Eugene Ostashevsky

Two Proofs of Absurdity: How the Geometric Method Sacked Saccheri and Hobbled Hobbes

This paper describes two concepts of absurdity observable in early modern philosophical engagement with Euclidean geometry and the geometric method.


If there is such a concept as absurdity, it cannot be known through itself but only through its opposite, rationality, as its negation. In other words, there can be no definition of absurdity that is not, simply, a definition of rationality with the logical complement sign in front of it (i.e. –p or ¬p, with p being “rationality”). Such an approach to absurdity is embodied in the reductio ad absurdum proof in geometry. You demonstrate that the contrary of what you want to prove contradicts something else you know as true. Therefore, at least in geometry, the negation of the assumption—the theorem you wanted to prove in the first place—must be true.

One famous reductio ad absurdum proof is that of proposition X.117 of Euclid’s The Elements, that the diagonal of a square is incommensurable with the side (i.e., there is no unit, no matter how small, that multiplied many times equals the diagonal, and another many times, the side). The proof of X.117 assumes that the diagonal is, on the contrary, commensurable to the side, but that, in such a case, the ratio of diagonal to side must include a number at once odd and even, a manifest absurdity. By demonstrating the absurdity of the contrary we establish the truth of the original proposition, here, of the incommensurability of the diagonal and the side of a square—or, which amounts to the same, the hypotenuse and the side of a right-angled isosceles triangle.1 That, given a proposition and its negation, one must be true and the other false, is the Law of Excluded Middle, formulated by Aristotle in On Interpretation and the Metaphysics.

Yet sometimes the absurd in mathematical proof turns out to have been not so absurd after all. Already in antiquity mathematicians tried to convert into a theorem the fifth postulate of Elements I, the so-called Parallel Postulate, since its equivalent states that, given a line and a point external that line, only one line may be drawn through the point that is parallel to the first line. The Baroque mathematician Girolamo Saccheri (1677-1733), proceeding by reductio ad absurdum, demonstrated that two possible negations of the postulate—that more than one line may be drawn through the point that is parallel to the first line, or that no such line may be drawn—contradict other rudiments of Euclidean geometry. By showing the absurdity of the contraries, Saccheri believed he transformed the Parallel Postulate into a theorem, making for a more elegant Euclid. In fact he had unknowingly discovered—and then discarded as obvious absurdities—some basic aspects of both elliptic and hyperbolic geometries that had to wait for about another century for the recognition of their possibility.2

La presomption est nostre maladie naturelle et originelle,” wrote Montaigne in the Apologie de Raimond Sebond. Hindsight renders Saccheri’s self-assurance both touching and risible. The construction of non-Euclidean geometries decentered and relativized Euclid, enacting a Copernican revolution in mathematics with deep ramifications for the theory of knowledge. From the science of real spatial relations arrived at by deduction out of absolute and self-evident truths—and consequently from the epistemic model that any knowledge of reality aspiring to sure and certain status would imitate (remember Spinoza’s Ethics)—Euclid became one geometry among several, its formerly self-evident truths about the world demoted to heuristics, assumptions necessary to a particular rational system, one that only described real spatial relations, and not in a very precise manner at that.

The moral we may perhaps draw from Saccheri’s reductio ad absurdum proofs is that, absurdity being derivative of rationality, and rationality being multiple—at least in the sense that multiple systems and definitions of rationality are possible—there might be no absurdity that would remain an absurdity relative to all possible rationalities, and that might not, in some perhaps unknown rational system, itself become rationality. In other words it is possible there does not exist an absurdity that is not ultimately subject to some type of rationalization, and an exhaustive rationalization at that.

Possible but, at least from this argument, not necessary. When Descartes circulated his Meditations on First Philosophy, many philosophers decried the cogito proposition as based on unstated and shaky assumptions (including Hobbes, for whom it entailed rather than disproved the union of mind and body). However, a far more questionable assumption on the part of Descartes is that the object of the mind’s rational investigation, whether it lie outside the mind (=the world) or be the mind turned in on itself, submits to rational investigation, and does so wholly, without any rationally intractable noise—in short, that the real is the rational, and that no irreducible absurdity may exist. As far as assumptions go, this is a tall order.

Nor is the truth of it so self-evident as to be universally accepted. Even Descartes’s predecessor in embracing skepticism to take it by the heel, the Baghdad philosopher al-Ghazali, crowns his systematic doubt with precisely that—doubt in the rationality of the real. After using reason to dislodge the evidence of his senses, al-Ghazali discards reason itself:

Who can guarantee you that you can trust to the evidence of reason more than to that of the senses? You believed in [the] testimony [of the senses] till it was contradicted by the verdict of reason, otherwise you would have continued to believe it to this day. Well, perhaps, there is above reason another judge who, if he appeared, would convict reason of falsehood, just as reason has confuted [the senses]. And if such [another] arbiter is not yet apparent, it does not follow that he does not exist.3

Since al-Ghazali’s passage gives us no ground to distinguish between the world of rationally irreducible absurdity and the “arbiter… above reason” who indicates such a world, is it an anachronism to identify al-Ghazali’s “arbiter” with both all of rationally irreducible absurdity and with God? For the “another judge” must, at least by suggestion of metaphor, be God. Indeed, this particular passage implicitly defines God as exactly that which cannot be rationalized (since “reason” is “falsehood”) and, consequently, as himself at least an example of irreducible absurdity. (The only distinction that might counter the proposition “God is everything that is rationally irreducible” here is, I think, that between God and the thoughts of God, which may perhaps be a specious one.) Note however that, in this particular passage at least, al-Ghazali does not at all affirm the existence of such a God. It may be, he says, that the real is exhaustively rational, but the contrary of this proposition may hold true as well. Only later, by abandoning skepticism and making the leap of faith, does al-Ghazali come to believe. We are not going to follow him.


Hobbes’s readers know he developed his approach to political theory after encountering Euclid. According to his friend and biographer John Aubrey,

Being in a Gentleman’s Library, Euclid’s Elements lay open, and ’twas the 47 El. libri I. He read the Proposition. By G—, sayd he (he would now and then sweare an emphaticall Oath by way of emphasis), this is impossible! So he reads the Demonstration of it, which referred him back to such a Proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps [and so on], that at last he was demonstratively convinced of that trueth. This made him in love with Geometry.4

The anecdote strikes one as symbolically loaded, Euclid I.47 being not just any old proposition, but that pearl of them all, the Pythagorean Theorem. Nonetheless, se non è vero, è ben trovato. The foundational significance of the axiomatic deductive method of The Elements, which Hobbes is portrayed as learning through reverse engineering, for rationalist philosophy in the looser sense of the word, is thereby underscored. Indeed, The Leviathan endorses geometry as “the onely science that it hath pleased God hitherto to bestow on mankind” [105].

Hobbes himself contemplates the spectacle of Euclidean proof in order to construct a model of rational thinking, and to find its basic instrument: “Speech,” which he regards as composed of “Names,” or nouns. It is common nouns, “as Man, Horse, Tree; every one of which though but one Name, is nevertheless the name of divers particular things,” that the author of The Leviathan identifies with the universals necessary for reasoning about not this or that one thing, but about many things of one kind at once, “there being nothing in the world Universall but Names; for the things named, are every one of them Individuall and Singular” [102]. In other words, “Names” for Hobbes being the counters with which rational operations are performed, it is only in “Speech” that we apprehend and reason about general laws governing particular instances. Conceiving of the language of geometry as a paragon of proper “Speech,” rather than as something different in kind, Hobbes illustrates his argument by two scenarios of geometrical problem-solving, first, as he thinks, without the help of “Speech,” and then with.

In the first scenario, the would-be geometer is one who is deprived of “Speech,” as Hobbes suggests, a deaf person:

A man that hath no use of Speech at all, (such as is born and remains perfectly deaf and dumb), if he set before his eyes a triangle, and by it two right angles (such as are the corners of a square figure), he may by meditation compare and find, that the three angles of that triangle, are equall to those two right angles that stand by it. But if another triangle be shewn him different in shape from the former, he cannot know without a new labour, whether the three angles of that also be equal to the same.

In the second scenario the would-be geometer is capable of “Speech”:

But he that hath the use of words, when he observes, that such equality was consequent, not to the length of the sides, nor to any other particular thing in his triangle; but onely to this, that the sides were straight, and the angles three; and that that was all, for which he named it a Triangle; will boldly conclude Universally, that such equality of angles is in all triangles whatsoever; and register his invention in these general termes: Every triangle hath its three angles equall to two right angles.

If you read the passage quickly, it appears to confirm that deaf men are incapable of constructing theorems (such as the one in question, Euclid I.325), but hearing and speech-enabled men have no such impediment. In broader terms, “Speech”-deprivation results in incapacity to reason with universal truths. The only knowledge allotted those without “Speech” is knowledge of particulars: presumably, of specific magnitudes and figures rather than of abstract relations concerning them. Such putative knowledge of particulars gets one nowhere in geometry. Yet the slower you read the passage the more problematic it becomes.

First take its apparent confirmation of the proposition that one needs “Speech” to reason. Is the passage a record of an actual experiment? In other words, did Hobbes set a deaf and a hearing person in front of a slew of right-angled triangles? No. Is it a thought experiment, then—like the thought experiments of Descartes? I don’t see how it can be. Hobbes, being “Speech”-endowed, can think through the relation of the deaf to Euclid only by assuming the truth of what he is trying to prove, i.e. that those without “Speech” can’t make the kind of judgments that are necessary in geometry. In other words, if this is supposed to be a thought experiment, it’s a case of petitio principii, begging the question.6 Which would not be out of character for Hobbes, whose objection to the incorporeality of Descartes’s cogito suffers from the same logical handicap.

Next consider the representation of the “Speech”-deprived person as deaf, or to be more literal, “such as born and remains perfectly deaf and dumb.” It may be historically unjust to take Hobbes to task over his audism, or hearing chauvinism that equates lack of sound with lack of language and even intelligence. It is equally anachronistic to charge him with its philosophical corollary, the epistemological privileging of voice that Derrida calls phonocentrism. The terms “audism” and “phonocentrism” are of recent coinage, and so is the position that the assumptions they stand for hurt one’s reasoning.7 Although in the seventeenth century there must have existed communities of deaf people who communicated among themselves in sign languages, little attention was paid to them by hearing scholars and, as a consequence, we know next to nothing about earlier forms of Sign language.8 Nonetheless, on purely logical grounds, why must the concept of “Name” refer to the concept of sound in order to be “Universall”? If we analyze the concept of linguistic sign, what is it about sound that would make it a necessary constituent? (The question may be especially pertinent since Hobbes’s example involves mathematical proof, whose symbolism seems rather removed from vocalization.)

Nonetheless, let us give Hobbes the benefit of the doubt. Let us grant him that a person “such as born and remains perfectly deaf and dumb” is indeed “Speech”-deprived in some nontrivial manner, in that he lacks access to signs that stand for categories of objects. For example, perhaps the deaf man was brought up in isolation outside any community that would enable him to develop Sign language skills. He lived in a small hamlet and never saw another deaf person. He used only a few rudimentary signs with his parents, and they were all deictic. Yes, let us imagine such a deaf Cratylus. As the philosopher imagines, our deaf Cratylus ascends to the attic and proceeds to “set before his eyes a triangle,” any triangle. Next to it, he draws two right angles, perhaps the angles of a square. But wait… What would Cratylus see if he were looking at a “triangle”? What would Cratylus see if he were looking at “angles”? How would Cratylus have the idea that these are “angles,” and those are “angles,” and that he can “compare” them?9 After all, the deaf man is by Hobbes’s fiat incapable of universal concepts!

How then can the deaf man set out to prove Euclid I.32? But is that what the deaf man is really doing? Let us look closely at Hobbes’s wording. After juxtaposing his triangle with the angles of a square, the deaf man “may by meditation compare and find, that the three angles of that triangle, are equall to those two right angles that stand by it.” Let us follow the deaf man:

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Can you “compare and find,” without recourse to language or proof, that the three angles of this triangle, if joined, are congruent to any two of the angles of this square? Frankly, strain as I might, I cannot. But let us say that the deaf man can apprehend this “by meditation.” Or let us even go farther than Hobbes, and admit that the deaf man establishes the equality of these angles taken together with those angles taken together (no universal concepts there!) not “by meditation” but by measurement (although why someone whose mind is capable only of particulars might want to measure angles, and how he would conduct the act of measuring, remains puzzling). What the deaf man can never do, according to Hobbes, is to conclude that the equality is not a particular case but a universal rule that applies to all triangles inasmuch as they are triangles. So, if you set another triangle before the deaf man, he must again juxtapose it with the square, and perform the same “meditation.” Like this:

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And so on all over again for every particular triangle, never to arrive at the universal proposition “Every triangle hath its three angles equall to two right angles.” It’s the “every” that deaf men have a particular problems with.

But how does the lucky “Speech”-enabled person prove Euclid I.32? That felicitous gentleman, brought up in the voluble company of other speech-makers, public speakers, lecturers, talkers, speechifiers, expounders, orators, declaimers, rhetoricians, haranguers; spokesmen, spokeswomen, spokespersons, mouthpieces; readers, lectors, commentators, broadcasters, narrators; (informal) tub-thumpers, spielers, spin doctors; (historical) demagogues, rhetors; (rare) prolocutors, and other volucres of that feather, soars over particulars by observing that the relationship of equality depends “not to the length of the sides, nor to any other particular thing in his triangle,” but on the very kind of figure the triangle is: that “that the sides [are] straight, and the angles three; and that that [is] all, for which he named it a Triangle.”

Let us slowly replay the situation. The hearing man looks at a triangle and says to himself, “Aha, this here thing corresponds to the name “Triangle.” But what about this here thing corresponds to the name?” Then he Googles “triangle.” “Well,” reasons the hearing man. “Let’s see. It must be either “(1) a plane figure with three straight sides and three angles” that I am looking at, or “(2) an emotional relationship involving a couple and a third person with whom one of them is also involved.” I will go with the plane figure as perhaps more palatable. The definition tells me that any figure with three sides and three angles is a triangle. Therefore, as far being a triangle is concerned, the lengths of sides is not important, nor is it the ratio of angles to one another.” Now the hearing man’s triangle is a true object of geometry rather than of measurement.

Unfortunately, that’s about all for geometry. The hearing man does not perform any action that operates with certain concepts called “geometrical” according to the axiomatic deductive method. Rather, as soon as he grasps that nothing matters in his particular triangle other than what answers to the name “Triangle,” the hearing man “will boldly conclude Universally,” explains Hobbes, “that such equality of angles is in all triangles whatsoever; and register his invention in these general termes: Every triangle hath its three angles equall to two right angles.” I find the phrase “boldly conclude” to be striking, but not in a good way. If Hobbes wanted his hearing man to construct a proof, he might perhaps have said so explicitly. But his phrasing, on the contrary, implies a leap from the particular case to the general rule, which we may analyze thus:

Step one. The hearing man perceives the equality of angles between his particular triangle and the two right angles juxtaposed with it. This step is the same as for the deaf man.

Step two. The hearing man realizes that the equality stems from whatever lets his particular triangle corresponds to the name “Triangle.” This step cannot be performed by the deaf man.

Step three. The hearing man “boldly conclude[s] Universally, that such equality of angles is in all triangles whatsoever.” This step cannot be performed by the deaf man either.

The hearing man’s procedure is not Euclidean because it is not a deductive proof. It bears no resemblance to any of the possible theorems proving Euclid I.32. If it involves induction, the induction seems entirely unjustified as based on a single case, not does the method of induction have any independent truth-value in mathematics. If the procedure is supposed to parallel the intuition of clear and distinct ideas in Descartes, the proposition “Every triangle hath its three angles equall to two right angles” seems too complex to follow from the (dictionary-induced!) intuition of triangle so self-evidently as to want no proof. It is almost as if Hobbes jettisoned the very concept of proof, and denies all distinction between Euclidean common notions, postulates and definitions on the one hand, and propositions needing theorems on the other. Therefore, the understanding of the hearing man must take the shape of a revelation or a mystic vision. As with all revelations, the resulting proposition can be communicated to others, but, there being no proof, its truth needs to be re-experienced by others by their own mystic vision every time they look into it.

From no possible angle can Hobbes’s comparison of two geometers be regarded as evidence for his claim that one needs “Speech” in order to reason. On the contrary, sometimes it seems that speech gets in the way. For the entire paragraph consists of nothing more than what Hobbes himself names “Absurdity, or senselesse Speech” that takes place “when we Reason in Words of generall signification, and fall upon a generall inference which is false” [112-13]. “The first cause of Absurd conclusions,” explains Hobbes, “I ascribe to want of Method; in that” people who speak absurdities “begin not their Ratiocination from Definitions” [114]. All the other “Absurd assertions,” as Hobbes defines them, derive from improper language use, such as “the giving of the names of the accidents of bodies without us, to the accidents of our own bodies; as they that say, the colour is in the body; the sound is in the ayre,” such confusion between what Descartes calls primary and secondary qualities apparently regarded as more a matter of language rather than physics or neuroscience.

Absurdity is to be distinguished from Error, which is what happens “when a man reckons without the use of words,” dealing merely with particulars [112-13]. Apparently, to make an error for Hobbes is not tantamount to doing something that is false, for, as he notes elsewhere, “where Speech is not, there is neither Truth nor Falsehood” [105]. Yet neither is there error in “Speech,” since to call a statement “absurd” is so much as to say that it is “without meaning”: that it consists of “words whereby we conceive of nothing but the sound” [113]. Indeed, when we begin our ratiocination with definitions, and proceed by carefully analyzing our names for their consequences—Hobbes glosses reason as “nothing but Reckoning (that is, Adding and Subtracting) of the Consequences of generall names” [111],—we cannot be wrong. “When we make a generall assertion,” explains Hobbes, “unlesse it be a true one, the possibility of it is inconceivable” [113]. Just as in geometry, where the reductio ad absurdum proof demonstrates a proposition when it shows that its complement, by implying a contradiction, cannot have geometrical meaning or being, so all that can be properly articulated for Hobbes must necessarily exist as true, and everything that cannot be properly articulated is entirely void of meaning, the equivalent of дыр бул щыл.

Hobbes believes that world is as rational as mathematics, and that “Speech” can be rendered as rational as the language of geometry, with the result that the rationality of world and word coincide. Hence it suffices to set words with correct meaning in a correct order to arrive at the unambiguous truth. It is not certain why he does not employ this laudable strategy in the section of The Leviathan under discussion.



1. Euclid X.117 is usually spoken of as proving the irrationality of the square root of 2, since that such is the diagonal of a square whose side is 1. Yet the idea of “irrational number” is an anachronism in Greek mathematics, which instead focused on incommensurability of magnitudes. For details of proof, see T. L. Heath, The Thirteen Books of Euclid’s Elements (Cambridge UP, 1908), vol. 3, p. 2.

2. For Saccheri on the Parallel Postulate, see Evert W. Beth, Mathematical Thought: An Introduction to the Philosophy of Mathematics (Dordrecht, Netherlands : Reidel, 1965), pp. 8-12.

3. Abu Hamid al-Ghazali, Munkidh min al-Dalal (Confessions, or Deliverance from Error), trans. Claud Field. Text taken from Internet Medieval Sourcebook: Online Reference Book for Medieval Studies, ed. Paul Halsall, Fordham University Center for Medieval Studies (, accessed January 6, 2010).

4. As quoted by C. B. Macpherson in his edition of the Leviathan (London: Penguin, 1985), pp 17-18, the edition used in my essay, and one to which all page numbers in square brackets refer.

5. The proposition of Euclid I.32 is: “In any triangle, if one of the sides is produced, then the exterior angle equals the sum of the two interior and opposite angles, and the sum of the three interior angles of the triangle equals two right angles.” See Heath, vol. I, pp. 316-22 for proofs and history.

6. For thought-experiment fraud, see “Kripke resigns as report alleges he faked results of thought experiments,” (February 22, 2012),

7. For ramifications of deconstruction to Deaf studies, see H-Dirksen Bauman, “Listening to Phonocentrism with Deaf Eyes: Derrida’s Mute Philosophy of Sign Language,” Essays in Philosophy (2008) 9:1:2 ( I would like to thank my wife, Oya Ataman, for directing me to Bauman’s work, as well as this issue in general.

8. Certainly at the Ottoman court, where the Venetian diplomat Ottaviano Bon observed that “that in the Serraglio, both the King and others can reason and discourse of any thing as well and as distinctly, alla mutesca, by nods and signes, as they can with words,” an English translation of his account published in 1625 (qtd. in M. Miles, “Signing in the Seraglio: Mutes, dwarfs and jesters at the Ottoman Court 1500-1700,” Disability & Society (2000) 15:1:115-34). A version of the article in paronomastic orthography available at

9. See Walter Ong, Orality and Literacy: The Technologizing of the Word (NY: Routledge, 2002), 49-54, on experiments carried out by Alexander Luria during collectivization with Central Asian farmers of varying degrees of literacy (published in English as Luria, Cognitive Development: Its Cultural and Social Foundations [Harvard UP, 1976]). When Luria showed his subjects geometrical pictures, summarizes Ong, “Illiterate (oral) subjects identified geometrical figures by assigning them the names of objects, never abstractly as circles, squares, etc. A circle would be called a plate, sieve, bucket, watch, or moon; a square would be called a mirror, door, house, apricot drying-board. Luria’s subjects identified the designs as representations of real things they knew. They never dealt with abstract circles or squares but rather with concrete objects. Teachers’ school students on the other hand, moderately literate, identified geometrical figures by categorical geometric names: circles, squares, triangles, and so on… They had been trained to give school-room answers, not real-life responses” (Ong, 50). There is no way Hobbes’s “Speech”-deprived person would recognize a triangle as such.