Challenging epistemology: Interactive proofs and zero knowledge

This essay explores what (if anything) research on interactive zero knowledge proofs has to teach philosophers about the epistemology of mathematics and theoretical computer science. Though such proof systems initially appear ‘revolutionary’ and are a nonstandard conception of ‘proof’, I will argue that they do not have much philosophical import. Possible lessons from this work for the epistemology of mathematics—our models of mathematical proof should incorporate interaction, our theories of mathematical evidence must account for probabilistic evidence, our valuation of a mathematical proof should solely focus on its persuasive power—are either misguided or old hat. And while the differences between interactive and mathematical proofs suggest the need to develop a separate epistemology of theoretical computer science (or at least complexity theory) that differs from our theory of mathematical knowledge, a casual look at the actual practice of complexity theory indicates that such a distinct epistemology may not be necessary.     

Formal Ontology and Mathematics. A Case Study on the Identity of Proofs

We propose a novel, ontological approach to studying mathematical propositions and proofs. By “ontological approach” we refer to the study of the categories of beings or concepts that, in their practice, mathematicians isolate as fruitful for the advancement of their scientific activity (like discovering and proving theorems, formulating conjectures, and providing explanations). We do so by developing what we call a “formal ontology” of proofs using semantic modeling tools (like RDF and OWL) developed by the computer science community. In this article, (i) we describe this new approach and, (ii) to provide an example, we apply it to the problem of the identity of proofs. We also describe open issues and further applications of this approach (for example, the study of purity of methods). We lay some foundations to investigate rigorously and at large scale intellectual moves and attitudes that underpin the advancement of mathematics through cognitive means (carving out investigationally valuable concepts and techniques) and social means (like communication, collaboration, revision, and criticism of specific categories, inferential patterns, and levels of analysis). Our approach complements other types of analysis of proofs such as reconstruction in a deductive system and examination through a proof-assistant.

     

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).

     

The logic of Simpson’s paradox

There are three distinct questions associated with Simpson’s paradox. (i) Why or in what sense is Simpson’s paradox a paradox? (ii) What is the proper analysis of the paradox? (iii) How one should proceed when confronted with a typical case of the paradox? We propose a “formal” answer to the first two questions which, among other things, includes deductive proofs for important theorems regarding Simpson’s paradox. Our account contrasts sharply with Pearl’s causal (and questionable) account of the first two questions. We argue that the “how to proceed question?” does not have a unique response, and that it depends on the context of the problem. We evaluate an objection to our account by comparing ours with Blyth’s account of the paradox. Our research on the paradox suggests that the “how to proceed question” needs to be divorced from what makes Simpson’s paradox “paradoxical.”     

Computers as a Source of A Posteriori Knowledge in Mathematics

Electronic computers form an integral part of modern mathematical practice. Several high-profile results have been proven with techniques where computer calculations form an essential part of the proof. In the traditional philosophical literature, such proofs have been taken to constitute a posteriori knowledge. However, this traditional stance has recently been challenged by Mark McEvoy, who claims that computer calculations can constitute a priori mathematical proofs, even in cases where the calculations made by the computer are too numerous to be surveyed by human agents. In this article we point out the deficits of the traditional literature that has called for McEvoy’s correction. We also explain why McEvoy’s defence of mathematical apriorism fails and we discuss how the debate over the epistemological status of computer-assisted mathematics contains several unfortunate conceptual reductions.     

What inductive explanations could not be

Marc Lange argues that proofs by mathematical induction are generally not explanatory because inductive explanation is irreparably circular. He supports this circularity claim by presenting two putative inductive explanantia that are one another’s explananda. On pain of circularity, at most one of this pair may be a true explanation. But because there are no relevant differences between the two explanantia on offer, neither has the explanatory high ground. Thus, neither is an explanation. I argue that there is no important asymmetry between the two cases because they are two presentations of the same explanation. The circularity argument requires a problematic notion of identity of proofs. I argue for a criterion of proof individuation that identifies the two proofs Lange offers. This criterion can be expressed in two equivalent ways: one uses the language of homotopy type theory, and the second assigns algebraic representatives to proofs. Though I will concentrate on one example, a criterion of proof identity has much broader consequences: any investigation into mathematical practice must make use of some proof-individuation principle.     

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.     

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.     

Supporting the formal verification of mathematical texts

The formal verification of mathematical texts is one of the most interesting applications for computer systems. In fact, we argue that the expert language of mathematics is the natural choice for achieving efficient mathematician–machine interaction. Our empirical approach, the analysis of carefully authored textbook proofs, forces us to focus on the language and the reasoning pattern that mathematician use when presenting proofs to colleagues and students. Enabling a machine to understand and follow such language and argumentation is seen to be the key to usable and acceptable math assistant systems. In this paper, we first perform an analysis of three textbook proofs by hand; we then describe a computational framework that aims at mechanising such an analysis. The resulting proof-of-concept implementation is capable of processing simple textbook proofs and constitutes promising steps towards a natural mathematician–machine interface for proof development and verification.     

The Zhuangzi on Coping with Society

Stories in the Zhuangzi detailing expert artisans and other extraordinary people are often read as celebrations of “skills” or “knacks.” In this paper, I will argue that they would be more accurately understood as “coping” stories. Taken as a celebration of one’s “skill” or “knack” they transform the Zhuangzi into an implicit advocate of conforming to, or even identifying with, one’s social roles. I will argue that the stories of artisans and extraordinarily skilled people are less about cultivating one’s talents so as to “find one’s calling,” better fulfill social expectations, or achieve oneness with Dao, than they are concerned with developing strategies for coping with natural and social contingencies. Read in this way, there is much to learn from the Zhuangzi when reflecting on contemporary social and political issues, especially those related to meritocratic hubris.     

Reciprocity and Rivalry: A Critical Introduction to Mimetic Scapegoat Theory

This paper presents a critical overview of René Girard’s mimetic theory, identifies several concerns about the adequacy of mimetic theory’s account of human agency and interdependence, and suggests ways this account might be clarified and enhanced. We suggest that mimetic theory tends to reify or hypostatize the core reality of mimetic desire, which sometimes is spoken of as a kind of trans-individual entity that directs human action, no doubt because of Girard’s concern to avoid any taint of individualism or subjectivism. We argue, however, that hermeneutic and dialogical philosophies like those of Hans-Georg Gadamer and Mikhail Bakhtin explicate profound human relationality and interdependence in a way that obviates individualism without overriding or obscuring personal responsibility. The concern of many Girardian theorists that hermeneutic philosophy covers over or even rationalizes conflictual and violent human dynamics, we contend, is unfounded. However, we insist that most contemporary social theory, including hermeneutic thought, fails to do full justice to the challenges posed by envy, enmity, and scapegoating violence in human affairs and to the struggle for a good or decent life that mimetic theory richly portrays. Thus, we explore some of the possibilities for a fruitful cross-fertilization of mimetic theory and hermeneutic/dialogical viewpoints. Similarly, we argue that recent work on virtue ethics in theoretical psychology contains rich resources for elaborating what would be involved in a so-called “positive” or “creative” mimesis that moves beyond the destructive kinds of mimetic entanglement upon which Girardian thought has tended to concentrate. Finally, we suggest that any effort of this sort to clarify what Girard terms our “interdividual” human reality and to investigate positive mimesis would be greatly assisted by Eugene Webb’s delineation of a fundamental kind of “beneficient motivation” or positive “existential appetite,” a third species of human desire in addition to the two that Girard clearly identifies, namely finite needs and the kind of insatiable, artificial craving that the term mimetic desire usually designates.     

What is Absolute Undecidability?†

It is often alleged that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability “absolute undecidability”. In this paper, I seek to understand what absolute undecidability could be such that one might hope to establish that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) if a mathematical hypothesis is absolutely undecidable, then it is indeterminate. I shall argue that on no understanding of absolute undecidability could one hope to establish all of (a)–(c). However, I will identify one understanding of absolute undecidability on which one might hope to establish both (a) and (c) to the exclusion of (b). This suggests that a new style of mathematical antirealism deserves attention—one that does not depend on familiar epistemological or ontological concerns. The key idea behind this view is that typical mathematical hypotheses are indeterminate because they are relevantly similar to CH.     

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.     

Recognition of proofs in conditional reasoning

Relatively little is known about those who consistently produce the valid response to Modus Tollens (MT) problems. In two studies, people who responded correctly to MT problems indicated how “convinced” they were by proofs of conditional reasoning conclusions. The first experiment showed that MT competent reasoners found accurate proofs of MT reasoning more convincing than similar “proofs” of invalid reasoning. Similarly, there was a tendency for MT competent reasoners to find an initial counterfactual supposition more convincing than did people who were less competent in MT. The second experiment showed that when individuals produced the correct MT response, and found correct MT proofs to be more convincing than “bogus” proofs, they were also less likely to find the conclusions to Denying the Antecedent, or Affirming the Consequent problems valid, compared to individuals who could not discriminate between valid and bogus MT proofs. These findings are discussed in terms of both their implications for the mental logic and mental models positions, and individual differences in System 1 and System 2 reasoning.     

Mathematics,science and ontology

According to quasi-empiricism, mathematics is very like a branch of natural science. But if mathematics is like a branch of science, and science studies real objects, then mathematics should study real objects. Thus a quasi-empirical account of mathematics must answer the old epistemological question: How is knowledge of abstract objects possible? This paper attempts to show how it is possible. The second section examines the problem as it was posed by Benacerraf in ‘Mathematical Truth’ and the next section presents a way of looking at abstract objects that purports to demythologize them. In particular, it shows how we can have empirical knowledge of various abstract objects and even how we might causally interact with them. Finally, I argue that all objects are abstract objects. Abstract objects should be viewed as the most general class of objects. The arguments derive from Quine. If all objects are abstract, and if we can have knowledge of any objects, then we can have knowledge of abstract objects and the question of mathematical knowledge is solved. A strict adherence to Quine's philosophy leads to a curious combination of the Platonism of Frege with the empiricism of Mill.     

Hilbert,logicism, and mathematical existence

David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to offer a new analysis of the emergence of Hilbert’s famous ideas on mathematical existence, now seen as a revision of basic principles of the “naive logic” of sets. At the same time, careful scrutiny of his published and unpublished work around the turn of the century uncovers deep differences between his ideas about consistency proofs before and after 1904. Along the way, we cover topics such as the role of sets and of the dichotomic conception of set theory in Hilbert’s early axiomatics, and offer detailed analyses of Hilbert’s paradox and of his completeness axiom (Vollständigkeitsaxiom).     

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.     

Virtue Ethics

I discuss a puzzle that shows there is a need to develop a new metaphysical interpretation of mathematical theories, because all well-known interpretations conflict with important aspects of mathematical activities. The new interpretation, I argue, must authenticate the ontological commitments of mathematical theories without curtailing mathematicians' freedom and authority to creatively introduce mathematical ontology during mathematical problem-solving. Further, I argue that these two constraints are best met by a metaphysical interpretation of mathematics that takes mathematical entities to be constitutively constructed by human activity in a manner similar to the constitutive construction of the US Supreme Court by certain legal and political activities. Finally, I outline some of the philosophical merits of metaphysical interpretations of mathematical theories of this type.     

The Mathematical Event: Mapping the Axiomatic and the Problematic in School Mathematics

Traditional philosophy of mathematics has been concerned with the nature of mathematical objects rather than events. This traditional focus on reified objects is reflected in dominant theories of learning mathematics whereby the learner is meant to acquire familiarity with ideal mathematical objects, such as number, polygon, or tangent. I argue that the concept of event—rather than object—better captures the vitality of mathematics, and offers new ways of thinking about mathematics education. In this paper I draw on two different but related theories of event articulated in the philosophies of Alain Badiou and Gilles Deleuze to argue that the central activity of ‘problem solving’ in mathematics education should be recast in terms of a problematic of events.