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Sunday, July 24, 2011


At this point, I must talk for a bit about Euclidean Geometry. When I was a boy, we actually studied Euclid's Elements in High School geometry class. [Just to give you some idea of how things have changed, the Chairman of the Math Department at Forest Hills High School, Dr. Frank, taught a special class in the morning, before school started, for a handful of us who were whizzes at math, in which he introduced us to the mysteries of Analytic or Cartesian Geometry -- x and y axes and the formula for a circle or parabola and that stuff. These days, I gather that is taught in kindergarten.] Euclid was the gold standard for math in Kant's day, and everyone was fully conversant with the definitions, axioms, postulates, and theorems in Euclid's Elements.

The dream of Leibniz, and of a great many famous philosophers and logicians since, was to derive all of mathematics from logic, thereby demonstrating that the propositions of mathematics, like those of logic, can be known with absolute certainty a priori. Kant was convinced that this was in fact false -- that mathematical propositions make assertions that go beyond what is contained in the definitions of the terms with which they are expressed. Thus, he believed, mathematical propositions are synthetic, not analytic. But he was also sure that we know the truth of mathematical propositions a priori, not a posteriori [as Hume actually thought, by the way, although that has nothing to do with this discussion.] This posed a real puzzle for Kant. How could there be propositions that, although synthetic, could be known with certainty, a priori? [The famous Kantian conundrum inaccurately described as "the problem of synthetic a prior propositions."]

Why did Kant think that Euclidean Geometry is NOT analytic deducible from the definitions, axioms, and postulates? Well, take an actual look at the theorems in Euclid's Elements, or at any modern version of the same material. Each Theorem asserts some proposition, about lines or angles or triangles or circles, and somewhere in the proof, usually near the beginning, there is a "construction." Euclid will tell us, for example, to describe a circle about a point [one of the axioms says we can do that]. Then we are to select a point outside the circle, and connect it by a straight line to the point that serves as the center of the circle. [Another axiom says we can always connect two points by a straight line.] Now, we are told, where the line thus produced intersects with the circle, label that point A..

How do we know that the line intersects the circle? What? How do we know? Just look at it! The center of the circle is a point inside the circle, and the point selected is, by construction, outside the circle. Of course a line connecting the two points must cross the circle somewhere. And there you have it. Nothing in the definitions, axioms, and theorems laid down at the beginning of the Elements implies that such a line must intersect the circle, but it is immediately and indubitably obvious that it must. Mind you, this is not a well-established empirical generalization, grounded in endless thousands of attempts to connect points outside circles with centers of those circles. It is immediately apparent to the mind by construction, whether one actually draws such a circle and line or merely considers it in one's mind.

Kant realized that some very powerful explanation was required for this familiar and often overlooked fact about Geometry. His solution was that space itself is not an independently existing "container" of things -- an unding or non-thing, as he rather dismissively characterized Newton's account -- but rather the form of our sensuous perception of things, a form lying ready in the mind that is imposed by the mind on its perceptual experience. When we do Geometry, we are simply spelling out the innate mind-dependent spatial structure of that form of intuition.

A few hasty clarifications for modern readers. First, Kant says little or nothing about algebra, or even about arithmetic. You might think that the mind-dependent perceptual form, time, bears the same relationship to algebra that space does to geometry, but Kant does not go that way. Second, anyone living today will immediately ask, "How do we know that everyone has the same innate forms of intuition? Could they evolve over time? Could they be culturally dependent?" These questions seem never to have occurred to Kant, or to his contemporaries, although a century later they would have occurred to everybody.

So the metaphysics of monads is certain, and knowable a priori, and true of things as they are in themselves -- including substances, God, and all that good stuff. Newtonian physics is not true of things as they are in themselves. It and the geometry on which it is based are true only of things as they appear to the human mind in the space and time that the mind imposes upon its experiences. Could there be other rational beings with different forms of intuition? Yup, though Kant is not really interested in that possibility [space travel was a long time in the future.]

Now, we may imagine some thoughtful reader of the Dissertation asking: I can see how we can known the truths of mathematics a priori inasmuch as they merely spell out the mind-dependent spatial form imposed by the mind on its perceptual experience, but how can I know the truths of metaphysics a priori, considering that they are asserted unconditionally and universally of things as they are in themselves?

Well you might ask, little grasshopper.

Almost immediately after delivering the Dissertation [and getting tenure -- a rare commodity in those days], Kant realized that he had no answer at all to this pressing question. How indeed is it possible to have knowledge a priori of the independently real? Kant concluded that it was in fact impossible, and that he would therefore have to give up for all time the ancient search for metaphysical knowledge. So much for Rational Theology, which had for two thousand years and more been the Queen of the Philosophical Disciplines, and for Leibnizean metaphysics, besides.

This was, as by now should be clear, a huge decision on Kant's part, a dramatic tilt in the direction of the position of the Empiricists, the Sceptics. Kant announced this break with the philosophical tradition in which he had been raised in a letter to his friend Marcus Herz, who had occupied the ceremonial position of Respondent to Kant's Inaugural lecture. Kant promised Herz that very shortly he would publish a book entitled "A Critique of Reason," in which he would present his new position to the world. But then, Kant was struck by a thunderbolt that transformed his life, his thought, and our entire philosophical tradition.

It came about like this. An irritating little man named James Beattie published in England an attack on people he considered rank heretics and sceptics -- among whom he included David Hume. The book, called An Essay on the Nature and Immutability of the Truth, was a series of "refutations" of such famous sceptics as Descartes, Locke, and Hume, and it appeared in 1770. It consisted of arguments roughly like this: "X says that y. But common sense tells us that not-y. So X is wrong." Naturally, it was a smash success, and went through annual editions in 1770, 71, 72, 73, 74, and 75. But, God bless Beattie, he included in his book lengthy extracts from the sceptics he thought he was eviscerating. Hume, who cared very deeply for his own literary reputation, and who was by now actually quite famous as the author of a six volume history of England, was stung by Beattie's contemptuous dismissal of his anonymous, juvenile work, A Treatise of Human Nature [sigh -- the greatest work of philosophy ever written in English]. In a new edition of his essays brought out in '72, he disavowed the Treatise as a work of youth and took umbrage at Beattie's extensive quotations from it. Beattie replied rather grandly by removing from the '73 edition and all subsequent ones the passages from the Treatise that he had included in the original edition, including passages in which Hume stated his devastating critique of causal judgments.

Kant could not really read English, but a translation of Beattie's work appeared in German in 1772, and thanks to the goodness of the gods [whose existence Kant had given up any hope of proving], the translator used the 1st edition, with the passages from Hume intact. Kant read the translation, and being of course light years smarter than Beattie or anyone else then alive save Hume, immediately recognized that Hume's arguments constituted a mortal threat to the new position he had just then taken up as a necessary retreat from metaphysics. Hume's arguments called into question even the knowability a priori [or indeed any other way] of Newtonian physics, which Kant thought he had made safe by restricting it to the realm of things as they appear to us in space and time. Clearly, a much, much deeper and more elaborate defense of science was required. Kant set aside his plans for the immediate release of a Critique of Reason and embarked upon the labors that resulted, nine years later, in the Critique of Pure Reason and the rest of the Critical Philosophy.

Why have I told you this story in such detail? Because I am the person who discovered the whole Beattie business fifty-five years ago. It is the only genuine scholarship I have ever done in my entire life, and I am inordinately proud of it. You may find it all laid out in glorious detail in the Journal of the History of Ideas, in an article entitled "Kant's Knowledge of Hume via Beattie."


Michael said...

After reading your post, I decided to see if an online version of your article is available. Turns out it is (under the title "Kant's Debt to Hume via Beattie") at

Chris said...

I must say I rather regret slacking off in HS and entirely blowing off my geometry course. Is Euclids Elements approachable for a basic virgin to geometry to begin restudying the matter?

Also I was reading some of Beattie and was quite impressed and horrified at one aspect I found in his book. Horrified that Hume was such an irrational racist, and impressed that Beattie rather took him to task for it!

Robert Paul Wolff said...

Quite true about Hume [and Kant, as it happens].

Don't bother with Euclid [save for antiquarian purposes.] Much better to study modern analytic geometry, which is quite elegant.

summortus said...

This may be silly (as maybe I am just pushing the intuitiv obviousness back a step), but it seems to me one can prove that a line from the center of a circle to a point outside the circle will intersect with the circle from any reasonably strong set of geometric axioms (I don't have Euclid's handy...).

If a circle is the set of points equidistant from the center, then we know that any point of distance r (=radius) from the center will be on the circle.

Now, by hypothesis, the line connecting the center of the circle to the point outside the circle has length more than r. And by the definition of a line segment (which I would loosely take to be a continuous subset of the points (x,y) that satisfy "y=ax+b" for a,b=constant, there'll be some point on that line segment which is length r from the center -- the continuity condition here being crucial, as it guarantees that if (letting x,y for the center be 0,0, is the x1,y1 such that the distance measure between the center and x1,y1 is greater than r, there'll be a x2,y2 with a distance measure equal to r

Thus, by the definition of a line segment and the definition of a circle and the stipulation that the point is outside the circle, you can analytically prove that the line and the circle intersect...

I think, in my experience, all of the experiences you're describing as "just look at it" in math either a) are false and illusory or b) can actually be rigidly and analytically proved. But I may be missing something...

Marinus said...

Summortus - it's one of those peculiar discoveries that pure mathematics allows us that in fact you can't prove that from merely the definition of line and circle - you also need something called the axiom of choice. This is because, since the amount of points on the circle is infinite, it is by no means obvious that you are able to select the point of intersection between the circle and the line. To use a beautiful analogy from Bertrand Russell, to do so requires the same skill as being able to choose a matching pair of socks out of a collection of infinitely many. Even if there is a sock to match the one you have already chosen, it is by no means certain that you'll pick its partner out in a finitely long time (which is the same as saying that perhaps you'll never find it). This is like the line intersecting the circle case where you have the point you suspect the line is equal to some x^2.y^2 = r^2 (one sock of a pair) and are looking for the point where the circle is equal to that same line (the second sock).

The axiom of choice is commonly assumed uncritically by mathematicians, but it is by no means obviously true.

Marinus said...

Chris - Contrary to Prof Wolff, I heartily recommend you study Euclid, if you at all have the opportunity. It is at times breathtakingly beautiful in its proofs and results, but, more importantly, it is probably also the most influential piece of technical writing in the Western tradition. For centuries Euclid's series of deductive proofs from self-evident axioms was taken as the gold standard for scientific explanation - a look at almost any page of Descartes or Hobbes will make that clear. Also, not everybody has gotten the memo yet about the painful lessons we learnt with logical positivism, that such a project could never describe our empirical sciences. Euclid's model and the issues that arise from it (basically, it's either tautologies or false) is an excellent object lesson in perhaps the most seductive method of investigation and its failure.

summortus said...

Yes, but that is not a problem of intuition! You might not like the axiom (and in this case, by the way, it's still not clear to me why the axiom of choice is necessary, but my ignorance is another question -- still, who said I needed to select the point -- I just know it's there, and then I say whereever it is, let's call it A), but the proof would still follow analytically from the definition, just with one extra axiom tossed in, no?

I thought the point of the Kant is to show us you need something more than than axioms, definitions, and what is clearly contained within them analytically. The axiom of choice cannot be the entry point for the synthetic here, for the simple reason that one can name it as a na axiom, stipulate it, and then ignore it.

J.Vlasits said...

I'm not sure that the Axiom of Choice is actually necessary here because I think you can run the proof so that the point of intersection is an algebraic number, in which case you will only be dealing with the countably infinite (an algebraic number is the root of a polynomial, e.g. squareroot(2). Cantor proved that these must be countable so the cardinality of the rationals plus the algebraics is also countable. It is only when you add the transcendental numbers like pi and e that you will reach uncountability.) I'm pretty sure that Tarski's axiomatization of Euclidean geometry does not assume the axiom of choice, although it does make assumptions about continuity.

Robert Paul Wolff said...

You are right that Tarski's axiomatization does not assume the Axiom of Choice. Back in 1952, when I was a senior at Harvard, I was actually told to do my honors thesis on his monograph [by Hao Wang, a wonderful but rather incomprehensible logician.] I bombed out and instead wrote on Ryle's CONCEPT OF MIND, after which it was downhill for me all the way to political philosophy -- The Rake's Progress.

Michael said...

Even if you're only dealing with countable sets, some kind of choice principle is needed. There is an axiom of countable choice ("Every *countable* collection of nonempty sets has a choice function") but this, like its stronger counterpart, is independent of the rest of the axioms of set theory. (And of course, it wasn't on Euclid's or Kant's radar at all.)

J.Vlasits said...

The Axiom of Countable Choice, although independent of ZF, does not as intuitively unobvious as AC because it does not entail things like the Banach-Tarski Theorem or create problems for Lebesgue measurability. I think that Tarski's axiomatization of geometry, however, still does not assume even weaker versions of AC. My point was that by setting up the problem correctly, you will be able to prove it without a lot of set theoretical machinery. Exactly how much is a question that I don't know how to answer.