▶︎ 0:01 I'd like to talk about the difference between two fundamentally different perspectives on infinity, and that is the distinction between potential infinity versus actual infinity. Everyone might agree that there are infinitely many numbers, to be sure, but mathematicians and philosophers sometimes disagree about what exactly that means because, according to the philosophy of potentialism, yes, there are infinitely many numbers. You can have as many as you like, more and more, but you will never have all of them. So the potentialist says the numbers are infinite because you can have more and more, as many as you like, but also the potentialist says you can't ever have them all as a completed totality. Whereas the actualist, who believes in actual infinity, believes that the numbers are infinite because we can form that infinite collection as a completed totality and then go on and proceed to do more with that collection, further constructions.
▶︎ 1:05 So maybe it's hard to imagine now, because almost all mathematicians these days are actualists. It's by far the most dominant philosophy in mathematics. But it wasn't historically true, and for thousands of years, almost all mathematicians, including almost all the great mathematicians, Gauss and so on, that you may have heard of, they were all potentialists. And going all the way back to Aristotle.
▶︎ 1:33 So Aristotle wrote, "For generally the infinite has this mode of existence. One thing is always being taken after another, and each thing that is taken is always finite, but always different." And so this was built into the classical conception of mathematics, that some things were infinite, but only in a potentialist sense. You never completed the infinite collection. You never had infinitely many things all at once. So I want to explore this a little bit.
▶︎ 2:02 Maybe it makes sense to consider the divisibility of a line segment. So we have some line segment. Then I can divide it in various ways. I could chop it into pieces. Now I've got a division. Or I could add more pieces to the division. I can gradually divide the segment into smaller pieces.
▶︎ 2:26 And it's infinitely divisible in the potentialist sense, that I can always divide further. No matter how much I've divided it, I can always divide it further. And that's a potentialist understanding of infinite divisibility. Whereas maybe you might think, well, what would it mean to divide the segment infinitely in the actualist account? You would have to actually have undertaken maybe all of the divisions.
▶︎ 2:52 And so it's connected with the conception, perhaps, of the line segment as constituted by the points on the line segment, which is very strongly conceived. It's part of our contemporary conception of the line segment as consisting of the points on the line segment, but that's really not how the mathematicians thought about it throughout most of the history. So there's another case.
▶︎ 3:20 Say the infinitude of a line in geometry. If we think about a line, a potentialist would say, "Yeah, the line is infinite because no matter how much of it you have, you can always extend it further." And it's interesting to read in Euclid's Elements, the ancient book of geometry, one of his axioms says, "Every line can be extended further." Which is strange from an actualist point of view, because the actualist point of view of the line is that if you have a line, it's already infinite. It doesn't make sense to extend it further, because it's already fully extended.
▶︎ 4:01 So to say that every line can be extended further is to admit that you have a potentialist understanding of the line. You have just a line segment, and you can extend it further as much as you like. So it's infinite, it's potentially infinite, because you can extend it further as much as you like. But you will never have the whole thing.
▶︎ 4:18 And if you think, of course, about the geometric tools of straightedge and compass, then the straightedge typically has a finite length, so you could never actually use the straightedge to create the actually infinite line. You can only extend it further, and that's what Euclid was talking about when he said that every line can be extended further. If you have two points on the line, you can just extend it, extend it, extend it as much as you want. So it's this potentialist conception. It's infinite because you can have as much as you want. You can always have more, but you will never have all of it.
▶︎ 4:54 Let's think about another case. In Archimedes' wonderful treatise, the Quadrature of the Parabola, he set himself the task. He had a parabola and a segment, and he wanted to calculate this area. And so what he did is, he put the largest possible triangle, the... let me draw it again. The largest possible triangle that fits here, so this height is the largest possible, and he proved that that actually occurs on the midpoint of this side.
▶︎ 5:39 So that's filling up most of this parabolic segment, and then he looked at these two spandrels here, and he put triangles in there also. So maybe this big triangle had area T, and he put two triangles on each side, like this, and he argued, actually those triangles here have a height that's exactly one-fourth of this height. And so, if we add those triangles in, we get T plus T over 4, because the total width of those two triangles, if you do one-half base times height and so on, then these two smaller triangles together add up to one-quarter of this triangle. And then we put more triangles in here, again, and those are one-fourth as big as the previous ones, and so on. And so the next term is going to be T over 16, and so on.
▶︎ 6:33 And so Archimedes argues that the area of the parabolic segment can be thought of as this infinite summation here. But he was a potentialist, and so he didn't have the idea that we would actually add up infinitely many terms, and yet, nevertheless, he was able to calculate that this is four-thirds T. So the area of a parabolic segment is four-thirds times the area of the principal triangle that's in there.
▶︎ 7:03 And if you think about what he meant by this infinite sum, it was inherently potentialist, and actually, our contemporary understanding of the value of an infinite sum is a potentialist understanding because we say an infinite sum has a certain limit. What it means is that no matter how close, for every epsilon, no matter how close you want to get, then if you take a sufficiently large finite number of terms, you will be that close. So, this is a potentialist understanding of what it means to add up an infinite sum. And I've argued that maybe we don't even actually have an actualist conception of what it means to add up infinitely many terms because we always define these infinite sums in terms of the finite partial sums and their behavior, which is a kind of potentialist understanding.
▶︎ 7:58 Let's talk about another classical mathematical result that exhibits this distinction between potential and actual infinity, and that has to do with the infinitude of primes. There are infinitely many prime numbers. We know the primes like 2 and 3, 5, 7, 11, 13, and so on. These are the numbers that have only themselves and one as factors.
▶︎ 8:28 Actually, that's quite a common definition, but it doesn't quite give the right answer in the case of the number one, which also has only one and itself as factors in the positive integers, and we don't generally want to count the number one as prime, even though it has this property of having only itself and one as factors. It's a little puzzle. Of course, we're free to make our definition of prime how we like to be convenient and so on, and mathematicians have decided that we don't want to count the number one as prime, even though it has the property of having only itself and one as factors.
▶︎ 9:05 And the reason we don't want to count the number one as prime is because it would ruin a lot of theorems. We would have to say, for every prime except one, and so on, or for example, in the uniqueness of prime factorization, every number can be factored uniquely into primes. It wouldn't be true anymore if the number one counted as prime because, for example, the number 12, well, that's 4 times 3, which is 2 times 2 times 3. But if one counted as prime, then I could write 2 times 2 times 3 times 1, or 2 times 2 times 3 times 1 times 1 times 1 times 1, I could add extra ones on, and that would destroy the uniqueness of the prime factorization. So this is one reason not to count the number one as prime.
▶︎ 9:43 One way of defining prime, therefore, is that a positive integer is prime if it has exactly two factors, namely one and itself, but they have to be different. Or sometimes people just say it's bigger than one and it has only one and itself as factors.
▶︎ 10:00 Let's talk about the infinitude of primes. There are infinitely many prime numbers. How do we know this? Well, let's suppose that we have some finite list of prime numbers. Let's make a list of prime numbers, P1, P2, P3, and so on, up to some PN. I've got some finite list of prime numbers, and what I'm going to do is multiply them together and add one. So I form the product P1 times P2 times P3, and so on, times PN plus one, and I'm going to think about this number.
▶︎ 10:43 Well, this number is not perfectly divisible by any of these prime numbers because it's a multiple of each of these prime numbers plus one. For example, if I multiply all these together times P1 plus one, I would have a remainder of one if I divided this number by any of those prime numbers. The remainder is one. But every number has some prime factor. We can prove this. Every number has a prime factor, at least one. Either it's prime itself or it's factorable into primes, so it has some prime divisor. But this number has a prime divisor, therefore, which is different from any of these primes. So therefore, there must be some other prime number Q. There's some Q that divides this number, and so I can extend the list.
▶︎ 11:36 There are infinitely many primes because any finite list of primes can be extended by adding at least one more prime, and then that's a finite list which can be extended and extended and extended. It's an inherently potentialist way to express the infinitude of primes. We don't say there are infinitely many primes as a completed collection. This is the classical way of understanding it. Rather, we say any finite list of primes can be extended to a larger finite list of primes. So therefore, the primes are infinite, potentially infinite.
▶︎ 12:09 Sometimes one sees a kind of actualist argument not like this. The argument I just gave is similar to how Euclid argues in The Elements for the infinitude of primes, but sometimes you might see the argument by contradiction. You'll see a mathematician say, "Well, there aren't finitely many primes because suppose towards contradiction, there are only finitely many primes. Put them on a list. Multiply those numbers together and add one. That's a number, and it has a prime factor that must be different from the numbers on the list, and that's a contradiction. Therefore, there must be infinitely many." So this is a kind of actualist understanding of the infinitude of primes.
▶︎ 12:51 There's a notable exception to this pervasive potentialist outlook in mathematics for thousands of years, and this is Galileo. Galileo argued against this potentialist orthodoxy, and he gave several arguments. So I'd like to discuss them. What he said is, "It's just not true that we can't conceive of completed infinities. We can't bring into actuality the completed case."
▶︎ 13:20 And so what he proposed to imagine was a sequence of polynomials. I mean, of polygons. So we start with, say, a square. And then maybe we have a hexagon. We're gonna have. Oh my gosh. I was drawing an octagon. So terrible. And then an octagon. Can I draw it? Let's see. There. I think that's an octagon. And so on. So we can have more sides, maybe a 40-sided regular polygon, and so on. So we're gradually increasing the number of sides in these polygons, regular polygons.
▶︎ 14:04 And Galileo said, "Well, I can imagine in a potentialist way, the polygon could have more sides, or more sides, or more sides. I can keep adding sides. Maybe I get a 1,000-sided polygon, 10,000-sided polygon, and so on. And is it conceivable to actualize the having infinitely many sides?" And Galileo said, "Yes. That's exactly what a circle is. A circle can be thought of," he argued, "as a polygon with infinitely many sides." It's the actualist case because, first of all, we can definitely talk about circles in geometry in an actualist way because a compass can draw them, and it's a basic concept in classical geometry, and we can conceive of this circle as an actualist case of a polygon with infinitely many sides. So this is one of the arguments he gave for actualism.
▶︎ 14:59 There's another quite interesting argument. So suppose you're a potentialist about the infinitude of primes. You think that you can have more and more, as many as you like, but you will never have all of them. So for example, you could have five primes in your possession, or you could have 100 primes in your possession, or 1,000 primes in your possession. So those possibilities are all different. Those are distinct possibilities, and you have those possibilities right now, not as realized, but as possibilities.
▶︎ 15:29 Those are distinct things that you have right now, possibilities, those different possibilities, and therefore, right now, you already have an actual infinity of different things. Namely, the possibilities of having those more and more primes. So this is an argument. Goes back to Galileo, and it is a sense in which every potentialist is committed to an actual infinity, an actual infinity of different possibilities, not as realized possibilities, but as abstract possibilities.
▶︎ 16:03 Let me turn now to a certain philosophical position in mathematics known as ultrafinitism. Ultrafinitism is the philosophical view that it is only the comparatively small or accessible numbers that actually exist. According to the philosophy of ultrafinitism, the various extremely large numbers that mathematicians conventionally take themselves to describe, such as 10 to the 100 or a googolplex, aren't actually meaningful. They don't actually exist as numbers, merely as descriptions of numbers. This is the view of ultrafinitism.
▶︎ 16:44 According to ultrafinitism, we can write down these expressions, but the expressions don't denote any number that exists. They're just finite strings of symbols that are ultimately meaningless and fail to refer. Maybe this position, this philosophy sounds a bit strange, but I just want to mention that there are definitely some mathematicians who hold this ultrafinitist philosophy and find it helpful for expressing their vision of what mathematics is about, although most mathematicians are not ultrafinitist.
▶︎ 17:15 Perhaps one might ask, "Well, if an ultrafinitist is denying the existence of, say, a googolplex, then are they committed to the idea of there being a largest number? Do they have to think that there's a largest number?" Some versions of ultrafinitism do commit to the existence of a largest number, whereas other versions of ultrafinitism do not. I want to talk about this a little bit. Maybe you think it's ridiculous to think that there should be a largest number because you have the idea that, look, if one were committed to a largest number, then you could imagine a number system that had one more number and wouldn't that be absurd?
▶︎ 17:56 But that criticism is a little bit too quick in light of these other positions on ultrafinitism that aren't committed to the existence of a largest number. And I'm reminded of a certain exchange once that was related, that I heard related by Harvey Friedman,
▶︎ 18:11 The logician Harvey Friedman, who described a conversation he had with Yesenin-Volpin, who is a famous ultrafinitist. And he put to Yesenin-Volpin the question, "Well, let's consider the numbers, 2, 2 squared, 2 cubed, 2 to the fourth, and so on, up to 2 to the 100." And one can imagine easily writing out these numbers from 2 to the 1 up to 2 to the 100. And it's this draw the line objection, and so what Harvey said is, "There's the obvious draw the line objection." So where do we stop having platonic reality in this list of numbers, 2 to the 1, 2 to the 2, 2 to the 3, and so on, up to 2 to the 100?
▶︎ 18:58 So what he says is, "I raised just this objection with Yesenin-Volpin," during a lecture of his, "and he asked me to be more specific. I then proceeded to start with 2 to the 1 and asked him whether this is real or something to that effect, and he virtually immediately said yes. I then asked about 2 squared, and he again said yes, but with a perceptible delay. And then 2 cubed, and yes, but with more delay. And this continued a couple of more times till it was obvious how he was handling this objection. Sure, he was prepared to always say yes, but he was going to take 2 to the 100 times as long to answer yes to 2 to the 100 than he would take to answering 2 to the 1, and therefore there was no way to get very far with that objection."
▶︎ 19:52 The picture of ultrafinitism that emerges from that kind of attitude maybe is that the numbers are getting blurrier as they get larger, and the existence of the numbers is being called more and more into question. Although, maybe if we wait long enough, they would come into focus. So what is the theory of ultrafinitism exactly?
▶︎ 20:16 It turns out to be somewhat difficult to pin down an exact theory that's expressing the idea of ultrafinitism, but perhaps part of the idea of ultrafinitism is motivated by the sense that there's something innocent about adding numbers or even multiplying them that's not so innocent about exponentiation that takes us into this realm of much, much larger numbers that we couldn't hope to reach by counting ever. One can write down formal theories of arithmetic that have this feature, that they prove that addition and multiplication is totally fine, whereas exponentiation is possibly not total. In other words, it's not always defined. It doesn't always give a meaningful result, even though, in those theories, there's no sharp cutoff for where the exponentiation fails.
▶︎ 21:10 If you think about one of the standard axiomatizations of arithmetic, it's called Peano arithmetic, and it puts forth, in the language of addition and multiplication and the order less than, one writes down the basic axioms about addition and multiplication and the recursive formulas that define multiplication over addition and so on. Then Peano arithmetic has what's called the induction scheme, which is the principle that says if you have a statement that's expressible in this language of arithmetic, and the statement is true of zero, of the number zero, and whenever it's true of a number n, then it's also true of the number n plus one, then the statement is true of all numbers. This is called the induction scheme, and using the induction scheme, one can prove all kinds of facts about number theory and essentially all of the elementary theory of number theory can be developed in this Peano system.
▶︎ 22:09 In particular, in Peano arithmetic, you can prove that exponentiation is total. You can prove that there's a function that satisfies the recursive definition of exponentiation. For example, you can undertake definitions by recursion, and you can define two to the zero is going to be the number one, and two to the n plus one is going to be the outcome of two to the n with one more two. This is a sort of recursive definition of the function two to the n that takes n to two to the n, and in Peano arithmetic, you can prove that all such recursive definitions are successful, and therefore, in PA, the theory of Peano arithmetic, you can prove that exponentiation exists.
▶︎ 22:58 This is not an ultrafinitist theory, but we can get an ultrafinitist theory by weakening the induction axiom. We have to weaken the induction axiom. If we're going to adopt an ultrafinitist theory that thinks addition and multiplication are fine but exponentiation is suspect, then we must deny induction, because as I just mentioned, with the induction principle, we can prove that exponentiation is total. Ultrafinitists must deny certain instances of induction, and it turns out that if we weaken the induction principle by only allowing the induction axioms when the statement being proved has a certain simple form.
▶︎ 23:40 One forms the theory that's known as I delta zero, which asserts the induction axiom, that's the I part, for delta zero assertions, which are assertions that involve only bounded quantifiers. If you limit the induction principle to assertions that only talk about relatively small numbers, smaller numbers than their arguments, then it turns out, in that theory, you can prove that addition and multiplication are totally fine. But you cannot prove that exponentiation exists, and so this is, in that sense, an ultrafinitist theory. It's an ultrafinitist formal theory of arithmetic in which addition and multiplication work out just as expected, but exponentiation does not. I want to mention
▶︎ 24:29 Briefly a distinction between two kinds of potentialism. That is, maybe one thinks that potentialism is an epistemological phenomenon versus an ontological phenomenon. In other words, epistemology is the study of knowledge and the nature of knowledge. And so to say that we are potentialists in an epistemological sense would mean that it's about our knowledge having a potential nature.
▶︎ 25:01 So, we can never know the entirety of the infinitude of primes, even though maybe in an ontological sense, in terms of actual existence of that abstract objects, there's no problem with the infinitude of primes being an existing completed object. It's about our knowledge being limited. We can only have this finite knowledge at a time. Let me turn now to a
▶︎ 25:25 A more contemporary understanding of the dispute between potential infinity and actual infinity. And the current philosophers looking into this are emphasizing the modal aspect of potentialism. And ultimately, in fact, they divorce the connection of potentialism and actualism dispute from the connection with infinity by finding the essence of potentialism isn't about infinity or being completed or not, but rather about the mathematical universe that one is discussing being completed or not.
▶︎ 26:06 The potentialist outlook is that it's wrong to think of the mathematical realm we inhabit as a completed, a finished totality, but rather we should have universe fragments, pieces of the universe that are getting larger. And maybe those universe fragments already have completed infinities inside them, but if they're not the whole mathematical universe, say in a set theoretic sense, then it would still be a potentialist point of view, even though it's allowing for completed infinities inside the universe fragments. So the emphasis is that the potentialist perspective is about the mathematical realm we inhabit being unfinished, and rather, we need to keep extending it to larger and larger realms, larger and larger realms, larger and larger universe fragments.
▶︎ 26:55 The picture here is that the potentialist outlook is that you have various universe fragments that are getting larger and larger, and none of them is completed, regardless of what kind of mathematical objects exist inside any of them. And so this is a kind of mathematical structure where you have possible worlds that are getting larger that's ripe for modal logic analysis. Modal logic is the study, the general study, abstract study, of possibility and necessity. And we can define a modal semantics with such a picture.
▶︎ 27:40 So if we're at some world, say U here, then we say that U thinks, this is the symbol that's used for something is true at world U, thinks that a statement is possible, phi is possible at U, if there is a larger V that thinks phi is true. So maybe there's some V up here where phi becomes true. So something is possible at a world if it becomes true in a larger world. And U thinks that phi is necessary at a world if all larger worlds think that phi is true.
▶︎ 28:26 So to be possible at a world means that it's true in some larger world, and to be necessary at a world means it's true in all larger worlds. So this is the basic idea of modal logic, and one recognizes in this case when the universe fragments are growing that this is a fundamentally potentialist perspective on the nature of mathematical truth. So let's look at a specific example, what
▶︎ 28:53 This might be called Aristotelian potentialism. Maybe we imagine the numbers zero, one, two, the natural numbers, say, increasing here. And I'm going to imagine the possible worlds are just the initial segments. The worlds are growing. You can have more and more numbers, as many as you like, but you will never have all of them. No world has all the numbers. Every world has only the numbers up to some point. And then we can provide this modal logic potentialist vocabulary for this understanding of the nature of potentialism.
▶︎ 29:38 For example, we might think about this statement, every number, every number N has a successor. N plus one. Well, it's not true. Actually, that statement is not true in any of these fragments, because in this world, there is no successor for two because you don't have it yet. You have to go to another world. So, if I think about this as, say, for every N, there is, let me say for every X there is a Y such that Y equals X plus one. This statement is not true in any world, because it's not true when X is the largest number in that world.
▶︎ 30:21 Rather, I should say it in a more refined way using the modal vocabulary. Namely, for every X, it's possible that there is a Y, such that Y equals X plus one. In other words, every N possibly has a successor, is a correct potentialist way of making the statement. But actually, we can go even stronger than this. Because not only is it true at every world that for every X possibly there's a Y which is the successor, but I can say necessarily. It's necessary. Every world thinks it's necessarily every individual possibly has a successor. Because, if I'm in any world, then no matter what other world I go to, in other words, necessarily, it's possible of every individual in that world that there's a successor.
▶︎ 31:17 This modal vocabulary leads you to be able to express in a finer way the nature of the potentialism that you're talking about. What are the potentialist principles expressed in this vocabulary of possibility and necessity? Or, do we have different kinds of potentialism that we might want to be talking about?
▶︎ 31:46 To give some more examples of this, if I want to say, every two numbers have a sum. So I want to say for every X and Y there is a Z such that Z equals X plus Y. This is saying for any two numbers X and Y, there's some Z which is their sum. In the Aristotelian potentialism, the initial-segment potentialism that I've described here, this is not true at any world, because sometimes Z is not going to be found in that world. I need to go to a larger world to find it.
▶︎ 32:21 And so really what I should say is for every X and Y, possibly there's a Z, and I can put a necessary here. So this is a slightly more correct way of saying it. Necessarily, any two numbers possibly have a successor. And I could also write product here. So maybe I want to be committed to the idea that necessarily any two numbers possibly have a successor, and I mean have a sum, and possibly have a product, but maybe they don't possibly have an exponential. Maybe this is part of the version of potentialism that you're going to be interested in.
▶︎ 33:01 This is about closure under addition and multiplication. What about the infinitude of primes? How could we express the infinitude of primes in this modal vocabulary? I want to say that there are infinitely many primes. One way of saying it is that, for every number, there's a prime above it. But I should really only say possibly there's a prime above it, so I should say necessarily, for every number, possibly there's a prime that's larger than that number.
▶︎ 33:34 That can express being prime, because a number's going to be prime. If I have a number and I'm in the Aristotelian potentialism case, then I also have all the smaller numbers. Therefore, the question about whether it's prime or not is just the question of whether it factors as X times Y for some smaller numbers X and Y, whether I can find a non-trivial factorization of it into smaller numbers. So I don't need any modal operators to express in the Aristotelian potentialism case whether a number is prime or not.
▶︎ 34:05 But let's talk about some other forms of potentialism that are not linear. This is Aristotelian potentialism, but maybe it's not quite what we want, because in light of this issue that we talked about in another lecture about, say, the number googolplex is very easy to describe, it's 10 to the 10 to the 100, but most of the numbers that are smaller than a googolplex are beyond our comprehension. We have no way of holding those numbers as an object of thought in our minds, because those numbers are basically random strings of decimal digits of length of a googol, and it would take us longer than the age of the universe to even pronounce those digits. So there's a sense in which maybe you think, look, if you're a potentialist about numbers coming into existence, then maybe we get to have a googolplex earlier in the potentialist framework than some of those smaller numbers.
▶︎ 35:10 That's a bit strange. But what I'm suggesting is that maybe the numbers don't necessarily arrive in order, because some of them are easier to talk about and describe. For example, a googolplex, we know a lot about it. It's even, for example, we know the prime factorization of a googolplex because it's 10 to the 10 to the 100, and 10 factors as two times five. So a googolplex is 2 to the 10 to the 100 times 5 to the 10 to the 100, and that's the prime factorization of it, and I can say a lot more. It's not a multiple of seven, therefore, and so on.
▶︎ 35:41 I can say quite a lot about a googolplex. I have quite a good understanding of the nature of a googolplex, and so maybe you might think that for a potentialist, we get to have that number earlier than many of the numbers that come earlier, whereas in the Aristotelian potentialism, we don't. Whenever you get a number then you already have all the smaller numbers, and so this opens the door to maybe what could be described as a nonlinear form of potentialism.
▶︎ 36:14 So maybe we have a version of potentialism where you have some numbers here, maybe the numbers up to 10 or something, and then you could have other numbers, but you could have other numbers in a different way, or then maybe you can have a collection of numbers that includes all of those and so on. It's not necessarily linearly ordered, although maybe you think that any two such universe fragments can be amalgamated together in a larger single fragment. So this kind of picture, this arbitrary set, arbitrary finite set potentialism in arithmetic has different modal features than the Aristotelian potentialism. Let's talk about some of those differences.
▶︎ 37:05 For example, let's think about how we would express x is even. For x to be even means there is a y such that y plus y equals x. This is one way of expressing the concept of even. And in the Aristotelian potentialism, we don't need to put any modal operators in here, because if we have x, then if x is even, we will already have the y that works because it's a smaller number. And if it isn't even, then there's no way that works.
▶︎ 37:44 But in the arbitrary set potentialism, maybe we have an x which happens to be even, but the y that would verify that, we don't have it yet maybe. In which case, this wouldn't express being even, because this number is even, even though we don't yet have the y that it's the double of. We don't have x over two yet. So we should really put a diamond here for this conception of potentialism. And so this shows you that depending on the nature of your potentialist conception of arithmetic, the way that you're going to translate your mathematical concepts into this modal way of thinking is going to be sensitive to the modal properties of your potentialist conception.
▶︎ 38:31 So one can start to write down the basic principles that maybe all of the potentialist conceptions are going to share. So let's imagine that we have a potentialist understanding, say, of arithmetic. We have a bunch of universe fragments, and they're included in one another, growing in a certain way. Not necessarily linearly ordered. But maybe we think that, look, if something is necessarily true, then it's true. So if phi holds necessarily, then phi holds.
▶︎ 39:10 And this principle expresses the idea that every world counts as a sub-world of itself. It's a reflexive relation. The accessibility relation is reflexive. If phi holds necessarily, then phi already holds, because for phi to hold necessarily means it holds in all worlds that you can get to from that world. But one of the worlds that you can get to from that world is that world itself, so phi would have to hold in that world.
▶︎ 39:36 So there's other principles. For example, if phi holds necessarily, then it's necessary that phi holds necessarily. And this is related to the idea that the accessibility relation on worlds is transitive. So if you think that phi holds in all worlds that you can get to from your current world, then if you go to such a world, then does it still happen to be the case that phi holds in all worlds that you can get to from that world? And the answer is yes, because if you're at a world U, and you know that box phi holds, phi holds necessarily. That means no matter which world you can get to, phi holds.
▶︎ 40:15 Now, if you go to a larger world and you ask, "Does box phi still hold there?" And the answer is yes, because if from this world you could go to a larger world, then that world is one of the worlds that you could have got to from the original world in two steps, and therefore we have necessary phi holding in those larger worlds. And so box box phi holds in the original world. So this is reducing to the transitivity of the accessibility relation. If you're at a world and you can access a larger world, and that world can access a still larger world, then the original world can access the still larger world.
▶︎ 40:52 And there's some other principles too. For example, there's this axiom known as K, for Kripke. It says that if you have an implication being necessary and you have the antecedent being necessary, then the conclusion is necessary. So if you know a certain implication holds no matter where you go from your current world, and you also know that the hypothesis of that implication holds no matter where you go, then you can deduce the conclusion is also going to hold in those other worlds.
▶︎ 41:29 So these axioms altogether constitute the modal theory known as S4. In modal logic, it's called S4. And so the basic picture here is that with only some very primitive requirements on the modal system, reflexivity and transitivity, we're going to get all of S4 being valid for our conception of potentialism.
▶︎ 41:55 But let's think about some of the other more specific principles that we had. Say, in the Aristotelian potentialism, the possible worlds, the universe fragments were linearly ordered. So given any two, one of them was above the other. And that's going to lead to some validities beyond S4. For example, let's think about the statement possibly necessary phi implies necessarily possibly phi. So let's suppose that I'm at a universe fragment, I'm in a world where possibly necessary phi holds. So I'm in some tiny world, and I have possibly necessary phi.
▶︎ 42:48 But this possibly means that I can enlarge it to a world where I have necessary phi. It's possible that necessary phi, so let's go to that place where necessary phi holds. And now that means in all larger worlds, phi is true. And now I want to ask about the original world. Is it true that it's necessarily possible? And the answer is yes, because starting from the original world, if you go somewhere, then no matter where you went, eventually you could enlarge it to include this world where necessary phi was holding. And so, for any world extending the original one, we can extend it further beyond this world, and so we're going to get a world where phi is holding. So therefore, I've argued that possibly necessary phi implies necessarily possible.
▶︎ 43:38 This principle is also true in the other potentialist conception that I mentioned, namely the arbitrary set potentialism, which is not linear. And maybe it's even a bit clearer, because if I have a world where I have possibly necessary phi, that means I can go to a world where necessary phi, so any larger world than this one will satisfy phi. And now if I consider that original world and consider any world extending it, then because of what we said about amalgamating, this world, there's a larger world that encompasses both of these worlds. And because it extends this one, it must satisfy phi, and therefore this world will satisfy possibly phi, and therefore this world satisfies necessarily possibly phi, because we said for any world you can extend it further to a world where phi is true. So, this is a case where this principle, which is not provable in S4, is nevertheless true for both of the potentialist conceptions that I had described. This is the central axiom of what's known as S4.2.
▶︎ 44:44 So, let's talk about one more principle that will distinguish these two cases. And this is the principle known as .3. These names are terrible. We're stuck with them because the first writers on modal logic had given these absurd names to the axioms and somehow it stuck, and so we have to be calling them by these silly names. So the axiom says the following. If phi is possible and psi is possible, so I have these two possible statements, then it's possible that the first is true and the second is possible, or it's possible that the first is possible while the second is true. So, in other words, if two things are possible in a larger world, then there's a larger world where one of them is true and the other one is still possible.
▶︎ 45:58 So, this axiom is expressing a kind of linearity, because if your worlds are linearly ordered, it will have this feature. If, imagine in the Aristotelian potentialism, the worlds are linearly ordered, and we have two statements that are true at some later point at least once. Well, one of them has to come first. If phi and psi are both true in some larger world, well, maybe phi happens before or at the same time as psi. One of them comes first. And whichever one comes first is going to realize that clause. If phi is true first, then psi will still be possible when phi is true. Or if psi comes first, then phi will still be possible when psi is true. So, in any linear system you get the validity of S4.3.
▶︎ 46:50 But it turns out that this axiom is not valid in the arbitrary set potentialism and you can use the fact that there's non-linearity to make it happen. You can make phi true if you go this way, for example maybe phi asserts that the number 17 exists but not the number 16, whereas maybe psi asserts the number 16 exists but there's no successor for it. Those things are both possible, but if you have one of them the other one's impossible because they're incompatible for one of them to be true and the other one to still be possible. So that's a violation of S4.3 in the arbitrary set conception of potentialism even though S4.3 is valid in the Aristotelian conception of potentialism.
▶︎ 47:34 So, what these modal assertions, it's a little bit technical, I agree, but the power of this way of thinking about potentialism is that it allows us to distinguish between these different kinds of potentialism. It's not just that there's potentialism and actualism. Rather, there's an incredibly rich variety of different potentialist conceptions and many of those distinctions between those conceptions can be articulated by means of this modal perspective.
▶︎ 48:08 So one of the standard conceptions of potentialism, and it's not just about numbers, but there's potentialism in set theory, say, about the set-theoretic foundations of mathematics. There's potentialism in any mathematical structure. So I have a paper with one of my graduate students, Vojtěch Voroušin, called Modal Model Theory, and we look at modal graph theory and modal group theory, modal field theory and so on. For any collection of mathematical structures in any theory whatsoever, you can form the collection of models of that theory and understand this as a potentialist framework for that subject.
▶︎ 48:50 Conceptions of potentialism that involve a linear hierarchy of possible worlds generally has S4.3, because of the reasons that I said. But there's other conceptions where maybe you have a possible world, and then you have another one growing one way, and another one growing another way, but then there's a still larger conception that encompasses both of them, and maybe still more growing one way or the other, but still we can encompass both. So this would be a convergent form of potentialism in the sense that given any world in the potentialist system, then if it accesses two worlds, then there's a further, larger world that encompasses both of them. And this conception of potentialism generally has validities S4.2. You can prove that this is always valid in such a convergent system.
▶︎ 49:48 But then finally, there's a much more radical conception of potentialism that's maybe more connected with ultra-finitism, but also with pluralist views in the foundations of mathematics and the philosophy of set theory, the multiverse view, and so on. And this is the radical branching perspective on potentialism, where you don't have convergence. So maybe you have a world and you have various extensions of that world, and they just don't come together again, so there's further branching that don't ever come together again, so these ones maybe stay separated. So the world is growing in such a way that depending on what happens, it affects future possibility.
▶︎ 50:36 And one can exhibit cases, for example, if you look at the models of Zermelo-Fraenkel set theory under end extension, then it has this kind of conception, and .2 in particular is not valid in that modal framework. And in that radical branching case, generally the validities are only S4, and in the case of the ZFC models, you get exactly S4 and nothing beyond that.
▶︎ 51:02 Let me talk a little bit about what's called the potentialist translation. We were already doing the potentialist translation before when I was translating assertions of mathematics into the modal framework, and the potentialist translation allows you. Given a statement phi in the actualist language, we produce a statement that I'll call phi diamond in the potentialist language, and basically, we replace existential quantifiers with possibly exists X, and we replace universal quantifiers with necessarily for all X. So, given any statement phi in a formal language, in the actualist language, I'm going to insert a diamond before every existential statement and a box before every universal statement, and that's how I get the potentialist translation.
▶︎ 52:24 And now we can prove that there's a kind of, in a convergent potentialist system, when the worlds have this feature that any two worlds that you got to, you can put them together again in a larger fragment. In any such convergent system, then let me erase this and make the statement. That potentialist translation is what we were doing when we said, for example, every number has a successor. What I said was necessarily for every X, possibly there's a Y such that Y equals X plus one. So I was exactly writing down the potentialist translation of the statement.
▶︎ 53:12 And the point is that if I have, if this is a potentialist system, and it's convergent in the sense that any two worlds you can get to, you can put them together in a larger world, then of course those possible worlds are, if I'm an actualist, I can union up all of those worlds to have this sort of limit model. So there's a limit model, M, they're converging to this limit model. In the case of radical branching, we're not converging to a limit model because if you go this way or that way, maybe totally different things happen and there's no way to talk about the limit model. But in the convergent potentialist system, no matter what you did, it's just a temporary thing on the side, and eventually it's all going to be assembled into this limit model M. Then you can argue that the limit model satisfies the statement if and only if the worlds in the system satisfy the potentialist translation of the statement.
▶︎ 54:25 So every number has a successor, that's an actualist statement. To say every number has a successor is equivalent to, in the actual model, is equivalent to saying of any fragment that necessarily every number possibly has a successor. Those are equivalent statements. And so this biconditional here is a kind of translation from actualism. This one is actualist. This is actualist truth, and this is a potentialist truth.
▶︎ 54:59 And so this shows you that for the convergent forms of potentialism, there's a translation between actualist assertions and potentialist assertions. We can just go back and forth. And I have argued that this is a sense in which convergent potentialism is implicitly actualist because there's nothing at stake in the dispute between potentialism and actualism if all the truth assertions that the actualist might want to make can already be made totally in the potentialist framework using only the potentialist ontology and the potentialist conceptions. So if you're a potentialist and you have a convergent conception of the nature of potentialism, then you can make assertions that exactly track the meaning of what I would want to make as an actualist. And therefore, that form of potentialism is implicitly actualist.
▶︎ 56:05 So what this means is that it's a kind of deflationary argument against the dispute between potentialism and actualism in the case where the conception of potentialism is convergent. This argument, this biconditional, does not work, we can prove, does not work at all in the radical branching form of potentialism where the nature of the potentialism can branch this way or that way and never come together again. Then there is no potentialist translation that works, that converts that kind of potentialism into actualism.
▶︎ 56:46 And so in a sense, my belief is that, if one is really committed to potentialism and to preserving potentialism, then maybe one should think more deeply about the nature of this radical branching potentialism as expressing a kind of fuller view of the nature of potentialism, because if you have the convergent view, then it's reducing to actualism in the sense of this biconditional and the potentialist translation.
▶︎ 57:16 Let me close by just mentioning more explicitly the sea change that occurred in mathematics really at the end of the 19th century. Before that time, almost everyone was a potentialist. But with the rise of set theory, I think particularly with Dedekind and Cantor, and the deeper understanding of infinity that grew out of that work, which is really an actualist understanding of infinity, it happened that mathematicians recognized it wasn't actually problematic to conceive of actually completed infinities, but rather this was a great source of insight into mathematical truth, and it was quite powerful and useful and eventually transformed the subject.
▶︎ 58:06 David Hilbert famously said, "Let no one cast us from the paradise that Cantor has created for us." And part of what he's expressing about that is the perspective of mathematics that allowed for actual infinity, completed totalities, and further constructions proceeding on top of those infinities. And by now, this is routine in mathematics. Almost everyone is an actualist, and we're building infinities on top of infinities on top of infinities, in a highly actualist manner. And so mathematicians these days find great inspiration from this actualist perspective.