For Kant , to set up the foundation of mathematics we need to start from the very initially step analysis of pure intuition. Kant means by a “pure intuition” as an intuition purified from particulars of experience and conceptual interpretation. i.e., we start with experience and abstract away from concepts and from particular sensations.
The impressions made by outward thing which is regarded as pre-established forms of sensibility i.e. time and space. Time is no empirical conception which can be deduced from experience and a necessary representation which lies at the foundation of all intuitions. It is given a priori and, in it alone, is any reality of phenomena possible; it disappears, but it cannot be annihilated. Space is an intuition, met with in us a priori, antecedent to any perception of objects, a pure, not an empirical intuition. These two forms of sensibility, inherent and invariable to all experiences, are subject and prime facts of consciousness in the foundation of mathematics.
Wilder R.L. issues that, for Kant, sensible intuition was necessary in the foundation of mathematics. According to Kant , the a priori character of mathematical judgments is synthetic, rather than analytic. It implies that the propositions of a mathematical theory cannot be deduced from logical laws and definitions. Space is represented as a pure intuition by showing that representation provides us with a way to structure empirical intuitions.
Shabel L. clarifies Kant’s notion of a particular feature of the concept of space i.e. the form of outer sense. It is able to account for the features of geometric cognition i.e. the synthetic a priori of geometric cognition. While, space, as form of sensible intuition, is able to account for the applicability of geometric cognition. If the pure intuition of space that affords cognition of the principles of geometry were not the form of our outer then the principles of geometry would have no role as a science of spatial objects.
According to Kant , mathematics depends on those of space and time that means that the abstract ex¬tension of the mathematical forms embodied in our experi¬ence parallels an extension of the objective world beyond what we actually perceive. Wilder R.L. points out the arguments for the claim that intuition plays an es¬sential role in mathematics are inevitably subjectivist to a degree, in that they pass from a direct consid¬eration of the mathematical statements and of what is required for their truth verifying them.
The dependence of mathematics on sensible intuition gives some plausibility to the view that the possibility of mathematical representation rests on the form of our sensible intuition. This conception could be extended to the intuitive verification of elementary propositions of the arithmetic of small numbers. If these propositions really are evident in their full generality, and hence are necessary, then this conception gives some insight into the nature of this evidence.
According to Wilder R.L., Kant connects arithmetic with time as the form of our inner intuition, although he did not intend by this to deny that there is no direct reference to time in arithmetic. The claim apparently is that to a fully explicit awareness of number goes the successive apprehension of the stages in its construction, so that the structure involved is also rep¬resented by a sequence of moments of time. Time thus provides a realization for any number which can be real¬ized in experience at all. Although this view is plausible enough, it does not seem strictly necessary to preserve the connection with time in the necessary extrapolation be-yond actual experience.
Wilder R.L. sums up that thinking of mathemati¬cal construction as a process in time is a useful picture for interpreting problems of constructivity the mathematical concepts. While, Palmquist, S.P. in “Kant On Euclid: Geometry In Perspective” describes that, as for Kant, space is the pure form of our sensible intuition. The implication of this theory is that the intu¬itive character of mathematics is limited to objects which can be constructed.
In other words , Kant's mature position is that intuition limits the broader region of logical existence to the narrower region of mathematical existence. There can be no doubt that it is clear to Kant that in geometry, the field of what is logically possible extends far beyond that of Euclidean geometry. Palmquist, S.P. (2004) states the following:
Under the Kant’s presupposi¬tions it is not only possible but necessary to assume the existence of non-Euclidean geometries because non-Euclidean geometries are not only logically possible but also they cannot be constructed; hence they have no real mathematical existence for Kant and are mere figments of though.
Palmquist, S.P. sums up that Kant's view enables us to obtain a more accurate picture of the role of intuition in mathematics. On the other hand, Wilder R.L. alleges that Kant went on to maintain that the evidence of both the principles of geometry and those of arithmetic rested on the form of our sensible intuition. In particular , he says that mathematical demonstrations proceeded by construc¬tion of concepts in pure intuition, and thus they appealed to the form of sensible intuition.
Other writer, Johnstone H.W. in Sellar W. ascribes that Kant’s sensible intuition account the role in foundation of mathematics by the productive imagination in perceptual geometrical shapes. Phenomenological reflection on the structure of perceptual geometrical shapes, therefore, should reveal the categories, to which these objects belong, as well as the manner in which objects perceived and perceiving subjects come together in the perceptual act.
To dwell it we need to consider Kant's distinction between (a) the concept of an object, (b) the schema of the concept, and (c) an image of the object, as well as his explication of the distinction between a geometrical shape as object and the successive manifold in the apprehension of a geometrical shape.
Johnstone H.W. indicates that the geometrical object is that the appearance which contains the condition of this necessary rule of apprehension and the productive imagination which generates the complex demonstrative conceptualization.
Bolzano B. (1810) acknowledges that Kant found a great difference between the intuition in which some sketched triangle actually produces, and a triangle constructed only in the imagination. Bolzano B. states that the first as altogether superfluous and insufficient for the proof of an synthetic a priori proporsition, but the latter as neccessary and suffi¬cient. According to Johnstone H.W. Kant’s sensible intuitions in mathematics are complex demonstrative thoughts which have implicit categorical form. Kant emphasizes the difference between intuitions on the one hand and sensations and images on the other. It is intuitions and not sensations or images which contain categorical form.
Johnstone H.W. highlights Kant’s notion that the synthesis in connection with perception has two things in mind (1) the construction of mathematical model as an image, (2) the intuitive formation of mathematical representations as a complex demonstratives. Since mathematical intuitions have categorical form, we can find this categorical form in them and arrive at categorical concepts of mathematics by abstracting from experience.
Meanwhile, Kant in “The Critic Of Pure Reason: APPENDIX” states :
It would not even be necessary that there should be only one straight line between two points, though experience invariably shows this to be so. What is derived from experience has only comparative universality, namely, that which is obtained through induction.We should therefore only be able to say that, so far as hitherto observed, no space has been found which has more than three dimensions
Shapiro claims that for the dependence intuition, ordinary physical objects are ontologically independent, not only of us, but of each other. The existence of the natural number 2, for instance, appears not to involve that of the empty set, nor vice versa. According to Shapiro , the dependence intuition denies that mathematical objects from the same structure are ontologically independent of each other in this way. The existence of the natural number 2, for instance, depends upon other natural numbers. It makes no sense to say that 2 could have existed even if 5 did not. Shapiro suggests that the natural number structure is prior to its individual elements, such that if one element exists, all do.
However, he admits that it is hard to give a satisfactory explication of the dependence intuition, since pure mathematical objects exist necessarily and the usual modal explication of ontological dependence gets no foothold. For on this explication, the existence of 2 no more depends on that of 5 than on that of the empty set. Shapiro stated the following:
There are two possible sensein which category theory could serve as a foundations for mathematics: the strong sense i.e. all mathematical concepts, including those of the current, logico-meta-theoretical framework for mathematics, are explicable in category-theoretic terms; and the weaker sense i.e. one only requires category theory to serve as a possibly superior substitute for axiomatic set theory in its present foundational role.
Bell argues that it is implausible that category theory could function as a foundation in the strong sense, because even set theory does not serve this function. This is due to the fact that set theory is extensional, and the combinatorial aspects of mathematics, which is concerned with the finitely presented properties of the inscriptions of the formal language, is intentional. Bell claims that this branch deals with objects such as proofs and constructions whose actual presentation is crucial.
Further, Shapiro claims that for the structuralist intuition, the Scarce Properties Intuition has probably been the primary motivation for the recent wave of interest in mathematical structuralism. This intuition says there is no more to the individual numbers “in themselves” than the relations they bear to each other. The numbers have no ‘internal composition’ or extra-structural properties; rather, all the properties they have are those they have in virtue of occupying positions in the natural number structure. A natural explication of the Scarce
Properties Intuition is that the natural numbers have only arithmetical properties and that, for this reason, science should be regimented in a many-sorted language, where arithmetical expressions form a sort of their own. Metaphysically , this would correspond to the claim that the natural numbers form their own category. Shapiro seems quite sympathetic with this explication.
One argument is that on Shapiro’s version of structuralism there is a plethora of mathematical structures that says “not only natural numbers but integers, rationals, reals, complex numbers, quaternions, and so on through the vast zoology of non-algebraic structures that modern mathematics provides”. Each of these structures has its own category. In contrast, there is no such proliferation of categories in the realm of the concrete. So there must be something special about pure mathematics that is responsible for this proliferation.
A second , complimentary, argument is contained in the third structuralist intuition. Shapiro draws upon that the properties of pure mathematical object are purely formal, unlike the substantive properties possessed by concrete objects. Shapiro called this the formality intuition. This intuition is captured by Shapiro’s claim that the subject matter of pure mathematics are structures, where a structure is said to be ‘the abstract form of a system’ of objects and relations on these objects. A structure can thus be instantiated by a variety of systems of more substantive objects and relations; for instance, the natural number structure can be instantiated by the sequence of ordinary numerals and by the sequence of strokes: |, ||, |||, etc.
Conversely , a structure can be arrived at by abstraction from a system of more substantive objects and relations. Shapiro makes a very interesting suggestion about what it means for a property to be formal, as opposed to substantive. Recall Tarski’s characterization of a logical notion as one whose extension remains unchanged under every permutation of the domain. Drawing on this idea, Shapiro suggests that a property is formal just in case it can be completely defined in a higher-order language. It uses only terminology that denotes objects and relations of the system.
-----, 2003, “Kant’s Mathematical Epistemology”,Retrieved 2004
2 ….., “Immanuel Kant, 1724–1804”, Retrieved 2004
3 Wilder, R. L. , 1952, “Introduction to the Foundation of Mathematics”, New York, p. 198
4 Ibid.p. 198
5 Shabel, L., 1998, “Kant’s “Argument from Geometry”, Journal of the History of Philosophy, The Ohio State University, p.20
6 Wilder, R. L. , 1952, “Introduction to the Foundation of Mathematics”, New York, p.198
7 Ibid. p.198
8 Ibid. p.198
10 Ibid. p. 198
14Palmquist, S.P., 2004, “Kant On Euclid: Geometry In Perspective”, Retreived 2004 < Steve Pq @hkbu.edu.hk>)
17Wilder, R. L. , 1952, “Introduction to the Foundation of Mathematics”, New York, p.198
18Johnstone, H.W., 1978, in Sellars, W., 1978 “The Role Of The Imagination In Kant's Theory Of Experience”, Retrieved 2003 < http://www.ditext.com/index.html>
20Bolzano, B., 1810, “Appendix: On the Kantian Theory of the Construction of Concepts through Intuitions” in Ewald, W., 1996, “From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Volume I”, Oxford: Clarendon Press, p.219-221
21Johnstone, H.W., 1978, in Sellars, W., 1978 “The Role Of The Imagination In Kant's Theory Of Experience”, Retrieved 2003 < http://www.ditext.com/index.html>
24Kant, I., 1781, “The Critic Of Pure Reason: APPENDIX” Translated By J. M. D. Meiklejohn, Retrieved 2003
25Linnebo, Ø., 2003, “Review of Stewart Shapiro, Philosophy of Mathematics: Structure and Ontology”, Retrieved 2004 < http://firstname.lastname@example.org>
27In Linnebo, Ø., 2003, “Review of Stewart Shapiro, Philosophy of Mathematics: Structure and Ontology”, Retrieved 2004 < http://email@example.com>
29In Linnebo, Ø., 2003, “Review of Stewart Shapiro, Philosophy of Mathematics: Structure and Ontology”, Retrieved 2004 < http://firstname.lastname@example.org>
38In Linnebo, Ø., 2003, “Review of Stewart Shapiro, Philosophy of Mathematics: Structure and Ontology”, Retrieved 2004 < http://email@example.com>
40Linnebo, Ø., 2003, “Review of Stewart Shapiro, Philosophy of Mathematics: Structure and Ontology”, Retrieved 2004 < http://firstname.lastname@example.org>