Speaker 1: Welcome to Stuff to Blow Your Mind from how Stuff
Speaker 1: Works dot com. Hey you welcome to Stuff to Blow
Speaker 1: your Mind. My name is Robert Lamb and I'm Joe McCormick.
Speaker 1: And today we're going to be revisiting a topic that
Speaker 1: came up in an episode last month. So last month,
Speaker 1: on March, we released a podcast on a fascinating subject
Speaker 1: that had been much requested by listeners in the past,
Speaker 1: known as the Boltzman Brain argument, and we wanted to
Speaker 1: revisit this topic for a couple of reasons. One is
Speaker 1: that in the original version of the episode we published
Speaker 1: on March fifteen, I said something that was completely wrong,
Speaker 1: not about Boltzman brains themselves, but about gay ord Cantor
Speaker 1: and the nature of infinity as a mathematical concept. And
Speaker 1: when we became aware of this, we edited and republished
Speaker 1: the episode so not to leave a mistake and assumption
Speaker 1: floating out there on the Internet. But ever since then,
Speaker 1: I wanted to come back to the topic because I
Speaker 1: know some of you out there probably downloaded the early
Speaker 1: version of the episode with an arrow left in. I
Speaker 1: wanted to acknowledge my mistake, make a clear correction, and
Speaker 1: then take that as an opportunity to talk a little
Speaker 1: bit about the amazing nature of mathematical infinity on its own. Yeah,
Speaker 1: I mean it's this is one of those topics that
Speaker 1: combines both our attempt to understand consciousness and what it is,
Speaker 1: and also the the this mind boggling concept well actually
Speaker 1: a number of different concepts regarding the nature of infinity.
Speaker 1: Uh So, you know, it stands to reason that that
Speaker 1: that there would be a few holes left to fill,
Speaker 1: that there would be room to revisit it in a
Speaker 1: future episode, And here we are absolutely So we're going
Speaker 1: to explore the thing that I mentioned that was wrong
Speaker 1: in the first episode, and we're gonna explore a lot
Speaker 1: of the actually mind blowing reality underlying that claim, and
Speaker 1: that that's going to have to do with can Her
Speaker 1: and the idea of accountable infinities and unaccountable infinities. But
Speaker 1: we're also just going to have a chance to explore
Speaker 1: the idea of Boltzman Brains a little more, including a
Speaker 1: little bit of feedback from listeners on the idea and
Speaker 1: so forth. So if you never listen to the original
Speaker 1: Boltzman Brain episode from back in March, you might want
Speaker 1: to go back and check that one out first. But
Speaker 1: I'd say it's not strictly necessary if you want to
Speaker 1: jump right in with us here if you did listen,
Speaker 1: but you're you're a little bit fuzzy on the concept
Speaker 1: because it's been a little while. Don't worry. We will
Speaker 1: give you a brief refresher on the Boltzman brain argument. Yeah,
Speaker 1: and if you're not even sure what the deal is
Speaker 1: with infinity, I'll try and roll through some of the
Speaker 1: basics there as well. But let's get back to these brains,
Speaker 1: these marvelous floating space brains that could destroy us. All. Okay, So,
Speaker 1: the standard Boltzman brain argument, as we explored the last time,
Speaker 1: states that if you assume yourself to be a typical observer,
Speaker 1: which you should, right, why wouldn't you assume yourself to
Speaker 1: be typical? Yeah, well, there's a danger in assuming that
Speaker 1: you were a privileged observer. Right. In fact, by definition,
Speaker 1: if you assume yourself to be a typical the odds
Speaker 1: are that you're wrong. Right, that's the definition of what
Speaker 1: it means to be a typical. If you assume yourself
Speaker 1: as you should, to be a typical observer, it is
Speaker 1: more likely that you are something like a disembodied hallucinating
Speaker 1: brain that randomly fluctuated into existence in empty space, floating there,
Speaker 1: hallucinating your life, your memories, your current sense experience, and
Speaker 1: all of that kind of stuff. It's more likely that
Speaker 1: you're one of those than that you're a normal mammalian
Speaker 1: organism that evolved on a rocky planet. This random isolated
Speaker 1: floating brain in space is known as the Boltzman brain,
Speaker 1: and normal conscious organisms that exist through biochemical evolution, like
Speaker 1: we assume ourselves to be, are called in these arguments
Speaker 1: usually ordinary observers. So you've got these these two different
Speaker 1: types of beings. You could be right and techly, you
Speaker 1: wouldn't know the difference whether you were one or the other. Yeah,
Speaker 1: And if you really stop and think long and hard
Speaker 1: about either possibility, they're they're they're both. They both feel
Speaker 1: kind of far fetched, you know, like they're they're they're
Speaker 1: kind of equally believable and equally fantastic, because it's just
Speaker 1: the the O O argument here that I'm just this
Speaker 1: mammal on this rock and I'm thinking about thinking is crazy.
Speaker 1: That's the core of the show. I mean, we spend
Speaker 1: most of our time on stuff to blow your mind,
Speaker 1: exploring the weirdness of what it means to be a
Speaker 1: mammal that lives on a rock floating in space, even
Speaker 1: though that's what all our best science tells us we
Speaker 1: actually are. Yeah, but according to the Boltzman brain argument,
Speaker 1: you should believe that it's more likely that you're actually
Speaker 1: a brain floating in space hallucinating everything about your life
Speaker 1: then that you're one of these creatures living on a
Speaker 1: rocky planet. And the way it works is pretty simple.
Speaker 1: You look at different models of the universe and you
Speaker 1: roughly calculate how many of each type of being you'd
Speaker 1: expect to find given the model of the universe that
Speaker 1: you're looking at. Now, if you're looking at our universe,
Speaker 1: you might still think, well, how could this be a
Speaker 1: place where there are more random space brains than animals
Speaker 1: on rocky planets? After all, there are at the very
Speaker 1: least billions of ordinary observers alive today here on Earth,
Speaker 1: and there have been billions more in the past, and
Speaker 1: there hopefully will be billions more in the future. And
Speaker 1: that's just Earth. I mean, we could think about millions,
Speaker 1: billions of other planets out there that are full of
Speaker 1: ordinary observers. To all those aliens, and so we have
Speaker 1: evidence of at the very very least billions of ordinary observers,
Speaker 1: and we don't have evidence of a single random brain
Speaker 1: floating in space assembled out of random particles ever existing
Speaker 1: in the history of the universe. So how could these
Speaker 1: random space brains out number normal evolved organisms That, on
Speaker 1: the face of it doesn't seem to make any sense, right,
Speaker 1: But this is where our old friend time comes in,
Speaker 1: and time, as we know kind of us, is up
Speaker 1: our ideas of probability with all kinds of things. Like
Speaker 1: if somebody tells you, well, I was drawing poker hands
Speaker 1: and I just drew five royal flushes in a row,
Speaker 1: you wouldn't believe them because that's impossible, that that will
Speaker 1: never happen in the history of Earth. But if you
Speaker 1: also know that they had been attempting to draw hands
Speaker 1: for I don't know, tend to the ten to the
Speaker 1: ten to the ten to the ten years, then okay, sure, yeah,
Speaker 1: it becomes a little more believable. Now. Right now, our
Speaker 1: best looking scientific model of the universe, the one that
Speaker 1: the most evidence today seems to be pointing toward, is
Speaker 1: the lambda cold dark matter model. And this model includes
Speaker 1: everything we know about the past and present, so it's
Speaker 1: got the idea that our local universe is about thirteen
Speaker 1: point eight billion years old, that it began in an
Speaker 1: incredibly hot, dense state, and then for about the past
Speaker 1: thirteen point eight billion years, it's been expanding and cooling.
Speaker 1: And if we look at what this universe is doing
Speaker 1: and what it's made of right now, we can actually
Speaker 1: make pretty decent predictions about what it's going to do
Speaker 1: in the future. And it appears that what the universe
Speaker 1: will do in the future is that it will continue
Speaker 1: to do what it's doing. It will expand and cool,
Speaker 1: and entropy will steadily increase, meaning that order is going
Speaker 1: to tend to disorder, and specialness and uniqueness will tend
Speaker 1: to unspecialness and equilibrium. An energy that can be used
Speaker 1: to do work will turn into useless ambient heat that
Speaker 1: can't do anything, and stars are going to use up
Speaker 1: their fuel and burn out and they'll collapse into black holes,
Speaker 1: and black holes eventually themselves will dissipate due to hawking radiation,
Speaker 1: and everything will just run down and cool and even out,
Speaker 1: until the universe eventually becomes a vast, undifferentiated, cold grave
Speaker 1: of thermal energy, no stars, no planets, no animals, and
Speaker 1: no ordinary observers going on into the future for more
Speaker 1: time than you can imagine, and it sounds like a
Speaker 1: grim outcome for things. But again, we're talking about very
Speaker 1: vast measurements of time. Yeah, I mean, you don't personally
Speaker 1: really need to worry about this. This is so many
Speaker 1: billions of years away that it's not going to be
Speaker 1: a problem for you. Yeah, it's more of a blow
Speaker 1: to one's sort of a loose worldview than it is
Speaker 1: an actual existential threat to you specifically. Yes, but I
Speaker 1: would guess more worldviews are actually compatible with this model
Speaker 1: of things, then you would guess at first blush. Well,
Speaker 1: I mean, I guess it certainly lines up with Ragnarok,
Speaker 1: but sure. But also, I mean, people it can feel
Speaker 1: kind of depressing when people say, like, oh, you mean,
Speaker 1: like human civilization can't exist forever. At some point all
Speaker 1: the usable energy would run out and we'd have to
Speaker 1: go extinct somewhere out there in the dark. Well, yeah,
Speaker 1: technically that does appear to be physically true, but on
Speaker 1: the time scales we're talking about, human civilization wouldn't be
Speaker 1: human civilization anymore. It would be some kind of thing
Speaker 1: evolved so far beyond what human civilization is now that
Speaker 1: you wouldn't recognize it at all. We would be will
Speaker 1: be so pretentious by that point that we're able to
Speaker 1: to glimpse these future post humans would say, oh, screw
Speaker 1: those guys often too, the twilight uh entropy with them. Well,
Speaker 1: I mean, if existence is endless variation on change, eventually
Speaker 1: one of the changes worth exploring might be the change
Speaker 1: of not existing. Yeah yeah, or or kind of to
Speaker 1: to invoke Ian and banks of sublime ng and taking
Speaker 1: this kind of alternate mode of existence that that that
Speaker 1: comes after you've you've you've had your full shot at
Speaker 1: the imperial conquest game. I like that sublime NG. That's nice.
Speaker 1: It's like a chemical process almost. Yeah. That in in
Speaker 1: his um uh constructed universe of the of the of
Speaker 1: the culture. That's what you see these super advanced elder
Speaker 1: civilizations doing. Uh. They they don't destroy themselves. They reach
Speaker 1: a point where they just sublime and they they just
Speaker 1: kind of leave everything that they've built behind. Well, as
Speaker 1: more and more of our existence tends to be uh
Speaker 1: trending toward being less physical and being more encoded as
Speaker 1: digital information, one wonder is if the ultimate transcendence is
Speaker 1: just to sort of like become a a radiation imprint
Speaker 1: on the background of the universe. Yeah, I mean that's
Speaker 1: kind of a beautiful afterlife in its own right. But anyway,
Speaker 1: back to Boltzmon brains. Um, So what what? What? What
Speaker 1: happens when the universe just goes on and on and
Speaker 1: on like that for periods of time you can't even
Speaker 1: begin to comprehend. Well, here's where the weird statistical argument
Speaker 1: comes in, that you're more likely to be a space
Speaker 1: brain than one of the kinds of creatures we're pretty
Speaker 1: sure we actually are. During all that vast time in
Speaker 1: the future, random fluctuations in this empty universe will occasionally
Speaker 1: depart from this dead equilibrium before returning to it again. Now,
Speaker 1: how will that happen? Well, Sean Carroll, the physicist who
Speaker 1: has written a good bit about Boltzmon brains, who we
Speaker 1: quoted in the last episode, will continue to quote today.
Speaker 1: He writes about this in his paper why Boltzmann Brains
Speaker 1: are Bad. Just to give one example of what's going
Speaker 1: on here, so he says, quote, there can be collisions
Speaker 1: between rare high energy photons or gravitons, which could pair
Speaker 1: produce electrons and positrons or protons and anti protons and
Speaker 1: so forth, and the general tendency of such pairs would
Speaker 1: be to re annihilate rather quickly. They would annihilate each other,
Speaker 1: he writes. But occasionally the new particles will have enough
Speaker 1: momentum to travel far apart from each other. Sometimes rarely,
Speaker 1: as should henceforth be understood, many such collisions will happen nearby,
Speaker 1: producing enough nearby matter to assemble itself into a macroscopic
Speaker 1: object such as a brain. Now again that sounds that's
Speaker 1: so improbable. So you're just saying random particles that are
Speaker 1: getting pair produced out there in space by random events
Speaker 1: are going to collect into enough atomic matter that they
Speaker 1: would form an object like a brain, like a specified
Speaker 1: object like that. Again, that is super improbable. If you're
Speaker 1: thinking that would never happen, you're right, Except you're never
Speaker 1: is not including enough time, and including enough time, it
Speaker 1: actually would happen. As I think we mentioned in the
Speaker 1: last episode, one is instantly reminded, of course, of the
Speaker 1: monkeys pounding on the typewriters to produce the complete works
Speaker 1: of William Shakespeare. Right, and in given normal time, that
Speaker 1: will never happen. But given enough time, if you just
Speaker 1: stretch the t variable out to arbitrary lengths that actually
Speaker 1: will happen. It's almost guaranteed to happen. But in fact,
Speaker 1: Carol writes that it it doesn't really matter exactly what
Speaker 1: the process of producing brains is as long as you've
Speaker 1: got two assumptions. And these two assumptions that give us
Speaker 1: the Boltzman Brand universe are that the universe either last
Speaker 1: forever or last for an extraordinary long time, much longer
Speaker 1: than ten to the ten to the sixty six power years,
Speaker 1: and that is a long long time. And then the
Speaker 1: second criterion is that it undergoes random fluctionations that could
Speaker 1: potentially create conscious observers. Now, as long as those conditions
Speaker 1: are met, randomly assembled brains or conscious computers or whatever
Speaker 1: other type of conscious agent you want to imagine, will
Speaker 1: eventually become more numerous in the history of the universe
Speaker 1: than normal conscious animals like us. And so Carol points
Speaker 1: out in his paper that the lambda cold dark matter model,
Speaker 1: the best current scientific model of the universe, looks like
Speaker 1: it fulfills exactly those criteria. It's laws of physics are
Speaker 1: going to allow random fluctuations that can randomly assemble particles
Speaker 1: into macroscopic objects. Like dust, like rocks, stars, planets, but
Speaker 1: also lobsters, blue rays of Jean Claude Van Damn, movies, uh,
Speaker 1: Christoph Lambert, even human brains. And it appears that the
Speaker 1: universe will go on existing for so long that there
Speaker 1: will be an unbelievably huge number of opportunities for random
Speaker 1: fluctuation objects like this to come and do existence. And
Speaker 1: so this gives us the argument there's going to be
Speaker 1: more time out there for random brains to as symbol
Speaker 1: out of particles in the void than there is going
Speaker 1: to be opportunities for ordinary observers to exist in the
Speaker 1: time space of the universe where there's usable energy to
Speaker 1: power things like stars and planets and evolution. Our evolution
Speaker 1: window is actually extremely small compared to the whole life
Speaker 1: of the universe, a lifespan that is so vast that
Speaker 1: from our finite perspective it might as well be infinite.
Speaker 1: And for all we know, we can't prove that it's
Speaker 1: not infinite. But of course, the question of what it
Speaker 1: would mean for time to be infinite is I guess
Speaker 1: kind of Uh, that's a vexing question itself, right, what
Speaker 1: would that mean in reality? All? Right, Well, let's take
Speaker 1: a quick break, and when we come back we will
Speaker 1: discuss the identity of infinity. Than alright, we're back, all right,
Speaker 1: So today in the podcast, we're gonna be discussing the
Speaker 1: concept of infinity and infinite sets, since this is where
Speaker 1: I made an erroneous statement in the first time we
Speaker 1: covered Boltzmann brains. The last time, we were in the
Speaker 1: middle of trying to illustrate the idea that it can
Speaker 1: be proved that some infinities are larger than others. Because
Speaker 1: if you've got a universe going on and on in
Speaker 1: an equilibrium, how can you compare how numerous two different
Speaker 1: sets of objects within them, like Boltzmann brains and ordinary
Speaker 1: observers are. If it just keeps going on, you'd want
Speaker 1: to have some way of saying, well, even though it
Speaker 1: keeps going on, one group within the universe is bigger
Speaker 1: than the other group within the universe. And so the
Speaker 1: idea here is that you look at relative frequencies. Right,
Speaker 1: So here's what I said last time. I said, some
Speaker 1: infinities are larger than others. And the example I gave
Speaker 1: was comparing two sets of integers. Those sets were even
Speaker 1: numbers everything divisible by two and numbers divisible by seven.
Speaker 1: So even though you could keep naming items in each
Speaker 1: of these sets forever, I claimed that there were more
Speaker 1: even numbers the numbers divisible by seven. And that seems
Speaker 1: quite intuitive, right, I mean, it made sense to me
Speaker 1: when I said it, and I thought I had actually
Speaker 1: read that in the past. But that turns out to
Speaker 1: be not true. Now on a sort of face value level,
Speaker 1: it's kind of hard to see how that's not true. Right,
Speaker 1: Even numbers happen more often if you just keep counting
Speaker 1: up the number line, you will hit even numbers more
Speaker 1: often than you hit numbers divisible by seven. Yeah. I
Speaker 1: think I kind of played off of this by saying
Speaker 1: with that Connor McCloud from Highlander, if he lived forever,
Speaker 1: he's going to cut off heads and he's going to
Speaker 1: use the bathroom, and he is going to use the
Speaker 1: bathroom more than he cuts off heads, right, right, And
Speaker 1: so that was that I had said, Yeah, when I
Speaker 1: when I when I take your example and I line
Speaker 1: it up with my Highlander heavy example, it seems to
Speaker 1: be true in both cases. We were right until you
Speaker 1: invoke the concept of infinity, and if you actually say
Speaker 1: this process extends infinitely, then everything gets ruined. Everything gets
Speaker 1: jammed up because, believe it or not, if the process
Speaker 1: mathematically were to go on for infinite time, and each
Speaker 1: set of events happens infinitely, then it's provable by something
Speaker 1: called set theory that you can show that those sets
Speaker 1: are of equal size. And that might be hard to understand,
Speaker 1: but we will explain that. It comes down to the
Speaker 1: work of a German mathematician named gay Org Canter that
Speaker 1: there are ways in which some infinite sets that seem
Speaker 1: like they should be of different sizes are actually the
Speaker 1: same size. And yet at the same time there actually
Speaker 1: are different sizes of infinite sets those so some infinities
Speaker 1: really are bigger than others, just not in the example
Speaker 1: that I gave of of integers. And I guess it's
Speaker 1: more arguable when you're talking about events like what Connor
Speaker 1: McCloud does, because then you're talking about things happening in
Speaker 1: the physical world, where the concept of infinity gets even
Speaker 1: more complicated. What would it mean for events to be
Speaker 1: recurring infinitely? Well, I think the problem is that by
Speaker 1: by dragging infinity into it, we're basically dragging God into
Speaker 1: the scenario, right, and you know how God reacts to
Speaker 1: mortals trying to to boss uh it around. Well, you know,
Speaker 1: infinity has historically been a concept of that's very, very
Speaker 1: wrapped up in ideas about theology and religion and the
Speaker 1: universe and the afterlife. Uh I just mentioned Georg Cantor.
Speaker 1: I mean, Georg Cantor had a lot of strange religious
Speaker 1: beliefs that kind of read to us as eccentric today,
Speaker 1: deeply wrapped up in his mathematical discoveries about the nature
Speaker 1: of infinity. Yeah. Yeah, the more you look at it,
Speaker 1: the more intertwined they are. Well, I guess we should
Speaker 1: come back to Cantor in a bit and talk specifically
Speaker 1: about the types of infinity that he dealt with, but
Speaker 1: more generally, maybe let's look at the concept of infinity
Speaker 1: and in cultures throughout history. All right, Yeah, so this
Speaker 1: is it's really fascinating when you think about it, because
Speaker 1: we obviously as humans live limited lives, finite lives of
Speaker 1: finite observation within a finite world. Yet we advanced to
Speaker 1: think beyond these limitations and in doing so envision to
Speaker 1: some degree infinity, or at least two toy with the
Speaker 1: concept of something being infinite. In fact, through the use
Speaker 1: of language. In particular, we used a to quote our
Speaker 1: old friend Julian James, finite set of terms that by metaphor,
Speaker 1: is able to stretch out over an infinite set of circumstances. Oh,
Speaker 1: I mean that's essentially the library of Babel, right. I
Speaker 1: mean that you have a finite set of encoding features.
Speaker 1: But those bits of code can make every statement in
Speaker 1: the world. Yeah, and we alone, of all the animals,
Speaker 1: are able to stand in wonder and horror of infinity.
Speaker 1: You look over your cat, your dog, you check out
Speaker 1: the smartest dolphin in the sea. They don't. They don't
Speaker 1: have any concept of infinity, and maybe they're happier for it. Well,
Speaker 1: I mean you might wonder if they have some kind
Speaker 1: of intuitive concept, not of in infinity, but of the
Speaker 1: idea of recurrence. There are different ways of picturing something
Speaker 1: is infinite. One of the things that I often think
Speaker 1: about is that if you go back into ancient times,
Speaker 1: you'll see these different ways of imagining, say time, even
Speaker 1: if you had the idea that there's not necessarily a
Speaker 1: beginning and end of time. Of course, many religions have that.
Speaker 1: They are totally different ways of emphasizing how that lack
Speaker 1: of beginning and end works. Is there a boundless infinity
Speaker 1: where time stretches all the way back forever and goes
Speaker 1: all the way into the future forever, or is there
Speaker 1: a bounded infinity where time endlessly repeats like along a
Speaker 1: loop that never stops going. You know, one of our
Speaker 1: probably the you know, the earliest ideas that gets tied
Speaker 1: up and eventually becomes adfinity is that, of course, of
Speaker 1: the ocean. And we still invoke this, uh, the old saying.
Speaker 1: You know, somebody breaks up with their high school breaks
Speaker 1: up with his girlfriend. He's he's he or she they're
Speaker 1: they're distraught about it. What's the nuggeta whizd them that
Speaker 1: is leveled at them? They're always more fish in the sea.
Speaker 1: But there's a finite number of fish in the sea, right,
Speaker 1: But there are so many fish that it's almost a
Speaker 1: kind of infinity, like there's a there's a boundlessness to
Speaker 1: it um And likewise, just the idea of the ocean
Speaker 1: itself is certainly a finite realm, but it is so vast,
Speaker 1: it is so hard, especially for for ancient people to
Speaker 1: comprehend that it was considered almost a kind of boundless realm.
Speaker 1: It was and it was tied up in these notions
Speaker 1: of primordial chaos. Well, it's one of those pseudo effective
Speaker 1: concepts that's useful to us in mathematics in the same
Speaker 1: way that our lives are full of useful pseudo random numbers.
Speaker 1: Like people cannot actually generate truly random sequences of numbers.
Speaker 1: We talked about this in the Eaching episode, But you
Speaker 1: can generate what feel like pseudo random numbers. You know,
Speaker 1: it's like at least seems kind of random, even if
Speaker 1: it's not truly mathematically random. And we use that kind
Speaker 1: of inference to you know, guess things and come up
Speaker 1: with a random solutions to things all the time in
Speaker 1: our life, in the same way we use pseudo infinities
Speaker 1: as a metaphor for understanding all kinds of things. Uh.
Speaker 1: You know, the stars in the sky or the fish
Speaker 1: in the sea were constantly picturing infinities that are not
Speaker 1: truly infinite, their finite numbers of things, but they're so
Speaker 1: big they stand in for infinity, which we can't conceive.
Speaker 1: But we've tried to conceive infinity. Uh, and we've been
Speaker 1: doing so for hundreds and hundreds of years from millennia. Uh.
Speaker 1: So I wanted to run through some sort of I
Speaker 1: guess you could say great moments in UH infinite naval
Speaker 1: gazing right here, just some of the UH, just a
Speaker 1: few of the important individuals and movements. One of the
Speaker 1: the earliest that's worth talking about, UH, it has to
Speaker 1: do with the religion of Jainism. Okay, So after the
Speaker 1: decline of the Vedic religion on in in on the
Speaker 1: Indian subcontinent around a four hundred b c. To other
Speaker 1: religions rose to prominence in India. You had you have Buddhism,
Speaker 1: which we've talked about on the show before, as well
Speaker 1: as Jainism. And it's a jain is um is really
Speaker 1: its own distinct religion. It does involve some uh properties
Speaker 1: and concepts that pop up in Hinduism or Buddhism, but uh,
Speaker 1: you shouldn't think of it as just a meteor offshoot
Speaker 1: of Hinduism or Buddhism. Its name derives from the Sanskrit
Speaker 1: verb to conquer. That's interesting because I tend to think
Speaker 1: of Jainism as embodying non violence to an incredible extent. Well, yes,
Speaker 1: but the thing you're conquering. Think of it in terms
Speaker 1: of conquering one's passion. It's the internal struggle. So James
Speaker 1: don't hold up a traditional founder. Uh. They have a
Speaker 1: number of key teachers that are called turth and Carras
Speaker 1: or ford Makers. Uh. So, for instance, UH, there was
Speaker 1: an important one named part of an author who may
Speaker 1: have lived in the seventh century BC, and he would
Speaker 1: have been the twenty three of these ford makers. But
Speaker 1: then the twenty four and last, Uh, turthen Cara was
Speaker 1: a man by the name of a vard Anna and
Speaker 1: known by the title Maja Vira or great Hero, and
Speaker 1: he would have been a contemporary of the historic Buddha.
Speaker 1: It was during his lifetime and the years immediately to
Speaker 1: follow some of the greatest contributions to Jane mathematics were made. Uh.
Speaker 1: And Jane philosophy is big into pondering the enumeration of
Speaker 1: large numbers. They classified numbers into three categories, innumerable, innumerable,
Speaker 1: and infinite. In fact, Jainis m recognizes five different forms
Speaker 1: of infinity. There's infinite in one and in two directions,
Speaker 1: there's there's infinite in area, there's infinite everywhere, and there's
Speaker 1: infinite perpetually. Infinity is also tied up in their metaphysics
Speaker 1: of the metaphysics metaphysics of the Jains soul. There are
Speaker 1: an infinite number of souls in the universe and liberated
Speaker 1: souls or siddhas, which are free from the cycle of
Speaker 1: samsara uh. They have in finite knowledge, infinite vision, infinite power,
Speaker 1: and infinite bliss. Now I wonder what it would mean
Speaker 1: to have infinite knowledge. This is something that's always been
Speaker 1: kind of interesting to me about the idea of omniscience.
Speaker 1: It seems to me that the idea of omniscience inherently
Speaker 1: invokes or involves things that appear to be contradictions in
Speaker 1: the same way that omnipotence does. Right, So people classically say, oh,
Speaker 1: if you've got an omnipotent soul or an omnipotent being,
Speaker 1: could it create a rock that it itself could not lift?
Speaker 1: You know that that's the classic one. The same thing
Speaker 1: would come through with omniscience, meaning could an omniscient being
Speaker 1: with all knowledge know what it is like to not
Speaker 1: know things well? Or would you know things if to
Speaker 1: be complete knowledge? Is that maybe just a complete absence
Speaker 1: of knowledge in a way, like you if you try
Speaker 1: and imagine a person that is defined, but you know
Speaker 1: what we have finite knowledge, finite vision, finite power, it's
Speaker 1: certainly finite bliss. If you ramp all those things up
Speaker 1: to the infinite, then do we just bleed into the
Speaker 1: fabric of reality? We vanished completely? We I mean maybe
Speaker 1: that that is that's the ultimate liberation right there. Well,
Speaker 1: all of these concepts that are being taken to the
Speaker 1: transcendent level of infinity here, like vision, power, bliss, knowledge,
Speaker 1: they're all in a way kind of defined by the
Speaker 1: bounds of limitation. What what would it mean to see everything? Then?
Speaker 1: Are you even really seeing? I mean seeing is an
Speaker 1: act of like focusing and an act of perceiving, And
Speaker 1: so if it's everything, I don't know what what it
Speaker 1: would necessarily mean. What is bliss of? Is it if
Speaker 1: it is experienced outside of the contrast with with suffering, right,
Speaker 1: or even something that's slightly less than bliss. Now I
Speaker 1: want to be clear, I'm not trying to like rag
Speaker 1: On Jane metaphysics or whatever. I mean. These are kind
Speaker 1: of beliefs that appear in all kinds of religions, but
Speaker 1: I think they highlight some of the inherent qualities of
Speaker 1: contem lating the infinite that makes it seem wholly and
Speaker 1: kind of mind boggling to begin with. It's exactly these
Speaker 1: types of contradictions that make it so attractive as an idea. Yeah, certainly.
Speaker 1: And of course the Greek scrappled with infinity as well.
Speaker 1: In fact, if we look back to the to the
Speaker 1: the ideas of an Aximander of Militiasts who lived six
Speaker 1: ten through five b c. Uh, he's considered the the
Speaker 1: earliest Greek philosopher to commit his ideas to writing. The
Speaker 1: only fragments of his work remain, and he introduced the
Speaker 1: principle of the apron or boundless, and this is uh
Speaker 1: what he would discussed as the original state of the universe.
Speaker 1: And he's certainly describing infinity that you can also see
Speaker 1: clear the clear influence of the Greek cosmological concept of
Speaker 1: primordial chaos here as well. Oh yeah, and this would
Speaker 1: be this would be an important concept that that subsequent
Speaker 1: Greek philosophers would toy with as well. One of the
Speaker 1: big names is of or says Zeno of Elia, who
Speaker 1: lived four ninety through b c. And he played with
Speaker 1: this concept of the apron uh, and he found that
Speaker 1: infinities breed paradoxes such as the fabulous paradox of Achilles
Speaker 1: and the tortoise. Uh and uh, this this is a
Speaker 1: this is a fun little thought experiment that I think
Speaker 1: will that lines up rather nicely with the Boltzmann brains
Speaker 1: thought experiment, because the idea here is that you have
Speaker 1: a tortoise and then you have the the the god
Speaker 1: blessed um hero Achilles. And the tortoise says, hey, I
Speaker 1: would like to challenge you to a foot race, and
Speaker 1: Achilles is like, what are you doing. I'm gonna I'm
Speaker 1: gonna of course, I'm gonna beat you. I am Achilles. Uh,
Speaker 1: you know, nothing stands in my way, and I'm certainly
Speaker 1: going to outrun a tortoise. And the tortoise says, well, uh,
Speaker 1: I would need a head start, of course, because I
Speaker 1: am a tortoise and you are Achilles. But I'll still
Speaker 1: beat you if you just give me a reasonable head start,
Speaker 1: and they agree on a head start something to the
Speaker 1: effect of like ten feet or so. There's nothing crazy,
Speaker 1: and Achilles is like, well, I'm still gonna be you,
Speaker 1: don't I don't understand, and the tortoise explains, well, think
Speaker 1: of it this way, You're gonna have to catch up
Speaker 1: with me. Before you pass me, and Achilles says, sure, yeah,
Speaker 1: I'm gonna have to I'm gonna have to run to
Speaker 1: where you start. And then the tortoise says, but by
Speaker 1: the time you reach my starting point, I'll be ahead
Speaker 1: of you, and uh, and then you're gonna have to
Speaker 1: catch up with where I've gotten to. And and he keeps,
Speaker 1: you know, carrying this out one step further, one step further,
Speaker 1: until he convince his Achilles that he can never catch
Speaker 1: up with the tortoise, and a Kelly's just concedes the
Speaker 1: whole race, and the tortoise wins. I remember reading about
Speaker 1: this paradox when I was younger, and I love stuff
Speaker 1: like this. It always like, I mean, it seemed right.
Speaker 1: It's one of those things that seems very true in
Speaker 1: the same way that it certainly seems like there must
Speaker 1: be more even numbers than there are than there are
Speaker 1: numbers divisible by seven, right, Yeah, yeah, I mean, like
Speaker 1: another version of this would be imagine how long it
Speaker 1: would take you to walk across a basketball court. Yeah, well,
Speaker 1: first you have to walk halfway across it. Before you
Speaker 1: reach that you gotta walk halfway across that distance, and
Speaker 1: then halfway across that distance. So you can take something
Speaker 1: finite and if you divide it up into infinite portions,
Speaker 1: then something that is very possible seems impossible. Uh. One
Speaker 1: of the take homes is that, according to the argument
Speaker 1: made by the tortoise, movement is impossible. Yeah, exactly. Now
Speaker 1: to move the analogy over to the Boltzmon brain argument,
Speaker 1: I would say, actually, if you wanna say, like the
Speaker 1: tortoise represents Boltzman brains and Achilles represents ordinary observers, all
Speaker 1: you don't really have to do is say, okay, tortoise
Speaker 1: does not get ahead, start, they start at the same time.
Speaker 1: Achilles runs, say a thousand times faster than the tortoise,
Speaker 1: and they both start the race and they just go.
Speaker 1: Except the difference is the tortoise is immortal, all right,
Speaker 1: So Achilles is gonna win the early part of the
Speaker 1: race by a long shot. Like tortoise isn't gonna come
Speaker 1: close until Achilles gets really exhausted and dies. I know,
Speaker 1: if if only, if only he was full God, he
Speaker 1: might have a shot. But then you've got eternity for
Speaker 1: the tortoise to just keep walking out ahead. Yeah, slow
Speaker 1: and steady right right, wins the race. Uh. The Great thinkers,
Speaker 1: of course, tackled infinity as well. Aristotle three twenty two
Speaker 1: BC he considered the the apperon as well. Um. Yeah,
Speaker 1: here he is in physics talking about the boundless quote.
Speaker 1: Everything has an origin or is an origin. The boundless
Speaker 1: has no origin, for then it would have a limit. Moreover,
Speaker 1: it is both unborn and immortal, being a kind of origin.
Speaker 1: For that which has become has also necessarily an end,
Speaker 1: and there is a termination to every process of destruction.
Speaker 1: Another thinker of note, h Thomas Aquinas, came along twelve
Speaker 1: seventy four. He focused on the quality of existence rather
Speaker 1: than the quantity. Of course he didn't uh. So he considered,
Speaker 1: you know, God is infinite in quality more so than
Speaker 1: in quantity. So he saw infinity as a mode of
Speaker 1: existence and identified a separation between mathematical infinity and religious infinity.
Speaker 1: So we see a curious principle emerge. Even an infinite
Speaker 1: God cannot create an infinite object. He talks about this
Speaker 1: at length in his work Assuma Theologica, uh, sort of
Speaker 1: taking both sides, like saying, all right, well if it
Speaker 1: if God is infinite, then X. If God is fine eye,
Speaker 1: then why But I think he says that nothing except
Speaker 1: God can be infinite, right right, Well, that kind of
Speaker 1: point of view is going to hold some sway for
Speaker 1: a while. Um. Another individual worth noting Nicholas of Cusa
Speaker 1: fourteen o one through fourteen sixty four. He argued that
Speaker 1: everything is within the infinite. The world itself must be
Speaker 1: within God. So he used mathematical examples to describe the
Speaker 1: relationship ship between God and the world. There's no circumference
Speaker 1: to God, and the center is everywhere. This is the
Speaker 1: concept of a maximum God, unlimited, transcendent, and also unreachable
Speaker 1: and unfathomable to a species that is defined by its
Speaker 1: finite limitations. So this say, is starting to sound kind
Speaker 1: of like the like God is the universe type belief.
Speaker 1: Then you have Spinosa comes around two through seven and
Speaker 1: he says, if God is infinite, then God is the
Speaker 1: one substance. The substance must have infinite attributes, and we
Speaker 1: must be modes of this one entity. So God is
Speaker 1: not the not a personal God. God and Nature are
Speaker 1: one and the highest ethic, according to Spinoza, is to
Speaker 1: live in accordance with the laws of nature, to be
Speaker 1: part of the infinite. So the ideas that were beings
Speaker 1: of mind and body, both of which are composed of
Speaker 1: this universal substance. That's an interesting idea that if if
Speaker 1: one thinks of God as infinite, then how could there
Speaker 1: be things that weren't God? Right? Because wouldn't God necessarily
Speaker 1: encompass those things? If God had no limit and went
Speaker 1: on forever, what what could be outside of him that
Speaker 1: would seem to suggest that there was a limit to him? Yeah? Yeah,
Speaker 1: what could what could be outside of the absolute, the
Speaker 1: maximum God? Yet again, I feel like infinity is one
Speaker 1: of those things where you start playing with language in
Speaker 1: a way, it's kind of a game that you you
Speaker 1: start using certain words thinking you know what they mean.
Speaker 1: But when you use words like infinity or everything or
Speaker 1: forever thinking you know what they mean, they end up
Speaker 1: kicking up conclusions that you couldn't expect because you you
Speaker 1: contemplate them more and more deeply all the time. Uh,
Speaker 1: and they tend to transcend the ways you originally invoked them.
Speaker 1: Does that make any sense? Yeah? Well, I mean I
Speaker 1: think that the big one is. Of course, it's one
Speaker 1: thing to talk about infinity in terms of just pure philosophy,
Speaker 1: but then when you start lining it up with mathematics, right,
Speaker 1: and when you start bringing the raw numbers in and
Speaker 1: crunching those numbers. Uh, that's where you get into some
Speaker 1: of these these real conundrums. That's where you you you
Speaker 1: hear these arguments of someone saying, well, you're talking about infinity,
Speaker 1: but you're not talking about mathematical and infinity. Because here's
Speaker 1: what happens when you throw the numbers in, right, And
Speaker 1: this brings us back to Gayard Cantor. So Gayard Canter
Speaker 1: was a I think I mentioned this earlier, but he
Speaker 1: was a German mathematician. He was born in Russia in
Speaker 1: eighteen forty five and he lived until nineteen eighteen, and
Speaker 1: he is the main person responsible for modern set theory
Speaker 1: and the theory of what are now called transfinite numbers,
Speaker 1: which was revolutionary and very controversial in its time, but
Speaker 1: in many ways is widely accepted now and has proven
Speaker 1: extremely useful. So in writing about infinity, Cantor made use
Speaker 1: of the idea of sets. Sets are both simple in
Speaker 1: the core principle and extra ordinarily complicated and powerful as
Speaker 1: a tool for mathematical reasoning. And so set theory is
Speaker 1: basically just it's a framework for grouping items into sets.
Speaker 1: So you've got some items, you could group them together,
Speaker 1: and then you've got a set of items, and you
Speaker 1: can treat that set as a thing. And one of
Speaker 1: the things set theory does is that it gives you
Speaker 1: a way of comparing the size of sets of things
Speaker 1: by matching the items in those sets in a one
Speaker 1: to one pairing off process. And the size of a set,
Speaker 1: meaning how many items it contains, is known as its cardinality. No,
Speaker 1: so I know that's a lot of terminology. We want
Speaker 1: to try to avoid getting too abstract here. But basically,
Speaker 1: if you think, okay, I've got a set with uh
Speaker 1: five objects in it, my five fingers on my right hand,
Speaker 1: and then I've got another set that has all my
Speaker 1: fingers on my left hand, are those sets equal in size? Well?
Speaker 1: I can check. I mean, obviously I can tell because
Speaker 1: I can count to five. But even if I couldn't
Speaker 1: count to five, I can check by pairing off the
Speaker 1: items in each set and seeing if they pair up
Speaker 1: in the same extent, or do I have leftover items
Speaker 1: in one set that can't go with the other. Now
Speaker 1: I have five fingers on each hand, so yes, I
Speaker 1: can pair up the sets. They match up all right,
Speaker 1: So you would say that the cardinality or the size
Speaker 1: of both of these two sets of fingers on each
Speaker 1: hand is five, and the sets are equal in size.
Speaker 1: But now let's go to the example that I that
Speaker 1: I got wrong in the last boltzmon Brain podcast. So
Speaker 1: that would be looking at the sets of all even
Speaker 1: numbers in one set and all numbers divisible by seven
Speaker 1: and the other set. Even though if we just count
Speaker 1: up the natural number line, we hit way more even
Speaker 1: numbers the number is divisible by seven. Cantor could show
Speaker 1: that the cardinality of these two sets is exactly the
Speaker 1: same because each item in each set can be paired
Speaker 1: one to one with an item from another set. So
Speaker 1: imagine you've got set A that's even numbers, and then
Speaker 1: you've got set B that's numbers divisible by seven. Well,
Speaker 1: let's pay off the first items in each set. You've
Speaker 1: got Set A is to set B is seven, and
Speaker 1: then the second item in each set that's Set A
Speaker 1: is four, set B is fourteen. Here's the question, what
Speaker 1: would stop you from counting forever this way? Would you
Speaker 1: ever hit a number in set A that did not
Speaker 1: have a corresponding number in set B. Clearly you wouldn't
Speaker 1: you can match them off until the end of time.
Speaker 1: Uh though, actually, if the end of time comes and
Speaker 1: stops you from counting more, then suddenly the even numbers
Speaker 1: set is much bigger. But because you know there's more frequency,
Speaker 1: you hit even numbers more often. But if you grant
Speaker 1: that they're infinite and you don't have to stop and
Speaker 1: tallly them up, but you treat them as ongoing sets,
Speaker 1: then they are in fact demonstrably the same size. Okay,
Speaker 1: I'm with you. I'm surprised you are, because that's it's
Speaker 1: messing with my brain. I mean, that seems wrong, right.
Speaker 1: There can't be the set of even numbers and the
Speaker 1: set of numbers visible by seven cannot be the same size.
Speaker 1: There's clearly more of one than the other. But cantor
Speaker 1: can prove that they're the same size. It's because we've
Speaker 1: invoked we've invoked infinity, and that changes things. It messes
Speaker 1: everything up. Suddenly all your intuitions go out the window
Speaker 1: and nothing makes sense anymore. Our listener Jim in New Jersey,
Speaker 1: who is a great, great email writer. He's always been
Speaker 1: hearing from from Jim for years. Uh, and Jim especially
Speaker 1: on like mathematical logical computer science type type topics. Jim
Speaker 1: has really great emails. He sent us an excellent email,
Speaker 1: uh gently correcting are my mistake in the first episode,
Speaker 1: and he had a really good analogy. He said, quote,
Speaker 1: think of this as a marathon race without a finish line.
Speaker 1: The hair will always be ahead of the tortoise, but
Speaker 1: they will always pass the same mile markers, just at
Speaker 1: different times. They both cover the same distance. It's just
Speaker 1: that one racer reaches each milestone quicker. They all reach
Speaker 1: the same milestones since there are an infinite number of
Speaker 1: them an infinite time. I think that's a nice way
Speaker 1: of picturing it. Slus. We got to work the tortoise
Speaker 1: in for like a third time here. I like it.
Speaker 1: How many tortoises have we done so far? Three? I
Speaker 1: guess because we talked about the tortoise and Achilles, and
Speaker 1: then you talked about the tortoise as boltzman brain, and
Speaker 1: now we have proper tortoise and hair example. So now
Speaker 1: it's tortoise as numbers divisible by seven, whereas the hair
Speaker 1: is even numbers. Yeah. So even though the hair goes faster,
Speaker 1: they eventually go the same distance if they go forever.
Speaker 1: But this is so weird. Right, because with this type
Speaker 1: of reasoning, I don't know, you might begin to sense
Speaker 1: some deeply troubling implications. One thing is, any infinite set
Speaker 1: of things that can be counted in order is equal
Speaker 1: in size to any other. Because if you can count
Speaker 1: them in order and it's clear what the order is,
Speaker 1: you can pare them off like this. And if you
Speaker 1: can pare them off like this, you can pare them
Speaker 1: off forever equally. I mean, I just keep coming back
Speaker 1: to the idea. If you're willing to accept the albeit
Speaker 1: loose idea of something going on forever that is just
Speaker 1: going to go on into infinity, then you can you
Speaker 1: can buy the fact that these two sets are equal.
Speaker 1: I guess so, I mean it's cutting me, man, It's
Speaker 1: well be because it comes back again to the idea
Speaker 1: that we are finite beings in a finite world. That
Speaker 1: is all we've ever evolved to be, and yet we
Speaker 1: imagine things that are beyond that, and so of course
Speaker 1: it breaks our our our our normal ability to to
Speaker 1: comprehend things. Yeah, let's keep imagining. So the implications of
Speaker 1: set theory and Cantor's work are obviously extremely profound. For example,
Speaker 1: here's one thing that feels pretty obvious which set is
Speaker 1: bigger All the natural numbers, and that means all the
Speaker 1: positive integers starting with zero, so all the counting numbers,
Speaker 1: you know, zero, what's bigger that set? Or all the
Speaker 1: rational numbers, which is another way of saying anything that
Speaker 1: can be expressed as a fraction with a quotation of
Speaker 1: U with positive eneger, So like one half or two
Speaker 1: over one or one third. Which of those sets would
Speaker 1: you think would be bigger? Well, I mean obviously it
Speaker 1: would seem to invoke the Akelles and Tortoise situation, right,
Speaker 1: you would think, well, if you just start dividing everything
Speaker 1: we've been talking about up, you can just you can
Speaker 1: you can just have infinite divisions that just just you know,
Speaker 1: blows that number out of proportion. Yeah, you think, obviously
Speaker 1: the set of rational numbers is bigger, right, that includes
Speaker 1: all the fractions, because the set of fractions includes every
Speaker 1: natural number. Every everything in the first set is also
Speaker 1: in the second set. One over one, two over one,
Speaker 1: three over one, So all the natural numbers are in
Speaker 1: the set of rational numbers, but it also includes every
Speaker 1: fraction in between every natural number. So you've got one half,
Speaker 1: one third, three, seventh, and nine tenths. Those are all
Speaker 1: rational numbers. So there must be more rational numbers or
Speaker 1: fractions than there are natural numbers. Right. Wrong, Yet again,
Speaker 1: Cantor used set theory to prove that these sets are
Speaker 1: of equal size. Now you might wonder again, how could
Speaker 1: you prove that? Like, remember from a minute ago that
Speaker 1: the key to being able to show that the items
Speaker 1: in accountable infinite set are equal is that you can
Speaker 1: arrange them in accountable order. You've got to be able
Speaker 1: to order it so that you can pare them off
Speaker 1: in an orderly way, making sure you're not missing anything. Right,
Speaker 1: So how could you count fractions? Right? You can't start
Speaker 1: with the lowest fraction and count up from there. That
Speaker 1: doesn't make any sense. You know. You can't say I'll
Speaker 1: start with one tenth and then go to two tenths
Speaker 1: because you have lower numbers than that. So what can
Speaker 1: you do? Well, here was a real stroke of genius.
Speaker 1: Cantor rearranged every possible fraction into an ordered table. Robert
Speaker 1: f a picture of this table in in our notes
Speaker 1: for us here, So what are we looking at? Well?
Speaker 1: I'll have to include a link to this image on
Speaker 1: the landing page for this episode stuffable you mind dot com,
Speaker 1: But for me, the first thing I think of is
Speaker 1: that if you ever did those charts in like P
Speaker 1: class that show the volleyball rotation plan when you're playing
Speaker 1: like team volleyball, that's kind of It's like, imagine, uh,
Speaker 1: a bunch of fractions got together to play some sort
Speaker 1: of strange team sport and they needed a guide to
Speaker 1: show how they're moving around. Well, it is like that
Speaker 1: in terms of how you navigate, but it's incredibly orderly.
Speaker 1: So here's how the table works. Is very simple. You've
Speaker 1: got a top line in a sideline. The top line
Speaker 1: is numbers at the bottom of the fraction, right, so,
Speaker 1: and that goes up with every digit one to and
Speaker 1: there's a column that goes down with fractions that have
Speaker 1: that number on the bottom. And then you've got a
Speaker 1: sideline that goes down with rows that are all the
Speaker 1: integers going on for a and those integers have rose
Speaker 1: beside them, where the top number in the fraction is
Speaker 1: that number. So you can actually move through this list
Speaker 1: in a diagonal back and forth pattern that allows you
Speaker 1: to make sure you're counting every possible fraction there could be.
Speaker 1: So it starts with one over one, then two over one,
Speaker 1: then one over two, then one over three, then two
Speaker 1: two three, one four one three two, and so forth.
Speaker 1: And you actually, through this method, could make a table
Speaker 1: where it was possible to count every fraction that you
Speaker 1: never actually get to the end, but you can organize
Speaker 1: them in a way that you know you're not leaving
Speaker 1: anything out. So it's basically volleyball, because in team volleyball
Speaker 1: everyone moves so that everyone has to serve during the
Speaker 1: course of the game. Um, you know, unless the gamings
Speaker 1: are like that, let's ignore that. Uh, everything must serve,
Speaker 1: everything must be counted. And that is exactly what kills
Speaker 1: its supremacy in terms of competing with the eyes of
Speaker 1: the set of natural numbers, because remember from before, if
Speaker 1: you can count them all in an order, then you
Speaker 1: can pare them off with natural numbers like one five. So,
Speaker 1: believe it or not, it can be mathematically shown that
Speaker 1: the size of the set of natural numbers and the
Speaker 1: size of the set of rational numbers is the same,
Speaker 1: even though that makes absolutely no sense to us. Cantor
Speaker 1: was actually writing about this to a mathematician friend of
Speaker 1: his named Richard did kind uh, And he said about
Speaker 1: some of his own discoveries that you know this this
Speaker 1: type of stuff. He was coming across j evoir me
Speaker 1: jana quapa, which means I see it, but I do
Speaker 1: not believe. Al Right, hold that thought, Joe. We're gonna
Speaker 1: take a quick break and when we come back back
Speaker 1: to infinity. And we're back to infinity. Now you can
Speaker 1: use the same logic to violate your intuitions lots of
Speaker 1: other ways. There are plenty ways to stab your brain
Speaker 1: with this. So here's an obvious one, which is bigger
Speaker 1: infinity or infinity plus one. Well, on the surface, it
Speaker 1: would sound like infinity plus one because it's all that
Speaker 1: plus one extra dude, exactly. I mean you remember this
Speaker 1: from playing games as a kid, right, yeah, Well, well
Speaker 1: I am you know, I am infinity strong. Well, I'm
Speaker 1: infinity plus one. Well, I mean it's it gets into
Speaker 1: not to summon the specter of the infinity hotel. But
Speaker 1: that's kind of the concept there. You have a hotel
Speaker 1: with infinite rooms and then infinite guests are staying in it.
Speaker 1: What do you do when one extra guest shows up?
Speaker 1: Is there room in the hotel? Of course there is.
Speaker 1: You just have everybody move one room over and it
Speaker 1: opens up one room for the new guests. And apparently
Speaker 1: that's how infinite sets work, because as crazy as it is,
Speaker 1: an infinite set plus one is still the same size
Speaker 1: as a regular infinite set. So you've got set A
Speaker 1: that's an infinite number of objects. Set B contains an
Speaker 1: infinite number of objects plus one extra object at the beginning.
Speaker 1: Can you still pair them off forever? Yep? Will you
Speaker 1: ever run out of items to pair off from each group? Nope?
Speaker 1: They are somehow still equal. And this is because, I mean,
Speaker 1: one of the things that's hard for us to remember
Speaker 1: is we often try to treat infinity as a number,
Speaker 1: and infinity is not a number. And infinity is a concept.
Speaker 1: You know, it's a mathematical tool, but it's not a number.
Speaker 1: Like if you add plus one to any number, it
Speaker 1: is more than that number. But plus one to infinity
Speaker 1: is not more than infinity because infinity wasn't a single
Speaker 1: finite number to begin with, right, and and and explaining
Speaker 1: this to my son, I don't tell him infinity is
Speaker 1: the largest number. I tell him numbers go on forever.
Speaker 1: That is infinity. That's good parenting. Well, I want him
Speaker 1: to have a proper you know, understanding of the boundless.
Speaker 1: But he's not, is he He's not? Kinna? Uh well,
Speaker 1: a proper understanding of the boundless. That's the kind of
Speaker 1: thing you spend your whole life trying to to wrap
Speaker 1: your head around. Yeah, well you ought to be careful.
Speaker 1: You might have a child that grows up to be
Speaker 1: a philosopher, mathematician. We'll see, we'll say, playing with fire there.
Speaker 1: All right, So where are we with with cantor though?
Speaker 1: What what is cant are saying at this point? Well,
Speaker 1: I mean, so we're at a weird place already, just
Speaker 1: with what we've talked about before, because especially when you
Speaker 1: take this and try to extrapolate it back to the
Speaker 1: more weird less mathematical, spiritual, theological, philosophical types of ideas
Speaker 1: of infinity, Like what does it mean to all these
Speaker 1: people who want to invoke infinity in their religion for
Speaker 1: us to discover that for some types of infinite quantities
Speaker 1: in set theory, a part of the whole is actually
Speaker 1: exactly the same size as the whole itself. Yeah, it
Speaker 1: doesn't leave much room for God. I mean, I guess
Speaker 1: it leads infinite room for God. That's the That's the
Speaker 1: confounding thing about all there you go. Uh So the
Speaker 1: other thing, though, is that Canter also showed it's absolutely
Speaker 1: true that some infinite sets are bigger than other infinite sets.
Speaker 1: Now we've just been messing everything up by violating our
Speaker 1: intuitions to show that sets that seem like they should
Speaker 1: be of different sizes if they're infinite, are actually the
Speaker 1: same size. How could it be possible that some infinite
Speaker 1: sets are bigger than other Cantor guides us again. Let's
Speaker 1: think about real numbers versus rationals. So last time we
Speaker 1: looked at rationals. That's fractions, right, anything that can be
Speaker 1: expressed as a normal fraction. Uh, then you've got real numbers.
Speaker 1: So real numbers include all of those numbers, but also
Speaker 1: include irrational and transcendental numbers, which are numbers that cannot
Speaker 1: be expressed as fractions, things like pie in the square
Speaker 1: root of two. You know you've seen people trying to
Speaker 1: calculate these out, like you can write out many many
Speaker 1: digits of pie three point one, four or so on forever,
Speaker 1: but you'll never get to the last decimal place of
Speaker 1: pie because it does not have a last decimal place.
Speaker 1: And if it did have a last decimal place, you
Speaker 1: would actually somehow be able to express it as a fraction.
Speaker 1: But despite the fact that it never terminates, pie is
Speaker 1: a real number, you can only write it with decimals.
Speaker 1: And there are presumably lots of numbers like this, But
Speaker 1: how many are there you might be guessing, give and
Speaker 1: what We've just been learning that there would be accountable
Speaker 1: infinity of these types of numbers. So maybe there's the
Speaker 1: same number of numbers that are rational versus numbers that
Speaker 1: are real, but no. Cantor showed that real numbers, including
Speaker 1: irrationals like maybe the square woradi of two, cannot be
Speaker 1: arranged into accountable list including all of the infinite possibilities.
Speaker 1: If you tried to make such a list. Cantor showed
Speaker 1: that you can always point out real numbers that don't
Speaker 1: appear on the list. So how would you show that? Well,
Speaker 1: for a simplified version, and this is a kind of
Speaker 1: dumb down version, Cantor was trying to do this with
Speaker 1: like binary numbers, but uh, for for a simplified version,
Speaker 1: imagine trying to make a list of all real numbers
Speaker 1: lining up their decimal values. So you've got like one
Speaker 1: point three, four, five, six, seven, eight, nine, ten, and
Speaker 1: then one point seven to five. What you know like
Speaker 1: that where you line up their digits in columns, and
Speaker 1: Cantor said, whatever is in that list, I can find
Speaker 1: a number that it's not already in it. And you
Speaker 1: can try to make a list like that that goes
Speaker 1: on forever. But no matter how you do it, I'll
Speaker 1: find numbers that are real numbers that aren't on the list.
Speaker 1: And the way he did this is is this genius
Speaker 1: thing where he went down the list diagonally. Now we
Speaker 1: already had a kind of diagonal squirmy line thing through
Speaker 1: the rational numbers, but this is a straight diagonal line
Speaker 1: through all these numbers that are listed. So what you
Speaker 1: do is you take the first decimal digit of the
Speaker 1: first number, and then the second decimal digit of the
Speaker 1: second number, and then the third decimal digit of the
Speaker 1: third number, and then take each of those digits and
Speaker 1: change them into something else, and then put them in
Speaker 1: sequence to create another decimal number. Now, by definition, this
Speaker 1: number is not on your list. It can't be, because
Speaker 1: you guaranteed at least one decimal place is off from
Speaker 1: every possible number on the list. So no matter what,
Speaker 1: an attempt to count real numbers fails, you can't possibly
Speaker 1: create an ordered way of listing them all. All this
Speaker 1: means they're not countable infinities. There's an uncountable infinity of them.
Speaker 1: And thus you can prove that some infinities actually are
Speaker 1: bigger than up than others. There are more real numbers
Speaker 1: than there are rational numbers or natural numbers. Now, let's
Speaker 1: try to bring this back to a concrete example, because
Speaker 1: I know we've we've been in the math for a while.
Speaker 1: I apologize for that, but I did want to try
Speaker 1: to set that straight here. So do you want to
Speaker 1: talk about Connor McCloud. Sure, yeah, How does Connor McLoud
Speaker 1: fit into office? Then let me get because at this
Speaker 1: point I'm no longer sure anymore. Certainly, I would think
Speaker 1: during the course of a normal immortals life, ah, he
Speaker 1: is pooping more than he is beheading. But how do
Speaker 1: we compare those two infinities now if he's living forever, well,
Speaker 1: that is a really difficult question. I am also in
Speaker 1: a in a strange place with you here now, because
Speaker 1: it's clear that the hair in this race, the one
Speaker 1: that accumulates faster, is that the highlander goes to the
Speaker 1: bathroom more often than he beheads. People. And if you
Speaker 1: just keep watching this process forever, any time you stop
Speaker 1: to check how many times each of these things has happened,
Speaker 1: the bathroom visits will be larger, and it will just
Speaker 1: keep getting relatively larger. It'll go on like that unless
Speaker 1: you say it goes on forever. Now, it's hard to
Speaker 1: say exactly what it would mean to say a sequence
Speaker 1: like that goes on forever, because you're talking about organisms
Speaker 1: that live in a universe that you know, even if
Speaker 1: you call him immortal, you'd think at some point he'd
Speaker 1: run out of usable energy in the universe. It's Highlander.
Speaker 1: With all things Highlander, it's best just not to ask
Speaker 1: too many questions about it. But we're in this weird conundrum,
Speaker 1: right because we've discovered that if it were actually possible
Speaker 1: for this to go on forever, whatever that means for
Speaker 1: events in real time, if it were to go on forever,
Speaker 1: then he would do those these different activities equally. But
Speaker 1: I don't know that it makes any sense to say
Speaker 1: events in the physical space go on forever in the
Speaker 1: same way that say a list of integers in an
Speaker 1: infinite set can go on forever. Yeah, I mean it,
Speaker 1: it comes back once again. To the idea that our lives,
Speaker 1: our experience, our world is finite numbers. This thing that
Speaker 1: we have either created to match up with the with
Speaker 1: the the universe, or that we have discovered in the
Speaker 1: fabric of the universe. These go on forever, These have
Speaker 1: true infinity. Yeah, I think that's a really great point.
Speaker 1: But then again, what if the laws of physics don't
Speaker 1: tell you that you can ever stop counting? You know
Speaker 1: what if you look at the laws of physics and
Speaker 1: you look at the universe and you say, I don't
Speaker 1: find anything that says time stops. So what do you do?
Speaker 1: Then you keep counting? I guess you keep pooping, you
Speaker 1: keep beheading, and you just and things balance out. Right,
Speaker 1: Then that seems like you've got to wonder, like, does
Speaker 1: that actually undercut the Boltzman brain argument then, because if
Speaker 1: you actually have a universe going on forever, obviously you
Speaker 1: have early universes where there is low entropy, and you've
Speaker 1: got all these stars and planets and we're we're all
Speaker 1: we're making aliens, you know, making all kinds of crazy
Speaker 1: organisms evolving through biological evolution, biochemistry, and you've got all
Speaker 1: these ordinary observers, and then you've got this long dead
Speaker 1: period where those things die out and you're generating boltzman
Speaker 1: brains out in space. But if it really all goes
Speaker 1: on forever, then it kind of doesn't matter how many
Speaker 1: of those things happen at either point, right, because it
Speaker 1: would just keep recurring. You'd eventually get an entropy fluctuation
Speaker 1: that would take you back to something like a big bang,
Speaker 1: and you just like started all over again, and then
Speaker 1: this would go on forever and you cannot compare them.
Speaker 1: That's true. So I actually wrote to Sean Carroll, did
Speaker 1: you ask him about Highlander infinity to poop? And it
Speaker 1: didn't mention a highlander. I don't know if he would
Speaker 1: have replied to me if I mentioned highlander, but he
Speaker 1: he was incredibly generous with his time to to respond
Speaker 1: to me. We really appreciate that. But yeah, he he
Speaker 1: basically acknowledged this. So Dr Carroll responded to exactly this
Speaker 1: kind of question and said that it's true that two
Speaker 1: accountable infinities are equal to each other. He said, however,
Speaker 1: it's also true that one is five times bigger than
Speaker 1: the other, or tend to a hundred times bigger than
Speaker 1: the other. That's how infinity works. And I think he's
Speaker 1: recognizing the problem there that like, you'll have these things
Speaker 1: where frequencies are obviously very different, but if you truly
Speaker 1: extend them to infinite sets of things, then they're accountably
Speaker 1: the same. So he says, quote, so comparing infinities is
Speaker 1: clearly not what you want to do. If what you
Speaker 1: care about is the relative frequencies of two kinds of events,
Speaker 1: you have two options. One throw up your hands and
Speaker 1: say there is no way to compare or to regularize,
Speaker 1: i e. Consider only a five night region of space time,
Speaker 1: so that all numbers are finite. Calculate the ratio of
Speaker 1: Boltzman brains to ordinary observers, and then take the limit
Speaker 1: as that region gets infinitely big. If you do the ladder,
Speaker 1: you will generally find Boltzman brains vastly outnumber ordinary observers.
Speaker 1: And and this is what they do in their papers. Now,
Speaker 1: to be clear, Carol is not saying he thinks we
Speaker 1: are Boltzman brains. He has physical arguments and to some degree,
Speaker 1: you might say, philosophical arguments for why we are not
Speaker 1: Boltzman brains, and we discuss those in the previous episode,
Speaker 1: and one of them was that he says, you know,
Speaker 1: assuming you are a Boltzman brain is cognitively unstable. It
Speaker 1: doesn't make any sense because what's your reason for thinking
Speaker 1: you're a Boltzman brain. It's like all this science and
Speaker 1: math and stuff that we only know because we think
Speaker 1: we have an accurate picture of the universe. If you
Speaker 1: were a Boltzman brain, you'd have no reason to think
Speaker 1: you had an accurate picture of the universe, and thus
Speaker 1: the method that you use for arriving at the conclusion
Speaker 1: that you're Boltzman brain would be totally worthless. So you're saying,
Speaker 1: there's a chance, do you everything about how subversive the
Speaker 1: slogan to infinity and beyond is? Yes? Unfortunately, um, it
Speaker 1: kept popping up during preparation for this. I'm like reading
Speaker 1: about these like that the Jane concepts of infinity, and
Speaker 1: it's just that that that stupid catchphrase from toy story
Speaker 1: sounding off in the back of my head, and and
Speaker 1: it gives me nothing. It provides no insight into what
Speaker 1: I'm and what into what I'm trying to wrap my
Speaker 1: head around. But it keeps going off like a like
Speaker 1: a malfunctioning and sprinkler or something that Yeah, I I
Speaker 1: can understand that. So I'm not one of these people
Speaker 1: who had like a deep emotional experience as an adult
Speaker 1: watching a Toy Story movie, because I don't think I
Speaker 1: ever saw anything after the first movie when I was
Speaker 1: a kid. But good, well, yeah I've heard that. But
Speaker 1: I remember that first movie and I didn't think about
Speaker 1: it back then. But now I'm thinking, wait, to infinity
Speaker 1: and beyond that's infinity plus one. Yeah, maybe he's he was.
Speaker 1: He was really contemplating infinity on a level that we
Speaker 1: just weren't prepared for. To infinity and incoherence to infinity
Speaker 1: and stay there. Well, anyway, I guess that's gonna have
Speaker 1: to wrap us up for today. I think we are
Speaker 1: we are out of time. Unfortunately, I wanted to get
Speaker 1: to some of the emails we've gotten about boltzmon Brains,
Speaker 1: but we do not have time to address the finite
Speaker 1: time in this podcast. We sadly do not have infinite time.
Speaker 1: And I'm sure also you have finite patients for for
Speaker 1: for infinity. You know, there's a fine line between things
Speaker 1: that are the most fascinating on Earth and things that
Speaker 1: really start to grade on your brain. It's it's like
Speaker 1: pulling on the tail of a of an of an
Speaker 1: infinite serpent, right You're never going to reach that point
Speaker 1: where you reach the you you find the creature's head
Speaker 1: and have a complete understanding of its anatomy. You're just
Speaker 1: gonna keep tugging and you will get pooped on. Yeah,
Speaker 1: you probably get pooped on because you really chose the
Speaker 1: wrong end to to tug. But I still think we
Speaker 1: we managed to have a good time here. We got
Speaker 1: to discuss uh infinity and a little more depth. We
Speaker 1: got to uh iron out some of the lingering questions
Speaker 1: remaining Boltzman brains, not all of them, because the mere
Speaker 1: concept of the Boltzman brain is kind of meant to
Speaker 1: uh to stir prolonged contemplation. Yeah, uh, And we I
Speaker 1: do not want you to walk away with the impression
Speaker 1: today that we're advocating the belief that you are a
Speaker 1: Boltzman brain. If you haven't heard that first episode, you
Speaker 1: should go back and listen to that where we actually
Speaker 1: talk about reasons you're not a Boltzman brain. Come on, yeah,
Speaker 1: don't uh don't lose any sleep over it. All right, Well,
Speaker 1: on that note, it's time for us to close out here.
Speaker 1: As always, I would like to direct you to stuff
Speaker 1: to blow your mind dot com. That's the mother ship
Speaker 1: that's we will find all the podcast episodes and links
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Speaker 1: too that if you want to support the show, a
Speaker 1: great way to do it is to rate and review
Speaker 1: wherever you get this podcast. Thanks so much, as always
Speaker 1: to our excellent audio producers, Alex Williams and Tarry Harrison.
Speaker 1: If you'd like to get in touch with us with
Speaker 1: feedback on this episode or any other, just to say hi,
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