Abstract
Benacerraf’s Problem about mathematical truth displays a tension, indeed a seemingly unbridgeable gap, between Platonist foundations for mathematics on the one hand and Hilbert’s ‘finitary standpoint’ on the other. While that standpoint evinces an admirable philosophical unity, it is ultimately an effete rival to Platonism: It leaves mathematical practice untouched, even the highly non-constructive axiom of choice. Brouwer’s intuitionism is a more potent finitist rival, for it engenders significant deviation from standard (classical) mathematics. The essay illustrates three sorts of intuitionistic deviations (weakening, refinement and outright contradiction), and goes on to sketch the technical tools and arguments that engender those clashes with classical mathematics and the philosophical principles underlying those arguments. Those philosophical principles coalesce into a “standpoint” no less unified and no less finitary that Hilbert’s. This intuitionistic standpoint is sufficiently detailed that it itself provides a benchmark for comparing all the rival positions; and it is sufficiently robust that it dissolves the Benacerrafian dichotomy. However, the intuitionistic refutation of the axiom of choice reveals two distinct sub-streams within that intuitionistic standpoint. Distinguishing these streams suggests perhaps that in spite of that robustness, intuitionism has an internal dichotomy parallel to the Benacerrafian split. The Benacerrafian split is a crack in the foundations of mathematics. But I shall argue that this internal intuitionistic dichotomy is not at all a foundational gap; it is rather a special insight about the nature of mathematical thought.
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Notes
- 1.
The occasion was a conference in New York City in March of 1976. The topic of our session was “Platonism and Mathematical Existence”. In addition to Mark and myself, the other panelist was Oswaldo Chateaubriand. The commentator was W.V. Quine. The conference was sponsored by the then government of Iran; its proceedings were never published. The title of my talk in that session was “Platonism and the Pre-Ontology of Mathematics.” However, Gila Sher has persuaded me that the expression Proto-Ontology much better captures the notion I have in mind. To be sure, the ideas that appear in the present essay have evolved in detail considerably since that first presentation; but the general outlook remains the same. I am grateful to Wim Veldman for comments on an early presentation of the material in Sects. 2 and 4, and to Yemima Ben-Menachem for comments on an earlier draft of the entire paper. The preparation of this paper was supported by Grant 1715–21 of the Israel Science Foundation.
- 2.
To be sure, Benacerraf doesn’t mention Hilbert by name, he speaks rather of the “combinatorial” approach. But this fits Hilbert to a tee.
- 3.
- 4.
The above version of the axiom will be most useful in what follows. However, one can get the same effect, and make the very same points as I do in this paper, by using the version found in Russell (1906) and Zermelo (1908): If U is a disjoint collection of non-empty sets, there is a set s which consists of one and only one element from each set in U. The problem will be needing but not being able to produce a “method” for choosing the required element from each set. Restricting the axiom to analysis, and even as we’ll do a bit later to functions taking values in the natural numbers, does nothing to reduce this affront.
- 5.
Hilbert (1926).
- 6.
Of course the causal conception is, as Burgess and Rosen (1997) say, “a perennially powerful picture”. Indeed they rephrase Benacerraf’s problem as “great gulf” between reality as a “system connected by causal relations and governed by causal laws” on the one hand, and a mathematical realm “teeming with entities unlike concrete entities and causally wholly isolated from them...” (p. 534).
- 7.
Bernays – who often served as Hilbert’s philosophical voice, explicitly speaks of this abstraction in (1923). Parsons (2008) §40 gives a detailed analysis of this aspect of Hilbert’s thought.
- 8.
Gödel’s representation theorem brings these together. Later devices, such as Kleene’s T predicate give numerical expression to the properties of recursive functions.
- 9.
In (1926), Hilbert speaks of the “finiten standpunkt.” The expression “standpoint” comes up from time to time in his writing and for instance in Hilbert and Bernays (1934) §1. However, it is probably Gödel’s (1958) – ironically, an ‘extension’ of the standpoint – that standardized the expression “finitary standpoint” as a term for the Hilbertian point of view.
- 10.
Never mind that Gödel showed this wrong. See Seig (2013) for a detailed study of the evolution of Hilbert’s thinking on this.
- 11.
See Colyvan and Resnik (this volume, §2) for a discussion of this sort of dispute.
- 12.
In Posy (1974) I call this “constructivism of the right”. Indeed, many mathematicians are surprised to hear that philosophically Hilbert was a constructivist.
- 13.
Here’s the argument that ~(p ∨ ~p) is contradictory even in intuitionism. Suppose ~(p∨ ~p) were the case for some p. Now for purposes of the argument assume p. This automatically gives (p ∨ ~p). So since (p∨ ~p) contradicts our initial supposition,~(p ∨ ~p), it means that our assumption of p leads to a contradiction. Thus, we must conclude ~p. However, for reasons identical to what we just saw, from ~p we once again conclude (p∨~p), again contradicting our initial supposition. So, we get that the initial supposition ~(p∨ ~p) leads to a contradiction. Q.E.D. Of course this in turn gives us ~~(p∨ ~p)
- 14.
Of course, this indeterminacy holds in general, but it will be useful below to restrict attention to [0,1].
- 15.
- 16.
Actually, Brower himself did not favor physical processes. However, such processes do come up in the intuitionistic literature from time to time; and recently there has been increased interest in them. (See Gisin, 2020).
- 17.
Sequences defined in this way indicate why Brouwer’s theory is often called the theory of free choice sequences.
- 18.
This is an idealized mathematician, to be sure: finite capacities but no upper bound on life. I believe that nothing rides on the apparent subjectivity – even solipsism – implicit in this notion. One can get the same effect by polling the entire mathematical community at regular intervals. That too is idealized: We assume that we can accurately poll the entire research community at preset times and that mathematical research will go on forever. Intuitionism adopts these idealizations.
- 19.
Brouwer speaks of “infinitely proceeding sequences p 1 , p 2 , …, whose terms are chosen more or less freely from mathematical entities previously acquired; in such a way that the freedom of choice existing perhaps for the first element p 1 , may be subjected to a lasting restriction at some following p ν , and again and again to sharper lasting restrictions or even abolition at further subsequent p ν ’s, while all these restricting interventions, as well as the choices of the p ν ‘s themselves , at any stage may be made to depend on possible future mathematical experiences of the creating subject; … Brouwer (1952), p. 142.
- 20.
In admitting species Brouwer differs from other constructivists such as Weyl (in his constructivist period) and Bishop.
- 21.
- 22.
Brouwer describes mathematical species as “properties supposable for mathematical entities previously acquired, and satisfying the condition that if they hold for a certain mathematical entity, they also hold for all mathematical entities which have been defined to be equal to it, relations of equality having to be symmetric, reflexive and transitive; mathematical entities previously acquired for which the property holds are called elements of the species.” Brouwer (1952), p. 142.
- 23.
Formally: x<y ➔ ∃k∃n∀m[(σy(n+m) − σx(n+m))>2−k]. I should note that the formal definition of [x,y] is somewhat complicated (See Posy, 2020, definition 2.5.2). That complication is not important for us now.
- 24.
I should note that intervals and real valued functions are higher order species, they have species as elements.
- 25.
For convenience, in my discussion and in the figure I am omitting the distinction between the ‘spread law’ and the ‘complementary law’ of a spread (See Posy (2020) §2.2.4).
- 26.
Here’s a more formal account:
-
s[0,1] admits the one-member sequence <0>
-
s[0,1] admits the two-member sequences <0,0> and <0, ½>
-
If s[0,1] admits <a0, …, an,> and an ≠ 0 then s[0,1] also admits each of
-
<a0, …, an, ,an>,
-
<a0, …, an, an+2−n>,
-
<a0, …, an, an−2−n>,
-
-
If s[0,1] admits <a0, …, an,> and an = 0 then s[0,1] also admits each of
-
<a0, …, an,>
-
<a0, …, an, an+2−n>.
-
-
No other continuations are admissible.
-
- 27.
See Posy (2020) Theorems 2.19 and 2.20.
- 28.
Here’s an example in practice. The sequence γ = <0, ½, ¾ ,7/8 … 1−2n, …> is a real number generator for 1. Applying \( \overline{f_2} \) as defined above gives the sequence \( \overline{f_2} \)(γ) = <0, 1,6/4, 14/8… 2−2n+1…>, which is an element of 2. I should point out that this implements our requirement that the value of f2 be independent of the representative real number generator for x. The real number generator, δ = <1,1,…1,…> co-converges with γ and is of course an element of the real number 1. \( \overline{f_2} \) (δ) = <2,2,2,...>, which co-converges with \( \overline{f_2}\left(\upgamma\ \right), \) and is also an element of 2.
- 29.
To be sure if the sequence were to be determined by a fair coin toss or some other random trial, then the probability of generating ½ would be 0. But in these circumstances, since probability 0 certainly does not mean impossibility, these considerations are irrelevant.
- 30.
In symbols: ~~\( {r}_{\sigma^{\pi }} \) rational does not entail \( {r}_{\sigma^{\pi }} \) rational.
- 31.
Once again, the fact that the collection of rational elements of [0,1] has measure 0 is simply irrelevant. In fact as we saw, in describing sequences generated by “free choice” Brouwer adds the possibility that the chooser might decide at some point along the way to restrict the freedom of choice, even perhaps to eliminate it, and thus to proceed algorithmically. (Brouwer, 1952). Should such happen, then the chosen algorithm might be one that guarantees the rationality of the corresponding real number; or it might be one that guarantees irrationality. These possibilities certainly override the current improbability of getting a rational real, or some particular irrational one.
- 32.
A spread, like s[0,1], that admits only finitely many continuations of any admissible finite initial sequence is called a ‘fan’. Hence the title of the lemma.
- 33.
This lemma itself is proved in a unique way: Brouwer assumes that \( \forall {\sigma}_{{\in}_{m_{s_{\left[0,1\right]}}}}\exists !{n}_{\in \mathbb{N}}A\left(\sigma, n\right) \) could only be true if there is a proof of that fact. He shows how to form an isomorphism between the proof itself (that is its tree of assumptions and inferences) on the one hand and the species of initial sequences of elements of S[0,1] by which the corresponding n has been determined. And he concludes the finiteness of that species of initial sequences from the finiteness of the proof (After all, graspable proofs must be finite). Since the species is finite, there must be a k such that all its elements are of length less than or equal to k. That k is the one we seek. This argument is actually a special case of a more general form of reasoning (presented in Brouwer, 1927a) that is not limited to fans. It is worth noting that Brouwer’s notorious Uniform Continuity theorem for the closed interval [0,1] follows directly from this lemma (See Posy, 2020, Theorem 2.22. For convenience both in the present case and in Posy (2020), the lemma is stated for a relation A between sequences and natural numbers. The original form of the Fan Theorem is stated for a relation between sequences and integers).
- 34.
Look at definition of the > relation in note 25. This, of course depends on the fact that x ∊ [0,1]. Though the least number principle doesn’t always hold in intuitionism (see Posy, 2020, pages 15–16), in this case we can show that there is a unique least such k.
- 35.
This doctrine is codified in the formal theory of the creating subject. Specifically, one of the axioms of that theory is \( p\to \sim \sim \exists n{\sum}_n^Ap \) (where \( {\sum}_n^Ap \) stands for “the idealized mathematician researching A has knowledge of proposition p at stage n of the research”). In the literature this axiom is called the Principle of Christian Charity.
- 36.
These assertability conditions are often called BHK conditions. “BHK” stands for Brouwer, Heyting, Kolmogorov. The conditions are implicit (and sometimes explicitly stated) in Brouwer’s work. Heyting set them out explicitly in (1934), and Kolmogorov (1934) set out an equivalent formulation. I prefer the more descriptive term ‘assertability conditions.’ The expression warranted assertability appears in Dewey [1938]. Kripke (1963) uses the term in describing the semantics of intuitionistic logic. Dummett famously developed this approach well beyond mathematics.
- 37.
The argument in note 12 for formula (6) (the double negation of excluded middle) illustrates the “when“ clause at work.
- 38.
This is just the Principle of Christian Charity in the form \( \left(\sim \exists n{\sum}_n^Ap\ \right)\to \sim p \). You might want to object that \( \sim \exists n{\sum}_n^Ap \) tells us only that p will in fact never be proved, not that it cannot be proved. But keep in mind the intuitionistic idealization: we are speaking here of ‘forever’. Any eventuality that is possible will ultimately be realized.
- 39.
Brouwer (1925) explicitly raises both of these aspects of negation.
- 40.
Dummett in (1976) emphasizes that the assertability reading of the logical particles constitutes a “theory of understanding” of our language. But it is not merely a theory of understanding it is a theory of truth as well (With one exception, Dummett tends to view truth as inherently non-epistemic, and thus he hesitates to put things in this way. The exception is Dummett (2004) chapter 2, “The Indispensability of the Concept of Truth”. There he does allow that what he calls the ‘intuitionist’ semantic theory will take something like warranted assertability as “truth”).
- 41.
A similar thought applies to the other cases we considered.
- 42.
Already in (1900) Hilbert speaks of “the conviction of the solvability of every mathematical problem”. This conviction resonates throughout his subsequent thought.
- 43.
You might want to say that knowing that we will eventually find a proof of either P or ~P isn’t enough to assert the disjunction. You might want to insist that we actually have a method which will when carried out will produce one or another of the proofs. But that in fact is simply a higher order version of myopia.
- 44.
- 45.
For, it means that for each x and each n we can find be a k, such that if x′ lies within 2−k of x, then f(x′) will lie within 2−n of f(x). This is sometimes called “Brouwer’s principle” or “Brouwer’s continuity principle”. It is popular to say that this principle is simply an arbitrary assumption, of Brouwer’s or of intuitionism in general. However, I am suggesting that that the principle follows straightforwardly from intuitionistic epistemology.
- 46.
To be sure in (1925) and other places Brouwer starts the ontology with the full series of natural numbers, and pays scant attention to individual finite natural numbers except as initial segments of that series. However, in (1948) and (1952) individual natural numbers play a more prominent generative role.
- 47.
I am differing here from Heyting. According to Heyting, (e.g., 1966) the intuitionist and his classical cousin are simply talking about different things; they have different domains of mathematical objects. The intuitionist is speaking about mental mathematical constructions, the classical cousin about some metaphysically obscure (Platonic) realm of transcendent objects. That would mean of course that there is no substantive dispute. The disputants are simply talking about different things.
- 48.
I mentioned that Brouwer has a phenomenological story describing how we get the initial basic ‘two-ity’. This is a pre-mathematical story. In strictly mathematical contexts he speaks only of the ‘cypher-sequence’ of order type ω.
- 49.
Extensional identity is certainly not decidable. Think of α1 and α′. But by allowing sequences whose rules may narrow over time, it turns out that intensional identity is not decidable either. This point goes against some remarks in the literature, e.g., Troelstra (1977). Kripke (1982) raises now famous problems for the decidability of identity for rules or procedures. See also Steiner (1996) and (2009).
- 50.
Brouwer sometimes spoke of rules for ‘fitting in.’
- 51.
For, it is straightforward to show that any real number generator co-convergent with α1 will also be co-convergent with α2, and vice versa.
- 52.
Brouwer, in (1981), emphasizes the importance of extensional identity in such cases.
- 53.
In the history of philosophy, one fairly constant ontological (actually proto-ontological) mark of a legitimate object is its predicative completeness. This is true for Leibniz (Discourse on Metaphysics), and for Kant (Critique of Pure Reason A573/B601)), for instance, and for Frege (“Concept and Object”). And the intuitionistic proto-ontology simply differs from this. In intuitionism one lives with objects that are incomplete in this way;
- 54.
Brouwer routinely acknowledges that the study of finite systems assumes full surveyability and with it classical logic (See for instance Brouwer, 1952).
- 55.
The grasp need not give me all the relevant information at one time, but it makes that information (finitely) accessible to me.
- 56.
- 57.
In (1917) Brouwer himself uses the expression “intuitionistic point of view”.
- 58.
Compound objects like formulas or proofs or even full formal system are in fact reducible – via Gödel numbering – to natural numbers.
- 59.
A formal system that provides a recipe for Hilbertian finitary reasoning will contain only inferences with a finite number of premises. In footnote 8 of (1927a) Brouwer points out that the general method from which the fan theorem argument is derived (see note 32 above) allows inferences with infinitely many premises (There’s no problem, recall, in finitely grasping such an infinite species). In this sense the intuitionistic notion of finitary reasoning clearly does not coincide with Hilbertian finitary reasoning.
- 60.
See, for instance, Brouwer (1952) page 141.
- 61.
In (1908) Brouwer coined the expression “the classical attitude.”
- 62.
See for instance B. Pourciau (2000).
- 63.
See, famously, Gödel (1947).
- 64.
See Maddy (1990).
- 65.
- 66.
Maddy’s situation is a bit more subtle: In (1990) she attempted to derive our intuition of numbers from actual perceptual intuition, but then went on to attribute our knowledge of the principles of more abstract mathematics from intra-mathematical considerations about the necessity of those principles. Subsequently, in a series of books (1997, 2007, 2011), further developed the ontology and epistemology involved in those strictly intra-mathematical considerations, and ultimately jettisoned the extra-mathematical derivations of (1990). One way or another, however, even were she to persist in those derivations, the fact remains that her views certainly are not finitary.
- 67.
Technically we are speaking of ℝℕ the “natural” real numbers.
- 68.
This argument is based on Troelstra and van Dalen (1988) Chapter 4, §2.1. It is the first case examined in the survey paper, McCarty et al. (Forthcoming).
- 69.
This is very different from more standard such spreads, but it serves our purposes here.
- 70.
Here’s a more formal account of \( {s}_{{\mathbb{R}}^{+}} \):
-
The finite sequence <0> is admissible
-
The admissible continuations of <0> are
-
<0,0> and
-
<0, ½>
-
-
The admissible continuations of a sequence of the form <0,a1, …, an−1, 0> are
-
<0,a1, …, an−1, 0,0> and
-
<0,a1, …, an−1, 0, 2−(n+1)>
-
-
The admissible continuations of a sequence of the form <0, ½, …, 2−n> are
-
<0, ½, …, 2−n, 2−(n+1)> and
-
<0, ½,… 2−n, n>
-
-
The admissible continuations of a sequence of the form <0, ½ , …, 2−n, n> are
-
<0, ½, …, 2−n, n>
-
<0, ½, …, 2−n, n− ½>
-
<0, ½, ¼, …, 2-n, n+ ½>
-
-
The admissible continuations of a sequence of the form <0, ½, …, 2−n, n, an+1, …, an+k> are
-
<0, ½ , …, 2−n, n, an+1, …,an+k, an+k>
-
<0, ½ , …, 2−n, n, an+1, …,an+k, an+k−2−k>
-
<0, ½ , …, 2−n, n, an+1, …,an+k, an+k+2−k>
-
-
There are no other admissible sequences.
-
- 71.
The model theory that allows such sequences is interesting, but I won’t go into it here.
- 72.
Brouwer in (1912) suggests a problem that may well be unsolvable. But he is careful – as he must be – to avoid saying that he has actually designated an unsolvable problem.
- 73.
Indeed, in (1900) Hilbert speaks of the “conviction of the solvability of every mathematical problem” as an axiom. He implicitly raises the possibility that this axiom itself might be subject to rigorous mathematical proof. Of course one may raise the abstract possibility that every mathematical problem can in fact be solved; but that this just happens to be the case. In either case, by Brouwer’s lights, the axiom or the fact will remain eternally unprovable.
- 74.
- 75.
With apologies to A Midsummer’s Night Dream (Act V, scene 1).
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Posy, C.J. (2023). Platonism and the Proto-ontology of Mathematics: Learning from the Axiom of Choice. In: Posy, C., Ben-Menahem, Y. (eds) Mathematical Knowledge, Objects and Applications. Jerusalem Studies in Philosophy and History of Science. Springer, Cham. https://doi.org/10.1007/978-3-031-21655-8_7
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