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Published online by Cambridge University Press: 01 January 2020
I regard the whole of arithmetic as a necessary, or at least a natural, consequence of the simplest arithmetical act, that of counting…
Richard Dedekind
Albert Casullo has argued that the propositions of arithmetic could be experientially disconfirmed, with the help of an invented scenario wherein experiences involving our standard counting procedures, as applied to collections of objects, seem to indicate that 2+2≠4. Our best response to this scenario would be, Casullo suggests, to accept the results of our standard counting procedures as correct, and give up our standard arithmetical theory. This suggestion, interestingly enough, is not as bizarre as it initially appears. But indeed a problem lies in the assumption, common to Casullo’s scenario and to his suggested resolution, that our arithmetical theory might possibly be independent of our standard counting procedures. Here I show that this assumption is incoherent, whether the independence at issue is supposed to make room for the genuine possibility that 2+2≠4, or the merely epistemic possibility that we could rationally believe that 2+2≠4: given our standard counting procedures, then (on pain of irrationality) our arithmetical theory follows.
1 Dedekind, Richard ‘Continuity and Irrational Numbers,’ in Essays on the Theory of Numbers, trans. Beman, Wooster Woodruff (New York: Dover 1963)Google Scholar, 4; originally published in 1888.
2 Casullo, Albert ‘Necessity, Certainty, and the A Priori,’ Canadian Journal of Philosophy 18 (1988) 43–66CrossRefGoogle Scholar. Future references to this article will be in the text.
3 Here and throughout I take our standard arithmetical theory to be 2nd-order arithmetic, as axiomatized by Dedekind or Peano; details of this theory will be discussed in a later section.
4 This conclusion goes only part of the way towards establishing that experience could not disconfirm the propositions of arithmetic. In Casullo's scenario, the participants are justified in believing first, that they have wits enough about them to count, and second, that the objects being counted are stable throughout the counting process. Scenarios in which both assumptions are rejected depart too far from the circumstances of our own experience to provide any illuminating grip on the question of disconfirmation. And as I'll show later, scenarios in which (just) the second assumption is rejected are susceptible to the arguments presented here against Casullo's scenario. But scenarios in which (just) our mathematical wits are called into question, such as those considered by Kitcher, Philip in The Nature of Mathematical Knowledge (New York: Oxford University Press 1985)CrossRefGoogle Scholar, are not susceptible to these arguments and may, for all I say here, represent live possibilities.
5 The arguments and results of this paper transcend this particular agenda, however, applying to any account of arithmetical propositions on which these could be disconfirmed (or refuted) by experience, under the general conditions of Casullo's scenario.
6 Prominent among those who have rejected an inductivist approach to mathematics on these grounds are empiricists such as Ayer, A.J. Language, Truth, and Logic (New York: Dover 1952), Ch. 4Google Scholar, excerpted and reprinted as ‘The a priori’ in Benacerraf, Paul and Putnam, Hilary eds., Philosophy of Mathematics (Englewood Cliffs, NJ: PrenticeHall 1964)Google Scholar and Hempel, Carl ‘On the Nature of Mathematical Truth,’ The American Mathematical Monthly 52 (1945) 543-56CrossRefGoogle Scholar, also reprinted in Philosophy of Mathematics who go on to argue that mathematical propositions are analytic, and so known a priori. See also Britton, Karl in ‘The Nature of Arithmetic: A Reconsideration of Mill's Views,’ Meeting of the Aristotelian Society 6 (1947)Google Scholar and Field, Hartry Science Without Numbers (Princeton: Princeton University Press 1980), Ch. 1Google Scholar.
7 Mill, John Stuart A System of Logic (New York: Harper 1867)Google Scholar. Mill's disconfirming scenario is discussed in chapter vi of Book II, and chapter xxiv of Book III.
8 Britton, ('The Nature of Arithmetic,’ 2–6)Google Scholar provides convincing textual evidence that Mill ‘half-acknowledges’ that these ‘two general conditions’ are assumed to be in place in his scenario, but it takes some doing.
9 Casullo does not argue explicitly for the inverse of (P2) - that if experiential evidence can disconfirm mathematical propositions, then it can confirm such propositions. Presumably his remarks here are intended to leave the inverse of (P2) open as a live possibility (in particular, for the inductive empiricist). Establishing this inverse would be a different project, and one that would have to respond to a priorist accounts that try to show that mathematical propositions are known a priori, in spite of being potentially disconfirmable by experience. Cf. Summerfield, Donna ‘Modest A Priori Knowledge,’ Philosophy and Phenomenological Research 51 (1991) 39–66CrossRefGoogle Scholar.
10 Strictly speaking, Casullo presents a version of (A) where one maintains only that the Correct Counting condition failed to hold, in spite of all evidence to the contrary. However, there seems to be no reason why one couldn't instead maintain that it was the Stability condition that had failed to hold.
11 Here Casullo is assuming an account of individuation of arithmetical theories according to which disconfirmation of even a single proposition of the theory disconfirms the theory as a whole. This is certainly true on an inductive empiricist account, given the foundational nature of the epistemically basic propositions at issue. But more generally, disconfirmation of the sort of elementary arithmetical propositions at issue here would likely render a sufficiently large tear in the fabric of standard arithmetical theory so as to render the theory disconfirmed as a whole, however one took that fabric to be woven.
12 Ayer, ‘The a priori,’ 318Google Scholar
13 Hempel, ‘On the Nature of Mathematical Truth,’ 378-9Google Scholar
14 Gasking, Douglas ‘Mathematics and the World,’ reprinted in Logic and Language, Flew, Antony ed. (New York: Anchor 1965), 430-1Google Scholar
15 Note that it would be enough for Casullo to establish that participants in the scenario could, consistent with the assumptions of the scenario, take their experience as disconfirming the proposition that 2+2=4; to block (P1) of the Irrefutability argument, he need not argue that they should, or even that they would, do this.
16 The above responses would also, for the same reason, fail to appropriately address the genuine possibility that might be at issue in Casullo's scenario.
17 Ayer, ‘The a priori,’ 319Google Scholar
18 Hempel, ‘On the Nature of Mathematical Truth,’ 379Google Scholar
19 Admittedly, the inductive empiricist could do more to make intelligible how the definitions of number terms might be amenable to experience. Kitcher, in ‘Arithmetic for the Millian,’ Philosophical Studies 37 (1980), 219CrossRefGoogle Scholar, attempts this on Mill's behalf: ‘I suggest that we read [Mill] as offering an epistemological thesis about definitions: to be justified in accepting the definitions on which arithmetic rests we must have empirical evidence that those definitions are applicable …. Mill would allow that certain sentences of our language are true in virtue of the connotations of the expressions they contain, and that we can defend our assertion of these sentences by citing our understanding of the language. However … our defense is adequate only so long as our right to use our language is not called into question. In particular, if experience gives us evidence that certain concepts are not well-adapted to the description of reality our assertion of sentences involving those concepts is no longer justified …. ‘ Kitcher supports his reading of Mill, in part, by reference to Mill's discussion of the term ‘acid.’ It was part of the original definition of this term that an acid had the property of containing oxygen, so that, at one point in time, the assertion of a sentence like ‘All acids contain oxygen’ could have been defended simply on grounds of understanding the terms involved. After the discovery of hydrochloric acid, consisting only of hydrogen and chlorine, both definition and defense were undermined. On this understanding, disconfirming scenarios can be seen as attempts to show that arithmetical relations stand to the definition of number terms as the property of containing oxygen stood to the original definition of ‘acid.'
20 Thanks to Mark Richard for this suggestion.
21 Gasking, ‘Mathematics and the World,’ 442Google Scholar
22 Of course (as per the epistemic possibility) if the participants are wrong about either the Correct Counting or Stability conditions holding, it might be the case that a given ‘counting to two’ failed to accurately reflect the number of objects being counted (although such inaccuracy would have to be persistent, systematic, and undetectable by participants in the scenario, who are justified in believing the conditions to hold). As it turns out, the accuracy of the counting results is irrelevant to the question of whether Casullo's scenario is coherent. For the moment, it is enough to note that Casullo's scenario is not designed to call counting results into question.
23 Casteneda, Hector Neri ‘Arithmetic and Reality,’ The Australasian journal of Philosophy 37, 2 (1959), 103Google Scholar. It is uncontroversial that something like these principles is involved in standard counting procedures. To see how Casteneda’ s principles could be applied to a concrete example (involving say, counting a collection of apples on a table), do the following:
(1) Take the set N of numbers in (C1) to be drawn from the ordered sequence of natural numbers (1, 2, 3, … ).
(2) Establish a 1-1 correspondence between the apples and the set N by pointing once and only once (mentally or physically) to each apple on the table, each time attaching (mentally or physically) the next number in the sequence of natural numbers (starting with 1). Note that here we would be carrying out (CS), version (i).
(3) When done, investigate N, and apply (C4). If (C2) has been satisfied (this will happen when there is at least one apple on the table), then either there will be a last number in N (this is just what it means to say that (C3) is satisfied), and this last number N will be the number of apples on the table, or the number of apples on the table is infinite. If (C2) was not satisfied, then there were no apples on the table.
24 In mathematical terms, a one-to-one correspondence (a.k.a. a ‘bijection’) is a relation R:S→ T that is 1-1 (distinct elements of S are R-related to distinct elements of T) and onto (for every element in T, there is some element in S that is R-related to it).
25 The axioms of 2nd-order Peano Arithmetic (PA2 ) (in whole numbers, for purposes of counting) are:
(PA1) 1 is a number.
(P A2) Every number n has an immediate successor, s(n), which is also a number.
(P A3) If two successors s(m) and s(n) are the same number, then m and n are the same number.
(P A4) 1 is not the successor of any number.
(PA5) Every property P is possessed by all numbers if 1 has P and if, when n has P, s(n) has P. (mathematical induction)
Peano's original formulation invoked 9 axioms, but 4 of these dealt with identity; these latter, being considered part of the assumed underlying logic of the theory, have been dropped from subsequent formulations. See Peano, Giuseppe ‘The Principles of Arithmetic, Presented by a new Method’ (1889) reprinted in Heijenoort, Jean van ed., From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931 (Cambridge: Harvard University Press 1967) 83–97Google Scholar.
Peano's 1889 axioms are directly based on Dedekind's 1888 definition of a simply infinite system (article 71) and theorem of complete (that is, mathematical) induction (article 80) in ‘The Nature and Meaning of Numbers,’ Essays in the Theory of Numbers 44-115). While Dedekind should be credited as such, I will follow current (inertial) usage in discussing the Peano (rather than the Dedekind, or Dedekind-Peano) axioms.
26 Thanks to Harold Hodes for making clear that this thesis underlies the present discussion, and for his generous assistance in investigating the question of definability of arithmetical operations in PA2 in what follows.
27 Those who are uninterested in how arithmetical operations may be defined in formalizations of PA2 can skip ahead to the paragraph starting ‘We might wonder, however, if another line of inquiry …’ without undue loss of continuity.
28 This usage of ‘implicit definition’ applies to interpreted languages, e.g., a language about the natural numbers as interpreted in a standard model. In this way it is weaker than current standard usage in model theory, which requires that every model for the definiens (the sentences doing the implicit defining) uniquely determines the interpretation of the definiendum (the constant[s] being implicitly defined). See Chang, C. C. and Keisler, H. Jerome Model Theory, 3rd ed. (The Netherlands: North-Holland 1990), 90.Google Scholar
29 In these formalizations, the first-order variables range only over natural numbers, and talk of first-order and second-order variables should be understood accordingly.
30 Peano inductively defines addition by introducing the constant ‘+’ and defining this constant as follows:
Take the base clause to be: n + 1 = s(n)
Take the induction clause to be: n + s(m) = s(n+m)
As an example, consider 2 + 2, i.e., s(1) + s(1). Apply the induction clause once to get
s(1) + s(1) = s(s(1) + 1)
Apply the base clause to get
s(s(1) + 1) = s(s(s(1))) = 4.
Peano's definition of addition does not explicitly call attention to the fact that he is introducing an (inductively defined) primitive symbol, and so effectively introducing two new axioms (corresponding to the base and induction clauses above), apparently due to his expressing the successor function in the original axioms as ‘x+ 1.’ (Two additional axioms are also needed to inductively define multiplication in Peano's system.) It is for this reason that later expositions of PA do not express the successor function using the ‘+’ symbol. As van Heijenoort ('From Frege to Godel,’ 83-4) remarks in his introduction to Peano's article, ‘From the outset, Peano uses the notation x + 1 for the successor function. He then introduces addition (section 1) and multiplication (section 4) as “definitions” …. Peano does not explicitly claim that these definitions are eliminable, but, just as he does for ordinary definitions … he puts them under the heading “Definition,” although they do not satisfy his own statement on that score’ (93), namely, that the right side of a definitional equation is ‘an aggregate of signs having a known meaning.'
31 This move allows the 1st-order axiom schema for mathematical induction (For P a property: (P1 & (Pn → Pn+1)) → ‘?'xPx) to be replaced by the 2nd-order closure of the schema ('?'P ((P1 & (Pn ∼ Pn+1)) ∼ ‘?'xPx)), and it is this replacement which rules out the unintended models.
32 This follows from Buchi's theorem (Siefkes, Dirk Buchi's Monadic Second-Order Successor Arithmetic [Berlin: Springer-Verlag 1970])CrossRefGoogle Scholar that the monadic second-order theory of zero and successor is a decidable theory; if addition and multiplication could be explicitly defined in this theory, it would include PA1 , which then would be decidable. But PA1 isn't decidable (by Church's theorem), so (by modus tollens) addition and multiplication cannot be explicitly defined in this theory.
33 If the 2“d-order quantifiers range over dyadic properties, we can explicitly define addition as follows: 
If the 2nd-order quantifiers range over monadic functions, we can explicitly define addition as follows: 
We can also provide explicit definitions in a way that utilizes the universal, rather than the existential, prefix.
34 In fact, Dedekind didn't transform his inductive definitions into explicit definitions, although the technique was semi-available at the time of his writing ‘The Nature and Meaning of Numbers.’ Frege used something like this technique in defining the ancestral of a sequence in his 1879 Begriffsschrift (reprinted in From Frege to Godel, 1-82), at the end of section 26.
35 Dedekind, ‘Continuity and Irrational Numbers,’ 4Google Scholar
36 Hempel, ‘On the Nature of Mathematical Truth,’ 382Google Scholar
37 This understanding of addition and multiplication plausibly explains why Peano did not take his ‘Definitions’ of addition and multiplication to be introducing new primitive constants (and hence, to be new axioms for his theory).
38 For an excellent discussion of how counting may be seen as a transformation of tallying, see Goodstein, R. L. ‘The Meaning of Counting,’ in Essays in the Philosophy of Mathematics (Leicester: Leicester University Press 1965)Google Scholar.
39 In fact, for purposes of fulfilling the requirements of Casullo's scenario, participants in the scenario need not have in hand an infinite sequence of cardinal representations, since the arithmetical propositions at issue (being epistemically basic propositions) involve only relatively small numbers. A sufficiently large finite sequence, isomorphic to some initial segment of the natural numbers, would do the trick. For conceptual and expository purposes, it is convenient to make the gesture towards infinity, but the arguments to follow would go through on an understanding of the tallying sequence as large, but finite.
40 Consider: beings with extremely fine-grained perceptual abilities might not find it necessary to move to symbolic representations of tally marks, if they were, in virtue of these abilities, capable of instantaneously grasping the cardinality represented by a given set of tally marks.
41 It should be clear that this argument holds against Casullo's claim (as does the argument to follow) even if the participants (misguided about one or other of the conditions’ holding) don't get it right about how many objects there were— that is, even if the results of their counting procedures aren't, in fact, correct. Casullo's claim is just the claim that given some counting results, the arithmetical relations holding between these results is ‘an open empirical question.’ Whether the counting results are correct is irrelevant to this claim, and to my arguments against it.
42 This argument can also be used to show that, under the assumption that the conditions in fact hold, only one outcome is genuinely possible. Just replace all references to justified belief with references to knowledge (or to what knowledge entails — namely, truth). The argument then goes: Since the Stability and Correct Counting conditions hold, no relevant difference can derive from either the objects being tallied, or from the participants carrying out the procedures — the objects are stable and the participants get it right in both cases. Most crucially, since tallying does not distinguish between accumulations and new starts, no relevant difference can be found in the procedures themselves … [as such] there doesn't appear to be any way for the results of the procedures to be different. If the result of the first procedure is known, then so is the result of the second procedure, and the participants (and we) can say this a priori.
43 Many thanks to Eric Hiddleston for the heart of this argument.
44 Here I am assuming that participants in the scenario could easily see that the composition of two one-to-one correspondences (where the range of the first is the domain of the second) will itself be a one-to-one correspondence. This seems right, since they can surely see (as we do) that this follows straightforwardly from even a non-technical understanding of the notion of a one-to-one correspondence (from the ‘transitivity of matching,’ as it were). In addition, attention to what participants would do with the results of counting/tallying procedures indicates their acceptance of compositionality. For example, given their justified beliefs that the Correct Counting and Stability conditions hold, participants would surely justifiably believe that the copying of a given tally onto another tablet would preserve the cardinality associated with the collection originally tallied. (After all, Casullo's scenario is not designed to call the results of countings/tallyings into question.) But to justifiably believe this is to justifiably believe that the compositionality of two one-to-one-correspondences (where the range of the first is the domain of the second) is itself a one-to-one correspondence.
45 As with the first argument, this argument can be adapted to show that, under the assumption that the conditions in fact hold, only one outcome is genuinely possible. Again, just replace all references to justified belief, and to what the participants ‘take’ to be the case, with references to knowledge (or truth).
46 Roughly: to multiply m by n, we (in tally notation) tally m, n times. That is, (keeping track, perhaps with another tally) we string together n tallies of the number m. For example, to multiply 3 by 2, we write one tally of
:
and then append a second tally of , to get
which member of the tallying sequence corresponds to the arabic numeral ‘6.’ Multiplication, it should be clear, just amounts to successive additions of a number to itself, so seeing this relation, too, as (we might say) supervening on the cardinal aspect of number, should not be surprising. And as in the case of addition, we can read off various properties of multiplication from a single tally. For example, we can draw purely conceptual marks in the previous tally as follows:
to see (after appropriate translations) that 2·3 = 6; that 2·3 = 3·2 (multiplication is commutative), and so on.
47 Both Casullo and Mill may have been drawn into thinking that the cardinality and arithmetical aspects of number were independent of each other, as a consequence of framing their discussions using the standard (arabic numeral) representations of the natural numbers. Using these numerals as representations of cardinality, it might seem that if we perform one ‘counting to two’ and then a second ‘counting to two,’ then we are still in the dark as regards the results of a counting of the objects taken altogether. Certainly placing the results of the first two countings next to each other (in this case, placing the second ‘2’ next to the first ‘2’ to achieve ‘22’) is in no way informative as to the result of the counting of the objects taken as a single collection. But by now it will be seen that this is an artificial feature of any representation of cardinality which, in counting, distinguishes ‘new starts’ from ‘accumulations.'
48 I would like to thank Carl Ginet, Eric Hiddleston, Mark Richard, Jeff Roland, Jason Stanley, Zoltan Szabo, Stephen Yablo, two anonymous referees for the Canadian Journal of Philosophy, and especially Harold Hodes, for invaluable assistance in improving this paper and my own understanding.
