37 - Hugh Benson on Aristotelian Method

Hugh Benson of the University of Oklahoma chats to Peter about Aristotle's views on philosophical method, and whether he practices what he preaches.

Press 'play' to hear the podcast: 

You are missing some Flash content that should appear here! Perhaps your browser cannot display it, or maybe it did not initialize correctly.

Adam's picture

Method

Hi Peter:

Firstly, thank you for investing in such a comprehensive endeavor. It must be mutually rewarding and cumbersome to take on “the history of philosophy without any gaps”; and, I for one have monumentally enjoyed the podcasts. However, despite the fact that I have found many of them stirring enough to comment and haven’t as of yet, recently I took time to listen to the episode on Aristotelian Method.  I know I am jumping the gun with upcoming western philosophers by putting forth the query to follow, but I cannot resist. At the start of the podcast, you and Hugh make reference to Plato and Aristotle possibly having an apprehension to “beginnings” with inquiry. This for me has long been a puzzle, as I have a strong background with scientific method; which may be a topic better suited for the collapse of the Aristotelian World View and thinkers such as Descartes, Galileo, Copernicus, Kepler, Newton and others; however, lately, method has struck me as a strange sort of thing (and maybe I have “Zen and the Art of Motorcycle Maintenance” by Robert Pirsig as a partial culprit to my notions, or maybe just Descartes, or a number of other factors). I cannot help but realize that any method is only as good as its premises, or hypothesis, or belief, or axis, or whatever term one chooses– more or less, the beginnings. And here, it seems to me, crossing into an inquiry of episteme is unavoidable. If the matter is this, let’s say: I know with certainty my premises; then, “demonstrating” or providing a proof of my belief is fairly simple using a method – whatever that particular method might be. However, if my premises are not so concrete, the use of any method is moot. It seems then that I have to know in order to know – and this is not inquiry at all – at least in my mind, and I may be wrong. For me, and please let me know if I am off, the point of inquiry is to start without knowledge of something, and in the end gain knowledge of that thing. I understand that a premise must be separate from a conclusion, and that, a conclusion must follow logically from its premises; however, my unrest lies more so with the gulf between foundationalism and the skeptic. So, to make matters more concise, do you believe that Aristotle may have had an epistemological problem with beginnings; in that, he must first know somethings with certainly in order to deduce other things using a method? And further, do you believe this plays some sort of role in Aristotle’s systematic categorization and definition of the world; namely, in an attempt to esoterically (or maybe rather bluntly) show that certain foundations must be established prior to the use of any method?  I would certainly appreciate your insight; thank you.

Cheers,

Adam 

Peter Adamson's picture

Beginnings

Dear Adam,

That's a pretty complicated question. But I would say the answer is basically "yes": when he talks about getting hold of first principles at the end of the Posterior Analytics, he can't just mean things like laws of logic. Because he talks about getting principles via sense-perception. So he must mean we are grasping certain "things" as you put it, like maybe certain first principles regarding horses if we are studying horses. An example might be something as banal as "horses are animals". This is useful because if you also know, for instance, that animals engage in nutrition, you can infer that horses engage in nutrition (or better: understand why they do, namely, because they are animals and all animals do this). It is banal, but a first principle, because you don't demonstrate that horses are animals on the basis of anything more fundamental. Possibly Aristotle does have something more robust in mind, for instance he might think that the principle is the whole definition of "horse" (which would include being an animal, but also some other things).

And you're of course right -- Aristotle makes this point too -- that if you take something false/shaky as a principle, that will infect the whole chain of inferences you draw from it.

Does that help?

Peter

Adam's picture

Beginnings

Thank you for the insight, Peter, it is quite helpful; and I apologize for the complexity of the first question. I think it is interesting that you use the word banal to describe some of Aristotle’s first principles. I have recently read Categories and On Interpretation, and, at times, it seems more like reinventing the wheel than the opposite – but maybe it is because it is easy to take some of his inferences for granted. Something I find remarkable though is about certain features being essential to objects or, if we are sticking with your example, horses; for instance: it would be essential that a horse be a quadruped, but it would not be essential that it be, let’s say, brown because that is accidental. So, in regard to method then, I guess it would make sense that Aristotle expects that one who is knowledgeable about the world would draw inferences from essential “qualities” (for lack of a better word) rather than accidental ones. Would this be a safe assumption? And if so, do you think Aristotle believes it common sense to be able to differentiate between what is essential and what is accidental – in that this can be derived easily from something like sense perception and understanding?  Or do you think that Aristotle believes it is a more difficult task to draw understanding from sensation – in that it is easy to have sensations but difficult to understand (have episteme)? Again, I appreciate any thoughts.

Cheers,

Adam

Peter Adamson's picture

Essential vs. accidental

Hi Adam,

That all sounds right to me -- I think it could be a matter of considerable difficulty to decide between essential and accidental features, actually. This is a potentially big problem for him: on the basis of induction how could you know that a feature which seemed essential wasn't accidental? You might, for instance, discover some ducks that don't live in water, having thought that living in water was essential to ducks. I think Aristotle is convinced that our minds are adapted to take on the natures of things around us, which means getting hold of their essential properties through experience. So that gives him reason for optimism (this may make more sense after the later episode on his theory of mind). But he doesn't need to say it is _easy_ to tell essential properties apart from accidental ones, only that it is possible given enough inquiry.

Peter

Michael P's picture

Aristotelean method in math(s)

 

A friend and I were debating about knowledge in mathematics and such, and the main points ended up aligning pretty closely with (what I now find) are the four methods that you and Benson discussed.

 

Mathematical Aporia are well-known open questions, and surprisingly initial results that lead to questions. Identifying patterns, making conjectures and useful definitions are types of Induction (in the sense of Aristotle, not mathematical induction). Proof of course is Demonstration, and for math it is both a part of inquiry as well as being our primary way to confirm knowledge although there are two problems: 1. the crisis of foundations, Godel, etc., and 2. distiguishing interesting/good math from true but irrelevant propositions, which aren't really math). Dialectic is the general process of mathematical inquiry and also describes the social aspect of determining when a published result is truly reliable.

 

In mathematics, demonstration is an essential part of mathematical inquiry and serves the related function of verifying knowledge (subject to caveats 1 and 2 above). As you mentioned in the podcast, the role of demonstration is less clear in Aristotle's philosophy.

 

If I understood correctly, one theory about Aristotle's view is that demonstration serves as a verifier (primarily?), much like proof in modern mathematics. This would be reasonable if they see mathematics as an example of what philosophy ought to be like (not unheard of elsewhere), or otherwise as being an inspiration for philosophy.

 

This leads me to wonder, what was seen as the role of proof within mathematics as it was practiced in Aristotle's time? And more generally, how did they think of mathematical inquiry? As a part of mathematics? Or not, but rather as a tool to gain knowledge of true mathematical statements, which themselves comprise mathematics? (Or something else.)

 

We know that they must have thought of mathematics somewhat differently. I'm under the impression that for them math=geometry, and that their response to the square root of 2 was to double down on pure, non-numeric geometry. Going out on a limb, perhaps this could be a motive for wanting to have a notion of Robust Knowledge, which would be delineated so as to not include irrationals. They had no crisis in foundations: Non-Euclidean geometry didn't occur to anyone until much later. And it seems like irrational numbers didn't affect their confidence in math's correctness. My guess would be that because of this, they would have been less wary of the limitations of proof than we are nowadays---and also maybe, the influence of mathematical proof on philosophy might have been stronger than would seem reasonable nowadays, with our modern view of proof.

 

Anyway, that's as far as I can get, with my near total ignorance of math history. But maybe there is more interesting stuff along similar lines.

Peter Adamson's picture

Aristotle and mathematics

Hello,

Thanks for this interesting post. In fact Aristotle mentions mathematics repeatedly in the Posterior Analytics, he clearly sees this as a paradigm case of the kind of demonstrative knowledge he is talking about (or, perhaps one should be more cautious and say that he thinks he can clearly illustrate points about demonstration by using mathematics). Given the work of Euclid at around the same time, and what we know was going on in Plato's Academy (also from examples used by Plato himself e.g. in the Meno to illustrate the method of hypothesis) we know that mathematical inquiry was never far from reflection on the possibility of knowledge. As I describe in episode 51, the immediate successors of Plato seem to have pushed this even further, to argue that reality must be in itself mathematical if it is to be a fit object of knowledge. Or at least that's one way of reading them; and some would say they were here developing themes from Plato himself.

I'll have a later interview episode on the topic of ancient philosophy and mathematics, actually, if all goes according to plan.

I think one difference between the way people nowadays think about mathematics, and the way Aristotle thinks about demonstration, is that mathematicians allow themselves to choose unargued starting points ("axioms") more or less arbitrarily and then study what follows from these. By contrast Aristotle thinks there is a privileged set of true first principles which are the basis for demonstration. I think he might see most (all?) of modern mathematics as dialectical in the sense that it is only arguing from agreed premises; but here I should probably admit that what I know about modern mathematics could be fit into a small isoceles triangle.

Thanks for listening and for the comment!

Peter

Anonymous's picture

modern math

Thanks for the reply. Sorry, but I can't resist a follow-up.

The axioms are not at all arbitrary. Actually, axioms are always carefully chosen to produce a recognizable piece of mathematics. For example, let's define Adamson arithmetic to contain constants a,d,m,s,o,n and a binary function + which has the following axioms:

1. Not(o+n=n+o)

2. x=y+z if xyz is a consecutive substring of "adamson"

3. (x+y)+z=x+(y+z)

That's it. Note that it's not good for anything (except providing an example of a consistent, uninteresting axiom system). No mathematician, modern or ancient, would call it mathematics in any sense. (Sorry, I had too much fun with that. Moving on...)

I agree that in modern mathematics we have mostly given up on a single set of axioms, but it's because we had to, because of Godel's work and also because we know of potential axioms where we're pretty sure that we can't justify always taking them or always omitting them. So, Aristotle had a view of mathematics that turned out to be wrong. For what it's worth, this informed my earlier comment.

Another point that I may be misunderstanding: I thought that Demonstration doesn't always need to proceed from correct First Principles; for example, in a Proof by Contradiction. If so, then I don't see why Aristotle would view modern math proofs as being only Dialectic, and not Demonstration.

Well, maybe once I listen to more podcasts - or even do the unimaginable, read Aristotle myself - I will ask less silly questions. Thank you for your patience.

Peter Adamson's picture

Arbitrary starting points

Yes, you're right that "arbitrary" may have been misleading but I think it is technically the right word. What I mean is that in modern maths one can simply choose axioms, and you're right that one chooses not at random but in such a way as to produce something interesting -- but still you get alternate systems which can be studied, e.g. different geometries. (So I basically was trying to say what you say in this latest post but not saying it as clearly as you have.) Whereas Aristotle as I read him thinks there is only one set of true starting points, the first principles from which all demonstrations ultimately proceed.

The question about proof by contradiction is very interesting. That style of argument becomes popular later in philosophy, for instance one sees it a lot in the Islamic tradition and this may be partially due to the influence of mathematics on philosophy. But I think Aristotle doesn't consider this to be a kind of "demonstration" in his sense, since for him a demonstration is a causal explanation of why something is the case (or rather, why some subject S has some predicate P). It's hard to see how that could be achieved by a reductio ad absurdum.

Peter
 

Michael P's picture

Maybe better to ignore the conclusion, such as it is

 

Right, I see. So a theorem of the form "p implies q" isn't really a theorem unless p is actually true... and now I remember that you or Hugh said as much during the podcast.

The misunderstanding about "arbitrary" is my fault.  In math "at random" has a technical meaning, so we say "arbitrary" in situations that people usually say "at random".

Different geometries could be (are now?) thought of as different definitions, which are based on the same axiom system (set theory) rather than alternative, incompatible axiom systems. I would argue more generally that axiom systems are mostly just a type of definition, and I want to discuss why I think so, why the ancient Greeks wouldn't agree with me, and what the implications of that would be.

Nowadays we have loads of important definitions that are built on definitions that are built on definitions, etc. (This is why it is typically difficult for a mathematician to describe their research to a non-expert). Some definitions are quickly forgotten, some become as an essential part of mathematics as points and lines, and there is a wide range in between. Because of this, we are accustomed to the idea that finding the right objects of study (definitions) is an essential part of mathematics. Furthermore, it's not at all obvious that there is a black and white distinction between those those that must be part of mathematics (like imaginary numbers or groups) to those that clearly are not (like Frontalot numbers). Ancient Greek mathematics, on the other hand (if I'm not mistaken) considers only their basic notions (points, lines, circles, angles, equidistant) or objects that are easily described using those basic concepts (like conic sections), so they haven't had the same experience, and would see no reason to believe that selecting good definitions is an issue at all.

So, there is something in the air now that challenges the mystique of a pure, single mathematics, something which did not exist back then. And if we're less certain that the eventual body of mathematics is a Platonic object, then it is no longer clear why one must have a unique axiom system. Also, in our previous comments, we agreed that an axiom system is only as good as the mathematics it can produce. If we are interested in different, not necessarily compatible bodies of mathematics (and isn't this how the ancient Greeks felt about geometry, integers, and/or magnitudes?), why would we think that it should all rely on a single axiom system?

Not sure I accomplished what I claimed I set out to do.  I guess that my conclusion is that even if we ignore the crisis of foundations and Godel and all that, there is still a difference in mathematics as practiced then and now that might lead to different views on the role of axiom systems (as discussed above)... and if that's true, might also lead to different views on the role of Demonstration, namely that it is too ambitious (to describe mathematics at least)... and that a more modest alternative interpretation or modification of Demonstration might work with my original attempt at mapping of Aristotle's four methods of philosophical inquiry to modern mathematical inquiry... and that such an alternative might not have seemed like a terrible idea to Aristotle if he knew what we now know about mathematics and nonetheless wished to preserve his four methods in some (necessarily modified) form.  

Well, I hadn't planned on cascading off into increasingly silly counterfactuals to support what is really a rather insignificant idea.

Anonymous's picture

modern math

Thanks for the reply. Sorry, but I can't resist a follow-up.

The axioms are not at all arbitrary. Actually, axioms are always carefully chosen to produce a recognizable piece of mathematics. For example, let's define Adamson arithmetic to contain constants a,d,m,s,o,n and a binary function + which has the following axioms:

1. Not(o+n=n+o)

2. x=y+z if xyz is a consecutive substring of "adamson"

3. (x+y)+z=x+(y+z)

That's it. Note that it's not good for anything (except providing an example of a consistent, uninteresting axiom system). No mathematician, modern or ancient, would call it mathematics in any sense. (Sorry, I had too much fun with that. Moving on...)

I agree that in modern mathematics we have mostly given up on a single set of axioms, but it's because we had to, because of Godel's work and also because we know of potential axioms where we're pretty sure that we can't justify always taking them or always omitting them. So, Aristotle had a view of mathematics that turned out to be wrong. For what it's worth, this informed my earlier comment.

Another point that I may be misunderstanding: I thought that Demonstration doesn't always need to proceed from correct First Principles; for example, in a Proof by Contradiction. If so, then I don't see why Aristotle would view modern math proofs as being only Dialectic, and not Demonstration.

Well, maybe once I listen to more podcasts - or even do the unimaginable, read Aristotle myself - I will ask less silly questions. Thank you for your patience.

Peter Adamson's picture

And a note from Hugh Benson

Here's a further thought from Hugh Benson, the interview guest on this episode: "the listener raises an important question which in my view those who are experts do not pay enough attention to, i.e. mathematical *inquiry*.  Everything that I read is addressed to proof and display - as though math was a finished product.  So Euclid gets a lot of attention.  But the process of discovery which results in the Elements is seldom addressed.  Of course, part of the problem here is the lack of evidence, but still..."

Michael P's picture

Thanks again

Thanks to both of you for your replies.  I would have more questions perhaps but I think it's about time that, in your shoes, I would ask the listener to trying reading a bit.

Best,

Michael