The paper critically examines an unpopular line of Frege’s view on numbers in the Foundations of Arithmetic. According to this view, which analyzes numbers in terms of properties and not in terms of extensions, numbers are properties of concepts vs. properties of objects. The latter view is held by Mill and is famously criticized in the Foundations. I argue that on the property account numbers cannot only be properties of concepts but they also have to be properties of objects. My (...) main argument rests on purely metaphysical grounds. It stems from the motivation that were numbers only properties of concepts we would not have been able to explain mathematical truths about the physical world or those truths would have been miraculous. On pains that we do have mathematical truths about the physical world that are not miraculous we cannot agree with Frege’s property line about the metaphysical nature of numbers. (shrink)

In recent years, neologicists have demonstrated that Hume's principle, based on the one-to-one correspondence relation, suffices to construct the natural numbers. This formal work is shown to be relevant for empirical research on mathematical cognition. I give a hypothetical account of how nonnumerate societies may acquire arithmetical knowledge on the basis of the one-to-one correspondence relation only, whereby the acquisition of number concepts need not rely on enumeration (the stable-order principle). The existing empirical data on the role of (...) the one-to-one correspondence relation for numerical abilities is assessed and additional empirical tests are proposed. In the final part, it is argued that the fact that the successor relation and the one-to-one correspondence relation can play independent roles in number concept acquisition may be a complication for testing the Whorfian hypothesis. (shrink)

Rips et al. argue that the construction of math schemas roughly similar to the Dedekind/Peano axioms may be necessary for arriving at arithmetical skills. However, they neglect the neo-Fregean alternative axiomatization of arithmetic, based on Hume's principle. Frege arithmetic is arguably a more plausible start for a top-down approach in the psychological study of mathematical cognition than Peano arithmetic.

Theories of analogical reasoning have assumed that a 1‐to‐1 constraint discourages reasoners from mapping a single element in 1 analog to multiple elements in another. Empirical evidence suggests that reasoners sometimes appear to violate the one‐to‐one constraint when asked to generate mappings, yet virtually never violate it when asked to generate analogical inferences. However, few studies have examined analogical inferences based on nonisomorphic analogs, and their conclusions are suspect due to methodological problems. We sought to elicit mixed inferences that could (...) result from combining information from 2 possible mappings. Participants generated 2‐to‐1 correspondences when asked for explicit mappings, but did not produce mixed inferences. Multiple correspondences appear to arise from multiple isomorphic mappings, rather than from a single homomorphic mapping. (shrink)

The Library of the Royal Society of London contains a large collection of manuscript material relating to Henry Oldenburg and his correspondents. Oldenburg was one of the two Secretaries of the Royal Society when it was founded in 1662. For many years he acted as intermediary between British and Continental philosophers: and scientists. He also edited the early volumes of the Royal Society's Philosophical Transactions . His contacts were accordingly very extensive. Nearly all the seventeenth-century pioneers of science were among (...) his correspondents. In his role of intermediary he was in the habit of sending extracts from some of his foreign letters to. (shrink)

This chapter articulates two concerns that Karl Jaspers raised (with Hannah Arendt) about the common practice of viewing moral evil as unintelligible. The first is that this involves exoticizing the act and/or perpetrator in such a way that moral condemnation becomes difficult. The second is that it can lead us to treat the perpetrator, place, or victim as tainted or stained by a force whose motives we cannot grasp; this in turn can lead to magical thinking about evil as somehow (...) contagious or contaminating. After distinguishing some of the main categories of evil discussed in the western tradition, I examine ways in which moral evil, in particular, has been characterized as unintelligible, and try to discern which of them raises these Jaspersian concerns. I argue that there are at least two conceptions of “radical evil”—not the Kantian one, but the ones articulated by Hannah Arendt—that do so. (shrink)

Within the framework of the contemporary discussions of the presuppositions of democracy, the author of this paper poses the question whether discussing justice primarily from the social rather than the personal aspect and level is, perhaps, more appropriate. This ties in with the question of the primary object of justice – is justice the trait of social institutions or individuals? Thus the question of what kind of justice matches democracy. The author explicates this network of questions through three underlying models (...) of the theory of justice – Plato, Aristotle and Rawls’s model. Plato’s model is about the realisation of a good life in a political community in view of the idea of justice that one should do that which is proper to one. Aristotle’s understanding of justice sets forth from a systematic analysis of the concepts that differentiate between the levels of general and particular justice. Rawls’s model of distributive justice rests on the mechanism of the forces of the institutional functional frameworks in well-organised societies. The author’s conclusion is that both forms of justice correspond well to the realisation of democracy. It is best when one’s personal disposition to justice provides strength and support to the social institutions of democracy. Without an adequate institutional framework there is, most certainly, no democracy. Without the personal support of citizens, which issues from fundamental consent, democracy can neither be sustained nor developed. In a democratic system, the private sphere of citizens cannot be fully separated from their public actions. Democracy always and necessarily contains in itself some of the political ideals, which citizens cannot, in their worldviews, relate to neutrally. The questions of the institutional and political shaping of relations, and the questions of the good and just life should be in the interest of the citizens. If they do not accomplish their democratic ideals through their own belief and virtue, then there is no substance that would shape identity for a just system of social institutions. If they do not relate to it as to their own common sense and good to be preserved, democracy has no prospects. (shrink)

A one-to-one correspondence is established between linearized space-time metrics of general relativity and the wave equations of quantum mechanics. Also, the key role of boundary conditions in distinguishing quantum mechanics from classical mechanics, will emerge naturally from the procedure. Finally, we will find that the methodology will enable us to introduce not only test charges but also test masses by means of gauges.

This article explores the existential conditions for a transition towards socioeconomic degrowth through an analysis of a paradigm shift between two extreme polarities of socio-ecological positioning: the Dominant Social Paradigm (DSP) and the New Ecological Paradigm (NEP). It is suggested that the transition from one to the other – understood as the first collective step towards degrowth – requires a transformation in the way we, in western capitalist society, define ourselves in relation to nature. This identity transformation corresponds with the (...) reconnection between humans and nature that ecopsychology has been calling for since its emergence in the 1970s. However, according to recent empirical studies in existential psychology, such a transformation contains potentially disquieting aspects, since it implies recognising and accepting the idea of our own mortality. Staging a dialogue between ecopsychology and critical degrowth theories, this article argues that the necessary transition towards degrowth requires new ways of dealing with these existential fears – namely, confronting them in a ‘reflexive’ rather than ‘defensive’ way – so as to develop relationships with nature that respect the limits of the biosphere. (shrink)

This work presents a version of the correspondence theory of truth based on Wittgenstein's Tractatus and Russell's theory of truth and discusses related metaphysical issues such as predication, facts and propositions. Like Russell and one prominent interpretation of the Tractatus it assumes a realist view of universals. Part of the aim is to avoid Platonic propositions, and although sympathy with facts is maintained in the early chapters, the book argues that facts as real entities are not needed. It includes (...) discussion of contemporary philosophers such as David Armstrong, William Alston and Paul Horwich, as well as those who write about propositions and facts, and a number of students of Bertrand Russell. It will interest teachers and advanced students of philosophy who are interested in the realistic conception of truth and in issues in metaphysics related to the correspondence theory of truth, and those interested in Russell and the Tractatus. (shrink)

Is the quantum state part of the furniture of the world? Einstein found such a position indigestible, but here I present a different understanding of the wavefunction that is easy to stomach. First, I develop the idea that the wavefunction is nomological in nature, showing how the quantum It or Bit debate gets subsumed by the corresponding It or Bit debate about laws of nature. Second, I motivate the nomological view by casting quantum mechanics in a “classical” formalism (Hamilton–Jacobi theory) (...) and classical mechanics in a “quantum” formalism (Koopman–von Neumann theory) and then comparing and contrasting classical and quantum wave functions. I argue that Humeans about laws can treat classical and quantum wave functions on a par and that doing so yields many benefits. (shrink)

On the correspondence theory of truth, a proposition is true if and only if it corresponds to fact. Criticisms of the correspondence theory of truth have argued that such a strict interpretation of the correspondence relation will not be able to account for the truth of statements about fiction or mathematics. This challenge has resulted in the introduction of more permissive correspondence relations, such as Austin’s correspondence as correlation or Tarski’s correspondence as reference satisfaction. (...) Recently, some mediated correspondence theorists of truth have proposed that the correspondence relation holds not only between thought and world but also between thought and language. In this paper, I argue that correspondence truth, direct or mediated, is not a monistic theory of truth, the view that there is one and only way for a proposition to be true. To argue for this position, I will have to show that each of the correspondence theories accept direct and indirect ways of understanding the correspondence relation as well as address potential objections to the view that correspondence theory is not singular and monolithic. (shrink)

Ever since the first meeting of the proponents of the emerging Logical Empiricism in 1923, there existed philosophical differences as well as personal rivalries between the groups in Berlin and Vienna, headed by Hans Reichenbach and Moritz Schlick, respectively. Early theoretical tensions between Schlick and Reichenbach were caused by Reichenbach’s Kantian roots, who himself regarded the Vienna Circle as a sort of anti-realist “positivist school”—as he described it in his Experience and Prediction. One result of this divergence was Schlick’s preference (...) of Carnap over Reichenbach for a position at the University of Vienna, and his decision not to serve as a co-editor with Reichenbach for the journal Erkenntnis that they jointly established in 1930. A second split rooted in different views on induction and probability, which culminated in the Hans Reichenbach’s refusal to serve as an invited author on probability within the International Encyclopedia of Unified Science series ed. by Rudolf Carnap, Charles Morris and Otto Neurath from 1938 onwards. In this regard it is remarkable that also Richard von Mises, who was the second leading figure of Logical Empiricism in Turkish exile, criticized the theory of probability put forward by his former Berlin colleague. In this paper I analyse this controversial exchange, drawing on the relevant correspondence and asking whether these philosophical differences were a typical feature of the pluralism inherent in Logical Empiricism in general. (shrink)

This book presents the first full translation of the correspondence of Leo Strauss and Gerhard Krüger, showing for each the development of key and influential ideas, along with seven interpretative essays by leading Strauss scholars. During the early to mid-1930’s, Leo Strauss carried on an intense, and sometimes deeply personal, correspondence with one of the leading intellectual lights among Heidegger’s circle of recent students and younger associates. A fellow traveler in the effort to “return to Plato” and reject (...) neo-Kantian conventions of the day, Krüger was also a serious student of Rudolf Bultmann and the neo-orthodox movement in which Strauss also took an early interest. During the most intense years of their correspondence, each underwent significant intellectual development: in Krüger’s case, through a penetrating series of studies of Kant and Descartes, respectively, ultimately leading to Krüger’s conversion to Catholicism; and, in Strauss’s case, through the complex stages of what he subsequently called his “reorientation,” involving what he for the first time calls “political philosophy.” Readers interested in tracing the development of Strauss’s thoughts regarding a theological alternative that he found helpfully challenging—if not ultimately compelling—will find this correspondence to be an accessible point of entry. (shrink)

In the context of modal logics one standardly considers two modal operators: possibility ) and necessity ) [see for example Chellas ]. If the classical negation is present these operators can be treated as inter-definable. However, negative modalities ) and ) are also considered in the literature [see for example Béziau ; Došen :3–14, 1984); Gödel, in: Feferman, Collected works, vol 1, Publications 1929–1936, Oxford University Press, New York, 1986, p. 300; Lewis and Langford ]. Both of them can be (...) treated as negations. In Béziau a logic \ has been defined on the basis of the modal logic \. \ is proposed as a solution of so-called Jaśkowski’s problem [see also Jaśkowski ]. The only negation considered in the language of \ is ‘it is not necessary’. It appears that logic \ and \ inter-definable. This initial correspondence result between \ and \ has been generalised for the case of normal logics, in particular soundness-completeness results were obtained [see Marcos :279–300, 2005); Mruczek-Nasieniewska and Nasieniewski :229–248, 2005)]. In Mruczek-Nasieniewska and Nasieniewski it has been proved that there is a correspondence between \-like logics and regular extensions of the smallest deontic logic. To obtain this result both negative modalities were used. This result has been strengthened in Mruczek-Nasieniewska and Nasieniewski :261–280, 2017) since on the basis of classical positive logic it is enough to solely use \ to equivalently express both positive modalities and negation. Here we strengthen results given in Mruczek-Nasieniewska and Nasieniewski by showing correspondence for the smallest regular logic. In particular we give a syntactic formulation of a logic that corresponds to the smallest regular logic. As a result we characterise all logics that arise from regular logics. From this follows via respective translations a characterisation of a class of logics corresponding to some quasi-regular logics where \ is the smallest element. Moreover, if a given quasi-regular logic is characterised by some class of models, the same class can be used to semantically characterise the logic obtained by our translation. (shrink)

In this paper, constructions of free algebras corresponding to multiplicative classical linear logic, its affine variant, and their extensions with -contraction () are given. As an application, the cardinality problem of some one-variable linear fragments with -contraction is solved.

Sex selection of children by pre-conception and post-conception techniques remains morally controversial and even illegal in some jurisdictions. Among other things, some critics fear that sex selection will distort the sex ratio, making opposite-sex relationships more difficult to secure, while other critics worry that sex selection will tilt some nations toward military aggression. The human sex ratio varies depending on how one estimates it; there is certainly no one-to-one correspondence between males and females either at birth or across the (...) human lifespan. Complications about who qualifies as ‘male’ and ‘female’ complicate judgments about the ratio even further. Even a judiciously estimated sex ratio does not have, however, the kind of normative status that requires society to refrain from antenatal sex selection. Some societies exhibit lopsided sex ratios as a consequence of social policies and practices, and pragmatic estimates of social needs are a better guide to what the sex ratio should be, as against looking to ‘nature’. The natural sex ratio cannot be a sound moral basis for prohibiting parents from selecting the sex of their children, since it ultimately lacks any normative meaning for social choices. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:RESPONSE TO BRUCE MARSHALL GEORGE LINDBECK There is an abundance of il'iches in Bruce Marshalrs essay. He makes me understand hoth myse1f and Aquinas hetter than I biaid done hefore; and, interestingly, it is chiefly hy his exegesis of St. Thomas that he does :bhis. If I had referred more to the Thomistic ideas he elucidates when I wirus writing Natrure of Dootrine1 it would have!been a better hook. (...) What he calls the " somewhat notorious " example of the crusadeT"s Christus est Dominus is a good oase in point. It would ihave helped if I had made dear that I was thinking in medieval [ashion of a an individual rutterance, the product of a pa•riticular :second 1aJct of the intellect, the ·acl of composing and dividing, of judging ·such and such rto ibe rthe ca,se. In Aquinas' intellectual setting, judgments, not sentences in ·abstraiction from 1 acls of affirmation, were propositions capruble of being true or f.a1se. Many of my ·reruders had a more modern or "Platonic " understanding of rpropositions, and therefore missed the force of the e:xJam:ple ·as Marshall so carefully and rightly explains it. Among his contributions, the major one, however, is system- •a1tically to introduce into the disCUJssion the distinction between the " truth " and " justification " of beliefs or pI'oposi- "'" tionS.- Once the point is made, it is evident that " alethiology" and " epistemology " (to mention a cognate, though not ~den­ tical distinction) ·are, 1 at least in some contexts, partially independent variables. There is no one-to-one relation between different meanings or theories of truth and the v:arious views as to how we know such ·and such is true. It was my failure to make this point explicit which confosed Fr. Colman O'Neill, of blessed memory, ias well as a good many other readers (1as Bruce Marshall quite rightly notes, though, with excessive 408 404 GEORGEl LINDBECK kindness to me, he!blames the,readers rather than the 'author for the misunderstanding). Once iclarfiied, as it has lbeen done hy Mr. Marshall, the issue turns on ·whether,a classical " correspondence " theory of truth can be ioombined,with, to empfoy O'Neill's terminology, the use of "coherentist" and " pragmatist " epistemological criteria in justifying rbelieis. I ·am not sure ·that this is possible for those who e:xdude any reference to an idea.I observer or knower (whether real or hypothetical) when defining truth, hut for any theist r.for whom God is prima veritas, as he was for Aquinas, the answer is dearly in the affirmative. In God, and only in God, are knowledge and reality, not only in correspondence, hut directly known to correspond. Only in him do truth·and knowledge of truth, 1 aleth:iofogy and epistemology, ooindde. In 1human knowledge in via, in contrast, there is always a gap. Our ibe1ieTs may correspond to reality, hut we are justified in holding that they do so, not by directly seeing the correspondence, 1but by some other means. That those other means might in part or w:hole be coherentist or pragmatist cannot :be e:xduded a priori. It is true that in the caise of an Aristotelian 1such 1as Aquinas, coherence •and rpmctise 1are not explicitly accorded major roles in the epistemological justification of natural kno·wledge or scientia, yet even here they are not e:xduded. Nothing can quali[y ais 1an item of knowledge unless it coheres with all other scientia, and right pra.ctice (i.e., training in virtue) is indispensible :llor ethical kno,wledge. Nor need one cease fo be an Aris,totelian, 1as far ·as I can see, if one accepts the contempOO 'ary oommonpJaice that sense experience itself is heavily dependent on linguistic and non-linguistic pmctices which are in part ·acquired rthrough variaible forim:s of aicculturation and not simply through the actualization of 1genetically encoded propensities. One could still, despite this increased emphasis on prwctiice.and on coherence 1 with webs of 1belief, ibe 'able to affum in good Aristotelian,fashion tha;t the knowledge naturally accessible to ra:t]onal animals such,as we a.re is primarily justified RESPONCE TO BRUCE MARSHALL 405 iby reference to... (shrink)

As the title “Doing without Concepts” suggests Edouard Machery argues that psychologists should stop using the notion of concept because: (1) the only interesting generalizations about concepts can be drawn at the level of types of concepts (prototypes, exemplars and theories) and not the level of concept in general; and (2) competences such as categorization or induction can rely on these different types of concepts (there is not a one to one correspondence between type of concept and competence). I (...) try to make the point that these two elements are not wholly compatible. If several types of concepts are used to perform a given competence (point (2)), then they have to be well regulated (e.g. which type is activated when, which type wins in case of conflict). These regulatory mechanisms can then be the basis for interesting generalizations (against point (1)). On the other hand, it is possible that point (1) applies to competences as well: that there are no interesting generalizations to be drawn about categorization in general. In which case different types of categorization are likely to be underlain by different types of concepts (against point (2)). Even though the arguments laid out in the book are forceful and well supported by empirical evidence, a more positive thesis might have been both more successful rhetorically and more interesting scientifically. (shrink)

Words refer to objects in the world, but this correspondence is not one-to-one: Each word has a range of referents that share features on some dimensions but differ on others. This property of language is called underspecification. Parts of the lexicon have characteristic patterns of underspecification; for example, artifact nouns tend to specify shape, but not color, whereas substance nouns specify material but not shape. These regularities in the lexicon enable learners to generalize new words appropriately. How does the (...) lexicon come to have these helpful regularities? We test the hypothesis that systematic backgrounding of some dimensions during learning and use causes language to gradually change, over repeated episodes of transmission, to produce a lexicon with strong patterns of underspecification across these less salient dimensions. This offers a cultural evolutionary mechanism linking individual word learning and generalization to the origin of regularities in the lexicon that help learners generalize words appropriately. (shrink)

According to widespread opinion, the meaning of body part terms is determined by salient discontinuities in the visual image; such that hands, feet, arms, and legs, are natural parts. If so, one would expect these parts to have distinct names which correspond in meaning across languages. To test this proposal, we compared three unrelated languages—Dutch, Japanese, and Indonesian—and found both naming systems and boundaries of even basic body part terms display variation across languages. Bottom-up cues alone cannot explain natural language (...) semantic systems; there simply is not a one-to-one mapping of the body semantic system to the body structural description. Although body parts are flexibly construed across languages, body parts semantics are, nevertheless, constrained by non-linguistic representations in the body structural description, suggesting these are necessary, although not sufficient, in accounting for aspects of the body lexicon. (shrink)

If true, this is one the the most important papers in the history of mathematics. the natural numbers are defined and one to one correspondence between the natural numbers is made precise. the paper deals with the very fundamentals of arithmetic and the logical principles differ quite markedly from those used by georg cantor.

According to widespread opinion, the meaning of body part terms is determined by salient discontinuities in the visual image; such that hands, feet, arms, and legs, are natural parts. If so, one would expect these parts to have distinct names which correspond in meaning across languages. To test this proposal, we compared three unrelated languages—Dutch, Japanese, and Indonesian—and found both naming systems and boundaries of even basic body part terms display variation across languages. Bottom‐up cues alone cannot explain natural language (...) semantic systems; there simply is not a one‐to‐one mapping of the body semantic system to the body structural description. Although body parts are flexibly construed across languages, body parts semantics are, nevertheless, constrained by non‐linguistic representations in the body structural description, suggesting these are necessary, although not sufficient, in accounting for aspects of the body lexicon. (shrink)

In 1992, I sent Milton Friedman a draft of my 1993 paper “Friedman's Theory of Corporate Social Responsibility.” He and I corresponded at length. My 1993 paper argues that Friedman's published formulations of his theory are not equivalent and that they prescribe different courses of action in many possible cases. In our correspondence, Friedman conceded that his two formulations of his theory are inconsistent and, at my suggestion, he endorsed a modified version of the view he presented in Capitalism (...) and Freedom as the preferred version of his theory. This modified theory is an important formulation of his position. In one of his letters to me, Friedman writes: “I agree that corporate executives might have duties to the general public which sometimes outweigh their duties to the shareholders.” I argue that this creates major problems for his theory. I also answer Friedman's published response to one of my criticisms. (shrink)

One major fault line in foundational theories of cognition is between the so-called “representational” and “non-representational” theories. Is it possible to formulate an intermediate approach for a foundational theory of cognition by defining a conception of representation that may bridge the fault line? Such an account of representation, as well as an account of correspondence semantics, is offered here. The account extends previously developed agent-based pragmatic theories of semantic information, where meaning of an information state is defined by its (...) interface role, to a theory that accommodates a notion of representation and correspondence semantics. It is argued that the account can be used to develop an intermediate approach to cognition, by showing that the major sources of tension between “representational” and “non-representational” theories may be eased. (shrink)

Correspondence analysis is Kooi and Tamminga’s universal approach which generates in one go sound and complete natural deduction systems with independent inference rules for tabular extensions of many-valued functionally incomplete logics. Originally, this method was applied to Asenjo–Priest’s paraconsistent logic of paradox LP. As a result, one has natural deduction systems for all the logics obtainable from the basic three-valued connectives of LP -language) by the addition of unary and binary connectives. Tamminga has also applied this technique to the (...) paracomplete analogue of LP, strong Kleene logic \. In this paper, we generalize these results for the negative fragments of LP and \, respectively. Thus, the method of correspondence analysis works for the logics which have the same negations as LP or \, but either have different conjunctions or disjunctions or even don’t have them as well at all. Besides, we show that correspondence analyses for the negative fragments of \ and LP, respectively, are also suitable without any changes for the negative fragments of Heyting’s logic \ and its dual \ and LP). (shrink)

One of the great challenges in word learning is that words are typically uttered in a context with many potential referents. Children's tendency to associate novel words with novel referents, which is taken to reflect a mutual exclusivity (ME) bias, forms a useful disambiguation mechanism. We study semantic learning in pragmatic agents—combining the Rational Speech Act model with gradient‐based learning—and explore the conditions under which such agents show an ME bias. This approach provides a framework for investigating a pragmatic account (...) of the ME bias in humans but also for building artificial agents that display an ME bias. A series of analyses demonstrates striking parallels between our model and human word learning regarding several aspects relevant to the ME bias phenomenon: online inference, long‐term learning, and developmental effects. By testing different implementations, we find that two components, pragmatic online inference and incremental collection of evidence for one‐to‐one correspondences between words and referents, play an important role in modeling the developmental trajectory of the ME bias. Finally, we outline an extension of our model to a deep neural network architecture that can process more naturalistic visual and linguistic input. Until now, in contrast to children, deep neural networks have needed indirect access to (supposed to be novel) test inputs during training to display an ME bias. Our model is the first one to do so without using this manipulation. (shrink)

Verifying an epistemic protocol involves creating a formalized version of the protocol in a suitable logical language, and next showing (i) that the steps of the protocol are in one to one correspondence with the steps in its formalized version, (ii) that the formalized version satisfies certain correctness conditions, and (iii) hence, that the original version also satisfies these conditions. We will show that DEL is a suitable medium for carrying out this program for an interesting example protocol.

According to the received view of practical decisions, ‘deciding to X’ is synonymous with ‘forming an intention to X’. In this article, I argue against the received view on the basis of both experimental evidence and theoretical considerations. The evidence concerns a case involving a side-effect action in which people tend to agree that an agent decided to X yet disagree that the agent had a corresponding intention to X. Additionally, I explain why one should expect decisions and intentions to (...) diverge in the case of certain side-effect actions. (shrink)

Philosophical theorizing about truth manifests a desire to conform to the ordinary or folk notion of truth. This practice often involves attempts to accommodate some form of correspondence. We discuss this accommodation project in light of two empirical projects intended to describe the content of the ordinary conception of truth. One, due to Arne Naess, claims that the ordinary conception of truth is not correspondence. Our more recent study is consistent with Naess’ result. Our findings suggest that contextual (...) factors and respondent gender affect whether the folk accept that correspondence is sufficient for truth. These findings seem to show that the project of accommodating the ordinary notion of truth is more difficult than philosophers had anticipated because it is fragmentary. (shrink)

This paper analyses the Marxian theory of exploitation. The axiomatic approach standard in social choice theory is adopted in order to study the concept of exploitation—what it is and how it should be captured empirically. Two properties are presented that capture some fundamental Marxian insights. It is shown that, contrary to the received view, there exists a nonempty class of definitions of exploitation that preserve the relation between exploitation and profits—called Profit-Exploitation Correspondence Principle—in general economies with heterogeneous agents, complex (...) class structures, and production technologies with heterogeneous labour inputs. However, among the main approaches, only the so-called ‘New Interpretation’ satisfies the Profit-Exploitation Correspondence Principle in general. (shrink)

The historical Molyneux’s question – roughly, whether congenital blind subjects can visually identify shapes in front of them right after being made to see – is having its renaissance in recent years (Ferretti and Glenney, 2021). While there have been many different formulations of it, and many attempted answers as well, no clear consensus has been reached. Moreover, although arguably both memory and imagination are involved in the process, their roles in the Molyneux’s task have not been adequately discussed. In (...) this paper, I focus on a specific version of Molyneux’s question, propose a specific hypothesis in relation to that version, and sketch the roles of structural correspondence, memory, and imagination in the relevant hypothesis. One key moral of this discussion is that while Molyneux’s question has primarily been regarded as a perceptual puzzle, other kinds of mental episodes such as memory and imagination have to be taken into account as well in order to have a more satisfying answer to versions of Molyneux’s question (Macpherson and Dorsch, 2018). (shrink)

Scholarly correspondence can be as insightful as scholarly work itself, as it often documents the motivating forces of its writers’ intellectual ideas while illuminating their lives more clearly. The more complex the authors’ scholarly works and the more troubled the eras in which they lived, the more substantial, and potentially fascinating, their correspondence. This is especially true of the letters between Alfred Schutz and Eric Voegelin. The scholars lived in incredibly dramatic times and produced profound, complex works that (...) continue to confound academics. The communication between these two giants of the social sciences, as they sent their thoughts to one another, was crucial to the work of both men. _A Friendship That Lasted a Lifetime: The Correspondence between Alfred Schutz and Eric Voegelin _demonstrates that Schutz and Voegelin shared a remarkable friendship: they first met as students in Vienna in the 1920s and found themselves great partners in discussion; years later they were pushed out of Europe by Nazi pressure and went to work at separate American universities. For twenty years they wrote each other, developing their respective scientific works in that dialogue. The letters bear witness to their friendship during the years they spent in exile in the United States, and they document the men’s tentative attempts at formulating the theories of “lifeworld” and “gnosis” associated with Schutz and Voegelin today. The entire collection of 238 letters was printed in German in 2004, but this edited volume is the first to present their correspondence in English and offers a selection of the most important letters—those that contributed to the thinkers’ theoretical discussions and served as background to their most significant thoughts. Editors Gerhard Wagner and Gilbert Weiss do not analyze Schutz’s and Voegelin’s works in light of the correspondence—rather, they present the collection to create a framework for new interpretations. _A Friendship That Lasted a Lifetime_ takes a unique look at two major social scientists. This volume is a valuable resource in the study of Voegelin’s political philosophy and Alfred Schutz’s contribution to American sociology and marks an important addition to the literature on these remarkable men. Showing how scholarly discourse and the dialogue of everyday life can shed light on one another, the book finally presents this correspondence for an American audience and is not to be missed by scholars of philosophy and sociology. (shrink)

Thomas Reid is now recognized as one of the towering figures of the Enlightenment. Best known for his published writings on epistemology and moral theory, he was also an accomplished mathematician and natural philosopher, as an earlier volume of his manuscripts edited by Paul Wood for the Edinburgh Reid Edition, Thomas Reid on the Animate Creation, has shown. The Correspondence of Thomas Reid collects all of the known letters to and from Reid in a fully annotated form. Letters already (...) published by Sir William Hamilton and others have been reedited, and roughly half of the letters included appear in print for the first time. Writing in 1802, Reid's disciple and biographer Dugald Stewart doubted that Reid's correspondence &"would be generally interesting.&" This collection proves otherwise, for the letters illuminate virtually every aspect of Reid's life and career and, in some instances, provide us with invaluable evidence about activities otherwise undocumented in his manuscripts or published works. Through his correspondence we can trace Reid's relations with contemporaries such as David Hume and his colleagues at both King's College, Aberdeen, and the University of Glasgow, as well as his engagement with the most controversial philosophical, scientific, and political issues of his day. If anything, the letters assembled here serve as the starting point for understanding Reid and his place in the Enlightenment. (shrink)

We thank the commentators for their thoughtful engagement with our paper.1 In different ways, they make the same substantial point: our suggested interventions are not enough to solve the problems of organisational failure. On this we wholeheartedly agree. Organisational failure in healthcare is complex and multifaceted, it cannot be solved by one intervention in medical education. We did not intend to imply that our proposals alone would solve organisational failure, and this positioning misconstrues the aims of our paper. We had (...) more modest ambitions, we wanted to shift the analytical emphasis away from the individual and explore the implications raised by our analysis for our piece of the jigsaw—medical education. Having stepped away from such grand claims, the commentators make some valid points to which we would like to respond more specifically. Jesudason makes the distinction that normalisation of deviance (NoD) is well suited to understand institutional misfeasance but not malfeasance.2 We are inclined to agree; NoD can do some things but not all. Likewise, …. (shrink)

The relationship between the One and the Multiple in mystic philosophy is a profound and central theme that explores the nature of existence, the cosmos, and the divine. This theme is present in various mystical traditions, including those of the East and West, and it addresses the paradoxical coexistence of the unity and multiplicity of all things. -/- In mystic philosophy, the **One** often represents the ultimate reality, the source from which all things emanate and to which all things return. (...) It is the absolute, the infinite, and the unchanging. The One is beyond all attributes and is often associated with the divine or the absolute truth. -/- The **Multiple**, on the other hand, represents the manifest world, the diversity of forms, and the realm of change and plurality. It is the world we experience through our senses, the domain of time and space, where differentiation and individuality are apparent. -/- The relationship between the One and the Multiple is not one of opposition but of emanation and unity. The Multiple is seen as a reflection, expression, or manifestation of the One. In this sense, the diversity of the world doesn’t contradict the unity of the One but rather demonstrates it in a myriad of forms. -/- Mystics seek to understand and experience this relationship through various practices and insights. They aim to transcend the illusion of separation and duality to experience the non-dual reality where the One and the Multiple are recognized as inseparable. -/- This concept can be illustrated by the metaphor of the ocean and its waves. The ocean represents the One—vast, deep, and all-encompassing—while the waves represent the Multiple—distinct, diverse, and ever-changing. Each wave is unique, yet it is not separate from the ocean. The wave’s existence is dependent on and made of the same substance as the ocean. In the same way, each individual entity in the Multiple is an expression of the One. -/- In summary, the relationship between the One and the Multiple in mystic philosophy is a dynamic interplay that challenges the conventional understanding of separation. It invites a deeper exploration of reality, where the apparent multiplicity of the world is a direct expression of the singular, underlying essence of all that is. -/- The relationship between the One and the Multiple in the context of mathematical philosophy is a profound topic that touches upon the very foundations of existence and knowledge. It’s a theme that has been explored by philosophers and mathematicians alike, often leading to the contemplation of unity and diversity within the structures of reality. -/- In mathematics, the concept of the One can be seen as the basis of unity from which all numbers derive. It’s the identity element in multiplication, the starting point in counting, and the foundation of dimension in geometry. The One is often associated with the concept of monism in philosophy, which posits that there is a single, underlying substance or principle that constitutes reality. -/- On the other hand, the Multiple represents the infinite variety and diversity of forms and numbers that arise from the One. It’s the embodiment of plurality and the complex interplay of different entities that mathematics seeks to understand and describe. This reflects the philosophical stance of pluralism, which acknowledges the existence of multiple realities or truths. -/- The interplay between the One and the Multiple can be seen in the mathematical concept of sets. A set can be thought of as a unity, a whole composed of distinct elements. Yet, each element within the set also retains its individuality, contributing to the diversity of the set’s composition. This duality is mirrored in the philosophical exploration of the universal and the particular, where the universal represents the One, and the particulars represent the Multiple. -/- In the realm of mathematical philosophy, this relationship often leads to questions about the nature of mathematical objects: Are they discovered as part of an objective reality (the One), or are they constructed by the human mind from a multitude of experiences (the Multiple)? This debate resonates with the philosophical inquiry into the nature of truth and reality. -/- Reflecting on the One and the Multiple can also lead to a deeper understanding of the self and the cosmos. Just as the number one is integral to the existence of all other numbers, the individual self can be seen as a unique expression of the universal whole. Similarly, the cosmos can be viewed as a grand unity composed of a multiplicity of forms and phenomena. -/- Gödel’s incompleteness theorems have a fascinating connection to the philosophical concepts of the One and the Multiple. These theorems, which are pivotal in mathematical logic and philosophy of mathematics, articulate the inherent limitations of formal axiomatic systems, particularly those sufficient to express the arithmetic of natural numbers. -/- Gödel’s incompleteness theorems have a fascinating connection to the philosophical concepts of the One and the Multiple. These theorems, which are pivotal in mathematical logic and philosophy of mathematics, articulate the inherent limitations of formal axiomatic systems, particularly those sufficient to express the arithmetic of natural numbers⁴. -/- The **first incompleteness theorem** reveals that within any such consistent system, there are propositions that are true but cannot be proven within the system itself. This reflects the idea of the Multiple in that there is an abundance of mathematical truths, a multiplicity that exceeds the unifying framework of any one system. It suggests that the realm of mathematical truth is more extensive than any single formal system can fully capture. -/- The **second incompleteness theorem** extends this by showing that a system cannot prove its own consistency. This relates to the concept of the One, as it implies that a system’s complete self-understanding, its unity and coherence, is unattainable from within. It must look beyond itself, to an external vantage point, to ascertain its consistency. -/- In the context of the One and the Multiple, Gödel’s theorems imply that the One (a consistent formal system) is inherently incomplete and cannot encompass the Multiple (the totality of mathematical truths). This resonates with philosophical discussions about the relationship between unity and plurality. Just as a single philosophical system cannot capture the entirety of truth, a single formal mathematical system cannot encapsulate all mathematical truths. -/- Moreover, Gödel’s work suggests that the pursuit of a single, unified theory of everything in mathematics—a One that encompasses all Multiples—is an inherently Sisyphean task. There will always be more truths (Multiples) than can be derived from any given set of axioms (the One). -/- In essence, Gödel’s incompleteness theorems provide a formal underpinning to the philosophical notion that the One cannot exist without the Multiple, and vice versa. They are interdependent, with the One giving rise to the Multiple, and the Multiple necessitating the One for its expression and comprehension. This interplay is a dance of limitations and possibilities, where the boundaries of logic, mathematics, and philosophy blur into one another. -/- The relationship between the One and the Multiple in the context of the mathematics of zero is a deeply philosophical inquiry that bridges the abstract world of numbers with the existential questions of being and non-being. -/- Zero, in mathematics, is a symbol of absence, a representation of nothingness, yet it holds a pivotal position as a number. It is the void from which all things emerge and to which they return. In the philosophy of mathematics, zero is the paradoxical junction where the One and the Multiple converge and diverge. -/- From the perspective of the One, zero can be seen as the origin—the singular point that precedes the existence of numbers. It is the empty set, the foundation upon which the edifice of mathematics is constructed. As the identity element in addition, zero maintains the integrity of numbers, for adding zero to any number leaves it unchanged, reflecting the immutable nature of the One. -/- Conversely, when we consider the Multiple, zero represents the infinite potentiality of creation. It is the canvas upon which the integers, both positive and negative, express their multitude. Zero is the balance point, the fulcrum around which the symphony of numbers dances. It embodies the plurality of possibilities, the beginning of the number line that stretches infinitely in both directions. -/- Philosophically, zero challenges our understanding of existence. It is both something and nothing—a number that quantifies the absence of quantity. This duality echoes the philosophical struggle to comprehend how the One gives rise to the Multiple. How does the unity of being manifest the diversity of the cosmos? Zero offers a mathematical metaphor for this mystery, as it encapsulates the transition from non-existence to existence, from the undifferentiated One to the differentiated Multiple. -/- In the realm of set theory, zero corresponds to the empty set—a set with no elements. This set is unique in that it is the only set that contains nothing, yet it is the foundation upon which all other sets are built. The empty set is the mathematical embodiment of the One, and all other sets, containing multiple elements, arise from it. -/- Reflecting on zero in the context of Gödel’s incompleteness theorems, we find a resonance with the idea that the One (a consistent formal system) cannot capture the entirety of the Multiple (the totality of mathematical truths). Zero, as the foundation of numbers, similarly suggests that from the nothingness of the One, the infinite complexity of the Multiple emerges—yet it can never be fully encompassed or expressed by any single system. -/- In conclusion, the mathematics of zero offers a profound reflection on the relationship between the One and the Multiple. It serves as a bridge between the abstract and the concrete, the known and the unknowable, challenging us to ponder the origins of existence and the nature of reality itself. -/- . (shrink)

Theories of Truth provides a clear, critical introduction to one of the most difficult areas of philosophy. It surveys all of the major philosophical theories of truth, presenting the crux of the issues involved at a level accessible to nonexperts yet in a manner sufficiently detailed and original to be of value to professional scholars. Kirkham's systematic treatment and meticulous explanations of terminology ensure that readers will come away from this book with a comprehensive general understanding of one of philosophy's (...) thorniest set of topics. -/- Included are discussions of the correspondence, coherence, pragmatic, semantic, performative, redundancy, appraisal, and truth-as-justification theories. There are also chapters or sections of chapters on the liar paradox, three-valued logic, Field's critique of Tarski, Davidson's program, Dummett's theory of linguistic competence, satisfaction, recursion, the extension/intension distinction, and an explanation of how theories of justification, properly understood, differ from theories of truth. -/- A persistent theme is that philosophers have too often failed to recognize that not all theories of truth are intended to answer the same question. When the various questions are made distinct, it is apparent that many of the "debates" in this field are really cases of philosophers talking past one another. There is much less disagreement within the field than has commonly been thought. (shrink)

In 1981, Takeuti introduced quantum set theory as the quantum counterpart of Boolean valued models of set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed subspaces in a Hilbert space and showed that appropriate quantum counterparts of ZFC axioms hold in the model. Here, Takeuti's formulation is extended to construct a model of set theory based on the logic represented by the lattice of projections in an arbitrary von Neumann algebra. (...) A transfer principle is established that enables us to transfer theorems of ZFC to their quantum counterparts holding in the model. The set of real numbers in the model is shown to be in one-to-one correspondence with the set of self-adjoint operators affiliated with the von Neumann algebra generated by the logic. Despite the difficulty pointed out by Takeuti that equality axioms do not generally hold in quantum set theory, it is shown that equality axioms hold for any real numbers in the model. It is also shown that any observational proposition in quantum mechanics can be represented by a corresponding statement for real numbers in the model with the truth value consistent with the standard formulation of quantum mechanics, and that the equality relation between two real numbers in the model is equivalent with the notion of perfect correlation between corresponding observables (self-adjoint operators) in quantum mechanics. The paper is concluded with some remarks on the relevance to quantum set theory of the choice of the implication connective in quantum logic. (shrink)

In one of the first attempts at autobiography, Rousseau1s Confessions not only retells the facts of the author1s life, but reveals his innermost feelings & details the strengths & weaknesses of his character.

Conventional wisdom holds that the von Neumann entropy corresponds to thermodynamic entropy, but Hemmo and Shenker (2006) have recently argued against this view by attacking von Neumann's (1955) argument. I argue that Hemmo and Shenker's arguments fail due to several misunderstandings: about statistical-mechanical and thermodynamic domains of applicability, about the nature of mixed states, and about the role of approximations in physics. As a result, their arguments fail in all cases: in the single-particle case, the finite particles case, and the (...) infinite particles case. (shrink)

The meaning of 'most' can be described in many ways. We offer a framework for distinguishing semantic descriptions, interpreted as psychological hypotheses that go beyond claims about sentential truth conditions, and an experiment that tells against an attractive idea: 'most' is understood in terms of one-to-one correspondence. Adults evaluated 'Most of the dots are yellow', as true or false, on many trials in which yellow dots and blue dots were displayed for 200 ms. Displays manipulated the ease of using (...) a 'one-to-one with remainder' strategy, and a strategy of using the Approximate Number System to compare of (approximations of) cardinalities. Interpreting such data requires care in thinking about how meaning is related to verification. But the results suggest that 'most' is understood in terms of cardinality comparison, even when counting is impossible. (shrink)

Much has been written of late concerning the relative virtues and views of correspondence and deflationary theories of Truth. What is troubling, however, is that it is not always entirely clear exactly what distinguishes different conceptions of truth. Characterizations of the distinction are often vague and sometimes vary from writer to writer. One central thing I want to do here is to diagnose the source of the difficulty in providing a clear characterization of the distinction. In light of this (...) diagnosis, I will argue that there is a simple distinguishing feature of such views. Roughly, the distinction depends on the modal status accorded to the T-sentences by the various conceptions. And finally, I will argue in favor of drawing the distinction in this way by showing that it yields a powerful method of arguing for or against a given conception of truth. (shrink)

Could each and every one of us, instead of interacting with actual objects, really be brains in a vat? In the first chapter of his new book, Reason, Truth and History, Professor Putnam raises this and related questions with the aim of undermining what he calls the “metaphysical realist” or “externalist” conception of reality. Putnam describes metaphysical realism as a view which holds that the world consists in “some fixed totality of mind-independent objects”; truth on this view amounts to a (...) correspondence between words or thoughts and these objects. Putnam contrasts metaphysical realism with a doctrine which he names “internalism” and calls on Kant to explicate its basic ideas. Putnam claims that Kant was the first philosopher to see clearly the difficulties inherent in metaphysical realism. The doctrine of transcendental idealism, says Putnam, represents Kant’s attempt to avoid externalism. But is Kant’s doctrine really just an early version of Putnam’s internalism? In this paper I shall argue that it is not. Although Kant obviously rejects the position Putnam calls “externalism,” I shall try to show that Kant’s metaphysical theory is distinct from internalism. Kant couples transcendental idealism with a second doctrine of empirical realism, and this enables Kant to circumvent both the internalist’s coherence theory of truth as well as a simple-minded correspondence theory. To be sure, one of Kant’s major concerns is to argue against the possibility of knowing things-in-themselves; Kant thus maintains, as Putnam correctly points out, that the nature of perceivers must figure in the nature of what is perceived. But this is not to say, as Putnam implies, that there is no mind-independent reality in Kant’s theory. On the contrary, Kant’s Refutation of Idealism—an argument that strikingly resembles the first horn of Putnam’s brain-in-a-vat dilemma—is designed to establish the existence of “matter,” which is Kant’s general term for what we might call the “external” or the “mind-independent.” Kant does argue that the perception of matter must accord with space and time, the two forms of sensibility, and this is the source of his transcendental idealism. But if we take this to mean that there cannot be anything independent of our minds or of our conceptual schemes, then we miss the point of Kant’s double doctrine of transcendental idealism and empirical realism, which is intended to avoid a coherence-only theory of truth while yet affirming that objects are knowable only as appearances. Putnam, by running together a number of Kantian notions, misrepresents Kant’s position and thus fails to see that Kant does offer an alternative to both metaphysical realism and internalism. I shall begin my defense of the claim that Kant does provide a third alternative by exploring more fully the debate between Putnam and the externalist concerning the BV dilemma. I shall then explore the Refutation of Idealism to support my claim that Putnam’s internalism is distinct from transcendental idealism. We shall see in section IV that in interpreting Kant on transcendental idealism Putnam makes two crucial mistakes which lead him to equate internalism with transcendental idealism. The result of these mistakes is discussed in section V. (shrink)

Thomas Reid is now recognized as one of the towering figures of the Enlightenment. Best known for his published writings on epistemology and moral theory, he was also an accomplished mathematician and natural philosopher, as an earlier volume of his manuscripts edited by Paul Wood for the Edinburgh Reid Edition, Thomas Reid on the Animate Creation, has shown. The Correspondence of Thomas Reid collects all of the known letters to and from Reid in a fully annotated form. Letters already (...) published by Sir William Hamilton and others have been reedited, and roughly half of the letters included appear in print for the first time. Writing in 1802, Reid's disciple and biographer Dugald Stewart doubted that Reid's correspondence &"would be generally interesting.&" This collection proves otherwise, for the letters illuminate virtually every aspect of Reid's life and career and, in some instances, provide us with invaluable evidence about activities otherwise undocumented in his manuscripts or published works. Through his correspondence we can trace Reid's relations with contemporaries such as David Hume and his colleagues at both King's College, Aberdeen, and the University of Glasgow, as well as his engagement with the most controversial philosophical, scientific, and political issues of his day. If anything, the letters assembled here serve as the starting point for understanding Reid and his place in the Enlightenment. (shrink)

Kit Fine’s book is a study of abstraction in a quite precise sense which derives from Frege. In his Grundlagen, Frege contemplates defining the concept of number by means of what has come to be called Hume’s principle—the principle that the number of Fs is the same as the number of Gs just in case there is a one-to-one correspondence between the Fs and the Gs. Frege’s discussion is largely conducted in terms of another, similar but in some respects (...) simpler, proposal—to define the concept of direction by laying down the principle that the direction of a line a is the same as that of line b if and only if lines a and b are parallel. Such principles have come to be known as abstraction principles. More generally, abstraction principles are ones of the shape: ∀α∀β ↔ α ≈ β), where ≈ is an equivalence relation on entities of the type over which α and β vary and, if the principle is acceptable, § is a function from entities of that type to objects. Since ‘the direction of’ is intended to stand for a function from objects of a certain sort to other objects, the Direction Equivalence is a first-order, or in Fine’s terminology objectual, abstraction; since ‘the number of’ is intended to stand for a function from concepts to objects, Hume’s principle is a higher-order, or as Fine says conceptual, abstraction. As is well-known, Frege himself abandoned the idea of defining number implicitly or contextually by means of Hume’s principle—adopting instead an explicit definition of the number of Fs as the extension of the concept concept equinumerous to the concept F— because he thought that an adequate definition should settle the question whether Julius Caesar, for example, is or is not the number of some concept, but that the proposed implicit definition cannot do so. But interest in abstraction principles has revived in the last couple of decades, largely as a result of Crispin Wright’s attempt to show that in spite of the many difficulties confronting it—including the notorious Julius Caesar problem— Hume’s principle can after all serve as a foundation for arithmetic. (shrink)

This chapter challenges Cantor’s notion of the ‘power’, or ‘cardinality’, of an infinite set. According to Cantor, two infinite sets have the same cardinality if and only if there is a one-to-one correspondence between them. Cantor showed that there are infinite sets that do not have the same cardinality in this sense. Further, he took this result to show that there are infinite sets of different sizes. This has become the standard understanding of the result. The chapter challenges this, (...) arguing that we have no reason to think there are infinite sets of different sizes. It begins with an initial argument against Cantor’s claim that there are infinite sets of different sizes and then proceeds, by way of an analogy between Cantor’s mathematical result and Russell’s paradox, to a more direct argument. (shrink)

The goal of this paper is to propose correspondence analysis as a technique for generating the so-called erotetic calculi which constitute the method of Socratic proofs by Andrzej Wiśniewski. As we explain in the paper, in order to successfully design an erotetic calculus one needs invertible sequent-calculus-style rules. For this reason, the proposed correspondence analysis resulting in invertible rules can constitute a new foundation for the method of Socratic proofs. Correspondence analysis is Kooi and Tamminga's technique for (...) designing proof systems. In this paper it is used to consider sequent calculi with non-branching, invertible rules for the negation fragment of classical propositional logic and its extensions by binary Boolean functions. (shrink)

Context: Second-order cybernetics and its implications have been understood within the cybernetics community for some time. These implications are important for understanding the structure of scientific endeavor, and for researchers in other fields to see the reflexive nature of scientific research. This article is about the role of context in the creation and exploration of our experience. Problem: The purpose of this article is to point out the fundamental nature of the circularity in cybernetics and in scientific work in general. (...) I give a point of view on the nature of objective knowledge by placing it in the context of reflexivity and eigenform. Method: The approach to the topic is based on logical analysis of the nature of circularity. Mathematics and cybernetics are both fundamentally concerned with the structure of distinctions, but there can be no definition of a distinction without circularity, since such a definition would itself be a distinction. The article proceeds by explicating the structure of reflexive domains D where the transformations of the domain are in one-to-one correspondence with the domain itself. Results: I show that every element of a reflexive domain has a fixed point. This means that eigenforms arise naturally in reflexive domains. Furthermore, a reflexive domain is itself an eigenform (at a higher level. This supports the context of a second-order science that would study domains of science as part of a larger cybernetic landscape. Implications: The value of the article is in its concise reformulation of the scientific endeavor as a search for eigenforms in reflexive domains. This new view of science is promising in that it includes the former worlds of apparent objectivity and it embraces those newer worlds of science where the theories and theorists become active participants in the ongoing process of creating knowledge. Constructivist content: I argue that the perspective of reflexive domains constitutes a new way to think about the practice of science, with observers deeply imbedded, and objectivity understood as the mutual search for eigenforms. (shrink)