(Cross-posted at M-Phi)

In his commentary on Euclid, the 5^{th} century
Greek philosopher Proclus defines indirect proofs, or ‘reductions to
impossibility’, in the following way (I owe this passage to W. Hodges, from
this paper):

Every reduction to impossibility takes the contradictory of what it intends to prove and from this as a hypothesis proceeds until it encounters something admitted to be absurd and, by thus destroying its hypothesis, conﬁrms the proposition it set out to establish.

Schematically, a proof by reduction is often represented as follows:

[~A]

.

.

.

⊥

------

A

It is well know that indirect proofs pose interesting philosophical questions. What does it mean to assert something with the precise goal of then showing it to be false, i.e. because it leads to absurd conclusions? Why assert it in the first place? What kind of speech act is that? It has been pointed out that the initial statement is not an assertion, but rather an assumption, a supposition. But while we may, and in fact do, suppose things that we know are not true in everyday life (say, in the kind of counterfactual reasoning involved in planning), to suppose something precisely with the goal of demonstrating its falsity is a somewhat awkward move, both cognitively and pragmatically.

Even in the relevant circles of specialists, quite a few people have issues with indirect proofs, most famously intuitionists who reject double-negation elimination – the crucial step which goes from the rejection of ~A to the assertion of A. It is also often said that Frege’s account of inference as going from true statements to true statements leaves no room for indirect proofs (but here is a recent paper by Ivan Welty countering this claim). So even within mathematics and logic, indirect proofs are somewhat controversial.

If we accept that indirect proofs are a bit of an oddity even within mathematics, it makes sense to ask how on earth this argumentative strategy might have emerged and established itself as one of the most common ways to prove mathematical theorems. Now, as some readers may recall, my current research project focuses on ‘the roots of deduction’, adopting the hypothesis that we need to go back to deduction’s dialogical origins to make sense of the whole thing (as discussed here, for example). And here again, it seems that the dialogical, multi-agent perspective offers fresh insight into the nature of indirect proofs.

Assume a dialectical context in which two participants are
disputing on a certain topic, and let us call them 1 and 2 and B to keep it
neutral. Then imagine that 1 wants to convince 2 of proposition A; how can she
go about? Well, she can propose ~A and see if 2 takes the bait. It is important that ~A be put
forward in the form of a question (which is indeed how such disputations often
began in ancient Greece, as attested for example by Aristotle’s *Topics*), so that by accepting ~A, 2 commits to
its truth *but not 1*; 1 has merely put
it forward as a question and thus has herself not endorsed ~A. 1 can now
proceed to show that something absurd follows from the acceptance of ~A, because this
is not *her* position; it is 2’s
position. By showing that something absurd follows from ~A, 1 in fact shows that it
was a bad idea for 2 to accept ~A in the first place. There is still the contentious last step
which goes from ‘accepting ~A is a bad idea’ to ‘accepting A is a good idea’. But 1 has not
done anything pragmatically incoherent because she herself never committed to ~A.

In legal contexts, reductio arguments are used in much the
same way. The prosecution may claim A (the defendant was at the crime scene),
and the defense may then show that, given additional background information, A
leads to absurdity (say, to the possibility of traveling between Paris and
London in less than 30 min). (Welty’s paper has a similar legal example.) So
what you show as entailing absurdity in a reductio argument is in fact the *position of your opponent*, not your own
position (not even your own assumption). The adversarial component is crucial
to understand what it means to prove something indirectly; it makes the
postulation of the strange speech-act of supposing precisely that which you
want to prove to be false superfluous. In a purely mono-agent context, in
contrast, she who formulates an indirect proof has to play awkwardly
conflicting roles. (Naturally, it is perfectly possible to formulate an
indirect proof on your own, but this is a consequence of what I describe as the
‘internalization of opponent’ by the method itself.)

I think that this multi-agent, dialogical account of
indirect proofs is conceptually appealing on its own, but within the Roots of
Deduction project, we (Matthew Duncombe, Leon Geerdink and myself) are also
investigating the historical plausibility of the hypothesis. For now, it is
interesting to notice that, in the *Prior Analytics*, Aristotle makes extensive
use of indirect proofs, as is well known, but also that he often uses
dialectical vocabulary to explain the concept of an indirect proof. (In fact,
he uses dialectical vocabulary throughout the text.) UPDATE: here is a subsequent post I wrote on indirect proof in the *Prior Analytics*.

(A cool coincidence is that just yesterday Mic Detlefsen invited me to present at his PhilMath Intersem colloquium in Paris in June, precisely on the topic of the history of indirect proofs. So there is much to be done on the topic for me, but for now this is my starting point.)

You can find reductios in the wild by doing a google search for "If that were true." I think the search undermines the hypotheses that all reductios are mathematical or legal, but it may confirm some of your other claims.

From, ultimately, a novel by Sherman Alexie

"I know, I know, but some Indians think you have to act white to make your life better. Some Indians think you become white if you try to make your life better, if you become successful."

"If that were true, then wouldn't all white people be successful?"

href="http://www.dailywritingtips.com/the-problem-with-grammar-check/">From a complaint about Word's grammar checker

They can compute, but they can’t think. Here’s where Word went wrong:

It assumed that the phrase “in this poem and without emphasis on them” was a compound phrase with the same structure as “on this page and on the next,” for example, and that this sentence could end with this phrase.

If that were true, “this poem has little to no meaning” would be an independent clause that could stand on its own.

From a discussion of the deficit

You know, if that were true...

That the tax cuts where the huge contributor to this, then raising taxes should fix it, but it won't. You cannot raise taxes on the rich high enough to cover the annual deficit.

Posted by: Mike Jacovides | 09 January 2013 at 03:49

I'm not saying that counterfactual, suppositional reasoning is not common, i.e. assume something you know not to be the case to see what follows from it. Even young children do something like that (e.g. the work of Paul Harris). But this is different from supposing something with the explicit goal of defeating it by showing that something absurd follows from it.

Naturally, if I define the concept of a reductio argument rather narrowly (which I do), then it will be less likely that one can find such things 'in the wild'. But I'm thinking specifically of how indirect proofs are formulated in mathematics. The point is of course that these reasoning practices of supposing something you know not to be the case are the precursors of what then became regimented within mathematical practice as indirect proofs.

Posted by: Catarina Dutilh Novaes | 09 January 2013 at 03:56

Catarina, reductios may also occur in the history of physics, sometimes by way of thought experiment. Galileo was a master at this (e.g., the "two stones tied together" thought experiment that reduce Aristotle's physics to absurdity).

Posted by: Eric Schliesser | 09 January 2013 at 08:16

Catarina: Mike Jacovides examples don't seem to be just cases of seeing what follows from a supposition. It seems the point is to show that what follows from the supposition is something that contradicts what we independently know (in the 1st example, that not all white people are successful), in order to refute the supposition. Or so it seemed to me. The speaker in Alexie's novel doesn't spell out only because it's clear what follows. No?

Posted by: Steven Gross | 09 January 2013 at 09:42

Though intuitionists reject negation elimination, they typically accept negation introduction:

H, A |- absurdity

--------------

H |- ~A

But this inference also requires supposing that something is true in order to conclude that it is false. So I don't think the controversy in mathematics over indirect proof really supports your thesis about the psychological oddity of indirect proof.

That said, it sure is difficult to get intro logic students to grok negation introduction and elimination. I'd find it a bit comforting if some of that difficulty was down to the psychological unnaturalness of the inferences.

Posted by: Joshua Brown | 09 January 2013 at 10:17

Thanks for this post Catarina. This is perhaps somewhat off topic, so my apologies in advance, but Piers Morgan's recent interview with Alex Jones has, at one level, the appearance of an indirect reductio argument. Alex Jones is the guy who launched the deport Piers Morgan campaign at the White House website in response to Morgan's advocacy of gun control in response to the Sandy Hook shooting. Since Morgan argues for G (gun control), by bringing Jones on his show for an interview he allowed him to present the non-G case, but then Jones is a conspiracy theory nutcase and any reasonably informed person would have quickly seen that his position was absurd, but absurd not in a formal contradiction sense (though there was plenty of that too) but rather as nonsensical, unsubstantiated, and outright loony from the perspective of common sense. So is the viewer of the interview suppose to conclude G? Probably not, even if this is what Morgan hoped for, which I doubt, for what Morgan probably wanted and got was for Jones to undermine his own credibility - a self-inflicted ad hominem so to speak - so that we could see the absurdity of his deportation request. I suspect reductios in the wild will be more complex and rely more on the perception of absurdity/violation of common sense response rather than the classical reductio in the logical sense - the case of the court room prosecutor and Eric's Galileo example are confined to fairly controlled environments - but perhaps there is a genealogical relationship between them.

Posted by: Jeff Bell | 09 January 2013 at 10:23

As the first commenter already pointed out, reductio occurs naturally in counterfactual constructions. Besides the "If that were true, then ..." variety, one finds things like the following:

--- "If he were (already) in London, he would have called by now. He didn't call. So he is not (yet) in London."

--- "If he had the measles, we would see red spots on his body. We don't see red spots. So he doesn't have the measles."

--- "If someone had walked on that beach, there would be footprints in the sand. There are no footprints. So no one walked there."

Indeed, I am tempted to think that this is one of the reasons we find counterfactuals useful: they allow us to conveniently express reductio arguments.

Are you saying that people "in the pub" would not use this style of argument?

Posted by: Ben | 09 January 2013 at 13:32

To my mind, these are not reductio arguments, these are occurrences of modus tollens: A --> B, not-B, so not-A. That's something quite different from an indirect proof (or proof by absurdity) as described above.

And as for modus tollens, as you may know the rates of success for modus tollens in reasoning experiments are typically well below those for modus ponens, so apparently it is not a reasoning pattern that everybody masters completely.

Posted by: Catarina Dutilh Novaes | 09 January 2013 at 13:49

My first reaction when reading your comment was to think of an ad hominem argument rather than a reductio. In fact, I think this illustrates nicely the differences between something that may look like a reductio in the wild, with all these different pragmatic levels, vs. what happens in an indirect proof in math.

Posted by: Catarina Dutilh Novaes | 09 January 2013 at 13:52

Fair enough, in fact this is the intuitionistic *definition* of the negation:

A --> bottom <==> not-A

Posted by: Catarina Dutilh Novaes | 09 January 2013 at 13:53

Yes, but I would count physics as a rather regimented, specialized context too. It would be interesting to see if such reductio arguments were used in physics before mathematics became the underlying methodology of physics.

Posted by: Catarina Dutilh Novaes | 09 January 2013 at 13:55

The Galileo argument I allude to is barely mathematical--it's really conceptual. (I think it is disputed if it really succeeds, although rhetorically it is very arresting.)

Posted by: Eric Schliesser | 09 January 2013 at 14:01

Often I know a mathematical theorem T is true (say, I read it in a textbook), but I don't know how to prove it, so I might think, "Well, let's see what happens if I suppose not-T, and if that turns out to lead to a contradiction, then I'll have a proof of T."

Would you consider that a proof by contradiction, or an instance of modus-tollens/counterfactual/suppositional reasoning?

Posted by: Richard | 09 January 2013 at 16:06

Also, suppose I *don't* know whether T is true or not, but I go through the same process (assume not-T and see what happens. If I get a contradiction, then I know that T is true). Would that count as an indirect proof or something else? And what (if anything) is the difference between this case and the former one I offered? Is it whether or not I know the truth of T beforehand that matters?

Posted by: Richard | 09 January 2013 at 16:11

The difference between modus tollens and reductio (in an informal argument) is subtle. In the first case, you reason to something you know is false, and in the second, you reason to a contradiction. But its a subtle difference because if you have the opposite of the false thing as a premise, then the false thing GETS you a contradiction right away. And in an informal setting, its not clear which of these two, exactly, is going on. So, Catarina: I think you are hanging a lot on a thin peg.

Posted by: Eric Winsberg | 09 January 2013 at 16:27

Although a speaker might introduce a supposition with the specific intention of showing it is false, I am skeptical that that intention (or the expression of the existence of such an intention) is a part of the speech act itself, because the linguistic competence that the listener employs in order to grasp and respond appropriately to that initial introduction does not depend on whether the argument ends up being a reductio or a long hypothetical deduction. Even the speaker might not know where the argument is going to go when the supposition is introduced.

Posted by: Dan Kervick | 09 January 2013 at 17:56

"It seems the point is to show that what follows from the supposition is something that contradicts what we independently know (in the 1st example, that not all white people are successful), in order to refute the supposition."

I independently know that ¬⊥, don't I? Or, in general, that ¬(some absurdity).

Posted by: Ben Wolfson | 09 January 2013 at 19:26

I suppose that in many cases you suppose that p and end up being forced to conclude that ¬p, rather than being forced to conclude that ¬q, where q is something else. But the vehicle is something independently known. E.g. to prove that all prime numbers are congruent to either one or five mod six, one step might be:

Suppose that p is prime, and that p is congruent to four mod six. Then there exists a number n such that 6n+4 = p. This may also be written as 2(3n+2) = p.

What do we say now? That therefore p is composite, hence not prime, and prime? Or that this contradicts something we independently know, namely that no prime number is composite?

This seems quite parallel with Jacovides' examples.

You can find more in a similar spirit by searching for "QED" along with "if that were true", e.g. here:

http://messages.yahoo.com/Religion_%26_Beliefs/Creation_vs._Evolution/threadview?m=tm&bn=1605812966&tof=4&rt=2&frt=2&dir=b&ri=131542&t=c

"""

robert: An organism constructed according to a complicated mathematical formula is evident of intelligence.

rex: If that were true, then everything would be designed, because nearly everything is constructed according to complicated mathematical formulas. That's what physics and chemistry are.

Do you think that humans "created" mathematics?

"""

These seem to be ordinary enough people engaged in reductio-style argumentation. (Not necessarily good argumentation!)

Or from the Straight Dope message boards:

"""

I read an article in a computer magazine the other day which said american cabling also could not deliver enough amps along with (because of?) the low voltage power design so a computer with two high end video cards, an overclocked processor etc would need two separate plugs into the wall to provide the wattage required. Is this true?

Steve

Q.E.D.

01-30-2007, 08:01 PM

Is this true?

No, not even close. If that were true, we wouldn't be able to use toasters or hair driers, since these can draw upwards of 1000 watts.

"""

Which one could continue: We can use toasters. So we can and can't use toasters. So ⊥. Hence it must be false.

We're more apt to continue it simply, "but we can, so it must be false".

Posted by: Ben Wolfson | 09 January 2013 at 19:40

Ben: Yes, you do know that. My point was that, in Mike's example, you could take the character to be reasoning as follows. From the supposition you get (it's claimed) that all white people are successful (no contradiction yet). The rest is left unsaid, but it's of course not the case that all white people are successful. So with *that* and the supposition, you can derive a contradiction; and then by reductio you can conclude the negation of the supposition. (Sorry if that was obvious and I'm missing your point!) --Catarina points out that you could construe the character as deploying modus tollens. On the other hand, as Eric points out, it's hard to know what to say about such a case of informal (not fully spelled out) reasoning. One might add that it's common, in formal systems, for modus tollens to be a derived inference rule, derived using reductio.

Posted by: Steven Gross | 09 January 2013 at 19:48

Anyone using 'QED', however inaptly (as in the examples you provide), is obviously trying to emulate mathematical discourse, which proves the point. I'd be interested to see if people having had no significant exposure to mathematical styles of argumentation would formulate similar arguments.

Posted by: Catarina Dutilh Novaes | 10 January 2013 at 01:29

I don't understand what you mean by 'you reason to something you know is false'. In a modus tollens, all assertions made (the premises) are presumed to be true: the conditional and the contradictory of the consequent of the conditional. So what is false here? In an indirect proof, the initial assertion/assumption cannot be put forward as true, as it is precisely what the agent wants to disprove. This is the (to my mind) fundamental difference that I am pointing out.

Posted by: Catarina Dutilh Novaes | 10 January 2013 at 01:36

"Anyone using 'QED', however inaptly (as in the examples you provide), is obviously trying to emulate mathematical discourse"

This strikes me as, at best, not even remotely obvious.

Posted by: ben w | 10 January 2013 at 01:43

Fwiw (not much, probably), that's pretty much what Wikipedia says about QED. Their example of an argument finishing with QED outside math is from Spinoza's Ethics, which avowedly emulates geometrical discourse.

http://en.wikipedia.org/wiki/Q.E.D.

Posted by: Catarina Dutilh Novaes | 10 January 2013 at 01:47

I wouldn't be surprised if a significant proportion of the colloquial uses of "QED" (to be honest, I don't know if "QED" is still used in mathematical discourse at all) come from people who haven't been exposed to mathematical styles of argumentation past high school, at which point, in the US, for the most part, you haven't really been exposed to proofs at all, though you might well have been exposed to people using "QED" to mean, more or less, "there's a nice, knock-down argument for you".

I would put only slightly less credence in the assertion that anyone talking about actualizing potentials is obviously trying to emulate Aristotelian discourse.

Posted by: ben w | 10 January 2013 at 01:50

"Fwiw (not much, probably), that's pretty much what Wikipedia says about QED. Their example of an argument finishing with QED outside math is from Spinoza's Ethics, which avowedly emulates geometrical discourse."

I quote from Wikipedia:

"""

In chapter six of The Hitchhiker's Guide to the Galaxy, by Douglas Adams, Q.E.D. is included in the following exchange:

The argument goes something like this: "I refuse to prove that I exist," says God, "for proof denies faith, and without faith I am nothing."

"But," says Man, "the Babel fish is a dead giveaway, isn't it? It could not have evolved by chance. It proves you exist, and so therefore, by your own arguments, you don't. QED."

"Oh dear," says God, "I hadn't thought of that," and promptly vanishes in a puff of logic.

"""

The example of Spinoza is specifically an argument outside math but within philosophy. There is no indication

even therethat anyone using "QED" is seeking to emulate mathematical discourse or mathematical reasoning, even if, in fact, that is what Spinoza was up to."One might add that it's common, in formal systems, for modus tollens to be a derived inference rule, derived using reductio"

I'm not sure how that cuts, since I read it as indicating that one can construe every use of modus tollens as a use of reductio, making the claim that the people Mike and I cite are using modus tollens and not reductio hard to make out. On the other hand, couldn't you also derive reductio from modus tollens? If I suppose p for reductio, eventually concluding q, which contradicts the already known ¬q, so ⊥, so ¬p, couldn't I also assume p for conditional introduction, eventually concluding p → q, and then using MT with the already known ¬q to conclude ¬p? (Maybe not! I don't know.)

Posted by: ben w | 10 January 2013 at 02:03

There is an interesting literature in math education (and the psychology thereof) pertaining to indirect proof (both proof by contradiction and proof by contraposition). Here is a recent exemplar, with some intriguing references and examples. I think you may find this useful. Catarina.

http://link.springer.com/article/10.1007%2Fs11858-008-0091-2?LI=true

Posted by: Branden Fitelson | 10 January 2013 at 02:09

That's very helpful, Branden, thanks! One of the things I want to do quite extensively in the Roots project is to look into the literature in math education, so that's spot-on

Posted by: Catarina Dutilh Novaes | 10 January 2013 at 02:12

That's what I figured, Catarina. Neat project! I will continue to follow it with interest (and admiration).

Posted by: Branden Fitelson | 10 January 2013 at 02:26

(I don't have time to think through the details concerning modens tollens above, but will just post this brief comment including some historical material and some shameless self-promotion - I hope you'll forgive me)

Proclus on reductio as "a kind of syllogism" (συλλογισμός τις), namely "the construction in accordance with the second kind of hypothetical [syllogism]" (ἡ πλοκὴ κατὰ τὸ δεύτερόν...τῶν ὑποθετικῶν) - and the second type is modus tollens. The example given (I quote Morrow's translation) is "if in triangles that have equal angles the sides subtending the equal angles are not equal, the whole is equal to the part. But this is impossible; therefore in triangles that have two angles equal the sides that subtend these equal angles are themselves equal" (In Eucl. 255-6). Of course this is not tollens, but Proclus apparently assumes that a disjunction (the sides subtending the equal angles are not equal or the sides subtending the equal angles are equal) is implied.

Philoponus is more elaborate (but not more useful, I think) when he discusses "demonstration through the impossible" and argues that it is always a combination of the categorical and the hypothetical. See his comments on An.Post. I 26, esp. 291, around n. 313 of the new English translation by Owen Goldin and Marije Martijn :) (http://www.amazon.com/Philoponus-Aristotle-Posterior-Analytics-1-19-34/dp/1780930909) This was a difficult passage to translate for us non-logicians (It seems Philoponus oversees the step from "not q, therefore not p", to "therefore not-p" - again, a disjunction is implied), comments are welcome!

Posted by: Marije Martijn | 10 January 2013 at 04:04

This is interesting, Caterina, but why tie agents 1 and 2 to participants in a dialogue, i.e. to individual people? Why not allow for the possibility that a single person considers several positions that can then operate as agents in her mind, more or less in line with Marvin Minsky's society of mind hypothesis? Then your dialogues could be monologues on a higher level of aggregation.

Posted by: Reinhard Muskens | 10 January 2013 at 05:45

Reinhard, this is pretty much *exactly* the idea of 'internalizing the opponent that I refer to above, but do not elaborate on (here). My general inspiration is the Vygotskian concept of internalization, which may (or may not, I don't know) be similar to the society of mind hypothesis that you mention.

Posted by: Catarina Dutilh Novaes | 10 January 2013 at 06:19

It seems to me that rather than saying that ordinary people don't use the reductio, we should perhaps say that what they lack is the clear distinction between the absurd in the sense of utterly false and the absurd in the sense of the formally contradictory.

Posted by: Zink | 10 January 2013 at 07:06

Thanks, Marije. I also hope to be able to look more closely into what the commentators say on the notion of proof to the impossible in the Prior Analytics (for example, at A 23), as it is a fascinating concept there too. It would be interesting to see if they (as Aristotle does) compare it to mathematical proofs.

Posted by: Catarina Dutilh Novaes | 10 January 2013 at 08:15

Following up on Marije Martijn's comment above, there's another remark by Proclus that might be of interest. At

In Euclid.69.76-77 (again quoting Morrow's translation) we read "[Euclild] also included reasonings of all sorts, both proofs founded on causes, and proofs based on signs". The word here translated as "signs" istekmeria. He brings this distinction up again at 206.15ff.Unfortunately, Proclus doesn't make it very clear what he's' getting at with this distinction. Apparently a proof based on construction (such as the simple proof that the sum of the angles in a triangle is a straight line) is somehow different from a proof based on definitions alone. But he doesn't really explicate why this is an important distinction.

Note also that

tekmerionis an Aristotelian rhetorical term. It's an ordinary Greek word meaning sign, mark, or token. In some contexts it can even be translated as "omen" insofar as what is read when the ashes of a burnt offering, or the entrails of a sacrifice, are interpreted is often described as atekmerion. Aristotle uses the term to mean "necessary sign". A little light googling to find the original text led me to this nifty page with text and translation (with some notes) of the relevant passage from theRhetoric: http://realisticthesis.blogspot.com/2011/12/i-rhetoric-216-signs-tekmeria-they-are.htmlSo, the question one might ask is whether a proof by contradiction, insofar as it introduces a statement to be refuted, counts as a "construction" analogous to the geometrical constructions one finds so often in Euclid? And does Proclus consider such a proof in geometry to be a sort of mathematical enthymeme?

Posted by: Cameron Majidi | 10 January 2013 at 10:02

I am very, very interested in this distinction between "proofs founded on causes vs. proofs based on signs", as it is potentially related to my previous work on formal languages. Do you have further references (secondary literature) on what Proclus may have meant? Thanks!

Posted by: Catarina Dutilh Novaes | 11 January 2013 at 05:34

In response to Marije's comment, I would add something concerning Alexander's commentary. It seems to me that Alexander makes a clear distinction between the reductio and the modus tollens. For Alexander the modus tollens is not useful for scientific reasoning (! cf. AlexAphr. In APr., p. 18, l.15ff), whilst the reductio is one of the three ways of demonstrating the validity of a given imperfect syllogistic mood. However, Alexander suggests that the best and more intuitive way of demonstrating the validity of an imperfect syllogism is by means of conversions. Ecthetic proofs and reductiones should be considered only because we cannot prove the validity of some syllogistic moods by means of conversions.

The reductio is made by a categorical syllogism which infers something impossible, and thus by means of some sort of stipulation we reject the premiss which we have assumed in this auxiliary syllogism. This sort of 'stipulation', which we already find in Ar.'s text, has perhaps a dialogical character, if I am correctly grasping Catarina's claim.

Hypothetical syllogisms are in some sense an appendix to syllogistic, because they do not demonstrate a predicative relation (cf. In APr p. 42, ll. 26-30). Therefore, they cannot be taken to be genuine syllogisms - since they fail to 'demonstrate something different from their premisses'.

As far as I remember, Philoponus in APr. regimented Alexander's presentation. Therefore, there should be a distinction between the Stoics' indemonstrables and the reductio in Phil.'s texts too, but I admit that I haven't checked them recently. And of course I very much look forward to reading the new English translation of Phil. on PostAn (which we haven't yet here in Leuven). :D.

Posted by: Luca Gili | 11 January 2013 at 06:17

Sorry about the delay in getting back to you, Catarina.

I strongly suspect that Proclus’ distinction between proofs from causes vs. proofs from signs is related to the implicit distinction in Euclid marked by which of two phrases is used to mark the end of a proof. The two formulas that terminate the proofs in Euclid are

hoper edei deixal(Q.E.D.) andhoper edei poiēsai(Q.E.F.).So, I think those are the two kinds of proofs he's referring to. But it's still somewhat mysterious to me why he characterizes the "poetic", or "constructive" proofs as being based on "signs", and why he uses that particular word for signs.

Also in the background to this is the question regarding whether geometrical propositions should be considered as “theorems” or “problems”. Proclus’ In Euclid. is one of the principal sources about this debate, which apparently took place in the period of the early Academy and is associated with the names Speusippus and Menaechmus. Proclus seems to imply that his (and Euclid’s?) position, since it admits of both kinds of proof (from cause vs. from signs), reconciles the stark opposition between the Speusippean and Menaechmian positions. As he puts it, both Speusippus and Menaechmus can be said to be right.

In terms of secondary literature, a good overview of the philosophical questions related to the issue of construction in geometry, and the philosophical implications of the differences in ancient and modern geometrical practice is the sadly obscure book

The Ethics of Geometry, by David R. Lachterman. I don't think Lachterman explicates the specific line in Proclus that I quoted, but he does discuss the Speusippus/Menaechmus debate in some detail. His perspective is a broad one, though and he doesn't go into details about what Proclus' own position is.A detailed description of Proclus' views on this subject can be found in

L’Architecture du divin: Mathématique et philosophie chez Plotin et Proclusby Annick Charles-Saget. There's an older book by Stanislaus Breton that's worth tracking down as well. I'm not going to be much help on newer secondary literature, as I'm somewhat out of the loop on such matters.There was a post on New Apps a few days ago that linked to a review of a recent book on Felix Klein and Husserl. David Lachterman, whose book I mentioned just now, was a student of Klein's and his book can be thought of as a continuation, in a sense, of Klein's work on the differences between ancient and modern mathematics. I don't know the details of your work on formal languages, but I suspect you'd find much of interest in Lachterman's work.

Posted by: Cameron Majidi | 14 January 2013 at 10:27