Arithmetical Certainties: A Few Exceptions Among Countless Knowledge-Statements

142 Thémata. Revista de Filosofía • nº 66 • julio-diciembre (2022) pp. 142-158 • ISSN: 0212-8365 • e-ISSN: 2253-900X • DOI: 10.12795/themata.2022.i66.07 José María Ariso 1 Universidad Internacional de La Rioja, España Recibido 21 abril 2022 ∙ Aceptado 19 junio 2022 1. [email protected] Resumen En este artículo discrepo de Kusch (2016) en tres cuestiones relacionadas con las expresiones de certezas aritméticas –en el sentido de Wittgenstein– y los usos regulares de las expresiones aritméticas. Específicamente, explico por qué los cálculos no se convierten en certezas por el hecho de haber sido probados; Argumen- to que los cálculos probados constituyen enunciados de conocimiento; y, por último, pero no menos importante, concluyo de esto que tales cálculos probados son decibles, mientras que las certezas aritméticas son inefables o indecibles. Palabras clave: Certezas, Conocimiento; Aritmética; Wittgenstein; Kusch. Abstract In this paper I disagree with Kusch (2016) on three issues concerning expressions of arithmetical certainties – in Wittgenstein’s sense – and regular uses of arithmetical expressions. Specifically, I explain why calculations do not turn into certainties by the fact that they have been proved; I argue that proved calculations constitute knowledge-statements; and, last but not least, I conclude from this that such proved calculations are sayable, whilst arithmetical certainties are ineffable or unsayable. Keywords: Certainty; Knowledge; Arithmetic; Wittgenstein; Kusch Arithmetical Certainties: A Few Exceptions Among Countless Knowledge- Statements . Certezas aritméticas: algunas excepciones entre innumerables enunciados de conocimiento. ESTUDIOS_ ARTÍCULOS DE INVESTIGACIÓN JOSÉ MARÍA ARISO ARITHMETICAL CERTAINTIES: A FEW EXCEPTIONS AMONG COUNTLESS KNOWLEDGE-STATEMENTS 143 Thémata. Revista de Filosofía • nº 66 • julio-diciembre (2022) pp. 142-158 • ISSN: 0212-8365 • e-ISSN: 2253-900X • DOI: 10.12795/themata.2022.i66.07 1 · Introduction One of the most characteristic traits of mathematics for traditional epistemologists is the lack of choice within this realm. As is the case with logic, mathematical truths are absolutely necessary, so that “2+2” inexorably equals “4”. It cannot be otherwise. This inexorability attracted the attention of Ludwig Wittgenstein, who remarked in his posthumous work On Certainty that the absolute certainty of arithmetical propositions like the multiplication tables is also characteristic of what appear to be empirical propositions such as “This is my hand” (OC §§448, 653) 2 , for they are all used within their respective language-games without any doubt or hesitation (cf. OC §447) 3 . McGinn (1989) and Moyal-Sharrock (2004) provided a non-epistemic rendering of Wittgenstein’s views on certainties and mathematical sentences according to which none of them are propositions because they are true in an empty sense; they can never be epistemically justified; they are in all senses prior to empirical knowledge; they cannot be uttered in a meaningful way; and they constitute grammatical rules. These five tenets were challenged by the insightful epistemic view developed by Kusch (2016) following Pritchard (2007), Williams (2004a; 2004b; 2007) and Wright (2004). As I will not discuss whether certainties should be regarded as propositions, in this paper I do not fully enter this debate; however, I will address three issues closely related to it. To begin with, and with the aim of providing a foothold for my subsequent reflection, I will show that we cannot be certain of the entirety of calculations. Later I will disagree with Kusch (2016) on three issues. Specifically, I will explain why calculations do not turn into certainties by the fact that they have been proved; I will argue that proved calculations constitute knowledge-statements; and, last but not least, I will conclude from this that such proved calculations are sayable, whilst arithmetical certainties are ineffable or unsayable. By this I do not mean that my contribution corresponds to Wittgenstein’s definite view, for, as Kusch rightly pointed out, Wittgen- 2However, Wittgenstein also considered some objections to this in OC §654. 3As Wittgenstein had previously indicated in his Philosophical Investigations, a “language-game” is “the whole, consisting of language and the actions into which it is woven” (PI §7). Examples of language-games include, among many others, “Giving orders, and obeying them”, “Reporting an event”, etc. (PI §23). JOSÉ MARÍA ARISO ARITHMETICAL CERTAINTIES: A FEW EXCEPTIONS AMONG COUNTLESS KNOWLEDGE-STATEMENTS 144 Thémata. Revista de Filosofía • nº 66 • julio-diciembre (2022) pp. 142-158 • ISSN: 0212-8365 • e-ISSN: 2253-900X • DOI: 10.12795/themata.2022.i66.07 stein’s remarks on mathematics “are full of expressions of hesitation and doubt” (122). Yet I think that my contribution constitutes an approach to the characterization of arithmetical certainties which may be useful to further develop the debate between the non-epistemic rendering of certainty and the epistemic one. 2 · Characteristics of arithmetical certainties Before analyzing the main features of arithmetical certainties, it is appropriate to present Wittgenstein’s notion of “certainty”. To this end, I will start by clarifying the categorial distinction that Wittgenstein established between knowledge and certainty (cf. OC §308). Specifically, he regarded knowledge as justified true belief: thus, if someone says “I know”, she must be able to demonstrate the truth by dispelling a doubt, so that her grounds must be surer than the assertion of what she believes (cf. OC §243). Thus, the assurance “I know” should only be given to someone who still has not become aware of the grounds of such knowledge-statement (OC §431). Conversely, certainty is ungrounded: though we might provide many grounds for supporting a certainty, none would be “as certain as the very thing they were supposed to be grounds for” (OC §307). Certainty can therefore be regarded as a spontaneous “attitude” (OC §404) which we daily and implicitly show in whatever we say and do (cf. OC §431) without the aim of removing any doubt. Countless certainties can thus be simultaneously shown without generating inconsistency, for they are far from being independent of each other: in fact, our certainties make up a “world-picture” (OC §§93-95) which constitutes the “background” against which we distinguish between true and false (OC §94). Most importantly, the possibility of making a mistake is “logically excluded” when it comes to certainty (OC §194). Wittgenstein was not referring here to the classical and context-independent conception of logic which is valid in all possible worlds (cf. OC §628), but to a peculiar logic that emanates from the rules that make up our language-games (cf. Author 2021). This indicates that Wittgenstein did not aim to provide a merely theoretical and decontextualized distinction between knowledge and certainty. JOSÉ MARÍA ARISO ARITHMETICAL CERTAINTIES: A FEW EXCEPTIONS AMONG COUNTLESS KNOWLEDGE-STATEMENTS 145 Thémata. Revista de Filosofía • nº 66 • julio-diciembre (2022) pp. 142-158 • ISSN: 0212-8365 • e-ISSN: 2253-900X • DOI: 10.12795/themata.2022.i66.07 Indeed, and following the priority given to descriptions characteristic of his Philosophical Investigations – where Wittgenstein intended to dissolve philosophical problems or conceptual confusions by describing language-games (cf. PI §109) – his interest in descriptions remained in On Certainty. For he did not aim here to state how people should behave; instead, he wanted to show how they do behave (cf. OC §284). That is why Wittgenstein does not indicate which errors should be excluded from our language-games; instead, he simply reveals that there are seeming mistakes that we do not accept as such within our language-games, which can be considered as a spontaneous expression of our very lives (cf. OC §559) 4 . While the mistake someone may make when uttering a knowledge-statement can only be made within a language-game whose rules indicate what is considered as an error and how it could be discovered (cf. OC §§196, 301), a seeming mistake concerning certainty could not be regarded as a mere – albeit huge – error but as an anomaly or irregularity that “happens as an exception” because there is no place for it in our language-games (OC §647). Such an anomaly would open up what Author (2013 137) called a “grammatical gap” between the concerned individual and her community of origin, for it would not be possible to discern what she “would still allow to be counted as evidence and what not” (OC §231). After making this brief introduction to Wittgenstein’s conception of “certainty”, I will now focus on arithmetical certainties. In order to shed light on this expression, I will begin by clarifying whether all calculations constitute arithmetical certainties. In this vein, it seems reasonable to assume that there exists an immutable mathematical realm in which all arithmetic operations are contained. According to Platonism, “2+2=4” constitutes an eternal truth inasmuch as it refers to a fact of a mathematical reality that is independent of human existence, as a result of which the mathematician must increasingly discover the mathematical relationships and properties that such reality contains. Wittgenstein, however, was far from being a Platonist (cf. Kober): instead, at least between 1929 and 1944, he seemed to be much closer to formalism (cf. Rodych). After all, Wittgenstein 4With very few exceptions – like language-games characteristic of military communication – our language-games arise and develop spontaneously, without anyone creating or modifying them at will. JOSÉ MARÍA ARISO ARITHMETICAL CERTAINTIES: A FEW EXCEPTIONS AMONG COUNTLESS KNOWLEDGE-STATEMENTS 146 Thémata. Revista de Filosofía • nº 66 • julio-diciembre (2022) pp. 142-158 • ISSN: 0212-8365 • e-ISSN: 2253-900X • DOI: 10.12795/themata.2022.i66.07 claimed that “the mathematician is not a discoverer: he is an inventor” (RFM II §2), for, in his opinion, “we can’t describe mathematics, we can only do it” (PR §159). Wittgenstein would therefore not regard arithmetic as a predetermined and immutable structure that the mathematician can discover and describe one part at a time; from his standpoint, mathematics merely constitutes a practice or language-game that has become established due to the concordance and consistency with which it has been used 5 . In view of the above, it seems strange that Wittgenstein claims in On Certainty that “[t] he propositions of mathematics might be said to be fossilized” (OC §657), and that they have “as it were officially, been given the stamp of incontestability” (OC §655). Seemingly, he refers to all arithmetical operations, as though all of them were certainties. It should be noted, however, that the examples of such operations provided by Wittgenstein are only two, i.e. “12x12=144” (OC §§43, 651, 653-654) and “the multiplication tables” (OC §658). The way I see it, it is highly significant that these examples do not concern complex calculations whose results must be estimated in order to be known, but a small group of arithmetical certainties shared by the large majority of educated adults: for, regarding these elemental calculations, it can be asserted without any hint of doubt that they have really been given the stamp of incontestability 6 . Of course, the shortage of textual evidence does not ensure that Wittgenstein only considered as arithmetical certainties multiplication tables and basic calculations of the kind of “12x12=144” 7 . But I should like to briefly 5Wittgenstein (OC §47) recommended focusing on our customary practice of calculating, thereby leaving aside “transcendent certainty”. As Weiberg (2020) pointed out, this transcendent certainty could be understood as “a rule from which it follows that there can’t have been a miscalculation here” (OC §44), but we might also “go wrong in applying it” (OC §26). 6According to Hand, there are two ways of imparting a belief to a student, that is, “by proving it to her or by indoctrinating her” (549). In this vein, Siegel warned that teachers are always obliged to offer reasons for all beliefs and practices: from this standpoint, a fully rational education should transform ungrounded certainty in grounded knowledge. However, “grounded knowledge will be unfeasible unless we previously rely on countless ungrounded certainties” (Author 2019a 403). 7It cannot be exactly indicated which calculations constitute arithmetical certainties, as that will depend, among other factors, on age, education, personal capacity for arithmetic, and, as I will show in section 2.3, personal experiences. As a guideline, it can JOSÉ MARÍA ARISO ARITHMETICAL CERTAINTIES: A FEW EXCEPTIONS AMONG COUNTLESS KNOWLEDGE-STATEMENTS 147 Thémata. Revista de Filosofía • nº 66 • julio-diciembre (2022) pp. 142-158 • ISSN: 0212-8365 • e-ISSN: 2253-900X • DOI: 10.12795/themata.2022.i66.07 analyze what the consequences would be if all calculations were certainties. In such a case, arithmetical certainties would owe their status to their belonging to an immutable order that is completely independent from people’s lives, which would result in at least two peculiar scenarios. On the one hand, it might then seem that there can be arithmetical certainties of which no one is certain or even aware. Yet, as we are considering Wittgenstein’s notion of “certainty”, arithmetical certainties, like any other certainty, must be shared – that is, they should be shown or reflected in whatever is said and done by people who play arithmetical language-games: for, as indicated above, the logical exclusion of mistakes, which constitutes the main feature of certainties in Wittgenstein’s sense, emanates from practices whose daily repetition by many people has given rise to consolidated language-games. On the other hand, it could then also be questioned whether such certainties are shared by all of us. But the answer is negative because we all, including the most notorious mathematicians, are only informed of a negligible number of calculations: it is therefore hard to understand how we could be certain of those calculations whose results we do not know until we compute them. Additionally, if someone were certain of the results of all possible calculations, she could not make mistakes or have any doubt about any of them, because such seeming mistakes or doubts would then be anomalies for which there would be no room in our arithmetical language-game. Nevertheless, this language-game admits errors and doubts concerning complex calculations – i.e. those calculations that are not established as certainties – but not regarding arithmetical certainties. At this point, it could still be argued that many people are certain that all possible calculations make up a perfectly harmonious system in which their results are unquestionable. Yet, though such certainty is widely shared, it should not be confused with the certainty concerning specific calculations. For the fact that someone is certain of the existence of an arithmetical system invariable throughout eternity does not entail that she will thereby be certain of the result of complex calculations: after all, she will not even know their results before she calculates them. be said that an educated adult will usually be certain of the results of at least additions, subtractions, multiplications and exact divisions with numbers lower than 11. JOSÉ MARÍA ARISO ARITHMETICAL CERTAINTIES: A FEW EXCEPTIONS AMONG COUNTLESS KNOWLEDGE-STATEMENTS 148 Thémata. Revista de Filosofía • nº 66 • julio-diciembre (2022) pp. 142-158 • ISSN: 0212-8365 • e-ISSN: 2253-900X • DOI: 10.12795/themata.2022.i66.07 Once explained why we cannot be certain of all calculations, I will now clarify why I disagree with Kusch on three relevant issues concerning arithmetical certainties. 2 · 1 · Are calculations certain by the fact of their being proved? I have just shown that arithmetical certainties do not include the entirety of calculations, but only a small group of them. In the present section I will go one step further by specifying which arithmetical operations are excluded from being certainties. To this end, I should like to start by noting that, according to Kusch, arithmetical certainties are “mathematical sentences that have been immunized by mathematical proofs”, in addition to which “they are certainties for us all” (128). Before explaining why I disagree with this, I should like to emphasize that I am referring to extremely basic arithmetical proofs, such as proving subtractions – for instance, a–b =c through c+b=a – or divisions – a:b=c through b·c=a – to give only two examples. Specifically, the distinction that Wittgenstein established between “experiment” and “calculation” in his Remarks on the Foundations of Mathematics may lead us to consider – in the context of this paper – “experiment” as the initial calculation we make in order to know an answer or result, while the paradigmatic “calculation” constitutes the further proof with which we can optionally check the result previously obtained: If we take a pile of 100 marbles and separate them into 10 groups of 10, that is an “experiment” in the sense of counting how many marbles are there (RFM I §36ff.). When we begin, we do not know the answer. But if we now film the procedure, regarding it as a paradigm of what can always be done, as a necessary feature of there being 100 marbles, then we regard the procedure as a paradigmatic “calculation” or proof: we have “unfolded” the implicit properties in the collection. We “see the necessity” in the procedure: the way of regarding typical of proof. (Floyd 57-58) Having said this, I will now explain why I disagree with Kusch when he claims that arithmetical certainties are “mathematical sentences that JOSÉ MARÍA ARISO ARITHMETICAL CERTAINTIES: A FEW EXCEPTIONS AMONG COUNTLESS KNOWLEDGE-STATEMENTS 149 Thémata. Revista de Filosofía • nº 66 • julio-diciembre (2022) pp. 142-158 • ISSN: 0212-8365 • e-ISSN: 2253-900X • DOI: 10.12795/themata.2022.i66.07 have been immunized by mathematical proofs”, in addition to which “they are certainties for us all” (128). Let us thus suppose that a number of mathematicians have converged in finding out the result of a complex calculation at the same time and independently from each other. Yet even though they agree that the proof is correct – and that the calculation has thereby received the stamp of incontestability – this does not mean that such calculation will automatically turn into a certainty. Regardless of whether the result has been proved, one should exert oneself to remember it: but, of course, certainty is not something that requires an effort for being remembered. Hence, in this case there would simply be a proved calculation that some people want to treat as a certainty. Yet Author (2016) has provided four arguments as to why certainties cannot be acquired at will. Now I will resort to those arguments in order to explain why a proved calculation cannot be converted into an arithmetical certainty solely on the basis of one’s own will. Firstly, certainty cannot be acquired through reasoning. Wittgenstein wrote: And now if I were to say “It is my unshakeable conviction that etc.”, this means in the present case too that I have not consciously arrived at the conviction by following a particular line of thought, but that it is anchored in all my questions and answers, so anchored that I cannot touch it. (OC §103) We cannot therefore touch or reach an arithmetical certainty through a conscious process such as a mathematical proof, for Wittgenstein regards certainties as ungrounded (cf. OC §§250, 307). The impossibility of reaching a specific certainty through reasons or proofs also concerns the world-picture, that is, the set of all our certainties: for the world-picture is not something we decide to adopt after reflecting thereon, as it constitutes our “inherited background” (OC §94). If a certainty could be acquired by the mere fact of finding out a reason or proof that supports it, then it would be an explicit learning; however, Wittgenstein warns that he does not “explicitly learn” such certainties: instead, he can only “discover them subsequently like the axis around which a body rotates” (OC §152, emphasis by the author). At this point it could be argued that the acquisition of a certainty is not completely external to the individual, for she can foster or avoid situations that facilitate its subsequent assimilation. By way of example, it will be more JOSÉ MARÍA ARISO ARITHMETICAL CERTAINTIES: A FEW EXCEPTIONS AMONG COUNTLESS KNOWLEDGE-STATEMENTS 150 Thémata. Revista de Filosofía • nº 66 • julio-diciembre (2022) pp. 142-158 • ISSN: 0212-8365 • e-ISSN: 2253-900X • DOI: 10.12795/themata.2022.i66.07 likely that someone ends up assimilating as certain a complex calculation like “7,765x3,267=25,368,255” if she repeatedly uses it in a variety of contexts; but this does not entail that the acquisition of such certainty depends on the individual’s will. Instead, and strictly speaking, it will be something that happens to her. Secondly, certainties are assimilated regardless of whether or not we are in a specific mental state. According to Wittgenstein, neither certainty nor knowledge are mental states like, so to say, “being sure” (OC §308). The main feature of certainty is the logical exclusion of mistakes thereon: thus, keeping in mind that such exclusion stems from our language-games, we should focus on them and not on our mental states in order to conclude whether there is room for a specific doubt or mistake (cf. OC §§601, 196). This is particularly clear in the case of arithmetic, for we do not check whether someone has calculated rightly by focusing on her mental state, but by seeing how she has used arithmetical rules (cf. OC §§30, 38). A certainty cannot therefore be attained by the mere fact of reaching a specific mental state. Thirdly, an arithmetical certainty cannot be acquired even when one is willing to accept the certainty or world-picture proposed by a persuader. Indeed, when two individuals or groups meet each other without sharing the same world-picture, none of them can be convinced about particular issues by being given grounds: in such cases, one party may attempt to persuade the other by giving one’s own world-picture (cf. OC §262). Churchill remarked that one “decides” whether she becomes persuaded (46); however, such a decision would simply constitute a mental state which can be regarded as “a mere willingness towards the conversion” (Author 2016 578). Yet the assimilation of a certainty does not depend on any mental state, so that persuasion is irrelevant to this end. Fourthly, a certainty cannot be attained by acting as if one had it, as an act of will shall then be necessary not to forget that one should not stop acting in tune with that certainty, or to clarify how to act according to it in particular cases. From this it follows that the concerned individual would be considering the possibility of making a mistake when manifesting a certainty; but Wittgenstein (cf. OC §§490, 494) remarked that there is no judgement of which one could be certain if she had the slightest doubt about a certainty. Thus, the decision to acquire an arithmetical certainty by acting JOSÉ MARÍA ARISO ARITHMETICAL CERTAINTIES: A FEW EXCEPTIONS AMONG COUNTLESS KNOWLEDGE-STATEMENTS 157 Thémata. Revista de Filosofía • nº 66 • julio-diciembre (2022) pp. 142-158 • ISSN: 0212-8365 • e-ISSN: 2253-900X • DOI: 10.12795/themata.2022.i66.07 Ariso, José María. “Can Certainties Be Acquired at Will? Implications for Children’s Assimilation of a World-picture”, Journal of Philosophy of Education 50 (2016): 573-586. Ariso, José María. “Some variations of the certainty of one’s own death”, Linguistic and Philosophical Investigations 14 (2015): 82-96. Ariso, José María. “Wittgenstein and the Possibility of Inexplicably Losing Certainties”, Philosophical Papers 42 (2013): 133-150. Ariso, José María. Wahnsinn und Wissen. Zu Wittgensteins Lage und Denkbewegung. Würzburg: Königshausen & Neumann, 2012. Boudry, Maarten and Jerry Coyne. “Disbelief in belief: On the cognitive status of supernatural beliefs”, Philosophical Psychology 29 (2016): 601-615. Churchill, John. “Wittgenstein: The Certainty of Worldpictures”, Philosophical Investigations 11 (1998): 28-48. Floyd, Juliet. Wittgenstein’s Philosophy of Mathematics. Cambridge: Cambridge University Press, 2021. Goody, Jack R. The Domestication of the Savage Mind. Cambridge: Cambridge University Press, 1977. Hand, Michael “Religious Upbringing Reconsidered”, Journal of Philosophy of Education 36 (2002): 545–557. Kober, Michael. Gewissheit als Norm: Wittgensteins erkenntnistheoretische Untersuchungen in ‘Über Gewissheit’. Berlin and New York: De Gruyter, 1993. Kusch, Martin. “Wittgenstein on Mathematics and Certainties”, International Journal for the Study of Skepticism 6 (2016): 120-142. McGinn, Marie. Sense and Certainty: A Dissolution of Scepticism. Oxford: Basil Blackwell, 1989. Moyal-Sharrock, Danièle. Understanding Wittgenstein’s On Certainty. Basingstoke: Palgrave Macmillan, 2004. Pritchard, Duncan: “Wittgenstein’s On Certainty and Contemporary Anti-scepticism”. Readings of Wittgenstein’s On Certainty, eds. Danièle Moyal-Sharrock and William H. Brenner. Basingstoke: Palgrave Macmillan, 2007. 189-224. Rodych, Victor. “Wittgenstein on Mathematical Meaningfulness, Decidability, and Application”, Notre Dame Journal of Formal Logic 38 (1997): 195-225. JOSÉ MARÍA ARISO ARITHMETICAL CERTAINTIES: A FEW EXCEPTIONS AMONG COUNTLESS KNOWLEDGE-STATEMENTS 158 Thémata. Revista de Filosofía • nº 66 • julio-diciembre (2022) pp. 142-158 • ISSN: 0212-8365 • e-ISSN: 2253-900X • DOI: 10.12795/themata.2022.i66.07 Ryan, Julie and Julian Williams. Children’s Mathematics 4-15: Learning From Errors and Misconceptions. Maidenhead: Open University Press, 2007. Shapin, Steven. “Proverbial Economies: How an Understanding of Some Linguistic and Social Features of Common Sense Can Throw Light on More Prestigious Bodies of Knowledge, Science For Example”, Social Studies of Science 31 (2001): 731-769. Siegel, Harvey. Educating Reason: Rationality, Critical Thinking, and Education. New York and London: Routledge, 1988. Verbin, Nehama K. “Uncertainty and religious belief”, International Journal for Philosophy of Religion 51 (2002): 1-37. Weiberg, Anja. “‘12 × 12 = 144’: Psychologische Sicherheit versus logischer Ausschluss des Irrtums in Über Gewißheit am Beispiel des Rechnens”, Wittgenstein-Studien 11 (2020): 271-282. Williams, Michael. “Wittgenstein, Truth and Certainty”. Wittgenstein’s Lasting Significance, eds. Max Kölbel and Bernhard Weiss. London: Routledge, 2004a. 247-281. Williams, Michael. “Wittgenstein’s Refutation of Idealism”. Wittgenstein and Scepticism, ed. Denis McManus. London and New York: Routledge, 2004b. 76-96. Williams, Michael. “Why Wittgenstein Isn’t a Foundationalist”. Readings of Wittgenstein’s On Certainty, eds. Danièle Moyal-Sharrock and William H. Brenner. Basingstoke: Palgrave Macmillan, 2007. 47-58. Wittgenstein, Ludwig. Philosophical Investigations. Oxford: Blackwell, 2001. (Abbreviated as “PI” throughout). Wittgenstein, Ludwig. On certainty. Oxford: Blackwell, 1997. (Abbreviated as “OC” throughout). Wittgenstein, Ludwig. Philosophical Remarks. Oxford: Basil Blackwell, 1975. (Abbreviated as “PR” throughout). Wittgenstein, Ludwig. Remarks on the Foundations of Mathematics. Oxford: Basil Blackwell, 1967. (Abbreviated as “RFM” throughout). Wright, Crispin. “Wittgensteinian Certainties”. Wittgenstein and Scepticism, ed. Denis McManus. London and New York: Routledge, 2004. 22-55.