Name: Chris Arledge. School: Ohio University. Mailing Address: 36 Poston Rd, The Plains, OH Phone:

Transcription

1 1 Name: Chris Arledge School: Ohio University Mailing Address: 36 Poston Rd, The Plains, OH Phone:

2 2 In his Critique of Pure Reason Immanuel Kant sets forth a philosophy of geometry within which the statements of geometry are considered to be both synthetic and a priori forms of knowledge. He thought that geometry is not only apodictic (hence a priori) but also contains propositions that are not derivable simply from the concepts included within the propositions themselves (hence synthetic). The decades that followed saw much assent to Kant s assertion concerning the epistemic and semantic status of geometry. However, during the 19 th century mathematicians developed various non- Euclidean geometries, which posed quite a threat to Kant s contention that the propositions of Euclidean geometry have a priori necessity. To make matters worse for Kant, the development of General Relativity in the early 1900s made use of these non- Euclidean geometries to describe the structure of space-time, thus weaving non- Euclidean geometry into the fabric of mathematical physics. Since then much work has been done arguing that the Kantian synthetic a priori must be rejected in light of modern physics and mathematics. 1 These arguments suffer from several deficiencies, however. First, these arguments misread what Kant is arguing in the Transcendental Aesthetic and as a result come to false conclusions concerning the Kantian conception of the nature of geometry, its relation to sensible objects and the visualization of geometrical figures. Second, the arguments from non-euclidean geometry and physics have already committed themselves to a metric concerning the physical world and thus necessarily must deny the Euclidean structure of space as well as 1 Cf. Nicholas Griffin, "Non-Euclidean Geometry: Still Some Problems for Kant."Studies in the History and Philosophy of Science 22, no. 4 (1991): ; Amit Hagar, "Kant and non-euclidean Geometry." Kant-Studien: Philosophische Zeitschrift 99, no. 1 (2008): 80-98; Hans Reichenbach, The Philosophy of Space & Time. (New York: Dover Publications, 1958).

3 3 the possibility of synthetic a priori knowledge as found in geometry. However, if a Euclidean metric is selected, the physical phenomena found in relativistic physics can be accounted for and thus the Kantian may be justified in asserting the possibility of geometry as a paradigmatic example of synthetic a priori knowledge. Finally, I shall argue that the arguments from non-euclidean geometry operate on a restricted notion of space. In his doctoral dissertation Carnap offers a demarcation of the concept of space into three distinct spaces: formal, intuitive, and physical space. 2 Each of these spaces has unique nuances that allow for a more thorough explication of the concept of space. Following Carnap, I contend that if space is demarcated into these three distinct spaces the arguments from non-euclidean geometry fail to refute the possibility of synthetic a priori knowledge. I Before delving into the arguments, it would be beneficial to look at what Kant actually says concerning the nature of geometry. In the introduction to the Critique of Pure Reason (B16/17) and the Transcendental Aesthetic (B65) Kant offers an argument for the synthetic status of geometrical propositions. He begins by claiming that, from mere concepts only analytic knowledge, not synthetic knowledge is to be obtained (B64/65). To put it another way one might say that if a propositions is analytic, then it must be derivable from the concepts contained within. Geometrical propositions, however, are unable to do this. Consider Kant s proposition that Two straight lines cannot enclose a space, and with them no figure is possible (B65). This proposition cannot be derived from the concept of straight line and the concept of two. Since 2 Rudolph Carnap, Der Raum: Ein Beitrag Zur Wissenschaftslehre. Unpublished Translation by Ran Cohen

4 4 geometrical propositions cannot be derived simply from the concepts contained within, such propositions must be analytic. If we accept Kant s argument here a question arises concerning the alleged a priori status of geometrical propositions and the recourse to intuition (B65) required for deriving these propositions. If indeed intuition of sensible objects is required for an intuition of space and if the intuition of space is required for an intuition of geometry, does that not ultimately ground geometrical propositions in sensible objects, making such propositions a posteriori? Furthermore, does Kant not even say in the Prolegomena that, Pure mathematics, and especially pure geometry, can only have objective validity on condition that it refers merely to objects of sense? 3 Kant does affirm sensibility as a necessary condition for the possibility of geometrical knowledge, albeit indirectly through the intuition of space. However, this does not commit Kant to geometrical propositions being a posteriori for the following reason. Kant makes it clear in the introduction to the first critique that the fact that all knowledge begins in experience in no way entails that all knowledge arises out of (is grounded in) experience. In order for us to affirm that geometrical propositions are a posteriori it must be conceded that geometrical propositions are derived from sensible objects and are thus grounded upon these sensible objects. This is not, however, Kant s assertion. Rather, the geometrical propositions express the structure of the pure intuition of space. This pure intuition of space is itself a priori because space is presupposed in the very possibility of any outer appearance and is not an empirical concept derived from outer experience (B38-39). It is true that the pure intuition of space requires representations of sensible objects, but this is only so because 3 Immanuel Kant, Prolegomena: to any Future Metaphysics that will be able to come forward as Science. 8th ed. (Indianapolis: Hackett Publishing Company, 1990),

5 5 space is nothing but the form of all appearances of outer sense and thus cannot be intuited directly itself. Nevertheless, space is the a priori condition for the possibility of any outer sense and since geometry is nothing but a description of the properties of this space, it too must be a priori. Having reconstructed Kant s arguments concerning the synthetic a priori nature of geometrical propositions it is possible to see where the arguments from non-euclidean geometry misunderstand Kant. The misunderstanding is located within the relationship between geometry and sensible objects. As was shown above, Kant conceives of geometry as the structure of the pure intuition of space. Thus, we experience geometry in the representations of sensible objects which themselves conform to space and these appearances can never contain anything but what geometry ascribes to them. 4 Since geometry is Euclidean for Kant he therefore considers all experience to necessarily conform to the Euclidean metric. Furthermore, since the structure of space is Euclidean, and since all outer sense conforms to the pure intuition of space, an experience that falsifies this Euclidean metric is impossible. If arguments from non-euclidean geometry are to succeed, then an experience is needed that will falsify the claim that appearances necessarily conform to the Euclidean metric as found in the pure intuition of space. The question must then be asked, is the experience of a non-euclidean figure possible? Philosophers Reichenbach and Hagar answer in the affirmative. Reichenbach offers a rather long attempt to prove the possibility of visualizing non-euclidean forms. 5 His conclusion is that the visualization of non-euclidean forms is possible if one rejects 4 Ibid Reichenbach, The Philosophy of Space and Time, 37-58

6 6 the Euclidean metric and refrains from attempting to translate non-euclidean relations into congruent Euclidean ones. 6 Hagar makes a similar claim regarding the presupposition of a Euclidean metric and the possibility of visual non-euclidean appearances. 7 From the conclusion it would seem that if one were to augment one s visualization techniques in the appropriate way one could visualize the various non- Euclidean forms. This suggestion, however, seems to be simply a thought exercise as to what is conceivable, rather than what is an actual state of affairs that might be given to human sensibility. Perhaps we can liken this suggestion to the suggestion that if we augmented our visualization techniques in the appropriate way we could visualize the 6- D space-time of a Calabi-Yau manifold. Given that we have the mathematical techniques to deal with these manifolds, it is only a matter of visualization technique that prohibits our visualization of these 6-D manifolds. While this example certainly is an extreme one the point is the same. 8 In both of these scenarios it is far from clear as to whether 1) such an adjustment is actually possible given the nature of human sensibility or 2) that such an adjustment would provide any reason for concluding that space has the specified metric. As Reichenbach himself admits it is the image-producing function of our visualization [that] so stubbornly rejects non-euclidean geometry. 9 To translate this into Kantian terms we might say that it is the necessary form of outer experience that so stubbornly rejects non-euclidean geometry. 6 Ibid Hagar, Kant and non-euclidean Geometry, A possible rejoinder to this analogy would be that our physiological makeup prohibits the visualization of such 6-D figures and thus the analogy is fallacious. However, often in utilizing non- Euclidean geometry in physical theories such as general relativity, one must make use of a 4-D manifold, which is not capable of being visualized in the same manner, as the 6-D manifold is not. Thus the analogy stands. 9 Reichenbach, The Philosophy of Space and Time, 58.

7 7 One caveat must be made concerning the above argument. Reichenbach makes it clear that he divides visualization into two separate faculties: the image-producing function and the normative function. 10 The image-producing function concerns the production of a particular object as an image. The normative function concerns the relations in a geometric system between the geometrical elements, such as lines, points, planes, etc. that constitute the images produced by the image-producing function. More specifically, the normative function expresses relations to which our image-producing function must conform. It is the normative function that Reichenbach attributes to Kant s synthetic a priori because on Kant s account we are compelled by the pure intuition of space to regard the structure of this intuition as conforming to Euclidean relations. Reichenbach thinks, however, that since we can determine the relations within various non-euclidean geometrical systems we can thus impose these relations on images produced by our image-producing function through an adjustment of our visualization techniques, thus proving that Euclidean geometry is not the necessary metric of space. Such a conclusion would lead to a rejection of Euclidean geometry as synthetic a priori knowledge. However, Kant does not seem to restrict his discussion to relations contained within a geometrical system. Rather, his discussion of the synthetic a priori status of geometry pivots on geometry s relationship with sensible objects through the pure intuition of space. An example of this would be the line from the Prolegomena in which Kant states, pure geometry, can only have objective validity on condition that it refers merely to objects of sense 11. This passage indicates that Kant is dealing with more than the normative function in his discussion of the status of geometry. There is an implicit 10 Reichenbach, The Philosophy of Space and Time, Kant, Prolegomena, 31

8 8 appeal to the image-producing function, which relates geometrical propositions to the objects located within the pure intuition of space. Thus, Reichenbach is mistaken to limit Kant s discussion to the normative function only and once we admit the image-producing function, non-euclidean geometry becomes much less visible. II A good many of the problems of the arguments from non-euclidean geometry, including the problem concerning the visualization of non-euclidean geometrical figures, is found in the arguments overly broad scope of the notion of space. For instance, Reichenbach s normative function of visualization properly belongs to the domain of intuitive space, as will be shown below; but had he demarcated the various conceptions of space, he may have seen the error of restricting Kant to the normative function of visualization. Carnap in his dissertation Der Raum did in fact demarcate space into three distinct spaces in an attempt to explicate the concept of space in its totality. Through an analysis of each of the three distinct spaces the conclusion shall be reached that while two types of spaces do indeed preclude the possibility of synthetic a priori geometry, one type of space maintains the possibility of synthetic a priori geometrical knowledge and thus, once again, the arguments from non-euclidean geometry fail in their attempts to eliminate the possibility of synthetic a priori knowledge concerning space. The first type of space Carnap outlines is what he entitles formal space. Formal space is a space consisting only of the relations between mathematical and logical elements. As Carnap notes, The advantage of this formal [space] lies, on the one hand, in its logical completeness and rigor, since it is free of non-logical elements (intuitive and

9 9 empirical). 12 An example of a formal relation might be given a mathematical proposition = 2, a relation equals 2 when added to 1 can be derived from the proposition. Another example is the relation of transitivity in which (a > b) & (b >c) therefore (a > c). Formal space is what we typically conceive of when we discuss mathematical proofs and completeness. Geometry can fit into this category as well, however, because of the relations that exist between geometrical elements, even though such elements are not themselves included within this space. Such a space cannot be the transcendental space Kant is speaking of when he discusses the nature of geometrical propositions for this space is devoid of intuitive content. This is due to the fact that propositions found within this spatial domain are necessarily analytic, since they express only the relations themselves and not the objects to which the relations might apply. Yet Kant is explicit in his denial of the possibility of deriving geometrical propositions from relations of concepts alone. Furthermore, formal space deals exclusively with the relations and not the elements that are needed to formulate complete geometrical propositions. Therefore, formal space is insufficient to house synthetic geometrical propositions. The second type of space explicated by Carnap is what he calls physical space. Physical space is very simply the space occupied by particular physical objects. Physical space is the space of physics. The physical geometry located within physical space necessarily contains synthetic propositions, which are a posteriori since such propositions express specific assertions about specific physical objects, such as the assertion that my cat is 10 feet from the stove. The notion of physical space is not especially important to 12 Carnap, Der Raum, 5.

10 10 this portion of the argument concerning non-euclidean geometry, but serves as a limit for the boundaries of the next type of space. The objects of physical space do, however, contribute to sensibility, thus allowing for geometrical propositions to be intuited through a pure intuition of space and therefore physical space will be dealt with in the subsequent section. The final type of space demarcated by Carnap is what he calls intuitive space. Such a space lies between the formal structure of relations (formal space) and the objects that produce appearances within the mind (physical space). Carnap defines such a space as the structure of relations between spatial shapes in the customary sense, hence between elements of line, plane, and space whose specific peculiarity we grasp on occasion of a sensual perception or even mere imagination. 13 If we translate Carnap s words into Kantian terminology the concept of intuitive space becomes clearer. Intuitive space is the space that is the structural condition for the possibility of all outer experience. It does not persist outside of the mind but, unlike formal space, cannot be intuited without sensibility either, giving it both an empirical and transcendentally ideal character. It is within intuitive space that we find the foundation for Kant s geometrical propositions mentioned in the Transcendental Aesthetic. For instance, the proposition three lines can enclose a space is not merely a relation, but is a relation between geometrical elements, namely the lines, and yet these lines are given no referent in physical space. Thus we see that propositions within intuitive space cannot be analytic, for the propositions are not derivable merely from the concepts employed. The propositions also must be a priori because they interact with elements that may be said to 13 Carnap, Der Raum, 4.

11 11 possess no particular nature (such as the things in physical space). 14 Prima Facie, then, it would seem that the synthetic a priori has been established. It is easy to see the problem with this argument. Consider the parallel postulate in Euclidean geometry, For any plane on which there is a line L and a point P that is not on L, there is one and only one line L, on the plane, that passes through P and is parallel to L. 15 Such a postulate belongs in intuitive space because it expresses a relation between generic geometrical elements. However, within both Riemannian and Lobachevskian geometries different, mutually inconsistent parallel postulates are given. For instance, in Riemannian geometry the postulate can be restated as For any plane on which there is a line L and a point P that is not on L, there exists no line that passes through P and is parallel to L. Likewise in Lobachevski s geometry there exist multiple parallels. The various postulates, though mutually inconsistent, are perfectly consistent within their system and meet the requirements needed to belong to the realm of intuitive space. It would appear that Kant has been defeated after all. There is, however, a possible rejoinder to the above objection that retains the possibility of geometry as synthetic a priori knowledge. In his dissertation Carnap offers a further demarcation within each of the three types of space that divides each space into three portions: projective, metrical, and topological space. For our purposes, projective space is irrelevant as it concerns the logical form of spatial structures. However, the demarcation between metrical and topological space is central to the argument and it is to these I now turn. 14 As will be shown below, it is not entirely true that all propositions located in intuitive space are a priori. For instance, the propositions in intuitive metrical space are in fact a posteriori. However, concerning propositions relating to Kant s pure intuition of space, such propositions are a priori. 15 Rudolph, Carnap, Philosophical Foundations of Physics; an Introduction to the Philosophy of Science. (New York: Basic Books, inc. 1966), 126.

12 12 Metrical space is the space of measurement. The structure of this metrical space is dependent upon what Carnap calls the metrical stipulation which is the choice of a metric between various competing metrics. For instance, since General Relativity utilizes Riemannian geometry in its mathematical formulations, the metrical stipulation is Riemannian. Intuitive metrical space cannot contain synthetic a priori propositions. Given that the metrical stipulation is chosen rather than necessitated, no a priori certainty can be granted to the geometrical propositions within. Reichenbach s conception of the normative function of visualization fits quite clearly into the fabric of intuitive metrical space. Since the normative function is concerned with the relations between geometrical elements it belongs in intuitive space. However, the manner in which these elements interact is determined by the metrical stipulation chosen by the measurer. This forces the determination that any geometry located within intuitive metrical space is synthetic, but not a priori due to the lack of the necessity of any particular metric. Thus, if Kant s geometry is found within intuitive metric space then his geometry would be synthetic a posteriori. However, metrical space is not the pure intuition that is considered to be the condition of the possibility of all outer experience. Metrical space succeeds this pure intuition when the cognizer chooses to augment his pure intuition with a non-euclidean metric for the purposes of measurement. Since Kant s geometry applies to the pure intuition of the condition for the possibility of all outer experience, his geometry cannot reside in intuitive metrical space. Topological space is the space that reproduces univocally what is present in our experience. 16 Topological intuitive space is therefore the space that reproduces 16 Carnap, Der Raum, 31.

13 13 univocally what is present in our experience of the structure of the condition of all possible outer experience (or the pure intuition of space). This demarcation of space retains geometrical propositions as synthetic because the space is intuitive, but it also restricts intuitive space to the transcendental structure of space that we experience in sensible intuition. Since, according Kant, we experience this space in an a priori manner, the propositions pertaining to the geometry of this space, which is Euclidean in the Kantian system, must also be a priori too. It might here be objected that it is rather unclear as to whether our experience of intuitive topological space takes on a Euclidean metric. However, as we will see below, any attempt to falsify the necessity of Euclidean geometry using empirical means always falls short, due to the mere possibility that non- Euclidean physical theories can be translated into Euclidean physical theories. While this does not establish the necessity of Euclidean geometry, it fails to show the contingency of Euclidean geometry and allows for the possibility that Euclidean geometry is in fact the necessary structure of our intuitive space. III As has been shown above, synthetic a priori geometry is located within intuitive, not physical space. Yet physical space offers up the sensible objects that are intuited, leading to the apprehension of geometrical propositions in intuitive space. For that reason much of the force of the argument against Kant is located in the (alleged) applicability of non-euclidean geometry to the physical world. For instance, with the formulation of the General Theory of Relativity, Riemannian geometry was integrated into the structure of space-time. Consider Einstein s field equation: + Λ=. The R, which is indispensible to the field equation, is the scalar curvature, present only in

14 14 Riemannian space. If indeed Einstein s field equation describes the world, then it would seem that the Kantian would have no choice but to reevaluate the status of Euclidean geometry within intuitive space. The problem with this sort of argument is that it remains far from clear as to whether Einstein s field equations or others containing non-euclidean geometrical elements actually describe the structure of physical space. If indeed the possibility remains open for a Euclidean physical space, then the arguments from non- Euclidean physical geometry fail to establish the impossibility of synthetic a priori geometrical knowledge in intuitive space. There is good reason to reject the assertion that non-euclidean geometry offers a true description of the structure of space-time. Poincaré argued that the choice between Euclidean and non-euclidean metrics within physics is merely a matter of preference. 17 Let s use the example of the bending of light rays near large celestial bodies. Within the theory of relativity it is fairly simple to account for the deviation in the position of light rays near a large body like the sun: these large bodies bend space-time, creating a gravitational field which then deflects the light ray toward the body. This solution, however, relies on a non-euclidean metric for it incorporates a geodesic path for the light rays which is found in Riemannian geometry. Yet it is possible to account for this phenomenon using a Euclidean metric, even if in order to do so one must introduce a variety of new, highly complex optical hypotheses. 18 For instance, take the example of Gauss experiment to prove the physical instantiation of non-euclidean triangles. Gauss used the peaks of three mountains as the vertices of his triangle. To construct the sides of his massive triangle Gauss shot beams of light from one vertex to another. Based on his 17 Henri Poincaré. Science and Hypothesis. (New York: Science Press, 1905), Carnap, Philosophical Foundations of Physics, 158.

15 15 calculations, the massive triangle did show a deviation in the sum of the degrees of the angles from 180. However, one can still account for the deviation of the sum of the angles from 180 degrees while maintaining a Euclidean metric, perhaps by postulating the bending of light, as Poincaré suggests. 19 Whatever the example chosen, it is clear that through an augmentation of physical laws and auxiliary hypotheses, the Euclidean metric can be retained in the face of its non-euclidean competitors. 20 Now it might be argued that Poincaré s suggestion was an interesting bit of theoretical philosophy of science, but has no bearing on the actual practice of physics. However, recent work in mathematical physics has indicated a very real possibility of formulating a theory of relativity using a Euclidean metric. 21 If we can replace Minkowski space-time with a Euclidean version of space-time without losing any empirical adequacy or predictive power, then we have what Carnap calls observationally equivalent theories. 22 Given two observationally equivalent theories, one Euclidean one non-euclidean, the possibility is presented that the Euclidean formulation is the metric of space-time and that the non-euclidean metric is merely a mathematical simplification. In the presence of such a possibility, the argument from non-euclidean physics fails to establish the impossibility of Euclidean geometry as a source of synthetic a priori knowledge. 19 Ibid If one were to find an example of a physical theory that is unable to be reformulated from non- Euclidean to Euclidean terms this would pose a significant problem to my argument. However, as of yet I am unaware of such a theory. 21 Cf. J.B. Almeida, "The alternative formulation of General Relativity in terms of 4-dimensional optics; a re-definition of time." Reading, Number, Time, Relativity from Lomonosov Moscow State University & Bauman Moscow State Technical University, Moscow, September 12, 2004; Hans Montanus, "Special Relativity in Absolute Euclidean Space-Time." Physics Essays 4, no. 3 (1991): ; Alexander Gersten, "Euclidean Special Relativity." Foundations of Physics 33, no. 8 (2003): Carnap, Philosophical Foundations of Physics, 150.

16 16 If we take into account both Poincaré s suggestion and the argument concerning visualization presented above (section I) we arrive at yet another problem for the argument from non-euclidean geometry. Carnap argues that Poincaré s suggestion implies a choice of metric prior to the measurement of whatever is being measured. Following the choice of metric, any measurement made would be conducted in such a way as to yield the geometry of the metric chosen. 23 Thus, if one accepts a non-euclidean metric, measurements will be designed to lead to a non-euclidean measurement. Therefore, arguments asserting the existence of non-euclidean appearances, such as Hagar s, fall victim to circularity in this regard. Such arguments have already assumed a non-euclidean metric. From here they note the existence of non-euclidean forms, which follow from using a non-euclidean metric. The problem comes when they try to use these non-euclidean figures to show that non-euclidean metrics represent the actual structure of space-time. Hence, if one begins with the assumption of the structure of space as non- Euclidean, then the empirical structure of space gathered through measurement will necessarily be of non-euclidean form. Such measurements by no means preclude the assumption of a Euclidean metric and therefore by no means show that space-time is indeed non-euclidean. All that is accomplished through such an argument is that given a non-euclidean metric, figures can be represented in a non-euclidean way. Given a Euclidean metric, the same figures can be represented in a Euclidean way, which maintains the possibility that Euclidean geometry is indeed the condition for the possibility of all outer forms of experience as Kant had claimed (though such an argument by no means proves this either!). 23 Ibid.160.

17 17 In sum, while the arguments from non-euclidean geometry have faired well in that past, such arguments fail in their task. Appeals to visualization, parallel postulates and physics all establish the possibility of a non-euclidean metric and the possibility of non-euclidean figures as part of our intuitive metrical space. Yet none of these arguments preclude the possibility of a Euclidean metric as the necessary structure of our intuitive topological or physical space (whether this is actual is a different matter) and fail to offer up the empirical falsification of Kant that is needed. Therefore, despite what has been argued in the past, arguments from non-euclidean geometry have failed to show the impossibility of synthetic a priori geometrical knowledge.

18 18 Bibliography Almeida, J.B. "The alternative formulation of General Relativity in terms of 4- dimensional optics; a re-definition of time." Reading, Number, Time, Relativity from Lomonosov Moscow State University & Bauman Moscow State Technical University, Moscow, September 12, Carnap, Rudolf. Philosophical Foundations of Physics; an Introduction to the Philosophy of Science. New York: Basic Books, inc Carnap, Rudolph. Der Raum: Ein Beitrag Zur Wissenschaftslehre. Unpublished Translation by Ran Cohen Gersten, Alexander. "Euclidean Special Relativity." Foundations of Physics 33, no. 8 (2003): Griffin, Nicholas. "Non-Euclidean Geometry: Still Some Problems for Kant."Studies in the History and Philosophy of Science 22, no. 4 (1991): Hagar, Amit. "Kant and non-euclidean Geometry." Kant-Studien: Philosophische Zeitschrift 99, no. 1 (2008): Kant, Immanuel. Critique of Pure Reason. Translated by Norman Kemp Smith. Reissued ed. New York: Palgrave Macmillan, Kant, Immanuel. Prolegomena: to any Future Metaphysics that will be able to come forward as Science. 8th ed. Indianapolis: Hackett Publishing Company, Montanus, Hans. "Special Relativity in Absolute Euclidean Space-Time." Physics Essays 4, no. 3 (1991): Poincaré, Henri. Science and Hypothesis. New York: Science Press, Reichenbach, Hans. The Philosophy of Space & Time. New York: Dover Publications, 1958.