Ontology and Semantics of Reductions of Arithmetic

Originally written for PHIL 1675 Set Theory, Spring 2023 with Prof. Eric Guindon.

If axiomatic set theory is supposed to be a foundation for all of mathematics, as it is often treated in mathematical practice, then there cannot be sui generis mathematical objects other than sets. That is, natural numbers must in fact be sets. The ontological identification of numbers with sets, or in other words the reduction of arithmetic to set theory, was challenged by Paul Benacerraf in his seminal 1965 paper “What Numbers Could Not Be.” Benacerraf’s objection, built principally on the idea that the existence of multiple possible reductions defeats all of them, has been widely considered to deal a fatal blow to the reductionist project and definitively demonstrate that numbers are not sets. Alexander Paseau’s 2019 chapter “Reducing Arithmetic to Set Theory” is one attempt to “overturn this orthodoxy” (Paseau 35) and vindicate reductionism from Benacerraf’s objection. Here, I will outline Benacerraf’s argument and Paseau’s response, and ultimately conclude that the original objection retains its force.

Grounding the full practice of mathematics in an axiomatic set theory such as Zermelo-Fraenkel set theory with Choice (ZFC) is clearly appealing. With a set theoretic foundation, mathematics is clean and economical, conservative in its commitments both in terms of axioms and objects. If there are to be no other mathematical objects other than sets, however, then one must provide an account of the other objects that seem to exist, such as the natural numbers. The question “what are numbers, really?” demands an answer, and in the reductionist paradigm just described the answer must be that they are some kind of set. Concretely, any particular number must be some particular set. There must be some set S such that 3 = S is true, and likewise for each other natural number.

Which sets are they? Any answer must be able to exhaustively account for all the arithmetic properties of numbers. That is, the answer must reduce all numbers, arithmetic operations, and numerical facts to statements about sets. One answer (the most popular one in mathematical practice) comes from the von Neumann ordinals: 0 is identified with ∅, the successor of a natural number n is identified with the set n ∪ {n}, the ordering relation < is identified with the subset relation ⊂, and some set S has cardinality equal to the number n just in case it can be put into bijection with the set n. In this way, every statement made in the language of (Peano) arithmetic can be translated into a set theoretic statement. Moreover, on the view generated by this von Neumann reduction, there is nothing to such arithmetical talk but its set theoretic translation as just given. The referent of the numeral 3 is the set {∅, {∅}, {∅, {∅}}}. This would be sufficient to reduce arithmetic to set theory. Such a reduction demands no changes in the language or practice of mathematics, it merely means that when mathematicians talk of numbers and arithmetic they are employing shorthands and nicknames, and that the true subjects of their discourse are sets.

However, the reduction given above is not the only one possible. Another way of translating arithmetic-talk into set-talk is via the Zermelo ordinals. On this picture, 0 is ∅ and successor of n is {n}. The < relation and the cardinality properties are less immediate, but on this reduction, too, they can be translated into set-talk. For example, we can recursively define the ordering relation by saying that, m < n⁺ (that is, m is less than the successor of n) just in case m = n or m < n. This picture, too, provides a complete reduction of all the arithmetic properties of the natural numbers to sets.

The natural numbers cannot simultaneously be reduced to the von Neumann ordinals and the Zermelo ordinals. If the numeral 3 denotes the set {∅, {∅}, {∅, {∅}}}, then it cannot also denote the non-identical set {{{∅}}}. The two reductions make claims that straightforwardly contradict: a von Neumann reductionist is committed to the claim that 2 is an element of 4, whereas a Zermelo reductionist is committed to this claim’s negation. The foregoing demonstrates that there are at least two candidate reductions of arithmetic to set theory, and that at most one of them can be correct. If no reduction is true, then reductionism itself is false, numbers are not sets after all, and our original goal of grounding arithmetic in set theory has been foiled. Thus, if reductionism is to be saved, it must commit itself to a particular reduction among the candidates.

For Benacerraf, such a choice of reduction must be non-arbitrary. “If the numbers constitute one particular set of sets, and not another,” Benacerraf says, “then there must be arguments to indicate which” (Benacerraf 58.) Benacerraf does not give a positive reason to think that no such argument is possible, but there don’t seem to be many possibilities for where to locate one. The linguistic practice of number words does not seem to decide the question, and mathematical practice seems to be of little help, too. Arguments about the ease of use for one reduction over another seem to be mere “stylistic preferences” (Benacerraf 62) that only have to do with which sets are in some way nicer, not which choice is more likely to actually be correct. Not only that, but many mathematicians take themselves to be working with sui generis (that is, irreducible) numbers when doing arithmetic, so simple deference to the experts does not seem available either. We have multiple candidate reductions that all perfectly fit the bill, and no principled way to decide among them. The issue can also be framed epistemically. If one particular reduction is correct, then in order for the true set-identity of numbers to be epistemically accessible there must be some way to come to know the truth of the one true reduction over all other candidates. That is, “if the number 3 is really one set rather than another, it must be possible to give some cogent reason for thinking so; for the position that this is an unknowable truth is hardly tenable” (Benacerraf 62.) Since no such cogent reason seems to exist, Benacerraf concludes the matter against reductionism.

Paseau hopes to save reductionism by countering Benacerraf’s demand for non-arbitrariness. For Paseau, “ it is more rational to make an arbitrary selection between two equally good set-theoretic objectual semantics than to refuse to reduce arithmetic to set theory” (Paseau 38.) He presents two metaphors that help develop this view. First, he calls on Buridan’s ass, a thought experiment originally critiquing the positions of 14th-century French philosopher Jean Buridan, in which a donkey that is equally hungry and thirsty is placed in the middle of a bale of hay and a bucket of water, equidistant from each. With no reason to prefer the food over the drink or vice versa, the donkey becomes hungrier and thirstier in exactly equal amounts until it dies of malnutrition. For Paseau, the non-arbitrariness concern over multiple reducibility is analogous. There are multiple good candidates for reductions of arithmetic to set theory. Abandoning the project of reduction just because there is no principled way to decide between them would be like dying like Buridan’s ass.

To be clear, Paseau does not take himself to be performing a reduction aimed at “meaning analysis” meant to “reveal what number-talk meant all along” (Paseau 37.) Instead, he takes his reduction as an “explication” of numbers where one’s own future number-talk is meant to stand for set-talk. This distinction is somewhat unclear, since Paseau does go on to use this explication to reinterpret essentially all number-talk, including that of mathematicians predating the choice of reduction. Regardless, Paseau seems to take the issue as essentially a semantic one.

To illustrate further, let us turn to his second metaphor, that of the identical twins. Paseau writes: “suppose a mother of identical twin girls is delivered of them at exactly the same moment. She had long ago settled on ‘Olympia’ for her first-born girl’s name and ‘Theodora’ for the secondborn; but as they were born simultaneously, she is in a pickle. What to do? An arbitrary choice is called for: say, name the baby on the left ‘Olympia’ and the one on the right ‘Theodora’” (Paseau 38.) To Paseau, the situation of arithmetic reduction is similar: the choice is akin to a naming, the issue is fundamentally semantic rather than ontological in character. If someone were to press hard on the name assignments, asking “which twin is Olympia really?” (Paseau 49) because of the arbitrariness of the naming decision, not only would there be no good story to respond with, but their question would border on the meaningless.

Paseau’s solution to the problem of multiple reducibility is thus to embrace the arbitrariness. We should simply pick a preferred reduction, decide that our future number-talk is in fact a nickname for the sets involved in that reduction, and when it comes to another speaker of number-talk, trade on her intention to “ mean by her number words what others, myself included, mean by them, and her wish for her arithmetic discourse to be interpreted as true” (Paseau 42) in order to also interpret her using the reduction. In this way, the explication reduces all “normal” number-talk to set-talk, and if there are speakers who hard-headedly mean sui generis numbers to be the referents of their talk, we adopt for them an error theory in which all they say is simply false because there exist no sui generis numbers to make it true. If there are reductionists around of a different stripe, and there are so many of them that “pressures of uniformity are sufficiently felt” (Paseau 44), then we are to reconcile with them perhaps by coordinating a shared semantics, or perhaps by deferring to them and changing our reduction to theirs. The idea is that mathematicians speak number-talk without strong ontological commitments, rather primarily with the intention of simply performing mathematics. Thus, we have the leeway to reinterpret them in light of our reduction.

Whether or not this semantic picture is fully coherent, Paseau neglects the significant ontological piece of the problem. The original motivation for reduction, even by Paseau’s lights, is “a gain in ontological, ideological and axiomatic economy” (Paseau 39.) Because of this, “what are numbers really” questions should not be brushed off as attempts to push too hard on arbitrary labelings, as in the case of the twins. A question like “is 2 really a member of 4?” is not akin to a question like “does this person really have a seven-letter name?”

To see why, consider a situation in which two isolated communities make differing semantic and linguistic choices. Community A accepts the von Neumann reduction of arithmetic, doing so arbitrarily but in accordance with the semantic picture laid out by Paseau, and also thinks that the left twin is Olympia. Community B instead accepts the Zermelo reduction, by an independent instance of Paseau’s process, and thinks that the left twin is Theodora. They differ on the answers to questions like “Is 2 a member of 4?” and “Does this girl have a seven-letter name?” We could fix their semantic disputes by performing an overall reconciliation in the vein of Paseau’s procedure, but in doing so should we worry that we are herding everyone to the wrong answers to these questions? In the case of the seven-letter name, we should only be worried if the name takes ontological priority over the girl. Since we think of names purely as a feature of our linguistic behavior, it does not matter much if we do some violence to them. Our primary ontological commitment is to the girl, and what we call her ultimately doesn’t matter. If, instead, we held some sort of folkloric magical position in which there are “true names” and the name is ontologically prior, we would not accept an arbitrary semantic realignment; the answer to the question about the length of her name would have real weight. In the case of arithmetic, we are faced with the same choice. If we think that the reduction is a mere labeling (as Paseau sometimes seems to) then indeed there is no harm done by an arbitrary realignment, and questions about which numbers are members of which others ultimately don’t matter. However, this attitude only coheres with an ontological attitude that accepts sui generis numbers. It only doesn’t matter which numbers are members of which others if the reduction is merely a linguistic practice regarding the true objects, numbers.

Of course, this arbitrary reduction does indeed show that the practice of mathematics is not inherently committed to the existence of sui generis numbers by providing a reinterpretation for that practice that contains only sets. However, it does not defend us from that commitment. In order to justify the choice of an arbitrary number-less ontology, there must be a background assumption of sui generis numbers, otherwise we would not be justified in nonchalance about questions of number membership. The dilemma for Paseau, then, is clear. If reduction is still supposed to be motivated by ontological economy, then we cannot afford to be arbitrary in our choice.

Another way to see that the ontological problem remains for Paseau’s semantics-first approach is by revisiting the epistemological framing. Although we might be able to resolve the issue of semantic ambiguity arbitrarily, we cannot do so for the epistemological issue. If the numeral 3 really denotes some particular set by virtue of our arbitrary semantic choice, how can we come to know that identification? It seems absurd to say that we are gaining knowledge by means of our arbitrary choice between reductions, but the only alternative is to contend that knowledge is irrelevant here. Once again, the same dilemma arises. If we are not gaining knowledge through our arbitrary choice, then the fact of the matter is irrelevant. Paseau is not presenting a view on which there are no sui generis numbers, just one in which we can consistently talk as if there aren’t. This demonstrates that our linguistic practices don’t precommit us to their existence, but it does nothing to provide support for the idea that they don’t actually exist. Thus, the view does not benefit from any ontological economy.

In sum, Paseau’s reply intentionally ignores the deeper ontological questions that reduction raises in favor of focusing on semantics. Paseau provides a pretense under which we can hide and pretend that there are no sui generis numbers, but he provides no reason to think that there actually are no such numbers. Our numbers might be sets, and we can talk as if they are, but we are equally in the dark as to whether they in fact are with or without Paseau’s account. Thus, Benacerraf’s original objection retains its force. We can commit ourselves to a particular reduction and play the game of pretending that it is the right one, but we still have no reason to believe its ontological underpinnings.

Works Cited

Benacerraf, P. (1965). “What Numbers Could Not Be.” Philosophical Review 74(1): 47–73.

Paseau, A. (2009). “Reducing Arithmetic to Set Theory.” In New Waves in Philosophy of Mathematics, ed. O. Bueno and Ø. Linnebo, pp. 35–55.