If mathematics is part of the cosmos, as some people believe, could there be other aspects of the universe that exist outside physical reality?
‘I think this allegory tells us that actually we can't know ultimate truth. We in our mortal condition at best can deduce what that is like from what we're seeing on the cave walls.’Transcript
David: Plato's Cave. The philosophers and scientists love telling us about Plato's Cave.
BO: Why? It's such a great, great story.
Ard: Do you want to tell us that story?
BO: Well there are many sides to the story. I want to know why they like the story first.
David: Well it's this sense that people like Roger Penrose particularly, or Greg Chaitin, the mathematicians, they say, ‘Look, there are the truths, there are things which are just absolutely true. Just because we can't see them, doesn't mean they're not there. We've just got to not get fooled by thinking that there’s just the flickering images we're seeing on the wall. That is to limit your imagination.’ And that's why they tell it.
BO: Shall I tell the story? Okay. My telling of the story is, of course, going to be imperfect. I think every storyteller… It's the beauty of story that we bend it a little bit in accordance with how we’re perceiving the world in that particular moment.
Anyway, these people have been in this cave. Let's think of them as prisoners, or they they've been in this cave for a while. The light comes from behind them, and they're in this cave in such a way that they can only see in the cave and see the images passing by on the cave wall.
And one of the things that I think people have taken from it is a great question of illusion and reality. The things that we're seeing, that they're seeing on the cave wall, is that truth? That's what you're saying. Or is that which creates the shadows, is that truth?
David: And what do you think of it? What does it say to you?
BO: I think it's one of the greatest stories, and it's a story rich with truth. It's one of those stories incredibly soaked in the truth of life, because that's what we do: we're constantly looking at things and taking them for absolutes. I think the story is one of the great… it is an allegory. It is an allegory of knowledge, but I think it's also an allegory of perception. And I think it's also an allegory of ultimate truths. And I think this allegory tells us that, actually, we can't know ultimate truth. We, in our mortal condition, at best can deduce what that is like from what we're seeing on the cave walls.
David: Yeah, that's what science says it does.
BO: If we were to turn round and walk out into what is creating that reality – what's creating the shadows on the wall – we now have to leave all of the orders of knowledge as we know it and enter into the orders of enlightenment of higher kinds of spiritual ecstasy, visionary states: things that don't belong to the conversation of science. And yet, at the same time, how is it strange that some of the great scientists were visionaries themselves?
Ard: And do you think in Plato's Cave we're looking at these shadows, and so that's all we see, and they can be distorted? Do you think that people find that sometimes a frightening thought? That all they are seeing are the shadows? Or do you think that people, maybe, sometimes also think that the shadows are all that there is?
BO: I think we have come to think that that is all there is. And I think atheism, at its worst, tells us that that is all there is, because you can see it. I mean, one of the great arguments of atheism keeps coming back to: ‘I can't see it. There's no evidence for it. I can't touch it.’ It goes back to evidence, and once we start to deal only with evidence, we're talking about shadows on the wall. We can see them; we can't see what's causing it. We can't deal with the source of the images on the wall. Evidence ends up being its own limitation.
'Let me put it like this: I am definitely sympathetic to all three of the Platonic ideals. I think beauty is a clear guide to truth. And then the moral I would see even more so, probably.'Transcript
David: If you're willing, as both of you do, say, look there's some kind of ideas, truths, in this case mathematical ones, which are woven into the fabric of the universe, could there be other kinds of truths? Like moral truths or aesthetic truths?
RP: Other kinds of truths? Well there certainly could be, yes, I'm not saying that mathematical truth is the whole of truth. That would be too arrogant a statement to make. We don't know of any other area which is so successful in describing the physical world. Now you see...
David: But could there be different truths altogether?
RP: Well, you see, there could be.
David: What's your feeling about that? Do you think the only truths that are in the world are maths?
RP: I think the trouble is the word truth. You can have things that people have a greater respect for. They might be great works of music, of architecture or painting, all sorts of things which have a value of their own, which you can't say you could reduce to mathematics, and I wouldn't necessarily want to do that.
Ard: Could there be other kinds of truths that you in effect discover rather than create?
RP: Well you might say... You see, I'm not against, I mean we're going to get sort of Platonism in some form. You see, the Platonic ideals, truth is only one of them, and you would say truth, beauty, if you like, and morality.
Ard: And goodness?
RP: The true, the good and the beautiful. Now I would be quite happy to give some kind of reality to all these things. Now the only thing that mathematics has to say in a clear way, I suppose, is the truth part, and it's talking about necessary truths. Well it's not talking contingent truths. We're not talking about something which might be here or might be there. We're talking about things which by their very nature are true or false.
It's also clear that there are inter-relations between, in particular, beauty and mathematics, and people very often talk in terms of a beautiful result. And it's certainly the case that if you have two alternatives where you worry about which is true, it's a better bet to think that the one which is more beautiful is more likely to be true. But this is always a very subtle issue. You might find there's a deeper reason that you hadn't realised before which makes the other one actually beautiful in a deep sense that you hadn't appreciated before.
Ard: So beauty is a guide to truth?
RP: I think beauty is a clear guide to truth.
Ard: But sometimes beauty… it's hard to be sure you've perceived it correctly.
RP: Yes, and beauty is, of course, a very personal thing, and people may have different views.
David: Has it been a guide for you in your work?
RP: Yes, certainly. I am definitely sympathetic to all three of the Platonic ideals. The truth one, which I'm taking as the pure, necessary truth, I think that's an absolute thing. And when it comes to beauty, well, you see, I would say there is a great subjectivity to beauty, and there's no doubt about that. But I would say there's a kernel of truth to all that which is independent of people. And I really sort of argue that great music can be great in itself, not just because people appreciate it.
David: And the moral?
RP: And then the moral, I would see even more so, probably. But, you see, this is an interesting question, because one of the things I spent a lot of time worrying about has been the issue of consciousness. And so I have these three worlds in a sense: you have the mathematical world, and then the physical world here. And the laws of physics seemed to be governed by mathematics, but it's only a part of the mathematical world, as far as we know, which governs the laws of physics. And it's only a part of the physical world, as far as we know, which has conscious experiences.
But if you are worrying about the other Platonic values, you see, do they have absolute existence as well? And the moral one seems to depend ‒ I'm not sure I think that this is entirely so ‒ but it seems to depend on the existence of consciousness. I mean, if there were no conscious beings around, the notion of morality somehow seems to evaporate. It has to do with conscious beings.
So I would say that truth and beauty are tied up together, and it's certainly a good guide to truth. In mathematics it's certainly true. And I would say also that the issue of consciousness is connected with this, because in order to understand what's going on in the mathematical world, I would argue that you need consciousness.
Ard: In some sense what you're saying is consciousness is needed to probe mathematical truth?
Ard: But conscious may also needed to probe goodness or moral truth?
RP: Well it’s all tied up with moral truth, because morality is tied up with consciousness.
Ard: Do you think these things like cruelty being wrong is something that we discover is true… like mathematical… like 1+1=2 is true?
RP: Yes, well I guess, in a sense, I am trying to take a view like that. It's a Platonic kind of view, but I think you have to be jolly careful with these things because there are always many different sides, and there is a danger, you know, if you're setting yourself up as saying this is the truth. Not just this is the truth, but this is right; this is what people should do.
Ard: But you might say, well, okay, some kinds of truths are…we know that they're there, but they're hard to access in a way that we're sure about, so we're more careful about them. Whereas if somebody comes around and says 1+1 does not equal 2, we can be fairly authoritarian and tell them that they're wrong.
David: I certainly wouldn't cross the bridge that they built.
Ard: That's right, that's a good point. Whereas I think the point is people worry about morals…that we're going to start imposing in the same way that when I mark an exam somebody doesn't get what 1+1 equals and says 1+1=3, I'll say it's wrong. They worry that if I think that moral truths are also there to be discovered that I'm going to do the same thing to them. But the fact is that morals, like beauty, there's a kernel there. There's definitely something about it which is much harder to put your finger on, but I think it's important to say it's more than just something that we can't quite put our finger on; it’s something that’s actually out there.
RP: Yes, I think would say that.
David: Why do you say that? I mean is it important to you? Why would you be led to think those ideas, to think those things?
RP: Very hard to know why one thinks something.
David: Because a lot of people might listen to you and say, well, look, wait a minute, I thought he was a mathematician and a scientist, and now he's veering off into these other things?
RP: That’s why I don’t like to talk about them. I think the danger there is it's much easier to get it wrong. You see in mathematics, that’s the whole beauty of the subject, if you like, that you can see who's right and who's wrong. And that is a big, important thing. Now that doesn't spread much into other areas, I mean it does to a degree. You can in physics, or in the real world, or in geography, you could say, okay, there's a continent over there, and you can go and find it, or not. You see that's… I mean there are...
David: But then why do you…? Since it is dangerous to have these ideas, where does this feeling or intuition come from? Do you think for you, such that you think, well, yes I just, I do think that there could be a moral...?
RP: It is true ‒ I don't think it is just a question of what works best in society. I mean you certainly get the view with people, often, that right and wrong is just a question of what makes society work.
David: It's a matter of fashion almost.
RP: It's a matter of fashion or a matter of…
RP: Convenience. It's a society that goes with…chugs away in a nicely oiled fashion.
David: But do you have a feeling it's...
RP: There is something much deeper.
David ...that some kernel of it is woven...?
RP: Yes, I do, yes.
David: But do you, how...?
RP: Without being religious, you see, I suppose that's what's a bit unusual, because you get people who would have a religion, and which they strongly believe in, and which they would argue is why they hold these beliefs.
David: But that's not the case for you?
RP: That's not what I would say, no. But if you… I mean, do you say Platonism is a religion of some sort? I don't.
‘The brain may be very good at encountering mind, but it may, and here I speculate completely, only be very good at encountering certain aspects of mind.'Transcript
DISCOVERING MUSIC, MATHS AND MIND
Ard: So another question that you were hinting at a little bit… We might unpack this one step at a time, so we’ll start with music. You’ve written a little bit about animals who do music, like humpback whales or nightingales, and you say they may be discovering music rather than evolution creating it itself. Did I get that right?
SCM: Well, thank you, yes. I need to first of all emphasise that the observations on the convergence of animal music, and potentially some of the metaphysical implications, derive directly from a wonderful essay written by Patricia Gray and colleagues which was published in Science some years ago.
In essence, what we know is many animals can vocalise, and there’s a major question whether our language is simply an extrapolation of that vocalisation. I rather think it’s not, but that’s very, very controversial, of course. But undoubtedly many animals can sing, and of course the songbirds are the most famous; the humpback whale, they too have a music in the males and so forth. And in many cases music is probably linked to sexual selection. In other cases, I believe, even with the humpback whale, it’s not entirely clear why they have this music. But the point about it, which as Patricia Gray and colleagues point out, is first of all this music is alarmingly analogous to our music, in as much as it uses melodies and harmonies, and they can exchange songs between different groups, for instance amongst the whales and so forth.
And what I found so extraordinary about this particular essay is that they then had this wonderful leap of imagination. I mean this in the most positive way. And they said, well, it’s all very well explaining these convergences in terms of vibration of columns of air. That’s fine – a bit like a bassoon or a trumpet. But supposing, as they said, there’s a ‘universal music’ out there, and in a sense the music is discovering this universal music.
And as I’ve said in many other contexts, this is not a universal ‘hmmmm’ – it’s the music of Mozart. And I’m not saying that birds and whales are Mozarts, or on their way to becoming Mozarts, but there’s something deeper about the music and why it’s so essential to us. It’s not just a tune.
David: Are you suggesting that certain kind of ideas, like music or mathematics, that they exist separately from the stuff of the universe.
SCM: That’s a possibility, yes.
David: Well, what do you mean by ‘it’s discovering’? To discover something makes it seem like it was there in some sense before. So here’s evolution blindly doing something, and you’re saying there’s things out there just waiting for evolution to get to, and it goes, ‘Ah ha!’
SCM: Well, I suspect that the processes of mathematics, of which I have no skill at all… the ability to play music, I can’t even manage a kazoo.
David: You and me both.
SCM: Those things are, in reality, realities: they are absolutes which we, in a sense, discover. The problem about describing this is two-fold. First of all, because we’re such spatial creatures, in the same way as ancient theologists thought God was just out there, somewhere behind a cloud, this is dealing with a different set of orthogonal dimensions.
David: You don’t mean it’s just past Australia.
SCM: Exactly, yes. Or it’s not on that carousel in the airport with the last bag going round and round at three in the morning waiting for someone to pick it up – precisely. And the other aspect about it is we may only be scratching the surface of what’s actually there. And that, I think, is very encouraging, because it gives us the sense that we’re dealing with unfinished business. Because the danger in science, too often, is to sort of clap your hands like that and say, ‘All sorted out, nothing to worry about. We got it right, give us the medals.’
Ard: So do you think evolution also discovers minds?
Ard: Perhaps, okay. Or are our minds simply created by evolution? Is that it? Or do you think that evolution is discovering something that’s already out there somehow?
SCM: My intuition, again, is that the mind is something which is discovered.
SCM: The general idea, of course, is that the brain and the mind are basically the same thing.
SCM: I mean, it’s pretty clear to all of us that you don’t have a mind without a brain.
SCM: But is the brain the entire explanation of what is mind? And you could point out, so far as we know, that when you die, then you stop thinking and so on and so forth. But on the other hand, you can regard the explanations of consciousness which depend merely on neurological complexity as woefully inadequate. The analogy which I prefer is rather than brains making mind: brains facilitate minds, brains filter mind. But the idea here is the mind, again, is part of an orthogonal reality.
David: Ah ha.
SCM: And once you’ve got a sufficiently complex nervous system, and once you begin to interrogate it, then it seems to respond to you.
David: So that gets back to your idea that there may be certain kinds of ideas or truths that are out there – that they were true before we came along. So, in other words, the brain is such that it can discover those ideas or those realms of ideas. Is that right? Is that what the mind is?
SCM: I think it is. But there is this danger that the brain may be very good at encountering mind, but it may, and here I speculate completely, only be very good at encountering certain aspects of mind. So our neurological substrate is such that there may be some things about which it is literally impossible to think. Which sounds very paradoxical, but of course as soon as you think about the impossible, then you allow about how you might think your way in to it. And this again gives me encouragement that the nature of mind is something far, far different than merely a secretion of a neuron.
David: Are you saying, getting back to your brain and mind, that evolution may have certainly crafted the mind? Are you saying that beyond that there’s a whole realm of ideas which you have to include in your story of what it is to be human? You can’t just pretend that they don’t have an influence?
SCM: I think first of all that evolution has crafted brain, and brain encounters mind.
David: Encounters mind?
SCM: Yes. And the point about the way in which evolution is understood is that in some circles it’s regarded more or less as a closed argument. And yet that’s actually rather curious, because of course in certain ways it depends on the physics and chemistry. And we know that our understanding of physics, in particular, is incomplete. So if we had a completely water-tight theory of physics, we might have more security that one extension of it, which we call life, is also entirely understood. And I think our problem at the moment is to confuse mechanisms, such as Darwinian mechanisms, versus, if you like, the substrate of possibilities.
To say there’s only one way of thinking would be a ridiculous way of compressing a huge area. But if mind is mind and we encounter mind, then potentially any life form would encounter one area of mind. But, again to stress, what we encounter may be an infinitesimally small fraction of what there is.
‘That space of abstract stuff was sitting waiting to be discovered and eventually minds reached a sufficient complexity that they could discover it.’Transcript
Ard: So that kind of raises another really interesting question, because there are these abstract truths, like the truths of mathematics, the world of ideas. Where do those come from?
David: You said where this time.
Ard: This time I said where. Where do they come from? I didn’t say, ‘where are they?’ But, ‘what is their cause?’
GE: What is the cause of those mathematical logical truths?
Ard: Yes, exactly.
GE: It’s the nature of logic is all I can say. That is the way it is.
David: You see, for you, it’s God.
Ard: I think it comes from God.
GE: God? Okay, well I’m prepared to say that that is one possibility. There’s an alternative possibility, which is that God has to obey…
Ard: Obey the law of logic.
David: God is one of the ideas in your realm of ideas.
Ard: Well no, there’s a long argument among theologians and philosophers…
GE: I’m sure there’s a long argument with theologians.
Ard: …whether the laws of mathematics are created by God, or whether God, in fact, has to obey the laws of mathematics. So, for example, you might say, even God can’t make a square circle because it’s a non-logical. It’s law of non-contradiction. But the interesting point is that here we have these abstract, non-physical realities that have causal powers in our world. We think of God as a non-physical, abstract entity who has causal impact on the world, so there’s some analogy there. And once you hold that there’s one kind of non-physical reality, then it’s not so strange to think there might be another kind.
GE: That’s this kind of theology which I avoid.
Ard: You avoid theology?
GE: Yeah, I avoid theology.
GE: From my view point, existence isn’t just physical existence: there’s these abstract existences. So then you should ask me in philosophical terms how do I justify the word existence? And I’ve got a very simple answer to that. I take the existence of physical entities, like we’re seeing in this room, as being real – that’s my starting point – and I take the hierarchy of this to be real. So, in other words, this thing is made up of a metal ball, which is made up of atoms, which is made up of quarks. I believe that the ball is real, as well as the atoms are real.
I think just because it’s made of atoms doesn’t mean that it isn’t a real ball. So that physical hierarchy is real. Then I say that anything else which has a demonstrable causal effect on here must also be real.
David: Ah, so your ideas?
GE: Otherwise you have uncaused entities in the world.
GE: I’ve got in my hand a pair of spectacles. Now, how did they come into existence? Someone had the idea of a pair of spectacles and then created these by a machine, and so on. If they hadn’t had that idea, this wouldn’t exist. So that idea has to be real too, even though it’s not a physical entity.
The generic way to think about this, the deep structure of cosmology, is possibility spaces. Now, physicists like to talk about physical laws, but you can talk about the laws, or you can talk about what is possible given those laws. And actually, in many ways, it’s better to talk about what’s called a phase space, or a Hilbert space.
Once you start the line of argument I’ve been giving, there’s a mathematical possibility space. It’s a space of possible logical arguments and outcomes. If you now keep pursuing this line of argument, we can only think a thought because it is possible to think the thought. That sounds like a meaningless tautology, but actually what it means is the following: there is a set of possible thoughts which is up there in some Platonic space. You can’t think a thought unless it’s one of the thoughts which can be thought because it’s a logical possible thought.
David: So that realm of ideas your talking about, you would say that came into existence in the Big Bang along with… along with the…?
GE: I wouldn’t necessarily say it came into existence. I think it might in some sense pre-exist the big bang.
David: Oh, okay. Pre-exist. But it exists, so then what natural selection is doing was creating more and more complicated minds, or brains, rather, and at some point they can access this realm?
GE: That is correct, and so that space of abstract stuff was sitting there waiting to be discovered, and eventually minds reached a sufficient complexity that they could discover it. But that space doesn’t need minds to exist, it’s there.
David: It’s there already.
GE: Yeah. There’s a wonderful book out of this by Paul Churchland called Plato’s Camera.
David: Yes, I’ve read it.
GE: And Plato’s Camera talks about, in detail, how the structuring of the mind as a neural network enables us to recognise Platonic patterns, and they then get incorporated into electronic patterns in the brain, and then they can go down and effect what happens in the real world. So I see this causal link from Platonic spaces into intelligent minds, into electrons. It’s a downward causation, and then into causing effects in the real world.
For instance Pythagoras’ theorem is used by surveyors and architects. Or the number Pi is discovered, and it’s then used by engineers and it changes what happens in the real world because it’s used in engineering design.
David: So ideas have some kind of existence in our universe. It’s just it’s not a, sort of, physical existence like wood or steel.
GE: Yes, that’s right.
'Plato’s Cave is wonderful. He talks about this allegory of the cave. The first point of his cave idea was that we live in an illusion, and that we have no idea what true reality is.'Transcript
David: You’ve mentioned being… You said you were a Platonist. What do you mean? Tell us about Plato’s Cave.
MG: Plato’s Cave is wonderful. He talks about this allegory of the cave. The first point of his cave idea was that we live in an illusion, and that we have no idea what true reality is. In effect, the senses are always telling you lies. And the way he illustrated that was to say, imagine that you have a group of slaves. They’ve been chained since birth to only look forward to a wall in a cave. So you have all these slaves. All they do is look forward to that cave. Now, they don’t know this, but behind them there is this big fire. And there are some people that hold up little figurines and little shapes to project shadows onto that wall. And the slaves look at those shadows and to them that is the world, their world, because all they can see is whatever is projected on that wall.
So the shadows on the wall are their reality, which basically means that we also are looking at some sort of fake reality, because our senses can betray our reason – that the only way you can really understand things is if you move away from the senses and you go into the realm of mind. So, to Plato, if you want to find truth, you don’t trust your senses, you trust your reason.
Ard: Because out there, the real thing, the true Platonic real thing, is out there, and what we see is just the shadow of it.
Ard: So if you have a triangle, we see the shadow of the true Platonic perfect triangle?
MG: Exactly. And so those objects of perfection, which are all mathematical objects of perfection – he called them the Perfect Forms – they only exist in your mind. And the true philosopher is the one that can understand those perfect forms and their mathematical beauty and purity, so that he or she can figure out how his God, which is the Demiurge, designed the world.
‘It leads us basically to the question where does goodness come from?’ ‘And where does it come from Martin?’ 'From God.'Transcript
David: Do you think mathematics… I mean, you are a mathematician… is it something we are making up?
MN: I believe…
David: Or something we are discovering?
Ard: Or do you think evolution created mathematics?
MN: I think mathematicians unveil eternal truth, you know, that exists independent of evolution. So you have something like prime numbers, numbers, they are just principles that exist. So, for me, the material world is an instantiation of fundamental principles.
David: Right. But if I’ve got those, then surely I don’t need God. I mean, in other words, there are these truths and they are the things which make the world the way it is, and they’re there before I thought about them, so I’m discovering them…
MN: But those fundamental principles, these eternal fundamental principles, for me, that is God.
David: Ah, okay.
Ard: Okay, here’s a question: if someone says to you, ‘Well mathematics is really science,’ what would you say?
MN: No. Mathematicians can make statements that are independent of any particular universe.
Ard: So give me an example.
MN: Well, some properties of prime numbers.
MN: There are infinitely many prime numbers.
Ard: And so these are true, they don’t need scientific experiments to…
MN: Yeah, they don’t need any particular universe to be instantiated to make them true.
Ard: And do you think there’s a parallel between these mathematical truths, which don’t need scientific proof, and theological truths or philosophical truths?
MN: Yes, very much, philosophical ways…
Ard: Or theological truths.
MN: So, for me, philosophy and theology, that’s very close, and the most convincing way to think about theology is always the philosophical one.
Ard: And so do you think that if something, just like mathematics, is true independent of scientific experiments, something like that could be true for philosophical truths or theological truths?
David: Or moral truths.
Ard: Or moral truths.
MN: Yes, definitely. So, for example, when Andrew Wiles proved Fermat’s last theorem, then it doesn’t need a scientific experiment to confirm it: it was just, this is now true.
Ard: Yeah. And could there also be, then, moral truths, for example, or other theological truths that are true the way Fermat’s last theorem is true?
MN: Yeah, again, you have to ask yourself, what is the meaning of goodness and where does goodness come from? And I would agree with Plato, that goodness is a form – is the highest form: it’s the form that illuminates all other forms, because all other forms that exist, it’s good that they exist.
Ard: And so goodness is something that exists independent of us?
MN: As the highest form, yes.
Ard: As the highest form. That’s really interesting.
David: If you see these truths as being out there, then when we think of them, the way a lot of people think is that, ‘I’m making this up,’ or ‘I’ve grasped it.’ If they’re there, then it’s more of an encounter, isn’t it? That you’re encountering something.
MN: So Socrates and Plato, they asked themselves, what is learning? And they came to the conclusion that all learning is only remembering.
David: In the sense that there is something there that was waiting for you.
MN: Yeah, in some sense. And also Augustine asks the question, actually, how do these concepts that you understand enter your brain? So if he understands the sense of touch, the sense of smell, the sense of seeing, of hearing, and there’s a certain concept, you know.
MN: And he then says, for that concept, maybe love, maybe goodness, it doesn’t come to you through any of those senses.
David: And where does it come from Martin?
MN: From God.
Ard: From God.
MN: From the fundamental principles. It’s like from the fundamental principles.
Ard: Martin and I agree.
David: Yeah, I know, I’m feeling in the minority. I don’t think I got mine from God, but I don’t know.
Ard: I think.. I think you did.
David: I know you do.