Logic of imagination. Echoes of Cartesian epistemology in contemporary philosophy of mathematics and beyond

Descartes’ Rules for the direction of the mind presents us with a theory of knowledge in which imagination, considered as an “aid” for the intellect, plays a key role. This function of schematization, which strongly resembles key features of Proclus’ philosophy of mathematics, is in full accordance with Descartes’ mathematical practice in later works such as La Géométrie from 1637. Although due to its reliance on a form of geometric intuition, it may sound obsolete, I would like to show that this has strong echoes in contemporary philosophy of mathematics, in particular in the trend of the so called “philosophy of mathematical practice”. Indeed Ken Manders’ study on the Euclidean practice, along with Reviel Netz’s historical studies on ancient Greek Geometry, indicate that mathematical imagination can play a central role in mathematical knowledge as bearing specific forms of inference. Moreover, this role can be formalized into sound logical systems. One question of general epistemology is thus to understand this mysterious role of the imagination in reasoning and to assess its relevance for other mathematical practices. Drawing from Edwin Hutchins’ study of “material anchors” in human reasoning, I would like to show that Descartes’ epistemology of mathematics may prove to be a helpful resource in the analysis of mathematical knowledge.     

Experimental mathematics, computers and the a priori

In recent decades, experimental mathematics has emerged as a new branch of mathematics. This new branch is defined less by its subject matter, and more by its use of computer assisted reasoning. Experimental mathematics uses a variety of computer assisted approaches to verify or prove mathematical hypotheses. For example, there is “number crunching” such as searching for very large Mersenne primes, and showing that the Goldbach conjecture holds for all even numbers less than 2 × 1018. There are “verifications” of hypotheses which, while not definitive proofs, provide strong support for those hypotheses, and there are proofs involving an enormous amount of computer hours, which cannot be surveyed by any one mathematician in a lifetime. There have been several attempts to argue that one or another aspect of experimental mathematics shows that mathematics now accepts empirical or inductive methods, and hence shows mathematical apriorism to be false. Assessing this argument is complicated by the fact that there is no agreed definition of what precisely experimental mathematics is. However, I argue that on any plausible account of ’experiment’ these arguments do not succeed.     

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).

     

Against Mathematical Explanation

Lately, philosophers of mathematics have been exploring the notion of mathematical explanation within mathematics. This project is supposed to be analogous to the search for the correct analysis of scientific explanation. I argue here that given the way philosophers have been using “explanation,” the term is not applicable to mathematics as it is in science.     

Ambivalence under scrutiny: Returning to the future

Abstract

This Special Issue on the study of ambivalent processes in psychology integrates issues from contemporary evolutionary, cognitive, and cultural psychology with new directions of formal models that are available in qualitative mathematics. Tribute is paid to the pioneer of the study of ambivalence—Else Frenkel-Brunswik. Her work antedates most of our contemporary efforts in this field. Becoming free from the limits of its obsession with numbers in lieu of “measurement”, psychology at our time faces the challenge of investigation of dynamic psychological complexity. Contemporary mathematics—which is qualitative in its nature—provides new opportunities for psychology. New mathematical models—based on topology (Morse functions) and from intuitionistic formal logic (theory of locales)—are shown to provide promising new directions for future research on ambivalence. The emphasis on mathematical tools as enablement devices for psychological theorizing leads psychology to the need to create new kinds of generalized understanding of complex psychological processes.     

Nominalistic content,grounding, and covering generalizations: Reply to ‘Grounding and the indispensability argument’

‘Grounding and the indispensability argument’ presents a number of ways in which nominalists can use the notion of grounding to rebut the indispensability argument for the existence of mathematical objects. I will begin by considering the strategy that puts grounding to the service of easy-road nominalists (“Nominalistic content meets grounding” section). I will give some support to this strategy by addressing a worry some may have about it (“A misguided worry about the grounding strategy” section). I will then consider a problem for the fast-lane strategy (“Grounding and generality: a problem for the fast lane” section) and a problem for easy-road nominalists willing to accept Liggins’ grounding strategy (“More on the grounding strategy and easy-road nominalism” section). Both are related to the problem of formulating nominalistic explanations at the right level of generality. I will then consider a problem that Liggins only hints at (“Mathematics and covering generalizations” section). This problem has to do with mathematics’ function of providing the sort of covering generalizations we need in scientific explanations.     

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.     

What Are the “True” Statistics of the Environment?

A widespread assumption in the contemporary discussion of probabilistic models of cognition, often attributed to the Bayesian program, is that inference is optimal when the observer's priors match the true priors in the world—the actual “statistics of the environment.” But in fact the idea of a “true” prior plays no role in traditional Bayesian philosophy, which regards probability as a quantification of belief, not an objective characteristic of the world. In this paper I discuss the significance of the traditional Bayesian epistemic view of probability and its mismatch with the more objectivist assumptions about probability that are widely held in contemporary cognitive science. I then introduce a novel mathematical framework, the observer lattice, that aims to clarify this issue while avoiding philosophically tendentious assumptions. The mathematical argument shows that even if we assume that “ground truth” probabilities actually do exist, there is no objective way to tell what they are. Different observers, conditioning on different information, will inevitably have different probability estimates, and there is no general procedure to determine which one is right. The argument sheds light on the use of probabilistic models in cognitive science, and in particular on what exactly it means for the mind to be “tuned” to its environment.     

A Virtue-Based Defense of Mathematical Apriorism

Mathematical apriorists usually defend their view by contending that axioms are knowable a priori, and that the rules of inference in mathematics preserve this apriority for derived statements—so that by following the proof of a statement, we can trace the apriority being inherited. The empiricist Philip Kitcher attacked this claim by arguing there is no satisfactory theory that explains how mathematical axioms could be known a priori. I propose that in analyzing Ernest Sosa’s model of intuition as an intellectual virtue, we can construct an “intuition–virtue” that could supply the missing explanation for the apriority of axioms. I first argue that this intuition–virtue qualifies as an a priori warrant according to Kitcher’s account, and then show that it could produce beliefs about mathematical axioms independent of experience. If my argument stands, this paper could provide insight on how virtue epistemology could help defend mathematical apriorism on a larger scale.     

“To learn healing knowledge”: Philosophy,psychedelic studies and transformation

“Philosophical learning” may be summarised in Sobiecki’s fitting catchphrase “to learn healing knowledge”. This catchphrase is taken from an article on the use of psychoactive plants among southern African diviners. In the spirit of this link, I aim to challenge contemporary negative attitudes to the topic of psychedelics, and argue that there are good reasons for philosophers to pay attention to the role that the psychedelic experience can play in promoting philosophical perception. I argue first that the results of some contemporary studies affirm the benefits of psychedelic use in an “orchestrated guided experience”. Secondly, I argue that the aims of such “orchestrated guided experiences” are consonant with the nature of philosophical learning. Philosophy, understood as a learning practice, has a strong historical precedent and ties to contemporary indigenous cultural practices. Here I cite research into the use of psychedelics and the Eleusinian Mysteries at the origin of Western philosophy. Numerous cultures, ancient and contemporary, venerate psychoactive substances as agents of learning, healing, and transformation. Thus, contemporary mainstream philosophy may have opportunities to learn, or relearn, from southern African indigenous cultural practices. Considering the positive light in which the topic of psychedelics will be painted, I will conclude by suggesting that psychedelics have the potential to play an important role in fostering the deeply transformative “philosophical learning” that is the condition for positive social change. This makes the topic of psychedelics worthy of philosophical reflection.     

On the status of proofs by contradiction in the seventeenth century

In this paper I show that proofs by contradiction were a serious problem in seventeenth century mathematics and philosophy. Their status was put into question and positive mathematical developments emerged from such reflections. I analyse how mathematics, logic, and epistemology are intertwined in the issue at hand. The mathematical part describes Cavalieri's and Guldin's mathematical programmes of providing a development of parts of geometry free of proofs by contradiction. The logical part shows how the traditional Aristotelean doctrine that perfect demonstrations are causal demonstrations influenced the reflection on proofs by contradiction. The main protagonist of this part is Wallis. Finally, I analyse some epistemological developments arising from the Cartesian tradition. In particular, I look at Arnauld's programme of providing an epistemologically motivated reformulation of Geometry free of proofs by contradiction. The conclusion explains in which sense these epistemological reflections can be compared with those informing contemporary intuitionism.     

Infinity in Mathematics and Theology

Abstract

“Infinity” and its derivatives are frequently used in mathematics and theology. Do these expressions denote the same thing in those distinct areas of scholarship? In this article the uses of “infinity” in mathematics and its uses in theology are examined and compared. One conclusion is that quite different concepts go under the heading of “infinity.” Although they must not be confused, there are some relations between mathematical and theological senses of infinity.     

No Reservations Required? Defending Anti-Nominalism

In a 2005 paper, John Burgess and Gideon Rosen offer a new argument against nominalism in the philosophy of mathematics. The argument proceeds from the thesis that mathematics is part of science, and that core existence theorems in mathematics are both accepted by mathematicians and acceptable by mathematical standards. David Liggins (2007) criticizes the argument on the grounds that no adequate interpretation of “acceptable by mathematical standards” can be given which preserves the soundness of the overall argument. In this discussion I offer a defense of the Burgess-Rosen argument against Liggins’s objection. I show how plausible versions of the argument can be constructed based on either of two interpretations of mathematical acceptability, and I locate the argument in the space of contemporary anti-nominalist views.     

Donna Haraway's Cyborg Touching (Up/On) Luce Irigaray's Ethics and the Interval Between: Poethics as Embodied Writing

In this article, I argue that Donna Haraway's figure of the cyborg needs to be reassessed and extricated from the many misunderstandings that surround it. First, I suggest that we consider her cyborg as an ethical concept. I propose that her cyborg can be productively placed within the ethical framework developed by Luce Irigaray, especially in relationship to her concept of the “interval between.” Second, I consider how Haraway's “cyborg writing” can be understood as embodied ethical writing, that is, as a contemporary écriture feminine. I believe that this cyborgian “writing the body” offers us a way of both creating and understanding texts that think through ethics, bodies, aesthetics, and politics together as part of a vital and relevant contemporary feminist ethics of embodiment. I employ the term “poethics” as a useful way to describe such a practice.     

An Inferential Conception of the Application of Mathematics

A number of people have recently argued for a structural approach to accounting for the applications of mathematics. Such an approach has been called “the mapping account”. According to this view, the applicability of mathematics is fully accounted for by appreciating the relevant structural similarities between the empirical system under study and the mathematics used in the investigation of that system. This account of applications requires the truth of applied mathematical assertions, but it does not require the existence of mathematical objects. In this paper, we discuss the shortcomings of this account, and show how these shortcomings can be overcome by a broader view of the application of mathematics: the inferential conception.     

Life,cognition and metabiology

At the biological level, what is innate is the result of an evolutionary process and is “programed” by natural selection. Natural selection is the “coder” (once linked to the emergence of meaning). This coupled process is indissolubly correlated with the continuous construction of new formats in accordance with the unfolding of ever-new mathematics, a mathematics that necessarily moulds coder’s activity. Hence, the necessity of articulating and inventing a mathematics capable of engraving itself in an evolutionary landscape in accordance with the opening up of meaning. In this sense, for instance, the realms of non-standard models and non-standard analysis represent, today, a fruitful perspective in order to point out, in mathematical terms, some of the basic concepts concerning the articulation of an adequate intentional information theory.     

The ontology of words: a structural approach

Words form a fundamental basis for our understanding of linguistic practice. However, the precise ontology of words has eluded many philosophers and linguists. A persistent difficulty for most accounts of words is the type-token distinction [Bromberger, S. 1989. “Types and Tokens in Linguistics.” In Reflections on Chomsky, edited by A. George, 58–90. Basil Blackwell; Kaplan, D. 1990. “Words.” Aristotelian Society Supplementary Volume LXIV: 93–119]. In this paper, I present a novel account of words which differs from the atomistic and platonistic conceptions of previous accounts which I argue fall prey to this problem. Specifically, I proffer a structuralist account of linguistic items, along the lines of structuralism in the philosophy of mathematics [Shapiro, S. 1997. Philosophy of Mathematics: Structure and Ontology. Oxford University Press], in which words are defined in part as positions in larger linguistic structures. I then follow Szabò [1999. “Expressions and Their Representations.” The Philosophical Quarterly 49 (195): 145–163] and Parsons [1990. “The Structuralist View of Mathematical Objects.” Synthese 84: 303–346] in further defining words as quasi-concrete objects according to a representation relation. This view aims for general correspondence with contemporary generative linguistic approaches to the study of language.     

Quine on the Nature of Naturalism

Quine's metaphilosophical naturalism is often dismissed as overly “scientistic.” Many contemporary naturalists reject Quine's idea that epistemology should become a “chapter of psychology” (1969a, 83) and urge for a more “liberal,” “pluralistic,” and/or “open‐minded” naturalism instead. Still, whenever Quine explicitly reflects on the nature of his naturalism, he always insists that his position is modest and that he does not “think of philosophy as part of natural science” (1993, 10). Analyzing this tension, Susan Haack has argued that Quine's naturalism contains a “deep‐seated and significant ambivalence” (1993a, 353). In this paper, I argue that a more charitable interpretation is possible—a reading that does justice to Quine's own pronouncements on the issue. I reconstruct Quine's position and argue (i) that Haack and Quine, in their exchanges, have been talking past each other and (ii) that once this mutual misunderstanding is cleared up, Quine's naturalism turns out to be more modest, and hence less scientistic, than many contemporary naturalists have presupposed. I show that Quine's naturalism is first and foremost a rejection of the transcendental. It is only after adopting a broadly science‐immanent perspective that Quine, in regimenting our language, starts making choices that many contemporary philosophers have argued to be unduly restrictive.