Clearly, the AntiGeometry is a generalization of Non-Euclidean Geometries. (shrink)

In this paper we extend Neutro-Algebra and Anti-Algebra to geometric spaces, founding Neutro/Geometry and AntiGeometry. While Non-Euclidean Geometries resulted from the total negation of a specific axiom (Euclid's Fifth Postulate), AntiGeometry results from the total negation of any axiom or even more axioms of any geometric axiomatic system (Euclidean, Hilbert, etc. ) and of any type of geometry such as Geometry (Euclidean, Projective, Finite, Differential, Algebraic, Complex, Discrete, Computational, Molecular, Convex, etc.), and Neutro-Geometry (...) results from the partial negation of one or more axioms [and without total negation of any axiom] of any geometric axiomatic system and of any type of geometry. Generally, instead of a classical geometric Axiom, one can take any classical geometric Theorem of any axiomatic system and of any type of geometry, and transform it by Neutrosophication or Antisofication into a Neutro-Theorem or Anti-Theorem respectively to construct a Neutro-Geometry or Anti-Geometry. Therefore, Neutro-Geometry and Anti-Geometry are respectively alternatives and generalizations of Non-Euclidean Geometries. In the second part, the evolution from Paradoxism to Neutrosophy, then to NeutroAlgebra and Anti-Algebra, then to Neutro-Geometry and Anti-Geometry, and in general to Neutro-Structure and Anti-Structure that arise naturally in any field of knowledge is recalled. At the end, applications of many Neutro-Structures in our real world are presented. (shrink)

Kant's arguments for the synthetic a priori status of geometry are generally taken to have been refuted by the development of non-Euclidean geometries. Recently, however, some philosophers have argued that, on the contrary, the development of non-Euclidean geometry has confirmed Kant's views, for since a demonstration of the consistency of non-Euclidean geometry depends on a demonstration of its equi-consistency with Euclidean geometry, one need only show that the axioms of Euclidean (...) class='Hi'>geometry have 'intuitive content' in order to show that both Euclidean and non-Euclidean geometry are bodies of synthetic a priori truths. Michael Friedman has argued that this defence presumes a polyadic conception of logic that was foreign to Kant. According to Friedman, Kant held that geometrical reasoning itself relies essentially on intuition, and that this precludes the very possibility of non-Euclidean geometry. While Friedman's characterization of Kant's views on geometrical reasoning is correct, I argue that Friedman's conclusion that non-Euclidean geometries are logically impossible for Kant is not. I argue that Kant is best understood as a proto-constructivist and that modern constructive axiomatizations (unlike Hilbert-style axiomatizations) of both Euclidean and non-Euclidean geometry capture Kant's views on the essentially constructive nature of geometrical reasoning well. (shrink)

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)

The mathematical developments of the 19th century seemed to undermine Kant’s philosophy. Non-Euclidean geometries challenged Kant’s view that there is a spatial intuition rich enough to yield the truth of Euclidean geometry. Similarly, advancements in algebra challenged the view that temporal intuition provides a foundation for both it and arithmetic. Mathematics seemed increasingly detached from experience as well as its form; moreover, with advances in symbolic logic, mathematical inference also seemed independent of intuition. This paper considers various (...) philosophical responses to these changes, focusing on the idea of modifying Kant’s conception of intuition in order to accommodate the increasing abstractness of mathematics. It is argued that far from clinging to an outdated paradigm, programs based on new conceptions of intuition should be seen as motivated by important philosophical desiderata, such as the truth, apriority, distinctiveness and autonomy of mathematics. (shrink)

In the chapter “The Geometry of Visibles” in his ‘Inquiry into the Human Mind’, Thomas Reid constructs a special space, develops a special geometry for that space, and offers a natural model for this geometry. In doing so, Reid “discovers” non-Euclidean Geometry sixty years before the mathematicians. This paper examines this “discovery” and the philosophical motivations underlying it. By reviewing Reid’s ideas on visible space and confronting him with Kant and Berkeley, I hope, moreover, to (...) resolve an alleged impasse in Reid’s philosophy concerning the contradictory characteristics of Reid’s tangible and visible space. (shrink)

We show the algorithmic unsolvability of a number of decision procedures in ordinary two dimensional Euclidean geometry, involving lines and integer points. We also consider formulations involving integral domains of characteristic 0, and ordered rings. The main tool is the solution to Hilbert's Tenth Problem. The limited number of facts used from recursion theory are isolated at the beginning.

Open Court began publishingThe Monistin 1890 as a journal“devotedto the philosophy of science”that regularly included mathematics. The audiencewas understood to be“cultured people who have not a technical mathematicaltraining”but nevertheless“have a mathematical penchant.”With these constraints,the mathematical content varied from recreations to logical foundations, but every-one had something to say about non-Euclidean geometry, in debates that rangedfrom psychology to semantics. The focus in this essay is on the contested value ofmathematical expertise in legitimating what should be considered as mathematics.While some (...) mathematicians urgedThe Monistto uphold disciplinary standards ofgeometrical reasoning, authors opposed tonon-Euclidean geometry aligned theirreasoning with practical applications, universal know-how, and nonhierarchicaldemocracy. As one contributorinquired,“How isthe professional expert betterfit-ted to see more lucidly in dealing with the elements of geometry than any otherperson of good geometric faculty?”. (shrink)

In this paper, we will make explicit the relationship that exists between geometric objects and geometric figures in planar Euclidean geometry. That will enable us to determine basic features regarding the role of geometric figures and diagrams when used in the context of pure and applied planar Euclidean geometry, arising due to this relationship. By taking into account pure geometry, as developed in Euclid’s Elements, and practical geometry, we will establish a relation between geometric (...) objects and figures. Geometric objects are defined in terms of idealizations of the corresponding figures of practical geometry. We name the relationship between them as a relation of idealization. This relation, existing between objects and figures, is what enables figures to have a role in pure and applied geometry. That is, we can use a figure or diagram as a representation of geometric objects or composite geometric objects because the relation of idealization corresponds to a resemblance-like relationship between objects and figures. Moving beyond pure geometry, we will defend that there are two other ‘layers’ of representation at play in applied geometry. To show that, we will consider Euclid’s Optics. (shrink)

In this paper, we review the history of quasicrystals from their sensational discovery in 1982, initially “forbidden” by the rules of classical crystallography, to 2011 when Dan Shechtman was awarded the Nobel Prize in Chemistry. We then discuss the discovery of quasicrystals in philosophical terms of anomalies behavior that led to a paradigm shift as offered by philosopher and historian of science Thomas Kuhn in ‘The Structure of Scientific Revolutions’. This discovery, which found expression in the redefinition of the concept (...) crystal from being periodically arranged to producing sharp peaks in the Bragg diffraction pattern, is analyzed according to the Kuhn Cycle. We relate the quasicrystal revolution to the non-Euclidean geometry revolution and argues that since “great minds think alike” there is a diffusion of ideas between scientific revolutions, or a resonance between different disciplines at different times. The story behind quasicrystals is an excellent example of a paradigm shift, demonstrating the nature of scientific discoveries and breakthroughs. (shrink)

Twentieth-century developments in logic and mathematics have led many people to view Euclid’s proofs as inherently informal, especially due to the use of diagrams in proofs. In _Euclid and His Twentieth-Century Rivals_, Nathaniel Miller discusses the history of diagrams in Euclidean Geometry, develops a formal system for working with them, and concludes that they can indeed be used rigorously. Miller also introduces a diagrammatic computer proof system, based on this formal system. This volume will be of interest to (...) mathematicians, computer scientists, and anyone interested in the use of diagrams in geometry. (shrink)