Abstract
In this paper, I critically discuss Frege’s philosophy of geometry with special emphasis on his position in The Foundations of Arithmetic of 1884. In Sect. 2, I argue that that what Frege calls facultyof intuition in his dissertation (1873) is probably meant to refer to a capacity of visualizing geometrical configurations structurally in a way which is essentially the same for most Western educated human beings. I further suggest that according to his Habilitationsschrift (1874) it is through spatial intuition that we come to know the axioms of Euclidean geometry. In Sect. 3, I argue that Frege seems right in claiming in The Foundations, §14 that the synthetic nature of the Euclidean axioms follows from the fact that they are independent of one another and of the primitive laws of logic. If the former were dependent on (provable from) the latter, they would be analytic in Frege’s sense of analyticity. But then they would not be independent of one another and due to their mutual provability would lose their status as axioms of Euclidean geometry, since according to Frege an axiom of a theory T is per definitionen unprovable in T. I further argue that only by invoking pure spatial intuition can Frege “explain” the (alleged) epistemological status of the axioms of Euclidean geometry completely: their synthetic a priori nature. Finally, I argue that his view about independence in The Foundations, §14 seems to clash with his conception of independence in his mature period. In Sect. 4, I scrutinize Frege’s somewhat vague, but unduly neglected remarks in The Foundations, §26 on space, spatial intuition and the axioms of Euclidean geometry. I argue that for the sake of coherence Frege should have said unambiguously that space is objective, that it is independedent not only of our spatial intuition, but of our mental life altogether including our judgements about space, instead of encouraging the possible conjecture that in his view it contains an objective and a subjective component. I further argue that for Frege the objectivity of both space and the axioms of Euclidean geometry manifests itself in our universal and compulsory acknowledgement of the Euclidean axioms as true. I conclude Sect. 4 by arguing that there is a conflict between the subjectivity of our spatial intuitions as stressed in The Foundations, §26 and Frege’s thesis in his dissertation that the axioms of Euclidean geometry derive their validity from the nature of our faculty of intuition. To resolve this conflict, I propose that in the light of his avowed realism in The Foundations Frege could have replaced his early thesis by saying that although we come to know the axioms of Euclidean geometry through spatial intuition and are justified in acknowledging them as true on the basis of geometrical intuition, their truth is independent not only of the nature of our faculty of intuition and singular acts of intuition, but of our mental processes and activities in general, including the inner mental act of judging. In Sect. 5, I argue that Frege most likely did not adopt Kant’s method of acquiring geometrical knowledge via the ostensive construction of concepts in spatial intuition. In contrast to Kant, Frege holds that the axioms of three-dimensional Euclidean geometry express state of affairs about space obtaining independently of our spatial intuition. In Sect. 6, I conclude with a summarized assessment of Frege’s philosophy of geometry.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Frege delivered many lectures on geometry during his entire academic careeer. See the list of lectures that he had announced at the University of Jena between 1874 and 1918 in Kratzsch (1979) such as: on the imaginary, analytic geometry of the plane, analytic geometry according to newer methods, synthetic geometry, projective geometry. See also the register of Frege’s scientific legacy, compiled by H. Scholz, in Schirn (1976, vol. I, p. 86 ff).
Cf. Hilbert (1899, §§9, 12). Frege was highly critical both of Hilbert’s consistency proof and of Hilbert’s independence proofs (cf. Frege 1976, pp. 71, 119, 148). In Frege’s view, these proofs rest on pseudo-axioms. In the course of criticizing Hilbert’s axiomatic method again in 1906, Frege tentatively proposes a method of proving mathematical independence claims (cf. Frege 1967, p. 317 ff.). However, four years later in his notes on Jourdain, Frege contends that the unprovability of the parallel axiom (of the other Euclidean axioms I suppose) cannot be proved. “If we do this apparently, we use the word ‘axiom’ in a sense quite different from the traditional one” (Frege 1976, p. 119). He makes a similar remark already in a letter to Liebmann of 29th July, 1900 (Frege 1976, p. 148): “I have reasons for believing that the mutual independence of the axioms of Euclidean geometry cannot be proved.” On Frege’s and Hilbert’s views on consistency see Dummett (1976), Blanchette (1996, 2007, 2012, 2016) and Schirn (2010, 2018a); on Frege’s and Hilbert’s views on independence see Tappenden (2000), Blanchette (2007, 2012, 2014, 2017) and Schirn (2018a).
Frege’s conception of synthetic a priori truths is akin to Kant’s but does not coincide with it. For a detailed discussion of Frege’s and Kant’s epistemological classifications of truths see Schirn (2018b).
von Staudt (1847) and von Staudt (1856–1860). Mark Wilson (1992) has argued that there is a remarkable connection between von Staudt’s definition of points at infinity and Frege’s definition of cardinal number. In private correspondence, Wilson drew my attention to the fact that Frege embraced both complexified projective geometry (in the sense that it includes a complex-valued metric) and Plücker-Lie dissections into alternative base elements, such as line geometry or the pairs of points that Frege uses in his 1883 lecture. For reasons of space, I must leave a discussion of this aspect of Frege’s views on geometry to another occasion; see in this regard the illuminating account in Wilson (2011).
Frege refers only in one place to a work by Helmholtz. In Frege (1903, p. 139 f., footnote 2), he finds fault with Helmholtz’s intention to provide an empirical foundation of arithmetic in the essay ‘Counting and Measuring from an Epistemological Point of View’.
Translations from Frege’s writings are by and large my own.
A purely geometrical treatment of imaginary elements was given by von Staudt (1847, 1856–1860) on the basis of elliptic involutions. One can define, for example, an imaginary point as an object that corresponds to a set of values of the coordinates when at least some of them are complex numbers. Through an imaginary point there is just one real line, the line joining the point to its conjugate; and on an imaginary plane there is just one real line, the line of intersection with the conjugate plane. Frege (1873, p. 1 f.) points out that the meaning of imaginary points is not confined to the field of algebraic analysis. Imaginary points can also be defined in purely geometrical terms by combining a circle with a straight line or by involution on a straight line. Imaginary points, he says, resemble points at infinity in that they too express something common to several forms. “Point at infinity”, he says, is only another expression for what is common to all parallels, which is what we usually call “direction”.
Intuitively speaking, a homeomorphism is a contiunuous stretching and bending of a topological space into a new shape. A square and a circle, for example, are homeomorphic to one another, but a sphere and a torus are not.
An anonymous referee (#2) asked me to explain why in this context I “suddenly” speak of the projective plane. The reason is that only by capturing the mathematical structure linked to this term have mathematicians succeeded in making sense of the point at infinity, that is, the point of intersection of any given family of parallel straights, which (i) is reached by indefinitely advancing along any of these in any direction until one goes beyond any point at a finite distance from the starting point, and (ii) which is therefore the point through which one can go from the right-hand extreme region of a line to its left-hand extreme region, without passing through any of the points that lie between both regions in the Euclidean plane.
(b) indicates a way of linking the projective plane to intuition insofar as a Möbius strip can be built with a pair of scissors and a piece of scotch tape.
An anonymous referee (#3) objects to my argument that what Frege has in mind is probably “a standard projection where the line at infinity is pulled inward to become a regular conic section”. According to the referee, I might be confusing here the projective line at infinity with the single point Riemann closure. (I have never come across the latter phrase, but presume that it refers to what perhaps more commonly is called the single-point compactification of the complex plane.) The referee goes on to observe that in any case “Frege is clear that his map for the complex points is only supposed to provide intuitive cover for a restricted portion of the complex projective plane a task that seems to have [been] pursued by many others in that time ...” Here is my tentative response: What I am trying to show is that there is no way in which one can define a bijective bicontinuous mapping of the Euclidean plane onto the real projective plane, let alone onto the complex projective plane (= a complex manifold described by three complex coordinates). To establish this claim, I need not discuss the projective line at infinity, nor do I see any risk of confusing it with “the single point Riemann closure”. A straightforward way of establishing my claim might be to remind the reader that the real projective plane is a compact non-orientable two-dimensional manifold, that is, a one-sided surface which cannot be embedded in three-dimensional space without intersecting itself. A standard projection, where the line at infinity is pulled “inward” to become a regular conic section, requires that we also pull “outward” a regular conic to become the line at infinity. If I am right, then in the projective plane there is no qualitative or quantitative difference between the line at infinity and the regular conic, nor is there a proper difference between “inward” and “outward”.
Frege represents the imaginary circular points at infinity by two straight lines, one of which lies in the plane E while the other lies in the plane at infinity. Every straight line in E represents the one imaginary circular point, while every straight line of the plane at infinity represents the other (cf. Frege 1967, §7).
Frege uses “sichtbar machen” and “veranschaulichen” in the same sense; cf. Frege (1967, pp. 26, 31, 37).
An anonymous referee (#2) pointed out to me that recent studies with people living in the Amazon region have shown that their geometric intuitions are not in complete agreement with ours. This might apply to people living in some other non-Western cultures as well. Thus the phrase “most Western educated human beings” appears appropriate in the relevant context.
Regarding the view that I attributed to Frege, namely that our geometrical intuitions present the structural facts of geometry to our intellect, an anonymous referee (#3) drew my attention to the following circumstance: “important consequences of these same facts can become more readily apparent to the human mind if intuitive geometrical mental representions get assigned to the subject matter of geometry in non-standard ways, such as picturing a circle’s tangent lines as if they were a swarm of points. This was standard fare in Frege’s time: the non-standard assignments allow us to look at familiar figures with a ‘fresh set of eyes.’ The hope is that new theorems will pop out at one when looked at through the ‘new eyes.”’ I agree with these observations, interesting as they are in the relevant context. On visual thinking in mathematics in general see the epistemology-based analysis in Giaquinto (2007).
Frege speaks of a creation of the concept of quantity by ourselves (cf. Frege 1967, p. 51). To the best of my knowledge, this is the only place in his entire work where he explicitly acknowledges a creation by ourselves of concepts (or numbers or logical objects). I am not sure how much importance we should attach to this remark. It is at variance with everything that Frege says in his later work about the formation of concepts in general and of mathematical and logical concepts in particular. Perhaps he only wanted to convey that the concept of quantity does not originate in intuition, but is rather something that we find only in rational, conceptual thinking.
Frege underscores that the earlier intuitive character of the notion of quantity was at bottom mere appearance. “Bounded straight lines and planes enclosed by curves can certainly be intuited, but what is quantitative about them, what is common to lengths and surfaces, escapes our intuition” (Frege 1967, p. 51). An anonymous referee (#3) calls into question my claim above, namely that in Frege’s view (at least around 1874) we lack a direct cognitive access to arithmetical concepts via intuition. According to the referee, Frege most likely held that the fundamental facts of arithmetic are supported in geometrical intuition, although “this dependency doesn’t adequately capture the wider applicability of numerical reasoning. Example: one can ground the basic algebraic facts concerning the complex numbers in the terms of spatial vectors arranged on the Argand plane, but that grounding doesn’t adequately explain why we trust the results of complex number calculations in other arenas (such as solving quartic equations).” (A note for the non-specialist: Jean-Robert Argand (1768–1822) developed a method of graphing complex numbers by means of analytic geometry and suggested that the value i can be construed as a rotation of 90 degrees in the complex plane, also called Argand plane. This plane has complex numbers assigned to each point on it.) I am not sure about the extent to which the idea of a close link between arithmetic and intuition-based geometry à la Frege can be said to mirror faithfully his reasoning about number systems at least at some stage in his career. If we disregard his late idea of providing a geometrical foundation of arithmetic, there seems to be relatively little direct textual evidence for such a close link. In The Foundations, §103, Frege concedes that what is commonly called the geometrical representation of complex numbers has at least the advantage over other proposals, which he had considered previously, that in it 1 and i do not appear as wholly unconnected and different in kind. However, he also points out that in the geometrical representation of complex numbers every theorem, whose proof must be based on the existence of a complex number, appears to depend on geometrical intuition and is therefore synthetic (which contravenes Frege’s logicism): “Whereby, then, should the fractions, irrational numbers, and complex numbers be given to us? If we take intutition to help, then we introduce something foreign into arithmetic...” (Frege 1884, §104). I recommend the reader to study Tappenden (1995) in this connection, especially section III entitled “Generality and relative priority: geometry and arithmetic” where he discusses the issue of the generality of arithmetic relative to geometry in Frege’s early work including The Foundations.
I quote from Blanchette’s comments on a draft on independence in Frege, which I had sent her in March 2017, with her permission.
I presume that in The Foundations, §14 Frege is tacitly appealing to the logical axioms of Begriffsschrift, §§13–22 when he mentions the primitive laws of logic.
In private correspondence, Blanchette expressed a reservation in this connection. While an axiom must of course be true, she does not yet see “that from this we can conclude that it would be incoherent to suppose for the sake of an argument that a given axiom is false. It would not, under this supposition, be an axiom (so: we would not be considering a situation in which a given thought is both an axiom and false; rather, we would be considering a situation in which a given thought, one that is in fact an axiom, is false (and so not an axiom).”
In a letter to Hilbert of 27th December, 1899 Frege explains (Frege 1976, p. 63): “I call axioms sentences that are true but are not proved because our knowledge of them flows from a source very different from the logical source, a source which might be called spatial intuition.” At first glance, this explanation—thesis (11)—might suggest that Frege intends to explain the non-provability of axioms in a theory T or the fact that they are not proved in T by pointing out that our knowledge of them flows from a nonlogical source of knowledge: from spatial intuition. However, at second glance there can be no doubt that in his explanation he has only geometrical axioms in mind—a point that he should have made explicit nonetheless to avoid ambiguity. Plainly, in the case of a logical axiom it would be pointless and even false, by Frege’s own lights, to explain its unprovability by saying that our knowledge of it derives from a source different from the logical source. Frege holds that axioms in general are true general thoughts which, thanks to their (assumed) self-evidence, do not stand in need of proof; nor do they admit of proof in a theory T once they have been chosen to form the deductive basis of T. Their truth is undeniable, beyond rational doubt. Moreover, axioms are said to contain real knowledge (cf., for example, Frege 1967, pp. 263, 265). It is the combination of these features—truth, generality, self-evidence, unprovability and epistemic value—that for Frege embodies the quintessence of axioms in the classical Euclidean sense, which he seems to endorse unconditionally throughout his career and declares as sacrosanct in his dispute with Hilbert on the foundations of geometry. (In contrast to a geometrical axiom of a theory S, for Frege a primitive truth of logic that is selected as a logical axiom of a theory T must satisfy the stronger condition of unrestricted generality, that is, its validity must extend to all areas of conceptual thought.) The upshot is that in his letter to Hilbert Frege should not even have said that “geometrical axioms are not proved because our knowledge of them flows from a source very different from the logical source”. For on the face of it, claiming the unprovability of the geometrical axioms or their property of not being proved in the way Frege does, might misleadingly suggest that, unlike geometrical axioms, logical axioms are considered to be provable because they flow from the logical source of knowledge, from a source which is crucially involved whenever inferences are drawn. But needless to say, he did not hold this view. In brief, the property of system-relative unprovability is one of the defining properties of axioms in general and as such entirely independent of the source of knowledge from which they stem, or to put it bluntly: an axiom of a theory T is unprovable in T simply because it is an axiom. It seems that Frege had at least a clue of the system-relative unprovability of axioms as early as in his Concept-Script. Possibly, he was even fully aware of it around 1879; cf. Frege (1879, §13). The issue of relative unprovability is discussed at some length in Frege (1914) (cf. Frege 1969, p. 221 f.). Note that in the logical calculus developed in Frege (1879) “\(a=a\)” is an axiom (cf. §21), while in the logical system of Basic Laws it is a theorem (cf. Frege 1893, §50). This shows that at least in the early 1890s Frege already knew that the choice of axioms for a theory T is strictly T-relative.
On the relationship between Frege’s views about geometry and those about real analysis see Heck (2011, p. 96). Heck claims that in order to reject the analyticity of geometry, Frege must reject the identification of geometrical objects with parts of Euclidean 3-space.
Blanchette suggested to me that there is still another possibility to understand Frege’s view regarding independence in §14: “that here he is giving merely a sufficient condition of, rather than a characterization of, independence. If so, then it is consistent with this text that independence is fundamentally a matter of non-provability.” See in this connection Blanchette’s most recent and illuminating analysis of the notion of independence by contrasting the views of Frege and Dedekind, Frege’s fellow combatant for logicism, in Blanchette (2017, section 2.2., especially 2.2.2). I agree with Blanchette that at least from Frege’s mature viewpoint sheer syntactic independence does not suffice for establishing independence between the thoughts of a given set of thoughts. One central point that she argues for is this: the provability of the Dedekind-Peano axioms from fundamental laws of logic in the spirit of Frege is enough to show that the former are provable from one another, and, hence, are not independent. “No core truth of arithmetic is, for Frege, proof-theoretically independent of any other. Contra Dedekind, each of the so-called ‘axioms’ of arithmetic is instead, once sufficiently-carefully analyzed, provable from the empty set, and so trivially provable from the remaining ‘axioms”’ (Blanchette 2017, section 2.2.2).
I ignore here Frege’s statement in 1910 that the unprovability of the parallel axiom—from the other Euclidean axioms, I assume—cannot be proved (cf. Frege 1976, p. 119). In my view, this statement has probably nothing to do with the (alleged) impossibility of coherently assuming the negation of a geometrical axiom, stressed as it was by Frege in his mature period.—In the course of presenting the formal development of his theory of real numbers in Frege (1903) (which was overshadowed by Russell’s Paradox and therefore remained a fragment), Frege points out that in order to frame a sustainable definition of the concept positive class the wider concept of what he calls a positival class must first be defined (§175). (A positive class is a positival class such the ordering < is dense and complete.) Frege emphasizes that in the definition of the concept positival class (\(\varPsi \)) he has taken pains to include only those clauses that are independent of one another. In the same breath, he admits that their mutual independence cannot be proved, and expresses the belief that especially the definitional clause, which says that the Relation composed of the Relation p and the converse of the Relation q belongs to the quantitative domain of the positival class \(\varSigma \), cannot be dispensed with. (Frege introduces Relations as double value-ranges of first-level relations.) In a footnote at the very end of his formal account (cf. Frege 1903, p. 243), he suggests, however, that his earlier claim that the mutual independence of the clauses of definition (\(\varPsi \)) is unprovable should not be construed in an absolute sense. For it is conceivable that one could find classes of Relations to which all conditions except one would apply “so that each of these would not apply in one of the examples”. Thus, if one managed to present a class of Relations satisfying all clauses of (\(\varPsi \)) with the sole exception of, say, the clause mentiond above, one would have succeeded in proving the independence of this clause from the other clauses. At the stage that he has reached in Frege (1903, §175) Frege doubts, however, that it should be possible to give examples of classes of Relations to which all clauses of (\(\varPsi \)) except one apply, without presupposing geometry, the rational and irrational numbers, or even empirical facts. Yet if geometry had to be presupposed in carrying out the independence proof, this would imply, by Frege’s own lights, an epistemological shift from the analytic to the synthetic (a priori). For further details concerning his “independence problem” in Frege (1903) see Adeleke et al. (1987) and Schirn (2013, 2014, 2018a).
von Helmholtz uses here the three-dimensional version of Felix Klein’s model of hyperbolic geometry.
Cf. Beltrami (1868). Incidentally, it is in this work that Beltrami succeeded in carrying out the first consistency proof for non-Euclidean geometry by constructing a Euclidean model of Bolyai’s and Lobachevsky’s plain hyberbolic geometry in the geometry of surfaces of constant negative curvature. A few years later, Felix Klein proved the consistency of hyperbolic geometry by constructing a model in projective geometry.
See in this connection Reichenbach’s arguments for the possibility of visualizing non-Euclidean geometries in Reichenbach (1928, §11). Reichenbach argues in great detail that whoever has successfully adjusted himself to a congruence differing from Euclidean congruence is able to visualize non-Euclidean structures as easily as Euclidean structures. Space will be visualized as non-Euclidean if we succeed in adjusting our visualization to non-Euclidean congruence, that is, if we manage to visualize the “interior curvature”, since the interior curvature is nothing but the deviation from Euclidean congruence. Reichenbach also deals with the question of whether we could, in principle, visualize a space of, say, four dimensions, which he denies is possible (cf. p. 329). So, both von Helmholtz and Reichenbach agree with Frege at least in one respect: when assuming a four-dimensional space we completely abandon the ground of intuition.
Cf. Carnap’s characterization of mathematical and physical geometry in his ‘Introductory Remarks on the English Edition’ of Reichenbach (1928, p. vi). Friedman (1992, p. 178) argues that our modern “distinction between pure and applied geometry goes hand in hand with our understanding of logic, and that this understanding simply did not exist before 1879 when Frege’s Begriffsschrift appeared.” This seems to be right if pure or mathematical geometry is seen as an axiomatic theory in the modern sense. However, from a historical point of view, it seems also correct to say that in a sense the distinction between mathematical and physical geometry did not emerge for the first time in the wake of Frege’s pioneering construction of first- and second-order logic, but much earlier. The distinction is at least hinted at or foreshadowed by Gauss as early as in 1830. In a revealing letter to Bessel, Gauss wrote: “we have to acknowledge humbly that if number is merely a product of our mind, space has a reality also outside of our mind to which we cannot completely prescribe its laws in an a priori fashion” (Gauss 1900, vol. VIII, p. 201). Needless to say, mathematical geometry as construed by Gauss was still far from being an axiomatic theory in the modern sense. On this issue, see the useful account in Kvasz (2011). Kvasz basically agrees with the view which I developed in Schirn (1991) and contrasts it with the view held by Michael Friedman.
In his main work Philosophiae Naturalis Principia Mathematica (Newton 1726), Newton speaks of an “absolute, unchangeable, rigid space” which exists independently of the existence of bodies in it.
Admittedly, this is only speculation, but, as far as I know, nowhere in his work and correspondence does Frege mention Einstein’s theory of relativity, nor even Einstein by name. According to Einstein’s general theory of relativity, gravitation is a consequence of the curvature of spacetime, and the structure of physical space differs in strong gravitational fields from that of Euclidean space.—An anonymous referee (#1) raised the question of whether Frege “could not have held that non-Euclidean geometries, though consistent, are simply false as theories of space? If not, why not?” I assume that the referee has physical space in mind and I take it that the question is whether Frege, who always emphasized the pre-eminent role of Euclidean geometry, could have rejected, on plausible grounds, both hyperbolic and elliptic geometry as theories that are false of physical space. My answer is roughly as follows: For certain reasons, Frege might have held this view, but I think that he could hardly have defended or justified it—not even before Einstein published his groundbreaking results in physics—unless precise measurements over very large cosmic distances would undeniably have revealed that the structure of physical space is indeed Euclidean and thus neither hyperbolic nor elliptic. (Clearly, the difference between a Euclidean universe of curvature 0 and a non-Euclidean universe of very small positive or negative curvature cannot be ascertained by carrying out ordinary measurements in physics.) Here is a simplified scenario which might help to shed some more light on the issue under consideration. Suppose that at the turn of the nineteenth and twentieth century experimental physicists had discovered that physical space has a non-Euclidean structure. More specifically, suppose that by repeatly carrying out, within the bounds of measuring accuracy, fairly precise measurements they had found out that the sum of the angles of a triangle between very distant fixed stars deviates significantly from \(180{^{\circ }}\). In the light of this result, they could basically have chosen between two options: (a) they could have used a non-Euclidean geometry for the description of the structure of physical space—hyperbolic geometry if the sum of the angles has a value \({<}180{^{\circ }}\), elliptic geometry if the value is \({>} 180{^{\circ }}\)—or (b) they could have retained Euclidean geometry for the description of physical phenomena by introducing new laws according to which all solid bodies expand and contract in certain ways. Moreover, they probably would have had to modify the laws of optics as well, because the deviation of the measured sum of the angles of the cosmic triangle from \(180{^{\circ }}\) could also be explained by a deflection of the light rays. In contrast to this second option in the scenario that I am imagining, Einstein’s choice of elliptic, Riemannian geometry for the description of the structure of physical space was presumably also guided by what one may call the criterion of the simplicity of the entire system of physics. According to this criterion, it makes sense to accept a considerable complication of the geometrical theory that is used to describe physical reality, if in this way one succeeds in retaining the standard laws of physics or even in simplifying them.
Kant (B72) makes it clear that an intellectual intuition seems to pertain only to the original being (Urwesen).
Frege’s incomparability claim regarding the spatial intuitions of different men is immediately reminiscent of remarks that he makes in connection with his distinction between the reference and the sense of a sign in ‘On Sense and Reference’: “When two persons represent the same thing, each still has his own idea. It is indeed sometimes possible to establish differences in the ideas, or even in the sensations, of different men; but an exact comparison is not possible, because we cannot have those ideas together in the same consciousness” (Frege 1967, p. 146). Frege makes a similar statement in ‘Logic’ I: “Strictly speaking, therefore, we can only form a superficial judgement of the similarity between mental processes, since we are unable to unite the inner states experienced by different men in one consciousness and so compare them” (Frege 1969, p. 4).
In ‘On Sense and Reference’ (Frege 1967, p. 145 f., fn. 3), Frege suggests that we may include with ideas (Vorstellungen) intuitions, “where sense-impressions and actions themselves take the place of the traces which they have left in the mind. The difference is unimportant for our purpose, especially since memories of sensations and actions presumably always go along with such sensations and actions to complete the perceptual image [Anschauungsbild]. Yet one may understand by intuition an object in so far as it is sensibly perceptible or spatial.” Frege’s characterization in this passage of what he understands by an intuition strikes me as awkward. A chair or a table is a sensibly perceptible, spatial object, but it is not an intuition in any strict sense of the term “intuition”. It does not make much sense to say that an intuition qua inner, mental act is sensibly perceptible. In ‘Logic’ II, Frege characterizes the difference between idea and intuition in a similar fashion as in ‘On Sense and Reference’: “By an idea we understand a picture that is called up by the imagination: unlike an intuition, it does not consist of present sensations, but of the reactivated traces of past sensations or actions” (Frege 1967, p. 142). By appeal to this characterization, Dummett (1982, p. 234) claims that in his mature period Frege no longer adhered to the concept of intuition which he had applied in The Foundations without providing any textual evidence for his claim. Moreover, he fails to mention that in ‘Logic’ II Frege characterizes intuition without any appeal to geometry. It seems to me that in his letter to Hilbert of 27th December 1899 (Frege 1976, p. 63) Frege conceives of the nature of spatial intuition as well as the role that it plays in geometry in basically the same way as in The Foundations. Although Dummett (1982, p. 254) mentions Frege’s letter to Hilbert of 27th December 1899, he does not attach much importance to Frege’s remark on spatial intuition there. Dummett’s observation that it is only a remark made in correspondence has little force; see in this regard also Frege’s letter to Hilbert of 6th January 1900 (Frege 1976, p. 70). For those readers who are interested in analyzing Frege’s conception of intuition in more detail, I list, in chronological order, all the places that I know where Frege uses the word “intuition”: Frege (1967, pp. 1, 31, 50 f., 87); Frege (1879, §§8, 23); Frege (1969, p. 36); Frege (1964, p. 109); Frege (1976, p. 164); Frege (1967, pp. 94, 262); Frege (1884, §§5, 12–14, 26, 31, 40, 42, 62, 64, 86, 89–91, 103 f., 109); Frege (1967, pp. 104, 145 ff., 262); Frege (1893, p. 1); Frege (1903, p. 97); Frege (1969, pp. 142, 298); Frege (1976, pp. 61–64, 70.)
In his review of Cohen (1883), Frege observes: “While geometrical objects may well be granted objectivity, they cannot very well be granted reality proper...” (Frege 1967, p. 102). Plainly, when in The Foundations, §65 Frege attempts to define the direction operator “the direction of line x” via a first-order abstraction principle, he introduces directions of lines as abstract objects. After having rejected this tentative contextual definition, due to the emergence of the “England problem”—one cannot, for example, decide by appeal to the criterion of identity for directions embodied in the definiens (= the equivalence relation of parallelism holding between lines) whether England is the same as the direction of the Earth’s axis—he finally defines the direction of line a as an equivalence class of parallelism and, hence, at least according to his mature view in Basic Laws of Arithmetic, as a logical object. Frege’s logical objects in his mature period (the two truth-values and value-ranges of first-level functions, and after the identification of the True and the False with their unit classes in §10 just value-ranges) form only a subset of the set of abstract (non-real) objects such as the equator or the centre of mass of the solar system (cf. Frege 1884, §26).
In my view, Dummett (1981, 1982) misinterprets §26 in one important respect. Contrary to what he seems to assume, for Frege the axioms of Euclidean geometry do not capture any features of our spatial intuitions; they deal exclusively with the features of objective space. This interpretation is not at odds with Frege’s view that our knowledge of the Euclidean axioms rests on spatial intuition. Furthermore, it is not quite clear which features of space Dummett has in mind when he speaks of those features which (in Frege’s sense) are independent of sensation and intuition. He probably appeals to those features of space that can be expressed in words possessing a meaning that is the same for all members of the relevant linguistic community. Yet I fail to see why Frege should acknowledge features of space that are not independent of our sensation, intuition, and imagination. In particular, it remains unclear what Dummett understands precisely by those features of our spatial intuitions that are said to be captured by the axioms of Euclidean geometry.
The content of a geometrical axiom qua sentence is objective since it is exactly the same for all who are capable of grasping it (cf. Frege 1967, pp. 45, 148 fn., 182, 353; Frege 1969, pp. 7, 115, 138, 145, 160, 223; Frege 1976, p. 128). To be sure, this characterization is intended to apply to every judgeable content or thought and, consequently, also to the content that is expressed by the negation of, say, the parallel axiom of Euclidean geometry. Moreover, like the parallel axiom itself, its negation contains unquestionably something conceptual and judgeable. It is solely the lawfulness that Frege would probably deny to the negation of the parallel axiom.
Note that this invites the objection that in order to interpret our visual experiences appropriately and to find our way in the world it suffices to assume that physical space is only in small portions approximatively Euclidean. I trust that Frege was aware of this but presumably thought that he could ignore this in his account.
Although Frege does not explicity say that judging is something subjective, he most likely thought that it is something subjective. I presume that all inner mental acts and processes are “indiscriminately” subjective for him. Thus, there is no room for a division between subjective and objective mental acts. If for Frege objectivity includes independence from judging —and I have argued that it does—then judging cannot be objective because it is not independent of itself. Nonetheless, it is correct to say along Fregean lines that judging qua mental, non-propositional act—a judgement is not performed by tacitly predicating “is true” of a thought—is linked to two objective items, namely to the thought that is acknowledged as true and the truth-value of this thought.
In his review of a book on analytic geometry by A. v. Gall and E. Winter, Frege compares projective with metrical geometry and says that projective geometry may be likened to a symmetrical figure where every proposition has a proposition corresponding to it according to the principle of duality (Frege 1967, p. 87). To be sure, projective geometry is much more bizarre than non-Euclidean geometry. In Bolyai’s and Lobachevsky’s hyperbolic geometry, for example, only Euclid’s parallell postulate is denied, while in projective geometry the relations of linear order and neighbourhood between the points of space are upset. Since distances and sizes are of no concern in projective geometry, it may be considered essentially non-metric. Nevertheless, in Klein (1921, vol. 1), Klein showed how metrical relations can be defined in projective space. In doing so, he relied on Cayley’s introduction of a general projective metric (Cayley 1889–1897; cf., for example, vol. II, pp. 592–605). Cayley wanted to prove that, by employing his general metric, ordinary Euclidean metrical geometry could be construed as a special part of projective geometry. Klein intended to provide, in addition, an account of the geometrical content of Cayley’s metric as well as to demonstrate that not only parabolic (that is, Euclidean) geometry, but also hyperbolic and elliptic geometries are special cases of this metric. Since in three-dimensional space Cayley’s metric just encompasses parabolic, hyperbolic, and elliptic geometry, it is tantamount to the assumption that parabolic, hyperbolic, and elliptic space is of constant curvature (cf. Klein 1921, vol. I, pp. 255, 258 f.). Klein addresses the issue of the independence of projective geometry from the theory of parallels for the first time in Klein (Klein 1921, vol. I, pp. 250 ff., 303 f). and examines it in greater detail in Klein (1921, vol. I, pp. 330 ff). He shows that any intuitively accessible region of space can be mapped one-to-one onto an open subset of the real projective manifold \(\mathbf{P}^{3}\) in such a way that the intuitive relations of neighbourhood and order are preserved by the mapping. On Klein’s demonstration see Bonola (1955), Appendix IV, §§7–9 and especially the lucid account in Torretti (1978), p. 143 ff. Regarding Klein’s views on geometry, see also Klein (1921, pp. 353–383) and Klein (1926). The first rigorous axiomatization of projective geometry is, however, due to Pasch (cf. Pasch 1874). On Pasch’s axiomatization see the instructive accounts in Torretti (1978, p. 210 ff.) and Schlimm (2010).
Frege would most likely not contend that a colour predicate such is “red” is everywhere sharply defined. By contrast, he would probably insist that the geometrical predicates “point” and “plane” satisfy the requirement of the sharp delimitation of the corresponding concept. For him, this requirement is only the law of excluded middle in a different guise.
Here is a different but somewhat related case regarding the objectivity of linguistic meaning (of sense and of reference) in a Fregean framework. In ‘On Sense and Reference’ (Frege 1967, p. 144), Frege contends that in the case of a proper name like “Aristotle” opinions as to its sense may differ. A person A may associate the sense of “the teacher of Alexander the Great and last great philosopher of antiquity” with “Aristotle”, while a person B may connect the sense of the coreferential definite description “the founder of the Peripatetic School” with “Aristotle”. In my judgement, Frege is clearly wrong in saying that outside science such variations of sense may be tolerated, so long as the object referred to by the name remains the same. As a consequence of the former contention mentioned above, speakers belonging to the same linguistic community may not only attach different thoughts to the same declarative sentence in which a proper name occurs, but they may also assign different truth-values to one and the same declarative sentence in which such a name occurs (cf. in this connection Frege 1976, p. 128). In the light of Frege’s distinction between customary and indirect reference, multiply embedded non-extensional sentences such as “It is possible that A knows that p” would be striking examples of this possibility. However, Frege’s claim that the sense of a proper name may fluctuate from a speaker A to a speaker B does not clash with his thesis that such a sense (and the sense of a linguistic expression in general) is objective, namely independent of our mental life and therefore communicable. Firstly, the senses of the two identifying descriptions of Aristotle which on assumption A and B connect with “Aristotle” are independent of their sensation, intuition and imagination. Secondly, A and B can perfectly well communicate about the different senses which they connect with “Aristotle”—these senses are intersubjectively accessible—while the sensations and ideas which A and B may associate with the name “Aristotle” and its bearer are, from Frege’s point of view, intrasubjective, not comparable with each other and incommunicable.
Frege does not always draw attention to a change of mind that he has undergone regarding his philosophical views, and rarely compares in detail an old view with a new, revised view. At any rate, his thesis from 1899 (Frege 1976, p. 63) that our knowledge of the geometrical axioms flows from spatial intuition is not intended to imply the view expressed in his early thesis (1) that their truth derives from the nature of our faculty of intuition; nor is his claim from 1924/1925 that the geometrical source of knowledge qua spatial intuition justifies our acknowledgment of the geometrical axioms as true, meant to imply the view that their truth depends on a specific cognitive faculty. Frege’s official view since, say, 1892 is that truth is independent not only of our cognitive faculties but also of our judgements and assertions. It is true that in his post-contradiction period he dissociated himself gradually from logicism but there is almost overwhelming evidence that he never abandoned his realist conception of truth.
Of course, on the assumption that all human beings acknowledge the same geometrical axioms as true, there may exist individual differences in their intuitive apprehension of geometrical configurations. It is possible, for example, that to one man geometrical configurations appear always coloured, while to another they always appear in black and white. Such differences in spatial intuition do not affect the common recognition of the geometrical axioms as true, though.
Teri Merrick (2006) argues that Frege uses the term “intuition” in two senses and that in his mature period he identified, probably under the influence of the neo-Kantian school of Cohen and Natorp, the geometrical source of knowledge qua spatial intuition with geometrical axioms in order to “depyschologize” it. Merrick believes that thanks to this identification Frege could have resolved the tension between his claim in Grundlagen, §26 that (spatial) intuitions are subjective and his contention that the axioms of Euclidean geometry, which are said to flow from spatial intuition, are objective. There is not a single recognition by Frege of Cohen’s and Natorp’s neo-Kantian approach to philosophy in general, nor any acknowledgement of their position regarding the acquisition and objectivity of geometrical knowledge in particular. To my mind, there can hardly be any doubt that neo-Kantianism à la Cohen and Natorp was not Frege’s cup of tea. In fact, I do not know of any substantial agreement concerning their philosophical positions; cf. especially Cohen (1871, 1883); on Frege’s review of Cohen (1883) (Frege 1967, pp. 99–102) see Schirn (2014, section 2.3). As to Merrick’s claim that Frege identified the geometrical source of knowledge with geometrical axioms (p. 69), she fails to provide conclusive evidence for that. Her appeal to Frege’s remark “...I only mean axioms in the original Euclidean sense, when I recognize a geometrical source of knowledge in them” is far from providing the requisite evidence. It seems to me that Frege is here only couching an earlier stated thesis, namely that the axioms of Euclidean geometry flow from the geometrical source of knowledge, in slightly different words; it may sound like an infelicitous choice of phrasing. If he made the identification in question, he would have to concede that in his view the Euclidean axioms flow from the Euclidean axioms. But this would hardly make sense. In contrast to Merrick, I assume that what in his late fragments Frege calls the geometrical source of knowledge comes very close to, if not coincides with our faculty of intuition known from his dissertation. The logical source of knowledge is likewise conceived of as a faculty of the human mind, namely first and foremost as the faculty of drawing logically correct inferences. Merrick (2006) comments also on some of my earlier views on Frege’s philosophy of geometry. I am grateful for her comments, but do not agree with all of them; see Schirn (2018a).
On Kant’s conception of geometrical knowledge see also Schirn (1991).
In hyperbolic geometry the sum of the angles of a triangle is always smaller than two right angles. Triangles of different size can never be geometrically similar. By contrast, in elliptic, Riemannian geometry the sum of the angles of a triangle is always greater than two right angles. This always follows from the assumption that there exist no parallels (cf. Hilbert 1899, p. 50). In elliptic geometry, triangles of different size can never stand to one another in the relation of geometrical similarity.
In B744-745, Kant outlines the Euclidean proof of this proposition; see Euclid, Elements, book I, proposition 32. Following more or less Kant, Klein considers geometric considerations to be impossible unless he has constantly before his eyes the relevant figure in question. Nonetheless, he regards spatial intuition as something that is essentially imprecise. By a geometrical axiom he understands the demand by virtue of which which he puts precise statements into the imprecise intuition. He adds that the content of the axioms then seems to be arbitrary in so far as it is consistent with the inexactness of our spatial intuition; cf. Klein (1921, vol. 1, p. 381 f.). Klein jettisons here Pasch’s idea, developed in Pasch (1882), that the geometrical axioms express the facts of spatial intuition in a way so complete that in our geometrical considerations, in particular when carrying out proofs, we need not appeal to intuition or representations or images, although from a purely pragmatic point of view it may be useful to do so; cf. Pasch (1882, pp. 45, 98). According to Pasch, representations or diagrams (he uses the word “Abbildung”) are devices that may facilitate our grasp of the relations that are expressed in a given theorem as well as of the constructions employed in its proof. Furthermore, the usage of images or diagrams is considered to be a fruitful means to discover such relations and constructions; cf. Pasch (1882, p. 45). In Pasch’s view, the Kernsätze or axioms of a geometrical theory are justified by observations. The axioms are said to spell out what has been observed by looking at very simple geometrical figures or configurations. In short, for Pasch the axioms of geometry are empirical. Unlike the axioms of a geometrical theory, the theorems must be justified—so Pasch argues—in a purely deductive way without invoking any images. And Pasch adds that it is the incontestableness of the proofs by means of which the theorems are reduced to the axioms, plus the experience-based evidence of the axioms themselves, that give mathematics the character of utmost reliability. Turning from Pasch to Hilbert (1899), it seems to me that Hilbert’s appeal to spatial intuition both in his Introduction and in §1 should probably not be taken in a strictly literal sense. Although it is true that almost throughout all the proofs that he gives of his sixty eight therorems in Foundations he provides geometrical constructions or diagrams for the sake of illustration, carrying out the proofs effectively does not crucially depend on drawing geometrical figures and on the spatial intuitions or intuitive representations that we may have of them. Thus, regarding both the role of geometrical illustrations and the axiomatic construction of geometry, Hilbert seems to be following in the footsteps of Pasch. While according to Kant—and also according to Klein—we essentially rely on our spatial intuition in a proof of a geometrical theorem, the derivations of the most important theorems from Hilbert’s or Pasch’s axioms could probably be carried out even by rational beings who lacked the faculty of spatial intuition altogether—and I think that both Pasch and Hilbert would agree with that.
See in this connection the account in Friedman (1992, p. 179 f.) and Shabel (2006, p. 11). Apropos Kant’s reasoning along Euclidean lines, an anonymous referee (#2) drew my attention to Manders’s analysis of diagram-based geometric practice in terms of exact and co-exact attributions and properties (cf. Manders 2008a, b, p. 91 ff.). According to this analysis, certain diagrammatic inferences are licenced, although the conclusions do not follow from the axioms or definitions of the relevant theory. Generally speaking, an assessment based on diagrams is seen to be unreliable chiefly for the following reasons (cf. Manders 2008a, p. 65 f.): First, drawn diagrams are inaccurate or imperfect; for example, drawn lines are not perfectly straight, drawn circles not perfectly circular, etc. Second, individual geometrical figures do not warrant the generality of the conclusions that we may draw in geometrical discourse. (It seems that these two objections could be regarded as directly applying to Kant’s description of how we can attain general geometrical knowledge a priori by considering a single figure that we have produced via an ostensive construction in intuition.) Third, there are different forms of geometry which differ regarding the conclusions we draw, so that a single diagram-based form of reasoning cannot represent them all. Fourth, there are geometrical elements “such as space-filling curves which utterly defeat diagram-based reasoning” (p. 66). Manders concedes that these objections are correct, but argues that within a logical theory for Euclidean diagram-based reasoning, as developed by him in Manders (2008b), they lose a good deal of their force. Moreover, he emphasizes that far from being obsolete, Euclid’s geometrical conclusions are subsumed by modern mathematics, notably in real analytical geometry, real analysis and functional analysis. As to the distinction between exact and co-exact attributions and properties, Manders points out that exact attributions can be expressed by algebraic equations. They license powerful inferences which play a key role in traditional geometrical discourse. Co-exact attributions are licensed directly by the diagram; they express the recognition of regions and their inclusions and contiguities in the diagram. In contrast to exact properties, co-exact properties are expressible by strict inequalities. Concerning the limits of reliable diagrammatic reasoning see also Giaquinto (2011).
According to common sense, imagination is the faculty of creating or reviving images in the mind’s eye, whereas according to Kant “imagination is the faculty for representing an object even without its presence in intuition”. If the synthesis of imagination is a performance of spontaneity, he considers the imagination to be a faculty for determining the sensibility a priori. Kant distinguishes the productive imagination from the reproductive imagination; the latter is subject solely to empirical laws of association; cf. B151-2 and A163/B 204.
Kant holds that the symbolic construction of concepts in algebra is also carried out in accordance with certain general rules in intuition. Algebra “achieves by a symbolic construction equally well what geometry does by an ostensive or a geometric construction ..., which discursive cognition could never achieve by means of mere concepts” (A717/B745).
In his influential essay ‘Frege’s Epistemology’ (Kitcher 1979), Kitcher not only overestimates the extent to which Frege and Kant agree about the nature of human knowledge in general and of mathematical knowledge in particular, but also misinterprets Frege’s epistemological concerns in more than one respect. See the discussion of Kitcher (1979) in Schirn (2018b).
Note in this connection that inThe Foundations and subsequent writings Frege does not use the word “objective” in exactly the same sense as Kant. For Kant, space is a subjective condition of our outer intuition; it is ideal as regards objects when they are considered in themselves through reason, but at the same time it is objective, that is, empirically real with respect to outer appearances. If that were not so, Kant would be at a loss, from his own point of view, to explain the universal and necessary validity of the geometrical truths.
When at the beginning of Hilbert (1899), §1 Hilbert states that each of the groups of the geometrical axioms expresses certain related facts fundamental to our intuition he does not endorse Euclid’s conception of axioms; see in this respect Frege’s comments on Hilbert’s conception of axioms in his letter to Hilbert of 27 December 1899, Frege (1976, p. 61 ff.). As far as I know, Euclid himself does not claim that his geometrical axioms express fundamental facts of our intuition. At any rate, Frege could not have made such a claim. For as we have seen, in Frege (1884, §26) he not only underscores the subjectivity of our spatial intuitions, but also appeals at least implicitly to the objectivity of the geometrical axioms and in line with this asserts explicitly our common and obligatory acknowledgement of their truth. To be sure, his thesis that the axioms of Euclidean geometry flow from the geometrical source of knowledge qua spatial intuition or that our geometrical knowledge rests on pure spatial intuition is not meant to imply that these axioms express basic facts of our spatial intuition. As to Hilbert, it is unclear whether he wants to be understood au pied de la lettre when he states that his axioms express certain fundamental facts of our intuition. In the light of what he says about the status, nature and role of geometrical axioms in the debate with Frege, I favour a negative answer, but due to the lack of conclusive evidence do not wish to vouch for it.
In his final report, referee #1 notes: “But should one instead say: ‘dependent only on...’? [my emphasis] Presumably, in some sense, all truths are dependent on the laws of logic—but the analytic ones would perhaps be those that are dependent on those laws and no others.” In a sense, I agree with this view. Accordingly, I have inserted the word “only” in the sentence under consideration.
I owe this formulation to an anonymous referee (#1).
References
Adeleke, S., Dummett, M., & Neumann, P. (1987). On a question of Frege’s about right-ordered groups. Bulletin of the London Mathematical Society, 19, 513–521.
Beltrami, E. (1868). Saggio di interpretazione della geometria non-euclidea. Giornale di matematiche 6, 284–312; reprinted in Beltrami, Opere matematiche, Ulrico Hoepli, Vol. I, Milan, 1902, 374–405.
Blanchette, P. (1996). Frege and Hilbert on consistency. Journal of Philosophy, 93, 317–336.
Blanchette, P. (2007). Frege on consistency and conceptual analysis. Philosophia Mathematica, 15, 321–346.
Blanchette, P. (2012). Frege’s conception of logic. New York: Oxford University Press.
Blanchette, P. (2014). Frege on formality and the 1906 independence test. In G. Link (Ed.), Formalism and beyond. On the nature of mathematical discourse (pp. 97–118). Boston & Berlin: Walter de Gruyter/Ontos.
Blanchette, P. (2016). The breadth of the paradox. Philosophia Mathematica, 24, 30–49.
Blanchette, P. (2017). Frege’s understanding of the role of axioms. In P. Ebert & M. Rossberg (Eds.), Essays on Frege’s basic laws of arithmetic. Oxford: Oxford University Press (forthcoming).
Bonola, R. (1955). Non-Euclidean geometry. A critical and historical study of its development; english translation of H. S. Carslaw, New York.
Burge, T. (2000). Frege on apriority. In P. Boghossian & C. Peacocke (Eds.), New essays on the a priori (pp. 11–42). Oxford: Oxford University Press.
Cayley, A. (1889-1897). The collected mathematical papers of Arthur Cayley (Vol. 13). Cambridge: Cambrige University Press.
Cohen, H. (1871). Kants Theorie der Erfahrung. Berlin: Ferd. Dümmlers Verlagsbuchhandlung.
Cohen, H. (1883). Das Prinzip der Infinitesimal-Methode und seine Geschichte. Ein Kapitel zur Grundlegung der Erkenntniskritik. Ferd. Dümmlers Verlagsbuchhandlung, Berlin: New edition with an introduction of W. Flach, Suhrkamp, Frankfurt/M. 1968.
Dummett, M. (1976). Frege on the consistency of mathematical theories, in Schirn 1976, Vol. I, 229–242.
Dummett, M. (1981). The interpretation of Frege’s philosophy. London: Duckworth.
Dummett, M. (1982). Frege and Kant on geometry. Inquiry, 25, 233–254.
Frege, G. (1873). Über eine geometrische Darstellung der imaginären Gebilde in der Ebene, in Frege 1967, 1–49.
Frege, G. (1874). Rechnungsmethoden, die sich auf eine Erweiterung des Größenbegriffs gründen, in Frege 1967, 50–84.
Frege, G. (1879). Begriffsschrift. Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle a. S.: L. Nebert.
Frege, G. (1884). Die Grundlagen der Arithmetik. Eine logisch mathematische Untersuchung über den Begriff der Zahl. Breslau: W. Koebner.
Frege, G. (1885). Über formale Theorien der Arithmetik, in Frege 1967, 103–111.
Frege, G. (1893). Grundgesetze der Arithmetik. Begriffsschriftlich abgeleitet (Vol. I). Jena: H. Pohle.
Frege, G. (1899). Letter to Hilbert of 27th December, 1899, in Frege 1976, 60–64.
Frege, G. (1899-1906). Über Euklidische Geometrie, in Frege 1969, 182–184.
Frege, G. (1903). Grundgesetze der Arithmetik. Begriffsschriftlich abgeleitet (Vol. II). Jena: H. Pohle.
Frege, G. (1914). Logik in der Mathematik, in Frege 1969, 219–270.
Frege, G. (1924/1925). Erkenntnisquellen der Mathematik und der mathematischen Naturwissenschaften, in Frege 1969, 286–294.
Frege, G. (1964). Begriffsschrift und andere Aufsätze. Edited by I. Angelelli. Darmstadt & Hildesheim: Wissenschaftliche Buchgesellschaft Darmstadt.
Frege, G. (1967). Kleine Schriften. Edited by I. Angelelli, Hildesheim: G. Olms.
Frege, G. (1969). Nachgelassene Schriften. Edited by H. Hermes, F. Kambartel & F. Kaulbach. Hamburg: F. Meiner.
Frege, G. (1976). Wissenschaftlicher Briefwechsel. Edited by G. Gabriel, H. Hermes, F. Kambartel, C. Thiel & A. Veraart. Hamburg: F. Meiner.
Friedman, M. (1992). Kant’s theory of geometry. In C. J. Posy (Ed.), Kant’s philosophy of mathematics. Dordrecht: Kluwer.
Gauss, C. F. (1900). Werke (Vol. VIII). Edited by Königliche Gesellschaft der Wissenschaften in Göttingen, Leipzig.
Giaquinto, M. (2007). Visual thinking in mathematics. Oxford: Oxford University Press.
Giaquinto, M. (2011). Crossing curves: A limit to the use of diagrams in proofs. Philosophia Mathematica, 19, 281–307.
Heck, R. G. (2011). Frege’s theorem. Oxford: Oxford University Press.
Hilbert, D. (1899). Grundlagen der Geometrie. Leipzig.
Kant, I. (1781/1787). Kritik der reinen Vernunft. Edited by R. Schmidt. Hamburg: F. Meiner, 1956.
Kitcher, P. (1979). Frege’s epistemology. The Philosophical Review, 66, 235–262.
Klein, F. (1921). Gesammelte mathematische Abhandlungen. Vol. I: Liniengeometrie. Grundlegung der Geometrie. Zum Erlanger Programm. Edited by R. Fricke & A. Ostrowski, Berlin. Reprint Springer, Berlin, Heidelberg, New York 1973.
Klein, F. (1926). Vorlesungen über die Mathematik im 19. Jahrhundert. Berlin: Teil I.
Kratzsch, I. (1979). Liste der von Frege zwischen 1874 und 1918 an der Universität Jena angekündigten Lehrveranstaltungen. In “Begriffsschrift”. Jenaer FREGE-conference, 7–11 May 1979, pp. 534–546.
Kvasz, L. (2011). Kant’s philosophy of geometry—On the road to a final assessment. Philosophia Mathematica, 19, 139–166.
Manders, K. (2008a). Diagram-based geometric practice. In P. Mancosu (Ed.), The philosophy of mathematical practice (pp. 65–79). Oxford: Oxford University Press.
Manders, K. (2008b). The Euclidean diagram. In P. Mancosu (Ed.), The philosophy of mathematical practice (pp. 80–133). Oxford: Oxford University Press.
Merrick, T. (2006). What Frege meant when he said: Kant is right about geometry. Philosophia Mathematica, 14, 344–375.
Newton, I. (1726). Philosophiae Naturalis Principia Mathematica. Edited by A. Koyré & I. B. Cohen (3rd ed.). Cambridge, MA: Harvard University Press.
Pasch, M. (1882). Vorlesungen über neuere Geometrie. Leipzig: B. G. Teubner.
Reichenbach, H. (1928). Philosophie der Raum-Zeit-Lehre, Berlin; Vol. 2 of H. Reichenbach, collected works in 9 volumes. Edited by A. Kamlah & M. Reichenbach, Braunschweig 1977. English translation: The philosophy of space & time, translated by M. Reichenbach and J. Freund, with introductory remarks by Rudolf Carnap, Dover Publications, New York 1957.
Riemann, B. (1868). Über die Hypothesen, welche der Geometrie zugrunde liegen. Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 13, 133–152.
Schirn, M. (Ed.). (1976). Studien zu Frege—Studies on Frege (Vol. I–III). Stuttgart-Bad Cannstatt: Frommann-Holzboog.
Schirn, M. (1991). Kants Theorie der geometrischen Erkenntnis und die nichteuklidische Geometrie. Kantstudien, 82, 1–28.
Schirn, M. (2010). Consistency, models, and soundness. Axiomathes, 20, 153–207.
Schirn, M. (2013). Frege’s approach to the foundations of analysis (1873–1903). History and Philosophy of Logic, 34, 266–292.
Schirn, M. (2014). Frege on quantities and real numbers in consideration of the theories of Cantor, Russell and others. In G. Link (Ed.), Formalism and beyond. On the nature of mathematical discourse (pp. 25–95). Boston & Berlin: Walter de Gruyter.
Schirn, M. (2018a). Frege on the Foundations of Mathematics. Synthese Library: Studies in epistemology, logic, methodology, and philosophy of science, editor-in-chief: O. Bueno, New York, London: Springer (forthcoming).
Schirn, M. (2018b). Funktion, Gegenstand, Bedeutung. Freges Philosophie und Logik im Kontext. Münster: Mentis (forthcoming).
Schlimm, D. (2010). Pasch’s philosophy of mathematics. Review of Symbolic Logic, 3, 93–118.
Shabel, L. (2006). Kant’s philosophy of mathematics. In P. Guyer (Ed.), The Cambridge companion to Kant and modern philosophy. Cambridge: Cambridge University Press.
Tappenden, J. (1995). Geometry and generality in Frege’s philosophy of arithmetic. Synthese, 102, 319–361.
Tappenden, J. (2000). Frege on axioms, indirect proof, and independence arguments in geometry: Did Frege reject independence arguments? Notre Dame Journal of Formal Logic, 41, 271–315.
Torretti, R. (1978). Philosophy of geometry. From Riemann to Poincaré. Dordrecht: D. Reidel.
von Helmholtz, H. (1968). Über Geometrie. Darmstadt: Wissenschaftliche Buchgesellschaft.
von Staudt, G. K. C. (1847). Geometrie der Lage. Nürnberg: F. Korn.
von Staudt, G. K. C. (1856-1860). Beiträge zur Geometrie der Lage. Nürnberg: Bauer & Raspe.
Wilson, M. (1992). The Royal Road from Geometry, Noûs 26, 149–180; reprinted with a Postscript in W. Demopoulos (Ed.), Frege’s philosophy of mathematics. Harvard University Press, Cambridge, MA, 1995, 108–159.
Wilson, M. (2011). Frege’s mathematical setting. In M. Potter & T. Ricketts (Eds.), The Cambridge Companion to Frege (pp. 379–412). Cambridge: Cambridge University Press.
Acknowledgements
I presented previous and shorter versions of this paper at the International Workshop “The Imaginary, the Ideal and the Infinite in Mathematics” in Pont-à-Mousson (France), 24.06.–27.06.2009; Munich Center for Mathematical Philosophy; University of Oxford (as one of my lectures on Frege’s philosophy of mathematics in Trinity Term 2014); Kyoto University; Hokkaido University (Sapporo). Thanks to the audiences, especially to Godehard Link, Hannes Leitgeb, Jamie Tappenden, Paolo Mancosu, Andrei Rodin, Daniel Isaacson, Yasuo Deguchi and Koji Nakatogawa. I am most grateful to Roberto Torretti for inspiring discussion of issues in the philosophy of geometry over thirty years, not least at his home in Santiago de Chile in the Spring of 2011. In our subsequent correspondence (2011-2017), he helped me, among other things, to clarify an important problem arising from Frege’s geometrical treatment of imaginary forms in the plane. I dedicate the present paper to Roberto Torretti. I am also most grateful to Patricia Blanchette for our recent discussion of the notion of independence in Frege, with special emphasis on his view in The Foundations of Arithmetic, §14. Thanks to this stimulating discussion, I managed to write an essential part of Sect. 3 of my essay. Thanks are also due to Mark Wilson for his interesting comments on an earlier draft of my paper and to Robert Thomas for his comments on the geometrical construction which I discuss in Sect. 2 of my paper. I am very grateful to three anonymous referees for their substantial and critical reports. Their reports helped me to improve my paper considerably and make it richer in content. Special thanks are due to referee #1 for checking meticulously the revised penultimate version and for his final list of typos and minor errors which I probably would not have detected before submitting the final version of my paper for publication in Synthese. My thanks go also to Daniel Mook for his help with this paper and to the editor of Synthese, Wiebe van der Hoek, for his special interest in the improvement of the paper, his encouragement and advice. Finally, I would like to thank Catherine Murphy and Palanimuthu Athimoolam (Production, Springer) for their help and Adittya Iyer for his advice and technical help in his capacity as JEO assistant for Synthese.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Schirn, M. Frege’s philosophy of geometry. Synthese 196, 929–971 (2019). https://doi.org/10.1007/s11229-017-1489-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-017-1489-6