of Kant's Geometry

ἀγεωμέτρητος μὴ εἰσίτω

Without Geometry, Enter Not.The sign over the door of Plato's Academy -- according to John Philoponus,

In Aristotelis De anima libros commentaria, Commentaria in Aristotelem graeca, XV, ed. M. Hayduck, Berlin, 1897, p.117,27.

Though there never were a circle or triangle in nature, the truths demonstrated by Euclid would for ever retain their certainty and evidence.

David Hume,

An Enquiry Concerning Human Understanding, Section IV, Part I, L.A. Shelby-Bigge, editor, Oxford University Press, 1902, 1972, p. 25.

To solve the problem of what is mathematical truth, Poincaré said, we should first ask ourselves what is the nature of geometric axioms. Are they synthetic

a priori, as Kant said? That is, do they exist as a fixed part of man's consciousness, independently of experience and uncreated by experience? Poincaré thought not. They would then impose themselves upon us with such force that we couldn't conceive the contrary proposition, or build upon it a theoretic edifice. There would be no non-Euclidian [sic] geometry.Robert M. Pirsig,

Zen and the Art of Motorcycle Maintenance, Bantam, 1974, p. 236.

As I read Kant, he does not say non-Euclidean geometry is logically impossible, but that is only because he does not claim that any geometry is logically true; geometry in his view is synthetic, not analytic. And Kant's belief that Euclidean geometry was true, because our intuitions tell us so, seems to me to be either unintelligible or wrong.

Jeremy Gray,

Ideas of Space Euclidean, Non-Euclidean, and Relativistic, Oxford, 1989, p. 85.

The fact that Euclidean geometry seems so accurately to reflect the structure of the 'space' of our world has fooled us (or our ancestors!) into thinking that this geometry is a logical necessity, or into thinking that we have an innate

a prioriintuitive grasp that Euclidean geometrymustapply to the world in which we live. (Even the great philosopher Immanuel Kant claimed this.) This real break with Euclidean geometry only came with Einstein's general relativity, which was put forward many years later. Far from Euclidean geometry being a logical necessity, it is anempirical observational factthat this geometry applies so accurately -- though not quite exactly -- to the structure of our physical space! Euclidean geometry was indeed, all along, a (SUPERB)physicaltheory. This was in addition to its being an elegant and logical piece of pure mathematics.Roger Penrose,

The Emperor's New Mind, Oxford University Press, 1990; Oxford Landmark Series, 2016, pp.204-205.

...Einstein's relativistic physics refuted Kant's claim that Euclidean geometry expresses synthetic

a prioriknowledge of space, thereby not only depriving Kant of an account of geometrical knowledge, but also, and more importantly, putting his entire account of synthetica prioriknowledge under a cloud of suspicion.Jerrold Katz,

The Metaphysics of Meaning, A Bradford Book, the MIT Press, 1990, p.292; an unfortunate confusion in a good philosopher.

I suppose infinity always dazzles us, and I have never been able to build up a good intuition about the concept. The problem is compounded here because there are actually two infinities competing with each other: there is the infinite volume of space, and there is the infinite shrinkage, or compression, represented by the big bang singularity. However much you shrink an infinite space, it is still infinite. On the other hand, any finite region within infinite space, however large, can be compressed to a single point at the big bang. There is no conflict between the two infinities so long as you specify just what it is that you are talking about.

Well, I can say all this in words, and I know I can make mathematical sense of it, but I confess that to this day I cannot visualize it.

Paul Davies and John Gribbin,

The Matter Myth, Touchstone, 1992, p.108.

Conventional wisdom says that the universe is infinite...

The question of a finite or infinite universe is one of the oldest in philosophy. A common misconception is that it has already been settled in favor of the latter...

One problem with the conclusion is that the universe could be spherical yet so large that the observable part seems Euclidean, just as a small patch of the earth's surface looks flat [a common idea in "inflationary" theories]. A broader issue, however, is that relativity is a purely local theory [!]. It predicts the curvature of each small volume of space -- its geometry -- based on the matter and energy it contains. Neither relativity nor standard cosmological observations say anything about how those volumes fit together to give the universe its overall shape -- its topology.

Jean-Pierre Luminet, Glenn D. Starkman, & Jeffrey R. Weeks, "Is Space Finite?"

Scientific American, April 1999, pp.90-92.

Kant's own claim about geometry came to grief: soon after he made it, Riemann discovered non-Euclidean geometries, and some one hundred years later, Einstein showed that physical space was in fact non-Euclidean.

Paul Boghossian,

Fear of Knowledge, Clarendon Press, Oxford, 2006, p.40.

Impressed by the beauty and success of Euclidean geometry, philosophers -- most notably Immanuel Kant -- tried to elevate its assumptions to the status of metaphysical Truths. Geometry that fails to follow Euclid's assumptions is, according to Kant, literally inconceivable.

Frank Wilczek, "Wilczek's Universe: No, Truth Isn't Dead,"

The Wall Street Journal, June 24-25, 2017, p.C4.

The human brain is wired in such a way that we simply cannot imagine curved spaces of dimension great than two; we can only access them through mathematics.

Edward Frenkel,

Love and Math, The Heart of Hidden Reality, Basic Books, 2013, p.2.

Revolutions in geometry in the nineteenth century had shattered Enlightenment philosopher Immanuel Kant's idea that Euclidean space is the form of all possible experience, and with it his idea of a kind of

a prioriknowledge that is not derived or acquired from experience. [Ernst] Mach stepped in with an alternative: there is no need to posit anything that sensory experience does not confirm or predict.Clare Mac Cumhaill & Rachel Wiseman,

Metaphysical Animals, How Four Women Brought Philosophy Back to Life, Doubleday, 2022, p.44.

If we can show that the denial of a proposition does not contradict the consequences of certain other propositions, we have then found a criterion of the logical independence of the proposition in question. In other words, the logical independence of this Euclidean axiom [the Parallel Postulate] of the other axioms would be proved if it could be proven that a geometry free of contradictions could be erected which differed from Euclidean geometry in the fact, and only in the fact, that in the place of the parallel axiom there stood its negation. That is just what Gauss, Lobachevski, and Bolyai established: the possibility of erecting such a noncontradictory geometry which is different from the Euclidean...

What is important to us here is this: The results of modern axiomatics are a completely clear and compelling corroboration of Kant's and Fries's assertion of the limits of logic in the field of mathematical knowledge, and they are conclusive proof of the doctrine of the "synthetic" character of the mathematical axioms. For it is proved that the negation of one axiom can lead to no contradiction even when the other axioms are introduced... And this was just the criterion that Kant had already specified for the synthetic character of a judgment: the uncontradictory character of its negation.

Leonard Nelson, "Philosophy and Axiomatics," 1927,

Socratic Method and Critical Philosophy, Yale, 1949, Dover 1965, pp.163-164.

The fifth book of the Bible is called "Deuteronomy," from Greek Δευτερονόμιον (Latin *Deuteronomium*, or *Liber Deuteronomii*), the "Second Law," or the "Repetition of the Law." In Hebrew, this was just , *D ^{e}bhārîm*, "Words," i.e. the words of Moses, which is from the first verse of the book. The translators of the Septuagint must have decided that a more informative title was in order, with the implication that it might be a bit redundant.

The theory of geometry of Immanuel Kant has been discussed extensively in these pages. It doesn't seem to have made much difference in philosophical, scientific, or documentary culture, where the same confusions and canards keep getting repeated. So I'll do some repeating myself and restate the terms of Kant's theory and the misconceptions involved in most of the critical remarks about it. Some of these confusions involve things about which the authors, often classical or academic philosophers, really should know better. Others are even failures of logic which could have been corrected by writers attending to their own reasoning.

The procedure here will be to discuss, one by one, the eleven quotations listed above (apart from the reference of John Philoponus to the sign over the door of Plato's Academy). These all have been featured elsewhere at this site, although not all have been individually and independently discussed. Mostly they involve some misunderstanding of Kant or false ideas about the issues involved in geometry.

For a good start and a hook for some background, we have a statement by David Hume. Most of the things for which Kant is wrongly accused are actually true for Hume, although this is rarely noted, probably because of the bias of Anglo-American philosophy for Hume. Thus, we find a combination of misunderstanding, embarrassment, and cover-up in the treatment of Hume by academic philosophers and others -- as I have noted under the main treatment of Hume. So Hume is left out of discussions of geometry, while what were actually his views end up paradoxically, even bizarrely, attributed to Kant:

Though there never were a circle or triangle in nature, the truths demonstrated by Euclid would for ever retain their certainty and evidence. [An Enquiry Concerning Human Understanding, Section IV, Part I, L.A. Shelby-Bigge, editor, Oxford University Press, 1902, 1972, p. 25.]

Hume is a very traditional philosopher here, and not a Skeptic in this matter, since he regards the axioms of geometry as self-evident, with truth entirely independent of experience, and even of the world. This characterization had been used since Aristotle; but Hume gives us an updated version of it. To Hume, the axioms of geometry are true as "relations of ideas," which means they make themselves true because, on Hume's own original criterion, they cannot be denied without generating a logical contradiction. Kant accepts this principle but provides a new and classic terminology. Propositions that cannot be denied without contradiction are true "analytically." So a simple logical criterion is all we need to evaluate "analytic truth." Frege proposed that the meaning of an analytic proposition be extended to all its logical consequences. This idea ended up collapsing, as explained by Jerrold Katz.

There had always been, however, some kinds of *difficulties* with the self-evidence of the axioms of geometry in Euclid. One is that they are not all called "axioms." Euclidean geometry begins with definitions, and then five "axioms" [ἀξιώματα, singular ἀξίωμα] and five "postulates" [αἰτήματα, singular αἴτημα -- related to αἰτία, "cause, charge, accusation, occasion, opportunity" -- the term for Aristotle's five "causes"]. We get a discussion about the meaning of "postulates" in Aristotle:

That which is in itself necessarily true and must be thought to be so [δοκεῖν ἀνάγκη, "thought necessary"] is not a hypothesis [ὑπόθεσις, plural ὑποθέσεις] nor a postulate [αἴτημα]; for demonstration [ἀπόδειξις], like syllogism [συλλογισμός], is concerned not with external but with internal [ἐν τῇ ψυχῇ, "in the soul"] discourse [λόγος]; and it is always possible to object to the former, but not always possible to do with to the latter. Thus any provable [nota bene-- δεικτὰ ὄντα, "things to be shown"] proposition that a teacher assumes without proving it, if the student accepts it, is a hypothesis -- a hypothesis not absolutely but relatively to the student; but the same assumption, if it is made when the student has no opinion or a contrary opinion about it, is a postulate. This is the difference between a hypothesis and a postulate; the latter is the contrary of the student's opinion, or any provable [sic] proposition that is assumed and used without being proved. [Posterior Analytics, I,x,76b,25-34; Loeb Classical library,Posterior Analytics and Topica, translated by Hugh Tredennick, Harvard University Press, 1960, 1966, pp.70-73]

If Euclid, who lived under the Ptolemies, was following Aristotle's definitions, then the Postulates of geometry are not "necessarily true" and need not "be thought necessary." But, note, he says they may be "provable." Whether they are even "postulates" and not "hypotheses" simply depends on the attitude of the student. But this is not a distinction that one really ever finds in subsequent geometry, probably because it involves no real logical difference. It is irrelevant for truth whether students believe a postulate or not, and this will be different with different students (as Socrates points out to Euthyphro that the gods apparently disagree, according to him, about what is just, beautiful, good, and pious).

So a "postulate" has, since all this, just been something we assume ("postulate"), and mathematicians do this frequently, just to see what happens. It is also part of scientific method, where something is assumed as part of a scientific theory, like Einstein's Equivalence Principle, and then we see if this leads to predictions that can be confirmed or falsified by observation.

We can, however, take it that "that which is in itself necessarily true and must be thought to be so" is going to be an "axiom." At the same time, all axioms, postulates, and hypotheses *may* fit Aristotle's definition of "first principles," ἀρχαί, *archaí* -- where the English expression shares a reduplication introduced in Latin, as *prima principia*. Whether postulates or hypotheses are first principles depends on whether they are actually provable [δεικτά] or not:

I call "first principles" [ἀρχαί, singlular ἀρχή] in each genus [γένος] those facts which cannot be proved [μὴ, "not," ἐνδέχεται, "is possible," δεῖξαι, "to prove"]. [x,76a,32-33]

Thus, we have a situation in Euclid where the Postulates are neither proven nor are thought to be necessarily true. This is a complication, overlooked by Hume, that is not well explained in the history of geometry. And it leaves things rather dangling, especially if we think that perhaps the Postulates are provable, after all -- which we might think after reading Aristotle. As a mathematician, Euclid, after a fashion, doesn't even need to care. But it is something that later mathematicians and philosophers will worry about.

The situation is made worse by the awkwardness of the Fifth Postulate, which is a complicated business about parallel lines, complicated enough that parallel lines are not even mentioned in Euclid's formulation. Restated by David Hilbert, it is a little clearer:

Letabe any line andAa point not on it. Then there is at most one line in the plane, determined byaandA, that passes throughAand does not intersecta.

With a line and a separate point, there is only one line through the point on the plane that is parallel to the given line. Simple enough. But it was a bad headache in the history of geometry. Given the ambiguity of the state of the Postulates, and the complexity of the Fifth Postulate, most mathematicians figured it was something that should be provable. And everyone tried, without success, over centuries. They kept trying, not because they thought the Postulate false, but because it did not seem self-evident and because there was *no other way* to construe Postulates as true but not self-evident or provable.

There is more to that story, which I have related elsewhere. For my purposes here, we should get on to the the next point.

Kant decided that Hume was wrong. The Postulates of Euclid, and perhaps the Axioms also, are not "relations of ideas." They can be denied without contradiction. Hume had a category for this, "matters of fact," and Kant introduced a new term: Propositions that can be denied without generating a contradiction are "synthetic"; and so we have a complementary logical criterion for "synthetic truth," which is that both the affirmation and denial of a proposition are *logically possible*. In terms of logic alone, they are equally likely to be true.

This clarifies the situation with the Postulates in Euclid. Whether or not the Axioms are self-evident, the Postulates now clearly, as *synthetic first principles*, are going to be *neither self-evident nor provable*. So what then makes them true? And how do we know they are? Aye, there's the rub.

We can approach an answer to that, with Kant, by asking about the method of the Greek geometers. The Pythagoreans, Plato, and then Euclid did not go out and observe space in nature. This wouldn't have done them much good anyway.

Laurent de La Hyre (1606-1656), Allegory of Geometry, 1649, Fine Art Museums of San Francisco, (Palace of the) Legion of Honor;one of the seven Liberal Arts |
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So what Greek geometers consulted, like their modern counterparts, was their imagination. It was not just Kant who called this "pure mathematics." We can visualize space, wherever we are and whatever we are looking at. The points and lines we imagine are subject to the definitions we have conceived, that the former are without extension and the latter with it only in one dimension. When the Empiricists like Hobbes, Berkeley, and Hume denied that there are such points or lines, they were in effect saying that we have no imaginations, that imagination can only reproduce things in perceptual terms, or that our imaginations are not probative or evidentiary.

But none of them were geometers, or even mathematicians. And they don't seem to have actually *known* any mathematicians, or respected their views if they did (as apparently Jim Holt, a "science writer," does not respect the views of the mathematician Edward Frenkel). It is said that some early Behaviorist psychologists were without visual imagination -- no dreaming of returning to Mandalay for them -- this is a genuine genetic mental condition, now called "aphantasia" -- and so were not troubled by their denial of the existence of internal mental states; but with geometry, it is hard to know what they or the Empiricists are saying that geometry would be about, since almost nothing of Euclid, which we have seen Hume warmly endorse, could exist on the terms that they demand for it. Indeed, elsewhere Hume says,

No priestlydogmas, invented on purpose to tame and subdue the rebellious reason of mankind, ever shocked common sense more than the doctrine of the infinite divisibility of extension, with its consequences; as they are pompously displayed by all geometricians and metaphysicians, with a kind of triumph and exultation. [ibid.p. 156]

Little does the young geometry student know, on being introduced to the definitions of the discipline, that he is being corrupted by the equivalent of "priestly dogmas," whose consequences, of course, are all the rest of geometry; but perhaps Hume does not realize that the infinite divisibility of extension, down to the extensionless point, is not just an essential feature of geometry but of the infinitesimal calculus and Newtonian physics, with whose terms Hume does not seem well acquainted (he still says "impetus" instead of "inertia"), but does express a kind of helpless admiration and endorsement -- like any of us who cannot do the math.

Our visual imagination of space Kant calls "pure intuition." And although it is really hard to know how we could function without it (as Hume says we cannot function without causality -- a point missed by many, perhaps even most, philosophers referring to Hume), whatever the Empiricists and Behaviorists say. Philosophers writing about Kant frequently have some difficulty understanding him and crediting the existence of their own imagination. We must ask them what Euclid did consult when constructing his system of geometry. He made diagrams, to be sure, but the objects of inquiry cannot be what is visible in diagrams, since their features are always imprecise and distorted -- especially since the Greeks often drew in sand, as we find Archimedes doing when he was killed by a Roman solider. In fact, the precise features of diagrams are irrelevant, except in so far as they illustrate the objects that are truly considered, whose abstraction disqualifies them from concrete presence in the world.

Our imagination also provides a venue for experiments, which might otherwise be called "thought" experiments, except that somewhat more than thought is involved. If the Axioms and Postulates of Euclid are constructed in the visual imagination, then such changes as are made to produce non-Euclidean geometries, if not constructed in the visual imagination, can perhaps at least be tested there. We shall see.

The next quote to consider, after Hume, is from a novel, *Zen and the Art of Motorcycle Maintenance*, by Robert M. Pirsig. Although Pirsig was not an academic philosopher, I was introduced to him by Paul Woodruff at the University of Texas in a graduate seminar on Aristotle, because Woodruff read the description by Pirsig's narrator of a graduate seminar in philosophy that he had attended at the University of Chicago. The room there was ugly and the experience did not sound enjoyable. I always suspected that seminars were ways for professors to get graduate students to do their research for them, without credit -- although in a seminar I attended on Hegel with Robert Solomon, he would have been well advised to solicit more criticism than he got. So Pirsig seemed to have some experience with academic philosophy and certainly had done extensive reading. The mathematician Jules Henri Poincaré (1854-1912) is not just someone that everyone knows about. Pirsig was persuaded by Poincaré's evaluation of Kant:

To solve the problem of what is mathematical truth, Poincaré said, we should first ask ourselves what is the nature of geometric axioms. Are they synthetica priori, as Kant said? That is, do they exist as a fixed part of man's consciousness, independently of experience and uncreated by experience? Poincaré thought not. They would then impose themselves upon us with such force that we couldn't conceive the contrary proposition, or build upon it a theoretic edifice. There would be no non-Euclidian [sic] geometry. [Bantam, 1974, p. 236.]

The idea, of course, that "we couldn't conceive the contrary proposition" is to contradict Kant's very own definition of a synthetic proposition. This is not unusual in treatments of Kant. Poincaré and Pirsig seem to think that adding the term *a priori* simply undoes the previous definition. Or perhaps they have not understood the previous definition.

In any case, Kant necessarily holds that the truth or falsehood of a synthetic proposition depends on the ground to which it refers. If there are synthetic propositions *a priori*, then indeed they must refer to something that is a "fixed part of man's consciousness, independently of experience and uncreated by experience." What this could be is what we must consider, as Poincaré and Pirsig apparently did not. With mathematics, and not just geometry, Kant says that the ground is in our "pure intuition," which involves space for geometry but more general intuitions of quantity for arithmetic [note].

With what "such force" does pure intuition "impose" itself on us? Well, no one doubted Euclidean geometry until the 19th century. That must have involved a fair amount of force. And then there is arithmetic. The modern hope, among those the most enamoured of logic, like Bertrand Russell, was that arithmetic could be derived from pure logic, with analytic criteria of truth, which would refute Kant's claim that arithmetic was synthetic. However, deriving arithmetic from logic didn't work out, and now the foundations of mathematics are found in the axioms of Set Theory, which, if they are not self-evident or provable, can only be synthetic propositions *a priori*. This is a development of the early 20th century, and Poincaré may actually have not lived long enough to see its full development. Logicians do worry, a little, about what makes the axioms of Set Theory true, but they never pause to give Kant any credit. That would open them up to all sorts of things, like actual metaphysics. This may have been something to avoid at all costs, although, as noted, mathematicians actually don't need to care. Just see what happens.

But with Poincaré and Pirsig, we never get that far, since they have failed to understand the more fundamental point. If the axioms of geometry or arithmetic are synthetic propositions, then by definition they can be denied without contradiction. Poincaré and Pirsig ignore this and wish to argue that *somehow* a contradiction would be *inconceivable* on the basis of Kant's theory. Thus, Kant, who is the first person in the history of mathematics or philosophy to suggest that the axioms of geometry could be denied without contradition, is paradoxically interpreted to mean that they cannot be. Poincaré doesn't explain this with any references to Kant, and, indeed, pure intuition, our visual imagination, is not a matter of concepts at all, except in so far as concepts (extension, division, etc.) are applied to it. Content with a mischaracterization of the nature of synthetic propositions, and of Kant's whole theory, Poincaré, as well as Pirsig also, is comfortable stopping there. Perhaps, in turn, this is the very *definition* of "half-baked."

The level of confusion in this continues right through the entire modern debate; and we might begin to wonder if it is a matter, not just of misunderstanding Kant, but of actual bad faith, where Anglo-American philosophers, with a personal and even *irrational* commitment to Empiricism, if not Nihilism, either deceptively or *self-deceptively* reverse the doctrinal positions of Hume and Kant, just because they cannot *imagine* or cannot *tolerate*, morally, how Hume could be wrong and Kant right -- imposing grotesque distortions on the interpretation of both philosophers.

The truth is that Kant's theory of geometry logically results in a *prediction of the existence* of non-Euclidean geometry, not its denial or contradiction. This is not a complicated point. For all the philosophers who did not realize this -- almost all -- we have a serious indictment of 19th and 20th century philosophy, which was so lacking in understanding and perspicacity that some of the fundamental terms of Kant's philosophy were continually and, we might even say, *traditionally* misconstrued, even by self-professed "Kantians."

The next quote has a little bit more going for it. Jeremy Gray, in his *Ideas of Space, Euclidean, Non-Euclidean, and Relativistic*, says:

As I read Kant, he does not say non-Euclidean geometry is logically impossible, but that is only because he does not claim that any geometry is logically true; geometry in his view is synthetic, not analytic. And Kant's belief that Euclidean geometry was true, because our intuitions tell us so, seems to me to be either unintelligible or wrong. [Oxford, 1989, p. 85.]

Although he seems hesitant and uncertain -- which in a way is odd in its own right -- Gray has gotten the idea about synthetic propositions. And he moves on to the next step, which is that our "intuitions" are the ground of Euclidean geometry. Why this would be "unintelligible or wrong," however, seems to me both unintelligible and wrong.

Where does Gray think that the Axioms and Postulates in Euclid came from, and the theorems subsequently constructed? What "told" Euclid that his geometry was true? Perhaps we don't need an imagination to construct the theorems, since they are supposed to follow from the axioms by logic alone, but is that likely? Didn't mathematicians have some notion of what they were looking for or what they might get? Didn't Carl Gauss (1777-1855) say that he always had his results before he was able to prove them? That bespeaks some kind of *method* that is not a function of logic alone -- something we also see in the famous *conjectures* in mathematics, which everyone expects to be true, which may take a very long time to prove, and may also remain unproven. But that is a lesser issue. The Axioms and Postulates of Euclid did not just leap out of the brow of Zeus. If Euclid's "intuitions" didn't tell him anything, as Gray seems to regard a reference to intuitions as "unintelligible or wrong," what did tell him? Gray gives us no clue, and he seems to have overlooked the relevant question to ask, how the Greeks constructed geometrical axioms in the first place. They didn't just make it up.

So let me pause for a moment and consider what our intuitions might be able to tell us. Non-Euclidean geometry was first developed by János Bolyai (1802-1860), Nikolai Lobachevski (1792-1856), and Carl Gauss. Their version involved denying the Parallel Postulate in a certain way, namely that there can be an infinite number of lines through a point that will not intersect a given line in a plane, not just one. Their system had some curious results, for instance that the interior angles of a triangle add up to less than 90^{o}, not the 90^{o} exactly as in Euclidean geometry. This was a curiosity, but no one seemed to get too excited about it.

What came next was perhaps more interesting. An earlier mathematician, Gerolamo (or Girolamo) Saccheri (1667-1733), had already understood that the Parallel Postulate could be denied in another way -- that there could be **no lines** through the point parallel to the given line. He was not trying to construct a non-Euclidean geometry, just to do an indirect proof to prove the Parallel Postulate (as everyone had been trying), which one does by assuming the contradictory of the conclusion and then deriving a contradiction. Through this technique, familiar since the Greeks, Saccheri, perhaps for the first time, introduced into mathematics what would become non-Euclidean axioms.

And notice, by using propositions that are denials of the Fifth Postulate -- that there are infinite parallels or no parallels -- Saccheri is already testing, before the fact, Kant's claim about geometrical axioms. If it was impossible for him to derive contradictions in his proofs, then this would prove Kant correct.

Although in his day non-Euclidean geometry wasn't even a twinkle in anyone's eye, and certainly not his, Saccheri derived one of the most important results in the entire history of non-Euclidean geometry. And it was this: **If there are no parallel lines, then lines are only of finite length.** This an astonishing enough result that Saccheri regarded it as the contradiction for which he was looking in his indirect proof -- we all know that straight geometical lines can be extended indefinitely, by which we can extrapolate to infinite lines. Or do we? Is it an *axiom* of Euclid that such lines are infinite?

Apparently not. So what ultimately followed was the construction by Bernhard Riemann (1826-1866) of a different kind of non-Euclidean geometry, one where there are no parallels and all lines are finite.

But what does this *mean* that all lines are finite? You can't just *run out of line*; and it is not in the least surprising that this possibility would not have occurred to Greek and Mediaeval mathematicians, or Saccheri, consulting their "intuitions" about lines. But there is more to it than that. They were consulting their *definitions*, not just their intuitions. We can see why when we consider something that is undoubtedly an *example* and a *case* of Riemann's non-Euclidean geometry.

**The surface of a sphere**. Here we can see *intuitively* that extending a line, without turning to the right or left, will ultimately result in the line coming around and joining onto itself. That's why it is finite. We don't run out of line, and we don't seem run out of space, or run up against some kind of barrier -- which were the sorts of things the Greeks could consider. What they couldn't consider was this: That such lines are straight. They obviously are not. They follow the curvature of the surface, and they close on themselves to form a circle -- with such circles being called "Great Circles" on a sphere. Thus, the whole development of "spherical trigonometry," which was essential for navigation, never suggested anything like non-Euclidean geometry to anyone.

It is sometimes said that spherical trigonometry was *already* non-Euclidean geometry. However, no one using spherical trigonometry ever thought to claim that the curved lines on a sphere were actually, in some sense, "straight," i.e. geodesics. That idea didn't exist. Also, spherical trigonometry is *not about lines at all*, but about angles. In nagivation, you may want to measure distances on the water, but your examination of the sky is in terms of arc. This is what a sextant measures. The angle of a star above the horizon tells you nothing about the actual distance of the star from, well, anything.

But now the philosophers and mathematicians can object: "But Great Circles *are* straight lines, because they are *geodesics* of the given space, i.e. the surface of the sphere." If we were denizens of Edwin Abbot's "Flatland" (from his book, *Flatland: A Romance of Many Dimensions*, by A. Square, 1884) on the surface of the sphere, we would not know that Great Circles are curved. They would look straight to us, since our visual imagination(!) would not include the *third dimension* of space into which the lines curve.

Fair enough. But there are problems with this. One is in terms of our intuitions. We are not being told that we have no intuitions here, because that would be "unintelligible and wrong," we are instead being instructed to *suspend disbelief* and suppose that what looks like a curved line, even a circle, and is in fact *called* a "circle," is actually, *ex hypothese*, a straight line. So here we have a manifest conflict between the geometry that we can *conceive*, and the geometry that we in fact visualize and imagine, which is not in the least "unintelligible," or perhaps even "wrong."

And there is more. In terms of the internal features of this non-Euclidean space, we have a curious fact. If we take a great circle, which in Flatland seems straight, and we cut out a segment, rejoin the ends, and shorten the line, we immediately get the miraculous result that the line becomes curved, even in its own intrinsic, non-Euclidean geometry. If we continue shorting the line, it shrinks down and down into increasingly small *circles* on the surface. Thus, the circular nature of a Great Circle, which is evident in three dimensions, becomes manifest even in two dimensions as soon as the line is shortened. If it is shortened enough, we can get down to the size of a dinner plate -- although in Flatland this does not provide a surface that we could use for food.

In the geography of the Earth, none of the Parallels of Latitude are actual parallels, because they are (intrinsically) curved, except the Equator, which is the only Parallel that is a Great Circle. The Parallel of Latitude of 89^{o}59' N -- 89 degrees 59 minutes North -- which is going to be one nautical mile (defined as a mean minute of Great Circle arc on the Earth's surface) from the North Pole, makes a nice circle, with that radius, around the North Pole. You could walk it -- in appropriate clothing. This is interesting terminology. The "Parallels" are called "parallels" because they do not intersect, but they don't intersect only because they are all, except for the Equator, curved. This obscures the circumstance that *there are no parallel lines* on the surface of a sphere. This is what makes it a non-Euclidean space.

Thus, despite the substitution of the technical term "geodesic" for "straight," we discover that the whole subject and terminology of non-Euclidean geometry is about *curvature*. The Lobachevskian space previously in question is "negatively curved" (or "hyperbolic"); and the Riemannian space in question is "positively curved" (or "ellipical") Yet calling them "curved" at all rather explodes the conceits and postulates of the business: ** Curvature is what is used to make the space intelligible or comformable to, of all things, our intuitions**. Otherwise, we are being asked to imagine something impossible.

And what do I mean, "impossible"? Let me put it this way. If we imagine a straight line, and we extend it indefinitely, how can it come back around and close on itself, which is what the structure of this Riemannian space requires? *Without*, that is, becoming, to our *intuitions*, manfestly *curved*? It cannot, and will not. What's more, no matter how large we imagine a vast Riemannian geodesic, we can always *scale it down* to the dinner plate, because *all geometry is the same at all scales*. So just losing track of the line in the vastness of our imagination is not going to help, with it unaccountably returning to its origin. We can't lose track of it, as we might lose track of a physical line in the great void of the cosmos.

Thus, it is tempting to imagine that we see a straight line head off into space, and then it goes over some kind of horizon -- perhaps the edge of the observable universe -- and disappears. After a while (like billions or trillions of years), we find the same straight line hitting us in the back of the head. That would be nice work, if you could get it. But it requires that we not imagine what the line does between its disappearance and its reappearance. Since we are not looking through a telescope, but through our "mind's eye," we can imagine all the space we need and cannot pretend that we have somehow "lost" the line in the cosmic distance. The only connection we can possibly make between the disappearance and the reappearance is that the line curves around -- it ceases being straight. Indeed, it turns out to be a kind of "Great Circle."

While a "positively curved" space has the most acute conflicts with our intuitions, representations or models of a "negatively curved" or Lobacheskian space always distort sizes, as in the Escher print at right. The figures shown are all, *ex hypothese*, the same size. But they are not, and if you move them around, they *change* size. This not the way it is supposed to be.

Of course, if we were looking at figures on a sphere that was projected into two dimensions, we would have the same problem. The difference is that we can look at an actual sphere in three dimensions, and the effect disappears. But we *cannot* look at an actual Lobachevskian space where the effect would also disappear. There is no such space available either for our inspection or for our imagination. We are stuck with nothing but the projection or the model.

Yet philosophers and mathematicians made a habit of ignoring this difference, talking about projections and models as though they were the space itself. It is beyond the point of suspending disbelief, it starts becoming confused, if not dishonest. It looks like they want to *refute* Kant, by any means necessary, even if these are misconceived, incoherent, or deceptive.

But for my purposes here, the most dramatic point already occurs with Saccheri. A space of finite lines conflicts, not just with Euclidean geometry, but with what we might even call common sense. And even in its own terms, the fact that geodesics become circles when shortened stands in stark contrast to an infinite Euclidean line, whose character doesn't change, in any respect, if we remove a segment and rejoin the ends. That is also true of lines in Lobachevskian space. An infinite line continues to be infinite, no matter how many finite segments are removed. The effect we get with Great Circles is different in kind.

The importance of the "positively curved" space, of course, and the reason why anyone cares about its bizarre features, is that Einstein used it to turn physics into geometry.

We get a reference to Einstein in a quote from Roger Penrose, in his remarkable and illuminating book, *The Emperor's New Mind*.

The fact that Euclidean geometry seems so accurately to reflect the structure of the 'space' of our world has fooled us (or our ancestors!) into thinking that this geometry is a logical necessity, or into thinking that we have an innatea prioriintuitive grasp that Euclidean geometrymustapply to the world in which we live. (Even the great philosopher Immanuel Kant claimed this.) This real break with Euclidean geometry only came with Einstein's general relativity, which was put forward many years later. Far from Euclidean geometry being a logical necessity, it is anempirical observational factthat this geometry applies so accurately -- though not quite exactly -- to the structure of our physical space! Euclidean geometry was indeed, all along, a (SUPERB)physicaltheory. This was in addition to its being an elegant and logical piece of pure mathematics. [Oxford University Press, 1990; Oxford Landmark Series, 2016, pp.204-205.]

In the history of geometry, of course, as Penrose knows well, geometers did not worry whether geometry reflected "the structure of the 'space' of our world." They weren't looking at the world, and the geometry they constructed, culminating in Euclid, was based on what seemed necessary truths in terms of logic and the manifest character of our spatial imagination. Nobody got "fooled" about anything. If Penrose now wants to say that there is evidence for *physical space* having a different character from the space of logic and human imagination, that's fine; but that isn't exactly what he says. He is wrong if he thinks someone like Kant regards Euclidean geometry as a "logical necessity" -- although Hume did -- and saying that "Euclidean geometry... is an *empirical observational fact*" is wrong enough, in terms of method, that his mathematician's union card should be revoked, or at least impeached. Geometry began, and always remained, even as non-Euclidean geometry, a matter of "pure mathematics," as Penrose ends up saying. He needs to straighten this out.

And Penrose's reference to Kant isn't exactly what Kant says. For Kant, Euclidean geometry necessarily applies to the *phenomenal world* because the phenomenal world is constructed according to the rules of our own minds. There is no "logical necessity" involved -- and if Penrose thinks that Kant believes such a thing, he, as others on this page, has misunderstood the meaning of "synthetic" propositions. But in modern terms, we could say that the phenomenal world reflects the hard wiring of the brain. Does Penrose really want to dispute that? He shouldn't.

I wrote Penrose about this passage, the only one in his book to mention Kant, even parenthetically; and he kindly wrote back, without, however, displaying much understanding of Kant. He said that if the postulates of Euclid are not logically "self-evident" tautologies, then what does it mean to say that they are *a priori*? As we have seen, there is a detailed answer to this; but if even academic philosophers are dismissive of it, or even unfamiliar with it, perhaps we can't really expect Penrose to do better. Indeed, if a reader of Kant does not understand that an *a priori* reference and ground can make a synthetic proposition necessary, then he has missed most of the point of the *Critique of Pure Reason*, since the *a priori* character of causality and other categories relies on the same principle. If Penrose ever realized that his original statement had demonstrated a poor understanding of Kant, he did not admit it to me.

So what happens with Einstein? General Relativity eliminates the idea of the "force" of gravity by substituting a geometry of *curvature*, a Riemannian geometry of positively curved space, which allows the paths of free falling bodies to *look like* they are responding to a gravitation force. But it is actually the curvature of space. Since all these non-Euclidean geodesics intersect (like the Meridians of Longitude), gravity brings all things together. This is an elegant and brilliant idea, and it accomplishes the beautiful union of physics and geometry, as Descartes had accomplished the union of algebra and geometry in Analytic Geometry, or Galileo the union of physics and mathematics -- for which he expected to be rebuked.

But does it mean that space is literally and physically a positively curved Riemannian "manifold"? That depends. To a Positivist, as Stephen Hawking at one point claimed he was (actually speaking to Roger Penrose), we don't care. It doesn't matter what physical space is like, or even if a physical space actually exists. All that matters is that our theories make predictions that we can check by observation. Positivism rejects all metaphysical claims, although Hawking had some difficulty following this consistently. Used consistently, Positivism would say that Einstein's theory is no more than a "device for calculation," the compromise, consistent with Aristotle's regard for mathematics, that was offered to Galileo to protect him from charges of heresy. But no, Galileo insisted *E pur si muove*.

If we want Einstein's theory to describe physical space, then there are some paradoxes. Looking at a sphere, we are told that Great Circles are really *not curved* in terms of non-Euclidean geometry. But looking at the orbits of planets, they obviously *are curved* in terms of our own "intuitions." So the application of non-Euclidean gometry here has the paradoxical effect of taking intuitively curved paths and reinterpreting them to "not really" be curved. This is the opposite of our problem that straight lines extended indefinitely are supposed to close on themselves. Looking at oribits, there is not a problem with them closing on themselves. We want them to.

The other problem turns up in cosmology, which I will consider later. But for the local effects of gravity, an issue is what it is that actually curves. Models for Relativity, which show up in museums, textbooks, or on documentaries, tend to show curved surfaces, usually with cones into which rolling balls fall. However, Einstein's theory is not just of space. Time is added in, to produce a volume of "space-time." But, if we think about it, the curvature of space-time is contributed by time, not by space. It is time that accounts for motion, not space. This is true quite generally. Models of balls rolling around cones and falling in presuppose motion, but there is no motion without time; and time is not actually represented in the model except by the motion of the balls. There is really nothing Relativistic about the balls rolling around in the museums. It looks like an arcade game.

But space-time, which contains a temporal dimension as the surface of cones does not, actually doesn't really display motion either; and it has even been called a 4-D "block" or "crystal," with what we would call "motion" frozen in the path inscribed in the temporal dimension. Time as merely another dimension of space is what the 4-D model dipslays; and, as it happens, this is exactly the way that Einstein and many since have thought of space-time. It has also been widely noted that this is a deeply fatalistic view of time. All that has ever happened and will happen is already frozen in the 4-D block. This eliminates, not only free will and genuine choice -- about which sophisticated physicists and philosophers may not care -- but also all genuine possibility and alternative futures -- something that seems at odds with the randomness generally celebrated as a wonderful and revolutionary feature of quantum mechanics. A realistic view of possibility and futurity can only be retrieved with a theory of multiple and alternative universes -- and so the "many worlds" interpretation of quantum mechanics. Traveling back in time and murdering your grandfather will not kill him in the universe where *you* exist. This contains its own paradoxes and blasts Ockham's Razor through the roof.

In a diagram of space-time, it is the curvature of time that accounts for the displacement of objects in space. I have never seen this considered or discussed by anyone, but it would radically transform the issue of the curvature of space. If space as such (the 3-D version) is not curved in Einstein's theory, then the traditional disputes over geometry are irrelevant. Since I haven't found anyone talking about it, there is no pursuing the matter on this page. But it leads into larger questions.

If we want to extend Einstein's appoach so that it accounts for all the forces of nature, then each force "sees" curvature in its own way; and the approach, as is now popular, of adding microscopic "rolled-up" dimensions doesn't really help when a force, like electromagnetism, has an infinite range just like gravity. A macroscopic dimension for gravity and a microscopic one for electromagnetism seem to miss the point. Forces with infinite ranges both require macroscopic dimensions; for otherwise we would not see the motion that they induce. The whole "rolled-up" dimensions business seemed arbitrary to Richard Feynman. We might consider taking him seriously.

Thus, anything with mass "sees" gravity and follows the curvature that represents motion in a gravitational field. Similarly, anything with electrical charge "sees" electrical charges and follows the curvature that represents motion in an electrical or magnetic field. Anything with the "color" of the strong nuclear force, "sees" color charge and follows the curvature that represents motion in the field of the strong nuclear force. But if time is a dimension just like space, then we have three different "spaces" here. One for mass, one for electrical charge, and one for "color" charge. This is a problem. And alternative universes won't help. Objects responding to gravity and responding to electrical or magnetic charge can be one and the same things, and certainly are visible in one and the same space. It is thus arguable, at the very least, that the existence of different forces in nature precludes the theory that time is a dimension just like the other dimensions of space. The brilliance of Einstein's theory was to eliminate the concept of "force" by substituting geometry. But this may genuinely, without Positivism, end up as no more than a "device for calculation," without an ontological reality behind it. The difference between gravity and electromagnetism is not two different times or two different spaces, but just, after all, two different forces.

Jerrold Katz was a very fine philosopher with excellent ideas and arguments on many issues. However, when it came to Kant and geometry, he shared the unfortunate confusions of many and made no efforts to unpack or reexamine the issues. In *The Metaphysics of Meaning*, he said:

...Einstein's relativistic physics refuted Kant's claim that Euclidean geometry expresses synthetica prioriknowledge of space, thereby not only depriving Kant of an account of geometrical knowledge, but also, and more importantly, putting his entire account of synthetica prioriknowledge under a cloud of suspicion. [A Bradford Book, the MIT Press, 1990, p.292]

First of all, this confuses Einstein's scientific theory in physics, which, on Popper's view of science, can only be falsified, not proven, with Euclidean geometry, which was a product of pure mathematics, based on our visual imagination and logic. Even if Einstein's theory were a metaphysically verified doctrine of physical space and time, this would leave the question open about its relationship to a pure mathematically theory. The Greeks assumed that what was required by pure geometry is what would reflect the character of physical space. They may have been wrong about that, but Katz is even less subtle here than Penrose, who, after some misdirection, sees Euclid as an "elegant and logical piece of pure mathematics." Katz fails to make this distinction, and Penrose indeed does not clarify how something can be "pure mathematics" which he otherwise refers to as a "physical theory." A "physical theory" requires empirical hypotheses, which is not the way that mathematicians have *ever* thought, even now.

As I have noted, according to Kant the application of geometry is restricted to phenomenal reality. Kant does not consider that there might be a *physical space* among things in themselves, whose character differs from Euclidean geometry, but then no one else, apparently, has considered that either. And, as I have discussed, the issue of how the Greeks constructed geometry, based on the manifest character of our visual imagination, is not only ignored but is implicitly contradicted by authors who don't seem to be aware of the questions they fail to consider. They make it sound like Greek geometry is based on something that is just made up out of whole cloth -- a procedure that should have produced axiomatically different geometries from the very beginning. No critical person is going to believe that the 72 translators of the Septuagint sat down and independently produced exactly the same Greek translation of the Torah. So different Greek geometers, coming up with their own axioms, should have produced multiple geometries. It is past coincidence that all their work should have led to the single system of Euclid. But it did.

In their book *The Matter Myth*, Paul Davies and John Gribbin stumble onto an issue of cosmology that arises in the application of General Relativity. Thus, when Relativity was new, and non-Euclidean geometry first introduced in a physical theory, the excitment in philosophy and philosophy of science was quite general, that this would allow for a universe that, in space, was *finite but unbounded*, i.e it would be finite in volume but would not have edges or boundaries to that volume. This would resolve the ancient dilemma of whether space was finite or infinite -- the dilemma of Kant's First Antinomy.

The problem with infinite space was that it was, well, infinite, which is really not something that we can conceive or visualize all at once. So Aristotle decided that, as such, it could not be. The alternative, however, a finite space, meant that it would need to come to an end at a boundary. Generally, this was good enough in Ancient and Mediaeval cosmology, but there was always a problem with it. The Atomists and others, who liked infinite space, objected that the boundary *implied* space on the other side, something we could easily imagine. And what could such a boundary consist of? Couldn't we recruit Heracles (or Arnold Schwarzenegger) to just go up there and punch a hole in it? What would be there to resist such a force?

That sounds a little silly, but it does highlight the circumstance that *logically*, as Kant sees it, there is no real advantage to either the finitude or infinitude of space. There is no good reason to prefer one or the other, so each violates the **Principle of Sufficient Reason**. But now perhaps Einstein has taken care of that. A positively curved Riemannian space is finite; but if we send Heracles or Arnold out there to punch out the boundary, he's not going to be able to find one. Problem solved.

This was fine as long as the geometry of Einstein's theory could be generalized to the whole universe. At first it wasn't. Einstein himself thought that the universe should be static, and so he introduced the "Cosmological Constant" into his field equation, which would balance gravity with an opposite force that would hold the universe steady. When Edwin Hubble discovered that the universe was expanding, Einstein dropped the factor out of his equation and declared it all a big mistake.

The result of this was to link the dynamics of the universe to gravity. Through the 1970's it all seemed simple. If the mass of the universe was dense enough, gravity would slow down its expansion and eventually cause it to collapse. This preserved the neat "finite but unbounded" feature of Relativistic cosmology. However, if matter was not dense enough, gravity would not be strong enough to stop the expansion; and the universe would expand forever. If the density was *just* low enough, then the geometry of the universe would be flat and Euclidean. If it was any lower, then the geometry of the university would be negatively curved and Lobachevskian.

Either of these latter possibilities creates a bad problem. While all this was going on, the idea that the universe originated in a Big Bang had caught hold (with actual evidence: the Cosmic Background Radiation). And the Big Bang went back to a "singularity," a geometrical point out of which exploded all the mass and energy and space itself that has subsequently existed (which Stephen Hawking initially gained fame by demonstrating). Uh, oh. If the university is actually infinite, whether Euclidean or Lobachevskian, then we cannot roll back time, shrink the universe, and get it down to a finite singularity. Shrinking infinity always leaves infinity.

To their great credit, Paul Davies and John Gribbin realize they have a problem here:

I suppose infinity always dazzles us, and I have never been able to build up a good intuition about the concept. The problem is compounded here because there are actually two infinities competing with each other: there is the infinite volume of space, and there is the infinite shrinkage, or compression, represented by the big bang singularity. However much you shrink an infinite space, it is still infinite. On the other hand, any finite region within infinite space, however large, can be compressed to a single point at the big bang. There is no conflict between the two infinities so long as you specify just what it is that you are talking about.Well, I can say all this in words, and I know I can make mathematical sense of it, but I confess that to this day I cannot visualize it. [Touchstone, 1992, p.108]

Some of this is a little silly. The problem is not that infinity "dazzles us," and no one has ever built a "good intuition" about it -- quite the opposite, since Aristotle, Kant, and modern "Intuitionist" mathematicians (e.g. L.E.J. Brouwer, 1881–1966) have based arguments on an *inability* to intuit it. Apart from those irrelevancies, it is indeed true that, "However much you shrink an infinite space, it is still infinite." This is the problem, and Davies and Gribbin have no answer to it. It doesn't matter whether we considered "any finite religion," because that isn't what creates the problem. You can always shink down a finite volume to any finite size you want. Nor does it matter whether Davies and Gribbin can visualize it. It is not a problem of visualization. We are not worried about the paradoxes of non-Euclidean geometry here. And this doesn't make any more "mathematical sense" than it does any other kind of sense. The problem is itself mathematical, which is how you would get from an infinite space, Euclidean or non-Euclidean, down to a singularity. You can't get there from here.

Philosophers and cosmologists, in general, resolutely ignore this problem, even if they are aware of it. But now they have a *way* to avoid worrying about it, thanks to a theory called "Inflation." Shortly after the Big Bang, the idea is that the universe suddenly and dramatically expanded. Faster than light, actually. This was to explain the uniformity of the Cosmic Background Radiation -- which would be evened out by the expansion. But now Inflation meant that only a part, perhaps a small part, of the universe is subsequently *observable* by us. Otherwise, as we look out in space, and the apparent expansion of the universe increases with distance, we get to a point that the expansion reaches the velocity of light; and nothing beyond that would be visible. Without Inflation, that boundary would itself be at the Big Bang, and we could see the whole universe, although we would see a lot of it only in its early stages. With Inflation, however, some part of the universe is beyond our observation.

So is the universe finite or infinite? There may be no way to ever know, and Kant's Antinomy is back with a vengeance. The Inflation of a infinite universe leaves us in a (relatively) small finite bubble, cut off from all the rest of space and time. Or the universe could be finite, leaving us uncertain *how much* of it our bubble actually occupies.

Some people are aware of these possibilities, and have tried to think of ways around it all, even looking for evidence that might reflect the character of the external universe. One cosmologist even thinks that the whole universe, not much larger than the observable one, has the shape of the Platonic dodecahedron. This does not sound like a serious theory; but I have not noticed that even the serious suggestions, or the reasons for them, are generally discussed in a way that makes its way to my attention. At the same time, with the discovery of Dark Matter, the observable density of matter in the universe looks like it is all but *exactly* at the Euclidean level. Space looks "flat." But everyone can still hope. Beyond the observable horizon, the universe might nevertheless be Riemannian, finite, and unbounded after all. We can always hope. Or hope against hope. And that is what it is. But it is a hope based in part on what may *necessarily* be an absence of evidence. And as such, to the extent that it is entertained or neglected, it begins to verge on the dishonest. It is as though the direct failure of the finite and unbounded universe now motivates us to hussle it off the stage, and leave it waiting in the wings until we can cook up some reason, any reason, to bring it back out.

Some of these issues were brought up by Jean-Pierre Luminet, Glenn D. Starkman, & Jeffrey R. Weeks an in article, "Is Space Finite?" in the April 1999 issue of *Scientific American*:

Conventional wisdom says that the universe is infinite...The question of a finite or infinite universe is one of the oldest in philosophy. A common misconception is that it has already been settled in favor of the latter...

One problem with the conclusion is that the universe could be spherical yet so large that the observable part seems Euclidean, just as a small patch of the earth's surface looks flat [a common idea in Inflationary theories]. A broader issue, however, is that relativity is a purely local theory [!]. It predicts the curvature of each small volume of space -- its geometry -- based on the matter and energy it contains. Neither relativity nor standard cosmological observations say anything about how those volumes fit together to give the universe its overall shape -- its topology. [pp.90-92]

There are some curious statements here. In 1999, I would have thought that the "conventional wisdom" was still yearning for a finite universe, as I suspect that Luminet, Starkman, and Weeks do, with them saying that it is a "problem" that "the universe could be spherical," i.e. Riemannian, finite, and unbounded. No, the *problem* is that the *evidence* is for flat space; and the possibility that the universe is finite consists of *speculative wishful thinking*. If they want to speculate, that's fine, but they should be presenting the situation as it is and not be using what is really biased language.

On the other hand, it is a remarkable assertion to say that "relativity is a purely local theory." Indeed, it would have been, if it had not been generalized to cosmological proportions. Which was done by Einstein himself, and which was a large part of the reason why everyone, like philosophers, got so excited about the whole business. A "purely local theory" is going to have no bearing on Kant's Antinomy of space and time.

If Luminet, Starkman, and Weeks want to unlink the Relativistic dynamics of the universe from its geometry, that is an avenue worth exploring -- especially since the expansion of the universe is now seen to be accelerating, under the influence of the mysterious "Dark Energy," for which there is currently no place in physics, except, ironically, Einstein's own Cosmological Constant. But if the universe has some sort of non-Euclidean geometry, and this is not related to Einstein's theory, then it is not clear how one would pursue the matter, especially when it is all invisible behind the observable horizon. If the evidence is for flat space, then anything else would be speculative metaphysics. That is what most philosophers had hoped Einstein would allow us to get away from. So, overall, the approach does not seem promising -- unless everyone indeed is willing to turn back to metaphysics, which would result in a lot of exploding heads in philosophy departments.

*Fear of Knowledge*, by Paul Boghossian, of New York University, is a pretty good book. But when it comes to Kant and geometry, it offers no more than nonsense; and Boghossian obviously devoted no more critical attention to this than did Jerrold Katz:

Kant's own claim about geometry came to grief: soon after he made it, Riemann discovered non-Euclidean geometries, and some one hundred years later, Einstein showed that physical space was in fact non-Euclidean. [Clarendon Press, Oxford, 2006, p.40.]

This is painfully wrong. The "discovery" of non-Euclidean geometry was made well before Riemann (Boghossian is forgetful, or shockingly ignorant, of Bolyai, Lobachevskii, and Gauss); this involved no "grief" for Kant, if we have properly understood that his theory predicts rather than precludes non-Euclidean geometry; and Einstein certainly did not "show" that space was "in fact" non-Euclidean. Right now, space in the observable universe looks flat and Euclidean, although no one would be so foolish as to claim that anything of the sort has been "shown" to be such "in fact." But we know who rushes in where angels fear to tread.

Talk about the "conventional wisdom" of the previous quote. Boghossian can only have written this by relying on the shallowest of half-baked stereotypes about Kant and Einstein, and this is no credit to his understanding of either. And how can anyone, in this day and age, say that a scientific theory, subject to falsification, can "show" anything to be "in fact" anything? It is a painful display, right up there with popular confabulations about the Fall of Rome.

Paul Boghosslian is not to be confused with Peter Boghossian, of Portland State University, a "new" atheist, but also a staunch defender of free speech. In September 2021 Peter Boghossian has resigned from Portland State University in protest over its Stalinist, politicized policies, violating free speech and academic freedom.

More recently we get Frank Wilczek, in a column, "Wilczek's Universe: No, Truth Isn't Dead," in *The Wall Street Journal*. Wilczek is a physicist rather than a philosopher, so he has more excuse than Jerrold Katz and Paul Boghossian, but he relies on the same kind of canards:

Impressed by the beauty and success of Euclidean geometry, philosophers -- most notably Immanuel Kant -- tried to elevate its assumptions to the status of metaphysical Truths. Geometry that fails to follow Euclid's assumptions is, according to Kant, literally inconceivable. [June 24-25, 2017, p.C4]

As we have seen, "literally inconceivable" is the opposite of Kant's theory. And we can reflect that, if non-Euclidean geometry exists, and it falsifies axioms in Euclid, then this is actually proof of their synthetic character. And since Kant introduced this terminology precisely to describe the nature of things like the axioms of geometry, it is a remarkable combination of irony, paradox, and confusion to then delabor Kant as denying the very thing -- deniability -- of which his theory consists. Frank Wilczek is very far from being responsible for this folly, and we might wonder what he has been reading and where he has gotten this stuff. He might even have been reading Paul Boghossian, although I doubt that *Fear of Knowledge* is on his reading list, and both of them have gotten it from a long tradition of confusion and misrepresentation.

We also might note here that *any geometries*, including non-Euclidean ones, if they are thought to represent physical space -- and we have not adopted Positivistic nihilism -- are going to touch on "metaphysical Truths," since the structure of space is an ontological issue. We might want to ask Frank Wilczek what he thinks a "metaphysical Truth" is, and if we discover that he began his study of metaphysics with Shirley MacLaine rather than Parmenides, Plato, and Aristotle, his informed familiarity with the subject may be in doubt.

Finally, we get someone who voices a key insight. Edward Frenkel is a mathematician; and in his book, *Love and Math, The Heart of Hidden Reality*, he says;

The human brain is wired in such a way that we simply cannot imagine curved spaces of dimension great than two; we can only access them through mathematics. [Basic Books, 2013, p.2.]

Here we have something that commentators on Kant, and other philosophers, have overlooked for a couple of centuries. What it is that we can imagine and visualize is something that is either thoroughly ignored or addressed with patently incredible statements -- such as Jeremy Gray's dismissal of Kantian spatial intuition as "either unintelligible or wrong." Frank Frenkel seems familiar with human spatial intuition as Jeremy Gray is not. Perhaps Gray is one of those Behaviorists without an imagination.

So Frenkel is aware that human imagination may be hard wired into the brain, which is something *so likely* from an evolutionary point of view, and supported by experimental evidence even on infants, that the difficulties of others with the thesis seem to bespeak some sort of personal reluctance, self-deception, or bias. But I have met physicists who seem to think that rotating things through multiple dimensions is something that everyone can do easily.

I think that they are confusing different things, which Frankel clearly distinguishes as the difference between imagination and the abstractions of mathematics -- or between the visualization of models and projections and the literal structure of things that can only be represented in equations. I think what often happens is that people begin to think of models and projections as the things themselves. The clarity and honesty of Frenkel is more than refreshing, but its rarity inspires no confidence in the ability of the tradition in philosophy, physics, or mathematics to deal with these issues.

What Frenkel allows us to consider is that Kant's understand of the foundations of geometry means that the human brain determines the character of space in the *phenomenal reality* that is represented in human consciousness. Kant did not believe that space and time even existed outside of consciousness, and peculiarities like the "non-local" nature of quantum mechanics might even suggest something of the sort. But it is no great leap to imagine that physical space does exist outside of consciousness, and that its character may be different from what we can imagine. Perhaps the universe does have a positively curved, Riemannian character. This would resolve Kant's First Antinomy, and it would be philosophically satisfying. With our observable universe confined to a possibly infinitesimal bubble, it may be that determining the character of the larger universe is beyond the capacity of human science. No one should be so foolish as to use words like "impossible," but the possibility of answering Kant's Antinomy, in terms of current science and evidence, is presently in serious doubt.

In their book about four women philosophers at Oxford, Clare Mac Cumhaill & Rachel Wiseman toss off a brief comment about Kant's geometry and Ernst Mach:

Revolutions in geometry in the nineteenth century had shattered Enlightenment philosopher Immanuel Kant's idea that Euclidean space is the form of all possible experience, and with it his idea of a kind ofa prioriknowledge that is not derived or acquired from experience. [Ernst] Mach stepped in with an alternative: there is no need to posit anything that sensory experience does not confirm or predict. [Doubleday, 2022, p.44]

The "alternative" priniciple of Ernst Mach, of course, is the reason why he rejected the atomic theory of matter, a century after John Dalton has found the evidence for it. Einstein even agreed with him -- until Einstein found further evidence in his examination of "Brownian Motion" -- revealed in one of his epic papers published in 1905. I'm not clear that Mach ever changed his mind, even then.

Meanwhile, geometry had been done for centuries -- and continues to be done -- without any reference to experience, "sensory" or otherwise. One wonders how Mac Cumhaill and Wiseman think that geometry is done.

At the same time, by "revolutions in geometry" Mac Cumhaill and Wiseman certainly mean the construction of non-Euclidean geometries. That was not based on experience either. Regardless of their *a priori* validity, they are certainly not "derived or acquired from experience," except to the kind of British Empiricists who denied basic features or geometry such as geometrical points. I don't know whether Mac Cumhaill and Wiseman see things that way or not. If they do, they won't have many friends among mathematicians; and they will have trouble understanding the Continuum Hypothesis.

While the very existence of non-Euclidean geometry is commonly taken to refute Kant's theory of geometry, the issue, as we have been seeing, is whether this means that the phenomenal *world* can be non-Euclidean or whether "all possible experience" can be non-Euclidean. The phrasing of Mac Cumhaill and Wiseman makes is sound like they accept that *experience* can be non-Euclidean. But, as we just saw with Edward Frenkel, it doesn't look like we experience or can imagine either multi-dimensional spaces or non-Euclidean spaces, except with models or projections.

Indeed, what it looks like is that Mac Cumhaill and Wiseman have devoted no serious or independent thought to this and rely on half-baked or poorly understood clichés about geometry -- much like most of the others I have been examining on this page.

Finally, we come to Leonard Nelson (1882-1927). In relation to the issues of geometry considered above, Nelson seems to have been the only philosopher in the 20th century to understand Kant's theory and realize the nature of its implications. This alone would make Nelson one of the great philosophers of the century; but then his neglect, and the continuing confusion and muddle about Kant and geometry among everyone else, serves to discredit almost the whole of 20th century philosophy.

Indeed, when we find it commonly said that the greatest philosophers of the century were Martin Heidegger, an unrepentant member of the actual German **Nazi Party**, and Ludwig Wittgenstein, who was at pains to show that philosophy shouldn't exist, it is hard to see how any sensible person should take philosophy seriously at all -- as many do not. And this was what Wittgenstein would have wanted. Indeed, when the public face of philosophy for many years after World War II was Bertrand Russell and Jean Paul Sartre, men whose personal moral confusion (public and private), philosophical shallowness, and political folly and naiveté were painfully evident, the evidence to discredit philosophy (as Witgenstein wanted) only piles up higher and deeper.

In his essay, "Philosophy and Axiomatics" [1927], Nelson says:

If we can show that the denial of a proposition does not contradict the consequences of certain other propositions, we have then found a criterion of the logical independence of the proposition in question. In other words, the logical independence of this Euclidean axiom [the Parallel Postulate] of the other axioms would be proved if it could be proven that a geometry free of contradictions could be erected which differed from Euclidean geometry in the fact, and only in the fact, that in the place of the parallel axiom there stood its negation. That is just what Gauss, Lobachevski, and Bolyai established: the possibility of erecting such a noncontradictory geometry which is different from the Euclidean...What is important to us here is this: The results of modern axiomatics are a completely clear and compelling corroboration of Kant's and [Jakob] Fries's assertion of the limits of logic in the field of mathematical knowledge, and they are conclusive proof of the doctrine of the "synthetic" character of the mathematical axioms. For it is proved that the negation of one axiom can lead to no contradiction even when the other axioms are introduced... And this was just the criterion that Kant had already specified for the synthetic character of a judgment: the uncontradictory character of its negation. [

Socratic Method and Critical Philosophy, Yale, 1949, Dover 1965, pp.163-164]

The clarity of this is remarkable, and in stark contrast to the muddle and falsehoods that we have seen in so many statements, by philosophers, physicists, and others, in all the time since. Of all the quotes about Kant here, only Poincaré, who had died in 1912, precedes Nelson's discussion -- and Nelson elsewhere addressed directly Poincaré's confusions about Kant.

So let me be clear myself: Kant's theory provides for a *prediction* of the possibility of non-Euclidean geometry, not a prohibition; and the later *existence* of non-Euclidean geometry in Bolyai, Lobachevski, Gauss, and Riemann actually *proves*, by falsification, that the Parallel Postulate is a synthetic proposition, as Kant said. The attempts to discredit Kant by referring to his belief in the *a priori* character of geometry discredit themselves by failing to consider the nature of Kant's theory of pure intuition and the fact, only appreciated here by Edward Frenkel, that the features of non-Euclidean geometry and multi-dimensional geometry, actually defeat our human imagination and visualization -- which vindicates Kant. But, of course, there is more to it than that. Kant's theory is vindicated, not just by the capabilities and incapacities of our imagination, but by the *logic* of cosmological conflicts between the finite and the infinite, or the natures of straight lines and geodesics. What other straight line becomes curved by being shortened? That only happens in a Riemannian space. Yet I do not see consideration of such things in public discourse or the philosophical works I have encountered.

Why this has happened this way is a ominous question for the whole future of philosophy. When the Nihilism (or actual evils) of Heidegger or Wittgenstein is held up as paradigmatic, there would seem to be no real future for the inquiries that we can trace back to Socrates and earlier. The wrong sorts of *people*, the sort already impeached by the cross-examinations of Socrates, are obviously involved. Sadly, there really is no way to get rid of them. As Schopenhauer discovered, the celebration of folly and idiocy may be the natural fate of *institutional* philosophy -- something now much easier to understand because of Public Choice economics. When philosophers become rent-seekers, there may be no remedy to their obscurantism and corruption. And this is before we even get to their *Trahison des Clercs* adherence to illiberal and totalitarian ideologies, which are now all the rage at American universities. This discredits, not just philosophy, but much of "higher education," certainly and tragically in the humanities -- as well as in the "social" (frequently called the "pseudo-") sciences. As for actual "schools of education," nothing less than demolition will improve them.

...daher es kommt, daß jedes Bessere nur mühsam sich durchdrängt, das Edle und Weise sehr selten zur Erscheinung gelangt und Wirksamkeit oder Gehör findet, aber das Absurde und Verkehrte im Reiche des Denkens, das Platte und Abgeschmackte im Reiche der Kunst, das Böse und Hinterlistige im Reiche der Thaten, nur durch kurze Unterbrechungen gestört, eigenlich die Herrschaft behaupten...Hence arises the fact that everything better struggles through only with difficulty; what is noble and wise very rarely makes its appearance, becomes effective, or meets with a hearing, but the absurd and perverse in the realm of thought, the dull and tasteless in the realm of art, and the wicked and fraudulent in the realm of action, really assert a supremacy that is disturbed only by brief interruptions.

Arthur Schopenhauer,

Die Welt als Wille und Vorstellung, Band 1, §59 [Reclam, 1987, p.457],The World as Will and Representation, Volume I [Dover Publications, 1966, E.F.J. Payne translation, p.324, translation modified], color added

Three Points in Kant's Theory of Space and Time

Einstein's Equivalence Principle

Philosophy of Science, Space and Time

History of Philosophy, Modern Philosophy

Synthetic propositions that are not grounded in something that would count as a "fixed part of man's consciousness, independently of experience and uncreated by experience," would precisely be those grounded on experience, Kant's "empirical intuition." These are not *a priori*, known before experience, but *a posteriori*, known *because* of experience, and so generated in time. While, in critiques of Kant and Fries (even by the quasi-Friesian Karl Popper), a lot of ink is spilled on the idea that they believed in the *certainty* or *infallibility* of these intuitions and the propositions based on them, this is not necessarily so, and, although Kant's views may be ambiguous, *certainly* not for Fries, for whom all propositional knowledge is *mediate, fallible, and corrigible*. If your proposition about experience turns out to be in error, you correct it by comparison, again, to the experience. This depends on the conceptual *recognition* of features in perception, something that we can now actually see the brain doing. That may be how you *discover* the error, or you may have doubts because of the testimony of others, or other evidence.

What is of metaphysical interest about empirical propositions is that they represent, not just states of affairs, but *facts*. The young Wittgenstein even said that the world consists of facts, not things. This is both paradoxical and false. Facts and things are different. Things are subject to change, and they can be created and destroyed. Facts are not subject to change, and while they come into being, and so in a sense are created, they cannot be destroyed. We might wonder sometimes whether Brutus really stabbed Caesar, but if he did, then it is fact that possesses its own kind of necessity, existing in a kind of abstract, atemporal, Platonic reality. In the absence of time travel and alternative universes, it cannot now be otherwise.

The necessity of facts is an artifact of the perfect aspect. This is generally ignored in metaphysics and in general philosophy, which these days may disdain metaphysics; but the issue began with Arisotle's discussion in *On Interpretation*, formulated as the problem of future contingency. Ideas about time in current physics seem to alternate between the denial of its very existence (the view of Kurt Gödel), and the fatalistic idea that all of time already exists all at once (with Einstein, who seemed to think that fatalism was emotionally reassuring), right now, with our own puzzling sense of directionality and the exclusive existence of the present. There is little in current physics that addresses the commonsense metaphysics of time, with the hard, indigestible lump of entropy preventing its complete, complacent dismissal. As with the proverbial atheist in the foxhole, many prayers are doubtlessly offered that the whole thing would just go away.

*Past, Present, and Future, A Philosophical Essay about Time*, by Irwin C. Lieb