# Experimental Mathematics, Armchair Physics

"Philosophers have traditionally classified mathematical knowledge as 'a priori' and scientific knowledge as 'a posteriori,'" Baler says. "In other words, mathematics can be done without leaving one's armchair, while physics cannot. In this lecture I challenge both sides of this traditional picture. On the one hand, are there such things as 'mathematical experiments' and what kind of role might they play in mathematics? On the other hand, can conclusions be justifiably reached about the nature of the physical world that do not depend on observation or experiment?" The lecture is intended for a general audience and no specific mathematical or scientific background is presupposed.

#### Audio Transcript

Sarah: Good afternoon and welcome to the Faculty Lecture Series. It's my pleasure to introduce Richard Eldridge, who will introduce today's speaker. Thank you.

Richard Eldridge: Gee. Thanks, Sarah. It's my pleasure to introduce my colleague, Alan Baker. Alan is a man of wide acquaintance and accomplishment both, acquaintance at the college. You may know him from playing in the orchestra with him, where he is a mainstay of the violin section. You may know him from the chess club, where he's the founding member and presiding spirit of the shogi section of the chess club, or it's an independent shogi-

Alan Baker: An independent.

Richard Eldridge: Yes?

Alan Baker: Mm-hmm (affirmative).

Richard Eldridge: He's also been for faculty members widely involved in discussion groups here at the college of various kinds. He's organizing together with Mark Goodwin this tri-college group in the Philosophy and Mathematics for next year. Since his time here, he's been a regular participant in the tri-college discussion group on complexity, emergence and chaos.

To give you the backstory to all of this, Alan took his undergraduate degree at Cambridge University. Following that, he went directly to Ghana to teach Logic for one year, two years? One year. Then, came back to Philosophy and entered the PhD program at Princeton where he completed his PhD in the Philosophy of Mathematics, principally riding on the indispensability argument for the existence of mathematical objects.

He moved to Xavier University for three years, and then we brilliantly stole him from there to come here. Last year, he held a Fulbright Fellowship at the University of what in ...?

Alan Baker: In Estonia.

Richard Eldridge: Tartu in Estonia? In Estonia, from where he traveled to meet people all over the Baltic states. He now has an international acquaintanceship in Philosophy and Mathematics circles with colleagues ranging from Australia to Estonia to Brazil and elsewhere. His work had centered in both general philosophy of science and in the philosophy of mathematics, particularly with arguments about the existence of mathematical objects, with collateral interests also in modern philosophy.

The interesting work we're hearing today brings together work in general philosophy of science and in mathematics. It's on, as you can see from the title, what sort of a science something called experimental mathematics might be. Alan Baker.

Alan Baker: Thanks very much. I'm going to start with this picture, because it's November the 5th and tonight is Bonfire Night in England. As I speak, my compatriots are setting off fireworks and burning effigies of the 17th-century ringleader of a plot to block the Houses of Parliament, Guy Fawkes, pictured in the middle of the bonfire there.

I also cleverly am segueing into my first quote of the evening. Now this is a famous philosopher's quote, an infamous quote, maybe, from the 18th-century philosopher David Hume, and it's going to introduce the main theme of my talk today. Hume was worried about what kinds of disciplines, what kinds of methods lead to knowledge or have content. He was particularly worried about, here, divinity and metaphysics. I'm not going to worry too much about divinity today. I'm going to worry about metaphysics.

His worry really is whether there's any space between abstract reasoning concerning quantity or number on the one hand or experimental reasoning on the other. In other words, if something isn't abstract and numerical and it's not experimental, is it doing any work for you at all? If not, we should just consign it to the flames.

The terminology I'm going to use is going to be slightly different. Today, rather than abstract reasoning and experimental reasoning, we have two other pieces of philosophical terminology that I want to introduce. I'm confusing you. They sound fairly similar, so I might write them up on the board at some point and leave them there.

What Hume is calling experimental reasoning, I'm going to use the term "a posteriori." A posteriori knowledge or an a posteriori claim is something that you can only know on the basis of experience. It's not true that it rained earlier today, I think. In order to know it's not true, I need to do more than just think about it really hard. I had to have a certain set of experiences or at least be informed by experience.

The claim that it rained earlier today would be an example of an a posteriori proposition. Notice it's going to be important for later on the "only" here, knowable only on the basis of experience. It's going to turn out that that's going to be important.

Then the other side of the dichotomy here is a "priori." This means that we don't need experience. The claim is a priori if we can know it just by pure thought of some sort. The claim either it rained earlier today or it didn't, if I happen not to have been outside or not really had any relevant experiences, the idea is that I will still be able to conclude that that is true. That would be an a priori proposition.

That's going to correspond roughly to Hume's experimental reasoning and abstract reasoning. Now the traditional view of it is that if we're going to make a rough distinction here, that mathematics is a priori. If we take a mathematical claim like there's a prime number between 5 and 10, that we do indeed know this is true, but that this knowledge is not based on any particular experience and this therefore should be classified as a priori. We can worry about this, but we'll start with the traditional view.

That's going to contrast with empirical science, including physics. You can have sentences that superficially have the same form. There is a planet between Mars and Saturn, and again, in order to know whether that's true or not, we need to have made some observations, done some experiments, had some experiences.

The question I want to ask then is, how firm, how fixed is this division between a priori mathematics and a posteriori science? Is there any sense in which the boundary could be blurred or even reversed? In other words, are the cases falling on the mathematical side that we're going to want to say that the reasoning is based on experience is a posteriori, and conversely, that reasoning on the scientific side that we're going to want to classify as a priori.

I'm putting this in terms of experimental mathematics, mathematics based on observational experiment. I'm pulling armchair physics, so we can generalize it to armchair science in general. These two questions are going to form the central part of my talk, and then towards the end I'm going to talk a little bit about metaphysics and where that might fit into the picture.

Now, one consequence of the definition of a priori knowledge and a posteriori knowledge is there's nothing to stop a particular person knowing an a priori proposition through experience, because a posteriori ... Let's slide this up here. A proposition is a posteriori if it's only knowable through experience, which leaves open things that I might happen to know by experience, but I don't have to know them that way.

What am I talking about here? In the mathematical case, I'm not a particularly sophisticated mathematician, but there's plenty of things that I think I know mathematically that I don't think I have an a priori justification for. Let me give you a well-known example.

I think that Fermat's Last Theorem is a theorem. I think it's most likely true, and I think I know it. I think I have good reasons for thinking it's true. Can I prove it? No. Do I have even the vaguest idea? Maybe the vaguest idea, but not much more than that. Very few people can prove it. Maybe a handful, if that.

I don't want to say that Fermat's Last Theorem is therefore not an a priori proposition. I think what I want to say here is that I have a posteriori access to it or a posteriori knowledge of it. How is it a posteriori? I've read newspaper reports, I've talked to mathematical experts, and I've got information, I've had experiences that lead me to think that someone else has proved this. I'm at the end of a chain that started off maybe with an a priori proof, but I'm at the end of a long a posteriori chain.

It seems like one way of putting this would be that Fermat's Last Theorem is a priori for the mathematical community, but not necessarily a priori for every member of that community, as long as someone in the community has proved it. We might want to relax the notion of a priori even more because there are other examples, but we might want to say that the mathematical community has a priori knowledge, has a proof where no individual member actually can or has proved it.

The one example that's relatively well known to mathematicians is the classification of a finite simple group. It doesn't matter where the finite simple group is for this purpose. What matters is that the classification of all of them, which was completed in the mid-'80s, ran to 500 general articles, about 10,000 pages of mathematics and about 100 different authors contributing to it over a 30-year period. This is sometimes called the enormous theorem for good reasons.

We might want to say that the mathematical community has proved the enormous theorem, and it may turn out you can make it a bit less enormous, but let's talk about when they first proved it. In that case, no single mathematician could prove it. It's too big. That would be a sense of the community having a priori knowledge of this thing, but actually no single person had had that knowledge.

I think there's more interesting things to say here, but I want to move on and take away from this a point that's going to come back on the other side of the divide when we talk about the armchair physics, which is that one way of thinking of this is that my cognitive limitations or cognitive limitations of individual human mathematicians is what leads to the reliance on a posteriori methods here.

When I'm not smart enough to prove something, I rely on someone who is, roughly. When the community as a whole isn't smart enough individually, they can sometimes as a community prove the thing and sometimes they won't even be able to do that, but the idea here that some had cognitive limitations placing limitations on an individual or a community's a priori access to something, but sometimes, because your pure reasoning powers are not good enough, you may end up with a posteriori knowledge of an a priori proposition.

What about on the other side of the divide? I'm going to get to the physics case in a few minutes, but you might think that a similar thing goes on with some our knowledge about the world, about the physical world. I think the world is round. I think the world's a sphere.

I'm trying to think. I don't think I've ever really had an experience that has persuaded me of that. I certainly haven't ever seen it directly as spherical. I think my experience is fairly consistent with it being flat, for sure. I think I can certainly organize my experience in a way that makes sense of it being flat, and of course that's why for a long time most people thought it was flat.

Why do I think it's spherical? Again, it's probably because experts on these things have told me and I've also seen images like this one. Of course, this is not observing the world to be spherical. This is an image. It could be a big conspiracy. I'm not introducing this to be skeptical about it, just to say that in a sense what I have here is probably indirect a posteriori knowledge in some sense. I haven't made direct observation of the thing.

It's not parallel with the mathematical case because here I'm still in the realm of the a posteriori. I'm just moving from direct to indirect. Maybe that's not a distinction we want to work with here, but there's not a sense here that, because I haven't seen it myself, I have to rely on a priori methods. What we're going to see later on is that may sometimes happen, so there may be a mirror image case here in the empirical sense.

It's important not to confuse the a priori/a posteriori distinction with whether something is actually relevant to the real world. In other words, we could argue that mathematics is a priori and also maintain it's very useful in predicting experience and telling us about the world.

One picture is this one. On the left-hand side, the M1 and M2 would be some sort of mathematical theory, sort of mathematical claims, and this is some physical situation. Say I'm swinging a pendulum and I want to know where it's going to be in three seconds' time. I could just wait and see, or I could model it mathematically and make a mathematical prediction and then map it back on to the physical situation. If my model is a good one and I've preserved the relevant things, that may give me a way of predicting something about the physics, but that says nothing about the a prioricity, and that's the adjective I think, of the mathematical case.

Take a simple example. 17 is prime. If I have 17 pieces of, I was going to say "cake," but we can slice those up, 17 wrapped-up pieces of Halloween candy and a certain number of small children I need to divide them among, knowing that 17 is prime will tell me something about how I can or can't divide these things. For example, if it's more than one child, I can't give them equal numbers of candy without having some left over. Does that mean 17 is prime is not a priori? No. It perfectly well could be a priori. It's just a useful thing to know in this candy-dividing situation.

Again, we could have numerical facts about number theory and facts about candy, and if they map on to each other in a relevant way, then the a priori facts can be useful. Whether that's a mysterious thing that needs explaining is a subject for another talk. I'm not going to say anything more about applied mathematics here, but just to say that I want to lay out those two issues.

Let's get on to the question here about what applied mathematics might look like, and I'm going to go through, I'm not going to talk about bubbles yet, I'm going to talk about Gauss. I'm going to take two approaches here.

One is to actually look at some possible examples of experiments that might have been done or could be done or are being done for mathematical reasons, because that's one way in trying to get a handle on experimental mathematics. Other examples of experiments with mathematical payoffs, if you like. The second approach will be to look at areas of mathematics that sound like they really should be a posteriori.

Let's take the first approach first. There's a story which I think is probably actually [a pocketful 00:16:26] of Gauss, a famous ... You know he's famous because he's on money. He went up one German mountain and triangulated, I guess, with light rays to two other German mountains to make a big triangle, and they weren't that close together, and the story is he measured the angles inside this triangle and saw if they added up to 180 degrees or not. In other words, he was working out whether we really are an Eucledian universe, because Eucledian geometry predicts the interior angles of a triangle at up to 180 degrees.

It runs out that's probably not what he was doing. Anyway, the experiment there is a way to, any curvature that you could measure here would be way too small. Imagine someone did this. You could do it with stars as well. You could do it with things that are much further away from each other and you'd actually have a hope of measuring it. Of course it turns out the space isn't Euclidean, but the point for this purpose is that whatever you find out here, let's go back to this thing, is not about the truth of Euclidean geometry. It's about whether it's applicable to the world.

What it's telling you is, is Euclidean geometry a good model for physical space? If your triangles and physical space don't add up to 180 degrees, it's probably not. If they do, it might well be, but it's not as if Gauss, the payoff here was purely mathematical. Whatever he was doing here, you're not going to say, "And therefore Euclidean geometry as a pure theory in itself is false."

I want to argue that regarding whether if you did this and what he was trying to do, that wouldn't count as a mathematical experiment because you're not really testing a mathematical theory. What you're testing is the applicability of a mathematical theory.

Second example. The Buffon's needle. Some of you may have come across this one. This is probably [inaudible 00:18:19]. Let's see. Turns out you can approximate pi by dropping needles or similar-shaped things on top of a grid, or not a grid, whatever you call this thing with parallel lines, and counting the number of times that a needle crosses at least one line. You imagine a piece of paper with two lines on it or a set of lines, dropping pins.

How does pi get involved? Intuitively, your chance of crossing a horizontal line depends on the orientation of a needle, and that's got to do with a circle and how it's orientated. I'm certainly not going to go through the mathematics here, but it turns out you can get a formula through the probability of it crossing a line that's based on pi, and therefore by doing the experiment enough times, I'll do it 10 times, down here is the approximation we're getting for pi. It's not terribly good here. It's in the ballpark. That's getting worse. It's getting better. That's 1,000 needle drops at 3.11. If you carried on, you'd get that.

Is this a mathematical experiment? The problem here I think is, aside from the statistical nature of the result, is that you're not learning anything new. You're certainly not learning anything that you couldn't prove just through a priori methods. We have a priori methods, traditional proof methods of getting pi to whatever accuracy we want. We don't need to drop needles. We're not discovering a new mathematical result here.

Again, I want to argue that that's not really experimental mathematics in a full-fledged sense. I think the closest example is this next one, which is a little grainy. In the 19th century, there were mathematicians interested in this problem of minimal surfaces. Given some set of points and space or some structure, what's the minimum surface area that's needed to bound it?

There was a blind Belgian physicist called Plateau who hit on the idea of, and he's not the first to hit on the idea of using bubbles, but in this particular case of using soap films. Turns out that soap likes to adopt the configuration to minimize the energy of the surface, which corresponds to minimizing the surface area, which is why bubbles are spherical, for example, but when you get these things suitably into soap, you actually get them into more surfaces if you're careful. They often turn out to be kind of weird and surprising.

This is actually, I think it's an inside cube with faces going onto the edge. This is the case for the cube. This method was used, as you might put it, to discover the answers to some of these minimal surface problems. Isn't this a mathematical experiment? This I think is the closest that we have so far. We are getting discovery. We're genuinely discovering new mathematical results, and we're doing it here by reversing the order of the traditional thing.

What I had up on the board before was this mathematical physics link where you use the mathematics to get the physics. Here we're using the physics to learn something about the mathematics. We're saying the relationship between bubbles and minimal surfaces, do your bubble thing or your soap film thing, and then you can make a prediction about the minimal surface, which is reversing the normal use of applied mathematics.

The reason I think it's not going to count as full-fledged experiment onto mathematics is now on the justification side, because it's actually quite easy once you have a minimal surface to prove that it's minimal. That's relatively easy. For these people, it was easy. Their problem was actually work out what the surface was in the first place. Having discovered this, you could go away and prove it. In a sense, your justification at the end of the day, you could still argue it was a priori. It was just based on proof.

Tentative conclusion from these sorts of literal experiments is that we not only capture ... We have the sort of Fermat's Last Theorem case for where cognitive limitations lead to reliance on maybe a posteriori methods. In that case, I'm not learning anything new for the community as a whole. In the case of the Buffon's needle case, we're perhaps getting a new way of justifying a known result, so a new way of justifying the rough value of pi, but it's not a new result.

Here we're actually getting a new result that we can still stay with an a priori justification. No single example if really capturing all aspects of mathematical method here, and also these cases are rare. I've more or less given you, maybe a couple of others, but that's about it for literal experiments or anything that approximates a literal experiment. It's really not central to mathematical method, these examples.

I want to move on to a second approach, which is a hermeneutic approach. Look for things in mathematics that sound experimental. Mathematicians have labs, which is, I want to have a Philosophy lab one day. I like labs. They also do stuff they call experimental mathematics. In fact, it's a growing field. Surely experimental mathematics is experimental, you might think.

If you notice what was in the labs, it was computers, I guess, strange-looking computers. It turns out that if you open your Experimental Mathematics book, it's all about computers and computer programs and what people have done with their programs. What's the link here then between computers and experimentation?

Let's look at a couple examples. Here's another famous still conjecture, because it hasn't been proved for 500 years old now? It's in the 400 years old, probably. Goldbach's Conjecture. Another frustratingly simple to state, has resisted terms of proof, every even number other than 2 can be expressed as the sum of two primes. Mathematicians think it's true. You can verify it for as long as you feel like it.

When you get tired, you can turn it over to a computer, which is currently being done. As of April of this year, they've verified it all the way up to 10 to the 18. They checked every even number from 4 to this number and every time you can find two prime numbers.

That's a lot of numbers, but there's a lot left. This is not a project with a future in terms of proof. This is long-term in the extreme. Nonetheless, it seems like it's evident. Once you've done 10 to the 18 of these things, I think you're pretty sure it's true.

The sense in which the computers are providing evidence, nearly all of these cases, human beings have done a few thousand cases, maybe up to 10,000, the rest have been done by computers. Nearly all the cases here have been done by computer in a way that you couldn't really turn over to human labor here.

We have here in a sense computer evidence, in some sense computer evidence for a mathematical conjecture, and it's not just restricted to evidence. Computers have proved things, or at least computers have been involved and approved. The famous example here is the Four-Color Theorem. This again for a long time was a conjecture. It was proved in 1976, I think.

The question is, what is the fewest number of colors to color a map on a plane such that no two regions that share a common border are colored the same way? It was long conjectured that you didn't need more than four colors. It turns out this may actually help you, ironically, to color real maps because you have to keep the seas all the same color.

The different seas are all blue, so actually you need more than four colors in theory for a real map, but it's still an interesting mathematical problem. Also the other problem is that countries sometimes have bits that are off them, so like Russia and Kaliningrad are the same country, you want them to be the same color, but they're actually two different regions.

Male: [Inaudible 00:26:52].

Alan Baker: Yes. That is continuous as well. This was proved in 1976. A big part of the proof was just going through a lot of different cases. They reduced it to something like 1,600 different cases of various degrees of complexity, and it basically said, look, if we can show that all these work, then the theorem's proved. That was the proof, reduced it to this thing. That stuff was done on a computer, several hundred hours of computer work, albeit 1976 computer work. In a way, they didn't seem replaceable again by human mathematical effort.

We have computers here involved in the justification of interesting and important problems, sometimes even involving the proof of those problems, but again, why think it's experimental? It's a good question because when you open up these Experimental Mathematics books, there's usually a chapter at the beginning where they talk about why it's experimental, but it's all fairly metaphorical.

Here's a typical quote: "Computing is to mathematics as the telescope is to astronomy." Or it should be computers is to mathematics. Anyway. "It might not explain things, but it certainly shows what's out there." This idea of computer as means of access to the mathematical realm, computer as instrument.

There is certainly an analogy. I showed you a math lab that it's full of computers. A science lab will be full of other sorts of equipment, maybe including computers, but microscopes and such things. There's certainly an idea, look, these are the machines that mathematicians use to augment their abilities, in a sense.

Another sense in which it's experimental, I guess, relatively is that the way we built computers and programmed them certainly depends on a posteriori knowledge. They're physical things and they depend on a knowledge of physics to actually be built. Then, as the quote I just gave you suggests, the sense in which the results are not explanatory, the raw data coming out in the same way that something coming out of an instrument. That's reasons to think you could consider it experimental.

The problem is, there are other reasons for thinking about ... This metaphor of seeing, I think it has to be metaphorical, when I see a water bottle in front of me, there's a sense in which that's because of the causal activity of the water bottle on me. Whatever's happening when a mathematician sits in front of a computer and runs some piece of software, there's not going to be a causal interaction with the numbers involved. Whatever the objects are of mathematics, it's not as if we're literally trying to causally interact with the mathematical realm.

The other thing is, what are we doing when we program computers? We're really trying to mimic certain aspects of human thought. You might think that these are really extensions of a priori capabilities. The kinds of things that computers are doing are really just pushing faster and more efficiently things that we're doing when we reason a priori.

Then, related to that, that given enough time and enough patience and enough people, you can in principle reproduce this. All right, 10 to the 18 is big, but it's ... Yeah, maybe it is too big, but given enough people and enough time, it's not too big. You could do that. This is in practice, in principle distinction.

A tentative conclusion I'm going to come down on another side to whether computers are experimental, at least in the sense of giving us a posteriori knowledge. I'm going to argue, at least provisionally, that computer mathematics is not experimental in any sense that should force us to see the results as a posteriori, but nonetheless, the boundary is being blurred here and it's being blurred because of cognitive limitations. What's been going on is cognitive ability is being augmented by the computer.

All right. I want to come at this now from the second direction, which is from the direction of armchair physics, possibly. Here we go. The connotation of armchair here, I'm using it slightly, I don't know how I'm using it, ironically, what I mean by armchair physics or armchair science is that you're doing armchair physics if you're not attempting to make the appropriate kinds of observations or coarsely interacting with the objects of your study. It's a takeoff from this idea of seeing.

I think in order to be respectable, we need a Greek word for it. Unfortunately, the Greeks didn't have armchairs as far as I can tell. They had couches, like reclining couches, and they had stools. If you put together arm and, a diphros is the standard Greek chair with the back, so brachiodiphrology, technically it's the study of armchairs, but it will serve my purposes.

The point about sitting in an armchair is not that you have no experiences, but you just maybe have an irrelevant experience. If you're doing armchair ornithology, unless you happen to be having your armchair in the middle of a jungle or something, you're not making the appropriate kinds of observations.

I did come across the armchair nanotube, but with the exception of that, there's not much you can interestingly ... The armchair connotation is meant to be, again, not having no experiences but having no relevant experiences. Can you find out how the physical world is through a priori reasoning, through just thinking hard enough?

A couple of quick points about how you can. You can work out what follows from what from your armchair. For example, that a certain scientific law implies another scientific law or this is a special case of that or ... There'll be things about the logical relations between scientific hypotheses that you can work out from the armchair.

Of course, you will only be interested probably in those hypotheses because of experiment, but the actual reasoning could be purely armchair reasoning. You can infer the properties of a hypothetical entity or an entity whose existence you didn't care or didn't know about, but again, I would be informed by information about the entity that you would have probably from experimental reasoning.

You can also do thought experiments. I'm not going to talk about thought experiments here, but that's the class of perhaps armchair reasoning. Again, the problem is going to be that the principles by which you evaluate the thought experiments are probably going to come from experience.

Take Newton's thought experiment, that firing a cannonball further and further in the diagonal direction, it falls further and further away, and as you fire it far enough, it starts orbiting right around the world, this idea of orbit as the extension of cannonball firing. Why we think the cannonball would do that is based on experience.

Those sorts of examples are not telling us really how the world is. They're telling us about deductive links between things, they're telling us about the hypothetical properties of things, were they to exist, and they're telling us about what's possible and impossible maybe in the case of thought experiments.

How about knowledge about how things are? Can we get that from the armchair? The example I want to consider is scientific theories which go beyond any available or even in principle available means of observation. The canonical example here is, oh, no, it's meant to be backwards, string theory.

There's been quite a lot of popular science press about string theory, also called super string theory, one of the theories of the boundaries of current physics. Why is the string theory so controversial? It's apparently because the leading string theories disagree only about things that are so remote from possible experiment that it looks like it's going to be impossible to actually experimentally distinguish between them.

Really roughly speaking, an 11-dimensional version of string theory and a 26-dimensional version of string theory, where the dimensions are rolled up really, really tiny, the ones you can't see beyond the three that we can see before. They seem to be so small that you would require such high energy as to see them that we can never actually experimentally distinguish. This is the argument.

Say this is true. It turns out to be a little bit more complicated than this. Say we had two theories, 11-dimensional and 26-dimensional. The argument is, in this case, we're not doing science anymore. We've got two theories where no possible experiments can be distinguished between them.

Commentators have argued, and often physicists, actually, have argued that once we go beyond experiments, we're back either in the realm of pure mathematics or maybe in metaphysics, but whatever we're doing, we're not doing science in a traditional sense. How worried we should be about that depends on the commentator.

I'm not going to talk about the string theory case, but what I want to argue is that actually, in much more mundane cases, you get the same kind of problem. We don't need to go all the way out to these esoteric theories. We get the same thing going on more or less whenever we do scientific reasoning, and the problem is ... Two problems.

One is the experimental data we have never conclusively implies what the right hypothesis is. You're doing some experiment and you get some nice points and you're lucky they're all lying on a nice line. It's a couple of things related to each other, and you try to decide what the actual relationship is between these things. You think, that's nice, they're all on a line, so that must be the right relationship.

Notice, this assumption which that I didn't have to draw the line that way. You might say, "Look, I did. I had to go through all the points," but this, except I can't draw it properly, but this is going ... [I believe 00:37:25] my point. That goes through all the points as well. Why isn't that any pretty good candidate for the relationship between these two things? Of course, I can draw infinitely many other bizarre-shaped curves.

One way of putting this is that the available evidence here doesn't determine what the right curve to draw here is. You might say, "That's not too bad because I can always do some more experiments, and now I want to decide between these two. I can look at something else, and if I get this, then I know my straight line was right and my curve is wrong." Nonetheless, any point where you stop observing and make a hypothesis, there are going to be an infinite number of different ways of drawing a curve.

What do we do in these situations? We tend to want to do ... The other thing is the situation is nearly always worse than this because we don't have things lying on a line. We have experimental error and we have things maybe not too far from being on a line, but this sort of thing ... and we're drawing some sort of line between them.

What we're trying to do here, we're trying to get as close to as many of these points as possible. We're also trying to be as simple as possible. No, we're not always just going to have a straight line, so if we get something like this, we are probably going to draw this, but there's going to be a balance here between how complicated your curve is and how many points you go through.

The point I want to make is that however you're deciding how to draw these curves and how to prefer one curve over another, that's not on the basis of the experiment level. It's not the experiment of the evidence that tells you this is better than this. If it's not the experimental evidence that's doing, then what is doing it?

What seems to be going on is that we had preferences as part of the scientific methodology for mathematically simpler relationships in these sorts of cases, and that we use those in hypothesizing from the data, but that happens, as I say, that we don't need to go to string theory to see that. That will happen anytime we make a scientific hypothesis on the basis of data.

We can make things worse. If we're allowed to cook up any old theories we want, we can create theories that are empirically equivalent. In other words, two theories that really just say all the same things. These are not equivalent. They make different predictions.

Bertrand Russell has an example a bit like this one. We dig up a bunch of fossils and we're trying to work out what the best theory is to why these fossils are there, and we have H1, the dinosaurs roamed the Earth millions of years ago, died and were fossilized, and then we have H2 which is, the world was created 500 years ago or five minutes ago, it doesn't matter, by an ironic deity who included rocks shaped like the bones of large reptiles.

The idea is these can be tailored to make all the same predictions about fossils. I assume we should prefer H1 to H2 and we think the rationale in so doing, but that's not because of any experimental difference between their predictions about fossils.

Or we can add things onto our theories. If I said, "Look, which is more likely to be true?" You may not like quantum mechanics, but which is more likely to be true, H3 or H4? You say, "Oh, H3." Whatever you're doing, you're not doing it because H3 makes better predictions, because H4 makes exactly the same predictions. This of course is not an invisible pixie, but this is what an invisible pixie would look like if it were visible.

We have another principle here maybe, which is, if you can do it with fewer things, don't add things. In other words, don't add things to your theories without doing any work, but hey, the world could have invisible pixies in it. Who's to say there aren't invisible pixies on every subatomic particle? We think there aren't, but again, it's not from having done a whole bunch of experiments.

We don't have any pixie-eliminating experiment. We have, we're eliminating any pixies. It's through these sorts of general principles about maybe they're not doing any useful theoretical work. Some philosophers said there's no social security in the physical world. If something's not paying its way, then we don't assume it. Maybe that's true of invisible pixies.

All right. This leads to the final few minutes of the talk, but I want to suggest that one thing we can do with this is argue that while there may be no purely armchair science, in other words, reasoning about how the world is that precedes from start to finish in the armchair, as it were, there's a sense in which there's an armchair component whenever we do, whenever we make substantive claims about the world, and the armchair component is reliant on these principles that seem not to be derived from or based on experience. Rather, they're principles that we use to actually organize and systematize and generalize from my experience.

What happens if we take these components? We have experimental evidence plus these principles of theory choice, which [I'd argue 00:42:57] they're non-empirical. What happens if we take away the experimental evidence? What happens if we take away the experimental stuff and we're just left with the principles.

One suggestion is, in so doing, we can actually then move from physics or science in general to metaphysics. That we can make room for metaphysics really is completely disconnected from experiment, but it just takes over these general theory principles.

I just want to talk for a couple of minutes about metaphysics, since this is one of the things that humans are worried about. Now, metaphysics is a dangerous term. If you go into Borders and look in the metaphysics section, you have books on aura photography and astrological charts and crystals and all this sort of stuff.

Maybe that's metaphysics, but it's not metaphysics in the sense that I'm interested in today. When you think about it, it is not constrained by anything at all, really. These are speculations that some of them could be true, but they're not really constrained.

This notion of constraint also goes over to something else that gets called metaphysics, which is the speculations of scientists about stuff, usually by famous scientists about stuff that they aren't actually famous for doing. Again, not to say this is value-less or wrong, but it is relatively unconstrained. Penrose on consciousness, Hawking on God.

A lesser-known example is Eddington. This is the largest number, specific number ever to figure in a claim about the world. Eddington is a famous physicist. In 1923, he decided he could work out how many protons there were on the universe through a priori means. He was doing metaphysics basically here. He based it on the value of the fine-structure constant, which he thought was one of 136.

Unfortunately, it turned out to be closer than over 137. He then redid his proof and managed to prove the same thing from that. That's what I mean by lack of constraint. In a sense, it was quite unimpressive, but it was not too constrained by the actual evidence here.

This is bad metaphysics. Is there room for good metaphysics? What I'm suggesting is the Cheshire Cat's grin, that we can take over these theoretical virtues that I want to argue scientists are using anyway, and they're not using on the basis of experiment, are using to organize their experimental evidence. This is not a definitive list. We can argue with these as well, and there are probably more.

Elegance would be the mathematical simplicity. Parsimony would be not having sprites all over the place. Fruitfulness, how does it lead to developing more theories, more powerful theories, interesting other theories, explanatory power, et cetera, et cetera. Consistency would be another one. We don't have inconsistent theory.

Could we do metaphysics to decide between theories that have no real experimental implications at all? In other words, here's my metaphysical theory, here's your metaphysical theory, we'll just look at this stuff. I'll give you a quick case study. We're running out of time.

Mereology is the metaphysics of parts and wholes. Philosophers argue ... This is not Eddington's worry. This is not how many things are on the universe. This is how many things are on your shopping cart when you check out the supermarket. When we have some things, we can also have things made up things. This table is a thing, but it has legs, wheels. When do some things make up a thing? When are a bunch of things a thing? That sounds like metaphysics.

Allism is the view that every collection of things is a thing. That table-like thing is a thing because it's made up of things. That seems okay, but also, this bottle and this piece of chalk is a thing. It's a thing, there's two parts is this bottle and this piece of chalk. Not only this, but this piece of chalk and the moon is a thing, and this piece of chalk and the moon and the Eiffel Tower is a thing. They don't have to be next to each other.

Anytime you collect things together, you get another thing. There's lots of things. Even if Eddington is right about the number of protons, you can't do the math, but one could do the math and, basically, you have a set of fundamental particles, and then every possible combination of them is going to generate a thing. That's allism. That's the technical name.

Noneism is no collection of things is a thing. In other words, whenever something has parts, it's not really a thing. That's not really a thing, like your chair is not really a thing, you're not really a thing, I'm not really a thing. Why not? Because we have parts that are things.

The only things in the world are whatever the fundamental particles are. Whenever you put them together, you don't really have things. We can talk about when it's convenient fiction. We can pretend they're chairs. It's useful to talk as if they're chairs and people and planets, but they're not really there.

Two extreme views, both of which sound insane. Then, the attempt to find the middle ground, Restricted Composition Principles. Look, so not everything, a collection of things is a thing, but some collections of things are a thing, so now we need a principle for when a collection of things is a thing. All right.

You might come up with some principle. For example, if they're stuck together, they're a thing. The bits of this table-like thing are a thing because they're stuck together whereas me and the Eiffel Tower are not a thing because we're not stuck together. Now we'd have to get clear on what stuck together meant.

Another view is that the only things there are are the fundamental particles and living creatures, organisms. Those are the only real things. You and I are things, but tables and chairs are not things. Anyway. Metaphysics, so the heavy-duty, hardcore metaphysics.

They don't make any experimental predictions. Allism, noneism. If I made an experiment to decide whether this table is a thing or not, you can't do it. There's no experiment. These are not disagreeing about experimental evidence. Nonetheless, there's lots of literature being produced by contemporary analytic metaphysicians on which of these is the best thing, which of these is right or what precisely the best principle of composition is.

They're constrained. They think of themselves as being constrained and they think of themselves as being constrained by these kinds of things. Is allism a more explanatory series and more fruitful? It's clearly not more parsimonious. Allism, there's a lot more stuff in the world than you thought, but maybe it does better on other things. Maybe it fits better with our infusions, et cetera, et cetera.

The idea is, again, this is the Cheshire Cat's grin that we can make room for working with these principles of theory choice in the absence of any experimental predictions at all. That might make room for some sort of metaphysics. Maybe it would apply in the supermarket.

On allism, you get up to 12 things pretty quickly. Say you took out four atomic things in your checkout line. It would be four things, plus this thing plus this thing plus this thing plus this thing, so that's four more, and then four sets of three would be four more, you're already up to 12. You've already got 12 things in your cart. There may actually be applications [inaudible 00:50:35].

All right, let me finish up. I've argued that the boundary between a priori and a posteriori, between mathematics and physics or science in general, can sometimes be blurred by limitations on both sides, that we're cognitively limited creatures. We're not logically omniscient. We can't see the consequences of everything we believe, and certainly not the logical consequences of everything we believe or the mathematical consequences.

We're also observationally limited. We can't see everything. If we could do both those things, then perhaps there would be a more firm boundary between mathematics and physics. On the mathematical side, we wouldn't need computers or any other form of augmentation because we could just set down our axioms and we'd know everything it followed from, we'd know what were theorems and what weren't.

On the observational side, we would never need to fit curves because we'd have observed all the data. We have all the points. It would be solidly, we'd already be a curve, but because we are limited in both these ways, there's a need for a priori speculation from what we've actually observed, so making predictions that go beyond the evidence, and there's a need, or at least the use for, a posteriori computer-based augmentation of our mathematical knowledge.

In each case, arguably we have an improvement of the guesswork. In each case, that's partly because of the constraints that in extending our abilities on each side, we are operating under constraints. One can wonder about the justification of those constraints. One can wonder, do we get the genuine knowledge? Perhaps we do, perhaps we don't. That's a big topic.

I guess what I want to leave you with is the idea that it's not clear. In Hume's opening remarks, he says anything that isn't experimental reasoning or reasoning with abstract quantity is sophistry and illusion, and I want to leave open that some blend, something in the middle here might at least have the potential for providing objective knowledge, both about mathematics and about the physical world. Thank you very much.