Here's another two, firstly Professor Edward Frenkel, a Russian-American
mathematician working in
representation theory,
algebraic geometry, and
mathematical physics. He is a professor of mathematics at
University of California, Berkeley:
And Professor George Ellis, the emeritus distinguished professor of cosmology and complex systems in the Department of Mathematics and Applied Mathematics at the University of Cape Town in South Africa, watch the videos here (see the 5th or 6th video on the list regarding the question of whether maths is eternal or invented):
George Ellis
Here's the transcript (GE is Professor George Ellis):
24:29 – THE ETERNAL TRUTHS OF MATHEMATICS
David: So this is something that Ard and I were discussing earlier. Are you saying that when I ask you what does two plus two equal, and you say four, it’s always seemed to me the reductionistic argument – when they say, well, consciousness doesn’t exist – is that somehow you come up with the answer four because you were forced to because electrons just got into that state? Whereas I’ve always thought that the reason you say the answer is four is because of the logic of mathematics. So, in other words, it’s the logic of mathematics which is pushing the electrons around, not the other way, where the electrons are forcing you to have a thought.
GE: No, you’re quite right. That’s exactly the way it is.
David: Okay, so he does agree with us. Because we were discussing this earlier, and then we thought, crikey, maybe we’ve both really misunderstood it.
GE: Do you want me to open this up to an even more mind-boggling place?
David: Go on then.
GE: Okay. Where does the logic of mathematics come from?
David: Oh dear.
GE: This is the old question: do we invent mathematics or do we find mathematics? And I’m an unashamed mathematical Platonist: we discover mathematics. Two plus two is four is too simple. Let’s take something more interesting like the fact that the square root of two is irrational. Now the square root of two is irrational no matter whether you’re an Ancient Greek or someone here or someone on Mars. The square root of two is irrational. It’s a timeless, eternal, unchanging mathematical truth. In other words it’s a Platonic kind of statement.
The ontology is the mathematics exists and is there and is unchanging. The fact that the square root of two is irrational is an eternal unchanging truth. What we understand about it is a historically contingent thing, and we didn’t know that 10,000 years ago and we do know it now.
David: But the thing which is true was always true?
GE: The thing which is true is always true and has been true since the beginning of the universe.
David: Right, so in other words it was true when there were only dinosaurs around, and it’s still true.
GE: It was true at the start of the Big Bang. It was true before, when there was just hot gas and nothing else.
Ard: I mean, if you think about it that way, it’s really hard to believe that wouldn’t be the case.
David: Except that if people, physicists, would say look, ‘I’ve got bosons and I’ve got quarks, you know, what is the particle that carries the idea?’ That’s what...
GE: Yeah, but physicists have great trouble telling you this famous question. Why does mathematics underlie physics? The famous thing that Galileo said that the nature of the universe is written in mathematics. And Wigner and Penrose and other people have pondered, why is it that physics can be written in mathematical terms? And that’s a deep philosophical question for which we don’t have a proper answer.
Ard: So the unreasonable effectiveness of mathematics?
GE: The unreasonable effectiveness of mathematics, yes.
27:21 – A SPACE TO BE DISCOVERED
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.
David: Right.
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.
