I criticize a recent paper by Thomas Tymoczko in which he attributes fundamental philosophical significance and novelty to the lately-published computer-assisted proof of the four color theorem (4CT). Using reasoning precisely analogous to that employed by Tymoczko, I argue that much of traditional mathematical proof must be seen as resting on what Tymoczko must take as being "empirical" evidence. The new proof of the 4CT, with its use of what Tymoczko calls "empirical" evidence is therefore not so novel as he (...) maintains. Finally, without attempting to give a full account of the notion of empirical mathematical evidence, I sketch a view showing how the use of calculation injects an empirical ingredient into proof. (shrink)

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

The Quine-Putnam Indispensability argument is the argument for treating mathematical entities on a par with other theoretical entities of our best scientific theories. This argument is usually taken to be an argument for mathematical realism. In this chapter I will argue that the proper way to understand this argument is as putting pressure on the viability of the marriage of scientific realism and mathematical nominalism. Although such a marriage is a popular option amongst philosophers of science and mathematics, in light (...) of the indispensability argument, the marriage is seen to be very unstable. Unless one is careful about how the Quine-Putnam argument is disarmed, one can be forced to either mathematical realism or, alternatively, scientific instrumentalism. I will explore the various options: (i) finding a way to reconcile the two partners in the marriage by disarming the indispensability argument (Jody Azzouni [2], Hartry Field [13, 14], Alan Musgrave [18, 19], David Papineau [21]); (ii) embracing mathematical realism (W.V.O. Quine [23], Michael Resnik [25], J.J.C. Smart [27]); and (iii) embracing some form of scientific instrumentalism (Ot´ avio Bueno [7, 8], Bas van Fraassen [30]). Elsewhere [11], I have argued for option (ii) and I won’t repeat those arguments here. Instead, I will consider the difficulties for each of the three options just mentioned, with special attention to option (i). In relation to the latter, I will discuss an argument due to Alan Musgrave [19] for why option (i) is a plausible and promising approach. From the discussion of Musgrave’s argument, it will emerge that the issue of holist versus separatist theories of confirmation plays a curious role in the realism–antirealism debate in the philosophy of mathematics. I will argue that if you take confirmation to be an holistic matter—it’s whole theories (or significant parts thereof) that are confirmed in any experiment—then there’s an inclination to opt for (ii) in order to resolve the marital tension outlined above.. (shrink)

For fifty years Hubert Dreyfus has done pioneering work which brings phenomenology and existentialism to bear on the philosophical and scientific study of the mind. This is a selection of his most influential essays, developing his critique of the representational model of the mind in analytical philosophy of mind and mainstream cognitive science.

The spectre of the repugnant conclusion and the search for a population axiology that avoids it has endured as a focus of population ethics. This is in part because the repugnant conclusion is often interpreted as a defining problem for totalism, while the implications of averagism and related views are taken to illustrate the theoretical cost of avoiding the repugnant conclusion. However, we show that this interpretation cannot be sustained unless one focuses only on a special case of the repugnant (...) conclusion: namely, the subset of instances of the repugnant conclusion where there is no portion of the population unaffected by the choice between population outcomes (as in Derek Parfit’s original illustration). To avoid an inappropriate focus on only this proper subset of instances of the repugnant conclusion, we formulate a general characterization of the repugnant conclusion. We then prove formally that all leading welfarist axiologies imply this conclusion, including averagism and Ng’s Theory X0 , including probabilistic and ‘very repugnant’ variants that involve the addition of negative lives. We then prove that the full range of axiologies considered by population ethics each imply an extended version of the repugnant conclusion, including axiologies that are incomplete, intransitive, rank-dependent, person-affecting, and/or pluralist. The upshot is that the repugnant conclusion does not ultimately tell against any approach to axiology, and the methodological requirement to avoid the repugnant conclusion should be dropped from population axiology. (shrink)

Persons are biological beings who participate in social environments. Is human sociality different from that of insects? Is human sociality different from that of a computer or robot with elaborate rules for social interaction in its program memory? What is the relationship between the biology of humans and the sociality of persons? I argue that persons constitute an emergent ontological level that develops out of the biological and psychological realm, but that is largely social in its own constitution. This requires (...) a characterization of the relationships between the bio/psychological and the social, and of the developmental process of emergence. It also requires a framework for modeling the bio/psychological level that makes any such emergence possible. Neither attachment theory nor information processing frameworks, for example, will do — the major orientations toward human sociality today make understanding that sociality ultimately impossible. Only an action framework, such as that of Peirce or Piaget2, suffices. (shrink)

To surmount the notorious difficulties of defining life, we should evaluate theories of life not by whether they provide necessary and sufficient conditions for our current preconceptions about life but by how well they explain living phenomena and how satisfactorily they resolve puzzles about life. On these grounds, the theory of life as supple adaptation (Bedau 1996) gets support from its natural and compelling resolutions of the following four puzzles: (1) How are different forms of life at different levels of (...) the vital hierarchy related? (2) Is there a continuum between life and non-life? (3) Does life essentially concern a living entity’s material composition or its form? (4) Are life and mind intrinsically connected? (shrink)

neglected aspect: knowledge of error, or ‘‘negative’’ knowledge. The development of knowledge of what counts as error occurs via a kind of internal variation and selection, or quasi-evolutionary, process. Processes of reflection generate a hierarchy of principles of error.

In her highly influential 1984 study Abortion and the Politics of Motherhood, Kristin Luker speculates that opposition to legal abortion among women was likely to be strongest among those who were full-time homemakers without a college education. But despite a marked decline in that demographic group and a well-documented rise in public support for gender equality since then, the rate of support for legal abortion has remained stubbornly fixed at between fifty and fifty-five percent. This tepid support has coincided (...) with a steep decline in abortion services in rural states, and ever more sweeping restrictions on abortion being tested in the courts. Karissa Haugeberg's Women against Abortion and Katie Watson's Scarlet A both seek to address this state of affairs, albeit in markedly different ways. Haugeberg provides a historical chronicle of the motives and strategies of certain key women activists in the fight against legal abortion, with an eye toward how their concerns “came to serve as blueprints to legislators and judges who continue to craft policies and laws that erode women's right to abortion”. Katie Watson draws upon her experience as an attorney and bioethicist to write a guide “intended to encourage and equip you to engage in respectful, productive, private conversation about your experience with, and opinion of, abortion”. Though both authors reveal their support for legal abortion, both are concerned to understand the motives and goals of those who fight against it. (shrink)

Formal arguments have proven that avoiding the Repugnant Conclusion is impossible without rejecting one or more highly plausible population principles. To many, such proofs establish not only a deep challenge for axiology, but also pose an important practical problem of how policymaking can confidently proceed without resolving any of the central questions of population ethics. Here we offer deflationary responses: first to the practical challenge, and then to the more fundamental challenge for axiology. Regarding the practical challenge, we provide an (...) overview of recent literature that explores the implications for public policy of the Repugnant Conclusion and related puzzles within population ethics, and shows that there is more commonality in the implications of different population axiologies than is generally recognized. The upshot is that uncertainty about population ethics presents no important problem for policymaking, and the importance of population ethics to policymaking has been overstated. We then turn to more fundamental issues about axiology, and describe a new series of formal proofs that undermine the idea that any plausible axiology could avoid the Repugnant Conclusion, including those that are assumed to avoid it in the literature. The philosophical upshot is that the Repugnant Conclusion cannot plausibly be avoided, and so it is a mistake to assume as a constraint on a plausible axiology that it must be avoided, as is assumed by most of the literature in population ethics, including by all of the fundamental impossibility theorems. (shrink)

Many people have been enraged lately by the enormous increases in certain generic prescription drugs. But free marketeers defend these prices by arguing that they simply represent what the market will bear, and in a capitalist society there is accordingly nothing wrong with charging them. This paper argues that such a defense is actually contrary to the very principles that free marketeers claim to embrace. These prices are not only unjust and exploitative, but government interference with them would not render (...) the market less free, at least if what it means for a market to be “free” is properly understood. Everyone, even free marketeers with a libertarian or neoliberal bent, should accordingly be against this kind of profiteering. (shrink)

ABSTRACT We consider several ways in which a good understanding of modern techniques and principles in physics can elucidate ecology, and we focus on analogical reasoning between these two branches of science. Analogical reasoning requires an understanding of both sciences and an appreciation of the similarities and points of contact between the two. In the current ecological literature on the relationship between ecology and physics, there has been some misunderstanding about the nature of modern physics and its methods. Physics is (...) seen as being much cleaner and tidier than ecology. When compared to this idealized, fictional version of physics, ecology looks very different, and the prospect of ecology and physics learning from one another is questionable. We argue that physics, once properly understood, is more like ecology than ecologists have thus far appreciated. Physicists and ecologists can and do learn from each other, and, in this paper, we outline how analogical reasoning can facilitate such exchanges. (shrink)

As with the problem of universals, late medieval thinkers were very concerned with the ontological status of relations, for they were central to numerous theological and philosophical problems. These relations were of various types: relations of identity, qualitative similarity, quantitative equality, causal relations, and intentional relations, such as those between knower and the object known. Each of these relations was taken to be an Aristotelian accident. Does it differ from the substance which is related? Broadly speaking, I have discovered four (...) alternative theories: as sentence of the form "a is related by a real relation R to b" is true only if the relation R is an extra-mental thing really distinct from a and b ; R only exists as a concept in a mind ; a and b exist in a certain real way ; the term "R" is connotative, signifying a directly and connoting b . Using printed sources and my critical edition of various codices, I ask the following thinkers whether a real relation is distinct from that which is related: Thomas Aquinas, Giles of Rome, Henry of Ghent, Richard of Middleton, John Duns Scotus, Henry of Harclay, Hervaeus Natalis, and Peter Aurioli. (shrink)

Time is perceived very differently from different vantage points. A year in the life of a primary-school student, for instance, is a very long time—somewhere between 1/5 and 1/ 12 of a primary-school child’s life. When you tlirow in the massive amount a child learns in any one year, compared with the diminishing returns that conspire against us later in life, a child’s year is more like a decade in adult years. But for a primary-school teacher, a school year is (...) just another ten or so months spent trying to remember names, delivering lessons, writing report cards, and endeavoring to shepherd students through the educational system. And, of course, the classroom is just one part of a teacher’s life. Teachers are juggling other serious and timeconsuming matters such as relationships, mortgages, further study, family and so forth. The best teachers, however, don’t let on about these asymmetries between the child’s world and their own; they conceal the differences in temporal perception and they give no clue that each student is just a small part of their life. Such sleight of hand can’t be easy, yet all my primary school teachers pulled it off to perfection. One, however, deserves special mention: Mr. Williams. In 1969 I turned 11 and was in 6th class at the Armidale Demonstration.. (shrink)

In this book Noel Curran suggests that considerations in the philosophy of mathematics—in particular, the proper interpretation of quaternions—leads to a “new” philosophy of space and time. According to Curran: space is Euclidean; time is absolute, flows and has a beginning; and God created the universe at the beginning of time.

This book is a must-read for researchers interested in taking a high-level, non-mechanistic approach to answering age-old fundamental questions in the brain ...

Values Education in Schools is a new resource for teachers involved in values and ethics education. It provides a range of 'practical philosophy' resources for secondary school teachers that can be used in English, religious education, citizenship, personal development and social science subjects. The materials include narratives to engage students in philosophical inquiry, doing ethics through the activity of philosophy, not simply learning about it.

With Fermat’s Last Theorem finally disposed of by Andrew Wiles in 1994, it’s only natural that popular attention should turn to arguably the most outstanding unsolved problem in mathematics: the Riemann Hypothesis. Unlike Fermat’s Last Theorem, however, the Riemann Hypothesis requires quite a bit of mathematical background to even understand what it says. And of course both require a great deal of background in order to understand their significance. The Riemann Hypothesis was first articulated by Bernhard Riemann in an address (...) to the Berlin Academy in 1859. The address was called “On the Number of Prime Numbers Less Than a Given Quantity” and among the many interesting results and methods contained in that paper was Riemann’s famous hypothesis: all non-trivial zeros of the zeta function, ζ(s) = ∞ n=1 n−s, have real part 1/2. Although the zeta function as stated and considered as a real-valued function is defined only for s > 1, it can be suitably extended. It can, as a matter of fact, be extended to have as its domain all the complex numbers (numbers of the form x + yi, where x and y √ −1) with the exception of 1 + 0i (at which point are real numbers and i =. (shrink)

It is an excellent companion to Plato's Dialogues.

In the 1990s a rise of populist or regionalist parties in Western democracies has challenged the ruling centrist consensus.1 There are, however, only a few similarities between them. Thus it is impossible to equate, e.g., Austria's Jörg Haider or France's Jean-Marie Le Pen with Canada's Preston Manning. Diverse political cultures produce different political figures, programs and ideologies. When all is said and done, Manning's Reform Party remains idiosyncratically Canadian, and it is necessary to examine the Canadian context within which it (...) arose in order to understand it. Generally speaking, the Reform Party offers little to traditionalists, while its “populism” often…. (shrink)

Anatomical studies propose that the primate auditory cortex contains more fields than have actually been functionally confirmed or described. Spatially resolved functional magnetic resonance imaging (fMRI) with carefully designed acoustical stimulation could be ideally suited to extend our understanding of the processing within these fields. However, after numerous experiments in humans, many auditory fields remain poorly characterized. Imaging the macaque monkey is of particular interest as these species have a richer set of anatomical and neurophysiological data to clarify the source (...) of the imaged activity. We functionally mapped the auditory cortex of behaving and of anesthetized macaque monkeys with high resolution fMRI. By optimizing our imaging and stimulation procedures, we obtained robust activity throughout auditory cortex using tonal and band-passed noise sounds. Then, by varying the frequency content of the sounds, spatially specific activity patterns were observed over this region. As a result, the activity patterns could be assigned to many auditory cortical fields, including those whose functional properties were previously undescribed. The results provide an extensive functional tessellation of the macaque auditory cortex and suggest that 11 fields contain neurons tuned for the frequency of sounds. This study provides functional support for a model where three fields in primary auditory cortex are surrounded by eight neighboring ‘‘belt’’ fields in non-primary auditory cortex. The findings can now guide neurophysiological recordings in the monkey to expand our understanding of the processing within these fields. Additionally, this work will improve fMRI investigations of the human auditory cortex. (shrink)

The dynamical patterns in mental phenomena have a characteristic suppleness&emdash;a looseness or softness that persistently resists precise formulation&emdash;which apparently underlies the frame problem of artificial intelligence. This suppleness also undermines contemporary philosophical functionalist attempts to define mental capacities. Living systems display an analogous form of supple dynamics. However, the supple dynamics of living systems have been captured in recent artificial life models, due to the emergent architecture of those models. This suggests that analogous emergent models might be able to explain (...) supple dynamics of mental phenomena. These emergent models of the supple mind, if successful, would refashion the nature of contemporary functionalism in the philosophy of mind. (shrink)

To form a coherent percept of the environment, our brain combines information from different senses. Such multisensory integration occurs in higher association cortices; but supposedly, it also occurs in early sensory areas. Confirming the latter hypothesis, we unequivocally demonstrate supra-additive integration of touch and sound stimulation at the second stage of the auditory cortex. Using high-resolution fMRI of the macaque monkey, we quantified the integration of auditory broad-band noise and tactile stimulation of hand and foot in anaesthetized animals. Integration was (...) found posterior to and along the lateral side of the primary auditory cortex in the caudal auditory belt. Integration was stronger for temporally coincident stimuli and obeyed the principle of inverse effectiveness: greater enhancement for less effective stimuli. These findings demonstrates that multisensory integration occurs early and close to primary sensory areas and—because it occurs in anaesthetized animals—suggests that this integration is mediated by preattentive bottom-up mechanisms. (shrink)

A leading economist and pioneering editor presents his writings in and on the discipline.