Hermann von Helmholtz’s geometrical papers have been typically deemed to provide an implicitly group-theoretical analysis of space, as articulated later by Felix Klein, Sophus Lie, and Henri Poincaré. However, there is less agreement as to what properties exactly in such a view would pertain to space, as opposed to abstract mathematical structures, on the one hand, and empirical contents, on the other. According to Moritz Schlick, the puzzle can be resolved only by clearly distinguishing the empirical qualities of spatial perception (...) from those describable in terms of axiomatic geometry. This paper offers a partial defense of the group-theoretical reading of Helmholtz along the lines of Ernst Cassirer in the fourth volume of The Problem of Knowledge of 1940. In order to avoid the problem raised by Schlick, Cassirer relied on a Kantian view of space not so much as an object of geometry, but as a precondition for the possibility of measurement. Although the concept of group does not provide a description of space, the modern way to articulate the concept of space in terms of transformation groups reveals something about the structure and the transformation of spatial concepts in mathematical and natural sciences. (shrink)

Starting from an analogy with Quine’s two dogmas of empiricism, I offer a critique of two dogmas of analytic historiography: the belief in a cleavage between the justification of a ph...

1. Does Hegel’s The Science of Logic (Hegel 2010) have any relation to or relevance for what is now known as ‘the science of logic’? Here a negative answer is as likely to be endorsed by many conte...

Logicism is the view that mathematical truths are logical truths. But a logical truth is commonly thought to be one with a universally valid form. The form of “7 > 5” would appear to be the same as “4 > 6”. Yet one is a mathematical truth, and the other not a truth at all. To preserve logicism, we must maintain that the two either are different subforms of the same generic form, or that their forms are not at all (...) what they appear. The historical record shows that Russell pursued both these options, but that the struggle with the logical paradoxes pushed him away from the first kind of response and toward the second. An object cannot itself have a kind of inner logical complexity that makes a proposition have a different logical form merely in virtue of being about it, nor can their representatives in logical forms be single things different for different forms, at least not without postulating too many such objects and thereby creating Cantorian diagonal paradoxes. There are only apparent objects which are actually fragments of logical forms, different in different cases. (shrink)

Relying upon a very close reading of all of the definitions given in Euclid’s Elements, I argue that this mathematical treatise contains a philosophical treatment of mathematical objects. Specifically, I show that Euclid draws elaborate metaphysical distinctions between substances and non-substantial attributes of substances, different kinds of substance, and different kinds of non-substance. While the general metaphysical theory adopted in the Elements resembles that of Aristotle in many respects, Euclid does not employ Aristotle’s terminology, or indeed, any philosophical terminology at (...) all. Instead, Euclid systematically uses different types of definition to distinguish between metaphysically different kinds of mathematical object. (shrink)

I propose a reading of Berkeley's Essay towards a New Theory of Vision in which Molyneux-type questions are interpreted as thought experiments instead of arguments. First, I present the general argumentative strategy in the NTV, and provide grounds for the traditional reading. Second, I consider some roles of thought experiments, and classify Molyneux-type questions in the NTV as constructive conjectural thought experiments. Third, I argue that (i) there is no distinction between Weak and Strong Heterogeneity theses in the NTV; (ii) (...) that Strong Heterogeneity is the basis of Berkeley's theory; and (iii) that Molyneux-type questions act as illustrations of Strong Heterogeneity. (shrink)

Hilbert’s choice operators τ and ε, when added to intuitionistic logic, strengthen it. In the presence of certain extensionality axioms they produce classical logic, while in the presence of weaker decidability conditions for terms they produce various superintuitionistic intermediate logics. In this thesis, I argue that there are important philosophical lessons to be learned from these results. To make the case, I begin with a historical discussion situating the development of Hilbert’s operators in relation to his evolving program in the (...) foundations of mathematics and in relation to philosophical motivations leading to the development of intuitionistic logic. This sets the stage for a brief description of the relevant part of Dummett’s program to recast debates in metaphysics, and in particular disputes about realism and anti-realism, as closely intertwined with issues in philosophical logic, with the acceptance of classical logic for a domain reflecting a commitment to realism for that domain. Then I review extant results about what is provable and what is not when one adds epsilon to intuitionistic logic, largely due to Bell and DeVidi, and I give several new proofs of intermediate logics from intuitionistic logic+ε without identity. With all this in hand, I turn to a discussion of the philosophical significance of choice operators. Among the conclusions I defend are that these results provide a finer-grained basis for Dummett’s contention that commitment to classically valid but intuitionistically invalid principles reflect metaphysical commitments by showing those principles to be derivable from certain existence assumptions; that Dummett’s framework is improved by these results as they show that questions of realism and anti-realism are not an “all or nothing” matter, but that there are plausibly metaphysical stances between the poles of anti-realism and realism, because different sorts of ontological assumptions yield intermediate rather than classical logic; and that these intermediate positions between classical and intuitionistic logic link up in interesting ways with our intuitions about issues of objectivity and reality, and do so usefully by linking to questions around intriguing everyday concepts such as “is smart,” which I suggest involve a number of distinct dimensions which might themselves be objective, but because of their multivalent structure are themselves intermediate between being objective and not. Finally, I discuss the implications of these results for ongoing debates about the status of arbitrary and ideal objects in the foundations of logic, showing among other things that much of the discussion is flawed because it does not recognize the degree to which the claims being made depend on the presumption that one is working with a very strong logic. (shrink)

Russell’s rejection in 1898 of the doctrine of internal relations — the view that all relations are grounded in the intrinsic properties of the terms related — was a decisive part of his break with Hegelianism and opened the way for his turn to analytic philosophy. Before rejecting it, Russell had given the doctrine little thought, though it played an essential role in the most intractable of the problems facing his attempt to construct a Hegelian dialectic of the sciences. I (...) argue that it was Russell’s early reading of Leibniz, in preparation for his lectures on Leibniz given at Cambridge in 1899, that most probably alerted him to the role the doctrine was playing in his own philosophy. Leibniz defended a similar doctrine and extricated it from difficulties like those faced by Russell by means of devices that were not open to Russell. Russell would have come across these views of Leibniz in writings by Leibniz that he read in the summer of 1898, just before he rejected the doctrine of internal relations. (shrink)

O presente manual tem como intenção constituir um guia para uma disciplina introdutória de filosofia da linguagem. Foi elaborado a partir da leccionação da disciplina de Filosofia da Linguagem I na Faculdade de Letras da Universidade do Porto desde 2001. A disciplina de Filosofia da Linguagem I ocupa um semestre lectivo e proporciona aos estudantes o primeiro contacto sistemático com a área da filosofia da linguagem. Pretende-se que este manual ofereça aos estudantes os instrumentos necessários não apenas para acompanhar uma (...) iniciação ao campo da filosofia da linguagem mas também para o estudo e a investigação autónomos posteriores. Para isso é percorrido um trajecto que conduz das intuições pré-teóricas acerca de linguagem, de que todos dispomos, até um conhecimento disciplinar específico, histórico e temático, da filosofia da linguagem. Em termos práticos, são considerados como precedentes da disciplina de Filosofia da Linguagem I – mesmo se, pelo existem actualmente precedências em sentido estrito – as disciplinas de Lógica I e II e de Filosofia do Conhecimento I e II. Os programas dessas disciplinas nos últimos anos estiveram presentes por trás da elaboração do manual. (shrink)

In this paper, we trace the conceptual history of the term ?metamathematics? in the nineteenth century. It is well known that Hilbert introduced the term for his proof-theoretic enterprise in about 1922. But he was verifiably inspired by an earlier usage of the phrase in the 1870s. After outlining Hilbert's understanding of the term, we will explore the lines of inducement and elucidate the different meanings of ?metamathematics? in the final decades of the nineteenth century. Finally, we will investigate the (...) earlier occurrences and come to the conclusion that the conceptual prehistory of the Hilbertian notion of metamathematics dates back to 1870, whereas the history of the word starts in 1799 at the latest. What is this: metamathematics? It is something amazing everybody, [?] since it makes the mind dizzy and withdraws thinking its sole fulcrum.1 1 Brunner 1898, p. 832: ?Was ist das: Metamathematik? Es ist etwas, was einen Jeden in das äusserste Staunen versetzt, [?] was das Gehirn schwindlig macht und dem Denken seinen einzigen Stütz- und Angelpunkt zu entziehen droht? (shrink)

Russell's philosophy is rightly described as a programme of reduction of mathematics to logic. Now the theory of geometry developed in 1903 does not fit this picture well, since it is deeply rooted in the purely synthetic projective approach, which conflicts with all the endeavours to reduce geometry to analytical geometry. The first goal of this paper is to present an overview of this conception. The second aim is more far-reaching. The fact that such a theory of geometry was sustained (...) by Russell compels us to question the meaning of logicism: how is it possible to reconcile Russell's global reductionist standpoint with his local defence of the specificities of geometry? * This paper was first presented at the conference ‘Qu'est ce que la géométrie aux époques modernes et contemporaines?’, organized by the Universität Köln and the Archives Poincaré. I would like to thank Philippe Nabonnand for having enlightened me about the issues relative to projective geometry. I would like also to thank Nicholas Griffin, Brice Halimi, Bernard Linsky, Marco Panza, Ivahn Smadja for their helpful discussions. Many thanks also to the two anonymous referees for their useful suggestions. CiteULike Connotea Del.icio.us What's this? (shrink)

The aim of this paper is to clarify the role of category theory in the foundations of mathematics. There is a good deal of confusion surrounding this issue. A standard philosophical strategy in the face of a situation of this kind is to draw various distinctions and in this way show that the confusion rests on divergent conceptions of what the foundations of mathematics ought to be. This is the strategy adopted in the present paper. It is divided into 5 (...) sections. We first show that already in the set theoretical framework, there are different dimensions to the expression foundations of. We then explore these dimensions more thoroughly. After a very short discussion of the links between these dimensions, we move to some of the arguments presented for and against category theory in the foundational landscape. We end up on a more speculative note by examining the relationships between category theory and set theory. (shrink)

Though scholarship has explored Karin Costelloe-Stephen’s contributions to the history of psychoanalysis, as well as her relations to the Bloomsbury Group, her philosophical work has been almost completely ignored. This paper will examine her debate with Bertrand Russell over his criticism of Bergson. Costelloe-Stephen had employed the terminology of early analytic philosophy in presenting a number of arguments in defence of Bergson’s views. Costelloe-Stephen would object, among other things, to Russell’s use of an experiment which, as she points out, was (...) first conducted by Carl Stumpf. Russell appeals to Stumpf's experiment in his attempt to prove that sense data are terms in logical relations, a thesis presupposed by the project of logical analysis outlined in Our Knowledge of the External World. A reformulated version of Costelloe-Stephen's argument put forth by this paper shows that Russell's argument fails to provide adequate proof for his thesis. Further modifications of the argument can also address a reconstruction (based on contemporary reports) of Russell's reply to Costelloe-Stephen. In his reply, Russell would use, already in 1914, the term ‘analytic philosophy’ in contrasting his and Moore’s approach to a continental one, exemplified by Bergson and Costelloe-Stephen. (shrink)

Introduction to the Special Volume, “Method, Science and Mathematics: Neo-Kantianism and Analytic Philosophy,” edited by Scott Edgar and Lydia Patton. At its core, analytic philosophy concerns urgent questions about philosophy’s relation to the formal and empirical sciences, questions about philosophy’s relation to psychology and the social sciences, and ultimately questions about philosophy’s place in a broader cultural landscape. This picture of analytic philosophy shapes this collection’s focus on the history of the philosophy of mathematics, physics, and psychology. The following essays (...) uncover, reflect on, and exemplify modes of philosophy that are engaged with these allied disciplines. They make the case that, to the extent that analytic philosophers are still concerned with philosophy’s ties to these disciplines, we would do well to pay attention to neo-Kantian views on those ties. (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)

H. G. Wells’ Time Traveller inhabits uniform Newtonian time. Where relativistic/quantum travelers into the past follow spacetime curvatures, past-bound Wellsians must reverse their direction of travel relative to absolute time. William Grey and Robin Le Poidevin claim reversing Wellsians must overlap with themselves or fade away piecemeal like the Cheshire Cat. Self-overlap is physically impossible but ‘Cheshire Cat’ fades destroy Wellsians’ causal continuity and breed bizarre fusions of traveler-stages with opposed time-directions. However, Wellsians who rotate in higher-dimensional space can reverse (...) temporal direction without self-overlap, Cheshire Cats or mereological monstrosities. Alas, hyper-rotation in Newtonian space poses dynamic and biological problems,. Controllable and survivable Wellsian travel needs topologically-variable spaces. Newtonian space, not Newtonian time, is Wellsians’ real enemy. (shrink)

Despite his well‐known deductivism, in his early (unpublished) writings, Popper held an inductivist position. Up to 1929 epistemology entered Popper's reflections only as far as the problem was that of the justification of the scientific character of these fields of research. However, in that year, while surveying the history of non‐Euclidean geometries, Popper explicitly discussed the cognitive status of geometry without referring to psycho‐pedagogical aspects, thus turning from cognitive psychology to the logic and methodology of science. As a consequence of (...) his reflections on the problematic relationship between geometrical‐mathematical constructions and physical reality Popper was able to get over a too direct notion of such a relationship, cast doubts on inductive inference and started conceiving in a new (strictly non‐inductivist) manner the relationship between theoretical and observational propositions. (shrink)

Review of Russell’s Philosophy of Logical Atomism 1897–1905, by Jolen Galaugher (Palgrave Macmillan 2013).

We focus on a branch of region-based spatial logics dealing with affine geometry. The research on this topic is scarce: only a handful of papers investigate such systems, mostly in the case of the real plane. Our long-term goal is to analyse certain family of affine logics with inclusion and convexity as primitives interpreted over real spaces of increasing dimensionality. In this article we show that logics of different dimensionalities must have different theories, thus justifying further work on different dimensions. (...) We then focus on the three-dimensional case, exploring the expressiveness of this logic and consequently showing that it is possible to construct formulas describing a three-dimensional coordinate frame. The final result, making use of the high expressive power of this logic, is that every region satisfies an affine complete formula, meaning that all regions satisfying it are affine equivalent. (shrink)

This article is about Avicenna’s account of syllogisms comprising opposite premises. We examine the applications and the truth conditions of these syllogisms. Finally, we discuss the relation between these syllogisms and the principle of non-contradiction.

Philosophers of physics and physicists have long been intrigued by the analogies and disanalogies between gravitational theories and gauge theories. Indeed, repeated attempts to collapse these disanalogies have made us acutely aware that there are fairly general obstacles to doing so. Nonetheless, there is a special case space-time dimensions) in which gravity is often claimed to be identical to a gauge theory. I subject this claim to philosophical scrutiny in this article. In particular, I analyse how the standard disanalogies can (...) be overcome in dimensions, and consider whether really licenses the interpretation of gravity as a gauge theory. Our conceptual analysis reveals more subtle disanalogies between gravity and gauge, and connects these to interpretive issues in classical and quantum gravity. 1 Introduction1.1 Motivation1.2 Prospectus2 Disanalogies3 Three-dimensional gravity and gauge3.1 gravity3.2 Chern–Simons3.2.1 Cartan geometry3.2.2 Overcoming obst-gauge via Cartan connections3.3 Disanalogies collapsed4 Two More Disanalogies4.1 What about the symmetries?4.2 The phase spaces of the two theories5 Summary and Conclusion. (shrink)

The empirical study of visual space has centered on determining its geometry, whether it is a perspective projection, flat or curved, Euclidean or non-Euclidean, whereas the topology of space consists of those properties that remain invariant under stretching but not tearing. For that reason distance is a property not preserved in topological space whereas the property of spatial order is preserved. Specifically the topological properties of dimensionality, orientability, continuity, and connectivity define “real” space as studied by physics and are the (...) spatial properties that characterize the physical universe as being an integral whole. By contrast the geometrical analysis of VS has taken little cognizance of its topology. Instead such properties have been presupposed a priori rather than being established a posteriori by empirical means, perhaps because these properties are self-evident. Applying the method of coordinative definition expounded by Hans Reichenbach for determining geometrical and topological properties of physical space, it can be shown that VS fulfills the topological criteria of being a “real” space sui generis. Though theorized to be produced by the brain, the topology of VS is not topologically equivalent with the structure and activity of the brain because, as will be shown, the topology of VS cannot be formed from the topology of the brain without tearing and/or cutting and pasting. (shrink)

Suppose we were to ask some students of philosophy to imagine a typical book of classical German philosophy and describe its general style and character, how might they reply? I suspect that they would answer somewhat as follows. The book would be long and heavy, it would be written in a complicated style which employed only very abstract terms, and it would be extremely difficult to understand. At all events a description of this kind does indeed fit many famous works (...) of German philosophy. Let us take for example Hegel's Phenomenology of Spirit of 1807. The 1977 English translation published by Oxford has 591 pages, and as for style here is a typical passage: Self-consciousness found the Thing to be like itself, and itself to be like a Thing; i.e., it is aware that it is in itself the objectively real world. It is no longer the immediate certainty of being all reality, but a certainty for which the immediate in general has the form of something superseded, so that the objectivity of the immediate still has only the value of something superficial, its inner being and essence being self-consciousness itself. The object, to which it is positively related, is therefore a self-consciousness. It is in the form of thinghood, i.e. it is independent ; but it is certain that this independent object is for it not something alien, and thus it knows that it is in principle recognized by the object. It is Spirit which, in the duplication of its self-consciousness and in the independence of both, has the certainty of its unity with itself. This certainty has now to be raised to the level of truth; what holds good for it in principle , and in its inner certainty, has to enter into its consciousness and become explicit fork. (shrink)

This essay considers the nature of conceptual frameworks in science, and suggests a reconsideration of the role played by philosophy in radical conceptual change. On Kuhn's view of conceptual conflict, the scientist's appeal to philosophical principles is an obvious symptom of incommensurability; philosophical preferences are merely “subjective factors” that play a part in the “necessarily circular” arguments that scientists offer for their own conceptual commitments. Recent work by Friedman has persuasively challenged this view, revealing the roles that philosophical concerns have (...) played in preparing the way for conceptual change, creating an enlarged conceptual space in which alternatives to the prevailing framework become intelligible and can be rationally discussed. If we shift our focus from philosophical themes or preferences to the process of philosophical analysis, however, we can see philosophy in a different and much more significant historic role: not merely as an external source of general heuristic principles and new conceptual possibilities, but, at least in the most important revolutionary developments, as an objective tool of scientific inquiry. I suggest that this approach offers some insight into the philosophical significance of Newton's and Einstein's revolutionary work in physics, and of the interpretation of their work by (respectively) Kant and the logical positivists. It also offers insight into the connections between modern philosophy of science and some traditional philosophical concerns about the nature of a priori knowledge. (shrink)

It is argued that Russell before 1905 saw no value in Frege's sense/reference distinction. This is clearest in the Mont Blanc correspondence. It is argued that Russell and Frege failed to engage because Frege lacked a grasp on the internal/external relations distinction. For Russell sense is either an external relation, objectionably separating out thought and reference, or an internal relation, so what is thought is altered such that we do not know what we are talking about. The novelty of the (...) present paper lies in the arrangement of the parts and the claim that Russellian propositions are not made up of the things themselves but of transparent representatives thereof. (shrink)

Beaucoup de « géomètres » du XIXe siècle - Bernhard Riemann, Hermann von Helmholtz, Felix Klein, Riccardo De Paolis, Mario Pieri, Henri Poincaré, Federigo Enriques, et autres - ont joué un rôle important dans la discussion sur les fondements des mathématiques. Mais, contrairement aux idées d'Euclide, ils n'ont pas identifié «l'espace physique» avec« l'espace de nos sens». Partant de notre expérience dans l'espace, ils ont cherché à identifier les propriétés les plus importantes de l'espace et les ont posées à la (...) base de la géométrie. C'est sur la connaissance active de l'espace que les axiomes de la géométrie ont été élaborés ; ils ne pouvaient donc pas être a priori comme ils le sont dans la philosophie kantienne. En outre, pendant la dernière décade du siècle, certains mathématiciens italiens - De Paolis, Gino Fano, Pieri, et autres - ont fondé le concept de nombre sur la géométrie, en employant des résultats de la géométrie projective. Ainsi, on fondait l'arithmétique sur la géométrie et non l'inverse, comme David Hilbert a cherché à faire quelques années après, sans succès. (shrink)

In this paper we briefly expose a project which could be summed up as ‘doing justice to Mario Pieri’. The main result of the project will be the publication of a book on him, in which we will mainl...

In recent years, several scholars have been investigating Frege’s mathematical background, especially in geometry, in order to put his general views on mathematics and logic into proper perspective. In this article I want to continue this line of research and study Frege’s views on geometry in their own right by focussing on his views on a field which occupied center stage in nineteenth century geometry, namely, projective geometry.

Richard Tieszen [Tieszen, R. (2005). Philosophy and Phenomenological Research, LXX(1), 153–173.] has argued that the group-theoretical approach to modern geometry can be seen as a realization of Edmund Husserl’s view of eidetic intuition. In support of Tieszen’s claim, the present article discusses Husserl’s approach to geometry in 1886–1902. Husserl’s first detailed discussion of the concept of group and invariants under transformations takes place in his notes on Hilbert’s Memoir Ueber die Grundlagen der Geometrie that Hilbert wrote during the winter 1901–1902. (...) Husserl’s interest in the Memoir is a continuation of his long-standing concern about analytic geometry and in particular Riemann and Helmholtz’s approach to geometry. Husserl favored a non-metrical approach to geometry; thus the topological nature of Hilbert’s Memoir must have been intriguing to him. The task of phenomenology is to describe the givenness of this logos, hence Husserl needed to develop the notion of eidetic intuition. (shrink)

Matthias Neuber’s book represents an important contribution to the relatively young discipline of the History of Philosophy of Science. Starting roughly in the 1980s, increasing attention has been devoted not only to the relationship between philosophy and the history of science, but to an accurate historical reconstruction of earlier projects within philosophy of science. One of the most outstanding results of these investigations has probably been the radical reshaping of the rather caricatural image of logical empiricism—for better or worse the (...) core of the philosophical heritage of many philosophers of science—summarized in the so-called ‘standard view’. By analyzing the historical, sociological, and philosophical questions surrounding logical empiricism new light has been shed on the sense of a cultural, social, and political mission that characterized it before its emigration from Europe to North America in the 1930s and 1940s. What came to be known as .. (shrink)

Both early analytic philosophy and the branch of mathematics now known as topology were gestated and born in the early part of the 20th century. It is not well recognized that there was early interaction between the communities practicing and developing these fields. We trace the history of how topological ideas entered into analytic philosophy through two migrations, an earlier one conceiving of topology geometrically and a later one conceiving of topology algebraically. This allows us to reassess the influence and (...) significance of topological methods for philosophy, including the possible fruitfulness of a third conception of topology as a structure determining similarity. (shrink)

Charlotte Angas Scott (1858–1932) was an internationally renowned geometer, the first British woman to earn a doctorate in mathematics, and the chair of the Bryn Mawr mathematics department for forty years. There she helped shape the burgeoning mathematics community in the United States. Scott often motivated her research as providing a “geometric treatment” of results that had previously been derived algebraically. The adjective “geometric” likely entailed many things for Scott, from her careful illustration of diagrams to her choice of references (...) and citations. This article will focus on Scott’s striking and consistent use of geometric to describe a reality of dynamic points, lines, planes, and spaces that could be manipulated analogously to physical objects. By providing geometric interpretations of algebraic derivations, Scott committed to an early-nineteenth-century aesthetic vision of a “whole” analytical geometry that she adapted to modern research areas. (shrink)

A Christian approach to scholarship, directed by the central biblical motive of creation, fall and redemption and guided by the theoretical idea that God subjected all of creation to His Law-Word, delimiting and determining the cohering diversity we experience within reality, in principle safe-guards those in the grip of this ultimate commitment and theoretical orientation from absolutizing or deifying anything within creation. In this article my over-all approach is focused on the one-sided legacy of mathematics, starting with Pythagorean arithmeticism (“everything (...) is number”), continuing with the geometrization of mathematics after the discovery of irrational numbers and once again, during the nineteenth century returning to an arithmeticistic position. The third option, never explored during the history of mathematics, guides our analysis: instead of reducing space to number or number to space it is argued that both the uniqueness of these two aspects and their mutual coherence ought to direct mathematics. The presence of different schools of thought is highlighted and then the argument proceeds by distinguishing numerical and spatial facts, while accounting for the strict correlation of operations on the law side of the numerical aspect and their correlated numerical subjects (numbers). Discussing the examples of 2 + 2 = 4 and the definition of a straight line as the shortest distance between two points provide the background for a brief sketch of the third alternative proposed (inter alia against the background of an assessment of infinity and continuity and the vicious circles present in contemporary mathematical arithmeticistic claims). (shrink)