# shabel2004

Interesting take on Kant’s argument in the Transcendental Exposition of Space in the Aesthetic. I think Shabel also rejects the position in this paper now, though I am not exactly sure why.

Shabel argues that the ‘argument from geometry’ is synthetic and not analytic, thus is not a ‘regressive’ transcendental argument, but rather an argument which starts from the conclusion of the Metaphysical Exposition (that space is a pure a priori intuition) and argues that it is the basis of geometric cognition. This is roughly the opposite of standard readings of the text. However, insofar as pure a priori intuition is supposed to explain the possibility of geometric cognition, it would seem like there remains at least the suggestion of a transcendental argument.

Further, her emphasis on the argument as a ‘bridge’ between the Exposition and the argument for the ideality of space seems somewhat trivial. It looks to me as if either the synthetic or analytic reading will constitute the necessary ‘bridge’ since both show that geometric cognition relies on the existence of a priori intuition. Kant can then move to his next point that the only possibility of an a priori intuition is if it is merely a subjective form (B41; Pr 4:281-2).

@article{shabel2004,
Author = {Shabel, L.},
Journal = {Journal of the History of Philosophy},
Number = {2},
Pages = {195-215},
Title = {Kant's "Argument from Geometry"},
Volume = {42},
Year = {2004}}


## Annotations: shabel2004.pdf

### ToC

1. The Argument Structure of the “Aesthetic” (p.198)

2. Euclid’s Geometry as an A Priori Science of Space (p.208)

3. The “Transcendental Exposition of the Concept of Space” (p.200)

4. Conclusion (p.214)

### Summary Text

“The “argument from geometry” does not analyze geometric cognition in order to establish that we have a pure intuition of space. Rather, the “argument from geometry” establishes that geometric cognition itself develops out of a pure intuition of space. The difference is subtle, but important: on the standard reading, our actual knowledge of geometry is traced to its source—namely, a pure intuition of space—in order to show that we must, therefore, have such a pure intuition. On my reading, our pure intuition of space is offered as both the actual source of our cognition of the first principles of geometry and the means for the production of further cognition based thereon.4 The “argument from geometry” so understood reveals a coherent and compelling philosophy of geometry that must be taken itself as a primary component of Kant’s project in the first Critique. More importantly, perhaps, the “argument from geometry” so understood serves to illustrate and clarify Kant’s notion of a synthetic deduction—a notion that has great interpretive influence over many subsequent arguments in the Critique. Note 4 – In this paper, I aim to show that Kant means to establish that a particular relation holds between our pure intuition of space and our cognition of geometry, as described above. An account of the details of this relation, that is, an account of exactly how our pure intuition of space affords us our cognition of the first principles of geometry, will require further work on the role of the faculty of productive imagination in constructing the objects of geometry.” (p.196)

“My overarching aim is to reinterpret Kant’s “argument from geometry,” which I take to show that Euclid’s geometry as Kant understood it is grounded in a pure intuition of space. My interpretation requires that I distinguish three Kantian claims that are typically conflated, any one of which understood in isolation might plausibly be taken to illustrate Kant’s doctrine of transcendental idealism, but none of which can plausibly be construed as the conclusion of the “argument from geometry”: [1] Space is a pure intuition; [2] Space is a pure form of sensible intuition; 1 Space is only as described in [1] and [2]. I take [1] to be equivalent to the claim that our representation of space is absolutely a priori and non-conceptual, or that space is represented to us as a pure intuition. I take [2] to be equivalent to the claim that the representation described in [1] provides us with a partial structure for cognizing empirical objects. I take 2 to express a further and independent claim that space is itself nothing over and above the structure described in [2], i.e. that space is transcendentally ideal.5 What I hope to show is that, rather than serving as an argument for any or all of the above claims, Kant’s so-called “argument from geometry” establishes the relation between our representation of space (as expressed in [1]) and our cognition of geometry, thus allowing geometry, the mathematical science of space, to play a role in subsequent arguments for [2] and 3. Note 5 – Notice that [1] and [2] do not include commitments to the nature of space itself, that is to whether or not space as we represent it (described by [1] and [2]) is space as it is independently of that representation.” (p.197)

“Accordingly, Kant gives arguments for [1] that are independent of considerations about geometry;6 he then proceeds to show (in the passages that constitute the so-called “argument from geometry"7) that the representation of space as described in [1] provides the foundation for pure geometric cognition. In the sub- sequent passage,8 he claims that the pure geometric cognition that is so founded is empirically applicable only if [2] holds. And in subsequent sections9 he provides direct support for 4, thus completing a series of arguments that together form a defense of the doctrine of transcendental idealism. On my interpretation, the “argument from geometry,” though providing only one step on the path to Kant’s arguments for the doctrine of transcendental idealism, nevertheless plays the crucial philosophical role of connecting Kant’s metaphysical theory of space as pure intuition with his mathematical theory of pure geometry. Note 6 – In the “Metaphysical Exposition of the Concept of Space.” Note 7 – In the second paragraph of the “Transcendental Exposition of the Concept of Space.” Note 8 – In the third paragraph of the “Transcendental Exposition of the Concept of Space.” Note 9 – In the “Conclusions from the Above Concepts.”” (p.197)

“On my view, then, Kant’s “argument from geometry” shows that our cognition of space affords us our cognition of geometry—roughly the reverse of what has long been alleged. That is, Kant’s reasoning is premised on the conclusion drawn in the preceding section—namely, that space is a pure intuition—and directed at an explanation of how that conclusion explains another body of cognition, namely geometry. Recalling the aim specific to this section, Kant asks whether the pure intuition of space does (and must) serve as a principle for gaining insight into the possibility of other synthetic a priori cognition. Having shown that space is a pure intuition, it is as if Kant asks: what are we representing when we represent space? What knowledge does our representation of space afford us, and how?21 Note 21 – Remember that at this stage of the argument, it is still an open question whether our representation of space is likewise a representation of features of things-in-themselves. Transcendental idealism has not yet been introduced or defended; so it is as yet unclear what Kant thinks a representation of space actually captures.” (p.202)

“Thus, the “argument from geometry” is meant to show that a pure intuition of space provides an epistemic foundation for geometry as a synthetic a priori science.32 The passage that follows the “argument from geometry” and concludes the “Transcendental Exposition” shows that the pure intuition of space that en- ables our geometric cognition is likewise the form of sensible intuition. It is only upon concluding the “Transcendental Exposition” that Kant explicitly introduces transcendental idealism, claiming that space represents no property of things in themselves and, equivalently, that space is nothing more than a subjective representation. So, while the “Metaphysical Exposition” gives us space as pure intuition (claim [1]) and the “Transcendental Exposition” gives us space as form of outer sense (claim [2]), Kant deploys further arguments to conclude that space is nothing more than a pure intuition and so nothing more than a form of outer sense (claim

### Important

**Important:**
“the mathematical demonstration is conducted with respect to an object—a triangular space and an immediately adjacent space—the construction of which is underwritten by pre-geometric principles given by a pure intuition of space itself. Further, the steps of the mathematical demonstration include only judgements made on the basis of what is, in Kant’s terms, an absolutely pure construction; thus the conclusion of the demonstration is cognized by us a priori. The mechanical demonstration, by contrast, is conducted with respect to an object that is empirically rendered and inspected; thus, reasoning on its basis leads us to a merely a posteriori conclusion. Moreover, the mechanical reasoning is itself underwritten by the mathematical : in the same sense in which an intuition of an ordinary object of experience is, for Kant, dependent on a prior pure intuition of space and time, so is an intuition of an empirically drawn triangle dependent on a prior pure intuition of euclidean space.” (p.213)

**Important:**
“the particular empirical features of the figure that accompanies the mathematical demonstration do not themselves play any role in the reason- ing; this figure is, on Kant’s view, a diagram of a mental act of construction and is rendered on paper for merely heuristic reasons.50

**Important:**
“The following important issue remains unresolved: what larger philosophical purpose can Kant’s account of geometric cognition serve? Recall that the “Transcendental Exposition” includes the claim that the cognitions that “flow from” the concept of space, namely the cognitions of geometry, “are only possible under the presupposition of a given way of explaining the concept” of space.” (p.215)

### Highlighted Text

**Interesting:**
“the standard interpretation of the “ar- gument from geometry” takes Kant to be arguing in the “analytic” or “regressive” style that he assumes in his Prolegomena. Such an argument begins with some body of knowledge already known to have a certain character, such as mathematics, in order to “ascend to the sources, which are not yet known, and whose discovery not only will explain what is known already, but will also exhibit an area with many cognitions that all arise from these same sources.”” (p.195)

**Interesting:**
“IN INTERPRETING THE IMPORTANT section of the Critique of Pure Reason entitled “Tran- scendental Exposition of the Concept of Space,” it has long been standard to suppose that Kant offers a transcendental argument in support of his claim that we have a pure intuition of space. This argument has come to be known as the “argu- ment from geometry” since its conclusion is meant to follow from an account of the possibility of our synthetic a priori cognition of the principles of Euclidean ge- ometry. Thus, Kant has been characterized as having attempted to deduce a theory of space as pure intuition from an assumption about mathematical cognition.” (p.195)

**Interesting:**
“first, the cognitions that are explained by exposition of the given concept must “actu- ally flow from” that concept. In the present context, this means that whatever cognitions are explained by an exposition of the concept of space are so explained because they “flow from” the concept of space.” (p.201)

**Interesting:**
“That Kant places these demands on his own exposition highlights the distinction he draws between a “metaphysical” exposi- tion of a concept, which isolates that which is given a priori, and a “transcenden- tal” exposition, which isolates what we cognize on the basis of that which is given a priori. Both such expositions describe a priori cognition of space: the first de- scribes the unique features of our passive capacity for spatial intuition, while the second describes the result of reflection on and employment of that very capacity.” (p.201)

**Interesting:**
“First, as we have noted, Kant’s goals in the current section differ from his goals in the preceding “Metaphysical Exposition,” where he takes himself already to have shown that space is a pure intuition. We should expect that his arguments in the separate sections—which have distinct aims—would not proceed toward the same conclusion.20
Note 20 – Henry Allison suggests that the arguments given in the “Transcendental Exposition” merely buttress arguments given in the “Metaphysical Exposition” (Allison 1983, 99). But, this does not account for Kant’s having made very clear that the sections have different aims.” (p.202)

**Interesting:**
“Kant is not reasoning on the basis of geometric cognition, as the above reconstruction suggests, but rather on the basis of that which has already been shown to be given a priori in our concept of space. That is, in the “Transcendental Exposition” Kant is reasoning synthetically from the pure intuition of space exhibited in the “Metaphysical Exposition” to the possibility of synthetic a priori geometric cognition. CPM – the fact that this is supported by Kant’s claim to be arguing ‘synthetically’ in the CPR seems to support Shabel here. " (p.202)

**Interesting:**
“Remembering the background claim that space is a pure intuition, Kant proceeds to ask: how does our represen- tation of space manage to afford us those cognitions that are the unique domain of the science of geometry? His question is not whether it does so, but how.” (p.203)

**Interesting:**
“Because the original representation of the object of geometry, namely space, is a priori (as has been established in the “Metaphysical Exposition”), the cognition of its proper- ties are likewise a priori. And because the original representation of the object of geometry, namely space, is an intuition (as has also been established in the “Meta- physical Exposition”) the cognition of its properties are synthetic. So, the syn- thetic a priori character of the geometer’s cognition of the properties of space is due to an ability to represent the object of geometry—space itself—in pure intuition” (p.203)

**Quote:**
“Now how can an outer intuition inhabit the mind that precedes the objects themselves, and in which the concept of the latter can be determined a priori? Obviously not otherwise than insofar as it has its seat merely in the subject, as its formal constitution for being affected by objects and thereby acquiring immediate representation, i.e., intuition, of them, thus only as the form of outer sense in general. (B41)” (p.205)

**Interesting:**
“In the first part of the “Transcendental Exposition,” he showed that space as pure intuition (claim [1]) accounts for the synthetic a prior- ity of geometric cognition; in this second part, he shows that space as form of sensible intuition (claim [2]) accounts for the applicability of geometric cogni- tion. Notice then that in both parts Kant argues from the concept of space to an explanation of geometry, and not vice versa. Notice too that if the pure intuition of space that affords cognition of the principles of geometry were not also the form of our outer, sensible intuition, then the principles of geometry would have no role as a science of spatial objects.” (p.206)

**Interesting:**
“the “argu- ment from geometry” provides a philosophical bridge from the “Metaphysical Exposition” to transcendental idealism by moving synthetically from our a priori representation of space through our a priori knowledge of the science of space to the empirical reality and transcendental ideality of space.” (p.207)

**Quote:**
“[as geometer I determine] my object in accordance with the conditions of either empirical or pure intuition. The former would yield only an empirical proposition (through mea- surement of its angles), which would contain no universality, let alone necessity, and propo- sitions of this sort are not under discussion here. The second procedure, however, is that of mathematical and here indeed of geometrical construction, by means of which I put to- gether in a pure intuition, just as in an empirical one, the manifold that belongs to the schema of a triangle in general and thus to its concept, through which general synthetic propositions must be constructed. (A718/B746)” (p.209)

**Reference:**
“An illuminating discussion of the relation between Newtonian and Cartesian conceptions of the geometrical (mathematical) and the mechanical is provided by Mary Domski in a manuscript entitled “The Constructible and the Intelli- gible in Newton’s Philosophy of Geometry.”” (p.210)

**Interesting:**
“This comparison is effected using what Wolff calls “Augen-Maß” (eye-measure);44 the geometer inspects the construction in order to gather data regarding the equalities of spatial magnitudes. Thus, the judgement that the second side of the second reconstructed angle “coincides” with the extended base of the triangle, CD, is an empirical assessment based on the features of the particular constructed triangle; the skill of the geometer who “carries” the arcs; and the precision of the tools used to do so. Note 44 – Christian Wolff, Anfangs-Gründe aller Mathematischen Wissenschaften (Hildesheim: Georg Olms Verlag, 1973), 161.” (p.211)

**Interesting:**
“In this demonstration, as in any other properly geometrical demonstration, “Augen-Maß” is not employed. On the contrary, any predication of equalities among spatial magnitudes proceeds on the basis of a prior predication of con- tainments among spatial regions. Further, any predication of containments among spatial regions proceeds on the basis of prior stipulations for constructing spatial regions. Finally, any construction of a spatial region (in this case a triangle, its interior and exterior angles) depends on the geometer’s original ability to de- scribe Euclidean spaces. My claim is that for Kant this description, ultimately, is available to us only via a pure intuition of space, that is, only via a singular, unique and immediate representation of infinite (Euclidean) space.” (p.212)

**Disagree:**
“the figures or “concrete intuitions” con- structed for the two demonstrations are very nearly identical despite the fact that, on Kant’s view, one is empirical and the other pure. How can they “look” the same? More importantly, what sense does it make to speak of the “look” of an absolutely pure intuition?” (p.213)

Note 50 – See Immanuel Kant, “On a Discovery According to which Any New Critique of Pure Reason Has Been Made Superfluous by an Earlier One” in _The Kant-Eberhard Controversy_, Henry Allison, ed. (Baltimore: The Johns Hopkins University Press, 1973), 127; Immanuel Kant, *Philosophical Correspondence 1759–99*, Arnulf Zweig, ed. (Chicago: The University of Chicago Press, 1967), 149 and 155." ([p.213][597i])


**Interesting:**
“we can accept Kant’s theory that our representation of space is absolutely a priori and non-conceptual, as well as his explanation of the role that this representation plays in the eigh- teenth-century science of Euclidean geometry, without concluding that space it- self is transcendentally ideal.” (p.215)

**Disagree:**
“Kant argues that geometric cognition relies essentially on principles that are accessible to us via a pure intuition of space. Cognition of these principles grounds the practice of Euclidean geometry; moreover, cognition of Euclidean geometry grounds a theory of the cognition of ordinary spatial objects. CPM – It is hard to read this claim and not think that Kant is thereby offering a transcendental argument concerning the status of space as a pure intuition.” (p.215)

1. ). … the “Transcendental Exposition,” and its included “argument from geometry,” is meant to show neither that space is a pure intuition, nor that space is only a pure intuition. The “Transcendental Exposition” provides two related explanations for the fact that we have synthetic a priori cognition of geometry that is applicable to objects cognized a posteriori: 1) geometry “flows from” the pure intuition of space; and 2) the pure intuition of space provides us with a form for intuiting real spatial objects.33 Note 31 – I am here disagreeing with Michael Friedman’s recent view on this matter: his claim that “the spatial intuition grounding the axioms of geometry is fundamentally kinematical” leads him ultimately to conclude that Kant’s theory of pure spatial intuition “does not provide, and does not attempt to provide, an independent epistemological foundation [for geometry]” (Michael Friedman, “Geometry, Construction and Intuition in Kant and his Successors,” in Gila Sher and Richard Tieszen, eds., Between Logic and Intuition: Essays in Honor of Charles Parsons [Cambridge: Cambridge University Press, 2000], 193). My emphasis. Note 33 – Were we to discard Kant’s eighteenth-century conception of geometry as about the “form and extension” of material, spatial objects we might also choose to discard his account of the applicability of pure geometric reasoning. Nevertheless, his theory of our representation of space and related theory of the pure science of that space still stands.” (p.207) “On standard readings of the “argument from geometry,” we fail to make a coherent connection between Kant’s theory of space as pure intuition and his theory of geometrical reasoning. This is primarily because, on standard readings, we suppose that Kant meant to defend his theory of space on the basis that it provides the best explanation for our knowledge of geometry. I have argued instead that Kant intended his so-called “argument from geometry” not as a defense of his theory of space, but rather as an independent account of the dependence of geometric reasoning on a pure intuition of space. On Kant’s view, our ability to construct geometric objects and investigate their properties depends on our having a pure intuition of space. In the “Transcendental Exposition,” however, Kant is not using this dependence to argue for our having a pure intuition of space; rather, he is offering an account of geometric knowledge that forges a connection between a prior theory of space as pure intuition and a future theory of space as form of sensible intuition. I am suggesting, then, that we not read Kant’s argument in the “Transcendental Exposition” as primarily a transcendental argument toward the conclusion that space is a pure intuition, but rather as an explanation of the role of space as pure intuition in our practice of geometry.” (p.214) ↩︎

2. ). … the “Transcendental Exposition,” and its included “argument from geometry,” is meant to show neither that space is a pure intuition, nor that space is only a pure intuition. The “Transcendental Exposition” provides two related explanations for the fact that we have synthetic a priori cognition of geometry that is applicable to objects cognized a posteriori: 1) geometry “flows from” the pure intuition of space; and 2) the pure intuition of space provides us with a form for intuiting real spatial objects.33 Note 31 – I am here disagreeing with Michael Friedman’s recent view on this matter: his claim that “the spatial intuition grounding the axioms of geometry is fundamentally kinematical” leads him ultimately to conclude that Kant’s theory of pure spatial intuition “does not provide, and does not attempt to provide, an independent epistemological foundation [for geometry]” (Michael Friedman, “Geometry, Construction and Intuition in Kant and his Successors,” in Gila Sher and Richard Tieszen, eds., Between Logic and Intuition: Essays in Honor of Charles Parsons [Cambridge: Cambridge University Press, 2000], 193). My emphasis. Note 33 – Were we to discard Kant’s eighteenth-century conception of geometry as about the “form and extension” of material, spatial objects we might also choose to discard his account of the applicability of pure geometric reasoning. Nevertheless, his theory of our representation of space and related theory of the pure science of that space still stands.” (p.207) “On standard readings of the “argument from geometry,” we fail to make a coherent connection between Kant’s theory of space as pure intuition and his theory of geometrical reasoning. This is primarily because, on standard readings, we suppose that Kant meant to defend his theory of space on the basis that it provides the best explanation for our knowledge of geometry. I have argued instead that Kant intended his so-called “argument from geometry” not as a defense of his theory of space, but rather as an independent account of the dependence of geometric reasoning on a pure intuition of space. On Kant’s view, our ability to construct geometric objects and investigate their properties depends on our having a pure intuition of space. In the “Transcendental Exposition,” however, Kant is not using this dependence to argue for our having a pure intuition of space; rather, he is offering an account of geometric knowledge that forges a connection between a prior theory of space as pure intuition and a future theory of space as form of sensible intuition. I am suggesting, then, that we not read Kant’s argument in the “Transcendental Exposition” as primarily a transcendental argument toward the conclusion that space is a pure intuition, but rather as an explanation of the role of space as pure intuition in our practice of geometry.” (p.214) ↩︎

3. ). … the “Transcendental Exposition,” and its included “argument from geometry,” is meant to show neither that space is a pure intuition, nor that space is only a pure intuition. The “Transcendental Exposition” provides two related explanations for the fact that we have synthetic a priori cognition of geometry that is applicable to objects cognized a posteriori: 1) geometry “flows from” the pure intuition of space; and 2) the pure intuition of space provides us with a form for intuiting real spatial objects.33 Note 31 – I am here disagreeing with Michael Friedman’s recent view on this matter: his claim that “the spatial intuition grounding the axioms of geometry is fundamentally kinematical” leads him ultimately to conclude that Kant’s theory of pure spatial intuition “does not provide, and does not attempt to provide, an independent epistemological foundation [for geometry]” (Michael Friedman, “Geometry, Construction and Intuition in Kant and his Successors,” in Gila Sher and Richard Tieszen, eds., Between Logic and Intuition: Essays in Honor of Charles Parsons [Cambridge: Cambridge University Press, 2000], 193). My emphasis. Note 33 – Were we to discard Kant’s eighteenth-century conception of geometry as about the “form and extension” of material, spatial objects we might also choose to discard his account of the applicability of pure geometric reasoning. Nevertheless, his theory of our representation of space and related theory of the pure science of that space still stands.” (p.207) “On standard readings of the “argument from geometry,” we fail to make a coherent connection between Kant’s theory of space as pure intuition and his theory of geometrical reasoning. This is primarily because, on standard readings, we suppose that Kant meant to defend his theory of space on the basis that it provides the best explanation for our knowledge of geometry. I have argued instead that Kant intended his so-called “argument from geometry” not as a defense of his theory of space, but rather as an independent account of the dependence of geometric reasoning on a pure intuition of space. On Kant’s view, our ability to construct geometric objects and investigate their properties depends on our having a pure intuition of space. In the “Transcendental Exposition,” however, Kant is not using this dependence to argue for our having a pure intuition of space; rather, he is offering an account of geometric knowledge that forges a connection between a prior theory of space as pure intuition and a future theory of space as form of sensible intuition. I am suggesting, then, that we not read Kant’s argument in the “Transcendental Exposition” as primarily a transcendental argument toward the conclusion that space is a pure intuition, but rather as an explanation of the role of space as pure intuition in our practice of geometry.” (p.214) ↩︎

4. ). … the “Transcendental Exposition,” and its included “argument from geometry,” is meant to show neither that space is a pure intuition, nor that space is only a pure intuition. The “Transcendental Exposition” provides two related explanations for the fact that we have synthetic a priori cognition of geometry that is applicable to objects cognized a posteriori: 1) geometry “flows from” the pure intuition of space; and 2) the pure intuition of space provides us with a form for intuiting real spatial objects.33 Note 31 – I am here disagreeing with Michael Friedman’s recent view on this matter: his claim that “the spatial intuition grounding the axioms of geometry is fundamentally kinematical” leads him ultimately to conclude that Kant’s theory of pure spatial intuition “does not provide, and does not attempt to provide, an independent epistemological foundation [for geometry]” (Michael Friedman, “Geometry, Construction and Intuition in Kant and his Successors,” in Gila Sher and Richard Tieszen, eds., Between Logic and Intuition: Essays in Honor of Charles Parsons [Cambridge: Cambridge University Press, 2000], 193). My emphasis. Note 33 – Were we to discard Kant’s eighteenth-century conception of geometry as about the “form and extension” of material, spatial objects we might also choose to discard his account of the applicability of pure geometric reasoning. Nevertheless, his theory of our representation of space and related theory of the pure science of that space still stands.” (p.207) “On standard readings of the “argument from geometry,” we fail to make a coherent connection between Kant’s theory of space as pure intuition and his theory of geometrical reasoning. This is primarily because, on standard readings, we suppose that Kant meant to defend his theory of space on the basis that it provides the best explanation for our knowledge of geometry. I have argued instead that Kant intended his so-called “argument from geometry” not as a defense of his theory of space, but rather as an independent account of the dependence of geometric reasoning on a pure intuition of space. On Kant’s view, our ability to construct geometric objects and investigate their properties depends on our having a pure intuition of space. In the “Transcendental Exposition,” however, Kant is not using this dependence to argue for our having a pure intuition of space; rather, he is offering an account of geometric knowledge that forges a connection between a prior theory of space as pure intuition and a future theory of space as form of sensible intuition. I am suggesting, then, that we not read Kant’s argument in the “Transcendental Exposition” as primarily a transcendental argument toward the conclusion that space is a pure intuition, but rather as an explanation of the role of space as pure intuition in our practice of geometry.” (p.214) ↩︎

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