Tuesday, October 3, 2017

Goat Product

I did a lot of maths at school, and while I’ve got a good memory and do use bits of it here and there, it’s been fifteen years and some of it’s a bit rusty. Partly in an attempt to remedy this, I recently I watched some videos on Youtube about calculus and linear algebra. They’re by someone called Grant Sanderson, whose account is called 3Blue1Brown, and they have a friendly style with lots of neat animations. The idea of the videos is that a lot of people are taught maths in a way that involves a lot of number crunching and rule following but doesn’t get them to understand the underlying concepts, and that this leaves a gap that can be filled by presenting the ideas using visual interpretations. For example, a 2x2 matrix represents a transformation of 2D space that turns every square of size 1 into a parallelogram whose size is the magnitude of the matrix’s determinant. (If it’s negative, that means things are mirrored.) It’s possible I learned this at school, but stressing it and showing a nice animation helps you really understand what the deal is with determinants. They’re cool videos, and if you like maths then I recommend them, although they’re not really designed to teach the topics from scratch. I also recommend this one by the same person, which shows how to give a topological proof of something that doesn’t really seem to be about topology. I watched it a while ago and it really is beautiful.

Anyway, as I was going through the linear algebra videos I wrote down some questions I still had so I could find answers to them later, and one of them was about the dot product (also known as the scalar product). The dot product of two vectors <a,b> and <x,y> is a dimensionless directionless scalar, equal to ax + by. Why? What’s it measuring? The video on the dot product gave it a couple of interpretations.

One was to project <a,b> onto the line <x,y> is on via the shortest route, and then multiply the length of the new vector by the length of <x,y>. That might be OK for giving a visual heuristic for calculating or estimating the dot product, but for me it didn’t get me much of a handle on what sorts of thing it represents. Multiplying the length of two vectors is something you’d do if they were the sides of a rectangle and you wanted to find its area, but if the vectors are on the same line, where’s the rectangle? And if it’s an area, why is the dot product a scalar? It makes sense for the determinant to be a scalar because it’s the scale factor for areas, but with the projection interpretation of the dot product I couldn’t really see which areas were getting scaled up.

The second interpretation replaced one of the vectors with a 1x2 matrix, and then visualized this as a transformation taking points in 2D space onto a number line embedded in that space. The 1x2 matrix <a,b> will take the point <x,y> in 2D space to the point ax+by on the number line. That’s great, and it shows you the nice duality between the dot product and this kind of transformation, and indeed that was the point of this part of the video. (The video was called “Dot Products And Duality”, after all.) But a 2x1 vector isn’t a 1x2 matrix, and the fact that that taking the dot product is dual to performing a process that outputs a scalar doesn’t really tell you why the dot product itself is a scalar.

There will be good reasons for the dot product to be a scalar, of course, but this video didn’t help me see what they are. I also thought it’d be good to have a better visual interpretation of the dot product, where I could point to something I could see and say “that’s the dot product”, instead of pointing at two things I could see and having to multiply them together to find the dot product, even though the multiplication didn’t correspond to anything in the picture. In the video series on calculus he actually flags up that he doesn’t have a nice picture for the derivative of 2t where we can point to it and say: “See? That part! That is the derivative of 2t!” He says he’d like one, but doesn’t know one. Perhaps there isn’t one. Although perhaps there must be one. I guess different people will have different views about how high our expectations about this sort of thing should be.

Anyway, I made a note of the question, and when I got to the end of the videos I still didn’t have an answer, so I googled things like “visual interpretation of dot product” and “why is the dot product a scalar”. And you know what? Nobody else seemed to have a decent answer either! I mean, I didn’t look very hard, but there seems to be a fair bit of mystification out there, both from people asking the question and people trying to answer it. On this Quora page about the question, one person said that’s just how it’s defined, which isn’t helpful, and another said it was the distance between points the two vectors point at, which isn’t true on any interpretation I could work out. On another page I found more helpful someone said it was like a booster ramp in Mario Kart, which boosts you best if you drive over it in the direction of the ramp, and doesn’t boost you so well if you drive over it obliquely. I quite liked that one, but it has some problems. One is that if you drive over it backwards it still boosts you forwards, so there’s nothing much that seems to correspond to negative dot products. The other problem is that it’s not obvious what corresponds to multiplication here. It seems to me that the ramp adds a certain amount of velocity, rather than multiplying a vector from the ramp by the vector given by your initial velocity. Also the result of going on a ramp seems more like a vector than a scalar. I think these three issues are related. I did still find it helpful, and maybe I just don’t have a good enough feel for how the booster ramps work (although I have played the game quite a lot), but I wanted something better. He had one involving the angles of solar panels too but he didn’t like that as much as the Mario Kart one, and I didn’t really either. (I couldn’t see how it produced negative dot products, for example.)

So anyway, I had a go at coming up with my own visualization, and I found one that I think works pretty well. You know how people share those pictures of goats standing on almost vertical surfaces, seemingly defying gravity? Well, imagine a goat standing on one of those. It could be a very steep one like the cliffs in the pictures, or it could be shallower like a hillside. Basically any inclined plane will do. We can represent the slope with a vector u, for upslope. The vector points up the slope and its magnitude is the gradient of the slope in that direction. <0,0> corresponds to a horizontal plane, which is fine, but no vector corresponds to a vertical one, which is just as well because even a goat can’t stand on a vertical plane. Now the goat sets off along the surface in a straight line, with its horizontal velocity given by a vector v. It may be easiest to picture v if you look down on the goat from above. Now the dot product is the rate of change in the goat’s elevation. Positive means the goat is going up, and negative means the goat is going down. It’s a scalar, not a vector, because it’s measuring how fast the goat is moving up or down, not measuring how fast it’s moving and which direction it’s moving. (Well, it’s measuring whether the goat is going up or down, but even scalars distinguish a positive and a negative direction. It’s not measuring the direction in three dimensions, or even two.)

The sums seem to work out. Think of u = <a,b> as giving the gradient a when moving north and the gradient b when going east, and think of v = <x,y> as giving the velocity’s northern component x and eastern component y. Negative gradients are downhill, and negative velocity north or east is movement south or west. So if it goes x north and then y east it will go a*x up and then b*y up.

It gives straightforward visual interpretations to some properties of the dot product too. To get maximum height for its horizontal displacement, the goat goes straight uphill, which means u and v have the same direction. Change in elevation for a given horizontal dispacement is proportional to gradient, and change in elevation for a given gradient is proportional to horizontal displacement. It’s clear when the dot product will be positive, negative or zero, depending on whether it’s moving uphill, downhill or along a contour. It’s a good visualization of how a difference in the angle between u and v makes less of a difference to the dot product when the angle is small than when it’s big. It’s also clear that if u = <0,0> the plane is flat and the goat won’t go up or down however much it moves horizonatally, and it’s clear that if v = <0,0> the goat is stationary so it won’t go up or down however steep the slope is.

Finally, we can think about how different vectors u can multiply by the same vector v to give the same dot product. To do that, imagine that the plane is one side of a V-shaped valley whose edges and base are horizontal, and the goat wants to get to the top or the bottom, depending on whether the dot product is positive or negative. The dot product represents the height to the top or bottom of the valley, and u represents the slope of the valley. Now imagine you’re looking at the goat from above, and it wants to get to the top. There are lots of ways it could get to its destination. It can run straight uphill, and then it won’t have gone so far horizontally. Or it can set off almost at right angles to the slope, and then it’ll go a very long way horizontally before it gets to the top. Or it can do something in between. But it can’t go at right angles or an obtuse angle to the slope, or it’ll never get to the top. And there’s also a minimum horizontal distance it has to go. Viewing from above, you can draw a little circle around the goat and see that if the circle is small enough everything inside it will still be in the valley.

So there’s my visualization for the dot product. I like it! Commutativity holds, as it must, but it doesn’t immediately drop out of the setup the way it might if u and v represented more similar sorts of things. I won’t try making a virtue of that. But I think it’s OK, and I like it better than the other ones I found, and assuming I’ve not made mistakes with it, I think other people would like it too. I’m sure there are other adequate ones out there, but not everyone is aware of them, and coming up with these things yourself is all part of the learning process anyway. I didn’t really get what the deal was with the dot product, and now I think I do. The dot product is the goat product.

Tuesday, September 12, 2017

They Do Things Differently There

Between 1962 and 1981, a scholar at Cambridge University called WKC Guthrie published a six-volume History of Greek Philosophy running from the early Presocratics to Aristotle. He was, it seems, something of an expert.  In the preface to the first volume (1962: xi) he says he had plans to go further: “It is my intention, Deo volente, to continue this history to include the Hellenistic period, stopping short of the Neoplatonists and those of their predecessors who are best understood in conjunction with them.” Six volumes is still a lot, of course, but don’t worry: he also wrote a much shorter book in 1950 covering the same period, called The Greek Philosophers: From Thales to Aristotle. It was based on a series of lectures aimed at undergraduates who weren’t taking classics and might not know any Greek. (His six-volume work was meant to be accessible for non-Greek-readers too.) I hardly know any Greek myself, and I’ve recently been taking an interest in Anaxagoras, so when I saw it in a second hand bookshop I thought I might like it. And I did! I’ve heard the basic story a few times before, but I’m always ready to read someone else’s take on these things, and there were a few things I found kind of interesting about Guthrie’s.


The first chapter is called “Greek Ways of Thinking”, and by page four he’s already got into a discussion about the meanings of words. He wants to stress that some of the key words ancient Greek philosophers used didn’t mean the same things to them as their usual English translations mean to us. He mentions the words translated as ‘justice’, ‘virtue’, and ‘god’ (or ‘God’), and the Greek word logos (λόγος). At the start of John’s Gospel, where it says “In the beginning was the Word, and the Word was with God, and the Word was God”, the word translated as “Word” is logos. It’s also where the “-logy” in “biology” and “geology” comes from. It’s hard to translate into English, and it was already kind of slippery in Greek, which noted slippery character Heraclitus apparently took full advantage of.


Ostensibly Guthrie’s talking about this because he’s writing (and had been lecturing) for an audience of non-classicists, and he doesn’t want people who don’t read Greek to get misled by the translations. I suppose it’s quite likely that a modern reader would be liable to bring some conceptual baggage to the table even if they were able to read the original Greek, but the less Greek you’ve read the less likely you’ll be to have a feel for what they meant by these words. He wants you to understand the mindset of the Greek philosophers, and he does this, at least in part, by trying to explain what their words mean.


It’s possible I was reading too much into it, but I kind of got the impression that casting this in terms of words was a result of the author being surrounded by British philosophers in the 1930s and 1940s. If the hip new thing is to think that philosophy is ultimately a matter of attending carefully to the meanings of words, then this is probably a smart way to present things. And of course Guthrie may have subscribed to some of this linguistic turn stuff himself too. I can vaguely recall linguistic-turn philosophers saying that while Plato, Descartes and the rest thought they were dealing with substantive non-lingistic questions they were really talking about the meanings of words, and as a result their insights can still have relevance to philosophy conceived as linguistic analysis. I wish I had an example of someone saying this for you, but I do not. Anway, assuming they actually did say this, I kind of think they had a point except that it’s the other way round: linguistic-turn philosophers were still blundering around in much the same insight-space with much the same moves available to them as the people who came before and after, and casting their insights about causes and knowledge as insights about the words “cause” and “knowledge” doesn’t stop us using them. There’s probably a limit to how far you can take this kind of ecumenism, and some of the linguistic-turn stuff probably can trace its badness to its conception of what philosophy is. Some of it can probably trace its goodness to that too! But I think that a lot of the time it doesn’t really matter, and similarly it doesn’t really matter that Guthrie casts his discussion of the Greek philosophers’ ways of thinking in terms of the meanings of words. Just to be clear, he thinks those ways of thinking were pretty different from those of twentieth-century British people.


Another thing I thought was interesting was how he focused on the political environment. Here’s how he starts the first of two chapters on Plato:


We shall probably understand Plato’s philosophy best if we regard him as working in the first place under the influence of two related motives. He wished first of all to take up Socrates’s task at the point where Socrates had had to leave it, to consolidate his master’s teaching and defend it against inevitable questioning. But in this he was not acting solely from motives of personal affection or respect. It fitted in with his second motive, which was to defend, and to render worth defending, the idea of the city-state as an independent political, economic, and social unit. For it was by accepting and developing Socrates’s challenge to the Sophists that Plato thought this wider aim could be most successfully accomplished.
The doom of the free city-state was sealed by the conquests of Philip and Alexander. It was these which assured that that compact unit of classical Greek life should be swamped by the growth of huge kingdoms on a semi-Oriental model. But they did no more than complete in drastic fashion a process of decline which had been going on for some time. (p.81)


I was quite taken aback by this. Obviously Plato’s most famous work is called The Republic, and in it he lays out a way for a city state to be organized, and my understanding is that you’re supposed to think some of these ideas are pretty good. (I’m told it’s a sort of centrally planned natural aristocracy with philosophers running the show and a covert eugenics programme for good measure, although I blush to confess I haven’t actually read much of it.) But Plato also talks about a bunch of other things, even in The Republic itself, and I’d always been given the impression that his political ideas were a bit of an eccentric sideline, and largely independent of his much more important stuff about forms, knowledge, truth, the soul, the Euthyphro problem and so on. I was already aware that Plato’s interest in politics wasn’t entirely theoretical, and that he’d been very disappointed by the Athenian democracy that killed Socrates, and also affected by the situation with the Thirty Tyrants, about which I don’t know very much. But a person can be interested in more than one thing. Guthrie seems to think that Plato’s more purely philosophical stuff is largely in the service of his politics, and Guthrie wasn’t some kind of oddball as far as I can tell, so it’s interesting to see him saying something like that. But perhaps I’ve got the wrong end of the stick.


The connection between politics and Plato’s less obviously political stuff is, according to Guthrie, related to the connection between religion and the state. They were, he says, closely bound together and in some ways identified. By Plato’s time the anthropomorphic paganism of Homer and Hesiod was getting challenged a lot, and the political order was getting challenged with it. The Sophists, who Plato wasn’t a fan of, were part of this. Plato wanted to find an alternative foundation for the political order, according to Guthrie, and that involved defending the idea of eternal principles of justice and a conception of the good life that would be best facilitated by city states, as long as they had plenty of central planning and a eugenics programme instead of this democracy nonsense that killed his buddy Socrates. Something like that, anyway. It makes a kind of sense, although I can't say I agree with Plato here.


A related issue is something Guthrie says about the relationship between virtue and self-interest in ancient Greek philosophy. These days we draw a pretty sharp distinction between the two. People disagree about the extent of our moral obligations to go out of our way to help others, about how much someone’s being a bad person is an intrinsic harm to them, and about the extent to which virtue and vice are rewarded and punished after death. But we’ve still got two pretty separate concepts of doing what’s right and doing what’s best for yourself, even if we might think they line up pretty closely in practice.


Now, utilitarians these days often say that for well-off people in well-off countries morality is very demanding indeed, and that the moral life involves a level of self-sacrifice that most people don’t come anywhere near, even people we ordinarily think of as moral exemplars. Basically the idea is that people should give away practically all their disposable income to charities. Utilitarians disagree over which charities, but the consensus among them is that the charities will spend it better than you will, and that means you should hand it over. This may not have always been the case. When utilitarianism was invented there was a lot of poverty about, but there may well not have been as much an individual could achieve by throwing money at the problem, because there wasn’t the same charity infrastructure in place. The situation where the more money you give the fewer people will get malaria is new, and Peter Singer’s [1972] problem is a New Moral Problem. You could probably make a case that it existed before to an extent, but it probably wasn’t as pressing before.


With this in mind, let’s look at what Guthrie says about the new socioeconomic realities of being an individual in a Greek city state:


In early societies, where communities are small and cultural conditions simple, no conflict is observed between moral duty and self-interest. As Ritter [1933: 67 or 57?] remarks: ‘He who in his relationship to his fellow men and the gods observes the existing customs is praised, respected and considered good; whereas he who breaks them is despised, disciplined and considered bad. In these conditions obedicence to law brings gain to the individual, whereas transgression brings him harm. The individual who obeys customs and law is happy and contented.’
Unfortunately this simple state of affairs cannot last. The Greeks had reached the more complex state of civilization where it was forced on their attention that acts of banditry, especially on a large scale - the banditry of the conquering hero - which successfully defied law and custom, also brought gain, and that the law-abiding might be compelled to live in very modest cicumstances or even under oppression and persecution. Out of this arose the sophistic opposition of ‘nature’ to ‘law’, and the conception of ‘nature’s justice’ as not only different from man’s but something greater and finer. [pp101-2]


Now, I don’t want to get into debating the anthropology here, and Guthrie doesn’t really defend it. But the idea that there being any tension between morality and self-interest was once a New Moral Problem is interesting. Huge if true, I guess.


I should probably clarify something here. Sometimes I’ve heard people saying that the ancient Greeks didn’t really distinguish virtue and self-interest, and they kind of rolled it all into one when they asked what the good life is, or how people should live. That’s not really what Guthrie’s saying. He’s not exactly saying the opposite either; it’s more that they were just beginning to develop concepts that could handle the distinction because political circumstances had only recently forced them to. Although Guthrie (and as far as I can tell, Ritter) also seem to say that Socrates and Plato argued that the divergence between virtue and self-interest was an illusion, that it was still in one’s interests to be virtuous, and that the illusion was created by the divergence of virtue (and thus self-interest) from doing what was immediately pleasant, and that where the Sophists had gone wrong was in identifying self-interest with immediate pleasure. They say Plato and Socrates say that when broadly enough conceived even pleasure can line up with virtue and self-interest, which I guess puts them in the same camp as the Epicureans, but not the Cyrenaics, who were more along the instant-gratification lines of Plato’s Socrates’s Sophist opponents.


But the big take-away here is that the apparent divergence of virtue and self-interest may once have been a New Moral Problem.


Close to the end of the book, Guthrie says that “Aristotle’s philosophy represents the final flowering of Greek thought in its natural setting, the city-state” (p.160). The idea is that once the political organization changed the philosophy changed with it, and so it makes sense to end the book there. I suppose this doesn’t completely square with the idea that his six-volume magnum opus ended in the same place because its author didn’t live long enough to end it later, and maybe Guthrie was making a virtue out of necessity in the face of space constraints. But given the other stuff he’s said about the interaction between Greek philosophy and its political environment, it hasn’t just come out of nowhere. He does think that Greek city states produced a distinctive kind of philosophy. (In the preface to the sixth volume, when he knew that his health wouldn’t allow him to write any more of them, he describes finishing with Aristotle as a pity but doesn’t seem to think it a catastrophe, because there were plenty of books on the subsequent period anyway, that period’s philosophy wasn’t as good as Aristotle’s, and Aristotle was ‘both the last of the ancient and the first of the modern philosophers’ [1981: ix].)


The other thing that leapt out at me was how Guthrie emphasized the continuity between Plato and Aristotle. According to the School of Athens Caricature, Plato is interested in transcendent, celestial stuff apprehended by reason, while Aristotle is interested in everyday, terrestrial stuff apprehended by the senses. There’s a temptation to take them as exemplars of the two sides of whatever debate we’re most interested in: empiricism vs rationalism, pluralism vs monism, steady-state vs big-bang, materialism vs dualism, nominalism vs, er, Platonism. But this is usually kind of anachronistic, and it doesn’t do justice to the amount Plato and Aristotle had in common.


Some people have gone the other way, and tried to make out that Plato and Aristotle were in agreement about everything important. This reached a bit of a highpoint with Iamblichus (c.245-c.325 CE), who apparently tried to make out that both of them were essentially just writing footnotes to Pythagoras. In the late second century, when Marcus Aurelius was establishing four chairs of philosophy in Athens, the chairs represented the four main schools: Stoic, Epicurean, Platonist and Aristotelian, which I guess means Platonism and Aristotelianism were still seen as separate then. But by somewhere in the third century, Aristotelians weren’t making an effort to distinguish themselves anymore:


Alexander [of Aphrodisias] was not the first but rather the last authentic interpreter of Aristotle. Although subsequent generations of commentators were profoundly influenced by Alexander, they were motivated by a very different exegetical ideal. Their primary aim was no longer to recover and preserve Aristotle’s thought for its own sake, but for the sake of finding agreement between Aristotle and Plato and presenting them as part of one and the same philosophical outlook. [Falcon 2017]


(I was a bit puzzled by this, but it seems what happened was that in the third century a bunch of charismatic Platonists, especially Plotinus and Porphyry, convinced everyone that Platonism was the bee’s knees. They still wanted to use Aristotle though, because he’s so useful. (They didn’t call his logical works “The Tool” for nothing. [Update 20/9/17: I recently read [Smith 2017: §2] that while they didn’t call it “The Tool” (Organon) for nothing, they also didn’t call it that because it was useful. They called it that to stick it to the Stoics. (Those aren’t Smith’s exact words.) The Stoics thought logic (which at the time included a lot of epistemology) was one of the three branches of philosophy (which at the time included a lot of stuff now counted as science), and the Aristotelians disagreed and thought logic (including epistemology) was just a tool used by philosophy (including a lot of science). Nowadays we tend to agree with the Stoics, but it’s still called the Organon. Oh well.]) Eventually Platonists solved the problem by writing a bunch of commentaries on Aristotle explaining how they were both basically on the same page, although this meant reading Aristotelian ideas into Plato as much as reading Platonist ideas into Aristotle. (Or in Iamblichus’ case, reading both into Pythagoras.) The resultant synthesis dominated philosophy in the West until the Renaissance, and had a pretty impressive run in the Middle East too. I was already kind of aware that something like this had happened, but I didn’t realize it had happened so quickly.)


Anyway, Guthrie doesn’t go to either of these extremes. He’s not interested in claiming that Plato had already anticipated all of Aristotle’s contributions; it seems that this kind of tosh had already gone out of fashion among serious scholars by Guthrie’s time. But he also isn’t interested in setting them up as two giants staking out the two main sides in a debate that the rest of us have been having ever since. And I think that’s kind of important. Even if you’re not going full bore with the School of Athens Caricature, you might think that the outlines of the big philosophical debates got laid out early on and the rest is just filling in the details. It’s refreshing to see that rejected, and a little challenging.


He starts off the discussion of Aristotle by talking about his life, and he emphasizes how Aristotle spent twenty years at Plato’s Academy, that he studied Plato’s work a lot, and that when Plato died and Aristotle left Athens he took the hardcore Platonist Xenocrates with him. Moving from the circumstantial to the more substantive, he says:


Fundamentally he remained on the side of Plato and Socrates. As Cornford put it: ‘For all this reaction towards the standpoint of common sense and empirical fact, Aristotle could never cease to be a Platonist. His thought, no less than Plato’s, is governed by the idea of aspiration, inherited by his master from Socrates - the idea that the true cause or explanation of things is to be sought, not in the beginning, but in the end’ [Cornford 1932: 89-90].
In other words, the question that both can and must be answered by philosophy is the question ‘Why?’ To answer the question ‘How?’ is not enough. To speak more strictly, we may say that the permanent legacy of Platonism to Aristotle was two-fold, though its two sides were intimately connected. What he took over and retained was:
(i) the teleological point of view;
(ii) the conviction that reality lies in form.
He could not give up his sense of the supreme importance of form, with which, as we have now seen, it was natural for the Greeks to include function. To know the matter out of which a thing had come to be was only a secondary consideration… The definition then must describe the form into which it had grown. [p.126-7, his emphasis]


It’s probably fair to say this is quite different from how a lot of metaphysics is done nowadays. Teleology is unfashionable except as metaphor surrounded by disclaimers. We love poking around in the fundamental constituents of reality out of which middle-sized things arise, and the forms of the middle-sized things themselves are often a bit of an afterthought if we talk about them at all. We’ll happily try to explain how intrinsic change is possible, but to try to explain what intrinsic change is for would seem decidedly weird to a lot of us. Guthrie’s take on Aristotle’s relationship with Plato reminded me of something Jonathan Schaffer said about the Quine-Carnap debate in meta-metaphysics:


Indeed, though the textbooks cast Quine and Carnap as opponents, Quine is better understood as an antimetaphysical ally of his mentor (c.f. Price 1997). The Quine-Carnap debate is an internecine debate between anti-metaphysical pragmatists (concerning the analytic/synthetic distinction, with implication for whether the locus of pragmatic evaluation is molecular or holistic). As Quine himself says:


Carnap maintains that ontological questions, ... are questions not of fact but of choosing a convenient conceptual scheme or framework for science; and with this I agree only if the same be conceded for every scientific hypothesis. [Quine 1966: 211]


The Quinean view of the task and method of metaphysics remains dominant. Indeed, the contemporary landscape in meta-metaphysics may be described as featuring a central Quinean majority, amid a scattering of Carnapian dissidents. Few other positions are even on the map. [Schaffer 2009: 349-50, his emphasis]


Schaffer is actually suggesting that we get a bit more Aristotelian (and not because we had previously been overly Platonist), but that’s not why I was reminded of it. It’s more just the structural point: what might seem like the two main contenders in a grand debate over an eternal cosmic question may really be two versions of a view that historically has been fairly niche. Framed in this way, perhaps it’s no wonder the late ancient Platonists were able to find so much common ground between Plato and Aristotle. (If you’re interested in when the grand synthesis of Quinean and Carnapian meta-metaphysics is coming, Carrie Jenkins is in the vanguard, responding especially to work by Amie Thomasson [2007, 2010]. Jenkins calls the resultant view Quinapianism. It involves the notion of a serious verbal dispute [Jenkins 2014]. The basic idea, as I understand it, is that it’s OK to justify metaphysical claims (such as if there are simples arranged baseballwise then there are baseballs) as being entailed by the way our concepts work, like Carnapians do, but that our concepts, and thus these entailments, are still subject to revision in the light of empirical investigation, as Quineans think everything is.)


In summary, there were three categories of thing that struck me about Guthrie’s book. First, he’s keen to emphasise that the ancient Greeks had different ways of thinking about things than we do, and he discusses this in terms of the meanings of their words. Second, he plays up the influence of the city-state political structure on ancient Greek thought up to and including Aristotle. Third, he thinks it’s reasonable to describe Aristotle as a bit of a Platonist.


There’s a unifying theme here: the ancient Greeks were a distinctive lot who were not like us, and this comes out in their philosophy. Now, often when we’re learning about a philosophical tradition we’re used to the people not being like us. When Westerners are taught about ancient Indian or Chinese philosophy, they expect to be presented with ideas that arise out of an unfamiliar mindset, and they expect to have to learn about the mindset to understand the ideas. We’re ready to find the similarities surprising and the differences exciting. I don’t think we tend to approach the Greeks the same way. We (by which I mean Westerners; I live in the UK) see ourselves as the Greeks’ intellectual heirs. Other influences on us are tributaries; ancient Greece is the source. We expect learning what Plato and Aristotle cared about to explain what we care about, not to challenge it. We think we’re coming from basically the same place, and this affects how we interpret them. We’re more prepared to find their ideas coming naturally to us, and we’re less curious and less charitable when they don’t. To an extent this attitude probably makes sense. There really is more Plato than Confucius in Western philosophy as it’s done today. But if you’re serious about engaging with ancient Greek philosophy, you should still expect a culture shock.


References


  • Cornford, Frances Macdonald (1932). Before and After Socrates. Cambridge University Press.
  • Falcon, Andrea, "Commentators on Aristotle", The Stanford Encyclopedia of Philosophy (Fall 2017 Edition), Edward N. Zalta (ed.), forthcoming URL = <https://plato.stanford.edu/archives/fall2017/entries/aristotle-commentators/>.
  • Guthrie, W. K. C. (1950). The Greek Philosophers From Thales to Aristotle. Routledge. (Page references to 1962 Methuen reprint.)
  • Guthrie, W. K. C. (1962-1981). A History of Greek Philosophy. Six vols. Cambridge University Press.
  • Guthrie, W. K. C. (1962). A History of Greek Philosophy: Volume 1, the Earlier Presocratics and the Pythagoreans. Cambridge University Press.
  • Guthrie, W. K. C. (1981). A History of Greek Philosophy: Volume 6, Aristotle: An Encounter. Cambridge University Press.
  • Jenkins, C. S. I. (2014). Serious Verbal Disputes: Ontology, Metaontology, and Analyticity. Journal of Philosophy 111 (9/10):454-469.
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Sunday, September 10, 2017

Closing The Altruism Loophole

The prisoner’s dilemma is one of the best known puzzles in game theory. Here’s a version of it.


Two criminals, Alice and Betty, have been captured and imprisoned in separate cells. The guards want them to talk. If one talks and the other doesn’t, the talker goes free and the non-talker gets a long sentence. If both talk, both get mid-length sentences. If neither talks, both get short sentences. Alice and Betty only care about the lengths of their own sentences. Should they talk?


Whatever Alice does, Betty does better if she talks. Whatever Betty does, Alice does better if she talks. So if they’re acting self-interestedly, talking is a no-brainer. But both talking works out worse for each of them than neither talking. The point is that it seems self-interest alone should be able to get them from the both-talk situation to the neither-talk situation, because it’s better for both of them. But it also seems there’s no rational way for them make this happen. That’s the puzzle.


One thing people sometimes suggest is that the solution is to be altruistic. The fact that if Alice talks and Betty doesn’t then Betty will get a longer sentence gives Alice a reason not to talk, if she cares about Betty. In a way, people bringing this up is annoying. It’s either a misunderstanding of the problem or a refusal to engage with the problem. Part of the set-up is that Alice and Betty only care about the lengths of their own sentences. But on the other hand, the prisoner’s dilemma is supposed to be structurally similar to some real-life situations, and in real life people do care about each other, at least a bit. Also, we sometimes like to do experiments to see how people behave in real-life prisoner’s dilemma situations. If the prisoner’s dilemma has self-interested subjects and our test subjects are somewhat altruistic, as people tend to be, then we’re testing it wrong.


There are at least three ways round the problem. One is to make Betty a less sympathetic character, who cares about something Alice doesn’t care about at all. One option I’ve heard is to make Betty a robot who only cares about increasing the number of paperclips in the world. Alice’s payoffs are money, and Betty’s payoffs are paperclips. But this introduces an asymmetry into the situation, and it also means we’re not dealing with two humans anymore; we’re dealing with a robot. And the robot doesn’t behave according to general principles of rationality; it behaves how we’ve programmed it to behave. If we can’t formulate a principle, we’ll struggle to program the robot to follow it. If we tell the robot to apply the dominance reasoning, it’ll talk. If we tell the robot to assume everyone picks the same option in symmetrical situations with no indistinguishable pairs of options, it won’t talk. (This principle is very close to what Wikipedia calls superrationality.) We don’t learn anything from this. It’d be better if we could test it with people.


A second way to try to avoid the altruism loophole is to set the payoffs so the participants would have to be very altruistic for it to affect what they did.




Betty



Talks
Doesn’t talk
Alice
Talks
Alice gets £2
Betty gets £2
Alice gets £7
Betty gets nothing

Doesn’t talk
Alice gets nothing
Betty gets £7
Alice gets £3
Betty gets £3


Suppose Betty talks. By not talking Alice would give up her only £2 to get Betty an extra £5. That would be awfully nice of Alice. Supposing Betty doesn’t talk, by not talking Alice would give up an additional £4 so Betty could keep her £3. That seems rather nice of her too. If Alice truly loves Betty as she loves herself, she probably won’t talk however we set it up, or at least she won’t know which to do because she doesn’t know what Betty will do. (Since the total payoff in nobody-talks is higher than in both-talk, not talking must increase the total either when the other doesn’t talk, when they do, or both.) But most people don’t love the other participant as they love themselves, and fiddling with the payoffs can make it so that more altruism is needed not to talk.


A third way is cleaner. I hadn’t heard it before, so when I came up with it I thought I’d tell you about it. The problem was that Alice might allow her behaviour to be affected by concern for what happens to Betty. To avoid this, we start by roping in three other people Alice cares about just as much as Betty (let’s just assume none of the participants know anything much about the others). There are four possible outcomes for Betty, so we randomly divide up the three outcomes Betty avoids among the three other participants. Since Alice is indifferent between Betty, Bertha, Bernice and Belinda, she doesn’t care which prize goes to Betty as it’s still the same four prizes distributed among the same four people. The only variable left for Alice to care about it what happens to her. Similarly, we divide the outcomes Alice avoids among Althea, Annabel and Albertine, so Betty will be indifferent between outcomes except insofar as they affect Betty. Alice and Betty won’t keep quiet out of altruism now, and if they can’t think of another reason to keep quiet, they’ll end up both talking and wishing neither of them had.

So, introducing the other people closes the altruism loophole. I guess it doesn’t close the justice loophole, if there is one. The problem there is that Alice might not talk because she is concerned that Betty might not either and she doesn’t want to punish Betty for doing her a favour. Or maybe Alice will think she has some special responsibility towards Betty as a fellow player. But at least we’ve closed the simplest version of the altruism loophole. If we haven’t tried testing the prisoner’s dilemma this way, I guess we should. Maybe we’ll get different results. Or maybe we’ll get results we’d previously attempted to explain through altruism, and we won’t be able to explain them away through altruism anymore. Of course, we may already be doing this. I don’t know. It's not my area.