Are there any problems with the modern theories of infinity? Does calculus really solve Zeno's paradoxes? If an infinite series does not have an end, can it still be completed? And is it reasonable to have any doubt about the foundations of modern mathematics? These are the questions I'm trying to answer on the 59th episode of Patterson in Pursuit. Hello my friends and welcome to another episode of Patterson in Pursuit, where this week I'm talking about some of my very favorite topics again, infinity and logic. I think this is the sixth interview that I've done on this topic with professors from at least four different countries. And if you're just tuning in for the first time, you should know that I've been trying to sort out in my mind the modern theory of infinities that was established around the turn of the 20th century. I've got some problems with the modern theories of infinities and I'm trying to make sense of them so I don't sound so presumptuous saying oh hey by the way I think the foundations of modern mathematics are based on kind of elementary logical error. But it turns out I'm not alone. Not only is there room for skepticism in this area, I think the arguments against the standard treatment of infinities in mathematics are overwhelming. But you'll have to make up your own mind. My guest this week is Dr. Michael Humer who has written a book very recently called Approaching Infinity where he tries to lay out a new theory of infinities because he too finds there to be some conceptual shortcomings in the standard treatment of infinities. Dr. Humer is the professor of philosophy at the University of Colorado at Boulder. He's also the author of three other books including the very popular The Problem of Political Authority. Now as you'll hear, this is really fascinating, Dr. Humer and I actually have radically different conceptions of mathematics. Totally opposite ends of the spectrum I think and how worth thinking about what mathematics is and what numbers for example are. And yet even on two opposite ends of the spectrum, we still agree hey, more work needs to be done in this area. A bit more critical of an eye needs to be focused on infinities. It's a really sticky and difficult topic and a paradox is abound. So in his book he lays out 17 different paradoxes and tries to resolve them with his proposed theory. If you want to pick up a copy of the book head over to the show notes page. This episode is steve-patterson.com slash 59. But we didn't just talk about infinities and mathematics in this episode. Near the end for about 10 or 20 minutes, I ask him questions about being a professional philosopher in academia because he wrote an excellent article which you can also read at the show notes page about the realities of publishing in the academic system about many of the restrictions that young intellectuals are going to face if they try to make a career in the world of ideas within the established system. This is obviously a topic very close to my own heart and my own career. And it's very relevant for the sponsor of this episode which is the company praxis. I have decided to build my career in the world of ideas outside of academia because I don't think the present American academic system is the place for future intellectuals. I think the internet renders a lot of the services provided by the formal academic system obsolete. I don't think it's necessary to go into college to get certified to be a productive citizen, a productive employee or productive intellectual. And the folks over at praxis agree with me. In fact, they are in the business of helping young enthusiastic competent individuals start their careers without the formal academic credential. The praxis program is three months of a professional boot camp that is followed by six months of a paid apprenticeship.  I think this is a far superior system and if you're interested in learning more go to steve-patterson.com slash praxis, P-R-A-X-I-S and you can learn more. So let's dive into the interview with Dr. Michael Humer who is a philosopher and author working out of the University of Colorado at Boulder. Dr. Michael Humer, thanks so much for coming on Patterson in pursuit. It's a pleasure to have you on the show. Thank you, thanks for having me. So I have had several episodes on the show and I've written a few articles on the theory of infinity and whenever I do I always get people that send me messages and send me emails saying you got to have Michael Humer on the show because he just wrote a book directly on this topic. It's called Approaching Infinity and it's one of my favorite subjects and so I want to kind of start the conversation off asking you about why you wrote the book and why you think we should be even discussing infinities in the 21st century. Haven't all of the conceptual loose ends been tied up when we're talking about theories of infinity? Right, yeah. Well I wrote the book because I was puzzled by infinity. It's a very puzzling subject. There are a number of paradoxes. I discussed 17 paradoxes in the book. There are a number of paradoxes that I was thinking about for many years and that don't have any generally accepted solutions. In many cases the common response is to just bite the bullet as they say in philosophy where you have some seemingly absurd consequence and a common response will be to just accept that, just accept the seemingly absurd consequence. And so I wanted a theory that would solve the paradoxes. I wanted to just sort of figure this stuff out for myself and I thought if I wrote it down it would be interesting to other people as well. At some point I eventually thought that I had a good theory that distinguishes possible from impossible infinities. Okay. And so I had a basis for writing a book. Okay, so generally when you talk to philosophers or even mathematicians about the subject and let's say we're talking about Zeno's paradoxes, the most famous paradoxes of infinity, there's a standard response. People say, oh, calculus solves it. That's the phrase. Oh, how can an infinite series of points be crossed while calculus solves it? Do you not find that a compelling resolution? No, I do not. I think that's confused. So it depends upon what you think Zeno's problem was. So Zeno is giving an argument that motion is impossible. And it might be open to debate what his argument was because we don't have his actual writings. We mostly have Aristotle's discussion of Zeno as the basis. Okay, so if you think that the problem was... So here's a possible argument that Zeno could have made. There's an infinite series of motions, which are all non-zero distance. And the sum of an infinite series of distances must be infinite. And therefore it's going to take an infinite amount of time to complete it. So you'll never get to the end. If that was his argument, then indeed the theory of infinite sum solves it. Although that would be a pretty silly argument because it's sort of stipulated at the beginning of the scenario that we're talking about a finite distance. But that's probably not the argument. And if you thought that that was the argument, you probably actually never read Aristotle's discussion of Zeno's paradox. The argument appears to be that it's impossible to complete an infinite series just by virtue of the meaning of infinite series. It's inherently impossible to complete a series just because it's infinite. And you have to complete the infinite series in order to reach your destination. That's not solved by calculus or the theory of infinite series. In fact, quite the opposite. The modern theory of calculus so far from showing that you can complete infinite series, it's basically founded on the assumption that you can't. Okay. So let's dive right into that because if I were to say that, people would say, Steve, you're crank, but I completely agree with everything you've just said. So what do you mean to say that this standard approach in calculus implies that an infinite series actually cannot be completed? Yeah. Well, so the way that these things are defined, so the way the sum of the infinite series is defined, the way that the derivative and the integral are defined in calculus is all in terms of these delta epsilon definitions. The whole point of these delta epsilon definitions is to avoid suggesting that you can get to the end of infinity, so to speak. So the definition of the sum of an infinite series is not the result that you get to after you add up all the infinitely many terms. The sum is defined as the number that you get closer and closer to as you add up more and more terms. Exactly. It's the limit of the finite sums. Right. And the whole point of doing that, of defining it that way, is to avoid assuming that you can complete the infinite sum. Right. So that you can only talk about finite sums and then you can define it. So I wrote a piece on this a little while ago and I used the example, if somebody's looking at a graph and they're talking about the function, you know, the f of x equals 1 over x. So the larger that x is on the left hand of the function, the smaller x is, you might say on the right end of the function. And you could think about it as it doesn't matter how large x is, it is by definition not going to be meeting the asymptote. It is by definition, it's 1 over that number, which means it's by definition not, it can't reach 0 and it doesn't matter how big. It could be the largest number you can possibly conceive of and it is by definition not meeting that y-axis. Right. There is no largest number that you can conceive of because for any number you can always say that number plus 1. But anyway, yeah. We'll have to come back to that. The limit of 1 over x as x increases is 0. That doesn't mean that 1 over x is ever equal to 0. Exactly. It just means that it gets closer and closer to 0. It gets as close as you like as you go further to the right. Exactly. Okay, now that is an argument that I would use and have used to say, just like Zeno, therefore it must be the case that motion is impossible if it's the case that reality or space is infinitely divisible. I actually buy his arguments. I just think he's wrong about the infinite divisibility of space. But how do you deal with that? Space is infinitely divisible. Sorry. We're going to have to live with it. Most people like the following argument for the infinite divisibility of space. If you think that space is only finitely divisible, then the distance between any two points should be a multiple of the minimum distance, an integer multiple. There's a minimum distance. Okay. It seems like there could be a square, for example. Suppose there's a square, the distance between one corner and the adjacent corner could be an integer multiple of the minimum distance, but then the distance between opposite corners cannot also be. Yeah. So I think that has to do with our theories of points. And maybe we can get into that a little bit later. I think there's actually a really wonky way that people have been conceiving of points for quite some time. And I know you cover this a little bit in your book. But maybe, so you would say then that, okay, so let me put it this way and see if you agree with this rephrasing, that some people get tripped up maybe by the language of mathematics when they talk about something like a sum of an infinite series. Let me say the sum of the infinite series is one. Most people think that that is a regular type of sum, that that means you add up every single individual element and then you get that number. But really when you're talking about summing infinite series, it means it's the limit. It means by definition it doesn't actually reach that number. Is that fair? Yeah, I think it's right. Okay, so for you, how then do you rescue the concept of motion if that is correct? Do you think that calculus in this respect is wrong or do you have another resolution to that? Oh, I see. So this might have been Zeno's argument. In order to move from one point to another, you have to complete an infinite series. It's impossible in general to complete an infinite series, so it's impossible to move from one point to another. And you're saying the first premise is false and I guess I'm saying the second premise is false. So objects do complete infinite series every time they move. Now, why would we think that you can't complete an infinite series? So I can give an argument for this, which I think is fallacious, but you can tell me if you think this is the argument, right? So an infinite series is a series with no end. To complete a series, you have to come to the end of it. You can't come to the end of a series that has no end, so you can't complete an infinite series. Yeah, I buy that argument, but what do you think is wrong with it? Yeah, I think it's an equivocation. So you can't complete an infinite series if that means coming to the last member of the series, where this is an infinite series with no last member. You can't get to the last member if there's no last member. That's true. But you could arrive at a point in time at which all of the members have been completed previous to that time. And that's the sense in which you can complete the Xeno series. So with normal series, with a finite series, it's a condition on completing it that you complete the last stage.   Yeah. You don't have to complete the last stage because there isn't the last stage, right? So this is my analogy. Suppose that I go away on vacation and I ask you to feed all of my pets, right? Okay. And I come back and I say, all right, so did you feed the lizard? You go, no. I say, I told you to feed all my pets. And you say, but there was no lizard. Okay. If I don't have a pet lizard, then you don't have to feed the pet lizard in order to feed all of the pets. Similarly, if a series doesn't have a last stage, you don't have to arrive at the last stage in order to complete all the stages. And that is the case with the Xeno series. So for every stage of the series, there's a time at which it gets completed. And there's a time at which, there's a time that's after all of the times that the steps get completed, right? So there's a time at which all of them have been completed. But there's no time at which the last stage gets completed because there's no last stage. Okay. Now that one's hard for me to conceptually wrap my head around. So let's use the analogy. This is the analogy I use in one of the pieces that I wrote on this. And I think it's a little bit easier to think about. And imagine we're trying to complete a whole pie. And the way that we are going to complete the pie is first we start with half a pie, and then we take a quarter of a pie, and then we take an eighth of a pie and a sixteenth and a thirty second and so on. With that method of completing the pie, are you claiming that it is possible to actually have a whole pie in a finite amount of time? Well, it depends on exactly how this is happening. But so if I may just sort of, there's a variation on Zeno series. So actually in Aristotle's discussion of this, this variation sort of comes out or is sort of obliquely alluded to. Suppose that at the end of each stage of the series, there's a pause, right? So like you move half the distance, stop, and then you move the next quarter of the distance and then stop and so on. If that's the way it goes, then you can't complete the series. Okay. Now the pauses, so if the pauses are of some non-zero time and if there are always at least some length of time, then the total is going to be infinite. Okay, but you might say, what if the pauses get shorter and shorter? Nevertheless, this is still uncompletable because if you have to stop each time, then you have to decelerate. Your speed has to go down to zero and then you have to accelerate back up to whatever your speed is during the next step of the journey and then you have to decelerate down again. And what that means is so each time you start and stop, you have to expend energy and force has to be exerted on you. And the force is going to increase without bound. The force required to accelerate back. And that means that at some point your body is going to be destroyed because of the unlimited force. Also an infinite amount of energy would be required to complete it. Now the story about baking the pie, if you're actually doing these as discrete steps where you do something analogous to stopping every time you get done with part of the pie, then you can never complete it, right? Because it's going to require an infinite amount of energy or something like that. Okay, but doesn't that imply that there's some, let's say placement of the pie which takes no time or takes no energy? Or to bring it back to the running circumstance, doesn't that imply that there's some distance which takes exactly zero time to cross? Well, in a sense there is, namely a distance of zero. A non-zero distance that takes zero time to cross. I'm not sure if I understood what you were saying there. Okay, so in the circumstance with the pie, you're saying that if I'm not talking about stopping it at, you know, I put the one piece of the pie down and then I stop, I'm talking it is one continuous process of putting the pie in place. To me it seems like what we've done is when we say, you know, we have like a formula for how we're putting the pieces of pie down. It seems like by the construction of that formula, we're saying every single piece of pie you put down, there by definition must be a little bit of pie left over. It has to be half, exactly half of the previous amount that you put down. So it seems like if you're saying by definition there has to be half the pie, you know, left over. You can never complete it. That seems like I don't understand how you could. It's the same thing with distance. There has to be some space left over. How could it, yeah. Well, yeah, so that definitely shows that within the series, you're never at the destination. If you're within the series, you haven't arrived. But that doesn't show that there will never be a time after the series is over, right, at which you had arrived. But how could the series be over if during the series it never, you never complete it? Well, the series is never over while it's going on. It's only over after it's over, you know, as they say it's not over till it's over. So with the, with the distance example, if I say at every point that I'm crossing, I am by, I necessarily must have some space left over, have to, by the construction of crossing the infinite amount of points. Yeah, at every point within the series, you have some space left over. At every point. In other words, could we rephrase it by saying, there is no point at which there is no space left over? Yeah, that's right. But again, there's no point within the series, right? So you arrive at the destination, when you arrive at the destination, destination is outside of the series, right? It's, the destination is the, is the first point after the series. So it's true that within the series, there's always some space left. I don't understand that if it's true at every point within the series, there is some space left over. That implies there is always space to go. Well, if you're still within the series, then there's always space to go, right? So what logically follows is that you can't get to the end while still being within the series. How does the series end then? How does that, how does that happen if you're still in the series? I mean, the time at which you're at the destination is identical to the time at which you're no longer in the series. So it sounds like there's kind of this magic popping that's like you're in the series and then like somehow you're outside the series and now there isn't any left over distance. Yeah, I mean, I don't know if it's magical, but I mean, you know, like logically, everything that you're saying just implies that you can't get to the destination while still being within the series, which of course is true because like the destination point is not part of the series. Like the number one is not part of the series one-half, three-quarters, seven-eighths and so on, right? Okay, so couldn't we say if Achilles is in the series, he's the guy in the series that then he would never actually complete the series? Doesn't that follow? So he's in the series, right? So how do you go from being in the series to being in the series? Well, he's in the series for some time and then at a later time he's outside it. What point does he get outside of it? Put it this way. So suppose that the person is traveling at a fixed velocity just for simplicity. If you know the velocity, you can calculate the time at which each member of the series gets completed. You can also calculate a time.  So say you're, you know, for simplicity, let's say the distance is one and your speed is one. Then, you know, provably every element of the series gets completed before time t equals one, before one unit of time has passed. There's a specific time for each of the members and that time is always before t equals one. So when t equals one happens, that means all of the steps got completed. In that scenario though, to say t equals one is one unit of time, I don't think that quite works. I think if there are, that would be there is one unit of time that has progressed. So my attempted solution at resolving Zeno's paradox is to say well it must be the case then that there's a base unit of time, there's a base unit of space. So a great analogy would be something like if you're watching motion on a screen, ultimately you have little base pixels. It looks like the motion is continuous, but it's not actually continuous. It is discreet. So you might have one unit of time elapsing, but there's no in-between by definition what we mean by one. That would be like the indivisible time unit. Right. Yeah, I mean, we could talk about the argument that space isn't discreet. Okay. Which again has to do with irrational numbers. Yes, I'd love to. Let's go right into that. Right, so the ratio of the diagonal of a square to the side is irrational, which means that they can't both be an integer multiple of some common base. Okay, so when we talk about irrational numbers I think we have to go right into the philosophy of mathematics, because this is when we talk about irrational numbers, real numbers, it presupposes a certain kind of conceptual framework that I think might have errors in it. There are other theories of mathematics which don't include real numbers just don't include irrational numbers where everything is discreet, nothing is infinite. And I think we have to grapple with those when we're talking about things like Zeno's paradoxes. So when you say like the diagonal of the square is irrational, what do you mean by that word? Standard mathematical definition, so it can't be written as the ratio of two integers. So let's talk then about what numbers are, let's talk about the metaphysics of mathematics, because there is one theory of math, a very, very popular theory, which says there are such things as infinite decimal expressions. So something like an irrational number 0.333 with a borrow read or 0.3333 and so on, that there are numbers that extend beyond where we're writing them down. And I don't think this is a correct theory of metaphysics of math. So I think that for example numbers are concepts, they're ideas we come up with and they don't somehow, just because you've written down a series of 3's with a borrow over them, doesn't somehow mean that you've referenced an actually infinite thing. So in your theory of numbers, how are you conceiving of numbers, what they are? So, yeah so you're thinking about there's a number that would have an infinite decimal expansion, and then you're thinking yeah, but that's not real. So, my view though is that that number is not infinite. It might be true that the way, the only way to write it down actually, I mean that wasn't a good example because you can just write 1 over 3. Right, exactly. But anyway, take an irrational number like pi, the only way to write it down using conventional decimal notation is to use an infinitely long expression, which you can't write. Okay, but that doesn't mean that the number itself is infinite. The number itself is a specific determinant quantity. It might just happen that we don't have notation that expresses it in a finite space, but that's just our expressive limitation, right? So that there are all of these numbers, all specific finite amounts. So you think pi is finite? It's a finite number, right? And it's not infinity. And it's a very specific quantity. It's just that there are infinitely many of these specific finite numbers. And, you know, it's like the appearance that there's something infinite about is it's only our notation that would be infinite if you try to write down the decimal expansion. Okay, so do you think it's the case that we can have a clear conception of the totality of the quantity that is trying to be expressed when we say pi, and we write down that symbol. Do you think that you can conceive of all of it? I mean, there's some sense in which I conceive of all of it and there's another sense in which I can't, right? So I can conceive of pi completely because I know the definition of it, right? Like the ratio of this circumference to the diameter of a circle. So that's a complete description of it. But that's a verbal linguistic description. That's not like a numerical understanding what the actual quantity is. Yeah, I mean, if you mean like the decimal expansion in some sense, I can't comprehend that, right? Now, if that's true, though, then why would we say such a thing exists? Doesn't that imply necessarily that numbers are these things that exist separate of our conceiving of them, that there's something out there in the world that we're trying to reference but we can't quite fully? Yeah, there is something out there in the world. So there are circles, right? And they have circumferences and diameters, right? And there's a ratio. Every circle that I've ever seen is a finite circle and it's constructed based on a certain amount of points. So analogy I love to give is just from my own career when I was doing animation work. And there's this computer program After Effects where you can make the most perfect circle you've ever seen in your life.  There's a, there's a, there's like a numerator over a denominator in terms of how many units make up the circumference, how many units make up the diameter, and there's no infinities invoked there. Yeah, so also the circle displayed on a computer screen won't be a perfect circle, nor will any other circle that anybody draws. So what is the perfect, because all the circles I encounter are, are, they don't have that. Yeah, they don't have, they're not perfect. Yeah, well there are, there are circular regions of space, you know, whether you can draw them or not. But I guess, I mean, more generally, there are specific quantities that things have and almost every physical magnitude, maybe this doesn't make sense. I was going to say almost every physical magnitude is an irrational quantity. Say there are two physical magnitudes that are not, not conceptually related to each other. Okay. And it's virtually guaranteed that the ratio of the two is irrational. So this I think assumes the the infinite divisibility of space or of space-time of physics in general. If things are discrete, then that wouldn't be the case at all. Yeah, I mean, that is, I'm saying that there, I'm, I guess assuming that there's infinitely many quantities, this doesn't have to be spatial. So it could be like the mass of two objects or, you know, energy of two objects or something. But I mean, I guess I didn't anticipate that in addition to proposing that space was discrete, you would think that the number series was also finite. So it comes from a particular metaphysical conception of what a number is. I don't think numbers exist separate of our minds. I think they're, I think if I say the word several, I think I'm talking about a concept in my head that if I didn't think of it, it wouldn't exist. I think that's the case if I say three X or three horses, I think that's a concept that we come up with. So in a sense in one sense of the term infinite, there's an infinite amount of numbers that I can think of something bigger than whatever I'm thinking of. But in another sense, there isn't an infinite amount of numbers because there's only a finite amount of concepts that anybody's conceiving of at any given time. I see. Yeah, so my view of numbers is that a number is a kind of property. Now, I don't know if you have this view about properties in general. So, you know, you see two horses. Let's say they're brown. They have the property of brownness. I assume you wouldn't say that brown is just an idea in our mind, right? I think that brown is a label of a description, a label on an experience that we're having. That's what I would say. I would reject the idea that brown is something in a property of the horse. I think it's a statement about an experience. Is there something that two brown objects have in common? Conceptually, yes. Metaphysically, I would say no. So if there were no people, nothing would be brown? Well, that's correct. So brown, the way that I'm looking at it, brown is a statement about a perceptual phenomena. It's a color. It's a qualia in our visual field. So, by definition, if there were no minds, there were no us, there would be no qualia. Well, okay. So, I mean, the nature of colors is controversial. So maybe we should take, you know, some primary quality that people don't have subjectivist theories about, right? Okay. So there's a square. I assume that's not a qualia in our mind, right? Squareness. So, I think shapes and objects are constructed phenomena. So, I seem to always have a water bottle next to me whenever I'm doing these interviews. I give the water bottle example all the time, but for example, I would say what the water bottle is, if I say the water bottle in front of me, I'm actually just referencing bits of matter that are arranged in a particular way, and I'm calling a water bottle. So without my mind, they wouldn't be boundaryed in some way. I think it's the same thing with a square. A square is a kind of composite object. There's an arrangement of the particles, right? What we call an arrangement, yes. I wouldn't say that's something in the world. I would say that's something that our mind is putting together. So, well, this is sort of getting to me interviewing you now. That's a conversation. I mean, I'm just going to assert that things have properties independent of us. Okay. And numbers are a kind of property. So, like, red is a property that tomatoes have in common with red roses. It's a thing that multiple physical objects have in common. Similarly, two is a property. It's a thing that multiple pairs of things have in common. So, whenever there's a pair of things, they jointly instantiate this property of tunis. There are multiple different examples of tunis being instantiated. So, tunis is what is in common between these different examples, right? And you can imagine the sort of examples I have in mind, right? You know, my hands and, you know, the Empire State Building and the Sears Tower. Those are examples of two. And without your mind, they would still have the same property? Yeah. So, in the sense in which any property could exist independent of the mind, numbers exist independent of the mind. So, that's about cardinal numbers. The real numbers, I think, are also properties, although they're different properties. So, there are physical magnitudes in addition to cardinal numbers. There could be a certain number of objects and then a different phenomenon is an object has a physical property that comes in degrees. And the real number sort of measures the specific degree of one of these physical magnitudes. Like the size of an object. Okay. Right. And so, like in my view these the numbers like what we're referring to is there's some underlying reality which could be in the physical world. Of course, numbers can also apply to mental things. So, it can also be in the mental world, right? But it's not necessarily in the mind. Like, there could be two ideas. There can also be two rocks. And in my view, there are as objective as anything. Right? If you think anything is outside the mind, you know, you should include numbers. Oh, okay. That's good. We'll put an asterisk by that one, but I want you to keep talking about your views as a very important claim that I think there can be some reasonable disagreement about. Yes. My views are not entirely uncontroversial, you might say. And I guess I will bring this around to infinity. So, what is infinity? There's sort of a long-standing debate about whether infinity is a number or as Kantor has it, whether there's a class of infinite numbers. In my view, infinity isn't a number because it's not a specific determinant quantity. So, the numbers are specific determinant amounts. Like, the cardinal numbers for is a determinant number of objects that you could have or the real numbers represent determinant magnitudes that a physical quantity can have. And infinity doesn't represent a determinant magnitude because it's something like being beyond all determinant magnitudes. Does this imply that you have a problem with the concept of an infinite cardinality? Sort of, yes, sort of no. That is, I think there are some categories such that there are infinitely many things of that category. That's correct. But there being infinitely many of them is not ascribing a number to them. Can you unpack that? There are infinitely many points in a line. Well, if points exist at all, there are infinitely many of them. But in saying that, I'm not assigning a specific number. What I'm saying is something like the points exceed every number. Yes, I like this quite a lot. Especially if we're talking about physical space. Okay, so it's kind of like a denial of whatever property you're talking about. So if you say there's an infinite number of these things you're actually saying it is not the case that there is some number that represents the totality of it. Sort of, right? That is implied, I think. What you're saying is sort of it exceeds every number. So when I say space is infinitely large, what I'm saying is really there are fairly large regions. That is, if you pick any definite size, there's a region larger than it. That's what it means that space is infinite. So let me ask you right on that topic then. Would you say that it is possible to talk about all of one of these infinite sizes? To say that you can have some totality that is an infinite size? I think the answer is no. That is I think my view suggests that you should deny that the term space refers to an object. So a traditional view of what it means to say space is infinite is there's an object called space and it has a volume. And its volume is equal to infinity. And that would be a particular number. And my view is no. What it means to say space is that there's this object called space. It's that there are regions and there are larger and larger regions without limit. And when you say space is infinite you're just referring to that fact. What do you mean by regions? Like a finite volume of space. So when you say to say it's not an object. I agree it's not one thing. I think it's kind of a way of talking but it sounds like even in that way it's something when you're talking about space. Well there's space the mass noun like this takes up a lot of space and then there's space conceived as a single object. So I don't think there's a single object that's all of space. Okay. There are just spatial regions and there are larger and larger ones. And there's no limit to them. So in that sense they're infinite. So like similarly what it means to say that a set is infinite side point I have arguments questioning whether sets even exist. But let's for now suppose that sets exist. What it means to say that a set is infinite is there are arbitrarily large subsets of it. That is if you take any natural number the set has a subset that contains more than that many elements. This is distinguished from the Kantorian conception where what it means to say the set is infinite is it's to assign a specific number as the number of elements in that set. And it's an infinite number. Is that correct though? Because I think that they might push back people who support the Kantorian notion might push back and say no all that it means is to say that there is one to one correspondence between each element of the subset and the set. There is not to say that there is a number that is infinity that represents the total size. They just say well all that it means is that there is this correspondence. Oh yeah so one attempted way of defining infinite set is it's a set that can be mapped one to one onto a proper subset of itself. However it was the view of Kantor and Frege and Russell that there are infinite numbers. And these infinite numbers are supposed to be numbers in exactly the same sense that 4 is a number. Right but isn't this from my understanding I might be wrong here but hasn't that idea been not debunked but I don't think that's the orthodox position anymore to say that there is such a thing as that infinite number like any other number. Are you saying that that's the mainstream view? I think it is yeah. They're supposed to be so like Elef Noel is supposed to be a cardinal number in the same sense that 4 is a cardinal number. It happens to be an infinite number instead of a finite number but that's okay it's still a number. I don't so I've had a few conversations with people on the show about this and I get a lot of push back when I say that because what they'll say is well no we're talking about cardinalities we're talking about numbers in a different sense they're infinite just like when they're talking about infinite sums the summation of an infinite series think well yeah we use the term sum but it actually means something different and that seems to be the case what they say is well we're not actually claiming there is this number it's kind of a way of talking but it means something different. Well I can only say that might be some people's view but I mean the people who are working in the foundations of mathematics including Cantor thought so I'll just tell you why they thought it was a number in the same sense because they had a theory of what a number was and it was that a number was a certain kind of set and you could find a set that's so that they thought that the number 2 was a certain kind of set something like it's the set of all two-membered sets something like that. In the way that you define the ordinary natural numbers you can define another object within set theory that looks just like that it's just like the finite numbers except that it's infinite so it's very natural to say that it's a number 2 so like the number 2 is supposed to be the set of all sets that can be mapped 1 to 1 on to the set that contains 0 and 1 something like that and LFNOL is the set of all sets that can be mapped 1 to 1 on to the set of all natural numbers so like you see that there's this pretty perfect parallel if you bought the original theory about the natural numbers then you should buy that the infinite numbers are also numbers. Okay, I'm this is interesting because I'm put in a position I haven't been in before usually I'm the one arguing with mathematicians about this and they say so I have to put another asterisk and say I don't agree with the orthodox theory I think it needs revision but I'm not sure that they would agree with that position so I can't defend them because I just agree with them but I would say they might push back a little bit on that at least some of them that I've spoken with. Yeah, well you know it's okay with me if the mathematicians reject Cantor's theory so I'll be happy with that I guess I've talked to philosophers a lot more than mathematicians so I can't really comment on what most contemporary mathematicians think. Okay. Yeah, but so my problem with this is I just don't buy the original theory about what a number is I don't think a number is a set. Okay. So I think it's a certain kind of property that a collection can have. Now it doesn't follow from that that infinity can't be a number but it also doesn't follow that it is a number. Let me ask you how important do you think metaphysics is to talking about infinities and mathematics in general because it seems like we have different competing theories about what numbers are that lead to radically different conclusions about you know theories of infinity. Yeah. Well I guess I would say metaphysics is important if you want to understand what the thing actually is. I mean it's possible to do a certain amount of work in mathematics without thinking about the metaphysics of the underlying objects that are being referred to. But if you I mean as a philosopher I want to understand what the things are that we're talking about. Yeah. It seems like that if the former situation might result in potential errors because for example in theology this happens all the time where theologians will be talking about God the properties of God what is implied by God and it might be the case that they're using some term that is supposed to have a metaphysical reference where they've actually made an error and that maybe their metaphysical reference doesn't make sense and so all of their theories are kind of built on a metaphysical error even though you might not see it in the theories and from somebody outside mathematics who is very skeptical of some of the claims in the foundations of math and the work of Cantor I think it actually is a pretty big deal because if numbers don't exist separate of our conceiving of them then it is definitely a case that we have to eject infinity from all of our thinking about mathematics because we can't conceive of all the infinity. So if numbers are mental then there can only be finitely many of them because they're only finitely many mental states which I gather that most people think is crazy and especially mathematicians would think that it's crazy that there are only finitely many numbers so like on your view there's a largest natural number. At any given time yes but think about it this way though it sounds crazy until you think okay well I think mathematics in general is a language that expresses concepts that can say true things about the world so I'd also say the same thing about sentences at any given time there is a largest sentence that is being conceived that doesn't mean that you can't add words to it it's the same thing with numbers you could think of a really big number sure that's the power of itself wow that's a really big number but that doesn't mean you can't conceive of a bigger one so it's at any given time it's finite which seems to make a lot of sense to me yeah yeah I only disagree with the starting point there I mean I guess like we would have to have sort of a more fundamental background debate about the nature of universals yeah I think so but I would love to have that conversation but I'll try to bring it back here to mathematics and infinity so you and I I think are in agreement that the orthodox treatment of infinities is not philosophically satisfactory that calculus maybe doesn't solve of Zeno's paradoxes my position is the extreme into the spectrum where I'm saying there is no such thing as an actual infinite because of my metaphysical beliefs you're saying there actually are circumstances of actual infinites so can you give some examples of what of all and in fact in in reading your book it seems like you think there are lots of actual infinite so can you talk a bit about where those are yeah there are lots of them so I draw a distinction between different kinds of quantities you might say so there are cardinal numbers and there are magnitudes so a cardinal number is an answer to a how many question so like four and three and 75 and then there are physical magnitudes like your height or your mass or the temperature of this room and so on among the among the magnitudes there are intensive and extensive magnitudes extensive magnitudes are roughly ones that are additive across the parts of a thing so length is extensive if you have if you have an object with a certain length the left half the length of the left half plus the length of the right half will have to be the length of the whole thing intensive magnitudes are not additive in this way so the temperature of this room it's not the case of the temperature of the room is the sum of the left half and the right half temperatures okay so in my view you can have an infinite cardinal number and you can have an infinite extensive magnitude but you cannot have an infinite intensive magnitude okay so examples well there are infinitely many natural numbers and there are infinitely many regions of space those are examples of infinite cardinality space is infinitely extended so that's an example of infinite extensive magnitude the future is infinite that is time is going to extend forever also I think the past is infinite but that's more controversial that again is an example of infinite extensive magnitude but there are no infinite intensive magnitudes so for example there can't be a thing with infinite temperature or there can't be infinite energy density in a particular region of space so you're making a claim that a lot of physicists also might object to because at least the way they talk maybe they don't mean this is literal this idea in physics that says in some circumstances maybe the center of a black hole you do have something like infinite density or maybe infinite temperature you say that would be logically impossible yeah I should say though that it is true that it's a prediction of general relativity that you can have infinite energy density in a black hole that's what a black hole is but it's also true that's widely regarded as an outstanding problem that is people are trying to figure out how to get rid of the infinities in a black hole or at the location of the Big Bang and is this for purely philosophic theoretical reasons well I think it's for sort of physics problem solving reasons a bunch of quantities go infinite and you can't figure out what's going on okay but the reason for this distinction is I can give an explanation of infinite cardinality where the explanation is not going to use infinity as a quantity and I can give an explanation for infinite extensive magnitudes like when I explain what it means I can explain it in terms of finite quantities but I cannot do that for infinite intensive magnitudes so for a set to be infinite means it has arbitrarily large subsets that is for any natural number you can find a subset with that many elements and notice that in saying that I only refer to natural numbers for space to be infinitely extended it has to be that any finite volume you pick there's a region of space with at least that volume and there I only refer to finite volumes but if I want to ascribe an infinite intensive magnitude to something I can't give that kind of explanation right so if something has an infinite temperature I can't say that that means that it has arbitrarily large finite temperatures right because its temperature isn't the result of a bunch of parts having temperatures I just have to say that the one temperature of that object is infinity I have to ascribe some one magnitude that's larger than all finite magnitudes so it sounds like this is directly related to the theory of sets in general the way that you're conceiving of of sets so do you want to talk about you left a cliffhanger there you said well this is assuming that sets exist do you want to talk about that for a few minutes yeah I'm a little skeptical about sets and my reason is not the reason that philosophers usually have which is philosophers usually say oh it would be better to make the world simpler or something like that my reason is that I just don't think anyone has given a good explanation of what these things are supposed to be right so if you look in books that are explaining a little bit about set theory sometimes they say a set is just a collection ok but it seems like the mathematical concept of a set is not the concept of a collection that we're familiar with so in the normal in the normal sense of collection if I don't have anything I don't have a collection like if I don't have a car I don't have a collection of cars but in the mathematical sense there's a thing called the empty set which you have when you don't have anything to collect also in the mathematical sense there's a thing called a singleton set which is distinct from its member but in the ordinary sense of a collection if I have one car I don't know if that counts as a collection of cars but it's certainly not that I have two objects the car and the collection of the car now isn't that kind of a way of talking though so it's almost in an ironic sense so I could say you know my set of all bank accounts which contain a million dollars or more and that's kind of a I could say that and it seems like it's communicating a concept in kind of a funny way because I don't have anything but you're saying that that doesn't work well yeah it depends upon what you say about that thing right so if you just say the set of bank accounts with over a million dollars owned by me is empty that's just a funny way of saying right right but there are other things that are said you know in set theory where you can't really paraphrase it away it has to be treating the empty set like it was an object or treating singleton sets as if they were distinct can you give an example so you know in pure set theory you're supposed to be able to construct this infinite hierarchy of numbers including an infinite hierarchy of infinite numbers all starting from the empty set right so when you get to the point where you're counting objects and you're getting infinitely many of them and they all started from the empty set I don't think there's any way of paraphrasing that away where you just say there isn't any of something right okay okay well let me try one what if it's the case that what a set is is a concept it's a conceptual boundary that we place usually around individual particulars so there's a little bowl of almonds in front of me if I were to say you know so I'll take three out and I've got three on the table here if I were to say the set of all almonds that are on the table in front of me that is an additional thing that is separate from the almonds and it is a conceptual boundary that I am placing around the actual individual units does that work yeah so by a concept you mean like a kind of mental state it's an idea yes I'm not exactly sure what concepts are but it's something like a mental state yeah so then there would only be finitely many of them right yes that's not a problem though for my particular theory but you know in set theory it's uncontroversial that there are infinitely many sets I see I agree that my theory is not compatible with the orthodox theory but you said earlier that there hasn't really been defined so you're skeptical that they exist what about the definition I've just given you so I guess I would say yeah I believe in concepts so like I believe that you could have a concept of the almonds on the table I just don't think that that's what people mean by a set right because people don't talk about sets the way that they would talk about ideas in the mind well if they're careful they can though I could say that the concept of this collection the thing that I'm treating as one object I could say that it is not spatially extended while the things that it's referencing the reference in the set do have spatial extension so I could if I wanted to be precise distinguish between the mental thing and the thing that it's referencing I guess if you're a mind body dualist you can say that right which I am a dualist then the idea might have an extension anyway leaving that aside yeah that's okay with me I'm not concerned to argue that the idea doesn't exist but I would say I don't think anyone else is going to accept that that's what sets are so then your position is you don't know whether they exist at all or do you have a different idea of what a set is I'm skeptical that they exist at all I mean we should think there's something very strange about this so I have alright so you have some almonds and then according to the standard theory just the fact that you have almonds automatically makes another thing exist which is not the almonds and it's not the myriological sum of the almonds either right so as it's not a physical object that has the almonds as parts it's just some other abstract object and by the way in standard theory there's not only that but you're going to get a whole bunch more objects in fact you're going to get an infinite number of things coming into existence just because you had three almonds here's another view when you have three almonds you just have three almonds you don't have to have an infinite collection of abstract objects isn't that what you're doing when you have the numbers that says numbers exist out there so in a sense you're doing this I think it sounds like you're doing the same thing when there are three almonds there is an additional thing which is three in addition to the almonds and which is separate from you well there are properties there are properties of things in the world you're saying the number is the property it's not this separate thing in addition to the almonds yeah I mean look I think that I can explain what the number is I can give examples of it I could say it's what these things have in common and then I give several examples of three-ness if somebody wants to explain what the set is it's really hard to explain Cantor gave this brief description at one point where it was something like it's a many that permits itself to be regarded as one but on the face of it you might think so many things are not in fact one thing so if introducing sets amounts to regarding many things as one thing isn't it just an error unless we say that that it's a concept actually I kind of like that definition because we could say something like we're treating it as one unit it's not obvious it's not one unit but we're treating it as one because it's an idea we've come up with you might think isn't that just an error like treating many things as one thing when they're not like why don't I introduce an object where if something is blue you can treat it as red then there's a new object that's red yeah well I mean okay I guess if we're gonna reject the idea that sets our ideas then I agree that it is kind of absurd but it makes a great deal of sense to me when we're talking about so if I were a shopkeeper and I were dealing with lots of units I wanted to come up with it makes a lot of sense to me to think instead of like if I have four plus four it makes sense to say I'm not gonna you know if I'm trying to add those what is four well it's you know four individual units I'm not gonna go one two three four five six seven eight to add those together I'm gonna say okay well I'm gonna chunk these four units as one thing and then just memorize four plus four equals eight it seems like it's a great shorthand yeah that sounds okay to me I mean I'm not I'm not prohibiting people from grouping things together right but I'm just worried about the sort of metaphysics when you treat that as another object right like when you think that they really is such an object and you know if you think about sets okay so if you go to sleep then something will happen to your idea well maybe the idea will still be there in a dispositional form if you if you die then the idea of the almonds will go away I assume that you're the only person who saw them so but you know nobody in mathematics would say that you can destroy a set right by like killing the person who's thinking about it so but we can easily say it's the same thing with sentences I mean if I'm conceiving of the sentence the cat jumped over the mat then when I'm not thinking about it it goes away but that doesn't preclude anybody else from thinking about those same words and when they're thinking about them well then it exists again yeah no that's right right but I mean like the sets are standardly treated as you know their mathematical objects right which you can't do anything to you can't interact with them they don't cause anything right well this is a great segue the last thing that I wanted to talk to you about but I don't think we're going to have much time you've said a few times that you know the standard treatment is what you object to and you and I are in agreement on that you wrote a few years ago a excellent piece about academic publishing about like people who maybe are students who haven't gone to college have a very romantic idea of what academic life is everybody takes the work of the professor seriously all the professors are really seriously engaged in the world of ideas but it seems like for somebody I'll just say for somebody like me who's outside the system with radical views if I were to come out as some some philosopher and say hey you know what I think the orthodox treatment of set theory is wrong for the last century and they've been making fundamental errors nobody's going to take that seriously so I found that I have a much more effective impact in the world of ideas just being outside the system and say okay well I'm not going to play that game I'm just going to do my own thing and so far so good but you wrote a fantastic piece which I'll make sure to link to in the show notes page where you essentially say hey look you know lots of people are interested in philosophy it's a great gig if you have a certain disposition but it's not so romantic the reality of academic publishing if you're interested in the world of ideas isn't so it's not as it might seem when you're undergraduate student so could you talk a little could you put a little bit meat on the bones here for somebody that is interested in the world of ideas and has a vision of what it would be like to be a professional academic I guess I want to preface this by saying it's not that there's some other thing that you can do that's going to be really great it's virtually impossible to influence the culture right like if you want to change the culture and like correct the bad ideas that are out there in our society it's almost impossible in fact you should basically assume that you won't be able to do it no matter what that's the spirit but anyway and it's not because nobody does that it's just that of the people who try to do that the percentage who succeed is so tiny that it's rational for you to basically assume that you won't be one of them right okay anyway and the academic world is it's a lot harder than you might think when you're an undergraduate student. Students don't realize I guess kind of don't realize how much like how many academics there are and how much material is already published and how much more is getting published every year and when you think about these things there are thousands and thousands of philosophy articles and books published every year so when I checked several years ago the philosophers index was getting 14,000 new records every year so you throw your paper into a pile of 14,000 papers and books right and it's almost impossible to get anyone to pay attention to it right if you're an academic you kind of you have to try to get into one of the top journals they reject 90 to 95 percent of all submissions and they reject them for reasons like one the referee thinks you're wrong number two the referee thinks you're right but it's obvious that you're right so there's no point in saying it just some people get an idea when you say the referee what is that who are you talking about yes you submit your paper to the journal and they send it out to some other professor who works on that subject and by the way you might be sending it to the person that your paper is criticizing that has happened or you know just somebody that you listed in your bibliography they probably send it to two if it's a good journal two referees and if one of the referees thinks that you're wrong there's a pretty good chance he's going to recommend rejection for that reason or if he thinks that you're obviously right you might recommend rejection because there's no point in saying this thing that's obvious or if he thinks it's too similar to things that somebody else have said somebody else has said or if it's not sufficiently tied to the literature okay so these four reasons together cover almost everything that you could write so it's almost impossible and the loophole is you have to take something that is so highly specialized that it hasn't been said before like going into some incredibly tiny part of the question right or take something that is so closely tied to the exact course of the recent discussion that it wouldn't have been said in the previous 2000 years of philosophy right and this is a large part of the explanation of why academic work is boring to most people right because just a whole lot of the work is just really tied to a specific discussion that happened like in the last several years right so that people who are not in that discussion it's not interesting to them well and on top of that these papers are literally not read by almost anybody I think I saw a study recently that said the vast majority of papers are read by exactly zero people and then I don't know 20% more than that I don't know what the exact numbers are two or three people so the amount of work that goes into writing on a topic trying to find something novel to say citing other people's work so that you get accepted commenting on an issue that might be totally irrelevant just so that you can put it on your resume because that's part of the credentialing process and you need that because it's published or perished and your work is not going to be read anyway that does not seem like a romantic intellectual life to me here's how you should estimate how many people are going to read your paper there are 14,000 philosophy articles and books what percentage of them are you going to read this year that's just 14,000 just this year what percentage of them are you going to read a fraction of a percent well that's probably a good initial estimate of the percentage of people who are going to read your article right what percentage of the profession is a fraction of a percent of the profession might read your paper so that is academic publishing in terms of what about teaching though so maybe the publishing you have the pressure to bend over backwards the published work that nobody is going to read it might not be that interesting but you got to do so but isn't the reward in teaching people who are interested in philosophy well there's a big difference between philosophy majors and non-majors but at most schools most of the students are there to get a credential you know more than they are there to learn about philosophy especially in the lower division courses they're there to you know they're satisfied they're jumping through the hoop so that they can get the degree at the end so in other words they're not particularly engaged with philosophy yeah now when you get to upper division classes then you get philosophy majors who are usually more interested in it and so you know that's fun and you might have a chance to influence people through your teaching although you probably won't know right you know one of the things that people don't realize about the academic world that I think I myself didn't think about at all was there are things that you're supposed to do besides research and teaching that is you're supposed to do annoying administrative tasks like for example the philosophy department has to admit new graduate students every year and we get between 100 and 200 applications somebody has to read them and decide who can get in right so you get on these committees doing these administrative chores that are really tedious you know which is supposed to be about 20% of our work okay so then the question is somebody who is interested, passionate about philosophy written a book that's been read by a lot of people about philosophy I'm not yet 27 I've got a podcast that's listened to by thousands of people I've got a website that's visited by tens of thousands of people making an impact I get feedback from people all the time but I'm not in academia is there a case to be made for the young interested philosopher in this system when there seems like there's a pretty awesome alternative which is you research what you want to research you write what you want to write how you want to write the audience is going to find you and you can do it without any of the bureaucratic nonsense you don't have to do any of the administrative stuff yeah I mean so like I can't I don't know very much about the alternative system I would guess though that it's extremely hard to make it is very difficult to make a living doing intellectual work outside of the academy yeah I guess also that it's harder to get people to listen to you even harder well that's definitely the case yes by the way like you know I mean after my sort of sounded like complaints about the academic world I should say like I've I don't regret it for a second I mean it's been awesome to me right I got a job at a major research school and I get to work on whatever I want and people pay me a pretty good amount of money you know for doing the things that I wanted to do right mostly however most people do not get a job in a research school so they wind up having to teach a lot more than they wanted to right and you know they get they get tired also don't like don't get the amount of attention that I have for my works right I've done pretty well like getting people to listen to me but most people do not succeed yes and there's definitely juicy jobs to be had in academia there's no question about it it's just for somebody that and I agree it is hard to get people to take you seriously I have lots of personal experience with that however on the other hand and again somebody being outside the academy my partially economic analysis is that there's too many professional philosophers there's too many professional academics out there that it's kind of a skewed market so actually to make it as an intellectual in general is really really hard and it might be the case that you have better chances in academia but maybe not because part of the difficulty when I talk to these people as I'm doing the show is people don't have a platform for their ideas I mean they'll be working in academia for 20 years and playing by the system trying to play by the rules and then immediately just coming on the show from guy you know who's got a microphone and access to the internet they get immediately get 10 times more exposure than they otherwise would have so it seems like you might actually have a better chance bypassing the system altogether if you can handle the criticism and people calling you a crank and oh you don't have credentials my guess is you probably actually have a better shot at it if you have what it takes to be somebody that's just professionally working in the world of ideas so this is most of the academics that you're thinking of didn't try to get a wide audience right is it the case that there's an academic that doesn't want a wide audience yes that is you might say well maybe they want it a little bit but they don't want it enough to make much effort there's a lot of people who are content to just like do their job in their school and you know and it's kind of I mean it's stable in a way that going out on your own is not like that is true yeah I could sit in this job okay I have tenure so I can just I can do whatever and I don't have to worry if somebody doesn't like what I'm doing I don't have to worry about losing customers or something because I can just stay here for 20 years and I have a steady income right that's true I guess maybe the crowd that I'm thinking about are those really passionate people that aren't necessarily looking for a comfortable career but are looking to really genuinely engage with and affect the world of ideas which maybe there aren't many but I think there are a lot that have been very disenchanted by their experiences in academia I would be one of them yeah but I think there really is an emerging opportunity online that the world has really never seen right now that you actually have a global voice just through new technology that has never existed and it seems just like the internet is changing a lot of market dynamics in other areas it seems like this is an area ripe for some significant shift yeah sounds fair I mean I would just say that you could still do this as an academic right so I think about people like Brian Kaplan who has a very popular economics blog and I would guess that his academic job has helped him because he doesn't have to worry about putting food on the table because he has the academic job he has a steady income and now he can just write about whatever he wants on his blog I think that's a great point and I think there are going to be some people yourself included who really are exceptional writers and communicators who it doesn't really matter if they're inside the academy or outside they're going to stand a good chance of making it but if they're inside the academy like you said it's guaranteed paycheck and it comes with a prestige as well of course only if you get tenure but yeah well this is a I really appreciate the conversation this is a great note to end on yeah do you have a website that you'd like to send people to if they're interested in more of your work yeah you could just google my name and find my website and you know and I have that book approaching infinity so everybody should buy it yes I'll make sure to put a link in the show notes page thanks so much Dr. Humer this has been fantastic alright thank you okay that was my conversation with Dr. Mike Humer hope you guys liked it I felt like with that interview I could have talked to him for probably a solid four more hours just trying to get down to some basic agreement about metaphysics really about the metaphysics of numbers mathematics and the connection between our concepts and the world and what his arguments are for thinking that numbers exist outside of our minds there's really more fundamental stuff that I feel like had to be established before we even get into talking about the infinities which is awesome this is one of the reasons I am in love with philosophy and I got sucked into it because whenever you're talking about a subject with somebody that knows what they're talking about immediately you start talking philosophy you get down to really fundamental things and it becomes very apparent oh hey we got all this other stuff we got to sort out before we even start talking about what we thought we needed to talk about so I hope that in the future I'll be able to have Dr. Humer back on the show there's also a bunch of other topics we've got these areas of mutual interest in epistemology and metaphysics and politics so hopefully he'll be back on the show and we'll try to sort those things out as well alright that's all for me this week enjoy the rest of your day


We often get letters from people who see our content on social media and they're grateful that we do give them that preview of what happens when someone commits they're able to see what the physical process of cleaning it looks like and what the family endures and so oftentimes people do reach out and say hey thank you we watched your content and because of that content we decided not to go through with this so that in itself is rewarding. I think that there is certainly a curiosity we're all curious about what happens to our bodies and so it's an opportunity for us to educate people you know the body is what it is it's flesh and blood and there is a decomposition process to it and it's good that we're able to give people that insight.


Ladies and gentlemen, we take you now behind the scenes of a police headquarters in a great American city Or under the cold glaring lights will pass before us the innocent the vagrant the thief the murderer This is the line up. I put love down in the first row. He's right over there. Thanks coin. Oh, you're in it Hello, how do you feel pretty good? I guess a little uncomfortable people. I'm up like this. Yeah How long do they have to keep the cast on about four months the doctor said Well, the chatter the bomb they had to put a new piece in that's a fun You think you got the guy? Well, it's the same when you're picked out Questions or identification please remember the number assigned to the prisoner at the end of each line when I ask for questions or Identifications call the number To your name will assist if they suspect so many to get a natural tone of voice They'll do not pay too much attention to their answers as they often lie Step out and talk up so the people in the back of the room can hear you The small guy that's the guy you're sure He's a one-off driving. Oh, well, he comes up. I want you to be positive. Okay You found any of the other men Third guy up there with no He is A couple of long time to you know, I got a memory 33 and a half North Lincoln Where do you work? I ain't been where did you work Friday used to be a fight a lightweight Little heavy now any weapons, huh? Any weapons on you when you were arrested? No, I just climbed into the place. I didn't want to hurt nobody Where are you from Pete Detroit? I've been here a couple of months. I guess I couldn't find nothing to do Can't fight no more got blind in this eye. Did you have a car? No, am I going to get a car? Okay, Pete step back. Yeah, okay Number three colleague price 774 and North Lexington hotel that team. What's your business colleague? I'm not working. You have any weapons on you? You sure that's him The office closer than this when he drove away sergeant Graham. Yes, sir. All number three for interrogation Quite a nash or are on duty? Maybe we can pick up a couple of others at the club. You're gonna lose Like the last time I was tired Hello, Kelly How are you? I'd like to make this short. Okay, it's up to you. I didn't do nothing. Are you just listening? Okay, the Spartan payroll was struck up. So what just listen for men wearing masks walked in and walked out with a $100,000 a guard was shot, but he got a good look at the driver to get away card Hmm Can I smoke nope? Okay, the God says you were driving the car. Ah, he's blind You don't want to make it short. I want to make it plenty short. Just take my word and let me go that's short He picked you out of the mug far before we hold you off that bus and so we made a mistake Okay, Kelly. Oh look, I'm telling you the truth. I wasn't in no rough time Yeah, one more lap you get life. Oh, you're going away, huh? You've been here before you know the routine Yeah, and it don't scare me. Where'd you get the $5,000 you had on you? I want it. Where? Oh, what's what horse? Sound off what great. I Race what track fifth what track? I don't know. I bet them all look it up Who are you doing the day before yesterday stayed home? Who's in a robbery with you? I wasn't in no robbery. Who are you driving the car for? I wasn't driving no car. How much did you bet on that horse? Huh? How much did you bet on that horse? Enough to win me five thousand. What were the odds? I don't remember. I bet every day. What day did you bet this horn? Yesterday we picked you up getting on the bus at 11 o'clock in the morning. No No, I wasn't yesterday before Yeah, yeah, I think so. Where did you bet it out of the books? I thought you said you stayed home I went out a guy can't stay in all the time. Who are the other four men in on the stick up? I don't know what you're talking about. You read the papers, don't you? You want to bet a lot of money to win five thousand? Yeah, where'd you get it? I had it, won it. I do pretty good on the race. What was the name of the book? Look, I don't want to get to no trouble. You were identified as the man driving the getaway car.  Where were you going on the bus? Out of town. Where? New York. Wasn't you sure you were? No. Who were the other four men? I don't know. Look, can I have a drink of water? No. You were in your apartment that day? Yeah, and you went out to the bookmaker. Yeah, yeah. What else did you do? I stayed all day. Yeah, yeah, I was tired. I listened to the police. Didn't you go out for food? Yeah, no. Who were they, Cully? Who were the other men? There wasn't any other men. I don't know what you're talking about. You'll get life, Cully. For what? For what I didn't do. We've got enough to convict you right now. All right, then do it. Do it. Go ahead. Where do you want the other four men? Give us their names. Nuts! Tell me some more about that horse you won $5,000. Come on, you've been working on me for four hours now.   Yeah, yeah, all the time. Yeah, look, can I, can I please have a drink, huh? How come you only got $5,000 out of $100,000 job? I, uh, I didn't get to try for my job. I got it, but I'm on, Cully. Fascio. David J. Lichman. Yeah, yeah. He just got out two months ago. Yeah, he, he, he, he figured the job. He would. I got a flat $5,000 and they split the take. All I did was drive. Give me a drink of water. I'm, I'm losing my voice. Not on APB. Yeah, yeah, descriptions on all of them. I hope we get Fleischman first. He's so smart he never argued. He'd just tell us who the rest were. You ready to get beat? I'm ready to find out if you can. Okay, ship it up.  I felt it tonight. Yeah, you can't even get out of your own way. Oh, wait. Oh, what's up?  Should I take? We'll come. We got time for a fast shower? You can take it after you get Fascio. For this, him, he's in a steam bath. Oh, he might even stick around for a rubdown. Yeah, the booze. Ain't got to be no shooting. He's getting a rubdown, isn't he? Yeah. I don't think he's got a gun with him. Quine, go check his clothes. Okay. Show me. Yeah, sure. That's very one, Mr. Fascio. Police. I had known I was going to get this kind of a workout. I wouldn't have suggested squash. Praise household word is the name of Bing Crosby. For years, Bing has set the styles in American music. You and your entire family are invited to join Bing every Wednesday evening over most of the same CBS stations. That's Wednesday night, Bing's night on the air. The Bing Crosby Show, an American listening tradition. You've been puttering in your garden, Fascio. You sure have a green thumb, huh? $18,000 we dug up, Fascio. In an old empty grapefruit can. Imagine our surprise. I don't trust banks. You're circular friends. I don't blame you. You've never been such a smooth operator, Fascio. How come your end was so much more than Cully's? Who's Cully? Oh, look, let's not play hide-and-seek anymore. I already got a blister digging in your devil grass. We've got a sign confession from Cully. It's only a matter of time until we get your playmate. Martin Fleischman, Lou Holstetter and David James. Right, Fascio? Now, do you still want to waste time? And I recommended Cully for the job. Five grand he got. He ain't made that in two years. Why, when I ran into him, he was tending bar in some sloppy saloon. I took him to see Marty, got him a suit of clothes, loaned him 100 and let him sleep on my couch. Had to get it cleaned, too. Crummy bum got hair oil all over the upholstery. Ouch! You'll have to stop talking through your nose for a while, Fascio. But don't stop talking. Where do we find the rest of the boys? I don't know. I don't get hasty. I just don't know. Holstetter was leaving the state. James, I never knew before. He was Marty's idea. What about Marty? Marty's smart. You'll have trouble finding Marty. Where'd you see him last? Look, Lieutenant, I'm not trying to be difficult. I know when to play it cozy.  Well, I'll have the doc look at it again, Fascio, after you tell us all you know. A tele-type from Illinois, man. They picked up Holstetter. Good. See, I told you he left the state. Take a couple of men, set him up on extradition. Right. Three down, two to go. Where's Fleischman, Fascio? I mean it, fellas. If I was you, I wouldn't know where to start looking for Marty. I'd cover it up good if I know Marty. Only he doesn't know we're looking for him yet. What are his habits? What does he do with his spare time? Well, he reads a lot. Ancient history he likes. I used to meet him at the library once in a while and drive him home. He ain't ever learned to drive. Which library? Big one in Central. He wouldn't go there now. Marty's too smart to go anywhere for a while. Matt, put a stake out on every library in town. Right. Ah, you're wasting your time, Marty, too smart. Aren't you like a smoke loop tenant? Wonder if it had hurt my nose. Oh, yeah. Try it and find out. Are you using a different brand of coffee? Well, it's that instant stuff. I thought I'd try it. They're dampening the grounds. You like it? That's not bad. Did I tell you that Molly's going away for a week? Yeah, where? Oh, visitor mother. It'll do her good to get away for a while. What are you going to do? Well, I thought we might... I'll bring him in. Last time she went away, you were like a lost puppy. Yeah. Hello, Halsteader. Sit down. They have you back. You should let us know when you leave town. Look, I'm not in the mood. Get to it, huh? All right, Halsteader. We want a full confession on the spot and hold up. Do we get it now or later? We've got Cully and Fashio signed and sealed. Your name is very prominent in the statements. So?  You don't need your confession. You just do yourself a favor.  That guard's still alive, fella. Not the way you planned it.  Go ahead, Halsteader. You were saying... Godfrey? Yeah? Are you sure? And don't take him unless he comes out before I get there. Yeah. Come on, Matt. You may have Fleishman.  This is Fleishman. You boys will have a fourth for Bridge. Sergeant Quinell in the tenure while you're going. Huh. I heard Fleishman all right. Trollman on the beat spotted him. I went up and took a look. He's in the ancient history section. Once I sent him. Well, it's not very crowded in there.  Get suspicious. I've got two men in there with him now. One man in the hall. Two men up there at the entrance. Anybody with him? Been spotting him. Just sitting there reading. At the end. You got the play squash? Oh, so. Is there something I can do for you? Please. I was just reading the rise and fall of the Roman Empire. You know what, Lieutenant? I read it in school. You should read it again. Very interesting. Let's go. I'm surprised to see you, boys. Very frankly, didn't expect it. Let's go. Is this an arrest? That's part and payroll. We've got everybody but David James. Oh, sorry, Miss. Oh, yes, certainly. I'd like to check it out, but I'm afraid I couldn't afford the overtime. Good day. Good day, sir. I'd like to find David James. Would it help me? What? I'm quite sure you can find him on his boat. His boat? He lives on it. David is a sort of a sea-going hermit. Doesn't care much for people. Where's the boat? The end of River Street. South Landing. Registered under Smith. All right, let's have your right wrist. Certainly. It rolls and fell. Rome? Fleischmann. You don't rent it. Which one is it? Ketch, one right down there. End of the slip. On the left? Yeah. Okay. You better get off the dock. Might be some trouble. Trouble? Please.  Let's go, Pop. You want Smith? They're all about it, Pop. Let's go. Pretty open. I'll just call him out. If he doesn't like the idea, give it to him. Maybe if we got him that other boat across the river. Yeah, yeah, that's a good idea. Police all over, my God. I'll keep your head down there. David James! James!  He doesn't want to cooperate. James! Whole area surrounded. You haven't got a... Everybody okay? Yeah. I got a face full of wood. Give me that Thompson. You'll throw slugs all over the bay. I'll let him know. James! We'll give you one more chance. Come out with your hands up. $6,000 catch. You ruined it. We had the bottom of a couple of hours. Look, friend, we did you a favor.  Or before you pass the innocent, the vagrant, the thief, the murderer. Listen again next week when we again bring you The Line Up. As Lieutenant Ben Guthrie and Wally Mayer as Sergeant Matt Grebb is written by Blake Edwards and Dick Kline with music by Eddie Dunstetter. Featured in tonight's cast were Hi Everback, Jim Nussar, Harry Lang, Eddie Maher, Junius Matthews, Ray Hartman, Dave Young, and Mary Schiff. The Line Up is produced and directed in Hollywood by Jaime Delvalle. Fills with narcotics and sales to teenagers. Be listening for Gregory Peck when suspense is heard this Thursday on most of the same CBS station.


Dear students, I would like to present to you the concept of the marginal probability mass function. Let us consider a discrete random vector. In other words, a vector whose entries are discrete random variables. Now, when we consider any one of these random variables in isolation, its distribution can be characterized in terms of its probability mass function, right? This one, when we have more than one random variables, this one which we just now considered, the one we considered in isolation of the other ones, it is called the marginal probability mass function. The first thing is that when you concentrate on another variable, you will use this term for its distribution, marginal probability mass function. And of course, this word marginal distinguishes this particular PMF from the joint PMF. Joint PMF, of course, is the one which characterizes the joint distribution of all the variables taken together. You must be wondering why are the words marginal? To understand the words of marginal, the example of discrete variables is very good, because you can then think about a table, a bi-variate table, for example, if you have two discrete random variables. And in the first column of that table, you have the numerical values of the first variable, and in the top row of that table, you have the numerical values of the other discrete random variable. And in the body of the table, you have all the probabilities for all those combinations of values of the first variable and the second variable, and the sum of all those is equal to one. Now, when you think about this table form, then it is very easy to see why we use the word marginal when we would like to concentrate on the probability distribution of either the first or the second variable in isolation of the other one. It is as simple as this, that even in the first column, you have values of the random variable X, and in the top row, you have values of the random variable Y, then to get the marginal distribution of X, you just have to add up all the probabilities, which are in the various rows in the body of that table. When you add them and you write them in the margin, then this first column of the values of X, and this last column of those probabilities, which are in that margin of that table, they are the ones we are talking about, and because they are in the margin, so it is absolutely correct to call them marginal probabilities. Just like this, on the top, you have the other variable Y, the values of the other variable. Now, if you want to find the probabilities of these values, you simply add up the probabilities in the body of the table column wise, and in the bottom margin of this table, you get those sums, which are the probabilities of those values of Y that you have in the top row. So, X and probabilities of X, Y and probabilities of Y. The probabilities of X are in the margin of that table, and the probabilities of Y are in the bottom margin. Is it not obvious then that it is appropriate to call them marginal probabilities? And after that, when this was developed in this way, then even for continuous variable, I suppose they must have thought that we should just maintain the same terminology, and so even if those variables are continuous, when you instead of summing, you are integrating over one particular variable, you are getting the marginal PDF of the other one. This is the concept of marginal probability distribution.