00:00:00.000 So this is part two of chapter five at the beginning of infinity, the reality of abstractions.
00:00:07.200 My better be called an appendacy because there's not much left for me to read of chapter
00:00:12.120 five, but there's a lot more to say to do with David Deutsch's views on the nature
00:00:18.400 And the nature of mathematics is the title of chapter 10 of David's first book, The
00:00:24.600 And I find that there's a lot of crossover here in the material.
00:00:28.160 There's a lot of new material in chapter five of the beginning of infinity.
00:00:32.600 But there are certain themes that are touched upon in the fabric of reality that don't
00:00:38.200 quite get a voice in the beginning of infinity.
00:00:41.520 David recently gave a talk at his Dirac Prize Medal Award ceremony in which he articulated
00:00:49.640 one of the most profound differences that David as a philosopher has with the mathematicians.
00:00:57.120 David has some really unique views when it comes to the nature of mathematics and it connects
00:01:02.080 directly to the reality of abstractions and to his work, of course, on quantum computation.
00:01:08.160 Now I want to try and explain why and I suppose in part this will be a personal reflection.
00:01:14.160 Like many people, I was inculcated with some supernatural ideas as a child.
00:01:19.400 You tend to learn about some aspects of religious mysticism and you learn about fairy
00:01:24.240 tales, sometimes these are presented as facts and this has the effect that when you begin
00:01:29.200 to take on a more scientific rational view of the world and a more reason to approach
00:01:35.440 to these matters, you tend to relinquish a whole bunch of this.
00:01:39.840 Some people not only relinquish these beliefs in the supernatural but they become actively
00:01:44.720 hostile to the notion that anything is supernatural at all or anything could possibly
00:01:50.320 And I think that's quite right, but this impulse, this reasonable rejection of the supernatural,
00:01:59.200 Now Richard Dawkins in the selfish gene speaks about genetic misfiring.
00:02:04.560 Some examples of genetic misfiring are things like the so-called altruism gene.
00:02:10.040 So the altruism gene is supposed to be the gene that causes social animals like ourselves
00:02:16.800 to help others, but sometimes this could misfire.
00:02:21.320 If it's all about the gene propagating through the environment, then if there is an altruism
00:02:25.760 gene, a gene that causes social animals to help one another, not all members of the
00:02:33.840 And so therefore if an altruistic member of the species decides to help to be altruistic
00:02:40.640 towards a member who isn't such an altruist, so if there is an altruism gene or
00:02:46.760 something like this, this can perhaps misfire, a misfire in human beings, let's say, as
00:02:51.400 it could cause you to help someone who refuses to help anyone else.
00:02:55.080 In other words, the altruism gene might in some situations actively work to cause itself
00:02:59.920 to be removed from the gene pool by helping those who help no one.
00:03:04.680 Altruism could be blind in that kind of way, or the gene that causes the desire for sex,
00:03:10.320 the whole purpose of which is to encourage reproduction of the species, or reproduction
00:03:17.040 This could misfire because it's still that the desire for sex is still there even in the
00:03:21.160 presence of contraception and even in homosexual members of the species.
00:03:25.720 So the desire doesn't actually help reproduction in some cases.
00:03:30.040 Memes rather than genes might do the same thing.
00:03:33.640 So rejecting the supernatural might be a meme that we all become inculcated with all
00:03:38.760 that we all learn at some point after we give up on, let's say, God and become atheists.
00:03:46.920 But a misfiring of that idea, a misfiring of the idea that one should reject supernatural
00:03:53.480 explanations, could be reject all non-physical explanations, or reject all explanations
00:04:00.800 cast in terms of things that include non-physical entities.
00:04:05.360 That's a misfiring, because not all non-physical things are supernatural.
00:04:09.960 And not everything in the universe is physical.
00:04:12.200 In particular, the laws of physics are not physical, numbers are not physical, mathematics
00:04:16.880 is not the study of physical things, but abstract things and so on.
00:04:21.360 All of these things are non-physical, parts of abstract reality, abstractions have a
00:04:26.080 reality, and this reality can have effects on the physical world as we saw in chapter
00:04:32.240 Now there is one kind of abstraction that is particularly interesting, and that is the abstract
00:04:40.200 Not all abstractions are mathematical, but the mathematical ones are the ones that often
00:04:44.520 And there are misconceptions on both sides, and David has recently spoken publicly about
00:04:49.240 what he calls the mathematicians' misconception.
00:04:52.880 And we'll come to this shortly, but essentially we have two groups of people once more.
00:04:57.240 On the one side we have people who think that mathematics is about abstract entities,
00:05:01.320 and we reason about those entities using mathematical intuition.
00:05:05.400 And this is somehow done in a kind of abstract space, and as such, what is able to
00:05:13.440 And so mathematics can deliver us with rock-solid, true proofs.
00:05:17.640 Mathematical knowledge on this account is of an altogether different kind than other kinds
00:05:23.040 Mathematical knowledge confers certain truths upon its conclusions on this view, or at
00:05:27.600 least truth of a kind that is more reliable or more robust or foundational than other
00:05:39.520 The other side of the debate is that everything in reality is physical.
00:05:43.920 There is only a physical reality and nothing else.
00:05:49.120 On this account, mathematics is a kind of useful fiction.
00:05:52.960 It happens to describe parts of this physical reality, but it cannot possibly have a reality
00:05:59.880 This is sometimes called physicalism or materialism.
00:06:04.200 Either way, this is of course a form of reductionism.
00:06:07.640 And sometimes seen as the uber-rational way to be.
00:06:10.840 It rejects everything except what science, supposedly, deems as real.
00:06:15.400 Usually through observations, so it's also a kind of empiricism.
00:06:20.120 So we have two quite rational doctrines here, it would seem, but yet they both seem to
00:06:27.160 Brilliant mathematicians who believe in perfect mathematical ideals that they can have
00:06:31.680 knowledge of through their mathematical intuition and that leads them down the path of
00:06:36.800 On the other hand, hard-nosed materialism of the scientific types that say only physical
00:06:42.760 So if you're a person who likes reason, you seem a little bit stuck.
00:06:47.280 Do you prefer brilliant mathematicians or do you like hard-nosed physics types?
00:06:51.240 So let me steal a famous line from David Deutsch here.
00:06:55.040 If you regard these two positions as being deeply informative of your rational worldview,
00:07:00.280 then quote, then they seem a little bit to conflict with each other.
00:07:04.360 But that does not prevent them from both being completely false, and they are.
00:07:10.480 So David walks the line between these two positions.
00:07:13.320 On the one hand, mathematics is indeed about perfect ideal forms, but that is not what mathematicians
00:07:23.920 Now they exist, but what we have are explanations of the laws of physics, imperfect.
00:07:30.840 Our knowledge of them is subject to change and improvement.
00:07:34.400 And on the other hand, as chapter five is argued at length, it cannot be the case that
00:07:41.880 Collections of atoms arrange themselves into patterns that are more than the sum of their
00:07:47.800 The reason the domino falls or doesn't in Hofsted as argument, go back to the last
00:07:51.800 video for this one, is because the input is either prime or it isn't.
00:07:56.600 It's nothing to do with the laws of motion acting on atoms in the void.
00:08:00.480 The explanation was 641 is prime, it's got nothing to do with quantum theory or general
00:08:06.240 Those are explanations of everything regardless, but cannot explain why in that situation.
00:08:12.640 So David's recent talk called the mathematicians misconception, given at the Dirac Prize
00:08:17.240 Award ceremony, was an explanation of how proof must be a physical process.
00:08:23.320 Now this is an amazing insight, and it's very counterintuitive.
00:08:27.280 It's even caused people to have strong emotional reactions, especially mathematicians.
00:08:38.360 Now this claim arises from mathematical intuition.
00:08:41.520 The claim that because mathematical objects are plotting ideals, they're not made
00:08:45.280 of atoms, so far this is true, that when we reason about them and reasoning on this
00:08:50.840 view is a non-physical process that what we get is some kind of mathematically ideal knowledge.
00:09:02.400 When a mathematician reasons what they are doing is thinking, they're using their brain,
00:09:08.840 Now as I've argued before, the mind is a kind of software, and it runs on the hardware
00:09:13.320 that is the brain, or if you don't like that or want some other language, reasoning, even
00:09:18.000 mathematical reasoning, is something that brains do.
00:09:21.280 Now brains are physical objects, they obey physical laws, there's no escaping this.
00:09:26.800 And so the repertoire of possible computations is in one-to-one correspondence with
00:09:35.360 And a universal Turing machine is the theoretical foundation underpinning what all really
00:09:44.960 So the repertoire of all possible computations that a brain can do is in one-to-one correspondence
00:09:50.280 with those of a universal Turing machine, there's nothing universal Turing machine
00:09:56.480 And there's nothing a person can do that a universal Turing machine cannot.
00:10:02.320 A universal Turing machine just has to be able to write, read, and erase symbols on a piece
00:10:09.600 Now a person can do that, a universal Turing machine just has to be able to write, and read,
00:10:15.400 and erase symbols on a piece of paper, and be able to move from one square to the next.
00:10:21.600 Now a person can do that, therefore a person is a universal Turing machine, done.
00:10:28.000 By the way, a person is more than just a universal Turing machine, but I don't need that
00:10:33.400 Now the other half of what I said is about whether or not there's anything a person can
00:10:40.280 Well, in terms of computation, a universal Turing machine is universal precisely because
00:10:47.800 Now, if a person can do something else in terms of thinking that a universal Turing
00:10:53.320 machine cannot, this means that thing is a non-computable thing.
00:11:01.120 They've a Deutsch proved that given computers of physical objects and obey the laws of physics,
00:11:08.000 then the repertoire of all the possible motions of a universal quantum computer must include
00:11:18.360 That is, a quantum computer can simulate the deepest laws we have, namely the laws of quantum
00:11:24.680 theory, and all matter obeys those laws that includes brains.
00:11:30.080 And so whatever a brain can do, because it's just made of matter, so too can a universal
00:11:37.440 And therefore we have the other half of our requirement that there's nothing a human brain
00:11:41.720 or a mind can possibly compute that a universal computer cannot.
00:11:46.800 And what a universal computer can compute is all the things that are possibly computable.
00:11:51.600 A mathematician's intuition cannot escape this.
00:11:55.080 The intuition itself is a computation of a kind, a mental activity must be, it could be
00:12:02.920 And what a quantum computer does, like everything else in the universe, is obey physical
00:12:08.360 And thus, if the laws of physics were different, what could be computed would be different.
00:12:18.720 They prove things about numbers, about abstract objects.
00:12:23.040 When they complete proof in their mind, it's their mind that has computed it.
00:12:28.080 But the possible proof cannot be of anything the laws of physics prohibit.
00:12:33.120 But the possible proof cannot be of anything that the laws of physics have prohibited
00:12:39.920 So there are many statements, indeed the overwhelming majority of mathematical statements,
00:12:46.160 This is where girdle's incompleteness theorems come in.
00:12:49.400 One of the incompleteness theorems say, there are statements that are true in mathematics,
00:12:57.360 And there's a proof of that, that's girdles, incompleteness theorems.
00:13:01.200 But that is only the case because the laws of physics are such, that what can be proved
00:13:06.440 prohibits us from ever finding the proof of those things.
00:13:11.760 Turing's version of this is that there are some statements that are not computable or
00:13:17.880 So there's a difference between what's true and what's provable.
00:13:23.640 That's the point of both Turing and girdles proofs.
00:13:25.840 And by the way, you can have systems such as so-called centencial or propositional logic.
00:13:35.120 And in these systems, the axioms allow you to prove everything that is true within that
00:13:41.920 Girdles incompleteness theorem is called the incompleteness theorem because if you have
00:13:45.200 a richer system of logic, something like simple arithmetic, then you end up being able
00:13:52.200 to generate true statements that cannot be proved true as such.
00:13:57.240 And this is the case for the overwhelming majority of logical systems that you can create.
00:14:02.320 Therefore, the overwhelming majority of mathematical claims you can make will not have proofs.
00:14:07.960 You won't be able to decide with the true or false or if they're true, you won't be able
00:14:13.760 The relevance of all this, for here and now, is that mathematical objects perfectly
00:14:18.040 true statements are things we only have access to through proof, which is a type of computation.
00:14:26.320 But computation requires a computer, and computers are physical objects obeying the laws of
00:14:30.760 physics, which mandate that we cannot ever hope to have error-free computation.
00:14:39.320 Now I should mention as an aside that David's great contribution here was a proof
00:14:43.400 of the Turing principle, namely that there can be a physical object whose repertoire
00:14:48.600 of possible motions contains the motions of all other objects.
00:14:54.520 But this is now called the Church Turing Deutsch Principle.
00:14:58.440 And David has said that really Ada Lovelace deserves credit for this also.
00:15:02.800 So, to lay with the point, the claim from some mathematicians, and here I'm drawing directly
00:15:09.080 on David's talk at his direct prize award ceremony, seems to be that out there in abstract
00:15:15.320 space, there is some other definition of proof, such that it's not physical.
00:15:22.120 There is another way to prove stuff in abstract space that's not physics.
00:15:26.040 But if some process that doesn't conform to that definition was a way of knowing about
00:15:30.760 some necessary truth, that process wouldn't be a proof of that truth.
00:15:35.840 The repertoire of integer functions that Turing machines and quantum computers are able
00:15:44.280 But we only know they're different speed from quantum theory.
00:15:48.320 And so when mathematicians attempt to say that they have an intuition that says that they
00:15:53.520 can prove things that in such a way that they're able to get their error frame, it's
00:15:58.560 not the method of proof is not in some way subservient to our knowledge of the laws of
00:16:05.120 They're wrong because all proof, even intuitions, that all mental processes, including
00:16:12.880 their own intuitions, arise from the activity of the brain, computing stuff.
00:16:18.120 And when you compute stuff, you're performing a physical process.
00:16:21.680 That's all that can be happening according to David Deutsch's proof of how quantum computation
00:16:26.080 works and the relationship between computation, which includes all intuitions, all
00:16:30.680 mental activities, all possible methods of proof.
00:16:33.640 The relationship between that and the laws of physics.
00:16:36.760 The laws of physics constrain what computation is and all mental activities, including
00:16:45.440 all intuitions and all methods of proof are physical.
00:16:50.320 Okay, so now let's return directly to chapter five of the beginning of infinity, the
00:16:58.120 And let's take note of the concluding remarks and how they synthesize with what I've
00:17:02.040 just been talking about primarily drawn from David's talk about the mathematicians' misconception.
00:17:10.200 And David writes near the end of this chapter, beauty, right and wrong, primality, infinite
00:17:17.280 They all exist objectively, but not physically.
00:17:22.520 Certainly they can affect you, as examples like Hofster to show, but apparently not in
00:17:29.720 We cannot trip over one of them in the street, however, there is less to that distinction
00:17:34.400 than our empiricism biased common sense assumes.
00:17:37.880 First of all, being affected by a physical object means that something about the physical
00:17:42.520 object has caused a change by the laws of physics or equipment or equivalently that the laws
00:17:47.760 of physics have caused a change via that object.
00:17:51.400 But causation and the laws of physics are not themselves physical objects.
00:17:56.880 There are abstractions, and our knowledge of them comes just as for all other abstractions
00:18:01.360 from the fact that our best explanations invoke them.
00:18:05.720 Progress depends on explanation, and therefore trying to conceive of the world as merely
00:18:08.920 a sequence of events with unexplained regularities would entail giving up on progress.
00:18:14.840 This argument that abstraction really exists does not tell us what they exist as.
00:18:21.160 For instance, which of them are purely emergent aspects of others and which exist independently
00:18:26.720 of the others, or pause there, so it tells us that abstract entities have an independent
00:18:36.960 existence, but we don't know what they are exactly.
00:18:44.760 So when people ask, what is the real nature of numbers?
00:18:52.240 Not though they are, it's very difficult to say anything more precise than that.
00:19:03.640 And if someone then asks, where do they exist, well the answer is simple.
00:19:08.720 They're not physical, so there is no where that applies to them.
00:19:16.280 And space is something that's part of the physical universe.
00:19:19.960 And so asking about where, for a number, is rather like asking what happened before the
00:19:25.640 beginning of time, grammatically, the sentence might make sense, but logically, it refers
00:19:34.880 David writes, what the laws of morality be the same if the laws of physics were different?
00:19:39.640 If they were such that knowledge could be obtained by blind obedience to authority, then
00:19:43.880 scientists would have to avoid what we think of as the values of scientific inquiry in
00:19:50.200 My guess is that morality is more autonomous than that.
00:19:52.440 And so it makes sense to say that the laws of physics would be immoral.
00:19:56.080 And as I remarked in chapter four, to imagine the laws of physics that would be more
00:20:02.800 Now if there's ever been, I'll pause there, this is me speaking, if there is ever
00:20:06.680 being a more non-relativist conception of morality, I haven't read it.
00:20:13.280 What David is saying there is that morality has an independent, possibly his guess is that
00:20:19.120 it has an independent existence, a side from the laws of physics.
00:20:23.720 So it doesn't matter what your laws of physics are.
00:20:26.320 It doesn't matter if you are some kind of super alien that has a bird's eye view of
00:20:32.240 the megaverse, a version of the universe with different physical laws or different kinds
00:20:39.760 Many universes with different physical laws that meant you would have to commit immoral
00:20:45.360 acts in order to make progress would be immoral.
00:20:48.360 That there is an independence to morality that stands apart from the laws of physics.
00:20:54.560 Now David's guess, and I'm sure this is David's guess, is that, we say, guess this
00:21:01.080 is David's guess, is that the laws of physics that we do have are not immoral.
00:21:08.080 That in fact, in order to make progress in this universe, that you have to be a moral person,
00:21:16.080 That immorality, in fact, is the way in which we don't make progress.
00:21:21.440 In fact, that must be the case because, for example, we need to be able to speak the truth.
00:21:27.800 We need to be able to tell the truth in order to make progress, that reality is connected
00:21:32.040 to truth in some way, and so our articulation of the truth is the thing that allows us to
00:21:36.200 make progress and solve problems, so our laws of physics are definitely moral.
00:21:43.120 The reach of ideas into the world of abstractions is a property of the knowledge that
00:21:46.360 they contain, not of the brain in which they happen to be instantiated.
00:21:50.280 A theory can have infinite reach, even if the person who originated it is unaware that
00:21:59.160 And there is a kind of infinite reach that is unique to people.
00:22:02.760 The reach of the ability to understand explanations, and this ability is itself an instance
00:22:08.440 of the wider phenomena of universality, to which I turn next.
00:22:15.800 I'm going to turn, in fact, to chapter 10 of the fabric of reality, because I'd like
00:22:20.640 to stay with this theme of mathematics and the theme of the reality of abstractions,
00:22:25.800 and really focus on that part of the reality of abstractions, that, personally, I find
00:22:30.720 most interesting, and a lot of people do as well, about abstract entities that are mathematical
00:22:39.040 How we can come to understand what mathematics is about, and how we can explain what
00:22:44.520 mathematical entities are, and a little more on the nature of proof.
00:22:49.520 So in the fabric of reality, chapter 10, page 222, at the very beginning of the chapter
00:22:55.880 day of the rites, now he's spoken about in the fabric of reality previously before
00:23:02.400 we get the chapter 10, obviously, quantum physics and evolution and knowledge.
00:23:11.800 And so what he writes here is, the fabric of reality that I have been describing so far
00:23:17.360 has been the fabric of physical reality, yet I have also heard freely to entities that
00:23:23.240 are nowhere to be found in the physical world, abstractions, such as numbers and infinite
00:23:27.840 sets of computer programs, nor are the laws of physics themselves physical entities in
00:23:34.760 So again, if you have bought the beginning of infinity, if you're a fan of the beginning
00:23:39.600 of infinity, you can see there's a lot of supplementary material here that really helps
00:23:45.960 So the fabric of reality, fantastic book, and everyone should get it.
00:23:51.080 So I'm not going to read the whole chapter here, I'm going to skip forward to the parts
00:23:57.840 Next part is David asks the question, do abstract non-physical entities exist?
00:24:05.320 I'm not interested here in issues of mere word usage.
00:24:08.720 It is obvious that numbers, the laws of physics and so on do exist in some senses and
00:24:14.720 The substantive question is this, how are we to understand such entities?
00:24:19.800 Which of them are merely convenient forms of words referring ultimately only to ordinary
00:24:25.240 Which are merely ephemeral features of our culture, which are arbitrary like the rules of
00:24:29.040 a trivial game that we need only look up and which, if any, can be explained only in
00:24:33.640 a way that attributes an independent existence to them.
00:24:37.040 Things of this last type must be part of the fabric of reality as defined in this book,
00:24:41.920 because one would have to understand them in order to understand everything that is understood.
00:24:46.840 Now, the next part that I won't read is David speaking about what's called Dr. Johnson's
00:24:52.160 criterion and Dr. Johnson's criterion is whether or not an entity kicks back and by kickback,
00:24:58.200 that would mean reacts in such a way or behaves in such a way or you're able to find
00:25:02.600 out something such that it was unexpected, that's surprising.
00:25:07.000 And this is why we regard numbers as being absolutely real, because merely by defining
00:25:13.960 a few axioms, the axioms of arithmetic, you can then go on to discover amazing things
00:25:20.040 that are completely unexpected, a very simple one is the distribution of prime numbers.
00:25:24.320 It's completely unexpected, we don't know what it is, we don't know what the next prime
00:25:31.120 So having defined these axioms, it then seems to reveal this reality of infinite complexity.
00:25:39.840 And so therefore numbers are of a kind of abstract reality, according to that last question
00:25:46.120 that David asked right there, he said, which if any can be explained only in a way that
00:25:51.200 attributes an independent existence to them, so that would be numbers, they have an independent
00:25:56.560 The next highest prime number that we find, no one knows when we'll find that next
00:26:01.240 prime number, and no one knows what it is, but can anyone doubt that it exists?
00:26:10.560 It just exists, not physically, it's not, in fact, the biggest prime numbers, the biggest
00:26:17.760 prime numbers don't kind of have any physical meaning, they can't possibly label anything
00:26:22.680 in physical reality, because they're so large that they, you know, orders of magnitude
00:26:28.080 bigger than all of the particle, the number of particles in the universe.
00:26:32.800 And so David writes, thus abstract mathematical entities, we think we are familiar with
00:26:37.600 can nevertheless surprise or disappoint us, they can pop up unexpectedly in new guises
00:26:43.040 or disguises, they can be explicable, and then later conform to new explanations.
00:26:49.440 So they are complex and autonomous, and therefore by Dr. Johnson's criterion, we must
00:26:56.000 Since we cannot understand them either as being part of ourselves or as being part
00:26:59.760 of something else that we already understood, but we can understand them as independent entities,
00:27:04.960 so we must conclude they are real independent entities.
00:27:08.760 Nevertheless, abstract entities are intangible, they do not kick back physically in the
00:27:14.000 sense that a stone does, so experiment and observation cannot possibly play quite the same
00:27:18.480 role in mathematics as they do in science, in mathematics, proof plays that role.
00:27:23.600 Dr. Johnson's stone, kicked back by making his foot rebound, prime numbers kick back
00:27:28.520 when we prove something unexpected about them, especially if we can go on to explain it too.
00:27:33.720 In the traditional view, the crucial difference between proof and experiment is that
00:27:37.360 a proof makes no reference to the physical world.
00:27:40.040 We can perform a proof in the privacy of our own minds, or we can perform a proof trapped
00:27:44.600 inside a virtual reality generator, rendering the wrong physics.
00:27:48.640 Provided that we follow the rules of mathematical inference, we should come up with the
00:27:54.440 And again, the prevailing view is that apart from the possibility of making blunders, when
00:27:58.880 we have proved something, we know with absolute certainty that it is true.
00:28:04.840 Mathematicians are rather proud of this absolute certainty, and scientists tend to be a little
00:28:09.640 For in science, there is no way of being certain of any proposition.
00:28:12.960 However, well one's theories explain existing observations at any moment someone will
00:28:17.160 make a new inexplicable observation that casts out on the whole of the current explanatory
00:28:22.480 Of course, someone may reach a better understanding that explains not only all existing
00:28:26.400 observations, but also why the previous explanations seem to work, but are nevertheless
00:28:33.520 Galileo, for instance, found a new explanation of the age-old observation that the ground
00:28:37.440 beneath our feet is at rest, an explanation that involved the ground actually moving.
00:28:42.680 Virtual reality, which can make one environment seem to be another, underlines the fact
00:28:48.560 that when observation is the ultimate arbiter between theories, there can never be any certainty
00:28:55.080 However, obvious, is even remotely true, but when proof is the arbiter, it is supposed
00:29:06.000 There we have David setting up the Mathematicians' misconception.
00:29:09.680 For the first, so he set this up decades ago now, but I'll skip a bit and David concludes
00:29:23.680 For the idea that Mathematicians yield certainties is a myth too.
00:29:29.480 Since ancient times, the idea that Mathematicians has a privileged status has often been associated
00:29:34.240 with the idea that some abstract entities at least are not merely part of the fabric
00:29:39.400 Even more real than the physical world, Pythagoras believed that regularities and nature
00:29:43.880 are the expression of mathematical relationships between natural numbers.
00:29:50.240 This was not meant quite literally, but Plato went further and effectively denied that
00:29:56.680 I'm skipping a bit and David talks a little bit more about Plato, and he then writes
00:30:03.080 However, the problem he posed of how we can possibly have knowledge that alone certain
00:30:11.320 In some elements of his proposed solution have been part of the prevailing theory of knowledge
00:30:16.400 In particular, the core idea that mathematical knowledge and scientific knowledge come
00:30:20.960 from different sources and that the special source of mathematics confers absolute certainty
00:30:26.600 upon it is to this day accepted uncritically by virtually all mathematicians.
00:30:32.040 Nowadays they call this source mathematical intuition, but it plays exactly the same role
00:30:40.520 They have been many bit of controversies about precisely which types of perfectly reliable
00:30:45.520 knowledge our mathematical intuition can be expected to reveal.
00:30:49.080 In other words, mathematicians agree that mathematical intuition is a source of absolute
00:30:53.600 certainty, but they cannot agree about what mathematical intuition tells them.
00:30:57.600 Obviously this is a recipe for infinite, unresolved controversy.
00:31:01.520 Inevitably, most such controversies have centered on the validity or otherwise of various methods
00:31:09.360 One controversy concerned so-called imaginary numbers.
00:31:12.960 The imaginary numbers are the square roots of negative numbers.
00:31:15.520 New theorems about ordinary real numbers were proved by appealing at intermediate stages
00:31:20.320 of a proof to the properties of imaginary numbers.
00:31:23.800 For example, the first theorems about the distribution of prime numbers were proved in
00:31:28.040 But some mathematicians objected to imaginary numbers on the grounds that they were not real.
00:31:33.600 Current terminology still reflects the old controversy, even though we now think that imaginary
00:31:40.600 I expect their school teachers had told them they were not allowed to take the square root
00:31:45.280 In consequently, they did not see why anyone else should be allowed to outline.
00:31:49.400 Not yet they called this uncharitable impulse mathematical intuition.
00:31:53.760 But other mathematicians had different intuitions.
00:31:55.840 They understood what the imaginary numbers were and how they fitted in with real numbers.
00:32:00.440 Why they thought should one not define new abstract entities to have any properties one likes?
00:32:07.360 Surely the only legitimate grounds for forbidding this would be that the required properties
00:32:13.760 Surely the only legitimate grounds for forbidding this would be that the required properties
00:32:20.240 That is essentially the modern consensus which the mathematician John Horton Conway has
00:32:24.560 robustly referred to as the mathematicians' liberation movement.
00:32:29.280 Admittedly, no one had proved that the system of imaginary numbers was self-consistent,
00:32:33.840 but that no one had proved that the ordinary arithmetic of the natural numbers was self-consistent
00:32:38.840 Now, I'm skipping a very substantial number of pages here, maybe one day I'll go back
00:32:46.520 I will do a series on the fabric of reality once I've finished the beginning of the infinity.
00:32:51.840 This will take many months and I've skipped forward a number of pages and then David writes.
00:32:59.360 Thanks to Girdle, we now know there will never be a fixed method of determining whether
00:33:04.040 a mathematical proposition is true anymore than there is a fixed way of determining whether
00:33:08.640 a scientific theory is true, nor there ever be a fixed way of generating new mathematical
00:33:14.240 Therefore, progress in mathematics will always depend on the exercise of creativity.
00:33:19.360 It will always be possible and necessary for mathematicians to invent new types of proof.
00:33:24.560 They will validate them by new arguments and by new modes of explanation depending upon
00:33:28.480 their ever-improving understanding of the abstract entities involved.
00:33:33.880 To prove them, yet to invent a new method of proof.
00:33:36.640 I said the method was based on the diagonal argument, but Girdle extended that argument
00:33:41.960 Nothing had ever been proved in this way before.
00:33:44.760 No rules of inference laid down by someone who had never seen Girdle's method could
00:33:48.880 possibly have been prescient enough to designate it as valid, yet it is self-evidently
00:33:58.520 It came from Girdle's understanding of the nature of proof.
00:34:01.400 Girdle's proofs are as compelling as any in mathematics, but only if one first understands
00:34:08.920 So explanation does, after all, play the same paramount role in pure mathematics as it
00:34:17.640 Being and understanding the world, the physical world, and the world of mathematical abstractions
00:34:23.240 is in both cases the object of the exercise, proof and observation, a merely the means
00:34:35.440 All the way back in the fabric of reality, 1997, we're getting a poor tent, really, of
00:34:41.280 what is to come with David's TED talks, what is to come with David's next book, and
00:34:45.640 what has to come with the vision and the worldview that he's gifted us with.
00:34:51.640 It's here in its nascent form in the fabric of reality.
00:34:55.000 In fact, more than nascent form, I mean, he's really articulating it clearly, there's so
00:34:58.880 much here, of course, that it's easy to have missed at night, met, but having read it, read
00:35:06.360 the fabric of reality for the first time when it was published in 1997, I didn't fully
00:35:11.600 get the centrality of explanation in science, and I didn't get it until the beginning of
00:35:18.320 So, I'm skipping forward a little more, and this is still in chapter 10, skipping some
00:35:22.640 more pages and David writes, it is often suggested that the brain, the human brain, may
00:35:27.920 be a quantum computer, and that its intuitions, consciousness and problem solving abilities
00:35:35.000 This could be so, but I know of no evidence and no convincing argument that it is so.
00:35:40.240 My bet is that the brain, considered as a computer, is a classical one, but that issue
00:35:44.560 is independent of Penrose's ideas, it's paused there, so he's just written a section
00:35:50.160 about Roger Penrose who does have this theory that it could be the case that the brain
00:35:56.320 is a quantum computer, or at least relies on some kind of quantum effects, and I think
00:36:00.960 there's a number of scientists including neuroscientists that kind of hint that they think
00:36:05.040 that this might be the case, but as David says, there's no relevance for that, and
00:36:08.480 this is problem of what's known as decoherence, in particular, in order for us to have
00:36:17.560 a quantum computer that works, we need to be able to have quantum systems, we need to
00:36:22.960 be able to have entanglement, and entanglement, so far as we can tell at the moment, only
00:36:28.000 works when temperatures are sufficiently low, such that noise doesn't affect the system,
00:36:36.000 and the kinds of technology that I've seen, the University of New South Wales here in Sydney,
00:36:41.960 where I am, is working on this, the kind of temperatures they need is very, very close
00:36:46.960 to absolute zero, I mean they're operating at some of the coldest refrigeration on the planet
00:36:54.560 in order to get these things to work, so until we have a new theory of how to build
00:37:01.480 quantum computers, really the argument for the human brain being and quantum computer
00:37:08.120 is unconvincing, but David continues to write, Penrose is not arguing that the brain is
00:37:14.320 some sort of universal computer, differing from the universal quantum computer by having
00:37:18.240 a larger repertoire of computations made possible by new post-quantum physics, he is arguing
00:37:22.880 for a new physics that will not support computational university, so that under his new theory
00:37:29.120 it will not be possible to construe some of the actions of the brain as computations
00:37:33.920 at all. David writes, I must admit that I cannot conceive of such a theory, however, fundamental
00:37:41.120 breakthroughs do tend to be hard to conceive of before they occur, naturally, is hard
00:37:47.000 to judge Penrose's theory before he succeeds in formulating it fully, so that's a very
00:37:53.040 generous reading of Roger Penrose, I'm going to skip forward a few more pages here and
00:38:01.760 David writes, Plato tells us that since we have access only to imperfect circles say
00:38:08.280 we cannot thereby obtain any knowledge of perfect circles, but why not exactly? One might
00:38:14.320 as well say that we cannot discover the laws of planetary motion because we do not have
00:38:18.560 access to real planets, but only to images of planets. The Inquisition did say this, and
00:38:24.720 as I have explained, they were wrong. One might as well say it as impossible to build accurate
00:38:29.600 machine tools because the first one would have to be built with inaccurate machine tools.
00:38:35.080 With the benefit of hindsight, we can see that this line of criticism depends on a very
00:38:38.560 crude picture of how science works, something like inductivism, which is hardly surprising
00:38:44.000 since Plato lived before anything that we would recognize as science. If say the only way
00:38:48.640 of learning about circles from experience, which will examine thousands of physical circles
00:38:52.960 and then from the accumulated data to try to infer something about their abstract Euclidean
00:38:56.800 counterparts, Plato would have a point. But if we form a hypothesis that real circles resemble
00:39:02.000 the abstract ones in specified ways, and we have to be right, then we may well learn something
00:39:06.560 about the abstract circles by looking at real ones. In Euclidean geometry, one often uses
00:39:12.160 diagrams to specify a geometrical pattern or its solution. There is a possibility of error
00:39:18.400 in such a method of description if the imperfections of circles in the diagram give a misleading
00:39:22.720 impression. For example, if two circles tend to touch each other when they do not. But if one
00:39:27.280 understands the relationship between real circles and imperfect circles, one can, with care,
00:39:31.760 eliminate all such errors. If one does not understand that relationship, it is practically
00:39:36.480 impossible to understand Euclidean geometry at all. The reliability of the knowledge of a perfect
00:39:42.080 circle that one can gain from a diagram of a circle depends entirely on the accuracy of the
00:39:47.440 hypothesis that the two resemble each other in relevant ways. Such a hypothesis, referring to a
00:39:54.480 physical object, the diagram, amounts to a physical theory and can never be known with certainty,
00:39:59.520 but that does not, as Plato would have it, preclude the possibility of learning about perfect
00:40:04.640 circles from experience. It just precludes the possibility of certainty. That should not worry
00:40:11.040 anyone is looking for certainty, but for explanations. That is great. How do we know about these
00:40:18.640 mathematical entities? Well, because we have theories, we have theories that arise from our study
00:40:24.000 of physically existing circles, let's say, and then we can prove things about those physically
00:40:29.600 existing circles that relate to ideal platonic forms in some way. This is true of all mathematical
00:40:36.240 platonic ideals. So mathematics is about these abstract entities. We only have access to
00:40:43.440 the physical versions thereof, but there is a relationship between, and we know what the relationship
00:40:49.120 is because we conjecture explanations between the perfect abstract entities and our physical
00:40:56.160 reality. David continues, skip a little, and he writes, let us re-examine another assumption of
00:41:02.960 Plato's, the assumption that we do not have access to perfection in the physical world.
00:41:07.120 He may be right that we shall not find perfect honor or justice, and he is certainly right
00:41:11.760 that we shall not find the laws of physics or the set of all natural members, but we can find
00:41:16.480 a perfect hand in bridge or the perfect move in a given chess position. That is to say,
00:41:22.000 we can find physical objects or processes that perfectly possess the properties of the specified
00:41:27.040 abstractions. We can learn chess just as well with a real chess set as we could with a perfect
00:41:32.720 form of a chess set. The fact that a knight is chipped does not make the check make, it delivers
00:41:38.240 any less final. As it happens, a perfectly Euclidean circle can be made available to our senses.
00:41:44.720 Plato did not realize this because he did not know about virtual reality. It would not be
00:41:48.400 difficult to program the virtual reality generators I envisaged back in a previous chapter.
00:41:54.240 Such that a user could experience perfect geometrical forms, skipping a fair bit more.
00:41:59.920 There is a lot of really good stuff here, but yes, I will have to return to it a later day,
00:42:05.520 but I just want to get to this mathematician's misconception and the relevant material about
00:42:11.440 the reality of abstractions. It is almost, I am reaching the punchline here,
00:42:17.680 and so I will read this. This is all our page, 246 of the fabric of reality and David Wright's.
00:42:23.520 A conventional symbolic proof seems at first sight to have a quite a different character
00:42:28.880 from a hands-on virtual reality sort of proof, but we now see that they are related in the ways
00:42:34.560 that computations are to physical experiments. Any physical experiment can be regarded as a
00:42:40.720 computation, and any computation is a physical experiment. In both sorts of proof, physical entities,
00:42:48.080 whether in virtual reality or not, are manipulated according to rules. In both cases,
00:42:53.920 the physical entities represent the abstract entities of interest, and in both cases,
00:42:58.880 the reliability of the proof depends on the truth of the theory that physical and abstract entities
00:43:04.400 do indeed share the appropriate properties. So, pausing there, a mathematical proof is
00:43:14.000 about abstract entities, but we use physical objects in order to represent those abstract entities.
00:43:26.320 We can also see from the above discussion that proof is a physical process. In fact,
00:43:32.800 a proof is a type of computation. Proving a proposition means performing a computation
00:43:38.960 which, if one has done it correctly, establishes that the proposition is true.
00:43:44.080 When we use the word proof to denote an object, such as an ink on paper text, we mean that
00:43:51.280 the object can be used as a program for recreating a computation of the appropriate kind.
00:43:58.080 It follows that neither the theorems of mathematics nor the process of mathematical proof,
00:44:03.680 nor the experience of mathematical intuition, confers any certainty. Nothing does.
00:44:09.920 Our mathematical knowledge may, just like our scientific knowledge, be deep, and broad,
00:44:15.920 it may be subtle and wonderfully explanatory. It may be uncontroversially accepted,
00:44:21.440 but it cannot be certain. No one can guarantee that a proof that was previously thought to be valid
00:44:27.520 will not one day turn out to contain a profound misconception, made to seem natural by a
00:44:33.040 previously unquestioned self-evident assumption about either the physical world or about the
00:44:38.720 abstract world, or about the way in which some physical and abstract entities are related,
00:44:44.640 skipping a little more. And I think that this is where, again, on a personal note,
00:44:49.840 I really had that sense on reading this chapter for the first time over the
00:44:56.400 ground falling out from underneath you, the sense of vertigo and falling that previously throughout
00:45:03.200 school and high school and university, I really was inculcated with the idea that mathematics
00:45:09.360 was this solid unchanging domain of certainty. And then I think I read this, David wrote,
00:45:19.760 A very similar misclassification has been caused by the fundamental mistake that mathematicians
00:45:25.360 since antiquity have been making about the very nature of their subject, namely that mathematical
00:45:31.360 knowledge is more certain than other forms of knowledge. Having made that mistake, no one has a choice
00:45:37.840 but to classify proof theory as a part of mathematics. For a mathematical theorem could not be
00:45:43.200 certain if the theory that justifies its method of proof, or itself uncertain. But as we have just
00:45:49.600 seen, proof theory is not a branch of mathematics. It is a science. Proofs are not abstract,
00:45:56.240 there is no such thing as abstractly proving something, just that there is no such thing as
00:46:01.040 abstractly calculating or abstractly computing something. One can of course define a class of
00:46:06.800 abstract entities and call them proofs, but those proofs cannot verify mathematical statements
00:46:12.720 because no one can see them. They cannot persuade any one of the truth of a proposition
00:46:17.440 any more than an abstract virtual reality generator that does not physically exist
00:46:21.520 can persuade people that they are in a different environment, or an abstract computer can factor
00:46:25.760 ize a number for us. A mathematical theory of proofs would have no bearing on which mathematical
00:46:30.960 truths can or cannot be proved in reality. Just as a theory of abstract computation has no bearing
00:46:36.880 on what mathematicians or anyone else can or cannot calculate in reality, unless there is a separate
00:46:42.240 empirical reason for believing that the abstract computations in the theory represent real
00:46:47.920 computations. Computations, including the special computations that qualify as proofs,
00:46:53.600 are physical processes. Proof theory is about how to ensure that those processes correctly
00:46:58.960 mimic the abstract entities they are intended to mimic. Goodell's theories have been hailed
00:47:03.600 as the first new theorem of pure logic for 2000 years, but that is not so. Goodell's
00:47:09.040 theorem is about what can cannot be proved and proof is a physical process. Nothing improved
00:47:14.240 theory is a matter of logical own. The new way in which goodell managed to prove general assertions
00:47:19.120 about proofs depends on certain assumptions about which physical processes can or cannot
00:47:24.720 represent an abstract fact in a way that an observer can detect and be convinced by. Goodell distilled
00:47:31.040 such assumptions into his explicit and tash at rules of his results. His results were self-evidently
00:47:36.720 justified, not because they are pure logic, but because mathematicians found the assumptions
00:47:41.600 self-evident, skipping more, and just getting to the part that I think I've quoted more from the
00:47:49.280 fabric of reality than any other section where David writes. That mathematicians throughout the
00:47:57.920 ages should have made various mistakes about matters of proof and certainty is only natural.
00:48:03.200 The present discussion should lead us to expect that the current view will not last forever,
00:48:06.800 but the confidence with which mathematicians have blundered into these mistakes, and their
00:48:11.600 inability to acknowledge even the possibility of error in these matters are, I think, connected
00:48:16.400 with an ancient and widespread confusion between the methods of mathematics and its subject matter.
00:48:23.040 Let me explain. Unlike the relationships between physical entities, relationships between
00:48:28.640 abstract entities are independent of any contingent facts and of any laws of physics.
00:48:32.640 They are determined objectively by the autonomous properties of the abstract entities themselves.
00:48:39.600 Mathematics, the study of these relationships and properties is therefore the study of
00:48:46.080 absolutely necessary truths. I'm just going to pause and repeat that sentence.
00:48:52.160 Mathematics, the study of these relationships and properties, is the study of absolutely necessary
00:48:59.440 truths. Mathematics is the study of absolutely necessary truths. Let me continue. In other words,
00:49:05.680 the truths that mathematics studies are absolutely certain, but this does not mean that our knowledge
00:49:11.440 of those necessary truths is itself certain, nor does it mean that the methods of mathematics
00:49:16.000 confer necessary truth on their conclusions. After all, mathematics also studies falsehoods and
00:49:22.000 paradoxes, and that does not mean that the conclusions of such a study are necessarily
00:49:26.000 false or paradoxical. Next is my favorite line of the entire chapter, and certainly up there with
00:49:34.160 one of my favorite lines out of both books, David Wright. Necessary truth is merely the subject
00:49:41.520 matter of mathematics. Not the reward we get for doing mathematics. I think I've quoted it a
00:49:49.840 hundred times before. Necessary truth is merely the subject matter of mathematics. Not the reward
00:49:55.360 we get for doing mathematics. The objective of mathematics is not, and cannot be mathematical
00:50:01.680 certainty. It is not even mathematical truth, certain or otherwise. It is, and must be,
00:50:06.720 mathematical explanation. So I'll just read the conclusion now of this chapter. I'm skipping a
00:50:12.480 little more in David Wright. There are physical objects such as fingers, computers, and brains,
00:50:16.880 whose behavior can model that of certain abstract objects. In this way, the fabric of physical
00:50:21.600 reality provides us with a window on the world of abstractions. It is a very narrow window and gives
00:50:26.800 us only a limited range of perspectives. Some of the structures that we see out there, such as
00:50:31.360 the natural numbers or the rules of inference of classical logic, seem to be important or fundamental
00:50:36.240 to the abstract world. In the same way as deep laws of nature are fundamental to the physical
00:50:40.720 world, but that could be a misleading appearance. For what we are really seeing is only that some
00:50:46.160 abstract structures are fundamental to our understanding of abstractions. We have no reason to
00:50:50.960 suppose that those structures are objectively significant in the abstract world. It is merely that
00:50:55.520 some abstract entities are nearer and more easily visible from our window than others.
00:51:02.320 Now, I'm just going to read also his terminology section that he has in the fabric of reality as he
00:51:06.560 does in the beginning of infinity. And so his terminology includes mathematics, which he defines as
00:51:11.680 the study of absolutely necessary truths. Again, mathematics is a study of necessary truth,
00:51:17.760 but it doesn't mean that what you are able to prove are necessary truths. It's just the study of them.
00:51:23.200 And so you get what you get is fallible explanations of those necessary truths.
00:51:30.560 He defines proof as, in the traditional way, a way of establishing the truth of mathematical
00:51:36.160 propositions, or a sequence of statements starting with some premises and ending with the
00:51:42.000 desired conclusion and satisfying certain rules of inference, or better yet, a computation that
00:51:48.320 models the properties of some abstract entity and whose outcome establishes that the abstract
00:51:52.800 entity has a given property. Mathematical intuition he defines as, in the traditional sense,
00:52:00.800 an ultimate self-evident source of justification for mathematical reasoning. And actually,
00:52:05.840 it's a set of theories both conscious and unconscious about the behavior of certain physical
00:52:10.000 objects, whose behavior models that of interesting abstract entities. And then he mentioned,
00:52:15.600 so what's called Hilbert's tent problem, and Hilbert's tent problem was whether or not we could
00:52:20.800 establish once and for all the certitude of mathematical methods by finding a set of rules of inference,
00:52:26.640 sufficient for all valid proofs, and then proving those rules consistent by their own standards.
00:52:32.720 And then Goodles in completeness theorem is a proof that Hilbert's tent problem cannot be solved.
00:52:38.240 For any set of rules of inference, there are valid proofs not designated as valid by those rules.
00:52:43.920 So that's it. That's the reality of abstractions, chapter five, the beginning infinity,
00:52:47.680 with some material from chapter 10 of the fabric of reality. Next, I move on to chapter six,
00:52:55.520 universally, but I hope you enjoyed that extended appendacy, I guess, to chapter five. See you next time.
