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The idea of construction is, for Kant, a constraint upon what might be called 'real' geometries, i. In this paper I shall emphasize one way of understanding the idea of construction in the critical writings, and link this with the doctrine of schematism. I shall then suggest that such a link provides a basis for a theory of algebra which Kant could have accepted 2.

That Kant's theory allows for non-Euclidean geometries is an interpretation which rests on two considerations, one direct, the other indirect.


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The indirect consideration is that since Kant insists on the synthetic character of the propositions of geometry, the replacement of the axiom of parallels by its contrary would generate no inconsistencies in the system as a whole. Since this is indeed the case, Kant must be correct in asserting the non-analyticity of the axioms and postulates of Euclidean geometry.

I have stated this argument baldly since I wish to concentrate on the direct argument that Kant employs, and I shall leave the indirect argument without further comment. Nothing I say in the sequel depends crucially on this first consideration being accepted as it stands 3. The second consideration rests principally on the following passage of the Critique:. Here Kant identifies mathematical 'existence' with the possibility of construction.

Critique of Pure Reason.

A mathematical object -here a geometrical figure -'exists' insofar as it can be constructed in pure intuition. Generally, Kant is taken to mean that space is given to us as being definitely and irrevocably Euclidean: what this really amounts to is the assertion that perceptual space - the space of any and every possible experience - could not be 're-constructed' such that non-Euclidean geometry provided the formal basis for intuitive, i.

What is involved in this idea of construction? The synthetic propositions of geometry are 'objectified' and thereby verified by constructing the 'object' of the concept in pure intuition, that is, by 'exhibiting' a priori the intuition which corresponds to the concept. The test of a 'real' geometry is this appeal to the possibility of constructing its figures -its objects -in pure intuition; more generally, the test is the possibility of intuitive construction. This means exhibiting particulars which manifest features that are true of a whole class of entities.

Pure constructions in space and time are symbolic instantiations. Hintikka has argued that a proper understanding of Kant's philosophy of mathematics depends upon recognising that there are two distinct but related notions of 'intuition' in Kant. The mature theory links it with sensibility directly, and this meaning tends to be confused with the more restricted, and original meaning of the term, found in the pre-critical writings and the Discipline of Pure Reason.

Here, 'intuitive' means that which represents an individual, and is contrasted with general concepts 5. It is not the spatial character of 'intuitive' constructions that is of crucial importance, but the fact that they can be employed as exemplars for a general class.

The constructed figure, a triangle for instance, is the spatial representation of the 'abstract' relations which constitute 'triangularity'. The figure is useful to us precisely because it embodies those relations which are less easily grasped independently of it. The a priori exhibition of a concept by means of an intuitive construction may consist in a simple empirical procedure such as making marks on paper or moving the beads of an abacus. A natural interpretation of Kant's meaning of construction - or 'exhibiting in intuition'- is by means of the logical procedure of existential instantiation.

The test of the meaningfulness of a concept - its 'real' rather than 'merely logical' possibility-is the construction of a figure. This is produced a priori -in a way that is somewhat analogous to syllogistic reasoning, which is determination of particular conclusions under general rules by means of the faculty of judgement - yet is at the same time 'an appearance present to the senses' 6. This idea -that Kant's philosophy of mathematics can be 'reconstructed' by appeal to quantification theory-is one of the main points of Hintikka's interpretation.

The use of the natural deduction rule of existential instantiation introduces new representatives of individuals - and this, on Hintikka's view, is what Kant's use of intuitive construction involves, and pre-dates the Aesthetic's use of 'intuition' where it relates to spatial intuition directly. According to Hintikka, the idea that the mathematical method is based on the use of general concepts in concreto - that is, in the form of individual instances -provides the starting-point for Kant's mature theory of mathematical reasoning 7.

Kant's view may be identified in a very general way with his claim against rationalist metaphysics that 'existence' is not a predicate:.

Kant: Making Reason Intuitive

We shall see later that for Kant the function of schemata is to 'particularise' certain concepts, that is, to present in intuition individuals which represent a general class. It should not be assumed that Kant thinks of this figurative construction as complete in itself; this would reduce the process of construction to an empirical procedure, valid for the presented spatial figure, but limited to it.

This would of course fail to yield the characteristics bound up, for Kant, with the recognition of mathematical truth, viz neceissity and universality. For instance, in order to obtain synthetic propositions about triangles it is not sufficient merely to consider the concept 'triangle:' such a procedure yields only analytic propositions. However, if we exhibit the triangle in intuition, i.

Clearly, this shows again that there must be something more in the notion of construction than merely the production of lines on paper or images in imagination. And Kant does indeed supply the required feature. So that the constructed figure may be 'adequate to the concept' Kant goes on to explain the procedure in terms of transcendental imagination, i. There must be an element in the procedure that is 'presuppositional': in this way, the empirical construction is given an a priori 'kick.

Kant's answer is that in employing the imagination to construct a triangle in pure intuition we uncover -by 'regressive analysis'- the a priori conditions by which imagination is itself bound in producing particular figures of this kind. This consideration of an act, presupposed in the empirical construction, supplies the necessary presuppositional element. I shall return to this below.

Mathematics, Kant insists, does not extend knowledge by analysis of concepts alone: verification in mathematics requires that it 'hasten to intuition'. In pure intuition the concept is instantiated and considered in concreto, yet non-empirically, since the construction is in pure, not empirical intuition. The concept is 'particularised,' i. In order to produce a particular construction which is adequate to the concept, we require some form of mediation between understanding-the faculty of rules which at the same time provides a priori concepts -and sensibility, in whose domain the constructions must be presented if they are to acquire existential significance, i.

In this way, we effect an isomorphism between the a priori truths which belong to the concept 'triangle' and the identifiable a priori conditions exemplified in the construction. In Kant's hierarchy of faculties, it is judgement that has the task of subsuming under rules, and is in general the procedure of moving from a major and a minor premiss of a syllogism to a particular conclusion. In this way Kant introduces -as part of the Transcendental Doctrine of Judgement- the idea of schemata of pure concepts of understanding:. It is the notoriously difficult Schematism chapter which expands the implications of mathematical construction and contributes to a less constricted understanding of Kant's philosophy of mathematics.

We have seen that the figure produced in intuition from which synthetic propositions, valid a priori, may be 'read off,' must in some matter be representative of all figures of that kind Any characteristic possessed uniquely by the 'empirical' figure can be abstracted and ignored in the reasoning process. How can a single figure perform such a task adequately?

As Kant admits, no image could be adequate to the general concept 'triangle' The answer lies in the notion of the transcendental schematism. At this point it will be helpful to rehearse some of the key ideas relating to 'synthesis' in the critical philosophy, as preparation for the use made of this concept in the chapter on the schematism itself. The idea of construction is much wider and of more general significance in Kant than focussing on his philosophy of mathematics may suggest.

It is initially located inside such a framework, but is generalised as the process of 'synthesis of the empirical manifold' This synthesis, as I suggested above, provides the presuppositional or transcendental element by means of 'imagination. This is the mirror-image of the specific case of mathematical construction.

In mathematical construction we procude, by means of an 'imaginative synthesis,' an image for a concept:. To subsume particulars under concepts is the task of the faculty of judgement in general, and the schematism in particular. The productive synthesis of imagination is a transcendental act:. Connection - synthesis of the manifold-is not a merely passive process undertaken by sensibility and intuition, but is an active procedure of the faculty of imagination.

Immanuel Kant (Stanford Encyclopedia of Philosophy)

Time, as formal intuition, demands synthesis of imagination qua transcendental act: as form of intuition, time is the undifferentiated phenomenon of lapse, and yields only the possibility of determinate succession The successive synthesis of the manifold - an act performed by means of the productive imagination -locates this whole problem for Kant within transcendental philosophy Geometry itself-'the mathematics of space [Ausdehnung]' -is grounded in the productive imagination in the generation of figures. It is on this basis that axioms are understood as conditions of a priori intuition in figurative construction.

In the transcendental deduction Kant had argued that the pure concepts of understanding apply to objects of intuition in general. However, such concepts are for this reason incapable of giving determinate knowledge of objects:. It is schemata that 'particularise' concepts in the required sense.

Only the schematism, qua transcendental act, can provide determinate knowledge of objects. The schema is a product of imagination. It is a universal procedure -an act-which provides an image for a concept.

The Critique of Practical Reason by Immanuel Kant (FULL Audiobook)

The schema of a 'sensibilised' concept - in this case a spatial figure - is a product of pure a priori imagination, through which, and in accordance with which, images - something empirical -first become possible. It is not, 'transcendentally' speaking, the constructed triangle as such which is the ground of a priori valid synthetic propositions, but rather the fact that it has been produced in accordance with the schema for 'triangle,' either as a figure on paper or imaginatively.

Images are connected with the concept by means of the schema which they designate This schema for 'triangle' is a rule of procedure for construction in intuition. Although he initially focused his studies on the classics, philosophy soon caught and held his attention. He returned to the University in to teach as a Privatdozent , which meant that he was paid directly by individual students, rather than by the University.

He supported himself in this way until Kant published many essays and other short works during this period. He made minor scientific contributions in astronomy, physics, and earth science, and wrote philosophical treatises engaging with the Leibnizian-Wolffian traditions of the day many of these are discussed below.

He finally succeeded in at the age of 46 when he completed his second dissertation the first had been published in , which is now referred to as the Inaugural Dissertation. However, this was anything but a fallow period for Kant. He realized that this response would require a complete reorientation of the most fundamental approaches to metaphysics and epistemology.

Although it took much longer than initially planned, his project came to fruition in with the publication of the first edition of Critique of Pure Reason. In addition to writing the Prolegomena to Any Future Metaphysics as a sort of introduction to the Critique , Kant wrote important works in ethics Groundwork for the Metaphysics of Morals , , and Critique of Practical Reason , , he applied his theoretical philosophy to Newtonian physical theory Metaphysical Foundations of Natural Science , , and he substantially revised the Critique of Pure Reason in Kant capped the decade with the publication of the third and final critique, Critique of the Power of Judgment Although the products of the s are the works for which Kant is best known, he continued to publish philosophical writings through the s as well.

The Religion was attended with some controversy, and Kant was ultimately led to promise the King of Prussia Friedrich Wilhelm II not to publish anything else on religion. Kant considered the promise null and void after the king died in During his final years, he devoted himself to completing the critical project with one final bridge to physical science.

Kant never married and there are many stories that paint him as a quirky but dour eccentric. These stories do not do him justice. He was beloved by his friends and colleagues. He was consistently generous to all those around him, including his servants. He was universally considered a lively and engaging dinner guest and later in life host. And he was a devoted and popular teacher throughout the five decades he spent in the classroom.

The most significant aspect of this distinction is that while the empirical world exists in space and time, things in themselves are neither spatial nor temporal. Transcendental idealism has wide-ranging consequences.