# “Surveyability” in Hilbert, Wittgenstein and Turing

## Abstract

**:**

## 1. Introduction

## 2. Surveyability at Work in the Grammar of “Computable”

§341. A dispute may arise over the correct result of a calculation (say, of a rather long addition). But such disputes are rare and of short duration. They can be decided, as we say, ‘with certainty’.

Mathematicians don’t in general quarrel over the result of a calculation. (This is an important fact.)—Were it otherwise: if, for instance, one mathematician was convinced that a figure had altered unperceived, or that his or someone else’s memory had been deceptive, and so on—then our concept of ‘mathematical certainty’ would not exist.

§240. Disputes do not break out (among mathematicians, say) over the question of whether or not a rule has been followed. People don’t come to blows over it, for example. This belongs to the scaffolding from which our language operates (for example, yields descriptions).

Perhaps I shall do a multiplication twice to make sure, or perhaps get someone else to work it over. But shall I work it over again twenty times, or get twenty people to go over it? And is that some sort of negligence? Would the certainty really be greater for being checked twenty times? (OC, §77)

## 3. Surveyability vs. Subitizing

## 4. Hilbert on Überblickbarkeit: What Is a Metamathematical “Foundation”?

If logical inference is to be certain, then these objects must be capable of being completely surveyed in all their parts [überblicken lassen], and their presentation, their difference, their succession (like the objects themselves) must exist for us immediately, intuitively, as something that cannot be reduced to something else. Because I take this standpoint, the objects [Gegenstände] of number theory are for me—in direct contrast to Dedekind and Frege—the signs themselves, whose shape [Gestalt] can be generally and certainly recognized by us—independently of space and time, of the special conditions of the production of the sign, and of insignificant differences in the finished product. [Note: In this sense, I call signs of the same shape ‘the same sign’ for short.] ([16], §25).

1+1

1+1+1,

and so on,

…and these number-signs, which are numbers and which completely make up the numbers, are themselves the object of our consideration, but otherwise they have no meaning [Bedeutung] of any sort” ([16] §§29). Thus “2”, “100” and so on are really abbreviations, “3 > 2” serving “to communicate the fact that the sign 3 (that is, 1+1+1) extends beyond the sign 2 (that is, 1 + 1), or that the latter sign is a part of the former” ([16], §29).

The chief requirement of the theory of axioms must … show that within every field of knowledge contradictions based on the underlying axiom-system are absolutely impossible ([56], §33).

- —there can be no general algorithm (or formalism) used to determine, for any arbitrary formalism, not only whether or not it is consistent (Gödel’s [87]) but even whether it will output this or that configuration of signs on a given input (Turing’s [7]). Hence, there are fundamental philosophical, mathematical and logical limitations to the Hilbert program.

Ingenuity and Intuition. I think you [Newman] take a much more radically Hilbertian attitude about mathematics than I do. You say ‘If all this whole formal outfit is not about finding proofs which can be checked on a machine it’s difficult to know what it is about.’ When you say ‘on a machine’ do you have in mind that there is (or should be or could be, but has not been actually described anywhere) some fixed machine on which proofs are to be checked, and that the formal outfit is, as it were, about this machine. If you take this attitude (and it is this one that seems to me so extreme Hilbertian) there is little more to be said: we simply have to get used to the technique of this machine and resign ourselves to the fact that there are some problems to which we can never get the answer. On these lines my ordinal logics [88] would make no sense. However, I don’t think you really hold quite this attitude because you admit that in the case of the Gödel example one can decide that the formula is true, i.e., you admit that there is a fairly definite idea of a true formula which is quite different from the idea of a provable one. Throughout my paper on ordinal logics I have been assuming this too ([9], p. 215).

- Note the Hilbertian tone of “optimism” in Turing’s remarks on “extreme” Hilbertianism: although we must resign ourselves, in the use of any one machine (formal system), to admitting that “there are some problems to which we can never get the answer”—i.e., we must face what Hilbert called the ignorabimus42 —this does not mean that, with “ingenuity”, i.e., the human use of informal steps, we may not come close enough to knowing answers. Moreover, Turing continues, if we think opportunistically about the potentialities of a variety of systems (“machines” now refers to literal machines, as well as routines calculated with by humans), we may well be able to circumscribe the very idea of “provability” more and more closely:

I am rather puzzled why you draw this distinction between proof finders and proof checkers. It seems to me rather unimportant as one can always get a proof finder from a proof checker, and the converse is almost true: the converse fails if for instance one allows the proof finder to go through a proof in the ordinary way, and then, rejecting the steps, to write down the final formula as a ‘proof’ of itself.

- The suggestion here is that one might still use “proof finder” systems of deductive logic to generate proof checking systems, so long as one realized this would not resolve all problems. Alluding to his own diagonal argument showing that there is no general decision procedure for logic ([7], §8) Turing notes, as he had shown, that a proof checking system cannot be assumed to be able to check any system, on pain of one being able to define a tautological machine in connection with its own behavior. The latter proof—as Wittgenstein later noted explicitly43—shows that if we imagine a single machine that could determine, Yes or No, the behavior of any arbitrary machine, it would collapse into tautological circularity when it ran into its own commands. Turing’s proof, and his suggestion to Newman, circles back to the Wittgensteinian idea that the limits of logic lie in tautological constructions of rules that cannot be followed as commands: in general, for logic human embedding in a particular context or form of life is needed “friction” (Wittgenstein [51], [PI] §107).

If we use contentual [non-finitistically regarded] axioms as starting points and foundations for the proofs, then mathematics thereby loses the character of absolute certainty. With the acceptance of assumptions we enter the sphere of what is problematic. Indeed, the disagreements among people are mostly due to the fact that they proceed from different assumptions ([16], p. 233).

Now the theorems at issue can in part be proved, in an absolutely certain and purely mathematical fashion, with the help of the present results, and they have therefore been removed from the dispute. Whoever wants to confute me must show me, as has always been customary in mathematics and will continue to be so, exactly where my supposed error lies ([16], p. 228).

- Certain symbols are a precondition of the application of logic. (They serve as parameters.)
- These symbols are extra-logical, discrete, and intuitively immediate before all thought.
- Logic’s certainty depends upon the surveyability [Überblickbarkeit] of these symbols in all their parts (simplicity).
- These symbols are irreducible and objects of direct intelligibility.

## 5. Turing 1936 and “Surveyability”

- (1)
- A direct appeal to “intuition”, i.e., something not mathematical.
- (2)
- A proof of the equivalence of two definitions (Turing’s with the λ-definable functions) “in case the new definition has greater intuitive appeal”.
- (3)
- Giving examples of large classes of numbers which are computable.

Turing’s computability is intrinsically persuasive in the sense that the ideas embodied in it directly support the thesis that the functions encompassed are all for which there are algorithms; λ-definability is not intrinsically persuasive (the thesis using it was supported not by the concept itself but rather by results established about it) and general recursiveness scarcely so (its author Goödel being at the time not at all persuaded)50.

- Turing alone, we could say, gave a surveyable characterization of surveyability, i.e., his characterization of “computable” incorporates our sense of human action into the model and gives us “direct intelligibility” in something like Hilbert’s sense. A Turing Machine has a double face: it is, from one point of view, nothing more than a little formal system, a set of equations. But from another point of view, it lives within a human form of life, and it is we who bring the dynamism and movement into the model of its “step-by-step” “actions”.

We suppose … that the computation is carried out on a tape; but we avoid introducing the “state of mind” by considering a more physical and definite counterpart of it. It is always possible for the computer to break off from his work, to go away and forget all about it. and later to come back and go on with it. If he does this he must leave a note of instructions (written in some standard form) explaining how the work is to be continued. This note is the counterpart of the “state of mind”.

- (1)
- Reproducibility: a proof must not be a one-off event, it must be able to be reproduced.
- (2)
- The reproduction must be an easy task. –We might add, following Turing, that the task must be able to be broken down into “easy”, “surveyable” steps.
- (3)
- We must be able to decide with certainty whether something is or is not the reproduction of a proof.
- (4)
- The kind of reproduction resembles the reproduction of a picture, or model.

- And so it is with Turing’s characterization of “computation”. Turing provided a surveyable picture of surveyability itself in just this sense. And it was this computational aspect of formal proof at which Hilbert had been aiming. Turing shows that even though there is one parameter for computation—the Universal Machine can do the work of all Turing machines, including itself—there are mechanical limits to our ability to construct mechanical procedures to resolve disputes in this particular way.

## 6. Phraseology, Types,“Logicism”: Turing 1948–1954

- (a)
- the anthropological stance of language-games as logic, pieces of human technology and procedure ([BrB]),
- (b)
- the extrusion of inner mental states from the analysis of logic as characteristically an embodied action of human beings operating according to fixed procedures with signs ([BlB]), and
- (c)
- the idea of using humans as machines by giving them short tables of symbols expressing step-by-step commands, i.e., “mechanical” procedures ([BrB])52.

- We need to ask, not whether Turing has actually given us necessary and sufficient conditions for computations—with quantum computing who really knows?—but instead whether his earmarks of “computation” make vivid a concept that has characterized human forms of life for thousands of years.

- A beautiful turn toward the notion occurs in the following remark, one of the earliest where Wittgenstein uses it:

The propositions of logic are “laws of thought” “because they express the essence of human thought”—but more correctly: because they express or show the essence, technique (Watson), of thinking. They show what thinking is and also ways of thinking ([19] [RFM] I §133; in original manuscript Wittgenstein [77], p. 396, FF §332).

- This swink ties together Frege, Hilbert and Turing in the following way. With Frege Wittgenstein acknowledges his willingness to conceive of the logical as connected with general features that are constitutive of certain aspects of human thinking, “forms” or “essences” of thought that run through it everywhere, but not in a psychological sense. With Hilbert he will “correct” Frege: Hilbert held that in metamathematics “the formula game is carried out according to certain definite rules, in which the technique of our thinking is expressed” ([97], p. 475). But with Turing—alluding to Alister Watson, whose discussions with Wittgenstein and Turing in the summer of 1937 had so impressed him55—Wittgenstein will correct Hilbert: logic shows us not merely what thinking is, but, now through a plurality of techniques, ways of thinking. And in his anthropological explorations of language-games in Remarks on the Foundations of Mathematics (Wittgenstein [19]) the point is pursued at length.

It has long been recognised that mathematics and logic are virtually the same and that they may be expected to merge imperceptibly into one another. Actually this merging process has not gone at all far, and mathematics has profited very little from researches in symbolic logic. The chief reasons for this seem to be a lack of liaison between the logician and the mathematician-in-the-street. Symbolic logic is a very alarming mouthful for most mathematicians, and the logicians are not very much interested in making it more palatable. It seems however that symbolic logic has a number of small lessons for the mathematician which may be taught without it being necessary for him to learn very much of symbolic logic.

In particular it seems that symbolic logic will help the mathematicians to improve their notation and phraseology, which are at present exceedingly unsystematic, and constitute a definite handicap both to the would-be-learner and to the writer who is unable to express ideas because the necessary notation for expressing them is not widely known. By notation I do not of course refer to such trivial questions as whether pressure should be denoted by p or P, but deeper ones such as whether we should say ‘the function f (z) of z’ or ‘the function f ’ ([38], p. 245).

- Philosophically Turing is realizing the Hilbert ideal of metamathematics, but in a newly dynamical way. The key would be to reflect on how notations could be developed that would provide surveyability. Here, a form of intelligibility does enter, but it is procedural: the point is to help the human being dealing with computational proofs formulate and discuss, so to speak “metamathematically”, the situations that arise, a kind of “programme”.

(i) an extensive examination of current mathematical, physical and engineering books and papers with a view toward listing all commonly used forms of notation.

(ii) Examine them to see what they really mean. This will usually involve statements of various implicit understandings as between writer and reader. But the laying down of a code of minimum requirements for possible notations should be exceedingly mild, avoiding the straightjacket of a logical notation.

(iii) Laying down a code of minimum requirements for desirable notations. The requirements should be exceedingly mild… ([38], p. 245).

- He also emphasized the need to make the deduction theorem central and to make “very clear statements of the fundamental nature of the symbols” ([38], p. 245). He adds, echoing Wittgenstein’s idea about the colorful mix of techniques, that

It would not be advisable to let the reform [of notation] take the form of a cast-iron logical system into which all the mathematics of the future are to be expressed. No democratic mathematical community would stand for such an idea, nor would it be desirable( [38], p. 245).

- Here we see a kind of dynamic, evolutionary attitude toward the role of notations and languages in human being’s implementation of Hilbert’s idea of metamathematics. This leads to the pluralism of techniques on which Turing’s 1939 discussions with Wittgenstein focused. It also points toward what Turing called the inevitable need for “common sense” (i.e., non-algorithmic uses of human “intuitions”, or hunches) in addition to “reason” (i.e., formal routines) in mathematics64.

The Masters [i.e., mathematicians] are liable to get replaced because as soon as any technique becomes at all stereotyped it becomes possible to devise a system of instruction tables which will enable the electronic computer to do it for itself. It may happen however that the masters will refuse to do this. They may be unwilling to let their jobs be stolen from them in this way. In that case they would surround the whole of their work with mystery and make excuses, couched in well-chosen gibberish, whenever any dangerous suggestions were made. I think that a reaction of this kind is a very real danger ([106], 496).

- Resonating with Wittgenstein’s remarks on certainty and surveyability, Turing’s point is that nonsense, the mucking up of surveyability in the use of language, would undercut the very possibility, not only of using computations to further mathematical research, but of mathematics itself. The point was of course wholly prescient, as we look at the problems of nonsense and disinformation at work in the world wide web.

The instrument which mediates between theory and practice, between thought and observation, is mathematics; it builds the connecting bridges, and makes them ever sounder. Thus it happens that our entire modern culture, in so far as it rests on the penetration and utilization of nature, has its foundation in mathematics ([68], §22, p. 1162)65.

There is the genetical or evolutionary search by which a combination of genes is looked for, the criterion being survival value. The remarkable success of this search confirms to some extent the idea that intellectual activity consists mainly of various kinds of search ([107], p. 516).

- This should be compared with Hilbert’s synthesis of all the sciences in his 1930, which also included biology ([68], §8, p. 1159).

The remaining form of search is what I should like to call the “Cultural Search‘… [T]he isolated man does not develop any intellectual power. It is necessary for him to be immersed in an environment of other men, whose techniques he absorbs during the first 20 years of his life. He may then perhaps do a little research of his own and make a very few discoveries which are passed on to other men. From this point of view the search for new techniques must be regarded as carried out by the human community as a whole, rather than by individuals.

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## Acknowledgments

## Conflicts of Interest

## Notes

1 | |

2 | See the published lecture notes and discussion of them by Sieg in Hilbert [1]. |

3 | |

4 | Bernays and Turing [8]. |

5 | Post wrote ([12], p. 284) about the lack of “surveyability” of formal logic at the time, noting “the forbidding, diverse and alien formalisms” that grew up by 1936; see Floyd [4], p. 107. The confluence of Post’s and Turing’s work on the concept of computability is discussed in Davis and Sieg [13] and Sieg [14]. |

6 | Hilbert began using the term in connection with logical foundations as early as 1917. His remarks, some to be elucidated below, appear in quotations from his lecture notes in Hilbert [1], pp. 48, 117, 490 as well as in Sieg [15], pp. 23, 32; Hilbert [16], §5; Hilbert [17], p. 383; Hilbert [18] §2, Sieg contributing excellent discussions of this material. See Wittgenstein [19] [RFM] III §2, and Wittgenstein [20] MS 122, p. 43r. Mühlhölzer notes ([21], p. 58 n. 2) that Wittgenstein does not write about the “surveyability” of proof until 1937, but the idea of surveyability in connection with the sense of a proposition, a totality of numbers, or “grammar” in recursive arguments occurs earlier, in, e.g., Wittgenstein [22] [PR] §§1, 121f., where the term is unfortunately sometimes unhelpfully translated as “bird’s-eye-view”. |

7 | |

8 | When “survey” occurs in Turing’s writings, it means, as in ordinary English, an organized presentation of types, forms, or results, e.g., the types of “ground forms” in Turing’s account of morphogenesis ([25], p. 824) or statistical surveys, interestingly dismissed by Turing as hopeless for the exploration of concepts such as thinking (Turing [26], §1). Though I shall in the concluding section speak of Turing as an “ordinary language” philosopher, I emphatically will not mean the “bad” ordinary language philosophy idea of simple statistical surveys, or fixed rules of grammar that are static, but something more normative and dynamic. |

9 | |

10 | |

11 | |

12 | |

13 | See Floyd [4], p. 124. |

14 | |

15 | Kennedy [39]. |

16 | After Gödel [40], p. 306, the most sophisticated such allegation may be found in Sieg’s work axiomatizing the “bounded locality” conditions involved in the concept of computation (see, e.g., Sieg [41]), work that brings Hilbert’s axiomatic method to bear, very beautifully, on Turing’s characterization of “computation”. I do not differ with this as a piece of genuine mathematico-logical and philosophical work. However, my reading makes Church, rather than Turing, the asserter of a full-fledged “thesis”, often deemed “unproveable”. On the issue of “proving” the Church thesis, see Black [42] and Folina [43]. |

17 | See Sieg [14]. |

18 | |

19 | |

20 | |

21 | |

22 | |

23 | As Mühlhölzer points out ([21], p. 61, n. 6) Hilbert [16] and Hilbert [17] do not use the stroke notation, whereas Wittgenstein did, at Wittgenstein and Waismann [57] (hereafter “[WVC]”), p. 84, Wittgenstein [58] [PR] §103ff, [58] [PG], pp. 329ff., 350, [19] [RFM] I §25ff., §45 §§64ff., §99, §169, III §10, §44, §§51ff. |

24 | |

25 | |

26 | Calculation in the head is an interesting phenomenon for Wittgenstein, because it is not so clear what the buckstoppers are. Presumably the certainty involved may be reproduced for easy cases “in the head” by humans who have mastered the usual written routines. This raises a question: in a case where one person has mathematical authority over another, and no pencil or paper are to hand, the buckstopper might be whatever that authority him or herself says. However, written materials may typically be used to check and undermine that authority. If not, we are bordering on the use of what an anthropologist would call an “oracle”, or priest. |

27 | |

28 | |

29 | Kreisel [63], p. 21, attributing the quote to Thomas Hewitt Key; compare Kreisel [64], pp. 158, 165n, 290. The actual quotation is: What is mind?—No matter. What is matter?—Never mind. |

30 | Bernays [65]. |

31 | Wittgenstein [55], [OC], §402. |

32 | |

33 | Gödel’s [73] did open with a lament at the relative lack of formal precision in Whitehead and Russell [74] and [75], a falling off from the standards set by Frege [76] and developed later on by Hilbert. Wittgenstein, in the Preface to PI (Wittgenstein [77]) states explicitly, alluding to Frege’s term in the Preface to his [76], that logic, and so the method of his book, cannot proceed in a “gap free” manner [Luckenlose]. This responds to the light shed for him on the nature of logic by Gödel [78] and Turing [7]. |

34 | For Russell in Principia Mathematica (Whitehead and Russell [74] and [75], Introduction Chapter II, section III) a “judgment of perception” in his “multiple relation” theory of judgment is also taken to be a successful judgment. By definition, to perceive that a singular judgment of true requires actual perception, hence, success in the sense that something perceived is in fact true. This is not to embrace self-evidence or dogmatism about truth, rather to frame a definition of truth that assumes we are at least capable of judging truths. For discussion see Floyd [82] and Floyd and Kanamori [83]. |

35 | |

36 | |

37 | |

38 | Wittgenstein [57], WVC, 147–148. |

39 | So long as it is a language of the “relevant” kind. On this see Kennedy [49]. |

40 | The anthropological idea of an “oracle” is also mentioned by Wittgenstein at Wittgenstein [37] [LFM] XI, p. 109, and in a general way adopts the anthropological quality of that work and Wittgenstein [58] [RFM] I, which is incipient in Wittgenstein’s Blue and Brown Books (Wittgenstein [61], BlB, BrB]). It was criticized by Post as merely “picturesque” ([43], p. 311, n. 23, discussed in Floyd [4], pp. 138ff). |

41 | See Copeland’s discussion in Turing and Copeland [9], pp. 135–145. |

42 | |

43 | |

44 | This is perhaps why, later on, Wittgenstein would conceive of metamathematics in terms of the idea of a “geometry of signs”, the development of “models” for reasoning about mathematics itself (Wittgenstein [19] [RFM] III §§46ff.) |

45 | Hilbert is quoted using this term in Sieg [15], pp. 24, 30 within a wonderful discussion of Hilbert’s programs. |

46 | Wittgenstein had emphasized the idea of a formal system as a “calculating machine” already in Wittgenstein and Waismann [57] [WVC], pp. 106, 136, and returned to the theme of human “mechanical” procedures in Wittgenstein [61] [Bl] and [Br]. Of course independently of Turing, Post [11] also adopted the human worker “mechanical” model of computation. On the relation to Turing, see Davis and Sieg [13]. |

47 | Wittgenstein [91], RPP I §§1096ff. |

48 | |

49 | |

50 | Again, compare Kennedy [49] for discussion. |

51 | See https://en.wikipedia.org/wiki/Machine, accessed on 28 October 2022. |

52 | |

53 | See Shapiro [96]. This does not imply that the step from “recursive” to “effectively computable” in the sense of Church, Gödel, Kleene and Herbrand is more determinate and not open-ended, since the relevant classes of functions are provably co-extensional, whereas the step from “effectively computable” to “recursive” is more conceptually involved and perhaps not yet proven. Compare Black [42]. |

54 | See Chapter 8, Floyd and Mühlhölzer [53] for discussion. |

55 | |

56 | |

57 | See Goldfarb [99] for a clear analysis of the situation with Frege, Russell and Poincaré. |

58 | Kennedy [50] explores the point in many directions, focusing on definability and “formalism freeness”. |

59 | |

60 | |

61 | |

62 | |

63 | |

64 | |

65 | One can hear a four-minute recording of Hilbert reading the lecture at https://www.maa.org/book/export/html/326610, accessed on 31 October 2022. |

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**MDPI and ACS Style**

Floyd, J.
“Surveyability” in Hilbert, Wittgenstein and Turing. *Philosophies* **2023**, *8*, 6.
https://doi.org/10.3390/philosophies8010006

**AMA Style**

Floyd J.
“Surveyability” in Hilbert, Wittgenstein and Turing. *Philosophies*. 2023; 8(1):6.
https://doi.org/10.3390/philosophies8010006

**Chicago/Turabian Style**

Floyd, Juliet.
2023. "“Surveyability” in Hilbert, Wittgenstein and Turing" *Philosophies* 8, no. 1: 6.
https://doi.org/10.3390/philosophies8010006