There are infinitely many numbers, to be sure, but mathematicians and philosophers of mathematics sometimes disagree about what this means exactly. What is the nature of existence of infinite collections or infinite objects?

According to the philosophy of potentialism, the natural numbers are potentially infinite—you can have more and more, as many as you like, but you will never have all the numbers as a completed infinite totality; according to potentialism, you will never have an actual infinity. The philosophy of actualism, in contrast, allows for the existence of completed infinities—actualists claim to form actually infinite completed collections and proceed further with them to construct other infinite mathematical objects.

Aristotle had emphasized the finiteness of everything that is—apparent instances of infinity are seen through a potentialist lens of becoming.

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. Again, 'being' has more than one sense, so that we must not regard the infinite as a 'this', such as a man or a horse, but must suppose it to exist in the sense in which we speak of the day or the games as existing things whose being has not come to them like that of a substance, but consists in a process of coming to be or passing away; definite if you like at each stage, yet always different.

(Aristotle, Physics, 206a27-29)

This potentialist understanding of infinity has been dominant for two thousand years—mathematicians from classical times until the dawn of the twentieth century have almost universally (but not quite universally) subscribed to a potentialist account of infinity. And yet, over the past century and half there has been a sea change. The actualist perspective has nowadays become dominant in mathematical practice. Let us explore the differences in these two philosophical perspectives. What is at stake in the choice between them?

Infinitely divisible segments

Take a line segment. A potentialist will agree that a segment is infinitely divisible, but this is a potential infinity of divisibility—you may divide the segment into two, and then four, and then eight; or you may divide it by some other proportion, and you can subdivide it again and again as finely as you would like.

But according to the philosophy of potentialism, we never achieve an actually infinite subdivision. Indeed, according to the potentialist perspective there is something downright incoherent in the idea of having an actually infinite subdivision of the segment.

After all, if every division that is made is further subdivided, then every segment produced during the process is itself destroyed, chopped finely up, and every constituent part of the dividing process thus evaporates into the mist of the limit of an actual infinity of subdivisions. What exists of the infinite subdivision in the end?

Infinitely extensible lines

Or take not a segment, but a line itself. According to the ancient geometers, a line is not a completed linear infinity, but rather a finite fragment of such a line, which might be extended further, as much as desired, by the use of the straightedge.

Euclid's second postulate, for example, asserts that any line can be extended indefinitely on either end. This is a potentialist manner of considering lines, since the view is that one has only finitely much of it at a time, and so it may need to be extended.

The actualist, in contrast, considers a line as a completed infinite totality, stretching out with infinite extent. For the actualist, it makes no sense to extend the line further, since the line consists already of its entire infinite extension.

Infinite exhaustion

Archimedes had used the method of exhaustion to determine the quadrature of the parabola, filling a parabolic segment with the inscribed triangles, whose total area he analyzed, showing that the parabolic segment was 4/3 the area of the principal inscribed triangle.

On the actualist account of the method of exhaustion, one considers the infinite exhaustion of the parabolic segment as a completed totality, arguing that the area of the triangles is the (actually) infinite sum

which can be proved equal to (4/3)T. And since the parabolic segment is completely exhausted by the inscribed triangles, this is seen as the area of the parabolic segment.

But Archimedes had undertaken a potentialist approach to exhaustion, without ever making reference to a completed process of exhaustion. Rather, he presented a double reductio argument, which required only arbitrarily large finite instances of the dissection by triangles. Namely, for the lower bound, he argued that the area of the parabolic segment could not be anything less than 4/3 of the inscribed triangle, because then some finitely many of the inscribed triangles would already exceed it. And conversely, for the upper bound, because the difference between the area of the parabolic segment and the finite triangle sums can be made as small as desired—it is reduced by at least half at each stage—it follows that the parabolic segment cannot be greater than 4/3 of the triangle, since if it were, then at the stage when the triangles became in total smaller than half that amount, the difference would not be cut in half.

Thus Archimedes mounts his analysis of the quadrature of the parabola as a potentialist construction without ever requiring the infinite exhaustion of the parabolic segment as a completed totality.

Infinitely many primes

Euclid proved that there are infinitely many prime numbers, but his presentation of this theorem has an explicitly potentialist character, for the proof shows precisely that any given finite list of primes can be extended with an additional prime.

Let us give the argument. Suppose that

are finitely many primes. Let n be the number obtained by multiplying them together and adding one.

Every number has some prime factor, but any prime factor p of this number n must be different from every prime p_i on the original list, since those primes all give a remainder of 1, but p, being a factor of n, has remainder zero. Thus, we have found a prime number p that does not appear on the original list. So there is a prime the list has missed, and thus it can be extended.

This is a potentialist approach to the infinitude of primes, for it shows explicitly that every finite list of primes can be extended to a larger list of primes. You can have as many primes as you like.

For your amusement, here also is a musical version of the proof, arranged and performed by Hannah Hoffman, lyrics written by me.

Galileo argues for actualism

Remarkably, Galileo resists the potentialist orthodoxy of his time and argues that actual infinities are possible.

He considers how one might bend a segment at angles to form a square or an octagon or a polygon of forty sides or more. This is not fundamentally different, he argues, from bending the segment into an infinite polygon, that is, to form a circle.

Salviati. If now the change which takes place when you bend a line at angles so as to form now a square, now an octagon, now a polygon of forty, a hundred or a thousand angles, is sufficient to bring into actuality the four, eight, forty, hundred, and thousand parts which, according to you, existed at first only potentially in the straight line, may I not say, with equal right, that, when I have bent the straight line into a polygon having an infinite number of sides, i. e., into a circle, I have reduced to actuality that infinite number of parts which you claimed, while it was straight, were contained in it only potentially? Nor can one deny that the division into an infinite number of points is just as truly accomplished as the one into four parts when the square is formed or into a thousand parts when the millagon is formed; for in such a division the same conditions are satisfied as in the case of a polygon of a thousand or a hundred thousand sides. Such a polygon laid upon a straight line touches it with one of its sides, i. e., with one of its hundred thousand parts; while the circle which is a polygon of an infinite number of sides touches the same straight line with one of its sides which is a single point different from all its neighbors and therefore separate and distinct in no less degree than is one side of a polygon from the other sides. And just as a polygon, when rolled along a plane, marks out upon this plane, by the successive contacts of its sides, a straight line equal to its perimeter, so the circle rolled upon such a plane also traces by its infinite succession of contacts a straight line equal in length to its own circumference.

Galileo, Dialogues concerning two new sciences

For Galileo, the points of the circle are just as well specified and definite and distinct from one another as are the folding points leading to the large regular polygons, and we might thus take ourselves to be in possession of the actual infinity of the sides of the circle just as much as we were in possession of the thousand-gon or the hundred-thousand-gon.

Is potentialism committed to an actual infinity of possibilities?

There is a further subtle point to be made here. The potentialist admits that whatever finite subdivision one currently has of a segment, one can always find a further finer subdivision. And thus, Galileo argues, even the potentialist must admit the actual infinity of the possibilities of refinement. If a potentialist truly believes, you can always have more, as many as you like, then there is an actual infinity of possible things you can have. In this way, potentialism might be seen to admit actual infinities, the infinities of possibility.

Ultrafinitism

Ultrafinitism is the philosophical view that only comparatively small or accessible numbers exist. According to ultrafinitism, the various extremely large numbers that mathematicians conventionally take themselves to describe, such as 10¹⁰⁰ or

do not actually exist. Ultrafinitism sees it as a kind of illusion to speak of such numbers, and indeed we often find ourselves unable to answer basic questions about them. For example, we have no tools for determining in general even whether an expression such as

represents an integer as opposed to a non-integer real number.

(It follows from Schanuel's conjecture in number theory that

and so on, are all transcendental and indeed algebraically independent and hence none are integers; but meanwhile, the truth of Schanuel's conjecture is an open question, perhaps intractable.)

According to ultrafinitism such expressions and others such as 5 ⇑ 5 or 3 ⤊ 3 are taken merely as formal terms, not necessarily representing any actual number in a meaningful way. The expressions themselves have a manageable size, to be sure, even if their supposed values are outsized, but when using the expressions ultrafinitists needn't speak of or conceive the number itself, rather than only of the expressions and the rules for manipulating them. Using the expression this way doesn't commit the ultrafinist to the view that there is actually a number being represented by them.

Must ultrafinitism commit to an absolutely largest number?

Since ultrafinitism denies the existence of very large numbers, one might naturally wonder where exactly the existence stops—is there a sharp cut-off where suddenly the numbers stop existing? Is ultrafinitism committed to the existence of an absolutely largest number?

The conclusion that there is a largest number strikes many as absurd, since for any given number, it seems possible to imagine a mathematical system with one number more, and so the numbers needn't have stopped just there, but could have gone on a bit longer. The claim that there is a largest number thus might seem at best contingent. Perhaps a critic hopes to refute ultrafinitism by burdening the position with what might seem a ridiculous commitment, the existence of a largest number.

But this criticism is frankly too quick, and the philosophical position of ultrafinitism is far more subtle than this. Most accounts of ultrafinitism do not take themselves to be committed to the view that there is an absolutely largest number. Rather, the view is that the arithmetic operation of adding one is innocent, whereas other constructions such as exponentiation and superexponentiation are seen as misguided, not necessarily leading to actual numbers. The question is whether one can coherently view addition and multiplication as unproblematic, while keeping exponentiation at bay like this, taking it as only partially defined.

No sharp cut off

Harvey Friedman described an exchange he had concerning the question of whether ultrafinitism is committed to a sharp end of the numbers.

I have seen some ultrafinitists go so far as to challenge the existence of 2¹⁰⁰ as a natural number, in the sense of there being a series of "points" of that length. There is the obvious "draw the line" objection, asking where in

do we stop having "Platonistic reality"? Here this … is totally innocent, in that it can be easily be replaced by 100 items (names) separated by commas.

I raised just this objection with the (extreme) ultrafinitist Yessenin Volpin during a lecture of his. He asked me to be more specific. I then proceeded to start with 2¹ and asked him whether this is "real" or something to that effect. He virtually immediately said yes. Then I asked about 2², and he again said yes, but with a perceptible delay. Then 2³, and yes, but with more delay. This continued for a couple of more times, till it was obvious how he was handling this objection. Sure, he was prepared to always answer yes, but he was going to take 2¹⁰⁰ times as long to answer yes to 2¹⁰⁰ then he would to answering 2¹. There is no way that I could get very far with this. (Friedman, Philosophical problems in logic, seminar notes, 2002, p. 4)

The picture of ultrafinitism here is perhaps one for which the numbers get a little blurrier and less definite as one ascends to higher realms, without there necessarily being any definite point where one transitions from existence to nonexistence.

But what is the theory of ultrafinitism exactly? Is there an axiomatic account of fundamental ultrafinitist principles? Many philosophical critics of ultrafinitism despair at the lack of a precise account of the position. It is difficult after all to criticise a philosophical view that has not been clearly articulated. Can one make precise what are the mathematical and philosophical commitments of ultrafinitism?

Nontotality of exponentiation

One of the core commitments of ultrafinitism seems to be the nontotality of exponentiation, that is, the exponentiation function is not defined on all numbers—many ultrafinitists deny the existence of exponentials even with comparatively small exponents, such as 2¹⁰⁰. The question is whether for every number n there is a number fulfilling the definition of what it would mean to be the exponential 2ⁿ, and similarly aⁿ with other bases a.

In the usual arithmetic theory of Peano, we can prove that exponentiation is a total function on the basis of the standard axioms. Specifically, for every number n there is a sequence of values 2^k for k ≤ n that fulfills the exponential recursion:

One proves by induction that for every n there is such a sequence. This is clear for the anchor case n = 0, and if it is true for n, then we can make the sequence for n + 1 simply by taking the sequence up to 2ⁿ and then multiplying by 2 to form 2ⁿ⁺¹, adding it to the end of the sequence. This argument involves an account of finite sequences of numbers, which requires some arithmetic resources, and also makes use of the induction scheme for existential assertions.

The main point here is that in order to deny to the totality of exponentiation an ultrafinitist will have to deny some of the axioms of Peano arithmetic.

Feasible numbers

One approach to ultrafinitism is based on the concept of feasible numbers. We would like somehow to express a realm of feasibility, which is closed under successor and perhaps addition and multiplication, but not under exponentiation. That is, we want a notion of feasible number such that 0 and 1 are definitely feasible, and if n is feasible, then so is n+1, and perhaps if n and m are feasible, then so are n + m and nm, but this closure property does not extend to exponentiation—there should be some feasible numbers n for which 2ⁿ is not feasible.

Of course, the feasible-number idea is challenged directly by the principle of mathematical induction. After all, if 0 is feasible and whenever n is feasible then also n + 1 is feasible, then it follows by induction that every number will be feasible.

In order to preserve ultrafinitism and develop a concept of feasible number, therefore, ultrafinitists will have to deny the principle of mathematical induction, at least for the notion of feasibility itself. One might hope to weaken the induction principle in such a way that a concept of feasible number can be formulated with the desired properties.

A formal theory of ultrafinitism

These observations about the conflict between ultrafinitism and induction point one way toward formulating a precise ultrafinitist theory. Namely, we can simply weaken the axioms of Peano arithmetic to allow only primitive instances of induction. For example, the theory known as IΔ₀ allows only instances of bounded induction:

That is, we include the axiom only when the statement φ has an especially primitive form, involving itself no unbounded quantifiers.

This provides an ultrafinitist theory, because it is known that the theory IΔ₀ is consistent with exponentiation being nontotal—there are models of IΔ₀ in which we can add and multiply numbers as usual, but not every number n has an exponential 2ⁿ. These models thus form a kind of arithmetic universe of numbers that accord with fundamental ultrafinitist ideas, one in which addition and multiplication are total functions, but not exponentiation, thereby establishing the basic coherency of the ultrafinitist position.

In these models, note that while exponentiation is not totally defined, nevertheless there will be no smallest number n that has no exponential, since if 2^k exists then so does 2^k+1, since this is just 2^k · 2. Because the induction scheme has been limited, however, one cannot prove from this that every number n has an exponential 2^n, since the assertion "number n has an exponential 2ⁿ" is too complex to fall under the induction axiom scheme. Denying the totality of exponentiation seems to require the denial of induction.

In this way we have found a precisely stated theory that seems to accord with many ultrafinitist ideas. And indeed there are many other such weakened arithmetic theories, generally weakening the induction principle to greater or lesser extents, which give rise to a wide spectrum of alternative accounts of ultrafinitism.

Meanwhile, I should like to mention that none of these approaches to finding a formal ultrafinitist theory are fully satisfactory for the most radical ultrafinitist claims, if one works in a non-ultrafinitist metatheory. This is because if one denies the existence of a largest number, then it follows by metatheoretic induction that the theory must admit the existence of 2¹⁰⁰ and

and 3 ⟰ 3 and so on, simply because these numbers are represented by (extremely large) terms in the formal language. The numbers n for which the exponential 2ⁿ does not exist in a model of these theories will be a nonstandard numbers, bigger than any standard number represented by an explicit term, and because all nonstandard numbers are necessarily very large, in this sense the models of the ultrafinitist theories do not feel very ultrafinitist.

Indeed it seems difficult or perhaps impossible to formulate a precise arithmetic theory that denies the existence of a largest number, while also denying the existence of 2¹⁰⁰.

For these reasons, ultrafinitism poses a special challenge in formalization, in that in order to formalize it successfully, we seem to require ultrafinitism also in the metatheory in which the formalization is carried out. We seem to have to know already what ultrafinitism is before saying what it is. Perhaps this explains why the more radical ultrafinitist positions lack a fully satisfactory formal foundation, and it can be seen as an open question to provide a precise formal account of what the commitments of ultrafinitism are exactly.

A modal conception of potentialism

Contemporary philosophical work has emphasized the modal aspect of potentialism, expressing various potentialist conceptions in a modal vocabulary of possibility and necessity. The main idea is to conceive of potentialism via realms of feasibility, in terms of the various possible finite universe fragments that one might have as actual at a given time. If the natural numbers are potentially infinite in the sense that you can have more and more of them, as many as you like, but never all of them as a completed totality, then let us consider the possible collections of them that you can have, considering each as a possible "world," which might become the actual world for a time, if only for a moment. We might have a small world u, contained in a larger world v, which is itself contained in a still larger world w, and so on. The potentialist outlook consists at bottom in the idea that any given world u might be successively expanded in this way to larger worlds.

This situation enables a natural modal semantics and formal language of potentiality. Namely, we say that an assertion φ is possible at a world u, denoted ◇φ, if there is some world v containing u at which φ becomes true; and φ is necessary at a world u, denoted □φ, if φ is true at all such worlds v. Meanwhile, for truth assertions at any particular world u, the quantifiers for exists (∃x) or for all (∀x) range over the values of x that are actual in the world u.

These modal operators provide a formal language capable of expressing fine shades of meaning with regard to potentialist conceptions. The potentialist can express not only what is true of a given finite part of the world already in possession, but also about what might later become true with a larger collection or what will be necessarily true, or necessarily possible, or possibly necessary.

Modal potentialism with finite universe fragments in the natural numbers

Let us imagine the potentialist system in which the possible universe fragments or worlds consist of all the numbers up to some arbitrary given number n, which provides a larger universe fragment when n is made larger, as large as desired.

What will be the nature of the modal potentialist assertions? Notice that a given world u will not satisfy the assertion

because if n is the largest number in u, the currently largest number at hand, then there is no actual n + 1 yet available. But it is possible that n + 1 exists, since there is another universe fragment that goes a bit longer. Thus, although the finite worlds of this potentialist system do not fulfill the assertion that every number has a successor, they nevertheless do fulfill the assertion

Indeed, since this will be true in every universe fragment, we can say that it holds necessarily:

Similarly we express the the potential existence of sums and products:

The modal conception of potentialism thus views many ordinary mathematical claims as modal in nature.

How do we express the infinitude of primes? One way to do so is to say that above any given number there is a prime number. But the potentialist modal context makes this assertion subtle in several respects. First, when saying "above any given number" we don't want to refer only to the currently actual numbers, but rather, to any possible number that we might have later with a larger universe fragment. So we should be saying necessarily, above any given number there is a prime. But a second issue is that saying it that way exactly wouldn't quite be right, since perhaps we extend to a universe fragment that has a number, but there is yet no actual prime yet above that number. What we want to say that is that it is possible that above that number there is a prime number. Putting all this together, the assertion that there are infinitely many primes can be expressed in the modal potentialist language as the assertion:

The reader is invited in the questions at the chapter's end to translate various other arithmetic assertions into the potentialist modal language.

Nonlinear potentialism

But that is not the only way to organize the potentialist system. Let us consider an alternative conception of potentialism, for which having a number as actual does not imply that all smaller numbers are also already actual.

This is a sensible idea, in light of the fact that we can often describe or define a very large number more easily than many of the numbers below it. For example, it is relatively easy to describe the number googol plex

In decimal notation, a googol plex has an initial digit 1 followed by a googol many zeros. Thus, a googol plex has about a googol digits, and most of the numbers less than a googol plex also have about a googol digits. But for such a number with essentially random digits, there will generally be no simpler means of describing the number exactly except by pronouncing the digits one after the other. But even if we pronounced one million digits every second since the beginning of time at the big bang, we will have hardly made progress on the googol many digits of our number. Most of the numbers below a googol plex are thus far more difficult for us to describe or even conceive than is a googol plex itself. In a precise sense, then, a googol plex is more amenable to thought than most of the numbers less than a googol plex.

In light of this, it seems reasonable to consider a form of potentialism in which the numbers might arrive in the order of the complexity of their descriptions, rather than in numerical order.

Or more generally, let us consider the potentialist system in which the universe fragments consist simply of any finite set of natural numbers—let us take literally the idea that you can have more and more, as many as you like, any finite set.

According to this conception of potentialism, we may have any finite set of natural numbers as actual, while any further finite set remains possible. Thus, a finite world u accesses the possibilities in worlds v₀, v₁ containing u, worlds which might be incomparable to each other, although both will be contained in a still larger world w. Let us use the symbols and ▣ to represent possibility and necessity for this potentialist conception.

This form of potentiality has a somewhat different structure than the earlier initial-segment formulation of potentialism, since those worlds were linearly ordered by inclusion, but here they are not. Nevertheless, with the arbitrary-finite-set conception, the worlds are directed, which means any two of them are contained in a common extension. This directed-but-nonlinear feature causes some differences in the manner of expressing valid modal potentialist assertions.

For example, to say that number x is even, one ordinarily would say that there is some y such that 2y = x. But if x is actual, with this arbitrary-set conception of potentialism, it doesn't follow that the y for which 2y = x will yet be actual, and so one should say that possibly there is some y for which 2y = x.

Similarly, to say that a number p is prime, one should refer to the possible factors rather than the actual factors.

Those contortions were not required in the initial-segment formulation of potentialism.

Philosophical varieties of potentialism

The modal conception of potentialism, expressed in terms of potentialist systems of possible worlds, enables us to analyze and express certain differing philosophical features that we seek to have in our various potentialist conceptions, as well as to express sweeping general principles of potentialism. So let us discuss several varieties of potentialism that arise naturally in this modal framework and use the modal resources to express those differences.

Linear inevitability

Consider first any form of potentialism, like the initial-segment form of potentialism we considered initially, in which the universe fragments of possible worlds are linearly ordered.

One moves from smaller universes to larger, and there is a kind of inevitability of the atomic structure of the objects that appear. Linear potentialism automatically validates certain kinds of potentialist assertions. For example, in linear potentialism every possibly necessary assertion is also necessarily possible.

The reason is that if world M thinks that φ is possibly necessary, then there is some larger world M₁ that thinks φ is necessary, true from that point on. But now every world N will access some world larger than M₁, and so every world N will think that φ is possible.

Similarly, linear potentialism also validates the following modal principle:

The reason is that if both φ and ψ are possible over a world M, then there are worlds M' and M'' containing M in which φ and ψ are true, respectively. But since the worlds are linearly ordered, either M' combines before M'' or conversely, and this therefore realizes one of the two possibilities in the conclusion of the principle, thereby establishing the validity.

Convergent potentialism

Consider next a form of potentialism in which the universe fragments of possible worlds are not necessarily linear ordered, but any two worlds can be amalgamated into a common extension. Thus, the universe fragments are altogether converging to an ultimate (actualist) limit structure.

Like linear potentialism, this convergent form of potentialism also validates the assertion that every possibly necessary assertion is also necessarily possible.

If φ is possibly necessary in world M, then there is a larger world M' in which φ is necessary, but for any other extension M'' of M, there is a common extension N of M' and M'', in which φ will be true since it extends v, and so φ was possible in M. Thus, φ was necessarily possible in M, as claimed.

But in general, convergent potentialism does not generally validate the linearity principle we considered earlier:

The reason is that a world M can access two worlds M' and M'' in which φ and ψ are true, but incomparable and never true again in larger worlds. In this situation, the premise of the implication will hold in M, but not the conclusion.

The potentialist translation

Convergent potentialism, which includes the case of linear potentialism, supports a certain translation from actualism to potentialism. Namely, if the universe fragments have a convergent nature, then from the actualist perspective, we may view them as converging to a certain limit structure.

The main idea of the translation is systematically to replace actualist existential assertions ∃x with possible existence ◇∃x, and actualist universal assertions ∀x with necessary universals □∀x. We had already performed this translation several times in our earlier account giving the potentialist interpretation of various statements in elementary number theory, such as the infinitude of primes. The point I should like to make now is that this is a completely general method of translating any statement from the actualist context to potentialism. Actualist assertions about the limit model are translated to potentialist assertions about the universe fragments of which it is constituted. Furthermore, one can prove that any actualist assertion about the limit model is equivalent to the potentialist translation assertion made about the universe fragments.

The conclusion is that for convergent potentialism, the potentialist seems to have all the resources in the purely potentialist ontology to refer to the objects, the structure, and even the truth assertions of the limit model. The potentialist, to be sure, holds that the limit model does not exist. And yet, he seems nevertheless to know everything about that actually infinite structure anyway.

Philosophically, we can take this observation two ways. On the one hand, it shows the power of the potentialist perspective, since in convergent systems the potentialist is nevertheless able to express all the truths and understanding that the actualist would want. But on the other hand, we could view the observation critically, taking it to show that convergent forms of potentialism are implicitly actualist—in convergent potentialism there seems to be nothing at stake in the potentialist claim that the limit model does not exist, since the potentialist has a complete account of it.

Radical branching potentialism

What remains is a more radical form of potentialism, in which the potentialist translation will fail and indeed the nature of possibility and necessity will depend on the particular manner in which the universe unfolds. As objects become actual, they may do so in such a way that precludes other possibilities from becoming actual, even when they might have been possible before. The issue is one of branching possibility, where extending a universe fragment M to M₀ or M₁ might exhibit certain features that cannot afterward be amalgamated.

In the world M, we have two potentialities M₀ and M₁, both as possibilities. But once one of them becomes actual, the other is permanently closed off as a possibility. Having moved to world M₁, say, we may find further possibilities M₁₀ and M₁₁, which were not possible over M₀.

For example, perhaps number-theoretic phenomenon are revealed in M₀ and M₁ that are incompatible with amalgamation. Perhaps a certain computational process halts with output 0 in M₀, but it halts with output 1 in M₁. There can be no common extension of these worlds, since the computational process would have to conform with both of the earlier worlds.

In my article The modal logic of arithmetic potentialism and the universal algorithm, I showed that the models of Peano arithmetic under end-extension fulfill the nonamalgamation properties of radical-branching potentialism. Each model of Peano arithmetic is an arithmetic realm of feasibility, with potentiality exhibiting this fundamental branching-possibility nature. Indeed, the branching occurs with computational processes in exactly the manner I just described.

Potentialist graph theory

Let me conclude the chapter by briefly mentioning how the potentialist perspective arises in other parts of mathematics, focussing on the case of potentialist graph theory. A graph is a set of points, called vertices, and lines drawn between some of them, called edges. The graph might have all edges between its vertices or just some of them, or none. The subject of graph theory is a rich domain of mathematical research, with its own theory and constructions. But graph theory also often happens to have applications, since many abstract relational structures can be fruitfully thought of abstractly as graphs.

Graph theorists consider a wide variety of combinatorial features a graph might exhibit, such as an Eulerian circuit, a closed path that traverses every edge exactly once, as shown here. A Hamiltonian circuit, in contrast, is a closed path following edges that visits every vertex exactly once.

Let us consider the collection of all finite graphs as a potentialist system, where we regard one graph G as accessing graph H as a possibility, when G is an induced subgraph of H, which means that the vertices of G are all also in H, but the edges of the two graphs are the same as far as vertices in G is concerned. In other words, you can add new vertices and edges, but not new edges between old vertices.

2-colorability is expressible in potentialist graph theory

One of the combinatorial properties that is commonly considered is the 2-colorability of the graph—can we color the vertices red or blue in such a way that no two adjacent vertices (vertices connected by an edge) have the same color?

It turns out that the 2-colorability of a graph is expressible in the modal language of potentialist graph theory. Namely, a graph G is 2-colorable if and only if possibly, there are adjacent nodes r and b, such that every node is adjacent to exactly one of them and adjacent nodes are connected to them oppositely.

Basically, if the graph is two colorable, then let us imagine that we have colored it, and then we create two new vertices r and b, and attach r to all and only the red vertices and b to all and only the blue vertices.

Connectivity is expressible

What I claim next is also that connectivity is expressible in the language. A graph is connected if any two vertices may be joined by a finite path that follows edges of the graph.

Let me observe first that vertex x is connected with vertex y if and only if necessarily, any c adjacent to x, with neighbors closed under adjacency, is adjacent to y.

That is, x is connected with y if in any extension of the graph, if there is a vertex c that is adjacent to x, written c ~ x, with the further property that any adjacent neighbor of a neighbor of c (except c itself) is a neighbor of c, then c is also adjacent to y.

If indeed x and y are connected, then this is clear, since c would have to be adjacent to every node on the path joining x to y, and thus it would be adjacent to y. And if x and y are not connected, then we can imagine an extension with a new vertex c that is adjacent to every node in the connected component of x, but not to y, making the property fail. So the property exactly expresses that x is connected to y. Using this, we can now express that the graph altogether is connected simply by asserting that every two distinct nodes are connected.

Share

Give a gift subscription

Questions for further thought

1. Present two detailed accounts of the quadrature of the parabola side by side, one following an actualist account of the exhaustion and one following a potentialist account. How do the arguments differ?

2. Assess the argument that potentialism, in asserting that you can always have more, as many as you like, is thereby committed to an actual infinity, namely, the infinite collection of possibilities.

3. Is our inability actually to count up to 10¹⁰⁰ a problem for the existence of this number? What is the connection between the limitations of human abilities in reasoning and computation and the ontological questions about the existence of large numbers?

4. Can one formulate a successful notion of feasible number and thus a form of ultrafinitism by saying that the feasible numbers are precisely those that one can count to in a reasonable amount of time? This would include all the smallish numbers, and if we can count to n in a reasonable amount of time, then it might seem also that we can count to n + 1 in a reasonable amount of time; and yet, we cannot count to 10¹⁰⁰ in a reasonable amount of time, since even saying a million numbers every second since the beginning of time at the big bang, we will have counted only a tiny portion of it.

5. Which of the following modal principles are valid for the finite-initial segment conception of potentialism?

   1. ◇□φ → φ

   2. □φ → φ

   3. □φ → □□φ

   4. ◇□φ → □◇φ

   5. □◇φ → ◇□φ

   6. ◇□φ ∨ ◇□¬φ

6. Give an example of a switch, a statement φ whose truth value can be turned on and off again endlessly as one ascends to larger universe fragments. Can you provide an infinite family of independent switches φ_n, meaning that any finite on/off pattern is necessarily possible over any given universe fragment?

7. Give an example of a unpushed button, a statement ψ that is not true in a given universe fragment but is possibly necessary over that world, becoming true from some point on in larger worlds. Can you provide two independent buttons ψ₀ and ψ₁, meaning that both are initially false in some world u, but there are extensions v₀ and v₁ such that in v₀, the assertion ψ₀ is necessarily true and ψ₁ is false, and in v₁, the assertion ψ₀ is false and ψ₁ is necessarily true? Does the answer depend on whether one uses the finite-initial-segment conception of potentialism or the arbitrary-finite-collection conception of potentialism? Can you make an infinite independent family of buttons for one of the potentialist conceptions?

8. Translate the following arithmetic assertion into the potentialist modal language, using the notion of possibility by which a possible world consists of an initial segment of the numbers { 0,1, …, n }. Express the assertion in natural language by using the words necessarily and possibly, but also in a more formal expression using the modal operators □ and ◇.

   1. There is no largest number.

   2. Every even number is the sum of two primes.

   3. There is always a prime between any number n and 2n.

   4. 7n² + 9n < n³ for all sufficiently large n.

   5. There are infinitely many prime pairs (primes differing by two).

   6. The square root of two is irrational.

9. Translate the assertions of the previous question into the potentialist conception by which a possible world is any finite set of numbers, not necessarily closed under initial segment. Use the modal operators and ▣ for this conception of possibility and necessity. What are the differences in formalization for this potentialist conception versus the initial-segment conception?

Further reading

• Galileo Galilei. 1914 [1638]. Dialogues Concerning Two New Sciences. Macmillan.

  Translated from the Italian and Latin by Henry Crew and Alfonso de Salvio.

  Available at Online Library of Liberty. https://oll.libertyfund.org/title/galilei-dialogues-concerning-two-new-sciences. Read Galileo's discussion of potential and actual infinity, with the example of bending a segment into polygons or a circle.

• Hamkins, Joel David, and Øystein Linnebo. 2022. The modal logic of set-theoretic

  potentialism and the potentialist maximality principles. Review of Symbolic Logic 15 (1): 1–35. doi:10.1017/S1755020318000242. http://wp.me/p5M0LV-1zC. A technical account of various kinds of set-theoretic potentialism.

• Hamkins, Joel David, and Wojciech Aleksander Wołoszyn. 2020. Modal model theory. Mathematics arXiv:2009.09394. An introduction to modal model theory, implements potentialism in diverse parts of mathematics.

Credits

The image of the infinite ladder was created by the author via DALL·E and is available in the collection at https://labs.openai.com/sc/QJC44RJTdvvy6dLDk62dTYLt.