Philosophy

One of the most prominent ethical concerns regarding emerging neurotechnologies is mental privacy. This is the idea that we should have control over access to our neural data and to the information about our mental processes and states that can be obtained by analyzing it. A key issue is whether this information needs more stringent protection than other kinds of personal information. I will articulate and support the view, underlying recent regulatory frameworks, that mental privacy requires a special treatment because of its relation to relevant aspects of personal identity. It has been suggested that this approach could be supported by the idea that mental privacy constitutes a fundamental psychological dimension of privacy. The connection between this psychological view of privacy and identity can be traced back to Irwin Altmanâs idea that privacy is an interpersonal boundary regulation process. However, it is not clear whether this notion of privacy can be associated with a conception of identity that is relevant in contemporary neuroethics. I will suggest that the narrative and relational approach to identity, a prominent view in recent ethical discussions of neurotechnology, lines up with key aspects of Altmanâs proposal. I suggest that if mental privacy is an essential component of identity, the latter could be affected by technological mind-reading.

OPINION published: 14 October 2021 doi: 10.3389/fnhum.2021.773441 Frontiers in Human Neuroscience | www.frontiersin.org 1 October 2021 | Volume 15 | Article 773441 Edited by: José M. Muñoz, University of Navarra, Spain Reviewed by: Roland Nadler, University of British Columbia, Canada *Correspondence: Abel Wajnerman Paz firstname.lastname@example.com; firstname.lastname@example.com Specialty section: This article was submitted to Brain-Computer Interfaces, a section of the journal Frontiers in Human Neuroscience Received: 09 September 2021 Accepted: 21 September 2021 Published: 14 October 2021 Citation: Wajnerman Paz A (2021) Is Mental Privacy a Component of Personal Identity? Front. Hum. Neurosci. 15:773441. doi: 10.3389/fnhum.2021.773441 Is Mental Privacy a Component of Personal Identity? Abel Wajnerman Paz* Department of Philosophy, Universidad Alberto Hurtado, Santiago, Chile Keywords: neurorights, privacy, mental privacy, identity, relational identity, narrative identity INTRODUCTION Motivated by the aim to minimize the significant suffering and economic impact caused worldwide by brain diseases and disorders such as dementia, chronic pain, depression, addiction, and autism, several national, and regional research projects emerged in the recent years to improve our understanding of brain function and dysfunction. These projects contributed to the development of neurotechnologies and techniques that have an unprecedented ability (both in terms of scope and reliability) to "read" mental states, in the sense of decoding information about mental states or processes by analyzing data about neural activity patterns, and "transcribe" mental states by modulating neural computation. These technologies and the knowledge obtained through them are critical to enabling novel therapies for brain disorders and therefore are ethically motivated by the bioethical principle of beneficence. However, it has been argued that they also entail new neuroethical challenges that have to be addressed at a regulatory level. Recently, Yuste et al. (2021) suggested that existing international human rights need to be further expanded and/or specified because they address certain ethically relevant dimensions of human life in very generic terms, often subject to interpretation, and regulating the ramifications of neurotechnology requires greater specificity. Following this idea, countries such as Chile, Brazil and Spain are stablishing specific rights for the regulation of neurotechnology, also known as "neurorights." Some of these proposals are inspired by the framework developed by the Morningside Group (an interdisciplinary group led by the neuroscientist Rafael Yuste), which introduces five key neurorights: the right to personal identity, the right to free will, the right to mental privacy, the right to equal access to cognitive enhancement technologies, and the right to protection against algorithmic bias. One of the most discussed neurorights is probablymental privacy. This is the idea that we should have control over access to our neural data and to the information about our mental processes and states that can be obtained by analyzing it. The "mind-reading" neurotechnologies that could put mental privacy at risk consist of a wide variety of applications, including not only the interpretation of neural activity patterns in isolation (e.g., determining what I am thinking about without using any external cue), but also the use of neural responses to consciously perceived stimuli (e.g., P300 signals) for identifying experiences of recognition (Rissman et al., 2010), and the use of subliminal stimuli for detecting sexual preferences (Wernicke et al., 2017) and empathic responses (Chiesa et al., 2017). A key issue is whether mental privacy is different from other forms of privacy. Specifically, we need to determine whether neural data and mental information needs more stringent protection than other kinds of personal information. I will articulate and support a proposal suggesting that mental privacy requires a special treatment because of its relation to relevant aspects of personal identity. Wajnerman Paz Is Mental Privacy a Component of Personal Identity? PRIVACY AS A PSYCHOLOGICAL CAPACITY A recent proposal posits that mental privacy requires special legal protection because it is intimately related to who we are. More specifically, the Morningside Group's framework, crystalized in the recent Chilean neuroprotection bill, affirms that neural data should be treated as organic tissue and thus protected by the laws for organ transplantation and donation. This approach consists of two main aspects. Firstly, people not only have a right to not be compelled to give up brain data but, crucially, brain data collection requires explicit "opt-in" authorization. Secondly, brain data cannot be commercially transferred and used but only donated for altruistic purposes. That is, the commercialization brain data is prohibited regardless of consent status (Yuste et al., 2017; Goering et al., 2021). The appeal of this proposal depends on being able to explain exactly how mental privacy is related to who we are. A possible conceptualization of mental privacy as a cognitive dimension of privacy seems to bring it closer to our identity, thus perhaps grounding the Morningside group's view. Given that all forms of privacy depend ultimately on mentally processing personal information, this ability can be regarded as the cognitive source of privacy (Ienca and Andorno, 2017; Goering et al., 2021; Wajnerman Paz, 2021). Privacy is partly defined by the control that persons have over the flow of information about them (i.e., being able to determine "when, how, and to what extent information about them is communicated to others," Westin, 1968). Most often, we say that information is under our control, at least in part, in a cognitive sense. For any piece of information, we can consciously grasp it, reason about its personal and social meaning and its potential applications, and finally decide whether, when, how, to whom and to which extent we want to share it. Privacy depends on this cognitive process of rationally filtering and selectively sharing information about us. Neurotechnological mind-reading may be specially threatening for privacy precisely because it bypasses this fundamental filtering process. Any given piece of information we have mentally considered and decided not to share will be anyway available to someone who has direct access to our mind. Technological mind-reading can potentially make the cognitive mechanisms that define privacy itself meaningless. Thus, by characterizing mental privacy as a psychological (and fundamental) dimension of privacy we can identify the risks it involves. However, a critical question is whether these risks affect some fundamental human dimension that is not threatened by other kinds of privacy violations. This kind of cognitive approach to privacy has been developed in the environmental psychology literature. Interestingly, a prominent view entails that this capacity is part of what constitutes our personal identity. One of the most influential approaches to privacy as a psychological capacity is perhaps Irwin Altman's idea that privacy is a boundary regulation process (e.g., Altman, 1975). Altman characterizes privacy as the regulation of social interaction, aimed at achieving an ideal level of interpersonal contact by avoiding two kinds of imbalances: being unable to seclude ourselves (e.g., crowding) and overachieving seclusion (i.e., isolation). Crucially, this equilibrium depends on control over social inputs (e.g., actively accepting or rejecting other's opinions) and social outputs (e.g., avoid or allow someone listen to one's opinions) (Altman, 1976; see Margulis, 2003 for a review of Altman's approach). Altman argued that the main function of this process is the construction of our identity, our understanding of ourselves. It includes "knowing where one begins and ends vis-a-vis others, what aspects of the physical and social environment are parts of the self and which aspects are parts of others" (Altman, 1976, p. 25). The notion of privacy implies a flexible barrier or boundary between the self and non-self, "such as the cell membrane becomes more or less permeable, in order to achieve a viable level of functioning" (Ibid, p. 13). However, it is not entirely clear whether this notion of privacy can be relevant for a conception of identity that may be useful in contemporary neuroethics. I will suggest that a recent and prominent approach to neurotechnological influences on identity seems to line upwith key aspects of Altman's proposal. DISCUSSION: MENTAL PRIVACY AND RELATIONAL IDENTITY There has been a debate in the recent neuroethics literature regarding how we should conceptualize identity in the situations in which it seems to be threatened by "mind-writing" neurotechnologies, such as deep brain stimulation (DBS) and other devices that can potentially modulate neural computation (e.g., see Klein et al., 2016; Goering et al., 2017; Gilbert et al., 2018). Mackenzie and Walker (2015) argued that these threats should be characterized as threats to a narrative and relational conception of identity. I will claim that such notion is closely related to mental privacy as a psychological capacity. Most identity-related concerns of DBS patients are plausibly not about the modification of our metaphysical essence or our individuation conditions, which are characterized by psychological and biological continuity theories of identity. The main worry seems to be better conceptualized in terms of practical identity, which articulates one's self-conception: one's defining beliefs and values, motives, emotions, etc. Furthermore, Mackenzie and Walker's view is that this self-conception is not discovered but rather created, there is no pre-existing real inner self awaiting to be found. We build our personal identities by developing self-narratives. These are cognitive structures through which we interpret our personal histories and psychological traits and shape our intentions and plans. Finally, Mackenzie and Walker (2015), characterize this selfcreation process in a relational way. Following Baylis, they affirm that we are "dynamic complex co-creations informed by the perspectives and creative intentions of others" (Baylis, 2012, p. 118). Constructing a narrative identity is therefore an ongoing negotiation between our self-ascriptions of identity and the interpretation (or mis-interpretation) and recognition (or rejection) of these ascriptions by others. Baylis claims that a Frontiers in Human Neuroscience | www.frontiersin.org 2 October 2021 | Volume 15 | Article 773441 Wajnerman Paz Is Mental Privacy a Component of Personal Identity? self-narrative is identity constituting only when it achieves a temporary stability between our perspective and that of relevant others, which depends on ongoing interpersonal communication. If we understand identity as the result of this kind of interpersonal communicative process, then it is clear how psychological privacy in Altman's sense can be considered a constitutive part of it. The inter-subjective equilibrium in the construction of an agent's self-narrative requires her to regulate a communicative output constituted by her projected self-narrative and an input constituted by her perceived narrative (the recognition and/or interpretation of her self-attributions by others) in a way that reflects Altman's boundary regulation. We must first open our interpersonal "output gate" to make our self-narrative accessible to others, and also open our inter-personal "input gate," being receptive to its interpretation and assessment by others, thus building a collectively agreed self-narrative. Crucially, when the self-narrative projected by an agent is disputed by others, she can achieve equilibrium through a variety of processes described by Baylis (2012), which also involve specific forms of inter-personal boundary regulation. She can seclude herself from dissonant interpretations by changing her community of belonging in favor of a community likely to be more accepting of her projected self-narrative, thus reshaping her inter-personal boundaries. Alternatively, the agent can change her output, trying to project her self-narrative more efficiently by sharing more information and being more open to others. Finally, she could be simply become more permeable to other's interpretations, accepting those that are inconsistent with her perspective and consequently revising her projected self-narrative. Thus, the regulation of interpersonal interaction that is constitutive of an agent's mental privacy seems to be part of the communicative process that constitutes her identity. This means that violations of a person's mental privacy, disrupting the cognitive control she has over what information about herself she shares or receives may actually affect the very process underlying the formation of her identity. CONCLUSION Mental privacy has emerged in the last years as a critical concern regarding emerging neurotechnologies. The threat is perhaps enhanced by the fact that these technologies increasingly exceed the clinical context. Many novel non-invasive and potentially ubiquitous consumer neurotechnologies have various educational, entertainment-related, work-related, and military applications, which are not fully explored or regulated by either national laws or international treaties. Some of the pioneering regulatory frameworks and their applications by policy-makers support a strong reading of mental privacy, as a distinctive privacy dimension closely related to our identity. In this article, I suggested that this conceptual connectionmay be a key to deepen our understanding of mental privacy and articulate it with broader discussions in neuroethics. If we understand mental privacy as the psychological basis of privacy, we can see how it underlies the construction of our relational identity. Thus, relational identity may become a concern which should be addressed not only in relation to mind-writing neurotechnologies but also to neurotechnological mind-reading. AUTHOR CONTRIBUTIONS The author confirms being the sole contributor of this work and has approved it for publication. FUNDING This work was supported by the Facultad de Filosofía y Humanidades, Universidad Alberto Hurtado and the Fondo Nacional de Desarrollo Científico y Tecnológico (Proyecto Fondecyt Regular #1200197 and #1210091). REFERENCES Altman, I. (1975). The Environment and Social Behavior: Privacy, Personal Space, Territory, and Crowding. Pacific Grove, CA: Brooks/Cole Publishing Company. Altman, I. (1976). A conceptual analysis. Environ. Behav. 8, 7–29. doi: 10.1177/001391657600800102 Baylis, F. (2012). "The self in situ: a relational account of personal identity," in Being Relational: Reflections on Relational Theory and Health Law, eds J. Downie, and J. Llewellyn (Vancouver, BC: UBC Press), 109–131. Chiesa, P. A., Liuzza, M. T., Macaluso, E., and Aglioti, S. M. (2017). Brain activity induced by implicit processing of others' pain and pleasure. Hum. Brain Map. 38, 5562–5576. doi: 10.1002/hbm.23749 Gilbert, F., Viaña, J. N. M., and Ineichen, C. (2018). Deflating the "DBS causes personality changes" bubble.Neuroethics 1–17. doi: 10.1007/s12152-018-9373-8 Goering, S., Klein, E., Dougherty, D. D., and Widge, A. S. (2017). Staying in the loop: relational agency and identity in next-generation DBS for psychiatry. AJOB Neurosci. 8, 59–70. doi: 10.1080/21507740.2017.1320320 Goering, S., Klein, E., Sullivan, L. S., Wexler, A., y Arcas, B. A., Bi, G., et al. (2021). Recommendations for responsible development and application of neurotechnologies. Neuroethics 1–22. doi: 10.1007/s12152-021-0 9468-6 Ienca, M., and Andorno, R. (2017). Towards new human rights in the age of neuroscience and neurotechnology. Life Sci. Soc. Policy 13, 1–27. doi: 10.1186/s40504-017-0050-1 Klein, E., Goering, S., Gagne, J., Shea, C. V., Franklin, R., Zorowitz, S., et al. (2016). Brain-computer interface-based control of closed-loop brain stimulation: attitudes and ethical considerations. Brain Comput. Interfaces 3, 140–148. doi: 10.1080/2326263X.2016.1207497 Mackenzie, C., and Walker, M. (2015). "Neurotechnologies, personal identity, and the ethics of authenticity," in Handbook of Neuroethics, eds J. Clausen and N. Levy (Dordrecht: Springer), 373–392. doi: 10.1007/978-94-007-4707-4_10 Margulis, S. T. (2003). On the status and contribution of Westin's and Altman's theories of privacy. J. Soc. Issues 59, 411–429. doi: 10.1111/1540-4560.00071 Rissman, J., Greely, H. T., and Wagner, A. D. (2010). Detecting individual memories through the neural decoding of memory states and past experience. Proc. Natl. Acad. Sci. U.S.A. 107, 9849–9854. doi: 10.1073/pnas.1001028107 Wajnerman Paz, A. (2021). Is your neural data part of your mind? Exploring the conceptual basis of mental privacy. Minds Mach. doi: 10.1007/s11023-021-09574-7. [Epub ahead of print]. Wernicke, M., Hofter, C., Jordan, K., Fromberger, P., Dechent, P., and Müller, J. L. (2017). Neural correlates of subliminally presented visual sexual stimuli. Conscious. Cognit. 49, 35–52. doi: 10.1016/j.concog.2016.12.011 Frontiers in Human Neuroscience | www.frontiersin.org 3 October 2021 | Volume 15 | Article 773441 Wajnerman Paz Is Mental Privacy a Component of Personal Identity? Westin, A. F. (1968). Privacy and freedom.Washington Lee Law Rev. 25:166. Yuste, R., Genser, J., and Herrmann, S. (2021). It's time for neuro-rights. Horiz. J. Int. Relat. Sust. Dev. 18, 154–165. Yuste, R., Goering, S., Bi, G., Carmena, J. M., Carter, A., Fins, J. J., et al. (2017). Four ethical priorities for neurotechnologies and AI. Nat. News 551, 159–163. doi: 10.1038/551 159a Conflict of Interest: The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Publisher's Note: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher. Copyright © 2021 Wajnerman Paz. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms. Frontiers in Human Neuroscience | www.frontiersin.org 4 October 2021 | Volume 15 | Article


Russell on Logicism and Coherence

Philosophy

According to Quine, Charles Parsons, Mark Steiner, and others, Russell's logicist project is important because, if successful, it would show that mathematical theorems possess desirable epistemic properties often attributed to logical theorems, such as a prioricity, necessity, and certainty. Unfortunately, Russell never attributed such importance to logicism, and such a thesis contradicts Russell's explicitly stated views on the relationship between logic and mathematics. This raises the question: what did Russell understand to be the philosophical importance of logicism? Building on recent work by Andrew Irvine and Martin Godwyn, I argue that Russell thought a systematic reduction of mathematics increases the certainty of known mathematical theorems (even basic arithmetical facts) by showing mathematical knowledge to be coherently organized. The paper outlines Russell's theory of coherence, and discusses its relevance to logicism and the certainty attributed to mathematics.

Russell on Logicism and Coherence Conor Mayo-Wilson Abstract According to Quine, Charles Parsons, Mark Steiner, and others, Russell's logicist project is important because, if successful, it would show that mathematical theorems possess desirable epistemic properties often attributed to logical theorems, such as a prioricity, necessity, and certainty. Unfortunately, Russell never attributed such importance to logicism, and such a thesis contradicts Russell's explicitly stated views on the relationship between logic and mathematics. This raises the question: what did Russell understand to be the philosophical importance of logicism? Building on recent work by Andrew Irvine and Martin Godwyn, I argue that Russell thought a systematic reduction of mathematics increases the certainty of known mathematical theorems (even basic arithmetic) facts by showing mathematical knowledge to be coherently organized. The paper outlines Russell's theory of coherence, and discusses its relevance to logicism and the certainty attributed to mathematics. Bertrand Russell famously claimed that mathematics is reducible to logic in the sense that (a) all mathematical terms (e.g. real number, integer, natural number, etc.) can be defined using only logical constants (e.g. equality, negation, etc.), and (b) all mathematical theorems can be derived from purely logical axioms. The conjunction of these two theses is called logicism. One purported consequence of logicism is that mathematical concepts and theorems inherit the important epistemological properties of logical concepts and theorems respectively. Why? If mathematical assertions are in fact abbreviations for logical assertions, and if all mathematical theorems can be derived from logical ones, then mathematical theorems must possess the important epistemological properties that are frequently attributed to logical theorems. In particular, one might argue that logicism implies that mathematical theorems are necessary, capable of being known a priori, certain, and/or self-evident, as these are properties typically attributed to logical truths. I call this thesis epistemic inheritance, and abbreviate it by EI. According to many philosophers, Russell's primary motivation for defending logicism was to defend EI. For example, in describing the motivations for reducing mathematics to logic, Quine attributes EI to the logicists: Ideally, the obscurer [mathematical] concepts would be defined in terms of the clearer ones so as to maximize clarity, and the less obvious laws would be proved from the more obvious ones so as to maximize certainty. Ideally, the definitions would generate all the 1 concepts from clear and distinct ideas, and the proofs would generate all theorems from self-evident truths.1 Similarly, Imre Lakatos, Charles Parsons, and Mark Steiner, amongst others, have all argued that EI was a central motivation for Russell's attempt to reduce mathematics to logic.2 As has been pointed out by Andrew Irvine and Martin Godwyn,3 however, Russell never believed EI, and moreover, EI contradicts Russell's views on the relationship between mathematics and logic as they are explicitly stated in several lectures, papers, and books, including even the introduction to Principia Mathematica, in which he writes: . . . the chief reason in favour of any theory theory on the principles of mathematics must always be inductive, i.e. it must lie in the fact that theory in question enables us to deduce ordinary mathematics. In mathematics, the greatest degree of self-evidence is usually not to be found quite at the beginning, but at some later point; hence, the early deductions, until they reach this point, give reasons rather for believing the premisses because true consequences follow from them, than for believing the consequences because they follow from the premisses. [my emphasis] If Russell never argued for EI, and if EI contradicts other views that Russell did hold, then the following question is immediate: Did Russell attribute any important epistemological consequences to logicism? More specifically, does logicism have important consequences for mathematical epistemology?4 According to Irvine and Godwyn, the answer to the above question is "yes." Under their interpretation, Russell attached importance to logicism not because 1See Quine (1969), pp. 70. Quine never mentions Russell in this passage, but later parts of the essay (See pp. 73-74) clearly indicate that Russell is one of the logicists who drew unacceptable epistemological conclusions from the reduction of mathematics to so-called logic. Also, Frege is not mentioned in the essay, which suggests that Quine is thinking primarily of Russell's logicist program. Of course, Quine thinks that logicism did not meet the epistemic goals he attributes to Russell, as foundational work showed that mathematics is reducible to set theory, not logic, and furthermore, that the axioms of set theory are far from certain and self-evident. 2See Lakatos (1978) pp. 16, Parsons (1967) pp. 193, 197, and Steiner (1975) pp. 14-17, 24. I learned of many of these passages from Irvine (1989). 3See Irvine (1989) and Godwyn and Irvine (2003). Douglas Lackey makes the same observation, in significantly less detail, in his editorial introduction to the 1907 lecture that provides Irvine and Godwyn with greatest support for their thesis. See Russell (1973), pp. 255. Another interesting, sociological question, which I will not try to answer, is "why did philosophers ever attribute EI to Russell?" Irvine (1989) speculates that, because Frege does endorse EI, the tendency to group together Frege and Russell's programs has led to a misunderstanding of Russell's motivations for logicism. This suggestion was also, independently, made to me by Paddy Blanchette. 4Note that Russell attributes many important consequences to logicism, some of which pertain to physics and philosophy broadly. See, for example, "The Philosophical Implications of Mathematical Logic" in Russell (1973), pp. 284-294. I will not discuss these features of Russell's views about the importance of logicism; I am only concerned with the importance of logicism to mathematical epistemology. 2 it increases the certainty that mathematicians attach to theorems of analysis, let alone basic arithmetical facts, but rather, because it improves mathematical explanation and facilities discovery. In reducing mathematics to logic, they claim, Russell's central motivations are to (1) show how seemingly different theorems (and their proofs) can be derived from a common core of axioms, thereby explaining said theorems, and (2) uncover techniques for proving theorems that have been stated but are currently unproven, and (3) suggest new, fruitful concepts and theorems in mathematics. I agree with Irvine and Godwyn's contention that Russell attached importance to the role of logic in improving mathematical explanations and in aiding discovery. In fact, later in the paper, I provide additional arguments and examples in support of this claim. However, the central thesis of this paper is that Russell did, in fact, believe that the reduction of mathematics to logic increases the certainty of mathematical theorems (even basic arithmetical facts). Like Irvine and Godwyn, I agree that this added certainty does not result from mathematical theorems inheriting the epistemological properties of logical theorems. Rather, in reducing mathematics to logic, the certainty attached to mathematical theorems is increased because mathematical knowledge is shown to be coherently organized, and for Russell, coherence is an indicator of truth.5 This paper contains two parts. I first argue that Russell never endorsed EI; here I follow Irvine and Godwyn, and analyze a few additional passages in which Russell discusses nineteenth century geometry to support their interpretation. In the second half of the paper, I then discuss Russell's theory of coherence and how it bears on logicism. Before beginning, however, two caveats are in order. First, in my analysis of Russell's work, I quote from articles and books that together span several decades of Russell's philosophical career. Because Russell's views on a number of important philosophical issues changed markedly during his life, one might wonder whether there is enough consistency in his views to support the reading I urge below. As I hope to show, however, Russell's views on coherence changed little between 1907 and the 1950's. Several passages of nearly identical timbre and wording occur in his writings over the course of forty years. Second, I wish to emphasize that Russell likely attributed many important philosophical consequences to logicism. My discussion of Russell's theory of coherence is meant only to deepen our understanding of the many reasons that Russell was motivated to his work in logic, and why such work might continue to be important. 5Thagard et. al., (2002) makes a passing remark to this effect, but his interpretation of Russell is not defended in any substantial way. Russell also argues that in reducing mathematics to logic, one also acquires a minimum set of postulates that are simpler (in virtue of the number of constituents they contain). Because Russell claims that simplicity is an indicator of truth of epistemology generally (See Russell (1997), pp. 23), one might infer that logicism implies mathematics is more certain because it shows how various theorems can be derived from simpler premises. Unfortunately, this is a point on which Russell seems to have changed his mind, as he claims that, in logic, a preference for fewer and simpler premises is "merely aesthetic." See "Epistemological Premises" in Russell (2009a), pp. 205. 3 1 Logicism, Explanation, and Discovery What is the importance of logicism, according to Russell? One standard interpretation, thoroughly debunked by Andrew Irvine and Martin Godwyn, is that logicism implies that mathematical theorems inherit all or some of the desirable epistemological properties standardly attributed to logical theorems, such as necessity, a prioricity, certainty, and self-evidence. Yet Russell repeatedly denies that axioms always possess these properties. First, Russell emphasizes that the primitive axioms, from which mathematical theorems are proven, are less obvious than their consequences. "The most obvious and easiest things in mathematics," he writes, "are not those that come logically at the beginning; they are things that, from the view point of logical deduction, come somewhere in the middle."6 It follows that, for Russell, mathematical axioms are not always self-evident.7 Second, as the search for mathematical axioms lies on frontier of mathematical knowledge, and concepts, such as that of a class, had led to contradictions in the recent past, Russell denied that logical principles are certain. He writes, "In mathematical logic, it is the conclusions that have the greatest certainty: the closer we get to the ultimate premises, the more uncertainty and difficulty we find."8 Third, the axioms of infinity and multiplicity, which assert the existence of particular classes, are not necessary, as one can imagine worlds, for example, in which only finitely many objects exist. In fact, Russell claims that no existential postulate is a logical truth: Among "possible worlds," in the Leibnizian sense, there will be one having one, two, three . . . , individuals. There does not even seem any logical necessity why there should be even one individual-why, in fact, there should be any world at all.9 Finally, Russell argues that mathematical axioms are justified by a pseudoinductive strategy, namely, whether they can be used to derive accepted the6Russell (2009b), pp. 2. Nearly identical passages occur in Whitehead and Russell (1962) pp. 1, and "The Regressive Method of Discovering The Premises of Mathematics," in Russell (1973), pp. 273, 294. In the 1907 lecture, Russell also explicitly claims that "Some of [Frege's] logical premises are not very obvious." See Russell (1973) pp. 279. 7According to Russell, some logical truths are self-evident, such as in the case of the law of non-contradiction. Russell claims that the law of excluded middle is also obvious, but this is a point on which he changed his mind. See "The Regressive Method of Discovering The Premises of Mathematics," in Ibid, pp. 279. 8See "The Philosophical Implications of Mathematical Logic," in Ibid, pp. 285. Similarly, Russell claims that by modifying Frege's axioms to avoid the paradoxes, one cannot be assured "with certainty" that the resulting axioms do not imply a contradiction. Ibid, pp. 280. Here, Russell's claim does not anticipate Gödel's second incompleteness theorem. Rather, he seems to indicate that one can only see whether a set of axioms imply a contradiction by repeatedly trying to prove one. 9In See Russell (2009b), pp. 203. This is another point on which Russell changed his opinion. One axiom of Principia Mathematics postulates the existence of at least one object, but in the above passage from Introduction to Mathematical Philosophy, Russell claims that this axiom is a "defect of logical purity." 4 orems of arithmetic and analysis. Hence, although the arithmetical theorems that they justify might be a priori, it's not clear that Russell's axioms are capable of being known a priori. Moreover, concerning the existential axioms that form a foundation for most of mathematics, Russell claims, "Existence theorems, where individuals are concerned, are now theorems as to existence in the philosophical sense; hence it is natural that they should not be demonstrable a priori."10 Additionally, because Russell often seems to identify a priori and necessary truths, his arguments again the necessity of certain logical axioms also likely indicate that he denied their status as a priori truths. Even if Russell had claimed that logical axioms always possessed the desirable epistemic properties discussed above, there is yet another reason to think that Russell never endorsed EI. Recall that logicism is the conjunction of the two theses: (a) all mathematical terms can be defined using only logical constants, and (b) all mathematical theorems can be derived from purely logical axioms. Call these the definitional and derivational theses respectively. Despite Russell's repeated insistence that mathematics and logic are identical, he admits that the truth of the derivational thesis is not settled by the work in Principia Mathematica. At the conclusion of his Introduction to Mathematical Philosophy, he writes: . . . We have sufficiently defined the character of the primitive ideas in terms of which all the ideas of mathematics can be defined, but not of the primitive propositions from which all the propositions of mathematics can be deduced. This is a more difficult matter, as to which it is not yet known what the full answer is. We may take the axiom of infinity as an example of a proposition which, though it can be enunciated in logical terms, cannot be asserted by logic to be true.11 Hence, although mathematical terms, like "real number", might merely be abbreviations for logical terms, Russell is unsure whether mathematical theorems, about real numbers for example, can be proven from purely logical axioms. Following the above passage, Russell argues that the derivational thesis of logicism might still be true, but proving it requires either (i) widening the concept of "tautology" so as to include existential assertions like the axiom of infinity, or (ii) proving standard mathematical theorems without the aid of existential assertions like the axiom of infinity. In retrospect, it is clear that (ii) is impossible: one can construct finite models of ZF minus the axiom of infinity (thus proving the axiom's independence from the remaining axioms), or models of ZF in which the axiom of choice fails. Furthermore, without such existential assertions, many standard theorems of mathematics are unprovable. If the derivational thesis of logicism were true, then such assertions must shown to be logical, which is what Russell explicitly denies in some of the passages cited above. This is an important point. Although Russell repeatedly endorses logicism, thus underscoring its importance, his later writings reveal a hesitation con10See Russell (1906) (65). 11See Russell (2009b), pp. 206. 5 cerning the derivational thesis. This hesitation suggests that there are two conclusions we ought to draw about how to understand Russell's logicism: (1) Russell attached great importance to the definitional thesis of logicism alone, and (2) the search for axioms for all of mathematics is independently valuable whether or not the axioms necessary for deriving mathematical theorems are purely "logical." What, then, is the epistemic value of logicism (if any)? According to Andrew Irvine and Martin Godwyn, there are at least two philosophically important consequences of logicism for mathematical epistemology that simultaneously explain Russell's explicitly stated goals for his work in logic and his admiration for nineteenth century geometry and set theory. First, in reducing mathematics to logic, one obtains better explanations of many mathematical theorems by unifying their proofs under a single set of axioms. Second, one discovers new, and fruitful concepts, such as that of equinumerosity and cardinal number, and one discovers new techniques for proving theorems. However, Russell never claimed that logicism implies that mathematical theorems inherit properties like necessity, a prioricity, self-evidence, and so on from logical theorems. So the oft-repeated interpretation of Russell, in which he is depicted as defending EI, is not supported by textual evidence. A central question, however, remains: is there any sense in which the reduction of mathematics to logic increases the certainty of mathematical theorems? In other words, even if mathematical theorems do not inherit self-evidence from the logical axioms from which they are eventually derived, is there any way in which the logical reconstruction of mathematics increases evidence one has for accepting a given theorem as true? Russell repeatedly suggests that the answer to this question is "yes," and the balance of the paper is dedicated to explaining why. 2 Coherence and The Epistemic Significance of Logicism For Russell, the law of non-contradiction is an a priori, necessary, self-evident truth. If a set of propositions implies a contradiction, then at least one of its members must be false. Therefore, for Russell, the existence of a potential contradiction or inconsistency in one's beliefs counts as evidence against such beliefs. Because the logical analysis of mathematics led to the discovery, and (one hopes) eradication of paradoxes concerning set construction principles, it follows that logicism, if true, would reduce the chance of error and evidence against the theorems of mathematics. This much seems to be agreed upon by Russell scholars. What is perhaps less well-recognized is that Russell also endorses the converse of the above thesis: the absence of a contradiction in a set of assertions counts as evidence in support of such assertions. In discussing Frege's premises for arithmetic, and their modification due to the discovered contradiction, Rus6 sell writes: . . . if we have seemed to discover precisely why our previous premises led to contradictions, so that what (apart from consequences) seemed reasonably true, now seemed obviously false, and if the whole kind of reasoning from which the contradictions sprang is ruled out by our new premises, we may have a reasonable confidence that we have at least made the right kind of modification, and if that more modification is required, it will be more of the same. . . . Thus, Frege's premises undoubtedly give a first approximation, and the exact truth must be very much like them [Russell's emphasis italicize; mine bolded].12 A similar assertion is made at the outset of Principia Mathematica, where Russell writes: The proof of a logical system is its adequacy and its coherence. That is: (1) the system must embrace among its deductions all those propositions which we believe to be true and capable of deduction from logical premises alone . . . and (2) the system must lead to no contradictions.13 (my italics) If by "proof of a logical system" we understand Russell to mean "truth of the axioms", then the two passages are very similar. Yet Russell's conclusion is odd. There are many consistent axiomatizations of mathematics, and in the absence of an argument that logical consistency of a set of propositions is, modulo some additional assumptions, evidence for their truth, Russell's conclusion that either Frege's axioms or those in Principia must be near to the "truth" seems unjustified. Russell does provide such an argument, and the first step in understanding the argument is recognizing that, despite his suggestion in the second passage above, "coherent" means more than logically consistent. In several essays, Russell argues that, when many propositions in a set bear some relation R to one another, then the set becomes more "obvious," "probable," or "credible," than each proposition individually. Russell later collects all such relations R together under the a single relation that he calls coherence. In the passage immediately above, the relation R is consistency. In his 1907 essay 'The Regressive Method of Discovering the Premisses of Mathematics", the relation R is derivability: Assuming the usual laws of deduction, two obvious propositions of which one can be deduced from the other become more nearly certain than either would be in isolation; and thus, in a complicated deductive system, many parts of which are obvious, the total probability may become all but absolute certainty.14 12See "The Regressive Method of Discovering the Premisses of Mathematics" in Russell (1973), pp. 280. 13See Whitehead and Russell (1962), pp. 12. 14See "The Regressive Method of Discovering the Premisses of Mathematics" in Russell (1973), pp 279. 7 By 1912, Russell realized that derivability is one of several ways in which one proposition might provide evidence for or against another. Thus, in a passage of nearly identical timbre to the previous one, Russell replaces "derivability" with "mutual coheren[ce]" and "obviousness" with "probability": In regard to probable opinion, we can derive great assistance from coherence, which we rejected as the definition of truth, but may often use as a criterion. A body of individually probable opinions, if they are mutually coherent, become more probable than any one of them would be individually.15 In his philosophical autobiography, Russell ordains this thesis with the name "the coherence theory of probability": I do not accept the coherence theory of truth, but there is a coherence theory of probability which is important and I think valid. Suppose you have two facts and a causal principle which connects them, the probability of all three may be greater than the probability of any one, and the more numerous and complex the inter-connected facts and principles become, the greater is the increase of probability derived from their mutual coherence.16 When Russell wishes to be careful in distinguishing the mathematical concept of probability from the concept of evidential support, he speaks of a coherence theory of "credibility" rather than probability: Given a number of propositions, each having a fairly high degree of intrinsic credibility, and given a system of inferences by virtue of which these various propositions increase each other's credibility . . . [we] arrive at a body of interconnected propositions having, as a whole, a very high degree of credibility.17 The above passages, similar in wording and spirit, appear in works spanning over fifty years of Russell's life. Moreover, the contexts in which the passages occur vary widely. The passages from Principia and the 1907 essay occur within the context of discussion of what might constitute evidence for mathematical axioms and a formal system more generally. In contrast, the passages from Problems occur within discussions of skepticism concerning the existence of physical objects, and they explain how Russell's coherence theory is relevant to philosophical theses in general. Finally, the passages from Human Knowledge and 15See Russell (1997), pp. 140. In an earlier passage (pp. 25-26), Russell makes a similar claim ". . . by organizing our instinctive beliefs and their consequences, by considering which among them is most possible, if necessary, to modify or abandon, we can arrive, on the basis of accepting as our sole data what we instinctively believe, at an orderly systematic organization of our knowledge, in which, though the possibility of error remains, its likelihood is diminished by the interrelation of the parts and by the critical scrutiny which has preceded acquiescence." 16See Russell (1959), pp. 204. 17See Russell (1948), pp. 395. 8 My Philosophical Development occur within discussions of the use of probability in inductive inference and discovery of causal relationships respectively. The frequency with which Russell discusses his coherence theory, the period of time over which the discussions occur, and the variety of subjects to which he found it applicable together suggest that Russell attaches importance to the theory. My central claim is that, in reducing mathematics to logic, Russell was attempting, in part, to show that mathematical knowledge can be coherently organized, thereby increasing one's evidence for mathematical assertions. Before defending this claim, it is necessary to explain what Russell's coherence theory asserts. There are three aspects of the theory that deserve clarification: (1) what does Russell mean by coherence?, (2) what relations do coherent propositions bear to one another? and finally, (3) of what epistemic properties of a set of propositions (e.g. obviousness, high probability, high credibility) does coherence provide evidence? First, it is important to ascertain what Russell means by coherence. In Problems, Russell explicitly defines coherence as follows: "Two propositions are coherent when both may be true, and are incoherent when one at least must be false."18 In the above passage, and others, however, Russell discusses the coherence of more than two propositions. How is the coherence of a set of many propositions defined? Although Russell does not say so explicitly, the above passage suggests that a set of propositions ("body of individually probable opinions") is coherent just in case any two such propositions are pairwise coherent ("mutually coherent"). Unfortunately, this definition does not seem to be the one employed when Russell discusses examples of incoherent beliefs. Consider Russell's discussion of dreams in the Problems, where he claims "If our dreams, night after night, were as coherent one with another as our days, we should hardly know whether to believe the dreams or the waking life."19 First, the phrase "were as coherent . . . as" suggests that coherence occurs in various gradations, whereas Russell's earlier definition renders coherence a binary variable: propositions cohere if and only if they are pairwise non-contradictory. Thus, Russell likely uses the word "coherence" to refer to a quantitative measure of how probable one proposition is, on supposition that another proposition (or set of other propositions) is true. Moreover, Russell's dream example makes little sense if one interprets coherence as pairwise non-contradiction. If one were to form one's beliefs on the basis of dreams from successive evenings, the resulting body of beliefs need be pairwise contradictory. In fact, such beliefs taken together might be satisfiable and/or fail yo imply a contradiction. Why? Different dreams may have entirely different content matter, and so, despite their fantastical character, they may not contradict each other in the slightest. Therefore, it is likely that Russell used the word "coherence" to refer to a quantitative measure of the evidential support that one proposition lends to another. This leads us to the second question: what relation might a proposition p 18Ibid, pp. 120. 19Ibid, pp. 140. 9 bear to proposition q such that p lends (or decreases) evidential support to q? In other words, what properties of a set of propositions Γ determine its coherence? A cursory glance at the above passages (in the context) suggests the following list: • Is Γ is satisfiable? • Is a contradiction derivable from Γ? • Are some propositions in Γ derivable from others? • How many propositions does Γ contain? • Are the propositions in Γ complex or simple?20 Furthermore, the passages in which Russell discusses inductive inference suggest that he thinks there might be probabilistic analogues of some of the logical notions in the list above.21 For example, when Russell speaks of the "interconnectedness" of propositions in a set Γ, he might be discussing whether, for any proposition γ ∈ Γ, the probability/credibility of γ conditional on Γ \ {γ} is greater than, equal to, or less than the probability/credibility of γ simpliciter.22 Russell is, however, not explicit on this point, and although conditional probability affords one way of making the notion of coherence precise, any number of measures might also be reasonable and lead to different assessments. Unfortunately, even among the relations that Russell explicitly claims constitute measures of coherence, a number are still vague. Consider, for example, Russell's claim that the number of propositions in a set is one criterion for judging whether the set is coherent. When represented in a formal language, any finite number of assertions can be written as a single assertion by forming a long conjunction. So, one might ask Russell, "can propositions be delineated so that there is a distinction between finitely many propositions and only one?" Similarly, one might wonder whether cardinality is an appropriate measure of "number" of propositions in an infinite set, if one's goal is to quantify the complexity of the set. The last aspect of Russell's coherence theory in need of clarification is the following: of what epistemic properties of a set of propositions (e.g. obviousness, high probability, high credibility) does coherence provide evidence? I claim that, although Russell repeatedly speaks of a coherence theory of probability, he likely uses the word "probability" to refer to a measure of evidential support that does not satisfy the axioms standardly used in probability theory today. 20Russell defines one proposition to be simpler than another if the former's constituents are a proper subset of the latter. 21In his later writings, Russell claims that even demonstrative (i.e. deductive) inference is probabilistic because long, intricate mathematical arguments require significant care in reading and constructing to avoid error. See Russell (1948) pp. 383. For the late Russell, then, logical inferences are the endpoint on a continuum of non-demonstrative inferences. 22Russell implicitly employs the concept of conditional probability when explaining how credibility of a proposition ought to be determined by probabilities (i.e frequencies), when such probabilities are known. See Russell (1948), pp. 385. 10 Why? Recall Russell's claim that "A body of individually probable opinions, if they are mutually coherent, become more probable than any one of them would be individually." If taken literally, the claim contradicts a simple fact about probability, namely that the conjunction of two or more propositions is less probable than either conjunct, i.e., p(A&B) ≤ p(A), p(B) for all A and B. However, Russell recognizes a far more general fact about probability, namely, that if φ entails ψ, then φ is less probable than ψ. In Problems, he writes For the probability that Socrates is mortal is greater, on our data, than the probability that all men are mortal. (This is obvious because if all men are moral, so is Socrates; but if Socrates is mortal, then it does not follow that all men are mortal.)23 This passage suggests that Russell uses the word "probability" in two different ways in his earlier writings, and perhaps this is why he felt it necessary to distinguish carefully between probability and credibility in Human Knowledge, which contains Russell's most sustained discussion of probability and its interpretations (e.g. frequentist and subjective Bayesian).24 Thus, Russell's theory of coherence is, unfortunately, incomplete. But the above sketch is sufficient for discussing its importance with respect to logicism. As the reduction of mathematics to logic eradicates contradictions in one's premises, finds logically simple premises, increases the number of propositions that one accepts by producing new theorems, and uncovers derivability relations between propositions, it follows that the logical analysis of mathematical theorems renders mathematical knowledge more coherent, and therefore, more probable according to Russell. We are now ready to ask the question, "According to Russell, why does coherence increase the credibility of a set of propositions? Why is coherence a mark of or truth?." In particular, one must answer the obvious objection that coherence is not a mark of truth because fictional novels, plays, and so on, can be coherent. The above three passages provide Ruseell's answer. All three passages suggest that coherence of a set of propositions is a mark of truth only when at least some of the propositions are independently supported by available evidence. In the first passage, Russell suggests that coherence is an indicator of truth only when one's beliefs are already "individually probable." In the second, Russell claims that coherence is an indicator of truth amongst mathematical and logical propositions when such propositions are already "obvious," and in the final passage, the coherence of the postulates Russell describes counts as evidence only because such postulates "are implied in inferences we all accept as valid" and they are used to derive "facts." A passage from Human Knowledge, Its Scope, and its Limits confirms this point: 23Russell (1997), pp. 80. 24There is one snarl for this interpretation. Notice that Russell again uses the word "probability" in the passage from My Philosophical Development quoted above, and that passage was written after Human Knowledge. I expect that the use of the word "probability" was a mere lapse of thought, as Russell carefully distinguishes between probability and credibility several pages before that passage. 11 Given a number of propositions, each having a fairly high degree of intrinsic credibility, and given a system of inferences by virtue of which these various propositions increase each other's credibility . . . [we] arrive at a body of interconnected propositions having, as a whole, a very high degree of credibility . . . The edifice of knowledge may be compared to a bridge resting on many piers, each of which not only supports the roadway but helps the other piers to stand firm owing to the interconnecting girders. The piers are the analogues of the propositions having some intrinsic credibility, while the upper portions o the bridge are the analogues of what is only inferred.25 In other words, Russell's epistemology mixes foundationalist and coherentist themes. Some justified beliefs are "probable," "obvious" or gain the status of "facts" because they are self-evident or directly inferred from observation. Others gain support from adding coherence to a set of independently obvious or probable opinions. Thus, a piece of fiction, for Russell, would lack coherence, as most of its claims would be ill-supported by evidence like direct observation or obviousness, and so there is no reason to believe the worlds constructed in novels, plays, and movies are real. In contrast, the laws of physics cohere with "facts" learned by direct observation of the world, and the axioms of mathematics cohere with the basic "facts" of mathematics described in elementary arithmetic and geometry. Thus, Russell anticipates the objection that the coherence is not always an indicator of truth. Only when some propositions in a set Γ are probable, selfevident, or obvious, Russell claims, do the propositions in Γ gain support from the coherence of the entire set. Still a question lingers: if coherence is not an indicator of truth generally, why does it ever provide additional evidence for a set of propositions? Although Russell does not answer this question specifically, his discussion of criteria for 'good' premises for mathematics and causal inference, I think, can be used to provide a Russelian-like response. Clearly, when one proves a complex mathematical theorem from simple premises, one gains greater evidence for the truth of the theorem. By exactly symmetric reasoning, proving that particular premises are necessary for deducing an obvious arithmetical fact provides greater evidence for the premises; one has, in essence, obtained a proof of the premises from an obvious arithmetical fact. That the assertions being proven are called "premises" and that the assumptions in such proofs are "theorems" makes no difference. This is what Russell means when he claims that a formal system ought to be judged by its "adequacy." One might object that accepted mathematical theorems might be deduced from infinitely many different sets of axioms, however, and so there are rarely axioms that can be proven to be "necessary" for deriving ordinary mathematics. Of course, certain axioms are necessary for proving certain facts modulo other axioms. I say more about this below. So I hypothesize that, for Russell, coherence is a measure of 'how necessary' the premises in an axiomatic system 25See Russell (1948), pp. 395-396. 12 are for deriving the consequences. This is why Russell emphasizes that the "interconnectedness" of propositions in a "complicated deductive system" is an indication of their coherence. To understand this point more fully, it is helpful consider Russell's analogy between mathematics and the sciences. In inductive inquiry, one can never show that a law or set of laws (e.g. Newton's laws) are necessary in order to explain particular phenomena (e.g. projectile motion). One can, however, gain confidence that if the true laws were to differ from hypothesized ones in specific ways, then one should not have observed the phenomena that he or she did. Hence, hypothesized laws become 'approximately necessary' for explaining observable facts. Russell explains: In induction, if p is our logical premise and q our empirical premise, we now that p implies q, and in a textbook, we we apt to begin with p and deduce q. But p is only believed on account of q. Thus, we require a greater or less probability that q implies p, or what comes to the same thing, that not-p implies not-q. If we can prove that not-p implies not-q, i.e. that p is the only hypothesis consistent with facts, that settles the question. But usually what we do is to test as many alternative hypotheses as we can think of.26 Similarly, in mathematics, one can show that a particular axiom is necessary for a proof modulo other axioms. For example, modulo ZF and classical logic, the axiom of choice is known to be necessary to prove any number of elementary mathematical results. The theorem that a countable union of countable sets is countable, for instance, is known to require the axiom of choice.27 Continuing with this example, although the axiom of choice might not be necessary for such a proof modulo some other set of axioms, if one thinks that the axioms one has used in showing the necessity of choice are few in number and simple, then one has gained greater evidence that choice is absolutely essential in explaining the mathematical 'phenomena.' This last hypothesis also suggests a relationship between Russell's views on simplicity and coherence. In short, coherence of a set of propositions can be an indicator of truth because it measures to what degree one's premises are 'approximately necessary' for proving those facts one does take as self-evident. Russell's theory of coherence, therefore, provides a partial answer to a desire for certainty in mathematics: even if the reduction of mathematics to logic is not completely successful (because certain axioms may not be logical), the increased organization of mathematical knowledge yields a coherent body of theorems that is of greater security than the disconnected collection of mathematical subjects with which one began. 26"The Regressive Method of Discovering the Premisses of Mathematics" in Russell (1973), pp. 274. 27I doubt that Russell would countenance this theorem as a basic fact, but it is one that is essential in modern mathematics. 13 Acknowledgements I am first and forement indebted to Teddy Seidenfeld for helpful comments on the earliest drafts of this essay. The arguments in the paper also benefited greatly from discussions with participants at PM@100 conference. 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