The desire to understand the mathematics of living systems is increasing. The widely held presupposition that the mathematics developed for modeling of physical systems as continuous functions can be extended to the discrete chemical reactions of genetic systems is viewed with skepticism. The skepticism is grounded in the issue of scientific invariance and the role of the International System of Units in representing the realities of the apodictic sciences. Various formal logics contribute to the theories of biochemistry and molecular (...) biology and genetics. Various paths of extension are invoked in these formal logics in order to express the information of biological apodicticism. Symbolizing the appropriate notations for invariant relations and for biological extensions of relations is fundamental to the exact generating functions of discrete algebraic biology. Aspects of philosophical perspectives of the relation scientific number systems are contrasted. The deep distinction between physical motion and biological motion is expressed in terms the roles of Aristotelian causes. The interior motion within perplex numbers is contrasted with the exterior motion of physical systems. The need for a new mathematics for biology is suggested. (shrink)

This chapter examines the pioneering work of Gottfried Wilhelm Leibniz (1646-1716) on various number systems, in particular binary, which he independently invented in the mid-to-late 1670s, and hexadecimal, which he invented in 1679. The chapter begins with the oft-debated question of who may have influenced Leibniz’s invention of binary, though as none of the proposed candidates is plausible I suggest a different hypothesis, that Leibniz initially developed binary notation as a tool to assist his investigations in mathematical problems that (...) were exercising him at the time, namely those concerning the divisibility of composite numbers, primality, and perfect numbers. The chapter then explores Leibniz’s development of binary, his little-known work on binary fractions and expansions, his use of binary as a symbol for creation in his philosophical-theology, and his response to the suggestion that there was a correlation between binary numeration and the hexagrams of the ancient Chinese divinatory text, the Yijing. The chapter then focuses on Leibniz’s work on other number systems, in particular his invention and exploration of hexadecimal as well as his work on duodecimal. The chapter concludes by revealing a hitherto unknown practical application of binary that Leibniz devised in the last year of his life. (shrink)

Clarke and Beck argue that the approximate number system represents rational numbers, like 1/3 or 3.5. I think this claim is not supported by the evidence. Rather, I argue, ANS should be interpreted as representing natural numbers and ratios among them; and we should view the contents of these representations are genuinely approximate.

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

The approximate number system (ANS) enables organisms to represent the approximate number of items in an observed collection, quickly and independently of natural language. Recently, it has been proposed that the ANS goes beyond representing natural numbers by extracting and representing rational numbers (Clarke & Beck, 2021a). Prior work demonstrates that adults and children discriminate ratios in an approximate and ratio-dependent manner, consistent with the hallmarks of the ANS. Here, we use a well-known “connectedness illusion” to provide (...) evidence that these ratio-dependent ratio discriminations are (a) based on the perceived number of items in seen displays (and not just non-numerical confounds), (b) are not dependent on verbal working memory, or explicit counting routines, and (c) involve representations with a part-whole (or subset-superset) format, like a fraction, rather than a part-part format, like a ratio. These results vindicate key predictions of the hypothesis that the ANS represents rational numbers. (shrink)

The significance of the complicated numbering of the propositions in the Tractatus has occasioned much speculation. Wittgenstein's own explanation has, following Stenius, been generally regarded as misleading. But an examination of the Prototractatus reveals that the numbering system was for Wittgenstein principally an aid in the composition of his work. It allowed him to mark out certain propositions which required further work or supplementation, without disturbing the basic structure of the treatise. But the reworking of the Prototractatus to form (...) the Tractatus considerably changed the original order in which the ideas were connected, and thus made it more difficult to understand the Tractatus. Several specific examples make it clear that it is essential to look back at the context of the Prototractatus in order to achieve a correct interpretation of many of the propositions in the Tractatus. (shrink)

We agree with Clarke and Beck that the approximate number system represents rational numbers, and we demonstrate our support by highlighting the case of the empty set – the non-symbolic manifestation of zero. It is particularly interesting because of its perceptual and semantic uniqueness, and its exploration reveals fundamental new insights about how numerical information is represented.

Number morphology (e.g., singular vs. plural) is a part of the grammar that captures numerical information. Some languages have morphological Number values, which express few (paucal), two (dual), three (trial) and sometimes (possibly) four (quadral). Interestingly, the limit of the attested morphological Number values matches the limit of non‐verbal numerical cognition. The latter is based on two systems, one estimating approximate numerosities and the other computing exact numerosities up to three or four. We compared the literature on (...) non‐verbal number systems with data on Number morphology from 218 languages. Our observations suggest that non‐verbal numerical cognition is reflected as a core part of language. (shrink)

Numbers are symbols manipulated in accord with the axioms of arithmetic. They sometimes represent discrete and continuous quantities, but they are often simply names. Brains, including insect brains, represent the rational numbers with a fixed-point data type, consisting of a significand and an exponent, thereby conveying both magnitude and precision.

The Approximate Number System (ANS) is a system that allows us to distinguish between collections based on the number of items, though only if the ratio between numbers is high enough. One of the questions that has been raised is what the representations involved in this system represent. I point to two important constraints for any account: (a) it doesn’t involve numbers, and (b) it can account for the approximate nature of the ANS. Furthermore, I (...) argue that representations of pure magnitude with vehicles that have an imprecision in the value of the unit of measurement (further clarified through a formal model from measurement theory) fit both these requirements. (shrink)

I argue that the human mind includes an innate domain-specific system for representing precise small numerical quantities. This theory contrasts with object-tracking theories and with domain-general theories that only make use of mental models. I argue that there is a good amount of evidence for innate representations of small numerical quantities and that such a domain-specific system has explanatory advantages when infants’ poor working memory is taken into account. I also show that the mental models approach requires previously (...) unnoticed domain-specific structure and consequently that there is no domain-general alternative to an innate domain-specific small number system. (shrink)

Wilfred Sieg and Dirk Schlimm. Dedekind's Analysis of Number: Systems and Axioms.

In this paper, we give an axiomatization of the ordinal number system, in the style of Dedekind’s axiomatization of the natural number system. The latter is based on a structure $(N,0,s)$ consisting of a set N, a distinguished element $0\in N$ and a function $s\colon N\to N$. The structure in our axiomatization is a triple $(O,L,s)$, where O is a class, L is a class function defined on all s-closed ‘subsets’ of O, and s is a (...) class function $s\colon O\to O$. In fact, we develop the theory relative to a Grothendieck-style universe (minus the power set axiom), as a way of bringing the natural and the ordinal cases under one framework. We also establish a universal property for the ordinal number system, analogous to the well-known universal property for the natural number system. (shrink)

Many commentators have dismissed Wittgenstein’s numbering system in the Tractatus as either incoherent or a joke. In this paper I offer a way to rehabilitate the system along the lines of Wittgenstein’s own instructions. Reading the Tractatus in this way not only offers a way to make sense of the numbering, but also offers a significant improvement in examining the meaning of the text.

Young children with limited knowledge of formal mathematics can intuitively perform basic arithmetic‐like operations over nonsymbolic, approximate representations of quantity. However, the algorithmic rules that guide such nonsymbolic operations are not entirely clear. We asked whether nonsymbolic arithmetic operations have a function‐like structure, like symbolic arithmetic. Children (n = 74 4‐ to ‐8‐year‐olds in Experiment 1; n = 52 7‐ to 8‐year‐olds in Experiment 2) first solved two nonsymbolic arithmetic problems. We then showed children two unequal sets of objects, and (...) asked children which of the two derived solutions should be added to the smaller of the two sets to make them “about the same.” We hypothesized that, if nonsymbolic arithmetic follows similar function rules to symbolic arithmetic, then children should be able to use the solutions of nonsymbolic computations as inputs into another nonsymbolic problem. Contrary to this hypothesis, we found that children were unable to reliably do so, suggesting that these solutions may not operate as independent representations that can be used inputs into other nonsymbolic computations. These results suggest that nonsymbolic and symbolic arithmetic computations are algorithmically distinct, which may limit the extent to which children can leverage nonsymbolic arithmetic intuitions to acquire formal mathematics knowledge. (shrink)

Many commentators have dismissed Wittgenstein’s numbering system in the Tractatus as either incoherent or a joke. In this paper I offer a way to rehabilitate the system along the lines of Wittgenstein’s own instructions. Reading the Tractatus in this way not only offers a way to make sense of the numbering, but also offers a significant improvement in examining the meaning of the text.

Belief in the divine origin of the universe began to wane most markedly in the nineteenth century, when scientific accounts of creation by natural law arose to challenge traditional religious doctrines. Most of the credit - or blame - for the victory of naturalism has generally gone to Charles Darwin and the biologists who formulated theories of organic evolution. Darwinism undoubtedly played the major role, but the supporting parts played by naturalistic cosmogonies should also be acknowledged. Chief among these was (...) the nebular hypothesis proposed by Pierre Simon Laplace in 1796, which explained the origin of the solar system as a natural development over extended periods of time. Ronald Numbers focuses on Laplace's theory as it affected American scientific thought. he first traces the history of Laplace's cosmogony chronologically, from its European inception to its demise about 1900. the last three chapters explore some of the theological and scientific consequences resulting from the acceptance of this cosmogony. Most significant was the change in the status of supernatural doctrine. When the nebular hypothesis lost credence at the end of the nineteenth century, those who had before tried to accommodate natural theory with supernatural doctrine no longer felt compelled to do so when faced with succeeding theories. The nebular hypothesis, it seems, had established natural law in the heavens. (shrink)

The distinction between non-symbolic and symbolic number is poorly addressed by the authors despite being relevant in numerical cognition, and even more important in light of the proposal that the approximate number system represents rational numbers. Although evidence on non-symbolic number and ratios fits with ANS representations, the case for symbolic number and rational numbers is still open.

It is known (see [1, 3.1.5]) that the order type of the nonstandard natural number system * N has the form ω + (ω * + ω) θ, where θ is a dense order type without first or last element and ω is the order type of N. Concerning this, Zakon [2] examined * N more closely and investigated the nonstandard real number system * R, as an ordered set, as an additive group and as a (...) uniform space. He raised five questions which remained unsolved. These questions are concerned with the cofinality and coinitiality of θ (which depend on the underlying nonstandard universe * U). In this paper, we shall treat nonstandard models where the cofinality and coinitiality of θ coincide with some appropriated cardinals. Using these nonstandard models, we shall give answers to three of these questions and partial answers to the other to questions in [2]. (shrink)

All humans share a universal, evolutionarily ancient approximate number system (ANS) that estimates and combines the numbers of objects in sets with ratio-limited precision. Interindividual variability in the acuity of the ANS correlates with mathematical achievement, but the causes of this correlation have never been established. We acquired psychophysical measures of ANS acuity in child and adult members of an indigene group in the Amazon, the Mundurucú, who have a very restricted numerical lexicon and highly variable access to (...) mathematics education. By comparing Mundurucú subjects with and without access to schooling, we found that education significantly enhances the acuity with which sets of concrete objects are estimated. These results indicate that culture and education have an important effect on basic number perception. We hypothesize that symbolic and nonsymbolic numerical thinking mutually enhance one another over the course of mathematics instruction. (shrink)

It has been argued that the approximate number system (ANS) constitutes a problem for the grounded approach to cognition because it implies that some conceptual tasks are performed by non-perceptual systems. The ANS is considered non-perceptual mainly because it processes stimuli from different modalities. Jones (2015) has recently argued that this system has many features (such as being modular) which are characteristic of sensory systems. Additionally, he affirms that traditional sensory systems also process inputs from different modalities. (...) This suggests that the ANS is a perceptual system and therefore it is not problematic for the grounded view. In this paper, I defend the amodal approach to the ANS against these two arguments. In the first place, perceptual systems do not possess the properties attributed to the ANS and therefore these properties do not imply that the ANS is perceptual. In the second place, I will propose that a sensory system only needs to be dedicated to process modality-specific information, which is consistent with responding to inputs from different modalities. I argue that the cross-modal responses exhibited by traditional sensory systems are consistent with modality-specific information whereas some responses exhibited by the ANS are not. (shrink)

This article accomplishes two goals. First, the paper clarifies Edmund Husserl’s investigation of the historical inception of the number system from his early works, Philosophy of Arithmetic and, “On the Logic of Signs (Semiotic)”. The article explores Husserl’s analysis of five historical developmental stages, which culminated in our ancestor’s ability to employ and enumerate with number signs. Second, the article reveals how Husserl’s conclusions about the history of the number system from his early works opens (...) up a fusion point with his investigations from his mature texts, The Crisis of the European Sciences and “The Origin of Geometry”. On the one hand, the essay shows that Husserl’s methodology was similar, as he sought in both his early and late writings to uncover the essence of the history of the formal sciences and was not executing mere intellectual history. On the other hand, the article discloses that Husserl’s insights from both time periods are strikingly analogous. Already in his early texts, Husserl saw that the sciences emerged from pre-theoretical experiences of the world and that the sciences are the result of a historical process, which involves the psychic activities of past individuals and the maintaining of discoveries over time by intersubjective communities. I conclude by showing how, in light of the analysis of this paper, we can rethink the evolution of Husserl’s philosophy. (shrink)

Nonsymbolic comparison tasks are commonly used to index the acuity of an individual's Approximate Number System (ANS), a cognitive mechanism believed to be involved in the development of number skills. Here we asked whether the time that an individual spends observing numerical stimuli influences the precision of the resultant ANS representations. Contrary to standard computational models of the ANS, we found that the longer the stimulus was displayed, the more precise was the resultant representation. We propose an (...) adaptation of the standard model, and suggest that this finding has significant methodological implications for numerical cognition research. (shrink)

For all of Brandom’s self-professed allegiance to Hegel, there is something perplexing about his fixation on semantic and epistemological issues at the expense of the type of social and political considerations that are at the heart of Hegel’s system. However, and although Brandom himself concedes that his work is circumscribed to a number of highly specialized and technical issues in the philosophy of mind and language, the truth is that his views often radiate to other philosophical fields, if (...) not always explicitly. My claim in this article is that at the heart of Brandom’s semantic theory, there are elements of a critical project, one that offers a normative standpoint to judge and improve our current practices. Moreover, these progressive features of Brandom’s normative pragmatics should be seen in the light of his adoption of a series of hermeneutic themes, ultimately culminating in his recollective conception of rationality and his edifying view of semantics. (shrink)

Rips et al.'s arguments for rejecting basic number representations as a precursor of the natural number system are exclusively based on analog number coding. We argue that these arguments do not apply to place coding, a type of basic number representation that is not considered by Rips et al.

Psychopathy fascinates. Modernist writers construct out of it an image of alienated individualism pursuing the moment, killing they know not why, exploiting in passing, troubled, if troubled at all, not by guilt, but by perplexity (Camus 1989; Gide 1995; Mailer 1957; Musil 1996). Psychiatrists and psychologists—even those who should know better—are drawn by it to take off into philosophical speculation about morality, evil, and the beast in man (Mullen 1992; Simon 1996). Philosophers succumb to the temptation of attempting to ground (...) speculation in the supposed facticity of psychopathy. -/- Psychopathy is a cultural construct with roots going back to the eighteenth century, but which only came to full flower over the last fifty years (Hare 1998; Lewis 1974). The currently fashionable constructions of psychopathy find wide acceptance among clinicians and academics. The problem is, however, whether the concept has connections that have been established to the behavior, psychopathology, or the structure and function of brain worthy of according even a modest level of scientific status. There is little point in basing an argument on some supposed scientific gold standard if what glitters is iron pyrites (fools' gold). -/- Neil Levy's opening paragraph contains the statement "Yet in other respects they seem quite irrational; so much so that the term moral insanity has sometimes been applied to them" (p. 129). This both asserts that there is a category "them," namely the psychopaths, and connects them to "moral insanity." Moral insanity entered the language of psychiatry when Pritchard chose to use the term to translate Pinel's category of mania sans deliré (Pritchard 1836). Pinel was attempting to conceptualize a range of disorders that bordered on insanity in the sense of severe disturbances of affect, judgment, and behavior, but that were not accompanied by hallucinations or delusions (Pinel 1798/1964). In today's terminology, the cases he described would probably fall into diagnostic groups, including major depression, bipolar disorder, epilepsy, and a range of personality disorders. A number of his more striking case histories involved antisocial and even murderous behavior, which probably reflected Pinel's role in assessing offenders for the criminal courts of Paris. Pritchard's use of the English word "moral" reflected an appeal first to its use at the time to designate something which was like or similar, but not identical (a usage the OED labels "vulgar"), and second moral in the sense of ethical, conforming to virtuous behavior. Anglophones seized on the latter meaning, although Pinel emphasized the former. Forty years later, Henry Maudsley, the most widely read psychiatrist of his generation, could assert that moral insanity was as real as Colour blindness, some people being Colour blind and some moral blind (Maudsley 1879). Moral insanity became connected variously to phrenology with its own bump over the underlying moral cortex, with the twisted genetics of degeneration, with atavistic theories of criminality, and with a range of disorders of brain from epilepsy to frontal lobe damage (Combed 1853; Ellis 1890; Pick 1989). Although it is anachronistic to use the terms "psychopathy" and "moral insanity" together, it is clear they occupied a similar social and psychiatric space, albeit in different era. This should caution against too ready acceptance of psychopathy which may turn out to be as empty and dangerous a conceptualization as moral insanity. Why dangerous? Dangerous to those labeled, because the morally insane and the moral imbeciles were placed in preventive detention in asylums and prisons, and even sterilized in some jurisdictions. Dangerous to society, because it gave a spurious scientific justification for viewing crime not as reflecting the social evils of deprivation, ignorance, and inequality but as the ravages of the born criminals. -/- Psychopaths are not, as Levy asserts, "(causally) responsible for … more than fifty percent of violent crime" (p. 129). A proportion of violent offenders, which varies widely between observers, are labeled psychopaths. This is often on the basis of Hare's psychopathy checklist, which is a subjective, albeit through training aspiring to be reproducible, and to some extent self-sustaining, evaluation system (Hare 1980, 2003). Part of the checklist gathers data on prior antisocial behavior; part constrains the examiner to subjectively estimate such matters as glibness and callousness, part requires judgments about whether the... (shrink)

A transcription and translation of Hermann Hankel's 1867 Vorlesungen über die complexen Zahlen und ihre Functionen, I. Theil: Theorie der Complexen Zahlensysteme, a textbook on complex analysis that played an important role in the transition to modern mathematics in nineteenth century Germany.

Robert Cummins [(1996) Representations, targets and attitudes, Cambridge, MA: Bradford/MIT, p. 1] has characterized the vexed problem of mental representation as "the topic in the philosophy of mind for some time now." This remark is something of an understatement. The same topic was central to the famous controversy between Nicolas Malebranche and Antoine Arnauld in the 17th century and remained central to the entire philosophical tradition of "ideas" in the writings of Locke, Berkeley, Hume, Reid and Kant. However, the scholarly, (...) exegetical literature has almost no overlap with that of contemporary cognitive science. I show that the recurrence of certain deep perplexities about the mind is a systematic and pervasive pattern arising not only throughout history, but also in a number of independent domains today such as debates over visual imagery, symbolic systems and others. Such historical and contemporary convergences suggest that the fundamental issues cannot arise essentially from the theoretical guise they take in any particular case. (shrink)

In this paper, we propose a method to design the pseudorandom number generator using three kinds of four-wing memristive hyperchaotic systems with different dimensions as multientropy sources. The principle of this method is to obtain pseudorandom numbers with good randomness by coupling XOR operation on the three kinds of FWMHSs with different dimensions. In order to prove its potential application in secure communication, the security of PRNG based on this scheme is analyzed from the perspective of cryptography. In addition, (...) PRNG has passed the NIST 800.22 and ENT test, which shows that PRNG has good statistical characteristics. Finally, an image encryption algorithm based on PRNG is adopted. In the encryption algorithm, the optimized Arnold matrix scrambling method and the diffusion processing based on XOR are used to obtain the final encrypted image. Through the evaluation of encryption performance, it is concluded that there is no direct relationship between the pristine image and encrypted image. The results show that the proposed image encryption scheme has good statistical output characteristics and security performance in line with cryptography. (shrink)

The intensity of the opposition to health reform in the United States continues to shock and perplex proponents of the Patient Protection and Affordable Care Act. The emotion and the apocalyptic rhetoric, render civil and evidence-based debate over the implications and alternatives to specific provisions in the law difficult if not problematic. The public debate has largely barreled down two non-parallel yet non-intersecting paths: opponents focus on their fear of government expansion in the future if PPACA is implemented now, (...) while proponents focus on the urgency and specifics of our health care market problems and the limited number of tools we have to address them. Frustration on both sides has led opponents to deny the seriousness of our health system’s problems and proponents to ignore the risk of governmental overreach. These non-intersecting lines of argument are not moving us closer to a desired and necessary resolution. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Feminist Studies 41, no. 3. © 2015 by Feminist Studies, Inc. 623 Jaye Cee Whitehead, Kath Bassett, Leia Franchini, and Michael Iacolucci “The Proof Is in the Pudding”: How Mental Health Practitioners View the Power of “Sex Hormones” in the Process of Transition In the United States today, popular discourse touts the power of “sex hormones” and hormone receptors in the brain to chemically produce gender expressions (manifested in (...) physical sex traits, behaviors, and attitudes) and identities (a sense of self as feminine or masculine). These accounts range from common assumptions that Western ideals of femininity (such as empathy understanding, cooperative orientation, nurturing behavior) and masculinity (rational/spatial understanding, competitive orientation, violent behavior) are produced by corresponding sex hormones, to studies that reduce complex gender identities and macrosocial economic phenomena to presocial, binary, hormonal processes.1 For sociologists, accounts that reduce complex identities to internal hormonal reactions are problematic for two primary reasons: first, they are inaccurate because they fail to examine how hormonal reactions are embedded in a complex cultural process of meaning and interpretation; second, they operate as bioreductive ideologies that obscure the social power inherent to the process of gender identity construction. In what follows, we extend both of these insights by examining mental health 1. For an example of a hybrid popular-scientific account of the power of sex hormones to destabilize economic structures, see Jon Coates and Joe Herbert, “Endogenous Steroids and Financial Risk Taking on a London Trading Floor,” Proceedings of the National Academy of Sciences 105, no. 16 (April 2008): 6167–72. 624 Whitehead, Bassett, Franchini, and Iacolucci practitioners’ divergent interpretations of the power of sex hormones in the process of gender transition.2 In doing so, we examine the cultural power of sex hormones in crafting gender identities without reproducing reductionist interpretations of trans embodiment. Perhaps what is most perplexing about the way we speak of sex hormones is that we know it is inaccurate to describe these hormones as sexed; scientists have acknowledged since the 1930s that they are neither sex specific in their function, nor in terms of their location in male or female bodies.3 In her extensive historicization of sex hormones, Anne Fausto-Sterling concludes that it is more accurate to call androgen- and estrogen-based hormones, “steroid hormones” as they have functions that are not confined to corresponding sexed bodies. In fact she urges scientists to “break out of the sex hormone straightjacket” and to look at steroids as just one of a number of components that are important to the creation of sex and gender, including environment and experience.4 Scholars also point to an inextricable link between the chemical operation of hormones and the social process of constructing meaning, both at the level of social interaction and macrocultural constructions of sex categories and gender ideologies.5 While brain organization/activation theory attributes sex and gender differences to hormonal interactions within the developing brain, Fausto-Sterling questions the distinction between activation and organization by pointing out how “the brain can respond to hormonal stimuli with anatomical changes…hormonal systems, after all, respond exquisitely to experience, be it in the form of nutrition, stress, or sexual activity (to name but a few possibilities ).”6 Drawing from Elizabeth Grosz, Fausto-Sterling argues that the power of hormones is best understood according to the model of the 2. “Gender transition” is a controversial phrase, as it implies that transgendered individuals who receive hormone therapy are changing identities. We are only using it for lack of clear alternative phrasing, not because we believe body modifications change gender identities. 3. See Anne Fausto-Sterling, Sexing the Body: Gender Politics and the Construction of Sexuality (New York: Basic Books, 2000); Rebecca Jordan-Young, Brainstorm: The Flaws in the Science of Sex Differences (Cambridge, MA: Harvard University Press, 2010); and Nelly Oudshoorn, Beyond the Natural Body: An Archeology of Sex Hormones (New York: Routledge, 1994). 4. Fausto-Sterling, Sexing the Body, 193–94. 5. Fausto-Sterling, Sexing the Body. 6. Ibid., 232. Whitehead, Bassett, Franchini, and Iacolucci 625 Möbius strip, wherein internal components of the self are always connected to and continuous with outer components such as culture and environment.7 In other... (shrink)

What is the Critique of Pure Reason about? The terminology of the work is so perplexing, its argument so obscurely expressed, that the ordinary reader may be forgiven if he puts it down at the end very much in the dark as to what it all means. He will have seen that in it Kant has attempted to establish certain conclusions: the subjectivity of space and time, the existence and objective validity of a number of a priori concepts or (...) categories, the falsity of the arguments used to defend the metaphysical system most widely favoured in German learned circles in the eighteenth century; but though he has grasped all this he may yet have failed to make sense of the work as a whole. It is the old story of not seeing the wood for trees; and in this case the fault is more excusable than in most, for the individual trees each demand so much attention and are so difficult to get round that it is all too easy to forget the very existence of the wood. At the worst, one may think that there is no wood at all; only a miscellaneous aggregate of individual trees which have nothing to do with each other. (shrink)