Abstract
The term ‘continuous’ in real analysis wasn’t given an adequate formal definition until 1817. However, important theorems about continuity were proven long before that. How was this possible? In this paper, I introduce and refine a proposed answer to this question, derived from the work of Frank Jackson, David Lewis and other proponents of the ‘Canberra plan’. In brief, the proposal is that before 1817 the meaning of the term ‘continuous’ was determined by a number of ‘platitudes’ which had some special epistemic status.
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Notes
See Reichenbach (1924) for another statement of the view.
Postulationists may disagree on the question of what exactly ‘coherence’ is. Perhaps the most obvious proposals are:
(a) A set of sentences \(\Gamma \) is coherent just in case there does not exist a proof of \(\bot \) all of whose premises are elements of \(\Gamma \).
(b) A set of sentences \(\Gamma \) is coherent just in case there is a model \(\mathfrak {M}\) such that each element of \(\Gamma \) is true relative to \(\mathfrak {M}\).
These views can differ significantly when applied to cases in which the mathematical language used allows second-order quantification. My own preference is for a view more like (b)—though see (Donaldson 2014) for details.
When using the stipulations and deductions method, one defines certain words by stipulating that certain sentences containing those words are true. One might prefer a variation of the method in which it is concepts which are defined. Following Rayo, I’ll stick to the ‘linguistic’ version of postulationism, simply because I prefer to avoid difficult questions about what concepts are.
Famously, Quine (1951) argued that notions like ‘definition’ and ‘synonymy’ are too obscure to be of any philosophical utility. He did, however, make an exception for stipulative explicit definitions:
There does, however, remain still an extreme sort of definition which does not hark back to prior synonymies at all; namely, the explicitly conventional introduction of novel notations for purposes of sheer abbreviation. Here the definiendum becomes synonymous with the definiens simply because it has been created expressly for the purpose of being synonymous with the definiens. Here we have a really transparent case of synonymy created by definition; would that all species of synonymy were as intelligible.
Some later, more extreme Quineans have thought that Quine should not have made this concession. See Juhl and Loomis (2010) for discussion, especially p. 214.
Rayo discusses the issue directly in Rayo (2013, ch. 8), though it would be not be too unreasonable to say that the whole book is a sustained criticism of the ‘metaphysicalist’ presuppositions of this particular objection to postulationism. For an exegesis of this Rayo’s position, see (Donaldson 2014).
One might protest as follows. Some of our mathematical theories entail that vastly many objects exist: Boolos has written that \(\hbox {ZFC}^{2}\) implies that there are, ‘by ordinary lights’, ‘(literally) unbelievably’ many objects (Boolos 1998). Now even if \(\hbox {ZFC}^{2}\) is coherent, there’s no guarantee that there exist that many objects, and so there’s no guarantee that there is any interpretation of the language of set theory on which all the axioms of \(\hbox {ZFC}^{2}\) are true.
I do not have space to explain Rayo’s reply. Instead, I will repeat a Rayovian metaphor. The objection fails, Rayo thinks, because it presupposes that we are ‘operating against the background of a fixed domain’ (Rayo 2013, p. 93). Instead, we should recognize that there are many ways of ‘carving up the world into objects’. Once we understand how this ‘carving’ works, Rayo thinks, we will see that so long as our purely mathematical theory is coherent, there will always exist a ‘carving’ which generates enough objects to verify the theory (Rayo 2013, §1.5).
I realize that this response is mere metaphor. (Donaldson 2014) contains (I hope!) a more satisfying discussion of the issue. I would like to thank an anonymous reviewer at Synthese for suggesting that I include a comment on this issue.
Rayo also makes a third suggestion, which is that one can establish that a system of axioms is coherent by giving a consistency proof for the system. I omit this suggestion because, as I hinted in footnote 2, I do not believe that proof-theoretic consistency suffices for ‘coherence’ in the relevant sense.
One indication of the epistemological significance of pictorial models in mathematics is this: one of the reasons that mathematicians are wary of Quine’s ‘New Foundations’ (Quine 1937) is that, so far, no pictorial model for the theory has been found. As Fraenkel, Bar-Hillel and Levy write:
From the point of view of the philosophy and the foundations of mathematics, the main drawback of NF is that its axiom of comprehension is justified mostly on the technical ground that it excludes the antinomic instances of the general axiom of comprehension, but there is no mental image of set theory which leads to this axiom and lends it credibility (Fraenkel et al. 1973, p. 164, emphasis added).
The postulationist should add that some mathematical beliefs are justified by non-deductive reasoning. For example, I take it that mathematicians today are justified in believing that \(\pi \) is normal, even though this has not yet been proven.
For example: Can there be a priori knowledge by testimony? See Malmgren (2006) for a recent discussion of this issue.
See Stedall (2008, Sect. 11.2.2) for Euler’s use of the term.
It is plausible that the modern definition of ‘continuous’ captures only one precisification of the term from 1800, and that on another precisification it meant something more like what is meant today by ‘uniformly continuous’. This is suggested by Lakatos (1976, p. 129, fn. 2).
There were informal definitions of ‘continuous’ prior to Bolzano’s paper. For example, in a monograph of 1814, Cauchy said that when a function is discontinuous, ‘insensible’ changes in the argument give rise to ‘brusque jumps’ in the value; this is a useful though very sketchy definition of ‘continuous’ (Grabiner 1981, p. 93). However, I assume that such informal definitions were not sufficient to determine the meaning of ‘continuous’ or to permit the demonstration of the important theorems about continuity.
I would like to thank Kate Manne for suggesting this response to me.
For example, at one point Cauchy says that it is ‘a fundamental axiom’ that ‘the sum of several numbers remains the same in whatever order we add them’ (Cauchy et al. 2009, p. 269).
I say ‘everyday’ to indicate that Lewis’s account is not supposed to cover technical terms from cognitive science.
See also Lewis (1970).
Lewis accepted that (a) some minority of the platitudes may be false. He also claimed that (b) the platitudes are ‘common knowledge among us’ (p. 256). How are these two claims to be reconciled?
Note that after making claim (a), Lewis wrote ‘let us set aside this complication for the sake of simplicity’ (p. 252). Later (p. 258) Lewis explained that he was ‘ignoring clustering for the sake of simplicity’. So my suggestion is that when Lewis made claim (b) he was engaging in simplification or idealization.
See Braddon-Mitchell and Nola (2009) for more recent work on the Canberra plan.
One might argue more directly that not all platitudes are true by claiming that (10) was a platitude in the mid-nineteenth century and observing that it is false. The argument is inconclusive, however, because it is not clear that (10) really was a platitude.
A terminological variant on this view is to insist on applying the word ‘platitude’ only to true platitudes, and then saying that all ‘platitudes’ are true, a priori and basic.
I risk being accused of over-intellectualization at this point. It can be argued that \((11')\) and \((12')\) are rather refined, theoretical claims—but platitudes are supposed to be boring, obvious truths. The first point to make in response is this: my topic in this paper is the mathematical knowledge of expert mathematicians, not the general public. And \((11')\) and \((12')\) are obvious for expert mathematicians. More importantly, despite the connotations of the word ‘platitude’, platitudes need not be obvious. Providing sufficient reason to believe that one’s platitudes are coherent can be very difficult; in such cases, the truth of the platitudes can be highly non-obvious. I would like to thank an anonymous reviewer at Synthese for bringing these issues to my attention.
Other examples are easy to find. When defending the principle of induction for natural numbers, mathematicians will sometimes show that it is entailed by the well-ordering principle. The claim that every non-empty set of reals that is bounded above has a supremum is today typically taken as an axiom of real arithmetic, but sometimes it is defended by deduction from the claim that the real line is Dedekind-complete, which is usually presented as a theorem. And so on.
I have reached this position by discussing the matter with mathematicians. Admittedly, this is not a rigorous procedure. Perhaps some experimental philosophy would be warranted here.
In some cases, the ‘new’ term will be a replacement for some pre-existing term—and may be pronounced and spelled in the same way as its precursor. For example, when Bolzano gave his definition of ‘continuous’, arguably he replaced a pre-existing term with a homonym. Carnap used the term ‘explication’ for this process:
The task of making more exact a vague or not quite exact concept used in everyday life or in an earlier stage of scientific or logical development, or rather of replacing it by a newly constructed, more exact concept, belongs among the most important tasks of logical analysis and logical construction. We call this the task of explicating, or of giving an explication... (Carnap 1947, pp. 8–9).
Explicit stipulative definitions of proper names, operators etc. can be dealt with in the same way. This is left as an exercise for the reader.
See Feferman (2000) for an illuminating discussion of ‘monsters’.
... or, indeed non-deductive arguments. Zermelo (1908) contains, in effect, a non-deductive argument for the axiom of choice, variants of which are still commonly used today. I believe that it is possible for the kitchen sinker to give an account of such arguments, but I won’t attempt this important task here.
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Acknowledgments
For helpful comments on this work, I would like to thank Agustín Rayo, Brian Weatherson, Jennifer Wang, Clay Cordova, Kate Manne, Andy Egan, Ernie Lepore, and two anonymous referees at Synthese. Most of all I would like to thank Zeynep Soysal and Antony Eagle, both of whom were amazingly generous with their time and provided tremendously helpful comments.
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Donaldson, T. Platitudes in mathematics. Synthese 192, 1799–1820 (2015). https://doi.org/10.1007/s11229-014-0653-5
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DOI: https://doi.org/10.1007/s11229-014-0653-5