The Skeptical Attack on the Hypothetical Method (M III 7-18)

As we've seen, there is some question about the meaning of hypothesis and archē in Against the Geometers, and the answer to this question will affect our understanding of what Sextus is doing in this book. Although Barnes gives reasons that suggest the

hypotheses Sextus attacks at M III 7-18 are geometric first principles, that is, their axioms and postulates, I think that the arguments are better interpreted as attacking the

geometer's hypothetical method. In this section, I will begin by arguing for this claim, and then I will interpret the arguments themselves in light of this.

There are a number of reasons to think that Sextus has the hypothetical method in mind when he attacks hypothesis in Against the Geometers. He begins the book, saying:

Since the geometers, when they see the multitude of difficulties [aporiai] that pursue them, flee for safety to a thing that seems secure and free from danger, namely, postulating from hypothesis the starting points [archai]³³³ of geometry, it would be well for us also at the beginning of the refutation against them to set down the argument about hypothesis. (M III 1)³³⁴

The first clue that Sextus wants to attack the geometric method is his claim that the

333Barnes (1990, 95) translates the word ἀρχὴ here as “first principle”, and it can certainly mean a foundational, indemonstrable principle of a science. However, it could also simply mean the first premise in a demonstration. Thus, I use the phrase “starting point” in order to capture the ambiguity of the term.

334Ἐπεὶ οἱ γεωμέτραι συνορῶντες τὸ πλῆθος τῶν ἐπακολουθούντων αὐτοῖς ἀποριῶν εἰς ἀκίνδυνον εἶναι δοκοῦν καὶ ἀσφαλὲς πρᾶγμα καταφεύγουσι, τὸ ἐξ ὑποθέσεως αἰτεῖσθαι τὰς τῆς γεωμετρίας ἀρχάς, καλῶς ἂν ἔχοι καὶ ἡμᾶς τῆς πρὸς αὐτοὺς ἀντιρρήσεως ἀρχὴν τίθεσθαι τὸν περὶ τῆς ὑποθέσεως λόγον. (M III 1)

geometers seek safety in an activity (i.e. postulating). He does not say that they think their hypotheses are secure; it is the postulating of claims that reassures them. Moreover, while the archai referred to here could mean the geometric axioms, as Barnes suggests, when Sextus refers to the geometric archai at M III 18, he goes on to talk about the conceptual starting points of geometry, namely, the point, line and plane. These are the very things that Sextus says have the aporiai:

While we proceed with what is next, let us teach that the starting points [archai]

of their art are false and unconvincing. Indeed, since there are many possible things to say for this purpose, as we said when we began the treatise, the matters with difficulty [aporiai] will be brought to them, those matters which, when they are destroyed, will also destroy together the remainder [of the craft]. So, since if the starting points are discredited, the particular demonstrations will not be able to make progress, we should recount the fitting [arguments] against the starting points. (M III 18 – my emphasis)³³⁵

If Sextus is talking about the same aporiai and the same archai at M III 1 and 18, then we must conclude that at M III 1 Sextus says the geometers recognize that there are

conceptual difficulties with the point or the line or the plane, yet they think they can avoid these problems by postulating them from hypothesis. My question is: What is it about positing something from hypothesis that makes them think they can avoid these difficulties?³³⁶ There must be something about postulating that helps geometers avoid the

335μετελθόντες δὲ ἑξῆς διδάσκωμεν ὅτι ψευδεῖς καὶ ἀπιθάνους αὐτῶν συμβέβηκεν εἶναι τὰς ἀρχὰς τῆς τέχνης. καὶ δὴ πολλῶν εἰς τοῦτο δυναμένων λέγεσθαι, ὡς ἐναρχόμενοι τῆς ὑφηγήσεως εἴπομεν, τούτοις προσαχθήσεται τὰ τῆς ἀπορίας ὧν ἀναιρουμένων καὶ τὰ λοιπὰ

συναναιρεθήσεται. ἐπεὶ οὖν τῶν ἀρχῶν διαβληθεισῶν οὐδὲ αἱ κατὰ μέρος ἀποδείξεις αὐτοῖς δύνανται προκόπτειν, λέγωμεν τὰ ἁρμόζοντα πρὸς τὰς ἀρχάς. (M III 18)

336Someone might object at this point that I am taking Sextus at his word and assuming that he speaks for the geometers. Nothing could be further from the truth. I am not at all concerned here with what actual geometers think. What I am trying to understand is Sextus' reasoning. Another way to put my question is this: Why would Sextus say that geometers think they can avoid the aporiai by postulating

hypotheses unless he thinks that they think the hypothetical method brings certainty. Of course, Sextus does not himself believe that hypotheses bring certainty, but he does not have reason to attack

hypothesis unless he thinks that the geometers take it to be a way of securing the truth. And he clearly does think that they think that.

difficulties, and I suggest that it is the fact that they think their assumptions will be validated; they intend to prove their assumptions through the joint method of analysis and synthesis.³³⁷

Further evidence for my view – that Sextus is attacking the hypothetical method and not the axioms and postulates of geometry – comes at the end of the section. When he sums up his arguments against hypothesis, Sextus says: “And from these [arguments]

it is sufficiently established that the professors do not do well when they accept the starting points [archai] of their demonstrations and of their theorems from hypothesis, saying 'let it be granted' [dedosthō]” (M III 17b).³³⁸ It is the word dedosthō that interests me here. This particular expression never appears in Euclid (this surprised me).³³⁹ However, it is an expression used by other geometers and astronomers.³⁴⁰ Archimedes uses it in some of his proofs. Take, for example, the following problem statement:

So similarly we shall prove that given two unequal magnitudes and a sector it is possible to circumscribe a polygon around the sector and to inscribe another similar to it, so that the circumscribed has to the inscribed a smaller ratio than the greater magnitude to the smaller. (de Sphaera et Cylindro 1.6, trans. Netz)

Archimedes begins this proof saying, “Let there be given [dedosthō] a circle A, and some area, B.” In this case, the posit is simply the existence of a circle and an area. Nothing is

337This helps make sense of the rest of Book III as well: The book is roughly divided into the section attacking the hypothetical method (M III 3-18), and the section attacking the presuppositions of the hypotheses (M III 18-107) with a coda attacking a couple of theorems (M III 108-113).

338Καὶ δὴ ὅτι μὲν οὐκ εὖ ποιοῦσιν οἱ ἀπὸ τῶν μαθημάτων ἐξ ὑποθέσεως λαμβάνοντες τὰς ἀρχὰς τῆς ἀποδείξεως καὶ ἑκάστου θεωρήματος, ἐπιφθεγγόμενοι τὸ ‘δεδόσθω’, διὰ τούτων αὐτάρκως κατεσκεύασται· (M III 17b)

339Sextus himself uses the expression several times; it is one of the ways that he concedes a point against which he has been arguing in order to go on and argue something further. For example, in Against the Rhetoricians, Sextus spends several sections (M II 89-105)developing arguments against the idea that rhetoric has three parts: the juridical, the deliberative and the laudatory. But then, he grants (δεδόσθω) that these are the parts of rhetoric at M II 106 in order to argue against the role of demonstration in rhetoric. See also M VII 381, VIII 183, 402, X 255.

340Ptolemy uses it several times in the Almagest 1,1.90,93, 241,385; 1,2.201.

claimed about these figures. Other examples in Archimedes are similar.³⁴¹ Although he does not use the term dedosthō often, Archimedes commonly begins a proof using terms like estō (“let it be...”), or ekkeisthō (“let it be set out...”) when he introduces a figure, and indeed, these are types of third person imperatives that Euclid utilizes to posit a figure at the beginning of a proof. Obviously, these posits are not geometric axioms. Rather, they represent the starting point of a proof that will solve a problem or demonstrate a

theorem.³⁴²

It seems strange to me – at least initially – that Sextus thinks his arguments against hypothesis apply to these suppositions. Consider Euclid's proof in Elements I 10,

“to bisect a line”: The proof begins “Let there be a finite straight line AB.”³⁴³ What is the supposition? It cannot be the imperative statement since Sextus claims that the skeptic will posit the opposite (M III 8), but we cannot make sense of the mode if he means the skeptic will demand “Do not let there be a finite straight line AB.” In what sense can conflicting imperatives be credible? Moreover, it would be strange to suggest that the skeptic would posit “there is not a finite straight line AB” to the geometers' “there is a finite straight line AB”, since, if the geometers assert anything, they only claim that we can conceive of such a thing. The geometric dedosthō is really a request for us to imagine a finite straight line. Put another way, the geometric proof at Elements I 10 is a

conditional proof. It shows that if there are any finite straight lines, then it is possible to bisect them. That is not to say that there are such things. In order to oppose the initial

341Archimedes de Sphaera et Cylindro 2.4; de Conoidibus et Sphaeroidibus 1.171, 198, 200; de Lineis Spiralibus 2.17, 19, 20, 21, 22; de Planorum Aequilibriis 2.111.

342On the distinction between problems and theorems, see Pappas Collection III 1 See also chapter 9 §4 in the introduction of Heath (1956, 1:124–129).

343I choose this example because Sextus argues that this is impossible at M III 109-111.

supposition, the skeptic must suggest that it is impossible for there to be a finite straight line; finite straight lines cannot exist (recall that Sextus explicitly argues that the

definition of a straight line is incoherent at M III 94-99).

Given all of this, I take it that the arguments against “positing something from hypothesis” at the beginning of Against the Geometers cannot be directed at the first principles of geometry (or against first principles, in general). Rather, the arguments are against a particular method of investigation used by the geometers to demonstrate their claims.

Now, let us turn to the arguments that Sextus makes. In what follows, I will claim that he uses a dialectical argument, which is meant to undermine the credibility of the hypothetical method. Sextus' arguments begin with a restatement of the mode of hypothesis:

Wherefore one ought to say straight away also that since those who accept something from hypothesis and without proof are satisfied by bare assertion [psilēi phasei]³⁴⁴ alone in relation to its trustworthiness, some one will ask them using a consideration of this sort. [8] Either the accepting of something from hypothesis is strong and firm with regard to its trustworthiness or it is

untrustworthy and weak. But if it is strong, then when the opposite [claim] is accepted by hypothesis, it will be credible and firm, so that we will posit

conflicting things. But if the opposite hypothesis is untrustworthy on the basis of its being accepted from hypothesis without proof, then [the original claim] is also untrustworthy on that basis, so that we will posit neither of them. So now [toinun], it is not the case that one ought to accept anything from hypothesis. (M III 7-8, cf.

344Another clue that Sextus does not have Aristotelian hypotheses in mind here is that he clearly thinks that the geometers accept their hypotheses as “bare assertions” (psilēi phasei): As I've already argued, Aristotelian first principles are not bare assertions, even if they are indemonstrable. They are claims which are understood through other means; in the case of Aristotle, they are known through eisagogē.

As I indicated, later philosophers had other theories about how we come to know the immediate foundations. But none of these theorists would claim to be “satisfied by bare assertions.” In contrast, the initial posits of the geometers are bare assertions. When they posit the existence of a finite straight line, they offer no reason to suppose that such a thing is possible (except perhaps the evident plausibility of the definitions which are never mentioned in the proofs). See Barnes' (1990) discussion of “bare assertions” at 97-98.

PH I 173, M VIII 370)³⁴⁵

This passage mirrors the description of the hypothetical mode in the Outlines (PH I 173), and it seems quite plausible that Sextus means to use the mode as the starting point of his dialectical argument against hypotheses. It begins with Sextus claiming that the skeptic can simply posit an opposing claim when the geometers hypothesize something. If the pragmatic interpretation of the modes is correct, then the hypothetical mode describes a dialectical move that the skeptic makes in order to raise the question regarding the credibility of the geometers' suppositions. What is interesting about its use in the passage above is that the constructive dilemma is built around the dialectical moves of the skeptic. What exactly is the conclusion of the argument supposed to be? Barnes suggests that the conclusion is that “one ought not accept anything from hypothesis.”³⁴⁶ But this does not follow immediately from what Sextus says about the mode. What he claims is that the skeptics will take action depending on what the dogmatic geometers say. If they think their hypotheses are trustworthy, then the opposite should also be trustworthy, “so that [hōste] we will posit conflicting things.” The “hōste” clause indicates a result, not an inference. And if the opposite is untrustworthy, then so will the original claim be, “so that [hōste] we will posit neither of them.” For each horn of the dilemma, Sextus tells us how the skeptics will respond. Barnes is right that the passage ends by suggesting that one ought not suppose anything from hypothesis, but the particle [toinun] – although it has an

345διόπερ εὐθὺς ῥητέον ὅτι καὶ ἐπεὶ οἱ ἐξ ὑποθέσεως λαμβάνοντές τι καὶ χωρὶς ἀποδείξεως ψιλῇ μόνον ἀρκοῦνται φάσει πρὸς τὴν ταύτης πίστιν, πεύσεταί τις αὐτῶν τοιούτῳ τινὶ χρώμενος ἐπιλογισμῷ. [8] ἤτοι ἰσχυρόν ἐστι καὶ βέβαιον πρὸς πίστιν τὸ ἐξ ὑποθέσεώς τι λαβεῖν ἢ ἄπιστόν τε καὶ ἀσθενές. ἀλλ' εἰ μὲν ἰσχυρόν, καὶ τὸ ἀντικείμενον ἐξ ὑποθέσεως ληφθὲν πιστὸν γενήσεται καὶ βέβαιον, ὥστε θήσομεν τὰ μαχόμενα. εἰ δὲ ἐπὶ τοῦ τὸ ἐναντίον ἐξ ὑποθέσεως λαμβάνοντος χωρὶς ἀποδείξεως ἄπιστόν ἐστιν ἡ ὑπόθεοις, ἄπιστος γενήσεται καὶ ἐπ' ἐκείνου, ὥστε οὐδέτερον αὐτῶν θήσομεν. οὐ τοίνυν ληπτέον ἐστὶν ἐξ ὑποθέσεώς τι. (M III 7-8)

346Barnes (1990, 100).

inferential force – does not indicate that Sextus has conclusively demonstrated anything.

Rather, the dilemma is meant to pressure the dogmatist to show why some hypotheses are credible and others are not, given that both P and not-P appear to be on equal footing insofar as they lack any support.

According to Barnes' interpretation, the remaining arguments in this section (with the exception of the next argument) all rely on the general principle that, if one can posit something without support, then one can posit anything, and that this is enough to undermine the geometer's archai.³⁴⁷ As I've just suggested, there is some truth to the idea that the skeptics posited an opposing hypothesis in order to raise the question about the need to justify the initial hypothesis. What I think Barnes has missed is the way Sextus presents the arguments as dialectical. This becomes clearer later in the passage in which Sextus suggests that his opponents will respond, for example, by raising objections to his arguments (e.g. M III 14).

Given the way that Sextus frames the discussion, and given that he includes remarks about how his opponent will respond, the arguments of this section are meant to operate in the context of a debate between the skeptic and the dogmatic geometer

regarding the question of where hypotheses derive their credibility. In order to explain this, first, I will outline the section. Then, I will go into each argument in detail. Here is how this section of M III is organized:

I. The hypothetical mode (M III 7-8): Asks where hypotheses gain their credibility.

II. Credibility of an hypothesis cannot come from:

347Barnes (1990, 102).

1. Its relation to the truth (M III 9-10).

2. Its place in the proof or its role in investigation (M III 11-12).

3. Its being (merely) posited (M III 13).

III. Final Dialectical Exchange:

1. Dogmatic Objection: Credibility comes from the conclusion reached (M III 14).

2. Skeptical Response: Credibility cannot come from conclusion (M III 14-17).

Once Sextus introduces the initial question using the hypothetical mode, he considers a number of responses by the geometers, and he argues that each possible source of credibility for the hypothesis is insufficient. The debate ends with a final dogmatic objection and skeptical rejoinder.

The first argument after the reiteration of the hypothetical mode responds to the claim that we should trust the geometric hypotheses because of their relation to the truth.

If the hypotheses are true, then they are credible; if they are false, then they are not.

Against this, Sextus argues that if we know the claim to be true, then we ought not hypothesize it insofar as supposing something to be true is only appropriate when its truth is unknown. Here is how Sextus puts it:

Also, the matter hypothesized is either true and and the sort of thing we

hypothesize it to be or else it is false. But if it is true, we should not posit it as a hypothesis, that is, fleeing for protection to a matter full of suspicion. Rather, we should accept it for itself since no one hypothesizes what is real and true, just as it is not the case that [we hypothesize] “it is now day” or “I am discussing, and I am breathing.” For the obviousness of these matters holds firm from itself as a claim and is not in doubt as a hypothesis. So that, if the matter is true, we should not posit it as if it is not true. [10] But if it is not like this, but has been established as false, no help will emerge from the hypothesis. For even if we hypothesize it

innumerable times, the conclusion will not follow from rotten foundations, as they say, since the investigation started from non-existent sources. (M III 9-10, cf. PH I 173, M VIII 371)³⁴⁸

Sextus is claiming that credibility [pistis] is not an absolute objective measure – like truth – but a relation. What is required for credibility is not that a given hypothesis be true, but that it be true and be known to be true. But, if it is known to be true, then there is no reason to suppose it as a hypothesis. Barnes says that he cannot make sense of this argument because he thinks it amounts to begging the question.³⁴⁹ If the issue is whether hypotheses can be trusted, the claim that hypothesizing is “a matter full of suspicion”

appears to beg the question.³⁵⁰ Moreover, Barnes adds, Sextus seems to distinguish between “hypothesizing” and “accepting it for itself” as if these both did not simply amount to supposing something to be true.³⁵¹

348καὶ μὴν τὸ ὑποτιθέμενον πρᾶγμα ἤτοι ἀληθές ἐστι καὶ τοιοῦτον ὁποῖον αὐτὸ ὑποτιθέμεθα ἢ

349Barnes (1990, 100–101). Dye and Vitrac (2009) agree that this argument is difficult to understand (197). They suggest that Sextus may be toying with the ambiguity of the term hypothesis, but they also develop a more promising interpretation: The claims that Sextus suggests are conspicuous and known from themselves are those which are part of the appearances – like “it is day” or “I am discussing and I am breathing.” These are not claims that require demonstration. But everything else – those things which are not part of the appearances – are among the “unclear things” (adēla) which must be demonstrated. Thus, Dye and Vitrac claim that Sextus' argument here assumes that the principles of geometry are open to dispute which is precisely what Sextus shows in remainder of the book (197-199).

350Of course, it is not begging the question to argue that one ought not hypothesize because suppositions are not to be trusted. But insofar as it is a substantive issue whether they should be trusted, one cannot use their suspicious nature as support. Note that if these arguments are meant to be directed against hypothesizing as a method of inquiry rather than against Aristotelian archai, as I've already argued, it helps make sense of why Sextus would consider it reasonable to assert that a hypothesis is full of suspicion. This does not explain why the argument does not beg the question, but it is yet one more reason to think that Sextus is not considering hypotheses as indemonstrable first principles.

351Barnes (1990, 101) claims that “hypothesizing” and “accepting” both amount to supposing something to be true. But surely we can distinguish between assuming something to be true for the purposes of discussion and actually thinking it to be true. For one, the latter will require assent and may also involve

But if my outline of the dialectic above is correct, the guiding question of this section is whence does the credibility of a hypothesis come? And if the geometer

responds that the hypotheses are true and known to be true, then a tension arises because the hypothetical method was a method of inquiry that moves from something unknown to something known. The geometer cannot claim that the hypotheses are known to be true and still use them as hypotheses. If they are known to be true, then one need not “grant”

responds that the hypotheses are true and known to be true, then a tension arises because the hypothetical method was a method of inquiry that moves from something unknown to something known. The geometer cannot claim that the hypotheses are known to be true and still use them as hypotheses. If they are known to be true, then one need not “grant”