 Hello my friends and welcome to the 52nd episode of Patterson in Pursuit. This episode has come to you from Brisbane, Australia and we're talking about the philosophy of mathematics. Supposedly, there are multiple sizes of infinity, in fact there's an infinite number of different sizes of infinity. How can we make sense of this? How can we make sense of one size of infinity? To help me answer I'm talking with Dr. Toby Meadows of the University of Queensland who is a philosopher that works in the philosophy of mathematics and he specializes in questions about set theory. We had a fantastic conversation on the topic as I'm trying to piece together and make sense of the orthodox mathematical claim that there are multiple sizes of infinity. I have a difficult enough time wrapping my head around one size of infinity. The sponsor for this episode is the company Praxis. 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So if you want to be a part of that, check out stevedashpatterson.com slash Praxis All right guys, so I hope you enjoy my interview with Dr. Toby Meadows talking about infinity and the philosophy of mathematics. First of all, I want to thank you for sitting down and speaking with me today, my pleasure. I've got a running theme on my show and that is trying to understand the nature of infinity and some of the claims in modern mathematics about the completed infinity and the infinity of infinities, the different sizes of infinity. So infinity seems to be this central concept in both mathematics proper and in the philosophy of mathematics, but I've got a really difficult time wrapping my head around it. So I was hoping we could start with the basic question, maybe a basic even just terminological question. When mathematicians talk about infinity, what exactly do they mean by the term? There's an intuitive sense of infinity, which is this idea of never ending or never completed or unbounded, something like that. Is that the correct way of thinking about infinity or the term infinity? One way of answering it is one standard sort of set-theoretic definition one might give is to say, if we took the natural numbers and we have a positive infinite collection, if I can make an injective map between the natural numbers and this collection, some injection into it, so to speak, then I have that other collection is infinite and by injection I mean it's a one-to-one map, so it's like tying a piece of string from every natural number to one and only one object in the other set and in that case you have an infinite set. But of course this might seem a little question begging and then I've already sort of wheeled the natural numbers in right at the beginning. So let me try to explain that, so let me try to rephrase that and see if I do a correct job and then correct me if I'm wrong. So the way that mathematicians talk about whether or not let's say a set is infinite is they say there's this concept of one-to-one correspondence that imagine you had, just imagine that you had an infinite set, so you have this set of numbers and for each element in that set you are mapping the one element to an element of another set, so you tie the piece of string, sometimes the example is used, imagine there's an infinite amount of people and an infinite amount of seats, there's one person for one seat and if there is this one-to-one correspondence each element is matched up then the mathematician concludes okay this additional set is also infinite, is that correct? Yes, if one of the comparative sets that we're comparing is the set of natural numbers which is sort of paradigmatically infinite. Okay so how do we get to the set of natural numbers, so we're not talking about the other infinite sets, we're saying the natural numbers, how can we understand that as an infinite? What does that mean? This is a good question, so I'm trying to think about the right way one might answer it. I mean in some sense I suppose it's obviously not finite in that there's no natural number along the natural numbers such that it has the same size as the totality of the natural numbers and it has this inductive property, I suppose that's probably the most natural thing to think of. So if we, well how are we for this? So I suppose maybe where we get, maybe one of the more philosophical we think about what the natural numbers are and try and think of it as being a kind of a given that we accept so that it's, so if we take one step toward the horizon then we can take another step toward the horizon then we take another step toward the horizon and then we can, if we consider that process so to speak coming to its metaphysical kind of limit then that would give us a kind of a model of the natural numbers in some sense as to, yeah, so I think, so I'm probably just really answering there I think why we have the natural numbers rather than answering why they're infinite. So there is another definition of infinite we could give which is known as, which is from Dettokind which is if I can take a set and inject it into itself then that set is infinite. So this is, so what do I mean by that, it's one of these one-to-one maps. Okay, so let's first convince ourselves that it doesn't work with finite cases. So if I have five sheep I can't make and then consider what we call proper subset so subsets of that set of sheep which are not the whole set, there's no way for me to do a one-to-one map back into it if I just tie the string so to speak back into the set I'll have to eventually tie, the two sheep I'll have to tie it to one sheep. So maybe if we imagine having sheep in two paddocks, we have five sheep in one paddock, four sheep in the other, if I try and tie a string from the five sheep paddock into the four sheep paddock at least in one instance I will have to have two sheep in the five sheep paddock lining up with one in the other paddock. So it's very simple to say, often easier to draw than saying. So that doesn't work in the finite case but in the infinite cases we do have this kind of property. So a classic example here would be I can tie a piece of string from every natural number into every even number. So this is something that infinite sets have. So we'd agree that some natural numbers are not even numbers so that we actually have a proper subset of the natural numbers here which is the evens and I can actually make an injection so I can tie a one-to-one, make this one-to-one map between the naturals and the even numbers. And so that's another way of saying we have even numbers. So that might be, using that definition we might feel perhaps more comfort with seeing that the natural numbers are infinite. That only works if it's the case that they are infinite though. So that would be like a property or a counterintuitive property of an infinite set is that you can take let's say every other element and still tie them in a one-to-one correspondence with every element. But that still only works when you get that infinite set in the first place. So with your horizon analogy, so it's every step you take you're not getting closer to the horizon. If we follow that line of reasoning intuitively I would think I'm down with that I think depending on what the metaphysics of numbers are but maybe we'll come back to that later. But wouldn't that mean you could never complete the infinite set? So maybe you could say something like there is not a finite amount of natural numbers but wouldn't that necessarily mean then you couldn't have a set of them, you could have all of them? So I see there are two things going on here so in the first explanation which I think is a good way of setting up what we think the natural numbers are but perhaps not a well not as grippy a way as describing why they're infinite. So I think that the datacind approach to answering the question of why they're infinite if we study the natural numbers and we talk about them we think it does have this property that I can inject it back into itself. So we might still worry about whether there are natural numbers but according to that definition of infinite we do seem to have something like infinity going on there. That said though so if we want to open up this other side of things so if I get this sort of metaphor of you know walking up to the horizon and I think you're right to say you know so there is a legitimate worry about completion of the natural numbers so to speak along this in this metaphor and you might wonder well can we talk of something as being a legitimate totality if it can't be completed? I think you know the standard sort of you know very sort of philosophical answer to this is it depends what you mean by completed. So in one very obvious sense yes not even in principle kind of you know like some ideal agent or you know machine you know get to the horizon and complete that process that's not what's going to go on. But maybe that's not what we need there's not the notion of completion that we need to be able to talk about collections for example and maybe that's what we want with a theory of sets so we want to be able to say well you know we've been talking a lot so far about these things called the natural numbers and you know this this collection of things and what is it that makes it that we can talk about them we can kind of give definitions of them sometimes I gave a kind of metaphoric definition we could we could get more technical and give a kind of axiomatic kind of definition but there does seem to be some sense in which we can pin down modulo some metaphysics whereas at least up to isomorphism anyway what we mean by the natural numbers and so that seems to mean we're talking about an infinite collection we've described it in some kind of fashion so maybe being able to describe it is the right kind of notion of completion to get this sort of thing moving it is a weakening though of completion so we should be careful with that. When you say it's an infinite collection for me I would say it depends on what you mean by collection because if we say something like there is not a finite amount of natural numbers then it seems to me that you can't actually be talking about well I guess I would say this any collection that you're talking about has to be a finite collection given what we mean by collection so if I say there's a there is a never ending amount or a never a never completed amount or a never totally you cannot totally encapsulate all of them then it would seem like we can't really talk about the natural numbers in the collection sense maybe we could talk about them as like a we could say there are these objects that have properties and we could talk about like the type of properties that the object has and we could say so you know that all the natural numbers have these particular property but we're not talking about all as a collection or all as some quantitative amount it's more like a property description of the thing that makes any sense yeah so one could be perhaps resistant to so there is a move that I'm making here so I'm kind of reifying the natural numbers when I talk in this kind of way I'm saying I'm comfortable enough talking about the natural numbers and the even numbers all of these classes of them that in some sense I've if we look just grammatically at how I'm speaking I've kind of I'm talking about them like they're objects of some kind yeah um whereas I feel like maybe you've got a kind of what we might think of as a metaphysically heavier or thicker notion of collection and I think mine's quite light I just really want to track our theoretical ways of speaking about these things and just be faithful to it being an object in the sense that because I'm a logician so you know I'm quantifying over these things what are the rules of this kind of game um you know that's the really sort of uh weak way of talking but yeah what's the one of the rules of talking about infinite collections and we do do it so how does how does this all come together we might still want to drive a kind if we're being you know put out you know metaphysics hats on we might still want to drive a wedge between a thicker more substantive notion of collection that involves some sort of very satisfying notion of completion like you know I counted them all I've got them here they're in this bucket you know yeah as opposed to well I can merely describe the properties that these things have in and so you know maybe I've got a criterion where I could uh if you offer me one of these objects I could tell you if it is in the set or not that kind of thing yeah that that might be a so that you know then we might move to something a little weaker so we could take us we could take some gentle steps toward the internet like so from having full completion we could think about the even numbers they're not that weird so I can talk about kind of rule for for capturing all in flight if I start with zero and then I add two and then I add two again and I add two again if I eventually get to that number with this sum up putative number of that process I can tell you it's it's in there if I get past it and it wasn't in it's not so that's a kind of we might sort of start thinking about different rules for there's a way of talking about infinite collections you know for rules for creating them this is this is still quite weak in comparison to set theory so let's talk about a little bit about that notion of the metaphysical weakness versus something with a little more I like that I think the term used was grippy I like that I'm gonna start using that if I were to ask a standard mathematician or logician what they are talking about when they're talking about a collection what is the answer is there something is it an actual something is a collection a concrete something or is it a way of talking I think there's a lot of variation in the mathematics community about this kind of thing this is and look I've never done any like studies of this I think this would be an interesting area for people to move into so my evidence is purely you know it's quite anecdotal I do talk to colleagues about this a bit um I mean there's what's the famous quote is that you know so mathematicians are from Monday to Friday are formless and then on Sunday they can pull out their happy Platonism and really believe in the natural numbers and I think this does speak of while I'm using it does speak of a certain kind of variation if you're trying to be careful yeah so you know so and when are people careful usually when philosophers talk to them so when you know these philosophers like to ask these sort of thorny questions and understandably mathematicians feel like that might be the the philosopher in question might be trying to corner them into a view that they don't like if you take a kind of formless sort of view where you think of it as well it's just a way of talking that happens to help me solve these sorts of problems it's kind of in some sense you can almost think of this sort of game that we play but there are pragmatic implications if it's not completely we do deserve research money you know but uh but it is um you know but we don't have to take it all too seriously so that's a and if you talk like that I think it's you often quite safe from a lot of problems but I think a lot of mathematicians have some quite deep commitments to the existence of natural numbers and to particular sets and what do they mean when they mean an infinite collection uh I mean I think they probably would tend to give you the kind of definition I'm you know something like datacines or a set theoretic definition like I gave first set theorists again have wide diversity of views about these sorts of things so I suspect if you're looking for the intersection of these views it probably gets relatively close to what the axioms and the textbooks kind of say and if when people stray further away from that things get a little bit more uh out of hand so would you say it's fair to to just make a statement about the discipline not necessarily a positive or negative way just depending on your perspective that there is no at least explicit metaphysical claim about the objects and mathematics that they are existent things that exist in some other universe or if maybe the objects and math are just ways of talking to solve problems would you say that they're the the standard uh if you can put that in quotes the the standard response as well it's we don't really make a claim either way yeah I think uh yes no I think in general that would be the standard response so one has to be careful so it means we mean standard yeah this is the safe kind of response better um but yeah if you're talking to a mathematician on the part at the pub on friday you know they might wax a little bit more rhetorical okay but uh but yeah I think that is the standard response I think that's um a good kind of answer because it is kind of safe and it does talk about you know it sort of just leans back into a kind of common framework but I think to get any further than this you do need to take kind of philosophical sorts of moves so um I mean a classic kind of approach to uh the ontology of say um so I I work mostly in set theory so the ontology of set theory would be loosely sort of quinian and so quin has this famous slogan to be is to be the value of a bound variable but to put which is pithy but the idea of it is that if I hold a particular theory really there's nothing more to what exists according to the theory than what the theory tells me exists we don't have to worry about whether numbers are like tables and chairs or things like that really this is just a manifestation um or a corollary so to speak of what my of what my theoretical commitments give but that's only one view there you know there are other more robust sort of Platonistic kind of views out there more constructivist views but it provided if we're all if we were all working in the same theory then a lot of these sort of meta ontological concerns really won't affect the actual mathematical practice and that's perhaps for the big big reason why mathematicians don't need to answer this question in some sense it won't really change things so if I were to say that from a metaphysical standpoint from a philosophical standpoint the concept of an infinite set is something that has no metaphysical corollary because there's some what we mean by a collection or what we mean by a set is something which is necessarily let's say bounded or finite or concrete or actual now if that's the position that that's the claim wouldn't that have implications on the mathematics though if it's true that you can't actually use that concept or if it's true that the concept of an infinite set is well I mean incorrect here we're getting into some of the deeper darker kind of metaphysics sort of stuff I suppose because I feel like just because you can't find a the principle that you're relying on here sounds like if you can't find some sort of metaphysical counterpart to my theoretical paraphernalia so you know this theory says there are natural numbers but there aren't any of those or there aren't sets of natural numbers around my theory doesn't even though my theory says that they don't exist if that's the case then you can't use such a theory now that that seems like a bold claim like can you give an example outside of mathematics would you say that there's any theoretical area of knowledge where the objects in a theory don't have to correlate to existing objects I mean I think I have to give mathematics a little bit of slack here in that even though there'll be wide divergence as to what abstract are so what abstract objects are most people would agree that mathematical objects are abstract objects there is of course this group of people called the nominalists who don't agree with this because we expect in philosophy of mathematics particularly that we'll have a lot of diversity here but so even if we grant the existence of abstract objects let's say I can at least entertain that as a coherent idea but what if I were to say an infinite set isn't even an abstract object because it implies some kind of a there's self-contradictory notions about collection and completed even if we say numbers are abstract objects so I mean if you're going to go if we're going to grant something like the principle that I think is underlying what you're saying and we really say that the infinite sets just aren't even abstract objects then I think maybe one would have grounds for saying that theories that admit them are just wrong and at this point you're going to end up in somewhere quite difficult like finitism or ultra-finitism which are so believing that there is a largest natural number and then you might ask the question can we add one to it and then things get difficult there so let me try to answer that yeah that is definitely my disposition is to be more of in the finiteist camp so for that question I don't think it follows that there is a largest natural number for this reason if we conceive of numbers as concepts as ideas in our head then it might be the case that at any given time there is a largest number that somebody is conceiving just like there's a largest sentence that has been written or anything like that but that doesn't mean that you can't still create a larger number or a larger sentence so it would seem weird to say there's a largest number but that's without including any actual affinities we could say there's no inherent limit to the size of number that you can conceive of yeah okay so this is a this reminds me of some so depending on we want so we probably don't want to think of as a specific time or something like that was we probably want it's it's a kind it seems a more like a modal kind of claim in the sense that it's like you're saying well perhaps there is a largest natural number but it's always possible maybe we could go to another world so to speak because it's a way philosophers like to talk about these sorts of things but it's a nice framework for it so it's possible that we could that there's always possible there's a large number is would that be an adequate way of paraphrasing what you're saying no i think it would be a much more it it would strictly be saying just like at any given time there's literally a largest sentence yeah that somebody is think you know there's a finite amount of words in that sentence yeah that doesn't mean that you can't take that sentence and make a larger one but at any given time it's still going to be finite so okay yeah so that's what i would say is the same thing with numbers there is literally at any given time the largest sure number that has you know grams number raised to a power of itself or something like that and somebody else could come up with a larger one but until they do that number there's no that number has no existence yeah all right so i wouldn't okay that's fine um i still feel like you're pushing a lot of there's there's a lot of possibility instead of like pushing this like so he's because your whole way of saying is it doesn't mean that someone couldn't make a larger number right so this is a you what you're doing is using possibility to to allow this expansion things and this is a common kind of move so this is so um probably the leading proponent of this sort of view at the moment is Kakua Ostein Lenevo in who's based in Norway um and he does this with set theory but uh a guy called his supervisor Charles Parsons used to do this with the natural numbers trying to develop a kind of a modal way of looking at infinity so the idea of looking at if we could go to an at a larger number maybe there's always a possible so i say possible well mainly just i don't want to i'm not super committed to what those are or anything like that but really this is kind of another theoretical framework but one of the interest the interesting idea here is i think using possibility or modality as philosophers always call it uh to analyze or um the notion of infinity maybe you could get rid of infinity if you just use possibilities so you could look at it as a kind of a vindication perhaps of Aristotle's idea that there aren't actual infinities um and this yeah this is a very interesting line of research and and yeah i mean i i do think infinity and possibly a kind of deeply embedded yeah okay so i i want to re-ask the question about using concepts that maybe don't correlate to any metaphysical existent so let's say that mathematicians get a special exception here that they get to do this is there any other area of thought that that can get away with this that we would say hey you can't you can't use a concept that doesn't correlate to anything i mean i think this is the the the reason why i'm trying to wheel in the abstract it as a kind of sort of a get out of jail card here i feel like um if we look at other areas of theoretical research say like uh theoretical physics or something like even though uh things get very complicated there there is still an idea that um these are pointing to you know carving nature so to speak at its joints yeah uh and even though it's very small and very difficult to conceive uh there's an idea that this is this is somehow representational and i'm not sure that is so um if the data point you know the overall data for mathematics is the same as it is in it's a physics in that kind of way uh and why i say that is i suppose if we consider radically different foundational viewpoints on mathematics that the ontology looks just so often looks so very very different i mean so an interesting feature is usually we end up with the natural numbers being in common so everyone you know it takes a long way to get down to there but even then if you consider some of the non-classical kind of uh para-consistence sort of approaches even the natural numbers can be wildly you know different and you know it's even very difficult to compare different ways of looking at at foundations you know what people think the landscape of mathematics is okay so yeah so that's maybe that's just to try and answer why i think mathematics might be a little different to other areas in this sort of sense so different theoretical frameworks can radically change the landscape in ways i think which are a little different do you think that there is a necessary reason to have that metaphysical almost agnosticism in mathematics because not being a mathematician and not knowing the huge amount of work that's in mathematics it would seem like you can you can have a mathematical framework and you can use math and physics and everywhere that you have applied mathematics it seems like you can have that metaphysical correlation so if an engineer is using mathematics they can use it to actually reference you know existing things in the world and their properties so and even in physics like you said there seems to be this representational part to mathematical physics is it's describing existing things so if you can do that then why would we what value is there in math that is disconnected from existent phenomena well i think there are there are probably two so particularly if we're thinking of infinite collections i think there are two kind of answers one could give so one's a kind of a practical answer so even though most of how we understand the physical world these days suggests that really it's finite collections of things bumping into each other and so we should just be able to you know so to speak model all of this with a computer because they're really finite many points to do this it actually turns out then a lot of cases that using the tools of analysis and calculus which involve which presuppose infinite collections actually makes things more efficient and simpler to calculate so there's a pragmatic answer one might have there a more principled answer might be to suggest that uh how we put it um we often need to make theories about infinite collections even though so for example the theory of the natural numbers so we want this so we don't want a theory that goes up to however you know n which is really really big and we'll probably never count that high because we always have this there's this in principle idea well there's still m plus one exists or maybe even that it's possible so because usually actually what happens is between these are modal ways of doing things and the the existential way of doing things they actually come out being the same theory in some sense you can inter translate between them so you might see this modal ways sort of being a more sort of philosophically satisfying but ultimately they're really going to do the same kind of work so what's in principle stopping you from considering this next possibility so if you really want to close out all of the possibilities then suddenly you're in this world of the infinite and so why how did you get there because in principle there was no reasonable place to stop and well if you can find it if yeah I suppose things if you can find an adequate answer to as to where we should stop then that that might be a way to answer that question so suppose if you may be um a strong physicalist and a nominalist you might just think the atoms in the universe is all are and we shouldn't talk about anything else and maybe then you could rule out the in principle uh solution along something along those lines but I think the pragmatic one probably still will stay for some time well this is this isn't my position but um something a number I would think of if somebody were going to take that um kind of physicalist approaches so you have the base unit in physical reality which is the the plank unit you have the absolute amount a number of those in the universe and then you could say something like a power set of all of those units so every possible combination with every other possible combination whatever that is you couldn't possibly I suppose that seems like a bit of a leap doesn't it in the power set that there aren't that many objects in the whatever so if the plank concept is the the smallest okay interesting then you so I would say you bring in set theoretic kind of content as soon as you bring this this because what you're doing here is you're considering all the possibilities that yeah of rearrangement and that's a modal notion or it's one of these things that gets it starts us off on the infinite kind of into the world of the infinite I suppose maybe one well this is a side note but it's actually directly related to this the reason I say that is I just had a conversation with the gentleman about consciousness and we were talking about this kind of this idea of emergence in this theory of you know the strict physicalism would you say that that kind of precludes this idea of radical emergence because if you could have emergence in addition to the the bits of matter then you would still have an additional thing or just after my head when you're saying that I think this is outside my area these days I yeah consciousness is a yeah I just have colleagues who work and who do good work and I can't weigh in okay okay fair enough so let's get into the standard a concept of infinities and this notion of the infinite number of infinities sometimes called the hierarchy of infinities so for my audience maybe that's unfamiliar with the standard way that mathematicians conceive of this I'm going to try to do it justice and if I make an error please correct me so there's this idea of countability in mathematics which says in principle you could if you had a list of infinite size then you could list out all of the natural numbers one two three four five and so on you could those are in principle listable but there are some numbers which are in principle not even listable that even if you had a list of an infinite size those numbers could be on the list and there's a supposed proof of this by a guy named Gayard Cantor who came up with the clever method which is called diagonalization which is essentially a way to point at a number that by its nature must not be included on any list even of infinite size so therefore because there are numbers which can't even be listed on your infinite list you have to have bigger sizes of infinity those numbers are out there but they're not even in the the set of natural numbers they're base level of infinity you have a larger size infinity that's where you get something like the real numbers let's say so is that a fair way yeah yeah yeah that's I mean that is fair here's maybe like a like a more palpable way perhaps of describing because you can't actually explain Cantor's there relatively quickly if you might have a quick search please yeah so imagine you have a collection of coins so I'm Australian so we'll have 10 cent pieces right and we'll but they'll go we'll have infinitely many of them we're very rich it's a very ideal situation so imagine them sort of going out to the horizon and they have a particular configuration of heads and tails okay so we've got one of them for every natural number so to speak we could write a natural number on each of them in ascending order as they go out to the horizon now we might then consider okay what are all of the different possible arrangements of heads and tails that we could have for those for this infinite sequence of coins so imagine that so it goes out to the horizon now imagine it's being at this giant table okay so it's a table with all of the different arrangements you might wonder if the the list of different arrangements could also be labeled using the natural number so another way of thinking of this is suppose it were then this table would be a giant infinite square so there's it's an odd way you know we have to be a little careful here but the approximation works well out of the well so then what Cantor says is okay well suppose you did have this giant square with all of the different arrangements on there I can show you an arrangement that isn't in your grid and so what he does is says go down the diagonal of the grid so just consider the first square in the first row and flip the coin then go to the next one the next diagonal so the second column in the second row flip the coin second third column in the third row flip the coin keep on doing this all the way down again to infinity it takes a while to do obviously so once you've done this though you have by definition a new row if you flatten this out so to speak so if you flatten out that diagonal and take it up consider that is a configuration which can't be in any of the ones we've got because it's different by definition of from each of them so we've shown though so what you show there is that actually if there were a square there would be something that had to be left out and so so we can't be a square so you can't represent all of the possibilities in a square so that means you can't list all of these infinite configurations of coins using the natural numbers and so these actually are like so what if we if you turn these into ones and zeros rather than heads and tails this is actually just a representation of all the infinite binary decimal place knobs these are and sets the reels between zero and one so these are what these often called logitians reels and yes so we can't enumerate these but these are all of the possibilities so to speak so and we take that principle and not only are there one size of infinity two size of infinity three size there's an infinite number of infinities yeah which seems very hard to wrap our head around now canter believed that there was such a thing as the absolute infinity the infinity which no greater could be conceived um and from my understanding canter also identified that with god he said that yes absolute infinity is god and as an interesting historical note he also believed that the absolute infinity god spoke to him and told him about the theory of the infinity of infinities which i think is an interesting it is interesting yeah maybe it happened maybe it didn't but when we think about sizes of infinity is it fair or is it correct to think of size in the way that we think about it finite things is it fair to take that concept and apply it to infinite size or is that is that is that a mistake so there's this concept in mathematics cardinality which makes a great deal crystal clear sense when i think about finite things but then when you talk about the cardinality infinite things i've spoken with a few people on this and they say well you cut that that intuition from finite sets breaks down and you can't there's it's something new i think i mean that that's all so yeah there's a famous old paradox called galileo's paradox which um where you try to you think of all the square numbers and you realize well it's a bit like what we're talking before and you notice that they are proper subsets so there are some numbers which aren't square numbers uh which so some natural numbers which aren't square numbers so you think well the square numbers must have less size than the natural numbers on that argument and then the other way of looking they go but for every natural number there's a square number so there must be as many square numbers as there are natural numbers so actually they must have the same there must be at least as big i hope i didn't there's a bit of negation there which i could go wrong but it's easy to look up so what do we what do we have here so we have two principles that work just fine in the finite case right so whenever you have a finite set and you consider a proper subset of a finite set so you have five ducks and you eat one then you have four ducks and and that is a proper subset and that means it has less cardinality that always works in the finite right similarly this idea of saying that we have the same size mathematics call this a bijection or this one to map one to one map so which is onto which means it exhausts the other collection so we tie a string from everything one set to the other set just so that we only have one object going to one object and we exhaust the other collection that means you have a bijection that's when we say we have the same cardinality that also works perfectly well in the finite case but they won't both travel into the infinite case so well so the proper subset one is not the one that mathematicians generally use we tend to say that two collections have the same two infinite collections have the same size if there is a bijection between them but we can we can have proper subsets of an infinite collection which have the same cardinality indeed that's the kind of that's the datacine definition that's a hallmark of an infinite collection you can take an injection of it of the collection into a proper subset of itself in fact bijection into a proper subset itself so some of the counterintuitive conclusions that follow from that be something like this infinite cardinality of the infinite set you can so in that size I guess for lack of a better word you can remove an element and you would still have the same size yep now that there's somebody that thinks in a finiteist way it seems like the concept of removing an element just necessarily implies exactly one less element in terms of the size of that set well it does imply one less element it certainly does contain one less element it's just that that is not what the measure of size that we're using here is is is taking into account okay so he so yeah so if we had the naturals and and let's get to the naturals where we take we throw out an element so this is you know make it palpable so we throw out the zero the zero of the stone so to speak and well can we make a bijection between them yeah I just put zero to one one to two so it's easy to do this again so if you're wanting this to live up to the the subset notion of cardinality it just won't do it so we have a fracture here of some kind you know that the concept is that works in the finite case has become broke you know it's pulled apart so this is what often happens in a lot of logical kind of studies so often we'll take a concept into a more difficult kind of scenario and it becomes more subtle the concept actually these things that were equivalent under very normal conditions are no longer equivalent later on and so you have to kind of make a choice there are actually people who do work on this sort of subset notion of cardinality these days so a colleague of mine Leon Horstin has done some work on this but it's so people do work it it's very difficult to get to work so there is something very elegant about the bijective way of dealing with things which and also which seems useful so people do play tend to use this one more often so perhaps arguably for pragmatic considerations pragmatic you know and simplicity yes the that's a consistent answer that I've heard both from logicians and mathematicians is that using this theory of infinities is massively practical where when you if you were to take the finitist approach the amount of work involved to essentially get the same answer is like way larger is that is that fair yeah I think I mean I'm not like I don't work on like a large scale I mean so if you were interested in finite data port analysis where you're doing fluid mechanics and stuff like that those guys would know more about this than me but um yeah my impression is definitely that it's using calculus and tools like this is it's efficient so it seems like the logic of working through the infinite sets and the other sizes of infinite sets works in so far as you accept that base level axiom that there is at least one infinite set if there is at least one infinite set well then you have an infinite number of infinite sets but that initial step for me I know is a hard is a sticking point I know for a lot of people is is really a sticking point and I think it does come down to metaphysics I'm glad you pointed that out that this this assumption that when you are dealing with particular theory that the objects and concepts in your theory have to have some meaningful metaphysical existence to them what would you say to somebody that would claim it is a it is reflective of a shortcoming of the orthodox way of doing mathematics that in order to get their theory to work it requires an acceptance of this initial axiom that has legitimate room for skepticism about it if you're skeptical about whether or not there are actual infinities or an infinite set as a reasonable position at least so what would you say to somebody that says well that's a problem that it takes it you got to get to that first step in order to get to all the other steps so I mean I I think we come back again to this sort of like either a pragmatic or a principled kind of defense and I think when you're facing skepticism you really I mean if there is reason for skepticism it's very rare to be able to mount an argument that will convince the skeptic otherwise this isn't to say we shouldn't try and I mean I do think one can provide interesting arguments for the existence of infinity I thought I think data kinds original one is a fun one but ultimately perhaps the best ground to to to respond to the skeptic in in these cases is something more pragmatic so something along the lines of pointing out that if you do have a finitistic theory well as far as we know there really aren't any there aren't many that we know to work very well so when you mean work based on what metric do you mean you don't you can't do calculus or something like that can't do calculus yeah that's a I mean we have to be careful so finitism is very low down in how things go we you know we might so you know maybe you might say well maybe if I can define a computer which could give you the rule for a particular infinite so I mean where I mean it depends where you're on this sort of spectrum do you do you worry about the even numbers do you worry when I talk about the even numbers because it didn't seem like it when I was talking so my own my own personal intuition is to think I'm very wary of even the language of mathematics when we say something like the even numbers are we describing a collection of objects or are we defining a concept that we're creating so if we're defining a concept then I would say it doesn't get you to any kind of infinities because as you can have as large a collection as you please and it's always going to be finite if we're talking about a collection of objects like the even numbers and I what I mean by that is as if I were referencing the chairs in this room that I'm talking about some external platonic object I'm also very skeptical of that because I when you deposit the existence of the platonic universe you're positing the existence of a great number of things sure I would like to have a really restricted ontology yeah yeah I feel like I'm not quite getting my so I'd like to get my finger a little more on what you mean when you when you're talking about the when I talk about the even numbers what do you is how are you paraphrasing me so one thing you could be Toby's talking about Santa Claus now you know so we'll just pass over this and not worry about but then I could say something that's I could say well you know for numbers visible by four then it's an even number and that's true so did I say something true or you know so what have I done there what what what what have I in my own yeah so here's what I would think because I'm very partial to this idea of the correlation of our concepts to existent metaphysical entities what I think numbers are are placeholders for concepts and what we're talking about things like you know 2 plus 2 equals 4 we're not talking about objects being combined to create a new object like if you had the abstract object of 2 and you somehow combined it with the other abstract of 2 you would abstract object 2 you would get this new entity of the abstract object of 4 I think what's going on is it's it's like um shorthand to describe necessary relations between any things so we have this amazing ability to abstract so on like on this table there are you know five items and what I mean by that is not there are the items in addition to the existence the number five which is also here on this table it's it's a way of talking about it's a way of stripping out the units to just talk about the like a quantifier on a on a unit so so it's still again just just surely be talking about the way that I can see the mathematics you still have a kind of logical certainty in the area of mathematics I'm not claiming as some of the intuitionists did that it's all just kind of made up like the laws of law just made up I don't think that's the case but I'm also not claiming a mathematical platonism where you it requires the existence of independent abstract objects out there I think it's things we come up with in our head that can perfectly correlate to the world and the cool thing about numbers is that they are as we're manipulating them in our minds they're unitless so I can talk if I wanted to tie my abstractions into the world I have to say you know one cell phone here one microphone here I have to have the one x I have to tie it to the thing but when we're just developing the theory of how quantity works we can yank the one away from the that underlying unit which is yeah I mean this reminds me of some logicism I suppose yes okay so yeah I mean so like this abstracting back from reality and this is what would kind of let with numbers yeah and I suppose maybe this is like a kind of a metaphysical kind of sort of shoot off or whatever they stand off at this point because I think I just have probably have weaker commitments to this sort of thing so I I'm not too interested perhaps in the distinction between concepts and genuine objects I'm just interested in what might the paraphernalia that my my theory needs me to talk about and so when I end up talking about natural numbers I posit some rules about how these things work and I and this allows me to then say things like well yeah you know the set of numbers divisible by four is a subset of the even numbers I can say that that makes a certain kind of sense there obviously I can paraphrase this back into something that into without the collections perhaps in some cases but eventually we'll get to places and this is why set here was invented where it gets harder to do this so we end up wanting to talk about sets and sets of sets of things and and this becomes quite a useful thing so we have to go and just dig into kind of analysis we want to do that and it's not trivial and so this is where Cantor originally came up with these ideas he's working I think there's some problems in Fourier series which aren't I haven't done that kind of thing in a long time so I can't speak about it he but so these did sort of crop up though out of real problems I suppose yeah but coming back to sort of the question though I feel like the metaphysics yeah I feel like yeah there's kind of maybe a metaphysical difference here I think you're maybe being I feel like maybe you're too committed to what a collection should be and maybe maybe mathematicians and set theorists aren't talking about collections in the sense that you're talking about maybe we are talking about what you would think of as concepts and maybe this is how you would you should you should translate into your metaphysics what a set theorist is talking about because the standard mathematical answer doesn't talk about these sorts of things they don't have to tell you you're wrong you know you can have a debate about this but this is strictly philosophy room kind of thing yeah well it's interesting when you say that that mathematicians aren't talking about what I'm talking about with collections but doesn't it go one step deeper is that they might not be talking about anything yeah I mean there's that that's a possibility like I mean that that that's for all we know yeah but that doesn't that doesn't I would I'm very biased in this and I would say well that's a demonstration of some error in the way that we're conceiving of a theory if if we accept this idea that well I'm talking about this thing but it's not a thing I would say well you got something's got to be revised because you got to talk about something well no we're not saying we're not we're not I'm not saying we're talking about a thing that's not a thing I'm saying that I'm I'm leaving a room some some fallibility here for what a thing is for what I mean here's an example so but even by my lights it could turn out that there are no sets as as I think they are because perhaps the theory that I use CFC is inconsistent in which case my mantra to accept what my theory tells me is is a bit broken here because my theory tells me everything is true now so I have to revise it so I have to use something different so yeah there are there are places where even though there are none that are so yeah what I'm getting is we're leaving open I'm just being honest about where these sorts of theories are are out and they you know that they that would be positive evidence that there are no sets if we could show it's inconsistent but it's very right very difficult to show these sorts of things but there's a certain asymmetry cut from girdle that means it's not showing inconsistency could happen but and we will never get positive evidence that it is consistent beyond using stronger theories which are even riskier so to speak but maybe this is a bit of a distraction to girdle that's very much relevant so as a general philosophic principle you would say that if we're if we're talking about sets we can describe them in a particular way we can describe properties and rules of them and be genuinely metaphysically agnostic on whether or not what we're talking about exists um no I'm no I'm not quite there what I'm saying is I'm saying that I'm using this is this is my best theory according and that I that so I'm running a this again I'm now running the mathematicians line so I've just been really careful um so but yeah I mean this is I don't know if I I have beliefs that the ontology is much deeper than this I suppose so I'm very wary of grippy ontological considerations I feel like they they pull at our intuitions that they probably just don't work in all contexts so when I say that certain things exist I mean that according to my best theory which comes from a variety of considerations my best the why I chose this is the best thing my best theory tells me that these things exist and so I say that they do am I telling you that it's impossible for this theory to be wrong no I'm not I'm not going to go that far I think there are even there are there are possibilities that could show up I don't think they will you know I believe this theory in the sense I think it's the best theory but I can't rule them out that could show that say the theory is inconsistent personally I'm less impressed by metaphysical considerations to try and show me so so because so you're arguing that you have a lot of difficulty thinking about infinite collections and I'm gonna say well okay maybe so my move here is to say maybe maybe I'm not talking about collections as you as you say I mean something else but but I think what I'm talking about here makes a certain kind of sense but and I think it's coherent at the very least and I think it's pragmatically useful so you would say that if I were to come along and say well infinite sets don't exist you're gonna say it's not really that that type of critique is yeah I'd be surprised if that kind of approach could be persuasive I feel like even if it's correct even if you said okay well I grant they don't exist oh no if I so can you fill out that question so if I were to come along and say I have some purely metaphysical proof based on the metaphysical metaphysical stats what numbers are the numbers are concepts in our head they do not exist separate of our conception therefore all sets must be finite because you can't conceive of all of infinity you would say yeah I don't think that would would sway me much at all it would probably make me think that you're talking about something really something something different we're talking past each other in some kind of way yeah okay so last question on this this has been an excellent interview and I really want to be able to know what you're talking about and what mathematicians are talking about because I've bumped up I keep bumping into this wall about conceiving of sets I think there are no sets and based on what I think numbers are based on what I think a set is so if we're talking past each other totally granted 100 languages ambiguous so this makes sense what are you talking about well I think coming back to I mean we don't have to get too deep into the sets of sets of sets or any of that kind of thing coming back to that kind of analogy for cantor is a good place to be so what I think and a lot of problems in that we still think about in in contemporary set theory are based around that kind of area so thinking about what it would mean to have this infinite sequence of coins going out to the horizon and then considering all the different possibilities that there are this is the continuum so this is the reals and the relationships that we have there are very difficult to kind of sort out so what I'm considering here is so the another way of looking at those heads and tails these are all these all represent different so if I consider if the fifth coin is heads and the seventh coin is heads we'll put that in a certain set so each of these rows actually represents a different set of natural numbers so if you accept this sort of thought experiment in some sense you're already into the realm of infinite collections so it's not that difficult to kind of smuggle them in in some kind of way and what we're interested in here so maybe where what set theorists are concerned with is it's not just even about what rules you could use to rearrange the coins we're interested in what really does it mean to have all of the different ways of arranging the coins and this maybe a more philosophically satisfying approach to analyzing this is modal but how to get a good understanding of what this range of natural of this range of different permutations is is a very difficult one and so this leads us to the problems like the continuum hypothesis where we so it's very easy to explain so we know what Cantor's theorem is we know that the reals are larger than the naturals the continuum hypothesis says that there's no size in between and you might think in some sense this is the first and easiest and most obvious question to ask at this point we now know after it took about 70 years for people to discover that no we that's EFC or ordinary theory couldn't actually can't actually figure out whether or not that's true and at this point now we still have no sort of palpable answer or real beyond sort of pragmatic ways of trying to approach this sort of problem it's so even at its beginnings we once you even invite infinity into the door into your house so to speak it ruins everything it makes things very challenging yeah yeah does it ruin everything it blows everything up maybe doesn't ruin it but it all of the standard intuition and all the the way that we think about everything in our lives really it just says you know I've been none of this applies well no I don't think it says none of it applies I think it says some of it applies and some of it doesn't and you have to kind of be very careful what is the when I think of all the you know when I think of additions subtraction multiplication sets I think of all the standard rules of arithmetic and how they correlate with infinity I think of the nature of numbers and how they would correlate with none of those seem to carry over the parts of them do okay I mean so I mean you can't be too strong I mean so the point is that as you move into infinite cases but I mean if this happens in other areas of mathematics too just so just because addition won't work in the same way in various areas of algebra as it will in the natural numbers so similarly moving into this space of the of infinite objects they don't work exactly the same like so some some laws that you hold really close and dearly to your heart they just don't work there in that kind of space anymore but still we can see something common about and there's a reason why we still think of it as being addition we can actually show that you know that they instantiate the same kind of pattern if so if you moved in sort of more category-theoretic approach so you can still see that it is addition of a kind but it just doesn't behave the same way okay okay I know I said one more question but I got to sneak one more question here because when you said that it just came up in my head so when I asked you know what is it that you're talking about and for me right the way that I'm conceiving of numbers is it's this abstraction from the existent things so the number the numbers are these abstractions but when you and the example you gave was the coin the impossibility of the coin in terms of in terms of metaphysics that infinite quantity or descriptor is it tied to existing things or are we saying that it's not tied to actually existing things because there's no actually there's no actually existing things so it's no longer the way that we can see the numbers is no longer abstracting from the concrete it's necessarily something that's outside the concrete right so if I if I think of a I guess that the analogy by I have this concept of colors and I come up with the concept of colors because of experience of color the concrete thing and so I have now I have abstracted from red and blue and green I'm thinking okay I got the concept of color but if you do that with numbers one two three four five and so on it doesn't get you to that infinity because it's always about abstracting from the concrete so then what is it what are you talking about when you're talking about the that infinite quantity that doesn't seem to be experienced or anything like that just a real softball yeah no no I'm trying to think about how on project I mean I suppose I'm tempted to try to go back to something like let's go back to ancient Greece and think of Zeno and think about like this the kind of so let's forget that so Achilles is racing the tortoise but just forget the tortoise and just recall that Achilles runs half a mile he runs a quarter of a mile he runs an eighth of a mile he runs a 16 etc etc and we know this thing converges at you know at one mile so you know he like but we this seems like a very palpable way to think of a process in the world that has infinitely many parts I know this doesn't like line up with with physics but that's not that physics doesn't line up with our intuitions either this is a very intuitive kind of of description of a process that that uh so I'm not saying you can complete it but you can get the idea of completing it really easily so you can you can see what it means to yeah but I can't I think Zeno was right I think his I think yeah I think the resolution is Zeno's paradox is that spacetime is finite it has to be discreet because otherwise you couldn't have motion okay yeah I'm um I don't I don't share that one with you but that um what's another way of trying to look at it then um this may or may not work so but I want to try and convince you that there's some use in in talking about sort of infinite processes and that we indeed actually pop up into the transfinite relatively easily um so you'd agree that two equals two all right okay yep that's true that two equals two okay that's true that it's true that two equals two that's true that it's true that it's true that two and this starts to get very tedious so one of the things that the sort of implicitly sort of surfacing here is that you think you probably think well why is this tedious because it's true that no matter how many times I say it's true that in front of two equals two that will still be true what I've done there though is I've talked about infinitely many so my statement there is talking about the pos every different possibility of adding finitely many it's true that's in front of two equals two and then I'm saying at the end of it I want to hold that whole thing in one statement I'm saying that thing's true so in some sense I've I'm covering so this is another way of thinking about data kinds thinking about thinking about um sort of waves his thought experiment for infinity is it so this is a way of doing it with truth and so I'm sorry to say that actually yeah there is a reason to want I want to be able to say that thing now I want to be able to say that it's true that no matter how many times you put truth in front of something it's still that thing remains true and to do that my theory somehow infinitely sort of extended it I've iterated infinitely many times but not actually infinitely because I'm talking about the if that that last statement is true then there can be no limit to so what I'm saying is yeah but at any given time as if you say it's true that it's true that it's true that it's true yeah doesn't matter how many you have that's that is true yeah uh but however many you have it's still going to be a finite amount you can't ever it doesn't I would say it can't even be conceived to say actually stretching infinitely so I I might even you could even use the term infinite but I wouldn't mean it and and take the example where it has an actually infinite amount of it is true that it is true that it is true that doesn't that kind of occur right that kind of occur the actual you can't so yeah I agree there is all I can think of is this is the first one it's true that zero equals zero it's true that it's true that zero this is sort of it goes off I claim to infinity but there is no adding infinitely many in front so the thing that approximates this this is me saying it's true that everything along that sequence is true when I do that I've captured I'm talking about infinitely many statements so this is so another way of thinking about this is the other kinds way so you know I have a thought I think about this thought I think about thinking about this thought and then I can think about thinking about it as many times as I want yes but that's still that's still a discrete amount you can think about it as many times as you want still doesn't get you to that completed no so yeah there's no debate I that you can complete this right so in this in in the palpable sense of like so in the strong sense of complete where you can you can get to the end of it no I'm what about wasina though wouldn't isn't that the claim that you actually can't complete the infinite process because Achilles actually reaches that end point yeah but each of these processes takes so this is the standard line right so it takes less and less time so we can you know the sequence of times converges that's it yeah I mean obviously we have a deep for disagreement which might be worth exploring but yeah unfortunately I don't think we'll have time on this interview but I really appreciate your time I really appreciate your patience by the way now this has been when you're when you're getting to really standard orthodox ideas I think people often get frustrated when they become base-level skepticism trying to pursue it so I really appreciate you taking that time oh my pleasure thank you all right that was my interview with Dr. Toby Meadows hope you guys enjoyed it as always I did and I must say I'm still not there I still would be considered a finiteist I've had several of these conversations with people who really know what they're talking about and I'm just not persuaded I don't think you can complete an infinity I don't think there is an infinite number of sizes of infinite sets I think a huge amount of revision needs to be done if we're going to base mathematics on sound foundations as always lots more to say and I got a fun episode for you coming up next week make sure to tune in and have a great week