00:00:12.240 This is another chapter that's largely about mathematics.
00:00:14.960 We had the reality of abstractions, which at least in part was about mathematics.
00:00:20.840 And in the window on infinity, we're really laying the groundwork for some of the material
00:00:28.080 So the motivation really is to speak about quantum theory and how to understand infinite
00:00:32.560 sets and how to count them and otherwise measure their number.
00:00:36.960 And we'll look at some really cool maths along the way, hence why whiteboard here.
00:00:42.720 So let me just begin the reading rather than having a huge preamble of such as I did in
00:00:46.120 the last episode and get straight into the book.
00:00:49.120 David Wright, the beginning of Chapter 8, mathematicians realized centuries ago that
00:00:53.840 it is possible to work consistently and, usefully, with infinity.
00:00:57.920 And let's say it's infinitely large quantities and also infinitesimal quantities also
00:01:03.160 Many of their properties are counterintuitive, and the introduction of theories about
00:01:09.120 But many facts about finite things are just as counterintuitive.
00:01:12.680 What Dawkins calls the argument from personal incredulity is no argument.
00:01:16.800 It represents nothing but a preference for parochial misconceptions over universal truths.
00:01:23.280 In Physics 2, infinity has been contemplated since antiquity.
00:01:26.960 Euclidean space was infinite, and, in any case, space was usually regarded as a continuum.
00:01:32.600 Even a finite line was composed of infinitely many points.
00:01:36.880 There were also infinitely many instants between any two times.
00:01:40.720 But the understanding of continuous quantities was patchy in contradictory, until Newton and
00:01:44.840 Leibniz invented calculus, a technique for analyzing continuous change in terms of infinite
00:01:52.960 The beginning of infinity, the possibility of the unlimited growth of knowledge in the future,
00:02:00.440 One of them is the universality in the laws of nature, which allows finite, local symbols
00:02:05.640 to apply to the whole of time and space until all phenomena and all possible phenomena.
00:02:13.800 Another is the existence of physical objects that are universal explainers, people, which
00:02:18.400 it turns out are necessarily universal constructors as well, and must contain universal
00:02:30.360 A person is a universal classical computer, or as David says there, a person contains
00:02:38.320 A person is a universal classical computer in a sense that they are that plus more.
00:02:43.720 They have all the capacities that a universal classical computer has, but unlike the universal
00:02:49.040 classical computers that, for example, are making this podcast and video upon, those computers
00:02:56.280 They do not have the capacity for explanation, for learning what we need in order for something
00:03:02.440 to be designated a person is a system that not only is universal in its capacity to
00:03:08.120 compute, but also university in universal in its capacity to explain one, the universal
00:03:14.520 capacity for explanation depends upon or has a prerequisite of universal computation, but
00:03:21.560 And universal computer doesn't need to be a universal explainer.
00:03:26.040 The next paragraph says, most forms of universality themselves refer to some sort of
00:03:32.040 infinity, although they can always be interpreted in terms of something being unlimited rather
00:03:38.440 This is what opponents of infinity call a potential infinity, rather than a realized one.
00:03:43.600 For instance, the beginning of infinity can be described either as a condition where progress
00:03:47.480 in the future will be unbounded, or as the condition where an infinite amount of progress
00:03:52.840 will be made, but I use those concepts interchangeably because in this context, there is no
00:04:00.320 There is a philosophy of mathematics called finiteism.
00:04:04.040 So it reads the next part, the next part is about the philosophical doctrine of finiteism.
00:04:07.840 I won't read that, I would encourage people to go to the book for the full understanding
00:04:12.440 of the problems with finiteism, but suffice it to say here, the question is, if one rejects
00:04:18.800 the reality of infinity, then one is forced in mathematics to conclude there must be a
00:04:24.400 large, largest possible number if you believe in finiteism, or if you think finiteism is
00:04:28.920 true, then there are not infinite sets including the infinite set of numbers, infinite
00:04:35.080 It must stop somewhere, so therefore there is an infinite there.
00:04:39.040 So therefore there is a finite number of numbers, however, in order to generate integers
00:04:43.920 in the first place, what we do is we add one to any number that we currently do have.
00:04:49.080 And so if the argument is there is a largest possible number, then we are contradicting
00:04:55.160 the rule that in order to get to the next number, we add one.
00:04:58.640 In other words, if someone comes up to you and says, if someone comes along and argues for
00:05:02.640 finiteism, then they are arguing for a large possible number, and they must be able to
00:05:06.720 answer the question as to why, if that large is possible number, whatever it is, cannot
00:05:13.720 So David spends a number of paragraphs, criticizing finiteism, criticizing finiteism as unreasonable,
00:05:22.080 and then he writes, the whole of the above discussion assumes the universality of reason,
00:05:27.960 the rich of science has inherent limits, so does mathematics.
00:05:31.840 So does every branch of philosophy, but if you believe that there are bounds under the
00:05:36.400 main in which reason is the proper arbiter of ideas, then you believe in unreason or the
00:05:43.960 Similarly, if you reject the infinite, you are stuck with the finite, and the finite is
00:05:49.640 parochial, so there is no way of stopping there.
00:05:53.240 The best explanation of anything eventually involves universality, and therefore infinity,
00:05:58.240 for reach of explanations, cannot be limited by fiat, and skipping just a little more,
00:06:07.320 Kantor found that the modern mathematical study of infinity, his principle was defended
00:06:11.920 and further generalized in the 20th century by the mathematician, John Conway, who whimsically
00:06:17.520 but appropriately, named it the mathematician's liberation movement.
00:06:22.360 As those defenses suggest, Kantor's discoveries encountered vitriolic opposition among
00:06:27.040 his contemporaries, including mathematical mathematicians of the day, and also many scientists,
00:06:34.600 Religious objections, ironically, were in effect based on the principle of mediocrity.
00:06:38.400 They characterised attempts to understand and work with infinity as an encroachment on the
00:06:44.360 In the mid-20th century, long after the study of infinity had become a routine part of mathematics,
00:06:49.160 and had found countless applications there, the philosopher Ludwig Wittgenstein still
00:06:53.720 contemptuously denounced it as meaningless, though eventually he also applied that accusation
00:06:58.680 to the whole of philosophy, including his own work, see chapter 12, or pause there, and
00:07:07.960 Yes, Wittgenstein and Popper had a great debate about whether or not there existed philosophical
00:07:15.640 Wittgenstein argued that there were not, there were only these things called philosophical
00:07:19.640 puzzles, that every single philosophical problem was an apparent problem because of our
00:07:26.520 misunderstanding of how to use language, so in other words, all philosophical problems came
00:07:30.520 down to language games, so this is the phrase that many people live in today, many philosophers
00:07:37.320 today still use, and Wittgenstein indeed said of his own philosophy that it was rather
00:07:43.160 useless, like the rest of philosophy and the rest of metaphysics, what he said of his own
00:07:47.800 philosophy was, it's kind of like a ladder that helps you to climb out of a dark well,
00:07:52.440 and once you are out of the dark well, you can dispense with the ladder, and that's
00:07:56.600 what you thought of his own philosophy, there was this heroic thing that allowed people to get
00:08:01.640 Now this is wonderful book, it's called Wittgenstein's Popper, I don't have it he with me now,
00:08:05.080 I'll put an image on the screen, it's sitting on my desk at my workplace actually,
00:08:08.840 and Wittgenstein's Poaker is about the sole encounter that ever happened between these two
00:08:15.400 big names in philosophy, Wittgenstein and Popper, and this great debate about whether or not
00:08:20.920 there were philosophical problems, it's called Wittgenstein's Poaker because apparently
00:08:25.640 during the debate, although accounts of what actually happened differ, apparently at some point
00:08:31.880 Wittgenstein picked up the Poaker out of the fireplace and pointed it at Popper in order to
00:08:36.600 emphasise a point, so there's some debate about whether or not that happens, so it's an interesting
00:08:41.640 sociological study of these two communities of philosophers or philosophies as well as the debate
00:08:49.880 itself, so I recommend this book, and I'll just say on the point it sounds like a very clever
00:08:56.520 thing to say, and I think it is one of the better things that possibly Wittgenstein did say,
00:09:02.760 it is clever, that his own philosophy is kind of like a ladder out of from which you use to escape
00:09:08.840 a well, it's a great analogy, but ultimately it is a false analogy, it's a false analogy because
00:09:15.880 you cannot escape from the problems of philosophy, the problems of philosophy are not like a well,
00:09:21.320 and the well is always there, you are forever climbing out of the well, and the ladder,
00:09:26.040 I suppose, is useful, this is the philosophical progress that you're making, but it is an infinite
00:09:30.120 climb, it is an infinite climb towards the light if you like, which is an infinitely far distance
00:09:33.960 off, so I suppose the ladder analogy works, but you just can't get out of the well, that's the mistake
00:09:41.000 that Wittgenstein made, skipping a little more, and then David writes, in mathematics,
00:09:46.680 infinity is stayed via infinite sets, meaning sets with infinitely many members,
00:09:51.640 the defining property of an infinite set is that some part of it has as many elements as the
00:09:57.080 whole thing, for instance, think of the natural numbers, now I'll pause here and David goes through
00:10:04.360 some examples, so I'll go through the example over here, we've got the counting numbers here,
00:10:10.840 I've added 0 to the numbers, so we've got integer numbers, so starting at 0, we go 1, 2, 3, 4,
00:10:18.040 okay, off to infinity, now I'm actually just taking part of that set, just the even numbers,
00:10:23.480 0, 2, 4, 6, 8, well it looks like this has half as many members, doesn't it,
00:10:28.600 but in fact it doesn't, it has exactly the same number of members, and the reason why
00:10:32.600 is because for every single number that I can write in the first set, I can write a number
00:10:37.880 in the second set, now I'll never run out of numbers in the second set, even though it doesn't
00:10:41.880 include all of the numbers that are in the first set, and so what we say is that they
00:10:46.120 these two set are in 1 to 1 correspondence, but every member we can write down another member in the
00:10:52.680 upper set, and so on if we did the three times tables, 3, 6, 9, 12, etc, if you take part of
00:11:01.320 this first set, and write it down here, we have a 1 to 1 correspondence, and say therefore the
00:11:07.560 size of the infinity is the same, they're both infinitely long, we'll never get to the end,
00:11:12.360 but we're going to see in a very short moment that there are kinds of infinity that are bigger
00:11:18.360 than other kinds of infinity, and the first thing I'll preface it to say is this one's accountable
00:11:22.840 infinity, we can literally count at 0, 1, 2, 3, 4, or this one 0, 2, 4, 6, 8, accountable
00:11:28.920 infinities, they're infinite, but they're not as big as other infinities that we're about to come to,
00:11:35.080 now so far I've skipped a actual substantial bit of the beginning of this chapter,
00:11:40.360 but now I'm going to read an extended bit of this chapter, I find it really entertaining as part,
00:11:44.920 and so let me just get straight into it, David writes, the mathematician David Hilbert
00:11:49.800 devised a thought experiment to illustrate some of the intuitions the one has to drop
00:11:53.800 when reasoning about infinity, he imagined a hotel with infinitely many rooms, infinity hotel,
00:11:59.960 the rooms are numbered with the natural numbers starting with one and ending with what?
00:12:04.760 The last room number is not infinity, first of all there is no last room,
00:12:08.920 the idea that any numbered set of rooms has highest numbered member is the first intuition from
00:12:15.240 everyday life we have to drop, second in any finite hotel whose rooms were numbered from one,
00:12:21.160 there would be a number whose room equaled the total number of rooms and other rooms whose
00:12:26.200 numbers were close to that, if there were ten rooms one of them would be room number ten
00:12:31.320 and there would be a room number nine as well, but infinity hotel where the number of rooms
00:12:36.040 is infinity, all the rooms have numbers infinitely far below infinity. Now imagine that infinity
00:12:44.360 hotel is fully occupied, each room contains one guest and cannot contain more, with finite hotels
00:12:50.680 fully occupied is the same thing as no room for more guests, but infinity hotel always has room
00:12:56.680 for more, I'll just pause there, so that's another strange intuition and it's kind of
00:13:03.000 it illuminates some of the struggle we have in trying to capture mathematical reality and
00:13:09.320 mathematical truth in normal natural language like English, so again he rides, if the hotel is
00:13:17.480 fully occupied in the case of a finite hotel, fully occupied means there's no room for any
00:13:23.480 else, that's what fully occupied means, but if you've got an infinitely large hotel, fully occupied
00:13:29.080 means there's still room, that seems like a contradiction, it's not and David's about to explain
00:13:34.120 why, so let me continue reading, one of the conditions of staying there is that guests have to
00:13:39.400 change rooms if asked by the management, so if and you guests arrives, the management just announced
00:13:45.320 over the public address system will all guests please move immediately to the room number one more
00:13:50.120 than their current room, thus the existing occupant room one moves to room two,
00:13:55.480 whose occupant moves to room three and so on, what happens at the last room, there is no
00:14:00.120 last room, and hence no problem about what happens there, the new arrival can now move into room one,
00:14:05.640 an infinity hotel, it is never necessary to make a reservation, evidently no such places that
00:14:10.760 infinity hotel could exist in our universe, because it violates several laws of physics, however,
00:14:15.640 this is a mathematical thought experiment, so the only constraint on the imaginary laws of physics
00:14:20.760 is that they be consistent, it is because of the requirement that they be consistent, that they are
00:14:25.720 counterintuitive, intuitions of an infinity are often illogical, it was paused there just to remark,
00:14:31.480 that we have to keep in mind throughout these thought experiments, that infinity hotel is not of
00:14:37.640 our universe, infinity hotel is in abstract space, so to speak, and so it can violate laws of physics,
00:14:45.720 that does not obey our laws of physics, this is absolutely crucial because it's going to eliminate
00:14:50.840 something about our laws of physics, okay, so let's keep on going, David writes, it is a bit
00:14:58.760 awkward to have to keep changing rooms, though they are all identical and are freshly made up
00:15:03.240 every time a guest moves in, but guest loves staying in infinity hotel, that's because it is cheap,
00:15:08.040 only a dollar a night, it extraordinarily looks luxurious, how is that possible, every day when the
00:15:13.560 management receive all the room rents of one dollar per room, they spend the income as follows,
00:15:19.000 with the dollars they receive from the rooms numbered one to a thousand, they buy complementary
00:15:23.320 champagne, strawberries, housekeeping services, and all the other overheads, just for room number one,
00:15:30.440 with the dollars they receive from the rooms numbered a thousand and one to two thousand,
00:15:34.520 they do the same for room two, and so on, in this way each room receives several hundred
00:15:39.960 dollars worth of goods and services every day, and the management make a profit as well,
00:15:44.520 all from their income of one dollar per room, word gets around, and one day an infinitely long
00:15:50.200 train pulls up at the local station containing infinitely many people wanting to stay at the hotel,
00:15:55.000 making infinitely many public address announcements will take too long, and anyway the hotel rules
00:15:59.880 say that each guest can be asked to perform only a finite number of actions per day, but no matter,
00:16:05.080 the management merely announced will all guests please move immediately to the room number
00:16:09.960 that is double that of their current room, obviously they can all do that, and afterwards the
00:16:14.920 only occupied rooms are the even numbered ones, leaving the odd numbered ones free for the new
00:16:18.920 arrivals, that is exactly enough to receive the infinitely many new guests, because there are
00:16:24.280 exactly as many odd numbers as there are natural numbers, and that's just as we can see here,
00:16:28.760 of course there's equally as many even numbers as there are natural numbers here, counting numbers,
00:16:35.640 and there will be equally as many odd numbers as well, so if person in room no one moves to two,
00:16:40.840 person in two moves to four, then we're left with room number one being free and room number three
00:16:45.800 being free, and so on. Okay, so in that thought experiment we've had an infinitely long train
00:16:52.520 turn up to infinity hotel that's packed to the brim with an infinite number of people,
00:16:57.560 but are still accommodated inside of the hotel, and the next thought experiment is even there,
00:17:03.960 so David writes, then one day an infinite number of infinitely long trains are over the station,
00:17:09.960 all full of guests for the hotel, but the managers are still unperturbed, they just make a slightly
00:17:14.920 more complicated announcement, which readers who are familiar with mathematical terminology can
00:17:19.080 see in the footnote, the upshot is everyone is accommodated, so even an infinite number of
00:17:24.120 infinite long trains with an infinite number of people on each train can still be accommodated in
00:17:28.280 infinity hotel, which is just to say there is a one-to-one correspondence between those two sets,
00:17:35.800 the set of infinitely long trains, the infinite set of infinitely long trains, each of which have
00:17:41.960 an infinite number of people in them, and the set of natural numbers here, but then David writes,
00:17:48.760 however, it is mathematically possible to overwhelm the capacity of infinity hotel. In a remarkable
00:17:54.600 series of discoveries in the 1870s, Kanto proved, among other things, that not all infinities
00:18:00.920 are equal, in particular, the infinity of the continuum, the number of points in a finite line,
00:18:08.280 which is the same as the number of points in the whole of space or space time, is much larger
00:18:13.000 than the infinity of the natural numbers. Kanto proved this by proving that there can be no one-to-one
00:18:19.480 correspondence between the natural numbers and the point in a line. That set of points as a
00:18:24.680 higher order of infinity than the set of natural numbers, and then David does an explanation
00:18:29.560 of Kanto's diagonal argument, and I'm going to do a slightly different version here,
00:18:34.600 it's kind of the version that you will see if you just look up Kanto's diagonalization argument.
00:18:39.240 Okay, so let's start again, and here we're going to use the binary system. In other words,
00:18:50.040 just the number 0 and 1, so if we were to write down all the different permutations,
00:18:56.680 different ways in which we could write an infinitely long sequence of zeros and ones,
00:19:02.520 let's see what we do. So maybe we could, if we were just going to write, we could write just zeros,
00:19:08.040 that would be an infinitely long sequence of nothing but zeros, boring, or infinitely long sequence
00:19:14.920 of just ones, okay, fine. And now if we start to combine, then maybe we could do an alternating
00:19:21.320 series of 1, 0, 1, 0, 1, 0, or we could do the opposite 0, 1, 0, 1, 0, 1. Oh maybe we could do
00:19:32.440 two ones and a 0, two ones and a 0, or maybe two zeros and a 1, two zeros and a 1,
00:19:45.320 or maybe it could be three ones and two zeros, one zero or three zeros, three ones and two zeros.
00:19:54.360 You can imagine all the different possible permutations of ways of writing zeros and ones,
00:20:01.800 all the infinite, all the different kinds of infinitely long sequences of zeros and ones.
00:20:09.160 Now it's at this point I realized my microphone had ceased working and so we're going to have
00:20:13.320 to continue this explanation in the very next episode where I can promise the audio is far better.
