Is there a historically plausible account of the real numbers?

Question

Frege's ill-fated program to define the natural numbers in terms of abstraction was intended as a genuine account of what natural numbers are, not just a way of encoding numbers in set theory like Russel and Whitehead did.

Russel and Whitehead identified, for example, the number 1 with the set containing the empty set. This can't be taken as a serious proposal for what the number 1 is and always has been. It is implausible that throughout history, when people have used the number 1, what they were really referring to is the singleton set containing only the empty set. How is this set connected to a set containing just one sheep or one urn of beer? Why would ancient people have had any use for a set containing just the empty set?

By contrast, Frege identified the number one with the set of all sets with one element. This is far more plausible as an account of what numbers are and always have been. There is a natural, intuitive relationship of membership between the set of all sets with one element and a set containing just one sheep and with a set containing just one urn of bear. Ancient people may not have been able to articulate that the number 1 is the set of all singleton sets, but they used it just like it was. They also couldn't articulate that the Morning Star was a planet roughly the same size as the entire earth, but that didn't stop them from referring to it.

Let me define a historically plausible account of an abstract object (such as a number) as an account that defines an abstract object in terms of other abstractions in such a way that the account explains how historical people were able to grasp the original object and operate on it without understanding exactly what it was. By this definition, Frege's account is historically plausible and Russel and Whitehead's is not.

Another example of a historically plausible account of mathematical objects is Whitehead's account of geometrical objects such as points and lines in terms of extensive abstraction.

Neologicists have taken up Frege's method and tried to fix the problems with Basic Law V as well as trying to account for other areas of mathematics such as real numbers. However, when they define real numbers, they define integers as pairs of real numbers, rational numbers as pairs of integers, and real numbers as sets of rational numbers. None of this strikes me as historically plausible.

So, I really have two questions: (1) Don't neologicists care about historical plausibility? Have they just discarded that part of Frege's work? (2) Has anyone tried to account for real numbers in a historically plausible way?

Answer 1

As far as I know, the modern concept of real numbers, as far as its historical semiotics go, in some places developed in the realization that an infinite series of simplest fractions, such as represented π, could have a semantically equivalent decimal form. But so we have this concept of a sequence of fractions, and so π, for example, does not have permanently repeating decimal digits because the counterpart pattern among its simplest-fraction format does not exist either. Is a sequence of fractions psychologically/historically equivalent to a set of them? In Cantorian terms, maybe, we would say that it was some sort of ordered set, notwithstanding the first-order patternlessness of π. So how far apart are the concepts of sequences and ordered sets?

Oddly enough or not, there's a Wikipedia article on multisets which includes a history section in which they mention a claim by one analyst that the concept of multisets has been immediately implicit in our thoughts going back to ancient times. (Immediate implication is near-surface mental knowledge in that they were implicitly expressing/applying the concept of multisets in their physical actions/semiotics. So besides how the Wikipedia article describes the claim, that's one positive way of trying to think through its being hypothetically true.)

Another set theorist makes the case that our negative epistemic states in general, when our understanding of them is applied to set theory, yields a "demand" for new forms/concepts of and in mathematics:

... the resolution of b-predicaments in a manner compatible with one’s position results in truly novel observations concerning V, thereby helping one to better situate one’s position. However, in coming to a resolution of a b-predicament, even one that ends up being resolved in a manner compatible with one’s own view, we may require a period of toleration of other foundational systems in tension with one’s current position. After all, it is precisely the nature of b-predicaments that we cannot yet formulate the relevant solution. Exploration of these issues thus requires novel techniques, some of which may conflict with our own view whilst being necessary to see the shape of a solution. Thus, somewhat paradoxically, one’s own view can be strengthened (both through understanding our knowledge/ignorance and our level of confidence in our justifications/explanations) by adopting a Pluralism towards foundational theory in order to draw out the contents of different ways of looking at the universe. The development of alternative frameworks yields information about one’s own preferred theory (indeed many other theories), and helps us to tell a better story of how different axiom systems stand with respect to confirmation. Fixation on a single theory masks this useful information, and obscures possible unorthodox pioneering insights.

POV: you're a "Platonic realist" about set theory: you say that the facts of mathematics are in some sense discovered, even if partly constructed/invented as well: the invention itself is discovered, so to say, but anyway, this always results in spiraling manifolds of new concepts, so mathematics is always historically transforming in various ways, based on this delicate interplay of partial construction and partial independent, primary reality, which is the "discoverable" domain proper.

In this context, you might interpret the development of the concepts of irrational numbers, algebraic reals, transcendental reals, and then arbitrary/forcing-theoretic reals (in higher set theory); interpret all this, that is, as discovering more about what the real numbers are, and better ways to notate the mathematics involving them. In set theory we are accustomed to often thinking of real numbers in terms of arbitrary subsets of countable infinity, but it was a pursued theme in the community that we could try Dedekind cuts or Cauchy sequences as good representations of the "real nature" of the real numbers.

Another far weirder but surprisingly stable option is to go to the surreal number system. This has both proper infinitesimals (with almost no conceptual effort! they appear as directly as possible in the notation) as well as real numbers, e.g. π (but also curious terms like ω_ω+21/π^π-ω₁₂). Some analysts claim that the form of the surreal Continuum is absolute in a way that its form in pre-surreal numbering, even in Cantorian/successor notations, is not. I think this is an at least moderately plausible conjecture as you can get surreal infinitesimal numbers for the form a/ω_b, for all a, b < ω_b, which is quite a lot of infinitesimals, after all. In fact, if the surreal Continuum is continuous in part because it sustains both infinitesimals (of any transfinite ordinal scale, perhaps) and real numbers, the surreal Continuum might be represented by some interleaving or other merging/unifying function for theories, applied to the dually surreal infinitesimal-real numbers.