Mlodinow’s Euclidean Equivocation

Leonard Mlodinow, who with Stephen Hawking co-authored The Grand Design, wrote the following passage in another book he wrote titled Euclid’s Window:

“By 1824, Gauss had apparently worked out an entire theory. On November 6 of that year, Gauss wrote to F. A. Taurinus, a lawyer who dabbled quite intelligently in mathematics, “The assumption that the sum of the three angles [of a triangle] is less than 180° leads to a special geometry, quite different from ours [Le., Euclidean], which is absolutely consistent, and which I have developed quite satisfactorily for myself. . ..” Gauss never published this, and insisted to Taurinus and others that they not make his discoveries public. Why? It wasn’t the church Gauss feared, it was its remnant, the secular philosophers.
In Gauss’s day, science and philosophy hadn’t completely separated. Physics wasn’t yet known as “physics” but “natural philosophy.” Scientific reasoning was no longer punishable by death, yet ideas arising from faith or simply intuition were often considered equally valid. One fad of the day which particularly amused Gauss was called “table-rapping,” in which a group of otherwise intelligent people would sit around a table with their hands placed in an arched position upon it. After a halfhour or so, the table, as if bored with them, would start to move or tum. This was supposedly some sort of psychic message from the dead. Exactly what message the ghouls were sending is unclear, although the obvious conclusion is that dead people like to position their tables against the far wall. In one instance, the entire Heidelberg law faculty followed for some time as their table moved across the room. One pictures a bunch of bearded, black-suited jurists pacing alongside, struggling to keep their hands in their appointed spot, attributing the locomotion to occult animal magnetism rather than their push. This, to Gauss’s world, was reasonable; the thought that Euclid had erred was not.”[1]

Ignoring the snide and historically fantastical notes about scientific reasoning ever having been punishable by death,[2] this passage is meant to set Gauss up as the champion of science, and his work is meant to signal the victory of science over philosophy. Mlodinow means to show how Euclid’s fifth postulate (concerning parallel lines), which was believed to be as philosophically secure as anything could be, was proven to be incorrect by the discovery, in modern physics, that space is non-Euclidean; in other words, it is not true that for any straight line L, and any point P not on L, there is only one line which can be drawn through P parallel to L. It may seem to be true, but our study of the physical world shows us that it isn’t (so the story goes). This, I will suggest, is just rhetorical slight of hand on Mlodinow’s part. In fact, it is worse; I believe that this represents a genuine antinomy in Mlodinow’s view of the nature of science itself, given what he has committed himself to in print elsewhere. I’ll develop this shortly below.

Shortly after the passage about Gauss trembling in fear of the indomitable secular philosophers (all of whom were apparently busy pushing tables around faculty lounges trying to communicate with the dead) Mlodinow introduces the character of Immanuel Kant. He chooses to portray Kant as the philosopher par excellence, and as the antagonist of scientific progress. He takes special care to note:

 “In Critique of Pure Reason, Kant calls Euclidean space “an inevitable necessity of thought.””[3]

It is worth saying a few words in defense of Kant, before we move on. Mlodinow provided no (precise) citation for the quote, so I searched through the Critique of Pure Reason for myself and could find no such statement. A quick glance at some Kant scholars indicates to me that there are some mixed signals here. On the one hand, never once does Kant refer to Euclid, or Euclidean geometry, or the fifth postulate, or even parallel lines (apart from one brief comment about immediately perceiving that the opposite angles of a parallelogram are equivalent) throughout the Critique of Pure Reason, so that the attribution to him of the saying above must, at best, be justified by reading between the lines. On the other hand, it is not unlikely that when Kant made statements about space, he was presupposing something like Euclidean space. It is true that Kant thought that the conception of space, like that of time, was an a priori intuition which was inalienable to the rational intellect. Instead of saying, with the empiricists, that our conception of space came come from the five senses, Kant thought that our conception of space was a precondition for our having intelligible empirical experiences at all. He writes:

“Space is not a conception which has been derived from outward experiences. For, in order that certain sensations may relate to something without me (that is, to something which occupies a different part of space from that in which I am); in like manner, in order that I may represent them not merely as without, of, and near to each other, but also in separate places, the representation of space must already exist as a foundation. Consequently, the representation of space cannot be borrowed from the relations of external phenomena through experience; but, on the contrary, this external experience is itself only possible through the said antecedent representation. Space then is a necessary representation a priori, which serves for the foundation of all external intuitions.”[4]

Although Kant’s student, Schultz, was apparently interested in the treatment of parallel lines, Kant never talked about Euclid’s fifth postulate in his published works at all (though there are comments sprinkled about in his unpublished works). In any case, what Kant says about our a priori intuition is hardly undermined by the discovery that the geometry of space is non-Euclidean (i.e., hyperbolic or elliptic). Andrew Janiak observes:

“Kant highlights the accepted fact that we represent space as an infinite Euclidean magnitude—this can be widely accepted, despite the dispute concerning space’s ontology. […] We do not have a sensation of an infinite Euclidean magnitude.”[5]

Wes Alwan also writes:

“If we discover that the universe is actually, objectively (in the Kantian sense) non-Euclidean when our spatial intuition suggests it is Euclidean, then there is a conflict here between the faculties of understanding and intuition. If you’ve studied non-Euclidean geometry you’ll readily see what this means: the denial of the parallel postulate violates our intuition (unless we model the new geometry within Euclidean geometry but as occurring on a hyperbolic surface); but it does not produce any logical inconsistency. And in fact this is the whole point of Kant calling our perception of Euclidean space “intuition”: I have no other basis for the parallel postulate — I cannot argue for it as following from a principle of logic or arithmetic; nor can I argue about it from some a posteriori discovery in physics about the nature of the world.”[6]

Mlodinow quizzaciously continues:

“Kant, noting that geometers of the day appealed to common sense and graphical figures in their “proofs,” believed that the pretense of rigor ought to be dispensed with, and intuition embraced. Gauss held the opposite view-that rigor was necessary, and most mathematicians were incompetent.”[7]

Nevermind that Kurt Gödel essentially reiterated the very same point as Kant’s, and offered (what he thought was) a demonstration of it in what we now know as the incompleteness theorem (though, to be entirely fair, Mlodinow acknowledges this later on in the book); the point, for Mlodinow, is for us to recognize in the confrontation between Kant and Gauss a microcosmic confrontation between philosophy and science. A conflict from which science emerged victorious over philosophy, physics over common sense, and observation over intuition. The empiricist’s wet dream could not have been better narrated.

Reminiscent of Friedrich Nietzsche’s famous statement that “God is dead!”[8] Mlodinow and Hawking write, in the opening passage of their book The Grand Design, that “Philosophy is dead.”[9] Somewhat ironically,[10] they mean the very same thing, which is that metaphysics is dead. Metaphysics has been confused for a great many things which it is not, so it is worth calling attention to its definition; metaphysics is nothing other than the study of the extra-mental, extra-linguistic, model-independent, objective nature and structure of reality. What the metaphysician wants to know is what the fundamental furniture of reality includes. Where the construction worker is happy to use bricks, metaphysicians want to know what bricks are really made of, and where the mathematician is happy to use numbers the metaphysician wants to know what numbers are. From the perspective of the metaphysician, all the physicist does is offer empirically adequate models of space-time phenomena. That’s it. No amount of empirical data can definitively settle the matter of whether those models are literally accurate, nor can any of those models be the complete story because scientific models (if interpreted literally) presuppose countless philosophical assumptions which cannot be scientifically explored.

Accordingly, Hawking and Mlodinow sketch out, in the book, a view which they call model-dependent realism about science. They write:

“According to model-dependent realism, it is pointless to ask whether a model is real, only whether it agrees with observation.”[11]

In fact, they boldly exclaim that one of the central conclusions of their book is that “there is no picture-independent concept of reality.”[12] In one of the most tantalizing passages of their book they give some indication of just how radical their view really is, and it is worth quoting at some length.

“Model-dependent realism can provide a framework to discuss questions such as: If the world was created a finite time ago, what happened before that? An early Christian philosopher, St. Augustine (354–430), said that the answer was not that God was preparing hell for people who ask such questions, but that time was a property of the world that God created and that time did not exist before the creation, which he believed had occurred not that long ago. That is one possible model, which is favored by those who maintain that the account given in Genesis is literally true even though the world contains fossil and other evidence that makes it look much older. (Were they put there to fool us?) One can also have a different model, in which time continues back 13.7 billion years to the big bang. The model that explains the most about our present observations, including the historical and geological evidence, is the best representation we have of the past. The second model can explain the fossil and radioactive records and the fact that we receive light from galaxies millions of light-years from us, and so this model—the big bang theory—is more useful than the first. Still, neither model can be said to be more real than the other.”[13]

Although somewhat cryptic, it is important that we do not gloss over what’s being said in this passage. What Hawking and Mlodinow are saying is that while some people believe that the big bang theory is true, and six-day creationism is false, and other people believe that the big bang theory is merely closer to being true than the story of six-day creationism, none of these people are correct. As a matter of fact, the big bang hypothesis happens to be a more useful model (given certain hypothetical goals) than six-day creationism and this is the only reason we adopt it in preference to the latter. Although they claim that their view circumvents debates between scientific realists and scientific anti-realists, in reality their model-dependent realism is just a thinly-veiled version of scientific anti-realism!

Now, my chief problem with Mlodinow is not his philosophy of science; he can be an anti-realist until the cows come home and it won’t be any skin off my back. My problem with him isn’t that he thinks we should be empiricists instead of rationalists with respect to objects of intuition like Euclid’s fifth postulate. My problem isn’t even that he thinks that science can license the claim that Euclid’s fifth postulate was literally incorrect (though I do find the suggestion annoying). My real problem with Mlodinow is that I see no way for him to put all of these beliefs together coherently. He cannot on the one hand say that science has shown us that Euclid’s geometry is objectively incorrect, and on the other hand say that no scientific model is ever objectively ‘real’ (by which he means model-independently true). The best he can do, I think, is argue that we ought to abandon Euclid’s fifth postulate when operating within models of geometry which better account for the curvature of space than Euclidean geometry. That’s it, end of story. He cannot even say that Euclid’s fifth postulate was wrong, because the parallel postulate is true within a Euclidean model of geometry! For Mlodinow to say anything which is one iota more philosophically committing than that we should abandon Euclid’s fifth postulate for the same reasons we should abandon Euclidean geometry is for him to wander into utter incoherence.

I want to finish by saying a word or two about this now typical attitude of dismissiveness, condescension, derision and contempt for philosophy among professional scientists, exemplified especially by people like Mlodinow. Although I have no doubt that Mlodinow is a great physicist, it is unfortunate that he has added his (incredibly shrill) voice to the cacophonous choir of scientists grossly overestimating their philosophical aptitudes. What makes his comments particularly irksome is not that I and other philosophers find them disagreeable, but that they are logically irreconcilable. That, to a philosopher, is like hearing the sound of forks scrapped across a chalk board. It really is true what they say; the man who thinks he has no need of philosophy is the one who will be in most need of it.[14] Einstein, whose best friend, it is worth remembering, was none other than Kurt Gödel, was absolutely right when he wrote:

“It has often been said, and certainly not without justification, that the man of science is a poor philosopher. Why then should it not be the right thing for the physicist to let the philosopher do the philosophizing?”[15]

There is one possible reprieve for Mlodinow; although he insinuates fairly strongly throughout his book that Euclid, Kant, et alia were literally wrong about the parallel postulate, he could perhaps backpedal and defend himself by insisting that he never committed himself to the statement that the parallel postulate is literally false. If this is the case, then I owe him an apology for today’s blogging exercise.

[1] Leonard Mlodinow, Euclid’s Window: The Story of Geometry from Parallel Lines to Hyperspace (New York: Simon and Schuster, 2010), 116.

[2] It is remarkably silly to say, as people often do, that Galileo was burnt at the stake, or that Marco Antonio Dominis was persecuted for his scientific ideas instead of his vitriolic attacks on the papacy, or that Cecco d’Ascoli was burnt alive for saying that there were people on the other side of the planet instead of his attempt to determine the nativity of Christ by reading his horoscope. There is a modernist myth that the man of science was persecuted in the age of the Church, but this sounds like a phantasmagorical persecution complex. It wasn’t only men of science who got into trouble with the church (it was also artists, writers, poets, theologians, and philosophers), and when men of science did get into trouble it was almost never on account of their scientific work (Galileo is the very notable exception; and serves as the exception which proves the rule). Notice that the same is not true for Theologians. It was, in fact, much more dangerous to do Theology than it ever was to do Science. It is approximately as puerile to say that scientists were, in general, afraid of the Church as to say that Gauss was afraid of secular philosophers.

[3] Leonard Mlodinow, Euclid’s Window: The Story of Geometry from Parallel Lines to Hyperspace (New York: Simon and Schuster, 2010), 117.

[4] Immanuel Kant, The Critique of Pure Reason,

[5] Andrew Janiak, “Kant’s Views on Space and Time,” in The Stanford Encyclopedia of Philosophy (Winter 2012 Edition), ed. Edward N. Zalta,


[7] Leonard Mlodinow, Euclid’s Window: The Story of Geometry from Parallel Lines to Hyperspace (New York: Simon and Schuster, 2010), 117.

[8] Friedrich Nietzsche, The Gay Science, section 125, transl. Walter Kaufmann (1974).

[9] Stephen Hawking and Leonard Mlodinow, The Grand Design, (Random House Digital, 2010): 5.

[10] Ironic because where Nietzsche as a continental philosopher treated systematic thinking with scorn, Hawking and Mlodinow, as scientists, revel in rigor.

[11] Stephen Hawking and Leonard Mlodinow, The Grand Design, (Random House Digital, 2010): 46.

[12] Stephen Hawking and Leonard Mlodinow, The Grand Design, (Random House Digital, 2010): 42.

[13] Stephen Hawking and Leonard Mlodinow, The Grand Design, (Random House Digital, 2010): 49-50.

[14] In particular because to think that one doesn’t need a philosophy is already to have a philosophy (though it is a very bad one).

[15] Quote from Howard, Don A., “Einstein’s Philosophy of Science”, The Stanford Encyclopedia of Philosophy (Winter 2015 Edition), ed. Edward N. Zalta; “Physik und Realität.” Journal of The Franklin Institute 221: 313–347. English translation: “Physics and Reality.” Jean Piccard, trans. Journal of the Franklin Institute221: 348–382. Reprinted in Einstein 1954, 290–323. Note that when taking this quote in its full context it isn’t nearly as complimentary of philosophers, but I’m not sure that Einstein was right about the rest of what he wrote on the matter; I’m merely saying that he was right about this comment.


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