00:00:00.000 Welcome to Topcast, and to the second part of my series on
00:00:07.400 kind of about a discussion between Sam Harrison, Max Tegmark,
00:00:11.300 that was had years ago on the making sense podcast.
00:00:14.400 In this part of the discussion, we're mainly focused on mathematics
00:00:19.300 and the philosophy of mathematics and how we come to know mathematics.
00:00:23.500 And a lot of what I would say are misconceptions from the mathematicians' misconception,
00:00:31.400 The mathematicians' misconception is that the brain or the mind rather of a human being,
00:00:37.400 and especially of the mathematician, has privileged access to mathematical certainty or mathematical truth in some way,
00:00:44.900 that the way in which mathematical knowledge is created,
00:00:48.400 or the theorems of mathematics come to be known to mathematicians,
00:00:52.900 is something other than this fallible method of conjecture and refutation,
00:00:57.900 that there is some other way of gaining insight into absolutely necessary truth,
00:01:05.400 This is the mathematicians' misconception that mathematics doesn't require a physical computation
00:01:15.900 All knowledge, mathematical, scientific, moral, political, whatever it happens to be,
00:01:23.900 which is running on a computer, which is obeying the laws of physics,
00:01:27.900 and the laws of physics mandate that you will have errors now and again.
00:01:32.900 That's just the way knowledge is generated via this method of error correction,
00:01:37.900 identifying the errors, and then correcting them.
00:01:39.900 And that's what motivates all of knowledge creation, finding some problem or error in the existing theories,
00:01:50.900 There's no way of tying up with a little bow what we have discovered in any area.
00:01:57.900 We always just at the beginning of infinity, beginning to scratch the surface of our understanding of reality.
00:02:02.900 Now, early on in the discussion, I think you can hear a sense of confusion coming through.
00:02:08.900 One thing I would say is that people struggle with this idea that number, mathematics,
00:02:14.900 the theorems, or whatever you want to call it, these objects of mathematics
00:02:18.900 have an existence, have an independent existence.
00:02:21.900 And when I say that, I just mean that, well, these things, these truths aren't made of atoms.
00:02:27.900 Material stuff is made of atoms, but if you are committed to being a materialist or a physicalist,
00:02:33.900 and this is the idea that you will only grant existence two things made of atoms,
00:02:38.900 or stuff in the physical world that is made of physical stuff.
00:02:45.900 and because you're talking about, let's say, the laws of physics, which you might grant exist,
00:02:50.900 They're not made of particles or anything like that.
00:02:58.900 They're the things that control what's going on in a certain sense.
00:03:01.900 So if you're a physicalist, you're automatically believing in,
00:03:04.900 or you're endorsing the existence of something that's not made of matter.
00:03:10.900 Some physicalists maybe deny that the laws of physics are real things as well.
00:03:13.900 You get some strange ideas among people who practice professional philosophy.
00:03:17.900 What I like to say is things like numbers exist.
00:03:26.900 And then people say, hmm, abstractly, it sounds like you're a platonist
00:03:30.900 that these objects of mathematics have some sort of existence outside of the physical world.
00:03:35.900 But where is this other realm of this platonic realism?
00:03:44.900 It makes as much sense to me as asking what I was doing decades before my grandfather was born.
00:03:51.900 I mean, grammatically, it's a correct sentence,
00:03:53.900 but it's referring to something that didn't exist.
00:03:59.900 It's illogical when you actually try and think through what's going on.
00:04:02.900 It's the same as what was happening before time was created.
00:04:06.900 Now, if you have an explanation of the beginning of time,
00:04:10.900 asking about what happened before time, likewise, doesn't make any sense,
00:04:14.900 asking what's outside of all of space, likewise makes the sense you're asking for
00:04:24.900 I would say that the same kind of thing is a mistake when we say,
00:04:35.900 The physical world consists of space time and the stuff in it.
00:04:44.900 And there are times in the physical world, if you like.
00:04:47.900 But the abstract world, that's not made of space time.
00:04:53.900 It consists of truths, abstractions drawn from that physical world,
00:05:02.900 We become to understand these abstract things using our minds,
00:05:09.900 So it's perfectly okay to say something has an existence,
00:05:14.900 without being able to place it in time and space.
00:05:20.900 In fact, you kind of get a sense that Sam is almost getting this.
00:05:23.900 At first, he seems to be suggesting that an idea of Platonism,
00:05:28.900 the idea that this abstract reality exists in some way.
00:05:33.900 Now, I'm not really sure exactly. It depends on who you ask.
00:05:39.900 Some of them will talk about a physical space of some kind,
00:05:44.900 When you start talking about physical spaces of abstract stuff,
00:05:47.900 well, how do you marry these two things together?
00:05:51.900 Abstract spaces are different to physical spaces.
00:05:54.900 Sam almost gets there because later on in the conversation,
00:05:59.900 He actually talks about an example that I invoke myself very, very often.
00:06:03.900 And that is the example of, what is the next highest prime number?
00:06:09.900 At the moment, we know you can look at Wikipedia and you can write out,
00:06:14.900 And they'll give you a number, a very, very large number.
00:06:16.900 Now, we happen to know as a matter of proof there are infinitely many primes.
00:06:20.900 Now, the next one, yet to be discovered, it exists.
00:06:23.900 It absolutely exists, and the one after that, and the one after that,
00:06:40.900 but they must exist because we can come to discover them.
00:06:45.900 We're not inventing them, we are discovering these things.
00:06:47.900 People are searching for them, literally searching, searching where,
00:06:50.900 searching through abstract space, the abstract space of numbers.
00:06:53.900 Computers are working hard, grinding away right now,
00:06:56.900 to try and locate in that space, that infinite space, mind you.
00:07:01.900 The next prime, number, the next highest prime number that will hold the record
00:07:05.900 until such times the one after that is discovered.
00:07:09.900 This continual discovering of this landscape out there.
00:07:14.900 Now, we exist in physical reality, and we are used to physical reality.
00:07:20.900 there are other landscapes, that we can come to understand more and more.
00:07:28.900 We can imagine a moral landscape of well-being by whatever definition.
00:07:34.900 I also like to invoke Jaron Leneier's concept of a disciplined dualist here.
00:07:40.900 I like to think of myself as a disciplined dualist,
00:07:43.900 that we can say that these abstract things exist, but not much more than that.
00:07:48.900 We can come to have a better and better understanding of this abstract reality,
00:07:54.900 And asking questions that are about the location in time and space.
00:07:59.900 In other words, where in physical space is the abstract stuff,
00:08:05.900 Now, some people are upset about that, but again,
00:08:07.900 it's making them a stake of presuming that there is only one kind of thing.
00:08:11.900 The stuff made out of matter that exists in space time.
00:08:16.900 There are such things as emergent stuff which are beyond merely matter,
00:08:25.900 We know they exist because they are invoked when we explain the world using our best explanations.
00:08:32.900 A thing exists insofar as it features in our best explanations.
00:08:38.900 And when we say it exists, we mean we know it exists.
00:08:44.900 And sometimes things that we think exist turn out not to exist,
00:08:48.900 like the force of gravity, and flogiston, and Elon Vital.
00:08:53.900 These things scientifically in the history were thought to exist.
00:08:58.900 We had a better theory that ruled those things out.
00:09:00.900 Instead, postulated other stuff, space time, and oxygen,
00:09:05.900 and information and evolution by natural selection.
00:09:12.900 We know absolutely for sure that it will never be able to turn any of that stuff.
00:09:16.900 No, but that's not what saying something exists means.
00:09:19.900 This is the distinction between ontology, what ultimately,
00:09:23.900 absolutely as a matter of final truth exists in some way, shape or form,
00:09:29.900 And epistemology, our knowledge of physical reality.
00:09:32.900 And we have access to physical reality and to abstract reality,
00:09:36.900 and all the other domains in which we might consider our knowledge applicable,
00:09:40.900 in which we can come to understand reality broadly,
00:09:50.900 that leaves more and more questions for us to continue to answer and make progress.
00:09:54.900 And this distinction between ontology and mathematics,
00:09:59.900 what really truly exists and our knowledge of what really truly exists,
00:10:03.900 this distinction here is also where the mathematicians' misconception comes in.
00:10:07.900 And one of my favorite lines from the fabric of reality sums up this entire notion.
00:10:12.900 And that line is necessary truth is merely the subject matter of mathematics.
00:10:18.900 Necessary truth is not the reward we get for doing mathematics.
00:10:23.900 So it's the distinction between, and I've said this before on topcast,
00:10:27.900 between whatever the fundamental constituents of matter really, really are,
00:10:31.900 and our knowledge of the fundamental constituents of matter at any given time.
00:10:36.900 Now, the history of that just shows that we continue to find smaller and smaller stuff.
00:10:41.900 Now, it used to be thought that atoms were the smallest thing,
00:10:43.900 and then we found out that there were subatomic stuff, protons, neutrons, and electrons.
00:10:47.900 And then we found out, well, even the protons and neutrons are made of these things called quacks.
00:10:50.900 And now people are saying, well, maybe the quacks are made out of strings.
00:10:54.900 And so our knowledge of the ultimate constituents of matter at any particular time is imperfect.
00:11:02.900 We do not know the final ultimate constituents of matter.
00:11:05.900 Particle physics isn't about finding the ultimate constituents of matter.
00:11:10.900 Instead, it's about finding our best explanation of what we know the constituents of matter are at any given time.
00:11:16.900 At any given time, we know the constituents of matter, but not perfectly, not finally.
00:11:20.900 That's not what no means. That's not what knowledge is.
00:11:25.900 It's about explaining what we know at any given time.
00:11:28.900 So to with mathematics, but this is very poorly understood.
00:11:33.900 In particular by mathematicians, and hence the mathematicians misconception that David Deutsch talked about,
00:11:38.900 it is mistaking, mistaking our knowledge of the necessary truths of mathematics with the necessary truths themselves.
00:11:45.900 If you could rock the necessary truths in and of themselves, that would be inherent.
00:11:49.900 But how would you get this in the errant knowledge?
00:11:52.900 How would you escape from the fallibility of the human mind?
00:11:55.900 How would you escape from the fact that whatever your mind is doing as a matter of physics is a physical process?
00:12:02.900 Whenever it's doing mathematics or performing a calculation or a computation, that is,
00:12:06.900 the action of neurons made of atoms, obeying laws of physics, the quantum laws of physics,
00:12:12.900 and the quantum laws of physics introduce error into the system uncertainty all the time.
00:12:17.900 That's just a law of physics and laws of physics are primary.
00:12:21.900 They rule over everything, including brains, including what brains do.
00:12:25.900 They can strain, how perfect, how knowledge of anything can be,
00:12:32.900 But the mathematicians' misconception is, oh, we can escape that when we're doing mathematics,
00:12:36.900 when we're doing mathematics, we can actually grasp ultimate truth.
00:12:42.900 Among other things, mathematics produces theorems via a method of proof,
00:12:48.900 and the method of proof must begin somewhere with the axioms.
00:12:55.900 Of course, some people will say, how could they be otherwise?
00:12:59.900 Well, I would just implore anyone listening to this who wonders about how could it be otherwise?
00:13:04.900 To go to my discussion with Naval, there at the Naval podcast,
00:13:08.900 where I talk about precisely this thing, and I invoke the idea of people thinking,
00:13:12.900 how could it be otherwise that two dots drawn on a piece of paper would have a unique line going through them?
00:13:19.900 How could Euclidean geometry possibly be otherwise?
00:13:22.900 How could we have a geometry different to this?
00:13:24.900 That's the history of mathematics of common sense, of logic of people,
00:13:29.900 even thinking they've found, absolutely secure foundations, absolutely unarguable.
00:13:41.900 The history of mathematics, philosophy, logic tells us this.
00:13:50.900 Already it's been about 15 minutes of just me talking.
00:13:53.900 But let me bring you a little bit of what Sam has to say,
00:13:58.900 and then some of what Max has to say on precisely this topic of,
00:14:02.900 the relationship between mathematics and the physical world.
00:14:05.900 And I'm going to pick it up where Sam is talking about that most famous of papers,
00:14:09.900 Eugene Wigner's, on the unreasonable effectiveness of mathematics and the physical sciences.
00:14:14.900 I want to linger on this question of the primacy of mathematics and the strange utility of mathematics.
00:14:25.900 At one point in your book, you cite the off-sighted paper by Wigner,
00:14:30.900 who I think he wrote in the sixties about, in a paper entitled,
00:14:35.900 The Unreasonable Effectiveness of Mathematics and the Natural Sciences.
00:14:40.900 And this is something that many scientists have remarked on.
00:14:45.900 There seems to be a kind of mysterious property of these abstract structures and chains of reasoning,
00:14:51.900 where mathematics seems uniquely useful for describing the physical world
00:14:57.900 and making predictions about things that you would never anticipate.
00:15:01.900 But for the fact that the mathematics is suggesting that something should be so.
00:15:07.900 And this is lured many scientists into essentially mysticism,
00:15:12.900 or the very least philosophical Platonism, and sometimes even religion,
00:15:19.900 or even pure mathematical concepts like numbers that exist in some almost platonic state beyond the human mind.
00:15:27.900 And I'm wondering if you share some of that mathematical idealism,
00:15:32.900 and I just wanted to get your reaction to an idea that I believe
00:15:41.900 I get Kenneth Craig who published a book in 1943,
00:15:50.900 but in passing, he tried to resolve this mystery about the utility of mathematics,
00:15:56.900 and he's simply speculated that there must be some isomorphism
00:16:00.900 between brain processes that represent the physical world
00:16:05.900 and processes in the world that are represented,
00:16:08.900 and that this might account for the utility of mathematical concepts.
00:16:12.900 I think he more or less asked, you know, is it really so surprising
00:16:15.900 that certain patterns of brain activity that are in fact,
00:16:20.900 what mathematical concepts are at the level of the human brain,
00:16:24.900 can be mapped onto the world of some kind of sameness of structure or homology there?
00:16:30.900 Does that go any direction toward resolving this mystery for you?
00:16:37.900 So it's a few things there I want to pick up on.
00:16:45.900 and Sam does this rather frequently, conflating brain with mind,
00:16:51.900 and then saying something like, well, mathematics at the level of the brain
00:17:04.900 It is, and our knowledge of mathematics is an understanding.
00:17:08.900 And an understanding is not just a pattern of neural firings.
00:17:12.900 For one thing, one day, in principle, it's possible to take the mind out of the brain,
00:17:18.900 so there will be no neurons whatsoever, and instead you'll be instantiated in silicon.
00:17:25.900 And then what do we say, the mathematics is identical to the transistors switching on and off?
00:17:29.900 I would still say, no, it's the wrong level of analysis.
00:17:33.900 It's like saying that the events of history are nothing but the movement of atoms.
00:17:39.900 Yes, movement of atoms were happening, but if you want to understand, understand,
00:17:44.900 not merely predict, as we like to say, then you have to distinguish between these things.
00:17:49.900 Neural firings are neural firings, and there are all sorts of neural firings.
00:17:54.900 Neural firings happen when you digest your dinner.
00:17:56.900 Neural firings happen when you think of artistic things, and when you think of mathematical things
00:18:03.900 But the fact that it's all neural firings doesn't mean that those things all reduced to neural firings,
00:18:11.900 Namely, that all just be neural firings, but they're not just neural firings.
00:18:15.900 There's something special about particular patterns of neural firings,
00:18:20.900 and it's the patterns, the abstractions, how those abstractions are represented in the neural firings.
00:18:25.900 Which is the really interesting thing, because you can instantiate mathematical truth,
00:18:30.900 mathematical explanation, just like you can with any knowledge in different physical forms.
00:18:36.900 As ink, does that mean the mathematics is identical to the scribbles of ink on the paper?
00:18:44.900 Does that make Pythagoras' theorem identical to the sound waves?
00:18:52.900 It's there in the squiggles when you write it on the piece of paper.
00:18:56.900 It's there in these sound waves when you speak the theorem.
00:18:58.900 It's there when you think it in neural firings.
00:19:03.900 And from those different ways, we abstract out the common thing,
00:19:10.900 Which has this independent existence, independent of each of these things.
00:19:14.900 In what senses an independent, because it's not identical to any one of those things.
00:19:21.900 Well, it exists in each of those forms, but not identical to those forms.
00:19:25.900 But where? Where is the actual Pythagoras' theorem?
00:19:36.900 And we only have access to that abstraction once we've come to an understanding of it.
00:19:40.900 The same is true of the next highest prime number.
00:19:43.900 The next highest prime number, we will be able to write down on paper,
00:19:47.900 if we like, or presumably be many thousands of pieces of paper we could write it down on.
00:19:59.900 That's all these different ways of representing this number.
00:20:03.900 And all these different representations, they have one thing in common.
00:20:06.900 They all represent the one thing, the abstraction, which is that prime number.
00:20:10.900 And the other thing there is this mystery about exactly why it is that
00:20:14.900 mathematics should be useful in describing the physical world.
00:20:24.900 Firstly, all the laws of physics are computable.
00:20:29.900 The functions are all computable, which is really interesting,
00:20:34.900 And so because the laws of physics as we know them,
00:20:37.900 we can use them to make predictions in the world.
00:20:40.900 Well then, that gives them a mathematical structure.
00:20:49.900 And he thinks that it doesn't necessarily need to be the case in the future.
00:20:53.900 Laws of physics necessarily have to have this mathematical character
00:20:58.900 Which is a really, there's a lecture out there on the difference between
00:21:04.900 It's quite a famous lecture you can find it on YouTube.
00:21:08.900 he actually makes some remarks about how he thinks the final law of physics
00:21:12.900 or something like that, you know, theory of everything, whatever.
00:21:17.900 or at least the mathematics, as we would think of it today.
00:21:20.900 I must say that there is possible, and I've often made a hypothesis
00:21:24.900 that physics ultimately will not require a mathematical statement
00:21:28.900 that the machinery ultimately will be revealed,
00:21:30.900 just a prejudice like one of these other prejudices.
00:21:33.900 It always bothers me that he in spite of all his local business.
00:21:37.900 What goes on in a tiny, you know, even no matter how tiny a region of space,
00:21:42.900 no matter how tiny a region of time, according to the laws as we understand in today,
00:21:46.900 takes a computing machine and infinite number of logical operations to figure out.
00:21:50.900 Now, how can all that be going on in that tiny space?
00:21:55.900 Why should it take an infinite amount of logic to figure out what one stinky, tiny bit of space time is going to do?
00:22:02.900 And so I made a hypothesis often that the laws are going to turn out to be in the end
00:22:09.900 simple, like the checkerboard, and that all the complexes are from size.
00:22:13.900 But that is of the same nature as the other speculations that other people make.
00:22:18.900 It's not good to be too prejudiced about this thing.
00:22:21.900 It's really curious, you know, maybe kind of like we think about constructor theory.
00:22:26.900 Now, maybe there will be a formalism of constructor theory in terms
00:22:30.900 of sophisticated mathematical apparatus, perhaps indeed there will be.
00:22:34.900 But if I'm in a sort of hinting act, well, you know,
00:22:37.900 ultimate greater grand a deeper theory might not need to have this same kind of mathematical character.
00:22:43.900 But the tradition in physics, of course, is that we go toward,
00:22:48.900 we tend in the direction of more sophisticated mathematics.
00:22:51.900 And so now we're at string theory, where the mathematics is exceedingly complex
00:22:55.900 and people struggle to understand it and to some extent many physicists are saying,
00:23:00.900 you know, well, they're not seemingly making any progress by the measure of generating testable predictions.
00:23:06.900 And this is why, you know, a whole bunch of them turn around and say,
00:23:12.900 Well, yeah, but testable predictions allows you to do things,
00:23:21.900 so actually make practical benefits for humanity, not to say that, you know,
00:23:25.900 mathematics and string theory couldn't possibly have any benefits to humanity.
00:23:30.900 Of course, it could, but problem solving requires solving physical problems.
00:23:34.900 Now, there is this problem with unifying general theory of relativity and quantum theory.
00:23:39.900 Yeah. And so string theory is an attempt to do that,
00:23:42.900 but the attempts have been going on for many decades now.
00:23:44.900 And out of that, well, where's the testable prediction?
00:23:49.900 What is the new piece of technology that a company can latch onto and to actually make profit out?
00:23:56.900 And it's one of the motivations for something like construct a theory,
00:23:59.900 a new approach, something different that makes some testable predictions
00:24:03.900 that might actually lead towards certain kinds of technology.
00:24:11.900 Let's give construct a theory a few decades, equal time.
00:24:16.900 The other issue of why is that mathematics should be so effective in the physical science as well.
00:24:23.900 And David George uses in the fabric of reality,
00:24:27.900 And we had a great hint of this in his lectures that he's given.
00:24:30.900 This idea that the human mind has this capacity to generate models,
00:24:37.900 models and explanations within itself that are of the rest of physical reality.
00:24:42.900 So you have this kind of one-to-one correspondence between explanations,
00:24:52.900 You know, David invokes quasars and the discussion of quasars.
00:24:55.900 That he is this object on the other side of the universe,
00:24:59.900 obeying the extreme limits of the laws of physics as we understand them.
00:25:04.900 You know, stars being swallowed whole by these black holes,
00:25:09.900 ripped apart and generating these huge jets that if they're directed in our direction,
00:25:16.900 Now, that physics, as David says, is someone like the physics of the human brain.
00:25:24.900 And so go to his TED talks to hear more about that from his perspective.
00:25:28.900 But the magnificent thing is that our understanding of quasars comes to resemble
00:25:33.900 what's really going on in quasars with increasing fidelity over time.
00:25:38.900 The one structure inside the mind, the model of the quasar,
00:25:42.900 comes to more faithfully represent the real physical thing out there in reality over time.
00:25:54.900 So let's hear what Max Tegmark has to say about all of this.
00:26:02.900 The argument that our brain adapts to the world,
00:26:11.900 So their processes in the world and their processes in the world
00:26:20.900 So I agree with the first part of the argument that disagree with the second part.
00:26:24.900 I agree that it's natural that there will be things in the brain
00:26:28.900 that are very similar to what's happening in the world.
00:26:31.900 Precisely because the brain has evolved to have a good world model.
00:26:34.900 But I disagree that this fully answers the whole question
00:26:38.900 because the claim that he made there that you mentioned
00:26:43.900 that brain processes of certain kinds is effectively what mathematics is.
00:26:47.900 That's something that most mathematicians I know would violently disagree with.
00:26:52.900 That math has something to do with brain processes at all.
00:26:55.900 They think of math rather as structures which have nothing to do with a brain.
00:26:59.900 Well, let's just pull the brakes there though, because clearly your experience of doing math,
00:27:05.900 your grasp of mathematical concepts or not, the moment something makes sense
00:27:10.900 or you persist in your confusion, your memory of the multiplication table,
00:27:16.900 your ability to do basic algebra and everything on up.
00:27:20.900 All of that is in every instance of it's being realized
00:27:25.900 is being realized as a state of your brain or you're not disputing that.
00:27:28.900 Of course, absolutely. I'm just crippling about how what mathematics is,
00:27:36.900 So they're both talking past each other, I would say,
00:27:40.900 and they're both talking past the most parsimonious way of just sorting out this confusion.
00:27:44.900 And the confusion is the distinction between the subject matter
00:27:48.900 and our knowledge of the subject matter, ontology and epistemology,
00:27:52.900 the real existence of mathematical structures, mathematical objects,
00:27:56.900 things like numbers, and our knowledge of those things.
00:27:59.900 Now, our knowledge of those things is going on,
00:28:02.900 or the creation of the knowledge of those things is going on inside of our minds.
00:28:05.900 Absolutely. That's where the creation is taking place.
00:28:08.900 That's what makes human beings, people, so unique,
00:28:12.900 so special in the entire universe that we can do this process of create explanatory knowledge.
00:28:20.900 It's a process of conjecturing, guessing about the reality that's out there.
00:28:24.900 And one of the realities that's out there is mathematical reality.
00:28:28.900 The reality of necessary truths of absolutely certain perfect number.
00:28:45.900 Now, it's whatever the thing is that exists, by the way,
00:28:49.900 that thing that exists might be matter, the material world.
00:28:57.900 We come to understand the truths of mathematics better.
00:28:59.900 But we're guessing at that. We're guessing at the material.
00:29:05.900 and we can come to understand that moral landscape of a better.
00:29:11.900 This meditation, this process of meditation is a process of coming to understand the mind.
00:29:20.900 Coming to understand your own mind and your problems
00:29:22.900 and your difficulties and whatever else in the action of thoughts as thoughts.
00:29:28.900 Just because you are there in your first person meditating,
00:29:34.900 best understanding of what the heck's going on now.
00:29:37.900 But sometimes you'll need to discuss with someone else,
00:29:40.900 because they can have better insight into exactly what you're thinking and why.
00:29:46.900 We tend to think we have this higher opinion of ourselves.
00:29:57.900 It's hard to keep track of everything you're thinking and why you're thinking it
00:30:01.900 and which thoughts you're thinking over and over again.
00:30:06.900 When I make these podcasts and I think I'm saying something really clearly
00:30:10.900 and I think I've spoken just exactly the way I want to speak,
00:30:17.900 I think I've said the word, let's say, mathematics.
00:30:22.900 And so I have to edit stuff and change my mind.
00:30:24.900 But if you had asked me immediately after having made the sentence,
00:30:27.900 I would have said something like, oh, no, mathematics is a domain of necessary truths.
00:30:31.900 And yet, and I would have insisted that this is what I really thought I said.
00:30:34.900 And then you play back the tape and I've actually said,
00:30:37.900 morality is the set of necessary truths or something like that.
00:30:41.900 I can be totally wrong about the contents of my own mind just a second ago.
00:30:47.900 And I think we can be systematically wrong about our own minds as well, by the way.
00:30:50.900 We can just continually tell ourselves we've got the right idea.
00:30:56.900 And this is one reason for talking to friends and talking to other people
00:31:00.900 and talking to counselors and talking to wise older people
00:31:03.900 and trying to come to a better understanding of your own ideas,
00:31:08.900 But what I'm saying here is that everyone understands seemingly this distinction.
00:31:18.900 And what Sam and Max would benefit from is simply recognizing this distinction
00:31:24.900 that the mathematicians, again, the mathematicians' misconception,
00:31:27.900 they think they've got access to this final ultimate truth
00:31:30.900 and that what's going on inside of their brains is the mathematics.
00:31:36.900 Now, you've come to an understanding of the necessary truths,
00:31:39.900 but it's going to contain error in misconceptions
00:31:43.900 that we can always come to a deeper understanding of these things.
00:31:47.900 Now, when I say that we can have errors and misconceptions in our knowledge,
00:31:56.900 people like to bring this thing up, you know, say,
00:32:00.900 How could you possibly have an error about that?
00:32:03.900 Well, when I say error, I just mean that you don't have the deepest possible understanding of it.
00:32:07.900 You think you know what 1 plus 1 equals 2 means.
00:32:10.900 I'm not saying that I know there's something wrong about that claim.
00:32:16.900 I can doubt that you have a complete understanding of what that really entails.
00:32:20.900 For one thing among many, it was a surprise to me at university,
00:32:23.900 learning that you could actually do this thing that proves 1 plus 1 equals 2,
00:32:27.900 and it takes you about an A4 page of handwritten argumentation from the axioms using penis axioms
00:32:37.900 and reaching the conclusion that 1 plus 1 equals 2.
00:32:44.900 Once you've done that, you have a deeper understanding in a sense of what 1 plus 1 equals 2 means.
00:32:51.900 And by the way, I like David Deutsch says, you know what does equals mean?
00:32:58.900 After all, just write down 1 plus 1 equals 2 and notice on the left hand side,
00:33:01.900 you've got this symbol 1, and you've got this symbol plus,
00:33:06.900 That's not exactly the same as the symbol you've got on the right hand side,
00:33:15.900 So there's all sorts of ways we can approach what the difference is between
00:33:20.900 an absolutely necessary truth, which is the subject matter of mathematics,
00:33:25.900 and our understanding of these things, which is where the confusion is,
00:33:29.900 I would say, a little bit between what Sam is saying, what Max is saying,
00:33:33.900 and they're not quite coming to what I think is a clear distinction.
00:33:38.900 And what other mathematicians, I would say, have on this point.
00:33:48.900 And all of our knowledge of mathematics is generated by our brains,
00:33:52.900 which is running on minds, which are obeying laws or physics,
00:33:55.900 which introduce errors as a matter of physical law.
00:34:00.900 OK, let's get going and listening to a little bit more from Max.
00:34:07.900 I'm just quiveling about how what mathematics is,
00:34:13.900 And I think it's interesting to take a step back and ask,
00:34:15.900 what do mathematicians today generally define math as?
00:34:25.900 they will often use math as just a bag of tricks for manipulating numbers
00:34:30.900 or maybe as a sadistic form of torture invented by school teachers,
00:34:36.900 Whereas mathematicians, they talk about mathematical structures
00:34:45.900 who has spent 10 years of his life studying this mathematical structure
00:34:57.900 And if I went and suggested to him that that thing on his wall
00:35:09.900 It was out there and he discovered that it was out there
00:35:12.900 and mapped out its properties in the exactly the same way
00:35:24.900 Similarly, if you look at something more familiar than E8,
00:35:27.900 you just look at the counting numbers, 1, 2, 3, 4, 5, etc.
00:35:35.900 Most mathematicians would argue that the structure,
00:35:39.900 this mathematical structure that we call the numbers,
00:35:42.900 is not the structure that we invented or invented properties of,
00:35:46.900 but rather that we discovered the properties of.
00:35:48.900 In different cultures, this has been discovered multiple times independently.
00:35:56.900 rather than discovered a different language for describing it.
00:36:13.900 you see that these are two equivalent descriptions of exactly the same structure.
00:36:19.900 What symbol you use to write the number two and three is actually different in the U.S.
00:36:24.900 versus in India today or in the Roman Empire, right?
00:36:27.900 But again, once you have your dictionaries there,
00:36:30.900 you see that there's still only one structure that we discover
00:36:37.900 To just drive this home with one better example,
00:36:40.900 he was really fascinated about these very regular geometric shapes.
00:36:49.900 That's what I would like to be able to say as well.
00:36:52.900 Just as clearly, you know, I endorse everything there.
00:36:57.900 What we would also add, he goes on to use the example of Plato.
00:37:01.900 He wasn't free to just invent more platonic solids once he'd discovered the ones that existed.
00:37:10.900 It has this independent reality that once you discover some part of it,
00:37:15.900 whether it's E8 or the platonic solids or just the natural numbers,
00:37:18.900 they have this autonomous set of properties that you come to understand over time
00:37:23.900 by making more and more discoveries, generating more knowledge about that particular thing.
00:37:30.900 We know this stuff exists by, as David Deutsch talks about in the fabric of reality,
00:37:41.900 Well, it does something kind of unpredictable that you product a little bit,
00:37:46.900 and it gives you back some kind of information about itself.
00:37:50.900 So in the case of, you know, just a normal natural counting numbers.
00:37:53.900 Well, it gives you back this weird, unexpected kind of thing of the prime numbers.
00:37:58.900 The distribution of prime numbers is kind of weird.
00:38:00.900 When the next one is going to crop up, it's kind of weird.
00:38:03.900 The last one at any given point, and then the next highest one,
00:38:08.900 So you start off with this simple set of assumptions,
00:38:15.900 and you end up with the prime numbers of infinite kind of complexity in a certain way.
00:38:21.900 By the measure that, they feature in our best explanations of reality.
00:38:28.900 Our explanations of reality include, well, the capacity to count stuff,
00:38:34.900 And so then we end up with the revealed richness of things like prime numbers, among other things.
00:38:40.900 Okay, so we'll go on and listen to what Sam has to say about this.
00:38:46.900 You know, what is the highest prime number above the current one we know?
00:38:52.900 Well, clearly there's an answer to that question.
00:38:54.900 You mean the lowest prime number above all the ones we know?
00:38:59.900 That number will be discovered rather than invented,
00:39:03.900 and to invent it would be to invent it perfectly within the constraints of its being,
00:39:10.900 So it's not wrong to call that pure discovery more or less analogous,
00:39:14.900 as you said, to finding Neptune when you didn't know it existed,
00:39:19.900 You know, it's Africa's there, whether you've been there or not.
00:39:23.900 So I agree with that, but it still seems true to say that every instance of these operations being performed,
00:39:33.900 every prime number being thought about or located, or having it.
00:39:37.900 Every one of those moments has been a moment of a brain doing its mathematical thing.
00:39:44.900 So I'm just more computer sometimes because we have a 15 large number of proofs now done by machines,
00:39:52.900 We're still talking about physical systems that can play this game of discovery in this mathematical space that we are talking about.
00:40:01.900 This fundamental mystery is that why should mathematics be so useful for describing the physical world
00:40:09.900 and for making predictions about blank spaces on the map?
00:40:14.900 Again, and I'm kind of stumbling into this conversation,
00:40:18.900 because I'm not a mathematician, I'm not a mathematical philosopher.
00:40:21.900 And so I'm sort of shooting from the hip here with you,
00:40:24.900 but I just wanted to get a sense of whether this could remove some of the mystery.
00:40:29.900 If in fact, you have certain physical processes in brains and computers and other intelligent systems,
00:40:35.900 wherever they are, that can mirror this landscape of potential discovery,
00:40:41.900 if that does sort of remove what otherwise seems a little spooky and platonic,
00:40:47.900 and represents a challenge for mapping abstract, idealized concepts onto a physical universe.
00:40:57.900 The answer you're going to get to that question will depend dramatically on who you ask.
00:41:02.900 There are very, very smart and respectable people who come down all across the very broad spectrum of views on this.
00:41:11.900 I chose to not say, this is how it is, but rather to explore the whole spectrum of opinions.
00:41:18.900 Some people will say, if you ask them about this mystery, there is no mystery.
00:41:23.900 There is, math is sometimes useful in nature, sometimes it's not.
00:41:28.900 That's it, there's nothing too mysterious about it, go away.
00:41:32.900 And then if you go a little bit more towards a platonic side,
00:41:37.900 you'll find a lot of people saying things like, well, it seems like a lot of things in our universe are very accurately approximated by math.
00:41:45.900 And that's great, but they're still not perfectly described by math.
00:41:50.900 And then you have some very, very optimistic physicists, like Einstein and a lot of string theorists,
00:41:57.900 who think that there actually is some math that we haven't maybe discovered yet,
00:42:02.900 but it doesn't just approximate our physical world, but describes exactly, and there's a perfect description of it.
00:42:09.900 And then finally, the most extreme position on the other side, which I explore at length in the book,
00:42:16.900 and that's the one that I'm personally adjusting on, is that not only is our world described by mathematics, but it is mathematics.
00:42:25.900 In the sense of the two are really the same, so you talked about how in the physical world, we discover new entities,
00:42:33.900 and then we invent language to describe them, similarly in mathematics, we discover new entities,
00:42:38.900 like, new prime numbers, the tautics, all of that, we meant names for maybe this mathematical reality,
00:42:47.900 and the physical reality are actually one and the same.
00:42:49.900 And the reason why, when you first hear that, and it sounds completely loony-toons, of course, you know, you look,
00:42:56.900 it's equivalent to saying that the physical world doesn't just have some mathematical properties,
00:43:01.900 but it has only mathematical properties, and that sounds really dumb when you, if you look at your wife,
00:43:07.900 or your child, or whatever, and this doesn't look like your bunch of numbers.
00:43:12.900 But, to me, as a physicist, and I look at them, of course, when I met Anaka your wife for the first time,
00:43:20.900 of course, he has all these properties that don't strike me as a mathematical.
00:43:23.900 Don't tell me you were noticing my wife's mathematical properties, but at the same time as a physicist,
00:43:29.900 you know, I couldn't help notice that your wife was made entirely out of quirks and electrons,
00:43:34.900 and what property does an electron actually have?
00:43:38.900 Well, it has the property minus one, one half, one, and so on, and we've made up nerdy names for these properties.
00:43:45.900 We physicists, such as electric charge, spin, and electron number, but the electron doesn't care what language we invent to describe these numbers.
00:43:54.900 The properties are just these numbers, just mathematical properties, and...
00:44:00.900 So, all is number. This is the Pythagorean idea.
00:44:04.900 But essentially, the true reality of the physical world we inhabit is actually reducible to mathematics in some way.
00:44:13.900 But I just think, well, this is metaphysics. This is a preferred metaphysical claim.
00:44:19.900 Now, saying there that the electron has these properties, described by numbers, minus one, one, and so on and so forth,
00:44:27.900 that is saying that the object is the properties, but I just think that's wrong.
00:44:33.900 I think that we can always come to a deeper understanding of these things. What is an electron after all?
00:44:38.900 Well, the physicists have refined their understanding of this thing over time.
00:44:43.900 It began as a particle, and they set a wave. Now, we think of this idea of fungible instances of the electron as particle.
00:44:48.900 I don't think we're going to get to the ultimate final answer as to what an electron is.
00:44:53.900 Now, as for saying that ultimately an electron reduces to nothing but the mathematics of an electron,
00:44:58.900 well, again, it's a metaphysical claim, and people have preferred metaphysics, but I don't know why they, in fact, have preferred metaphysics.
00:45:06.900 And why physicists want to say, well, I think it's just a cell box.
00:45:11.900 Maybe that's what it comes down to. It's like, you make a strong claim that everything's reducible to mathematics,
00:45:17.900 and the cell more books, because people want to argue against you.
00:45:21.900 Now, I don't have a strong opinion on this. I just think that it's one of these claims where there's little point
00:45:27.900 arguing about it, because there's no way of testing at the moment, the truth of this matter.
00:45:33.900 Well, one thing I would say is that it just doesn't comport with what I understand about the rest of physical reality,
00:45:38.900 which is that we come to a deeper and deeper understanding of it over time.
00:45:42.900 That if, for example, string theory describes what fundamental particles are,
00:45:47.900 so-called fundamental particles, they wouldn't be fundamental anymore.
00:45:52.900 Then we would have a deeper understanding of the electron.
00:45:54.900 And vibration of a string, but then you would say, well, what's the string made out of?
00:45:59.900 And people of the future, physicists of the future, I would imagine,
00:46:02.900 would come down in one some way, shape or form, in saying, well,
00:46:06.900 actually the strings are made out of this other stuff.
00:46:09.900 And so on it goes, because, like we say here, we're at the beginning of infinity,
00:46:13.900 we're at the beginning of our understanding of things.
00:46:15.900 But Max is kind of postulating, and many physicists do this.
00:46:28.900 The theoretical physicists have this fixation on the final completed physics.
00:46:35.900 Unify a few more things, unify the forces, and then one or two steps beyond that,
00:46:45.900 I think there's going to always be open questions.
00:46:48.900 There's always going to be a question as to why.
00:46:54.900 which we call mathematics, the knowledge of mathematics itself is abstract,
00:46:58.900 the mathematical things themselves are abstractions.
00:47:03.900 if you're saying that's what physics ultimately is,
00:47:10.900 And then the question about where this abstract space is,
00:47:17.900 We occupy an abstract, not a physical space anymore.
00:47:20.900 But then we just have, we're presented with the same list of questions.
00:47:24.900 Now, you're still going to have the question about the origins of the universe.
00:47:29.900 You're still going to have the question about dark energy.
00:47:32.900 You're still going to have questions about what the fate of the universe was.
00:47:36.900 Are there other logically possible universes that we can access in some way?
00:47:40.900 Just for a few more minutes with Max and then I'll say a few more things.
00:47:45.900 And if you take seriously that everything in both your life and in the world
00:47:53.900 is made of these elementary particles that have only mathematical properties,
00:48:12.900 Einstein discovered that also has some more properties.
00:48:14.900 It has some more properties called curvature and topology,
00:48:18.900 And if both space itself and all the stuff in space have only mathematical properties,
00:48:25.900 then it starts to sound a little bit less ridiculous idea that maybe everything is completely mathematical.
00:48:32.900 And we're actually part of this enormous mathematical object.
00:48:41.900 I think it's the same mistake that was being made earlier.
00:48:43.900 I think it's just confusing knowledge of something,
00:48:46.900 a mathematical knowledge of physical structures,
00:48:52.900 That our descriptions, our use of these mathematical labels for certain properties
00:49:01.900 Now, saying that the electron and the quark differ only in these properties
00:49:07.900 is just to say our way of describing differences between these things involves numbers.
00:49:18.900 where certain quarks have a charge of plus 1 third.
00:49:28.900 Well, I don't think so, because if you're saying that,
00:49:31.900 then you're also saying, well, the mass of the electron is what it is,
00:49:43.900 That part of the explanation is obscured from you,
00:49:47.900 if you're just going to say, well, they're nothing but these numbers.
00:49:50.900 Then explain why there should be attraction between things or not,
00:49:57.900 Now, Feynman gave this great explanation as to why, for example,
00:50:02.900 electrons when brought close together tend to repel one another
00:50:05.900 because they're emitting photons towards each other.
00:50:08.900 Well, that's an explanation, but it's not reducible just to the number properties.
00:50:12.900 You need natural language in order to come to an understanding of this stuff,
00:50:19.900 And the numbers are just being invoked as part of the explanation.
00:50:23.900 But to say, to then go that step further and say,
00:50:25.900 well, the explanation, these parts of the explanation,
00:50:28.900 are not only just explaining what's going on here.
00:50:30.900 It is what's going on here, as a matter of final fact,
00:50:36.900 It's confusing knowledge of with the thing in itself.
00:50:40.900 The thing in itself, you don't have a perfect understanding of.
00:50:44.900 We should know now that we don't have a perfect understanding of particle physics.
00:50:48.900 For one thing, particle physics is part of quantum theory.
00:50:51.900 Quantum theory doesn't quite mesh with general relativity.
00:50:57.900 Okay, but that's where I think we'll leave it today.
00:50:59.900 We'll move on from mathematics next time to some other part of the discussion.
00:51:04.900 I don't want to go through the entire discussion.
00:51:06.900 I think you get a sense now of the way in which I'm approaching this.
00:51:12.900 I would suggest that just picking up the books,
00:51:15.900 if I break a reality in the beginning of infinity,
00:51:17.900 do give you this as I like to talk about this coherent worldview.
00:51:20.900 And so you're able to spot misconceptions and spot errors
00:51:24.900 and kind of feel in a sense a little bit more clear in your thinking
00:51:28.900 about certain things that where others tend to encounter
00:51:34.900 This is certainly not to say we have all the answers.
00:51:37.900 It's just that when you have a way of meshing together,
00:51:40.900 the philosophy, epistemology, morality, mathematics,
00:51:43.900 physics, computation, and all of these other things,
00:51:51.900 Even if you're not necessarily a top-notch expert,
00:51:54.900 but not at regard myself as a top-notch expert in physics.
00:52:03.900 I'm not a top-notch expert in astronomy or anything else,
00:52:06.900 but I tend to be able to spot the errors, spot the errors in thinking
00:52:14.900 which we know is the case because of our best existing explanation.
00:52:26.900 what is known as a matter of our best explanation,
00:52:31.900 and when you know that the view they're espousing
00:52:33.900 has already been refuted in some way by some other thinkers,
00:52:36.900 then it's an interesting way to sort of look at some
00:52:40.900 But it's still fascinating, still very, very interesting,
