Geometrical conventionalism and the Theory of Relativity: can we know the true geometry of space?

Can We Know the True Geometry of Space? by

Paul Jacob Mueller

Bachelor of Arts, University of Guelph, 2006 Bachelor of Computing, University of Guelph, 2006

A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of

MASTER OF ARTS in the Department of Philosophy

 Paul Jacob Mueller, 2010 University of Victoria

All rights reserved. This thesis may not be reproduced in whole or in part, by photocopy or other means, without the permission of the author.

Supervisory Committee

Geometrical Conventionalism and the Theory of Relativity: Can We Know the True Geometry of Space?

by

Paul Jacob Mueller

Bachelor of Arts, University of Guelph, 2006 Bachelor of Computing, University of Guelph, 2006

Supervisory Committee

Audrey Yap, (Department of Philosophy) Supervisor

Jeffery Foss, (Department of Philosophy) Departmental Member

Carrie Klatt, (Department of Philosophy) Departmental Member

Abstract

Supervisory Committee

Audrey Yap, (Department of Philosophy)

Supervisor

Jeffery Foss, (Department of Philosophy)

Departmental Member

Carrie Klatt, (Department of Philosophy)

Departmental Member

The central question which will be addressed in this paper is: can we know the true geometry of space? My answer will be in the negative, but not first without heavy qualification. The thesis concerns the notion of truth in mathematical science, i.e. physical science for which mathematics (particularly geometry) is integral, and will ask whether we can know with certainty, or via some empirical test, which geometry is an accurate description of the actual universe. It will be a fairly historical approach, but hopefully not entirely so. We will begin with a 17th century debate on the nature of space between Newton and Leibniz and how Kant proposed to resolve the debate, and then move on to the views of the late 19th century mathematician Poincaré, but we will end with Einstein's Theory of Relativity - a theory which uses a very different geometry to which most of us are perhaps accustomed. In general, the goal will be to better understand the nature of geometry and its role in scientific theory; specifically, however, it will be an attempt to answer, in the negative, the central question before us.

Table of Contents

1.1 The 18th Century Debate: Newton and Leibniz ... 10

1.2 Newton’s Absolute Space ... 10

1.3 Leibniz’s Relational Space ... 14

1.4 The Kantian Critique... 18

1.5 The Mathematical Views of Newton and Descartes (The ‘Mathematicians’) .. 19

1.6 The Mathematical View of Leibniz (The ‘Metaphysician’) ... 22

1.7 The Kantian Alternative ... 25

1.8 Introduction to Non-Euclidean Geometries ... 28

1.9 Kant and Geometry ... 31

Chapter 2 ... 35

2.1 Reasoning by Recurrence: Poincaré’s Synthetic A Priori ... 36

2.2 The Creation of the Mathematical Continuum and Measurement ... 41

2.3 The Truth of Geometry, or is Euclidean geometry true? ... 46

2.4 Geometrical Space and Representative Space ... 52

2.5 Poincaré’s Thought Experiment ... 57

2.6 Friedman Summary and Critique ... 61

Chapter 3 ... 70

3.1 Friedman Summary and Response... 70

3.2 Einstein’s Perspective ... 72

3.3 Einstein’s Measuring Device ... 76

3.4 Relativity in Euclidean Terms ... 79

3.5 Can We Know the True Geometry of Space? ... 83

3.6 The First of Three Responses: Kant ... 88

3.7 The Second of Three Responses: Poincaré ... 89

3.8 The Last of Three Responses: Einstein ... 93

3.9 Conclusion ... 96

Introduction

The central question which will be addressed in this paper is: can we know the true geometry of space? My answer will be in the negative, but not first without heavy qualification. Indeed, it would seem, before we can even begin to consider the question, there is much that needs to be said about the very meaning of the terms. Underlying the entire thesis will be the question of truth in scientific theories; more specifically, it concerns the notion of truth in mathematical science, i.e. physical science for which mathematics (particularly geometry) is integral, and will ask whether we can know with certainty, or via some empirical test, which geometry is an accurate description of the actual universe. It will be a fairly historical approach, but hopefully not entirely so. We will begin with a 17th century debate on the nature of space between Newton and Leibniz, and then move on to the views of the late 19th century mathematician Poincaré, but we will end with Einstein’s Theory of Relativity – a theory which uses a very different geometry to which most of us are perhaps accustomed. In general, the goal will be to better understand the nature of geometry and its role in scientific theory; specifically, however, it will be an attempt to answer, in the negative, the central question before us.

There are two extremes that one can consider when thinking about the relationship between mathematics and nature. We can think that mathematics is a pure product of reason – that it is somehow inextricably bound to the way that we think, somehow rooted in logic itself. This intuition seems natural for anyone who has encountered Euclidean geometry. Its axioms are very simple, yet undeniable principles on which a massive edifice is built. Since these basic foundations are undeniable truths

which everyone must affirm, it is then thought that the mathematics derived using only these self-evident truths is likewise undeniable. The result is an entire system which is therefore thought to be universally and necessarily true. And since it is universally true, no matter what part of the universe we go to, mathematical truth will always be the same – true in spite of any experience we might have.

The problem with accepting this view of mathematics is that when we try to apply it to elements of our experience, there is (or perhaps should be) a lingering doubt whether these truths are all merely ‘in our mind’. That is, perhaps it is the case that these mathematical structures are merely ‘creative constructions’ that some very smart people have invented over the years. It would perhaps not be an issue if these constructions remained in textbooks, but this is not the case. Rather, these constructions are turned around and applied to nature. And indeed, more often than not, it would seem as though they are not only applied to nature, but are ascribed to nature, in the way that we might say that the universe is Euclidean (as with Newtonian physics) or that the universe is a four-dimensional manifold (as with the General Theory of Relativity). But the lingering doubt is, if these mathematical constructions are simply human creations, then what can assure us that nature in any way resembles these products of human thought? Are these constructions actually true of reality – necessarily reflective of what nature is really like? There is another approach, however, which might dispel worry. It is the thought that perhaps we are simply reading mathematics off of nature. That is, that perhaps all of our mathematical concepts had their origin in experience, as something we learned. Take the simple example: we have one rock; we have another rock; put them together and we have two rocks. It may be thought that numbers, lines, circles, and so on, are all concepts

which are the progeny of experience. This approach seems to ensure that whatever complex system arising from such thoughts will ‘automatically’ have a correlate in reality.

However, it might be asked, may it likewise be thought that one rock and another rock will produce three rocks? Or, if we add an even number of rocks to another even number of rocks, we will have a total which is an odd number? Perhaps it is the case that in a different part of the universe mathematics works very differently from how it does here; so does that mean mathematics requires empirical testing? We certainly have not seen/experienced/tested most of the universe, so why are we so convinced that our mathematics must be the way that it is? Simply put: what guarantees its truth as

necessary? If the answer is something akin to ‘it is simply unthinkable that the

mathematics could not be the way it is’, then it seems as though the earlier view does not seem so bad, that is, that mathematics has its origin in the rules of how we think – but then we are back at where we started.

So, on the one hand, if we think that mathematics is a result of ‘reason’ or ‘pure thought’, then there is the inevitable question of its possible application to nature – what guarantees that our constructions correspond to reality? But if, on the other hand, we are of the mind that mathematics is learned from nature, then it seems possible that some new experience could contradict a supposed ‘truth’ of mathematics.

These questions are relevant to an 18th century debate between Newton and Leibniz regarding geometrical truth and the nature of the universe. The debate will be presented in terms of two conditions that Immanuel Kant believed ought to be demanded of mathematical science. We will return to these ideas later in this chapter, but briefly:

the first condition is apriority – the idea that science should contain an element which is necessarily true, regardless of the content; the second condition is applicability – the idea that science should be universally applicable to all objects of experience. Kant thinks that Newton meets the condition of apriority by his inclusion of the absolutely necessary truth of Euclidean geometry in his physics. Leibniz realizes the condition of applicability because he thinks mathematical truths are nothing more than abstractions from relations between objects (and since experience is the source of such truth, turning around and applying it to objects of experience should never be a problem). However, although Kant thinks that Newton and Leibniz each meet one of the demands, i.e. Newton meets apriority and Leibniz meets applicability, neither thinker meets both.

So Kant set out to develop an account of mathematical science that would satisfy the demands of both apriority (i.e. necessity) and applicability to nature. To this end, he used a new category of thought: the synthetic a priori1

1 _{There are traditionally distinctions made between the ‘analytic’ and ‘synthetic’, and the ‘a priori’ and the ‘a}

posteriori’. Analytic is usually identified with the a priori, and synthetic with the a posteriori. Kant was the first to introduce a new category called the synthetic a priori. This will be explored in more detail below.

. Since he thinks space is the very form of all of our intuitions, i.e. there is no outer experience we could ever have which is not spatial – it is the necessary form that all of our outer experiences must take. It therefore satisfies both of Kant’s demands: on the one hand, the space of all outer experience is Euclidean space (Euclidean geometry was thought to be the paradigm of a priori thought); and, on the other hand, since it is the form that all our outer experiences must take (including those made in science, i.e. empirical claims), it is also decidedly applicable to nature. Kant believed he had accomplished what he saw lacking in his fellow natural philosophers, namely, he had given science a firm foundation (insofar as it

is a priori), but one which is capable of growth without fear of losing that foundation (insofar as it is synthetic).

As we shall see, the Kantian victory was relatively short-lived. The trouble for Kantians began in the early 19th century when a type of geometry was discovered which was not Euclidean. Upon the introduction of this non-Euclidean geometry, many who might have otherwise been sympathetic to Kant began to turn. Indeed, it was not just on Kant that many turned, but since Euclidean geometry was seen to be the utmost pillar of what science should be (not to mention the fact that Newtonian physics was based on Euclidean geometry, and indeed often seen as an extension of it), many began to question the truth of science itself. Despite the fact that it was based on a mere handful of postulates, propositions and definitions, Euclidean geometry had nevertheless lasted unchallenged for millennia, all the while constantly growing and expanding. When it was discovered that alternative geometries were possible – that Euclidean geometry was not unique, but only one of many – the doubt that resulted was not reserved simply for geometry, but for the whole of science. After all, if what was thought to be the surest and most certain science could be put into question, then what hope might the rest of science have?

This then is the milieu in which the late-19th century French mathematician Henri Poincaré found himself. His situation is not at all dissimilar to the problem faced by Kant. They both without question saw the tremendous value in the scientific project and both seemed to take it upon themselves to show how it was – regardless of any naïve interpretation that comes its way2

2 _{He provides a particularly poignant “naïve” interpretation in the Preface to Science and Hypothesis (xxi). He}

states that to the superficial observer, certainty in science is found in the following chain of reasoning:

benefit of hindsight, had an advantage over Kant. He was well aware of the developments in geometry which had taken place before him and, given his ‘philosophical disposition’, proved to be the ideal person to comment on their significance.

The second chapter will largely trace the origin of geometry, according to Poincaré, in order to get a better understanding of its nature and its role in mathematical science. Geometry for Poincaré has a different nature than it does for Kant. For Kant, (Euclidean) geometry is simply given to us as the very form of our intuition, as a kind of ‘necessary precondition’ for experience as such. For Poincaré, geometry is not simply ‘given’ to us in such an immediate form, but is rather one step in a ‘hierarchy’ of thought, with the most necessary truths serving as the foundations leading up to the least necessary. Geometry occupies a special place in the middle, but this does not mean that it is without a secure foundation (i.e. that it lacks ‘rigour’). Indeed, Poincaré thinks that its very strength comes in tracing its origin (i.e. back to the ‘ground’ of the hierarchy), and it is only in ascertaining its original source that we can gain any epistemic security in its use.

What will be uncovered are Poincaré’s reasons for saying that our use of one geometry over another comes down to nothing but a conventional choice, a position (which he is credited as originating) called geometrical conventionalism. The heart of the argument lies in the fact that different geometries are intertranslatable between one another – that each is really just a different ‘expression’ of the same information (i.e. a

mathematical truths are to be found in a few self-evident propositions which follow by flawless reasoning; they are imposed on us, but equally to nature; the Creator is therefore limited to a relatively small number of choices when determining the universe; so we, the scientists, need make very few experiments to determine what choices have been made; and after each experiment, a series of consequences will follow which will finally see nature exposed. This, to the superficial observer, is the apparent origin of certainty.

different way to ‘lay out’ or model the data). We can compare the situation to language, in the way that two different words in two different languages can have the same basic meaning. Seen in this way, the use of one particular geometry over another comes down to a conventional choice (i.e. ‘choose something and stick with it’). There is also, therefore, no ‘true’ geometry: no empirical test, or piece of data, will ever force us to use one geometry over another (in the same way that a French word in a French-German dictionary will not be ‘more correct’ than its German translation3

It is a compelling argument, one which had a considerable impact on 20th century analytic philosophy

).

4

Now it would seem as though if we were forced to choose between Poincaré and Einstein, given the high degree of confirmation of the latter’s theory, the prudent choice would be to simply discard Poincaré. However, arguments in the third chapter will show that it is not quite such a binary choice. And, indeed, it will be shown that Einstein himself was greatly influenced by Poincaré – that the two actually share a very similar view concerning the role of geometry and physical laws in theoretical physics. They . However, the problem for those who might otherwise be convinced is that the General Theory of Relativity was published a few short years after Poincaré’s death. Scientific advances would not normally be a problem for Poincaré’s position (that is, according to his position), except that Relativity Theory uses a geometry which is explicitly described by Poincaré as being incompatible with his views. And so, for reasons which will be further explicated, the second chapter will present criticisms from Michael Friedman arguing that the two positions are completely irreconcilable.

3 _{Although, as with our language analogy, the expression (of some equation, for example) might be in a far}

more complex form after translating into a different geometry.

4 _{This relationship between Poincaré and the logical positivists (most notably Shlick, Reichenbach and,}

perhaps to a lesser extent, Carnap) has been often noted in the philosophical literature, but it will not be explored in this paper.

essentially agree on (what we might call) the ‘epistemological status’ of geometry and physical laws – that is, that both have the nature of being conventional – that geometry is meaningless without physical laws, and vice-versa. They essentially disagree, however, on which elements must be included in any theory; and the easiest way to insure that something is a part of a theory is to simply begin with it.

The choice of where to begin is of tremendous importance and, it would seem, tends to show off which elements of a theory a scientific theorist considers to be the most important – i.e. something which they think ought to be included in the theory. Poincaré, although he thinks that our choice of geometry is completely conventional, does not think that it is completely arbitrary either. He thinks that factors that are not inherent to the geometrical system play heavily in our decision, and chief amongst these factors is simplicity (i.e. we will always choose the simplest way of representing the data). And it is because he thinks we will always want to have the simplest system of representation possible, we will therefore always choose to include Euclidean geometry5

What will be made clearer in the third chapter is that Einstein disagreed with Poincaré in practice, but not in principle – particularly regarding the relationship between geometry and physical laws. Both thought that the only real restriction on a theory is that it be free from contradiction; what this means is that, if any contradiction in a theory

. Einstein, on the other hand, chose rather to begin with a few physical principles which he considered of fundamental importance – indeed, more important than always to retain Euclidean geometry – and which ultimately led to the adoption of a non-Euclidean geometry.

5 _{We will see how this picture is slightly more complicated, since Poincaré thinks that our choice of geometry}

is restricted to geometries of constant curvature (i.e. Euclidean, hyperbolic and elliptic), and this restriction does not include the Riemannian theory of manifolds which is used by Einstein. It is therefore not simply the case that Einstein is merely choosing a different geometry as a convention; rather, Einstein’s system has the appearance of being fundamentally opposed to Poincaré’s system.

occurs (due to some new experiment, etc.), the theorist is completely free to change/revise/update the physical laws and/or the geometry in order to restore ‘internal coherence’.

The third chapter will also contain examples of Euclidean formulations of General Relativity. This will be in order to show that it is not only logically possible, but practically possible as well. However, this will not be to ‘prove’ that Euclidean geometry is in fact true of the world, thereby ‘disproving’ Riemannian geometry; on the contrary, it will serve merely to support the truth of the epistemological claim (i.e. the relationship between, and the ‘truth status’ of, geometry and physical laws) that we see coming from Poincaré, and inherited by Einstein. It will be on this fundamental agreement that I will draw conclusions about the relationship between geometry and nature. Or, perhaps more accurately, it will let me draw conclusions about drawing conclusions on the truth of geometry and nature. In other words, we will return to the central thesis question from an epistemological perspective. With support from Kant, Poincaré and Einstein, although perhaps for different reasons, we will answer negatively the question at hand: Can we know the true geometry of space?

Chapter 1

1.1 The 18th Century Debate: Newton and Leibniz

A debate raged in the early 18th century between Newton and Leibniz concerning the true nature of space. Newton, some decades earlier having written the Principia, thought that space was ‘absolute’ – that there was an absolute frame of reference, independent from all matter in the universe, but which serves as a kind of ‘immutable reference point’. It is something like the intuitive or ‘normal’ view that many of us have when we think about matter in the universe, i.e. that the ‘stuff’ in the universe has a definite place relative to the universe as a whole (almost that space is like a giant container6). Leibniz, on the other hand, believed space to be ‘relational’, i.e. there is no absolute point of reference (no ‘giant container’), but only ‘stuff’. And so ‘stuff’ can only possibly reference other ‘stuff’, and it never references anything ‘absolute’. The idea of space, then, is something that we abstract away from individual bodies and their relations to other bodies. It is completely ideal – space does not exist in nature itself, only in our abstractions.

1.2 Newton’s Absolute Space

While Leibniz seems to have stuck with metaphysical arguments7

6 _{The giant container analogy is Sklar’s and, depending on one’s interpretation of Newton, may be}

contentious. But the general idea here is that, for Newton, relationalism (i.e. the idea that there are only relative frames of reference) cannot account for certain physical effects (see below).

, Newton attempted to make a case for absolute space based on observational evidence. His

7 _{Though, it should be mentioned, he employs rather foundational principles (if one will grant them as}

argument is three-staged: first, from certain observational facts, we are forced to accept absolute acceleration (and derivatively absolute motion); second, from absolute acceleration we can show that relationalism is inadequate; and third, with the issue finally settled, we can build a proper theory (Sklar, 159). The clear problem, so to speak, for any proponent of absolute space is that space is unobservable – not only is it completely invisible, but it is immutable as well. What this means is that there is no experiment which could ever alter it, and this makes it very difficult indeed to test hypotheses about it. But while space is not directly observable, Newton still believed there to be indirect evidence in support of his view. Before considering his evidence, however, we should clarify his view.

Absolute space is probably best thought of as a giant container and, as such, it seems like the “natural” position8: it is a three-dimensional container that persists, unchanged, through time; it is eternal, absolute and immutable9; and space is distinct from bodies, existing independently of all matter. Further, space is “uniform and unlimited stretching out in length, breadth and depth”; it is an “entity that is infinitely extended, continuous, motionless … but that is not itself corporeal and that can be conceived as empty of bodies; moreover, space is a unified whole of strictly contiguous parts” (Shabel 39). Bodies, then, occupy space in all three dimensions and, as such, each body possesses an ‘absolute place’ in this container, where material objects are ‘point-coincident’ with absolute space10

8 _{Though, it is “natural” for us at least partly because of its long-standing importance in our physics.}

(Sklar 162).

9 _{This is so for interesting reasons having to do with God’s own creation and existence, which seems basically}

the view that God created his own ubiquity (Newton 11) – a point on which Leibniz would challenge him.

10 _{It should be mentioned again here that this is one interpretation of Newton, particularly advocated here by}

Sklar, and may or not be contentious depending on one’s interpretation of Newton. The really important point to note is that absolute space is immutable – nothing can affect it, especially not in the way that we

One of the most important features of absolute space is the fact that is wholly Euclidean11: “Space can be distinguished in every way into the parts the common limits of which we are accustomed to call surfaces; and these surfaces can be distinguished in every way into parts, the common limits of which we name lines; and in turn these lines can be distinguished in every way into the parts which we call points” (Newton 8). Matter is not a necessary precondition of space (since space can be empty), but space is a necessary precondition for matter since any postulation of a body requires first a postulation of space. The importance of this conception of space made it quite convenient for the mathematical physicist, because not only did physics ground itself in certainty (i.e. the apparent certainty of Euclidean geometry) but it also meant that any geometrical advancement could have a direct impact on the empirical science12

Newton argues that for any material object, we must first postulate space. But this necessary but invisible space seems to present a problem for the true empirical scientist (i.e. the reductionist) who will argue that the only meaning comes from observational consequences, or at least consequences which are observable in principle (Sklar 157). Newton did indeed believe that certain observational consequences of/about space were possible (though perhaps only in principle

. And indeed the invention of analytic geometry by Descartes and its development by Newton provided the foundations of the new physics.

13

find, e.g., matter affecting space-time in the general theory of relativity – and yet, it is real (‘real’ in the sense of there actually existing an absolute frame of reference).

) which would force us to

11 _{This should not be too shocking given it was the only geometry available at the time.} 12

This point foreshadows a number of issues which will arise later in this chapter: first, this notion of ‘apriority’ and ‘applicability’ will come up again with Kant; and later, the status of ‘geometrical certainty’ will be questioned.

13 _{As will be discussed, his two thought experiments require us to ‘abstract away’ all other objects in the}

conclude, or necessarily postulate, that absolute space exists. What he had in mind is the ‘intimate connection’ between absolute acceleration and observable forces (Sklar 184); that is, the fact that observable consequences are produced only when an object is accelerating with respect to absolute space (i.e. not necessarily when it is accelerating relative to another object). His argument from observational consequences somewhat ironically comes to us in the form of two (similar in kind) thought experiments:

The first is the Spinning Bucket experiment. Imagine a half-filled bucket of water attached to a rope, and consider the possible ‘states’ that this bucket, the water and the rope can be in: (a) neither the water nor the bucket is in motion – the water is at rest with respect to the bucket, and the water is flat; (b) rotate the bucket so as to twist up the rope, and let it go so the bucket begins to spin – there is a relative difference of motion between the water and the bucket, and the water remains flat; (c) as friction increases, the water will begin to spin along with the bucket, eventually reaching an equilibrium (the water with the bucket) – it is as this point that the water is once again at rest with respect to the bucket, but this time the water is concave (pushed up the sides of the bucket). The question he poses is: we know the water is accelerating (since it is accompanied by physical forces), but relative to what? In both (a) and (c) we have the same relative motion (between the bucket and the water), but there are quite different physical effects. Leibniz thinks that it is only relations between objects that matters, but this thought experiment shows for Newton that there are certain physical effects which cannot be accounted for by Leibniz’s viewpoint – in other words, there has to be more than simply the relations between two objects.

If one is still unsatisfied with this thought experiment14, Newton offers another which attempts to remove all doubt. It is similar, but simpler: consider an empty universe containing only two weights, or globes, attached together by a string, rod or rope. There are two imaginable situations: (a) they are not rotating and there is therefore no tension in the rope; or (b) they are rotating (about the centre) and there is tension in the rope (Sklar 183). In both cases, however, each weight will be at rest with respect to one another, and yet there is a detectable difference in the amount of tension in the rope connecting them – so how does this force arise? Since there are no other objects in the universe and since motion must be relative to something (since change of motion, i.e. acceleration, is the explanation of the force), it must be relative to space itself. This discrepancy of noticeable effects marks the difference between accelerations which are ‘real’ (when the objects are really accelerating with reference to absolute space) and those which are merely ‘apparent’ (when there is only the appearance of acceleration, without any observational consequences).

1.3 Leibniz’s Relational Space

Leibniz was dissatisfied with this explanation. One major problem is that there is simply no way to test it. No experiment (unless we include controversial thought experiments) is devisable to settle the question. And to make matters worse for the absolutist, there is no way (even in principle) to determine what the absolute place or the absolute velocity of an object is, since absolute space is immaterial (cannot be seen) and

14 _{There seem to be many good reasons to remain unsatisfied, for example: the experiment relies on}

Newtonian mechanics (which already takes absolute space as one of its assumptions) in order for the experiment to work; testing this scenario in an ‘empty universe’ except for a few objects really strains the limit of a ‘testable hypothesis’; and so forth.

all parts of space are isotropic (exactly alike). And because absolute space is not directly observable, it is interesting to note that both parties, absolutists and relationists alike, from a pragmatic perspective actually must use relative reference frames. Since absolute velocity and acceleration cannot be determined, a suitable reference frame must first be chosen in order to perform calculations. Yet absolutists still seem to consider it nothing more than a practical matter of measurement.

In addition to the lack of direct evidence available, Leibniz considered the need for a separately existing ‘space’ simply unnecessary; rather, space was something “merely relative” (Sklar 168). The only things that exist are material objects and space is something completely ideal which does not truly exist in reality. Material objects and their ‘displacements’ from each another are wholly contained in their relations to other objects – not in some absolute sense, but in a merely relative sense. There is no ‘absolute position’ (and therefore no absolute velocity or acceleration) but only positions relative to other objects15. Rather, space can be seen as “something merely relative, as time is, taking space to be an order of coexistences, as time is an order of successions. For space indicates … an order of things existing at the same time, considered just as existing together, without bringing in any details about what they are like”16

15 _{He calls these kinds of relations ‘space-point relations of displacement’ and says that these ‘space-points’}

are the unextended constituents of objects (Sklar 167).

(Leibniz-Clarke 9); or, another way of putting it, space is an “order or set of relations among bodies, so that in the absence of bodies space is nothing at all except the possibility of placing them” (Leibniz-Clarke 10, my italics). For Leibniz, what is essential is not the spatial distribution as such but the logical order in which these objects are placed. One of his

16 _{Sklar thinks there are two ways of interpreting “in terms of possibility”: 1) relations to empty space if there}

reasons is that there is no absolute reference frame (like Newton’s absolute space) but rather that judgments of distance, motion and acceleration are to be made only after choosing a relative frame of reference17

Assertions concerning the separate existence of an absolute space, thinks Leibniz, are completely unnecessary. Bodies and their respective displacements exist as a ‘brute fact’. There is no need to assert the existence of something which is real and eternal yet immaterial and completely unobservable when enough ‘information’ is contained in the ‘objects’ themselves. Sklar qua Leibniz gives the example of a relation between two brothers, ‘brotherhood’, and that asserting absolute space is very similar to asserting the genuine existence of three things: two brothers and brotherhood, as really existing objects – it is tantamount to misinterpreting the scientific language (Sklar 167). Space is simply an ideal system of relations.

. And without matter, there would be no “situation of bodies” and therefore no (relational) space.

The absolute/relational debate seems to revolve around two key points. First, space in Euclidean geometry (whether physically or ideally) is by nature featureless and so it is impossible to distinguish (whether experimentally or in principle) one point in space from another. Second, and similarly, it is impossible to determine the velocity of an object in absolute terms (Hugget 161). In his correspondence with Clarke (Newton’s mouthpiece), Leibniz poses two thought experiments of his own which use this idea18

17 _{The special theory of relativity at least partially vindicates Leibniz on this point.}

: first, take the entire universe and move it to another part of absolute space; second, increase the velocity of all parts of the universe simultaneously, instantaneously, and by an equal amount. In both cases, there would be no noticeable difference, nothing would

experience a change of force, nor would anything change. The ‘absolute place’ of a thing does not matter, only their place relative to other things.

Consider further Leibniz’s principle of the identity of indiscernibles (PII). If we are to grant him this basic logical principle19

The problem (or, at the very least, the shortcoming) of the relationalist view is the inability to fully account for the empirical observation of forces, regardless of the reference frame. At the heart of Newtonian mechanics is the principle of inertia, according to which accelerations produce forces. The effects of these forces are generally detectable, so the question of whether an object is accelerating or not is determinable empirically. Absolute space serves as an absolute reference frame for acceleration. The proof of this, for the absolutist, lies in the fact that, even in certain reference frames wherein the object might appear to be at rest (take, e.g., a reference frame which is ‘in tandem’ with the object in question), it is in fact accelerating (as detectable with, e.g., an accelerometer

, i.e. that there “is no such thing as a pair of individuals that are indiscernible from each other” since “this [one] thing has two names” (Leibniz-Clarke 16), then the thought experiments of a ‘shifted universe’ (above) are all the more poignant. In both scenarios, the resultant universe is completely indiscernible from the original one. The situation lacks the very possibility of detection between the original and resultant universes – in other words, it is not even observable in principle. It is therefore, according to the PII, the same thing under two names.

20

19 _{Hugget also points out that the PII relies heavily on the principle of sufficient reason (PSR), so the PII is}

only as strong as the PSR is (Hugget, 164).

). In other words, there is a kind of ‘absolute measure’ of acceleration which is quantified by the degree of inertial force applied. So,

while one might find the relationalist’s arguments convincing, they still might have a difficult time reasoning away the (thought) experiments concerning observational consequences proposed by Newton.

1.4 The Kantian Critique

During his pre-critical period, the younger Immanuel Kant held relationalist views similar to Leibniz (Torretti 29). He thought that forces (presumably the same as the forces in Newtonian mechanics) exist in the materials themselves and that these forces were somehow responsible for the creation of space and time21

There was a problem latent in regarding the relationship between mathematics and reality, and it seemed that there were two options available

. By 1768, his opinion changed dramatically, because he began to think that the relationalist view is untenable because the very concept of ‘spatial things’ logically presupposes space itself. (Or, in spatiotemporal terms, that events already presuppose both space and time – we cannot conceive of an event positioned outside of space and time.) But he also considered the absolutist view (i.e. the Newtonian view about the reality of absolute space) to be untenable since the need to ascribe to space an absolute reality is an inferential leap we are simply not licensed to make (Torretti 29). He thus took it upon himself to develop a third option.

22

21 _{Though it is not entirely clear how exactly he thought that the structure of space was determined by matter}

interacting, Torretti reports that Kant did believe the three-dimensions of space were the result of “forces which are inversely proportional to the square of the distance” (and interestingly, that a different law of interaction of particles would result in a different structure of space – that is, greater or less than three-dimensions) (Torretti 29).

. Either, one could

22 _{To provide some historical context, Shabel argues that the character of the debate in the early modern}

period is perhaps best understood within the context of the mathematical development of the early 17th and 18th century. Mathematics, even in the 18th century, dealt with magnitude, but the very notion of magnitude was still entangled with that which it was meant to represent. In other words, ‘numbers’ were not yet

adopt the position that mathematics is ‘read off’ reality – that ‘numbers’ really do have their source in ‘numbered things’, originally found in empirical entities – but this would seem to strip them of the necessity, universality and apriority that had long been ascribed to them. Or, one could adopt the metaphysical position that mathematics is simply a product of ‘pure reason’ – true because that is simply how we think, as a brute fact – but this leads us to wonder, if it did not have its origin in empirical entities, how we are able to turn around and apply mathematics to those entities? For Kant23, each position failed to meet one of the two criteria necessary for any ‘good’ science, applicability and apriority: Newton (the ‘mathematician’) failed to meet the applicability demand and Leibniz (the ‘metaphysician’) failed to meet the apriority demand.

1.5 The Mathematical Views of Newton and Descartes (The ‘Mathematicians’)

Both Newton and Descartes, despite their specific differences, consider space to be something real. And since space is the space of Euclidean geometry, mathematical ‘objects’ have correlates in reality24

disambiguated from ‘numbered things’, i.e. numbers were not independent entities in their own right (as they might be conceived today) but were still defined in terms of the ‘numbered’ entities (Shabel 30).

. In his Meditations, Descartes’ argument for such a view goes roughly as follows: God exists and is not a deceiver; any idea we perceive as clear and distinct is true (since God would not deceive us); we have a clear and distinct idea of substance; the essence of substance is extension; since extension is the essence of material objects, we therefore have qualitative and certain knowledge about material objects (Descartes, Meditations, 1641).

23 _{That is, according to Shabel’s account of Kant.} 24

This led to certain problems. For instances, Cartesians had a very difficult time dealing with negative numbers, denoting them as “absurd”, “privations of true”, etc. (Shabel 34).

Although Newton agrees with Descartes concerning the reality of a subsisting space, he does not share Descartes’ metaphysics of space (i.e. that space does not exist apart from bodies). He considers space to be “infinitely extended, continuous, motionless, eternal, and immutable” but he also considers space to exist independently from the bodies which are contained in it. Newton’s conceptual argument for the independence of space provides a revealing look at his own epistemology. He argues that since we can conceive of extension as existing independently of body, it therefore exists independently: “we can clearly conceive extension alone without existing in some subject, as when any extra-mundane spaces or places are imagined empty of bodies; and we believe (it) to exist anywhere we imagine (that) there is no body” (Newton 7). So, space is ‘pure extension’ since it is conceptually possible to conceive it as such, and “Newton [thus] follows Descartes in positing a faculty of understanding as the real source of mathematical cognition, a tool with which we can comprehend the eternal and immutable nature of extension, which he conceives as infinite space” (Shabel 39).

At first glance, it might seem as though ‘the mathematicians’ have satisfied both of Kant’s demands: apriority, through the certainty of mathematical truths, as given to us via the ‘faculty of understanding’ (for Newton and Descartes25

25 _{Although for Descartes this justification of certainty comes about in a similar (and similarly mysterious)}

way, via the ‘natural light’. The ‘natural light’ will be returned to below.

); and applicability, since “geometrical cognition of space” can be identified with “knowledge of bodily extension”. Kant certainly seems to appreciate the mathematician’s view, but he also sees a problem with it. On the one hand, the mathematician gives a ‘perfect’ a priori account of the empirical world: all of the geometrical axioms hold for physical space since space is essentially geometric (regardless of whether existing co-extensively with [Descartes], or

independently from [Newton], matter). But, on the other hand, space is (and must be) something posited – that is, the mathematicians must assume that there exists an eternal and self-subsisting entity called ‘space’26

It is a problem, firstly, because this ‘entity’ is essentially the type of thing which is independent of all bodies (i.e. matter) and is thus always beyond the possibility of experience. And secondly, because certainty is only gained after we have accepted the assumption that space is a really existing entity. That is, our a priori knowledge of the

natural world (i.e. the world of spatial objects) comes only after we have accepted the

features of the supernatural world (i.e. the world over and above the world ‘of appearances’ – the very container in which the geometrically spatial objects are located) (Shabel 47). So, on the one hand, it provides us with a mathematical account of the

entire world of geometrical objects (i.e. the whole of the empirical world, or the world of

‘appearances’); but, on the other hand, our a priori knowledge must first be extended “beyond the domain of appearances without explanation or justification” in order for the former knowledge to be justified (Shabel 47). This, for Kant, is an unacceptable position, since it violates a condition for genuine knowledge, namely, that it is outside of the realm of ‘possible experience’

. This necessary presumption of space carries at least two problems with it.

27

.

26 _{Time is the second such necessarily posited ‘entity’, though we shall largely ignore it (at least in this}

section).

27 _{It would be tempting to ascribe to Kant a similar argument as one that Leibniz gave with respect to absolute}

space – that is, that it contradicts the an early formation of the principle of verifiability inasmuch as we would never be in a position to ever experience absolute space (i.e. it is beyond any empirical test). However, Kant gives quite different reasons for thinking so, as will be outlined below.

1.6 The Mathematical View of Leibniz (The ‘Metaphysician’)

But what about the account of the metaphysician? Leibniz’s account of the fundamental character of reality states that there is nothing in existence except simple, sizeless entities called monads – a realm of being which is essentially logical (non-spatial and atemporal)28. The ‘world of appearances’ is not the real world, and indeed even space and time are not real properties in the real world – they are merely abstractions from relations. Mathematical entities likewise fall into this category, as they represent pure abstractions, and thus the ‘monadic realm’ cannot be explained using mathematics29

Let us first take a look at the source of mathematical knowledge. He explains that we have an internal “common sense” when it comes to mathematics and number. This “common sense” is where we receive many things that come from the senses – color, sounds and tactile qualities (indeed, it is from these sources that we perceive shapes) (Shabel 41). It is also from the “common sense” that we get clear and distinct ideas which are the objects of both pure and applied mathematics. This is the function of two ‘faculties’, the imagination and the understanding. The imagination “operates on” ideas . But a real and genuine worry arises for anyone wanting to make truth claims about mathematics: what exactly are these mathematical entities? If we are merely talking about abstractions (i.e. abstractions from relations between things), then what guarantees that these relations are absolutely certain (i.e. necessary and universal)?

28 _{I am quite certain that a full understanding of Leibniz’s view of reality will not be achieved here. Perhaps}

let it suffice to say (not as argument, but as statement) that, for Leibniz, mathematical entities are merely an abstraction from the relations between things. They are, therefore, neither real in themselves nor do they reflect the ‘true reality’.

29 _{This puts him into sharp distinction with the Cartesians who, as mentioned above, had a difficult time}

explaining things like ‘negative numbers’; Leibniz felt free to create anything he wanted (e.g. infinities) without having to worry about its ontology because he didn’t think it real in the first place.

that emerge from what is common to more than one sense, while the understanding is further necessary for building the sciences from these notions (Shabel 41).

Even though mathematical objects arise from experience (i.e. their origin is in the senses), the understanding (i.e. reason) is responsible for assuring us of their truth. For Leibniz, then, “mathematical truths describe features of our sensible experience despite finding their justification in the understanding alone” (Shabel 41). Thus the understanding gives us a good sense of what ‘must be’ and what ‘could not be otherwise’ – that is, the understanding gives us truths such as the principle of non-contradiction and the axioms of geometry (or more generally, mathematical axioms and the laws of logic) – and these truths are “are a priori because they are all founded on and deduced from universal and necessary truths known to us by the ‘natural light’” (Shabel 43). So Leibniz seems to satisfy the criterion for apriority (i.e. since the understanding or the “natural light” is the ground for certainty) and seems to explain the applicability of this a priori science (i.e. since the source of mathematics lies in experience, there should be no problem applying the same mathematics to objects of experience). In other words, this seemingly a priori science does well at describing empirical phenomena. It does not seem to bother Leibniz that mathematics is not describing the ‘really real’ stuff (i.e. monads) nor that mathematics are ‘pure abstractions’ (indeed, even in some cases, merely “useful fictions”, such as with negative magnitudes and infinitesimally small magnitudes [Shabel 42]); rather, he seems satisfied in thinking that we nevertheless have a useful tool for describing the relations between appearances.

Kant appreciates this view as well, but he still thinks it is inadequate. Space30 is found in the ‘relations of appearances’ by way of abstraction, which means that space is not something real and self-subsisting on its own. Further, Kant considers the relationalist to hold that space is still something completely ‘real’ insofar as the relations between objects are real – the relations are waiting to be discovered and therefore space is waiting to be discovered (as a ‘form’ of the relation). The limits of what is experientially possible are already ‘prefigured’ in experience, since experience is the original source for mathematical knowledge (i.e. ‘naturally’ delimited by experience itself). Mathematical knowledge is thus applicable to all objects of experience31

The metaphysician must still come up with an answer to the apriority of mathematics – what gives mathematical truths their ‘apodictic’ certainty? Under this view, mathematical knowledge is gained through experience but its truth is somehow assured to us by the ‘natural light’ – how is this possible? The question can be rephrased like this: how is it that we could ever gain a priori knowledge, a posteriori? Since the

origin of our knowledge is experience, our knowledge of mathematical truths must

therefore be a posteriori (Shabel 46). The metaphysician might reply that some of our mathematical truths are being supplied a priori, that is, through the understanding (laws, axioms and derivable theorems) while others are given a posteriori (e.g. “knowledge of the mathematical features of real objects to which those theorems might be thought to . But while he thinks that this is a great strength of the view, it is only as far as applicability is concerned.

30 _{… (and time) …}

31 _{In other words, the applicability demand is met nearly immediately given that experience of the ‘world of}

appearances’ is the very source for mathematics, so it seems obvious that we can turn around and use mathematics to describe this very same world.

apply” [Shabel 46]). But there exists a gap in this argument. According to Kant, their view has an essential discord between what is necessary (i.e. found in the understanding) and what is empirical (i.e. found in experience). What is found in the understanding belongs to a different domain than what could be applied to experience with any degree of certainty: “[a] priori mathematical truths can only be about ‘useful fictions’ and not ‘real things’” (Shabel 46). Since this a priori knowledge cannot be applied to the real objects of possible experience (i.e. no principle of connection), Kant thinks it fails the apriority demand.

1.7 The Kantian Alternative

Both the relationalist and the absolutist theories ultimately fail to meet the standards that Kant thinks ought to be in place for genuine science. Having dismissed both theories as inadequate, Kant was forced to forge a new answer. Both theories fail, in Kant’s eyes, because both fail to appreciate what space actually is. If we consider a simple thought experiment we can begin to see his position: if we try to imagine anything at all, the first thing that Kant gets us to realize is that it must be spatially located. We can never have an (outer) experience which is outside of space32

32 _{Or, properly speaking, which is outside of space and time. Space is necessary for ‘outer’ intuitions and time}

necessary for ‘inner’ intuitions.

– space is a necessary condition for the possibility of experience. We can never represent to ourselves ‘no space’ – only perhaps space devoid of matter. Space exists for us as a necessary condition to represent at all: “space is nothing other than merely the form of all appearances of outer sense, i.e., the subjective condition of sensibility, under which alone

outer intuition is possible for us” (Kant 177). Space therefore cannot be ‘learned’, for in order for us to have an experience in the first place, space is, and must be, presupposed.

This is only one piece of the puzzle. Kant still has “no doubt whatever that all our cognition begins with experience” (Kant 136); but, not all cognitions have their original source in experience. There are a two pairs of distinctions that Kant uses. The first is between the a priori and a posteriori. These are fairly standard divisions in early modern period philosophy and I believe Kant uses them in a manner which is fairly consistent with that period33

The second distinction of terms is between analytic and synthetic. An analytic judgment is one to which nothing new is added (i.e. the predicate is not new, but is already contained in the subject) – for example, when we judge that all bodies are extended, or that all unmarried men are bachelors. Synthetic judgments are ‘ampliative’, in that the judged predicate does not already exist in the subject (i.e. something new is added which is not found in the original concept). So take, for example, the idea that all bodies are heavy – it is not contained in the concept of ‘body’ that it is heavy, but rather it is something new that is being added to the concept of ‘body’.

. An a priori idea is understood as logically ‘prior to’ experience, in the sense that it must be true always and everywhere and does not depend on experience for its justification. An a posteriori idea is understood as coming after experience, i.e. is “post-” experience, in the sense that this type of knowledge is contingent and requires experience for its justification. Examples of the former are usually thought to be things like ‘reason’ and mathematics and so forth, whereas the latter are usually associated with the senses, like colours and so on.

33 _{He does employ different kinds of a priori and a posteriori notions, such as a priori concepts and a posteriori}

representations, etc. But only the whole, I believe the basic meaning of a priori and a posteriori is the same as his near-contemporaries.

The a priori and the analytic are usually associated and so are the a posteriori and the synthetic, but Kant makes an argument that there are judgments which are both synthetic (something new is added) and a priori (necessarily true). For Kant, all judgments of experience are always synthetic, in that, experience is always ampliative – something is found which is not contained ‘in the concept’. In other words, for all judgments based on experience, something is predicated of the subject which is not contained in the subject concept, but is instead provided by experience. But he also notices that experiential judgments also contain an a priori element, in that outer experience is only possible if space is presupposed. So consider again the statement ‘all bodies are heavy’: it contains both a synthetic element, i.e. it is not necessarily contained in the concept ‘body’ that they are ‘heavy’; and an a priori element, i.e. that the very notion of a body presumes a situation in space.

Space is the source for synthetic a priori claims, since space must be presupposed for any experience to be possible – space grounds some synthetic a priori claims. In other words, space is the form or structure of any representation of outer experience that we might have – any given representation ‘makes no sense’ outside of space. And since space is the necessary form of our outer representations, space necessarily determines objects of experience insofar as they will necessarily be spatial – but does not necessarily reflect how things are in themselves (i.e. ‘outside’ of human cognition).

What are things really like in themselves? According to Kant, we cannot really say, since we do not have epistemological ‘access’ to what objects are like ‘outside’ of our representing them spatially and we thus do not have license to make any knowledge claims about them beyond our representation of them. However, since space is the

necessary form of all outer experience (and furthermore, the necessary form for every human cognizer), it is therefore ‘objectively valid’ for all places and all times. In Kant’s words, space is ‘empirically real’ (i.e. applies to – or is the basis of – every representation we could possibly have) yet it is also ‘transcendentally ideal’ (i.e. the same a priori rules are true for every representation).

Also geometry (i.e. Euclidean geometry) is essentially the (pure) science of space (since geometry does not require empirical observations, but only the form of representation as such), and thus geometrical propositions are likewise grounded in the a priori necessity of space (i.e. geometrical propositions likewise ‘make no sense’ outside of space). This fact – that the space of Euclidean geometry is the same space of all outer experience – satisfies for Kant the two demands he put on science: on the one hand, geometry is grounded in the a priori necessity of space (i.e. apriority); and, on the other hand, geometrical space is the same as experiential space (i.e. applicability). What this means is that the rules which govern outer experience (the ‘rules’ of space, i.e. geometry) must be true for any outer cognition to be possible, since space is a necessary precondition for experience, and are thus valid for all times and places. Furthermore, since space served as the basis for all of the natural sciences, Kant believed he had established a secure foundation for science, one which is necessarily true and universally applicable.

1.8 Introduction to Non-Euclidean Geometries

There was a problem for Kant, however, which did not emerge until nearly 40 years after his Critique of Pure Reason came out. The geometry that Kant describes is

wholly Euclidean. However, in the early 19th century, geometries were discovered which were not Euclidean, and yet consistent. And the simple fact that there now existed geometries which were not Euclidean was enough for some to bring the entire Kantian thesis into question. Indeed, upon the discovery of these geometries, it was the defenders of the Kantian orthodoxy that were the first to dismiss these geometries as, although interesting, nevertheless not descriptive of the true science of space (Torretti 33). Unfortunately, Kant did not have the benefit of hindsight. Euclidean geometry had been the only option available for millennia and in that time had only grown. But with Euclidean geometry stripped of its necessity, this seemed to strip Kant of his secure foundation for scientific truth as well.

In order to better understand the problem, we need to take a look at the nature of geometry and a brief glance at its more recent history. Geometry is a system which has a ‘bottom-up’ design. At the ‘bottom’ are the most basic elements of the system: the axioms, postulates and definitions. The definitions are the ‘weakest’ since they function as a kind of “heuristic device” – that is, they connect the technical terms with the intuitive spatial objects, e.g. ‘a line is an element with length but no breadth’ (Sklar 14). More important are the axioms and the postulates. They are the ‘simple truths’ which are merely asserted as true. They do not require justification since they are considered self-evident (e.g. ‘things equal to the same thing are equal to each other’ and ‘a straight line may be extended in either direction by a straight line’).

Given the small number of ‘simple truths’ (five axioms and five postulates), an incredibly complex system was built around them. Euclid’s Elements provided a perfect science, one which served as exemplar to philosophers and which gave a model for a

system in which complex truths are derived from self-evident truths. It is a perfect science because it is a pure science of actual space: pure because it depends solely on the necessary form of empirical experience of space; but also a science of actual space because it applies to all such experience. And moreover, the system was never considered exhausted – there were always new developments, e.g. Descartes’ introduction of analytic geometry (i.e. equations ‘standing in’ for spatial figures) and, building on analytic geometry, Leibniz and Newton’s invention of differential and integral calculus (Sklar 16). But no development was seen as a challenge to the basic foundation; rather they were merely seen as an augmentation, just more ‘confirmation’ of its truth.

However, there was a blemish which for a long while stood to question the purity of Euclidean geometry. If one takes a look at Euclid’s original axioms and postulates, all of them are presented in a very clear way, and they all have a kind of simplicity and intuitiveness that one would expect from a set of self-evident truths (e.g. ‘all right angles are equal’ or ‘equals added to equals yield equals’) – all, that is, except one, the fifth, or parallel, postulate. The fifth postulate is nearly three times as long as any other postulate and is commonly charged as being far less intuitive. It can be stated, in its derivative form (for simplicity’s sake), as: “if a straight line falling across two straight lines makes the sum of the interior angles on the same side less than two right angles, then the two straight lines intersect if sufficiently extended, on that side” (Sklar 14)34

34 _{This is not actually Euclid’s fifth postulate, it is Playfair’s. Euclid’s original postulate has to do with}

interior, complimentary angles between two parallel lines and a third which intersects them both. The two are logically equivalent, however, and make the transition to non-Euclidean geometry much easier to understand (Greenberg 16).

. It is tantamount to saying that if you take a line (A) and a point not on that line, there is only one line (B) going through that point which does not intersect the first line (A); or, put another way, it

is a condition for parallelism. Many attempts were made to prove the fifth postulate, by trying to derive them from the others.

Although many attempts were made, they all ultimately failed35. In the early 19th century, however, there were simultaneous discoveries of a different kind of geometry which, rather than attempting to defend Euclid’s fifth postulate, instead chose to deny it. In the 1820s, a friend of renowned mathematician Gauss sent him a system of geometry that his son had discovered. In it, the young Bolyai described a system which did not depend on the uniqueness of parallel lines, but which instead postulated that for every line, there is at least one parallel – a kind of many-parallel postulate. Nearly simultaneously, another young thinker named Lobachevsky had developed his own system using a similar principle of parallelism36. The discovery of a non-Euclidean (yet logically consistent37) geometry marks an important step for the question put forth in this paper.

1.9 Kant and Geometry

To understand why this marks an important step let us return to Kant for a moment. For Kant, space is not a concept – that is, any given ‘space’ should not be regarded as a particular instance of some general concept of ‘space’. Rather, space is a

35 _{The best attempt was done by Saccheri in 1733. He showed that there could not be no line which is}

parallel, and that t

here must be at least one line, but could not prove that there is only one line (Sklar 17).

36 _{Gauss had apparently also invented the basic components of such a system himself. He was apparently, for}

whatever reason, afraid to publish it.

37 _{The logical consistency is thought to have been established by a kind of relative consistency proof –}

relative, that is, to Euclidean geometry. If we ‘abstract away’ all of the content of the system, and it is shown to be logically equivalent to the known consistency of Euclidean geometry, then we gain consistency.

‘singular representation’ in that all things which are regarded as spatial must fall under it, in the way of a part to a whole38

So, for Kant, space is not a concept, but rather it exists in the spatial representations themselves, as their very form: “that space has only three dimensions, that there is but one straight line joining two given points, that a circle can be drawn on a plane about a given point with any given radius, etc., these facts cannot be inferred from some universal notion of space, but can only be perceived concretely in space itself” (Torretti 31). Thus geometrical propositions are not analytically true, since they are grounded in the spatial representations themselves – and they can therefore be denied without fear of logical contradiction. But – and this is the important part – if someone does come up with relations different from those given in ‘space’ (i.e. in spatial representations), they must first presume Euclidean space in “support of his fiction” (Torretti 31). In other words, the spatial relations are precisely those found in Euclid’s

Elements and any experience is therefore Euclidean because any spatial representation

must first presume Euclidean space.

. But our idea of space is an intuition which does not depend on that which it represents, but is rather the very form of the representation. This is why he does not think that we can properly say that space belongs to the things themselves – it is the form of our intuition, or in other words, it grounds the very possibility of spatial cognition.

What does the discovery of non-Euclidean geometry mean for Kant’s theory of space? And a fair and related question is, given the later developments in geometry, why

38 _{In the first section of the Transcendental Aesthetic (Of Space) in the Critique of Pure Reason, he writes:}

“Space is not a discursive or so-called general concept of the relations of things in general, but a pure intuition. For … we can imagine one space only and if we speak of many spaces, we mean parts only of one and the same space.”

should does Kant’s view of space now matter, particularly if it appears as though he was wrong about the uniqueness of Euclidean geometry? It seems as though there is still division on just how to interpret the discovery of non-Euclidean geometry and its impact on Kant’s theory of space. It seems there are two ways we can look at it. Either, we might think, he was simply wrong to think that the only representation we are able to have of space is Euclidean. Given that the Critique of Pure Reason begins with his critical examination of space (i.e. in the Transcendental Aesthetic), and because it in a sense serves as foundation for much of the Critique (at least in terms of the ‘system of representation’ described), then it might be tempting to take a hostile position towards the rest of his theory. In other words, since his theory of space was wrong, perhaps we are free to discard the rest (an extreme position which some feel is justified).

But I think the approach to outright dismiss Kant is disingenuous to say the least. There is an alternative move which gives Kant the benefit of the doubt. The edifice of Euclidean geometry had stood the test of time, with only ‘misgivings’ about the fifth postulate. It seemed to be generally assumed that Euclidean geometry was a unique and necessary science. Kant should not be out-of-hand dismissed, following this reasoning, for not having anticipated the discovery of other forms of geometry which are just as consistent. And there are some who think that the Kantian system, with modification, can incorporate different formulations of geometry. Torretti, to give a brief example, has commented that intuition merely gives the form of space (i.e. it provides the possibility for spatial experience), but does not determine it – its determination (e.g. as Euclidean) is the work of the understanding – and so this could, at least on the face of it, leave room for alternative geometrical constructions arising from within the understanding (Torretti 33).