Immanuel Kant
“For, first, one can only represent a single space, and if one speaks of many spaces, one understands by that only parts of one and the same unique space. And these parts cannot as it were precede the single all-encompassing space as its components (from which its composition would be possible), but rather are only thought in it. It is essentially single; the manifold in it, thus also the general concept of spaces in general, rests merely on limitations. From this it follows that in respect to it an a priori intuition (which is not empirical) grounds all concepts of them. Thus also all geometrical principles, e.g., that in a triangle two sides together are always greater than the third, are never derived from general concepts of line and triangle, but rather are derived from intuition and indeed derived a priori with apodictic certainty. … Space is represented as a given infinite magnitude. A general concept of space (which is common to a foot as well as an ell) can determine nothing in respect to magnitude. If there were not boundlessness in the progress of intuition, no concept of relations could bring with it a principle of their infinity.”
Critique of Pure Reason, A25/B41
This is one of the most well known and criticized - even over criticized - passages in Kant. Part of the problem is that Kant’s language is characteristically poorly chosen. For instance the word ‘magnitude’ in “Space is represented as an infinite magnitude.” (Or ‘gegebene’ in “Der Raum wird als eine unendliche gegebene Größe vorgestellt.”, I suppose) is hardly the same ‘magnitude’ as Kant’s analysis of spatial magnitudes later. Anyway, we can continue his discussion by cutting away to a much later passage.
“Give a philosopher the concept of a triangle, and let him try to find out in his way how the sum of its angles might be related to a right angle. He has nothing but the concept of a figure enclosed by three straight lines, and in it the concept of equally many angles. Now he may reflect on this concept as long as he wants, yet he will never produce anything new. He can analyze and make distinct the concept of a straight line, or of an angle, or of the number three, but he will not come upon any other properties that do not already lie in these concepts. But now let the geometer take up this question. He begins at once to construct a triangle. Since he knows that two right angles together are exactly equal to all of the adjacent angles that can be drawn at one point on a straight line, he extends one side of his triangle, and obtains two adjacent angles that together are equal to two right ones. Now he divides the external one of these angles by drawing a line parallel to the opposite side of the triangle, and sees that here there arises an external adjacent angle which is equal to an internal one, etc. In such a way, through a chain of inferences that is always guided by intuition, he arrives at a fully illuminating and at the same time general solution of the question.”
Critique Of Pure Reason A716/B744
The above figure illustrates the mathematical content of the quoted passage. Kant’s proof of the angle sum property of triangles is simply Euclid’s proof. The ray CD is parallel to the line segment BC and the ray CE is parallel to the line segment BA. Of course, the proof that the angle between the line segment AC and the ray CE is equivalent to the parallel postulate.
Kant of course does not mean to supply a ‘synthetic’ proof of the parallel postulate (he isn’t Hobbes). I would recommend Cassier’s book Einstein’s Theory Of Relativity for more on that front. Kant is highlighting a different aspect of this proof. You see, though it is characteristic of “modern philosophy” to contrast the philosopher with the scientist to the philosopher’s detriment, it is uncharacteristic of Kant to be so sarcastic. From whence this sauciness?
David Hume
This sauciness is typical of Hume. I attempted to quote at length, but really one should just read Treatise On Human Nature, Book I, Part III, Section I.
Kant in the quoted passage attempts to show the superiority of his system by developing how geometry can be as precise a science as it seems to be. Kant flips Hume’s sarcasm against himself: the philosopher goes from being an errent Scholast to being Hume himself. If Hume cannot reason about a chiligon because of a defect of his eyes, then he will find those issues already in a trigon. Tobias Dantzig (George Dantzig’s father) in Henri Poincaré , Critic Of Crisis puts the issue:
“How little reliance can be placed on these direct perceptions was forcefully brought home to me some years ago, when I was attending a dinner of executives and engineers of a large ball bearing corporation. Some practical joker produced a steel ball and passed it among those present with the request that each of us record his estimate on a card provided for the occasion. Now, most of those present had been handling steel balls for a great number of years; still, the estimates ran all the way from 7/8 of an inch to 1 inch and a quarter, and even the average estimate departed substantially from the measured value, which, we were informed later, was one inch in diameter.”
Coming back to Kant, after that sarcastic interlude, Kant explains symbolic procedures work as well for demonstration as geometric ones. Finally, Kant asks
“What might be the cause of the very different situations in which these two reasoners find themselves, one of whom makes his way in accordance with concepts, the other in accordance with intuitions that he exhibits a priori for the concepts? According to the transcendental fundamental doctrine expounded above, this cause is clear. At issue here are not analytic propositions, which can be generated through mere analysis of concepts (here the philosopher would without doubt have the advantage over his rival), but synthetic ones, and indeed ones that are to be cognized a priori. For I am not to see what I actually think in my concept of a triangle (this is nothing farther than its mere definition), rather I am to go beyond it to properties that do not lie in this concept but still belong to it.”
Critique Of Pure Reason, A717-8/B745-6
Put oversimply and in modern terms: the theoretical interest is not in a triangle as an object in itself, but what it reveals about the space it is embedded. To talk about a chiligon is not to talk about an object alone, but the object as it is in space. At the top we see one example - the triangle inequality shows the triangle is embedded into a metric space at all - and in the sarcastic passage we see another - the angle sum property shows the triangle is embedded into a flat space.
Henri Poincaré
At the top, I said part of the problem in understanding these passages is Kant’s opaque language. But the main problem with interpreting this today is that we approach the problem of axioms differently than Kant. We tend to view axioms and postulates as ‘disguised definitions’ of the relations involved rather than ‘intuitive truths’.
Poincaré puts the issue nicely in Why Space Has Three Dimensions:
“It is therefore possible to use these axioms with the condition that it has been proved that they are not contradictory, and it will be possible to base on them a geometry in which figures will not be needed, and which could be understood by a man who possesses neither sight, nor touch, nor muscular sense, nor any of the senses, and which would be reduced to pure understanding. Yes, this man would probably understand in the sense that he would very well realize that the propositions are logically deduced one from the other; but the collection of these propositions would seem artificial and baroque to him, and he would not understand why this collection is preferred instead of a multitude of other possible collections.
If we do not experience the same astonishment, it is because for us axioms are not really simple definitions and arbitrary conventions, but truly justified conventions. … As to the axioms of order, it seems to me that there is something more; that they are true intuitive propositions, relating to analysis situs. We see that the fact that a point C is between two other points on a line is connected with the method of cutting up a one-dimensional continuum with the aid of cuts formed by impassable points.
…
I shall conclude that there is in all of us an intuitive notion of the continuum of any number of dimensions whatever because we possess the capacity to construct a physical and mathematical continuum; and that this capacity exists in us before any experience because, without it, experience properly speaking would be impossible and would be reduced to brute sensations, unsuitable for any organization; and because this intuition is merely the awareness that we possess this faculty. … And yet this faculty could be used in different ways; it could enable us to construct a space of four just as well as a space of three dimensions. It is the exterior world, it is experience which induces us to make use of it in one sense rather than in the other.”
As can be seen, in this essay, Poincaré ends up siding with Kant. For Kant and Poincaré some aspects of space - the triangle inequality for Kant and that the betweenness relation is an order for Poincaré - are a priori and some are a posterori. But still there is an important question.
Jacques Hadamard
The question concerns the order of explanation: do we accept the order axioms of betweenness because we know geometric betweenness orders points on a line, or vice versa? As Hadamard points out, evidence for the first even reading Hilbert (who is often taken as the exemplar of vice versa): there are diagrams on nearly every page of Foundations Of Geometry. Hilbert himself actually said similar things in Geometry And The Imagination.
But, in my opinion, it’s not really the a prioricity of parts of geometry that bothers people. Rather Kant’s claim is essentially that the explanation of geometry depends on two somewhat mysterious concepts: intuition and synthesis.
Kant is extremely clear about the notion of intuition: ‘All intuitions are extensive magnitudes’, he says. This geometrical idea of theory can be contrasted with the mechanical view prevailing in Kant’s time. Newton before Kant and Maxwell after him were criticized for offering models of the universe which could not have mechanical interpretations.
It is typical of a mind as precise as Kant’s to prefer the geometrical to the mechanical. As the magnificently moustacioed G B Halstead said of Gibbs:
“Professor Gibbs was much inclined to the use of geometrical illustrations, which he employed as symbols and: aids to the imagination, rather than the mechanical models which have served so many great investigators; such models are seldom in complete correspondence with, the phenomena they represent and Professor Gibbs's tendency toward rigorous logic was such that the discrepancies apparently destroyed for him the usefulness of the model. Accordingly he usually had recourse to the geometrical representation of his equations, and this method he used with great ease and power.”
G B Halstead, Josiah Willard Gibbs
Moving back to Kant, each intuition depends on how such a magnitude psychologically effects the agent, and thus all concepts on their function in altering said affections. Nice examples of spatial intuition come from visual perception: we perceive a visual field of continuous magnitudes in space and time despite visual blind spots and blinking.
With respect to mathematics, intuition is complicated, not complex. Thus it must be the notion of synthesis that really trips people up. As is characteristic, Poincaré’s discussion brings this out nicely: if it follows from ‘We can intuit a continuum at all.’ that we can synthesize a continuum of any dimension, then we better know what ‘synthesis’ is.
Going back to the discussion of the angle sum of a triangle above, there is an unnecessary impediment to understanding by not pointing out any properties of the triangle accessible to the philosopher. For instance, the triangle’s self-duality is accessible to the philosopher: the three sides correspond exactly to the three angles.
This runs into the issue which I think most trips up young readers of Kant. We established earlier that Kant’s Copernican revolution has brought him back from Hume to ordinary opinion. This is to be expected: Kant’s system does not create new ways of thinking but reorganizes old ways of thinking. Now Hegel actually did try to innovate new ways of thinking, but not Kant.
Herbert Simon
Perhaps this all seems terribly outdated. I think with regards the parallel postulate, this discussion is in fact old fashioned. Not only has the shock of non-Euclidean geometry long since cooled, but also even as Kant discourse it is terribly dated. Because the parallel postulate was not a major mathematical research interest until there was reason to doubt it, and Kant in particular never discussed it, the a prior parallel postulate interpretation of Kant is an example of what historians of science call “Whig History” in the negative sense.
Rather than comments on the parallel postulate, the above passages quoted concern two senses of universal quantification. The first is negative: no triangle has side lengths a+b<c. The second is positive: all triangles have angles a+b+c=180. Kant is, to oversimplify, arguing we can understand truths of universal quantification by delimiting the background (synthetically) as well as from mere definition (analytically).
But Kant has a third, much more interesting and important understanding of universal quantification. Anyway, to return to the text:
“The affinity of the diverse, notwithstanding the differences existing between its parts, has a relation to things, but a still closer one to the mere properties and powers of things. For example, imperfect experience may represent the orbits of the planets as circular. But we discover variations from this course, and we proceed to suppose that the planets revolve in a path which, if not a circle, is of a character very similar to it. That is to say, the movements of those planets which do not form a circle will approximate more or less to the properties of a circle, and probably form an ellipse. The paths of comets exhibit still greater variations, for, so far as our observation extends, they do not return upon their own course in a circle or ellipse. But we proceed to the conjecture that comets describe a parabola, a figure which is closely allied to the ellipse. In fact, a parabola is merely an ellipse, with its longer axis produced to an indefinite extent. Thus these principles conduct us to a unity in the genera of the forms of these orbits, and, proceeding farther, to a unity as regards the cause of the motions of the heavenly bodies—that is, gravitation. But we go on extending our conquests over nature, and endeavour to explain all seeming deviations from these rules, and even make additions to our system which no experience can ever substantiate—for example, the theory, in affinity with that of ellipses, of hyperbolic paths of comets, pursuing which, these bodies leave our solar system and, passing from sun to sun, unite the most distant parts of the infinite universe, which is held together by the same moving power.
The most remarkable circumstance connected with these principles is that they seem to be transcendental, and, although only containing ideas for the guidance of the empirical exercise of reason, and although this empirical employment stands to these ideas in an asymptotic relation alone (to use a mathematical term), that is, continually approximate, without ever being able to attain to them, they possess, notwithstanding, as à priori synthetical propositions, objective though undetermined validity, and are available as rules for possible experience. In the elaboration of our experience, they may also be employed with great advantage, as heuristic principles. A transcendental deduction of them cannot be made; such a deduction being always impossible in the case of ideas, as has been already shown.”
Critique Of Pure Reason, A661-2,B690-1
You see type Kant calls Newton’s methods “heuristic”, meaning “for discovery” and ultimately from the Greek “εὑρίσκω”. This passage is, in fact, the source of the term heuristic in the modern sense.
Kant’s interest in Newtonian astronomy drew on Kant’s deep philosophical analysis of the methods of conic sections - what we now call ‘algebraic geometry’. Kant was fascinated by the way the theory of conic sections could be developed referencing only the special case of the circle and extended by a special limiting process.
‘All conic sections are in one family, bound together by invisible but continuous parameters. This is not a method of proof, but of discovery.’. Though mathematicians still often speak this way, as a theory this is an appropriate 18th century concern, thus have exorcised the problem of Whig History.
But there is something we have done much stronger than this, we have the key to one of the most important transitionary passages in Kant, where he begins to endorse biological evolution as a metaphysical research program. Eventually this would lead Kant to his mature political, ethicaland anthropolgical position in Perpetual Peace, Metaphysics of Morals and Anthropology from a Pragmatic Point of View. Let us close this piece giving this most important passage by Kant the last word:
“The same [dialectic between diversity and unity] is the case with the so-called law discovered by Leibnitz, and supported with remarkable ability by Bonnet—the law of the continuous gradation of created beings, which is nothing more than an inference from the principle of affinity; for observation and study of the order of nature could never present it to the mind as an objective truth. The steps of this ladder, as they appear in experience, are too far apart from each other, and the so-called petty differences between different kinds of animals are in nature commonly so wide separations that no confidence can be placed in such views (particularly when we reflect on the great variety of things, and the ease with which we can discover resemblances), and no faith in the laws which are said to express the aims and purposes of nature. On the other hand, the method of investigating the order of nature in the light of this principle, and the maxim which requires us to regard this order—it being still undetermined how far it extends—as really existing in nature, is beyond doubt a legitimate and excellent principle of reason—a principle which extends farther than any experience or observation of ours and which, without giving us any positive knowledge of anything in the region of experience, guides us to the goal of systematic unity.”
Critique Of Pure Reason, A668/B696