Putnam, Peano, and the Malin Génie: could we possibly bewrong about elementary number-theory?

This article examines Hilary Putnam's work in the philosophy of mathematics and - more specifically - his arguments against mathematical realism or objectivism. These include a wide range of considerations, from Gödel's incompleteness-theorem and the limits of axiomatic set-theory as formalised in the Löwenheim-Skolem proof to Wittgenstein's sceptical thoughts about rule-following (along with Saul Kripke's ‘scepticalsolution’), Michael Dummett's anti-realist philosophy of mathematics, and certain problems – as Putnam sees them – with the conceptual foundations of Peano arithmetic. He also adopts a thought-experimental approach – a variant of Descartes' dream scenario – in order to establish the in-principle possibility that we might be deceived by the apparent self-evidence of basic arithmetical truths or that it might be ‘rational’ to doubt them under some conceivable (even if imaginary) set of circumstances. Thus Putnam assumes that mathematical realism involves a self-contradictory ‘Platonist’ idea of our somehow having quasi-perceptual epistemic ‘contact’ with truths that in their very nature transcend the utmost reach of human cognitive grasp. On this account, quite simply, ‘nothing works’ in philosophy of mathematics since wecan either cling to that unworkable notion of objective (recognition-transcendent) truth or abandon mathematical realism in favour of a verificationist approach that restricts the range of admissible statements to those for which we happen to possess some means of proof or ascertainment. My essay puts the case, conversely, that these hyperbolic doubts are not forced upon us but result from a false understanding of mathematical realism – a curious mixture of idealist and empiricist themes – which effectively skews the debate toward a preordained sceptical conclusion. I then go on to mount a defence of mathematical realism with reference to recent work in this field and also to indicate some problems – as I seethem – with Putnam's thought-experimental approach as well ashis use of anti-realist arguments from Dummett, Kripke, Wittgenstein, and others.     

Wittgensteinian Pragmatism in Humean Concepts

David Hume’s and later Ludwig Wittgenstein’s views on concepts are generally presented as standing in stark opposition to each other. In a nutshell, Hume’s theory of concepts is taken to be subjectivistic and atomistic, while Wittgenstein is metonymic with a broadly pragmatistic and holistic doctrine that gained much attention during the second half of the 20th century. In this essay, I shall argue, however, that Hume’s theory of concepts is indeed much more akin to the views of (post-Tractarian) Wittgenstein and his epigones than many, including Wittgenstein himself, probably might have suspected. As I try to show, Hume anticipates many themes central to Wittgenstein’s writings on language and meaning, and actually takes initial steps towards both an anti-subjectivistic and anti-atomistic psychology and epistemology.     

Wittgenstein and the Real Numbers

When it comes to Wittgenstein's philosophy of mathematics, even sympathetic admirers are cowed into submission by the many criticisms of influential authors in that field. They say something to the effect that Wittgenstein does not know enough about or have enough respect for mathematics, to take him as a serious philosopher of mathematics. They claim to catch Wittgenstein pooh-poohing the modern set-theoretic extensional conception of a real number. This article, however, will show that Wittgenstein's criticism is well grounded. A real number, as an ‘extension’, is a homeless fiction; ‘homeless’ in that it neither is supported by anything nor supports anything. The picture of a real number as an ‘extension’ is not supported by actual practice in calculus; calculus has nothing to do with ‘extensions’. The extensional, set-theoretic conception of a real number does not give a foundation for real analysis, either. The so-called complete theory of real numbers, which is essentially an extensional approach, does not define (in any sense of the word) the set of real numbers so as to justify their completeness, despite the common belief to the contrary. The only correct foundation of real analysis consists in its being ‘existential axiomatics’. And in real analysis, as existential axiomatics, a point on the real line need not be an ‘extension’.     

Conceptual engineering for mathematical concepts

In this paper I investigate how conceptual engineering applies to mathematical concepts in particular. I begin with a discussion of Waismann’s notion of open texture, and compare it to Shapiro’s modern usage of the term. Next I set out the position taken by Lakatos which sees mathematical concepts as dynamic and open to improvement and development, arguing that Waismann’s open texture applies to mathematical concepts too. With the perspective of mathematics as open-textured, I make the case that this allows us to deploy the tools of conceptual engineering in mathematics. I will examine Cappelen’s recent argument that there are no conceptual safe spaces and consider whether mathematics constitutes a counterexample. I argue that it does not, drawing on Haslanger’s distinction between manifest and operative concepts, and applying this in a novel way to set-theoretic foundations. I then set out some of the questions that need to be engaged with to establish mathematics as involving a kind of conceptual engineering. I finish with a case study of how the tools of conceptual engineering will give us a way to progress in the debate between advocates of the Universe view and the Multiverse view in set theory.     

On Certainty,Change, and “Mathematical Hinges”

Annalisa Coliva (Int J Study Skept 10(3–4):346–366, 2020) asks, “Are there mathematical hinges?” I argue here, against Coliva’s own conclusion, that there are. I further claim that this affirmative answer allows a case to be made for taking the concept of a hinge to be a useful and general-purpose tool for studying mathematical practice in its real complexity. Seeing how Wittgenstein can, and why he would, countenance mathematical hinges additionally gives us a deeper understanding of some of his latest thoughts on mathematics. For example, a view of how mathematical hinges relate to Wittgenstein’s well-known river-bed analogy enables us to see how his way of thinking about mathematics can account nicely for a “dynamics of change” within mathematical research—something his philosophy of mathematics has been accused of missing (e.g., by Robert Ackermann (Wittgenstein’s city, The University of Massachusetts Press, Amherst, 1988) and Mark Wilson (Wandering significance: an essay on conceptual behavior, Oxford University Press, Oxford, 2006). Finally, the perspective on mathematical hinges ultimately arrived at will be seen to provide us with illuminating examples of how our conceptual choices and theories can be ungrounded but nevertheless the right ones (in a sense to be explained).

     

Wittgenstein’s challenge to enactivism

Many authors have identified a link between later Wittgenstein and enactivism. But few have also recognised how Wittgenstein may in fact challenge enactivist approaches. In this paper, I consider one such challenge. For example, Wittgenstein is well known for his discussion of seeing-as, most famously through his use of Jastrow’s ambiguous duck-rabbit picture. Seen one way, the picture looks like a duck. Seen another way, the picture looks like a rabbit. Drawing on some of Wittgenstein’s remarks about seeing-as, I show how Wittgenstein poses a challenge for proponents of Sensorimotor Enactivism, like O’Regan and Noë, namely to provide a sensorimotor framework within which seeing-as can be explained. I claim that if these proponents want to address this challenge, then they should endorse what I call Sensorimotor Identification, according to which visual experiences can be identified with what agents do.

     

The Middle Wittgenstein’s Critique of Frege

ABSTRACT

This article aims to analyse Wittgenstein’s 1929–1932 notes concerning Frege’s critique of what is referred to as old formalism in the philosophy of mathematics. Wittgenstein disagreed with Frege’s critique and, in his notes, outlined his own assessment of formalism. First of all, he approvingly foregrounded its mathematics-game comparison and insistence that rules precede the meanings of expressions. In this article, I recount Frege’s critique of formalism and address Wittgenstein’s assessment of it to show that his remarks are not so much a critique of Frege as rather a defence of the formalist anti-metaphysical investment.     

WITTGENSTEIN AND STRONG MATHEMATICAL VERIFICATIONISM

Wittgenstein is accused by Dummett of radical conventionalism, the view that the necessity of any statement is a matter of express linguistic convention, i.e., a decision. This conventionalism is alleged to follow, in Wittgenstein's middle period, from his 'concept modification thesis', that a proof significantly changes the sense of the proposition it aims to prove. I argue for the assimilation of this thesis to Wittgenstein's 'no-conjecture thesis' concerning mathematical statements. Both flow from a strong verificationist view of mathematics held by Wittgenstein in his middle period, and this also explains his views on the law of excluded middle and consistency. Strong verificationism is central to making sense of Wittgenstein's middle-period philosophy of mathematics.     

Technology Games: Using Wittgenstein for Understanding and Evaluating Technology

In the philosophy of technology after the empirical turn, little attention has been paid to language and its relation to technology. In this programmatic and explorative paper, it is proposed to use the later Wittgenstein, not only to pay more attention to language use in philosophy of technology, but also to rethink technology itself—at least technology in its aspect of tool, technology-in-use. This is done by outlining a working account of Wittgenstein’s view of language (as articulated mainly in the Investigations) and by then applying that account to technology—turning around Wittgenstein’s metaphor of the toolbox. Using Wittgenstein’s concepts of language games and form of life and coining the term ‘technology games’, the paper proposes and argues for a use-oriented, holistic, transcendental, social, and historical approach to technology which is empirically but also normatively sensitive, and which takes into account implicit knowledge and know-how. It gives examples of interaction with social robots to support the relevance of this project for understanding and evaluating today’s technologies, makes comparisons with authors in philosophy of technology such as Winner and Ihde, and sketches the contours of a phenomenology and hermeneutics of technology use that may help us to understand but also to gain a more critical relation to specific uses of concrete technologies in everyday contexts. Ultimately, given the holism argued for, it also promises a more critical relation to the games and forms of life technologies are embedded in—to the ways we do things.     

Is Intuition Necessary for Defending Platonism?

G?del asserts that his philosophy falls under the category of conceptual realism. This paper gives a general picture of G?del’s conceptual realism’s basic doctrines, and gives a way to understand conceptual realism in the background of Leibniz’s and Kant’s philosophies. Among philosophers of mathematics, there is a widespread view that Platonism encounters an epistemological difficulty because we do not have sensations of abstract objects. In his writings, G?del asserts that we have mathematical intuitions of mathematical objects. Some philosophers do not think it is necessary to resort to intuition to defend Platonism, and other philosophers think that the arguments resorting to intuition are too na?ve to be convincing. I argue that the epistemic difficulty is not particular to Platonism; when faced with skepticism, physicalists also need to give an answer concerning the relationship between our experience and reality. G?del and Kant both think that sensations or combinations of sensations are not ideas of physical objects, but that, to form ideas of physical objects, concepts must be added. However, unlike Kant, G?del thinks that concepts are not subjective but independent of our minds. Based on my analysis of G?del’s conceptual realism, I give an answer to the question in the title and show that arguments resorting to intuition are far from na?ve, despite what some philosophers have claimed.     

The Structure’s Legacy: Not from Philosophy to Description

In the paper I consider how empirical material, from either history or sociology, features in Kuhn’s account of science in The Structure of Scientific Revolutions and argue that the study of scientific practice did not offer him data to be used as evidence for defending hypotheses but rather cultivated a sensitivity for detail and difference which helped him undermine an idealized conception of science. Recent attempts in the science studies literature, appealing to Wittgenstein’s philosophy, have aimed at reducing philosophy to multifaceted empirical research in relation to science. I discuss how this turn which is at odds with Wittgenstein’s philosophy, cannot be a continuation of Kuhn’s project which bears similarities to Wittgenstein’s.     

Wittgenstein on Set Theory and the Enormously Big

Wittgenstein's conception of infinity can be seen as continuing the tradition of the potential infinite that begins with Aristotle. Transfinite cardinals in set theory might seem to render the potential infinite defunct with the actual infinite now given mathematical legitimacy. But Wittgenstein's remarks on set theory argue that the philosophical notion of the actual infinite remains philosophical and is not given a mathematical status as a result of set theory. The philosophical notion of the actual infinite is not to be found in the mathematics of set theory, only in a certain associated philosophy – what Wittgenstein calls a certain kind of “prose”.     

Wittgenstein and religious dogma

It is well understood that Wittgenstein defends religious faith against positivistic criticisms on the grounds of its logical independence. But exactly how are we to understand the nature of that independence? Most scholars take Wittgenstein to equate language-games with belief-systems, and thus to assert that religions are logical schemes founded on their own basic beliefs and principles of inference. By contrast, I argue that on Wittgenstein’s view, to have religious faith is to hold fast to a certain picture of the world according to which one orients one’s actions and attitudes, possibly even in dogmatic defiance of contrary evidence. Commitment to such a picture is grounded in passion, not intellection, and systematic coherence is largely irrelevant.     

Informational Realism and World 3

This paper takes up a suggestion made by Floridi that the digital revolution is bringing about a profound change in our metaphysics. The paper aims to bring some older views from philosophy of mathematics to bear on this problem. The older views are concerned principally with mathematical realism—that is the claim that mathematical entities such as numbers exist. The new context for the discussion is informational realism, where the problem shifts to the question of the reality of information. Mathematical realism is perhaps a special case of informational realism. The older views concerned with mathematical realism are the various theories of World 3. The concept of World 3 was introduced by Frege, whose position was close to Plato’s original views. Popper developed the theory of World 3 in a different direction which is characterised as ‘constructive Platonism’. But how is World 3 constructed? This is explored by means of two analogies: the analogy with money, and the analogy with meaning, as explicated by the later Wittgenstein. This leads to the development of an account of informational realism as constructive Aristoteliansim. Finally, this version of informational realism is compared with the informational structural realism which Floridi develops in his 2008 and 2009 papers in Synthese. Similarities and differences between the two positions are noted.     

Wittgenstein,Quasi-fideism,and Scepticism

In the discussion of certainties, or ‘hinges’, in Wittgenstein’s On Certainty some of the examples that Wittgenstein uses are religious ones. He remarks on how a child might be raised so that they ‘swallow down’ belief in God (§107) and in discussing the role of persuasion in disagreements he asks us to think of the case of missionaries converting natives (§612). In the past decade Duncan Pritchard has made a case for an account of the rationality of religious belief inspired by On Certainty which he calls ‘quasi-fideism’. Pritchard argues that religious beliefs are just like ordinary non-religious beliefs in presupposing fundamental arational commitments. However, Modesto Gómez-Alonso has recently argued that there are significant differences between the kinds of ‘hinges’ discussed in Wittgenstein’s On Certainty and religious beliefs such that we should expect an account of rationality in religion to be quite different to the account of rational practices and their foundations that we find in Wittgenstein’s work. Fundamental religious commitments are, as Wittgenstein said, in the foreground of the religious believer’s life whereas hinge commitments are said to be in the background. People are passionately committed to their religious beliefs but it is not at all clear that people are passionately committed to hinges such as that ‘I have two hands’. I argue here that although there are differences between religious beliefs and many of the hinge-commitments discussed in On Certainty religious beliefs are nonetheless hinge-like. Gómez-Alonso’s criticisms of Pritchard mischaracterise his views and something like Pritchard’s quasi-fideism is the correct account of the rationality of religious belief.

     

LOT 2: The Language of Thought Revisited

A main thread of the debate over mathematical realism has come down to whether mathematics does explanatory work of its own in some of our best scientific explanations of empirical facts. Realists argue that it does; anti-realists argue that it doesn't. Part of this debate depends on how mathematics might be able to do explanatory work in an explanation. Everyone agrees that it's not enough that there merely be some mathematics in the explanation. Anti-realists claim there is nothing mathematics can do to make an explanation mathematical; realists think something can be done, but they are not clear about what that something is.

I argue that many of the examples of mathematical explanations of empirical facts in the literature can be accounted for in terms of Jackson and Pettit's [1990] notion of program explanation, and that mathematical realists can use the notion of program explanation to support their realism. This is exactly what has happened in a recent thread of the debate over moral realism (in this journal). I explain how the two debates are analogous and how moves that have been made in the moral realism debate can be made in the mathematical realism debate. However, I conclude that one can be a mathematical realist without having to be a moral realist.     

Wittgenstein, Moorean Absurdity and its Disappearance from Speech

G. E. Moore famously observed that to say, “ I went to the pictures last Tuesday but I don’t believe that I did” would be “absurd”. Why should it be absurd of me to say something about myself that might be true of me? Moore suggested an answer to this, but as I will show, one that fails. Wittgenstein was greatly impressed by Moore’s discovery of a class of absurd but possibly true assertions because he saw that it illuminates “the logic of assertion”. Wittgenstein suggests a promising relation of assertion to belief in terms of the idea that one “expresses belief” that is consistent with the spirit of Moore’s failed attempt to explain the absurdity. Wittgenstein also observes that “under unusual circumstances”, the sentence, “It’s raining but I don’t believe it” could be given “a clear sense”. Why does the absurdity disappear from speech in such cases? Wittgenstein further suggests that analogous absurdity may be found in terms of desire, rather than belief. In what follows I develop an account of Moorean absurdity that, with the exception of Wittgenstein’s last suggestion, is broadly consistent with both Moore’s approach and Wittgenstein’s.     

Brown

In Remarks on Colour Wittgenstein discusses a number of puzzling propositions about brown, e.g. that it cannot be pure and that there cannot be a brown light. He does not actually answer the questions he asks, and the status of his projected ‘logic of colour concepts’ remains unclear. I offer a real definition of brown from which the puzzle propositions follow logically. It is based on two experiments from Helmholtz. Brown is shown to be logically complex in the sense that the concept of brown can be ‘unpacked’ or resolved into simpler concepts. If my solutions to Wittgenstein's puzzles are the right ones, then science does bear upon the ‘logic of colour concepts’, and the contrast between logic and science which Wittgenstein sets up is a false one. At best it will appear as the contrast between the demands of logic and the demands of a particular kind of scientific theory and a particular mode of scientific theorizing. The solutions to the puzzles about brown are distinguished from psychological explanations, and the paper ends by suggesting what it was in his own doctrine that prevented Wittgenstein from answering the questions he had set himself.