We give a geometric realization of space-time spinors and associated representations, using the Jordan triple structure associated with the Cartan factors of type 4, the so-called spin factors. We construct certain representations of the Lorentz group, which at the same time realize bosonic spin-1 and fermionic spin- $${\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0em}\!\lower0.7ex\hbox{$2$}}$$ wave equations of relativistic field theory, showing some unexpected relations between various low-dimensional Lorentz representations. We include a geometrically and physically motivated introduction to Jordan triples and spin factors.

Usually the Lorentz transformations are derived from the conservation of the spacetime interval. We propose here a way of obtaining spacetime transformations between two inertial frames directly from symmetry, the isotropy of the space and principle of relativity. The transformation is uniquely defined except for a constant e, that depends only on the process of synchronization of clocks inside each system. Relativistic velocity addition is obtained, and it is shown that the set of velocities is a bounded symmetric domain. If (...) e=0, Galilean transformations are obtained. If e>0, the speed 1/√e and a spacetime interval are conserved. By assuming constancy of the speed of light, we get e=1/c 2 and the transformation between the frames becomes the Lorentz transformation. If e<0, a proper speed and a Hilbertian norm are conserved. (shrink)

There is not, of course, a one-to-one relation between policy conclusions and the conclusions of positive economics; if there were, there would be no ...

This paper defends the Intention Account of Inquiry. On this account, inquiry is best understood by appeal to a ‘question-directed intention’ (QDI), an intention to answer a question broadly construed. This account’s core commitments help meet recent challenges plaguing extant approaches to characterizing inquiry. First, QDIs are the type of mental state central to inquiry, not attitudes like curiosity or wonder. Second, holding a QDI towards a question and acting in service of it constitutes the start of inquiry. Third, controversial (...) norms which mandate a rational inquirer's ignorance towards the answer to her question can be reformulated and defended by appeal to rational constraints on intention. Fourth, instrumental pressures inquirers face are the standard pressures of plan-rationality. In defending these theses, I show that the Intention Account provides compelling explanations to standing challenges, in ways competitors cannot. It does so by advancing understanding of how our epistemic and practical agency are intertwined. (shrink)

A contribution to the feminist discussion on moral theory, exploring the debate between moral impartiality and the partiality that characterizes personal relationships, the ethic of care and its relation to justice in a gender asymmetrical society, and the role of intimate friendship in an era of the dissolution of both extended and nuclear families.

In this paper I explore the contours of a picture of normative epistemology that speaks centrally to the question of how to inquire rather than just the question of what to believe. What if normative epistemology were expanded to encompass inquiry in full? I argue that while a 'zetetic epistemology' builds on traditional normative epistemology in many appealing ways, it also faces some challenges.

Capitalism is the best. It's free enterprise. Barter. Gimbels, if I get really rank with the clerk, 'Well I don't like this', how I can resolve it? If it really gets ridiculous, I go, 'Frig it, man, I walk.' What can this guy do at Gimbels, even if he was the president of Gimbels? He can always reject me from that store, but I can always go to Macy's. He can't really hurt me. Communism is like one big phone company. (...) Government control, man. And if I get too rank with that phone company, where can I go? I'll end up like a schmuck with a dixie cup on a thread. (shrink)

Perhaps this is due to the belief that technology has a value-neutral nature, and that issues of value are better left to philosophers.

Historians of philosophy, science, and mathematics explore the influence of Kant's philosophy on the evolution of modern scientific thought.

What do we owe those with whom we inquire? Presumably, quite a bit. Anything beyond what is necessary to secure knowledge? Yes. In this paper, I argue for a class of ‘zetetic rights.’ These are rights distinctive to participants in group inquiry. Zetetic rights help protect important central interests of inquirers. These include a right to aid, a right against interference, and a right to exert influence over the course of inquiry. Building on arguments by Fricker (2015), I defend these (...) rights, and explore cases of their violation: zetetic wrongings. I argue that zetetic wrongings constitute a distinctive, ubiquitous, and significant form of injustice in our epistemic and zetetic lives. To improve our inquiries and avoid epistemic injustice, we need to contend with zetetic wronging. Making this case helps show that a complete picture of epistemic life must account for the deep influence of our cooperative ties to one another. (shrink)

A basic problem of daily life is determining who owns what. One way that people may solve this problem is by relying on a ‘first possession’ heuristic, according to which the first person who possesses an object is its owner, even if others subsequently possess the object. We investigated preschoolers’ use of this heuristic in five experiments. In Experiments 1 and 2, 3- and 4-year-olds inferred that an object was owned by the character who possessed it first, even though another (...) character subsequently possessed it. Two-year-olds also showed this bias, but only when the object was placed between the characters when children were asked about ownership. Experiment 3 ruled out the possibility that children’s bias to select the first possessor results from a tendency to select the character first associated with the object. Experiment 4 showed that 3- and 4-year-olds have difficulty disregarding the first possession heuristic, even when provided with evidence that the character who first possessed an object is not its owner. But Experiment 5 found that children can disregard the heuristic in at least some situations. These five experiments suggest that the first possession heuristic guides children’s ownership inferences. The findings provide the first evidence that preschoolers can infer who owns what, when not explicitly told, and when not reasoning about objects with which they are personally acquainted. (shrink)

Debates about how to incorporate the severely cognitively disabled into liberal theory typically focus on John Rawls’s assumption that citizens choosing the principles of justice should be understood as full social cooperators. In this paper, we argue that social cooperation is not the fundamental barrier to the inclusion of the severely cognitively disabled. We argue that these persons are excluded from the entire project of liberal legitimacy in virtue of the apparent inability of a severely cognitively disabled person to understand (...) and evaluate the legitimacy of political principles for herself. Severely cognitively disabled persons lack a kind of access to political principles that is crucial, according to liberal theory, for political principles to be legitimate to someone, and not simply for someone. (shrink)

What I call the dynamics of reason is a post-Kuhnian approach to the history and philosophy of science articulating a relativized and historicized version of the Kantian conception of the rationality and objectivity of the modern physical sciences. I here discuss two extensions of this approach. I argue that, although the relativized standards of rationality in question change over time, the particular way in which they do this still preserves the trans-historical rationality of the entire process. I also make a (...) beginning in extending my historical narrative from purely intellectual history (both philosophical and scientific) to the wider cultural context. (shrink)

We introduce insertion domains that support the placement of new, higher, vertices into finite trees. We prove that every nonincreasing insertion domain has an element with simple structural properties in the style of classical Ramsey theory. This result is proved using standard large cardinal axioms that go well beyond the usual axioms for mathematics. We also establish that this result cannot be proved without these large cardinal axioms. We also introduce insertion rules that specify the placement of new, higher, vertices (...) into finite trees. We prove that every insertion rule greedily generates a tree with these same structural properties; and every decreasing insertion rule generates (or admits) a tree with these same structural properties. It is also necessary and sufficient to use the same large cardinals (in the precise sense of Corollary D.25). The results suggest new areas of research in discrete mathematics called "Ramsey tree theory" and "greedy Ramsey theory" which demonstrably require more than the usual axioms for mathematics. (shrink)

In “Popper, Weber, and Hayek,” I claimed that the economic and political world governed by social democracy is too complex to offer hope for rational social‐democratic policy making. I extrapolated this conclusion from the claim, made by Austrian‐school economists in the 1920s and 30s, that central economic planning would face insurmountable “knowledge problems.” Israel Kirzner's Reply indicates the need to keep the Austrians’ cognitivist argument conceptually distinct from more familiar incentives arguments, which can tacitly reintroduce the assumption of omniscience against (...) which the Austrian economists rebelled. Callahan's Reply illustrates the need to keep in mind the hypothetical, conditional nature of economic arguments. Notturno's Reply changes the subject by asserting the intrinsic value of social‐democratic participation, regardless of its outcomes—even while, like Talisse's Reply, ignoring the evidence that public ignorance and elite dogmatism probably can't be mitigated by more information or by appeals to be open minded. And Hill focuses on the very real imperfections of market and other private‐sphere decision making, without weighing the shortcomings of the political alternatives. Markets and other private‐sphere decision arenas don't have to be perfect for them to improve upon public‐sector processes. In the latter, cognitive demands tend to be much greater, because of the divergence of modern social problems from the conditions that would have faced the hunter‐gatherers whose cognitive apparatus we have inherited. (shrink)

We define a class of functions, the descent recursive functions, relative to an arbitrary elementary recursive system of ordinal notations. By means of these functions, we provide a general technique for measuring the proof-theoretic strength of a variety of systems of first-order arithmetic. We characterize the provable well-orderings and provably recursive functions of these systems, and derive various conservation and equiconsistency results.

(2005). Ernst Cassirer and Contemporary Philosophy of Science. Angelaki: Vol. 10, continental philosophy and the sciences the german traditionissue editor: damian veal, pp. 119-128.

Based on the relevant logic R, the system R# was proposed as a relevant Peano arithmetic. R# has many nice properties: the most conspicuous theorems of classical Peano arithmetic PA are readily provable therein; it is readily and effectively shown to be nontrivial; it incorporates both intuitionist and classical proof methods. But it is shown here that R# is properly weaker than PA, in the sense that there is a strictly positive theorem QRF of PA which is unprovable in R#. (...) The reason is interesting: if PA is slightly weakened to a subtheory P+, it admits the complex ring C as a model; thus QRF is chosen to be a theorem of PA but false in C. Inasmuch as all strictly positive theorems of R# are already theorems of P+, this nonconservativity result shows that QRF is also a nontheorem of R#. As a consequence, Ackermann's rule γ is inadmissible in R#. Accordingly, an extension of R# which retains its good features is desired. The system R##, got by adding an omega-rule, is such an extension. Central question: is there an effectively axiomatizable system intermediate between R# and R##, which does formalize arithmetic on relevant principles, but also admits γ in a natural way? (shrink)

Kant's original version of transcendental philosophy took both Euclidean geometry and the Newtonian laws of motion to be synthetic a priori constitutive principles—which, from Kant's point of view, function as necessary presuppositions for applying our fundamental concepts of space, time, matter, and motion to our sensible experience of the natural world. Although Kant had very good reasons to view the principles in question as having such a constitutively a priori role, we now know, in the wake of Einstein's work, that (...) they are not in fact a priori in the stronger sense of being fixed necessary conditions for all human experience in general, eternally valid once and for all. And it is for precisely this reason that Kant's original version of transcendental philosophy must now be either rejected entirely or radically reconceived. Most philosophy of science since Einstein has taken the former route: the dominant view in logical empiricism, for example, was that the Kantian synthetic a priori had to be rejected once and for all in the light of the general theory of relativity. (shrink)

There have been numerous results showing that a measurable cardinal κ can carry exactly α normal measures in a model of GCH, where a is a cardinal at most κ⁺⁺. Starting with just one measurable cardinal, we have [9] (for α = 1), [10] (for α = κ⁺⁺, the maximum possible) and [1] (for α = κ⁺, after collapsing κ⁺⁺) . In addition, under stronger large cardinal hypotheses, one can handle the remaining cases: [12] (starting with a measurable cardinal of (...) Mitchell order α ) , [2] (as in [12], but where κ is the least measurable cardinal and α is less than κ, starting with a measurable of high Mitchell order) and [11] (as in [12], but where κ is the least measurable cardinal, starting with an assumption weaker than a measurable cardinal of Mitchell order 2). In this article we treat all cases by a uniform argument, starting with only one measurable cardinal and applying a cofinality-preserving forcing. The proof uses κ-Sacks forcing and the "tuning fork" technique of [8]. In addition, we explore the possibilities for the number of normal measures on a cardinal at which the GCH fails. (shrink)

I discuss three potential sources of evidence for truth in set theory, coming from set theory’s roles as a branch of mathematics and as a foundation for mathematics as well as from the intrinsic maximality feature of the set concept. I predict that new non first-order axioms will be discovered for which there is evidence of all three types, and that these axioms will have significant first-order consequences which will be regarded as true statements of set theory. The bulk of (...) the paper is concerned with the Hyperuniverse Programme, whose aim is to discover an optimal mathematical principle for expressing the maximality of the set-theoretic universe in height and width. (shrink)

The continuum function αmaps to2α on regular cardinals is known to have great freedom. Let us say that F is an Easton function iff for regular cardinals α and β, image and α<β→F≤F. The classic example of an Easton function is the continuum function αmaps to2α on regular cardinals. If GCH holds then any Easton function is the continuum function on regular cardinals of some cofinality-preserving extension V[G]; we say that F is realised in V[G]. However if we also wish (...) to preserve measurable cardinals, new restrictions must be put on F. We say that κ is F-hypermeasurable iff there is an elementary embedding j:V→M with critical point κ such that H)Vsubset of or equal toM; j will be called a witnessing embedding. We will show that if GCH holds then for any Easton function F there is a cofinality-preserving generic extension V[G] such that if κ, closed under F, is F-hypermeasurable in V and there is a witnessing embedding j such that j≥F, then κ will remain measurable in V[G]. (shrink)

Assuming the existence of a weakly compact hypermeasurable cardinal we prove that in some forcing extension א ω is a strong limit cardinal and א ω+2 has the tree property. This improves a result of Matthew Foreman (see [2]).

The notion of interpretation is absolutely fundamental to mathematical logic and the foundations of mathematics. It is also crucial for the foundations and philosophy of science - although here some crucial conditions generally need to be imposed; e.g., “the interpretation leaves the mathematical concepts unchanged”.

The debate over Green and Shapiro's Pathologies of Rational Choice Theory sustains their contention that rational choice theory has not produced novel, empirically sustainable findings about politics?if one accepts their definition of empirically sustainable findings. Green and Shapiro show that rational choice research often resembles the empirically vacuous practices in which economists engage under the aegis of instrumentalism. Yet Green and Shapiro's insistence that theoretical constructs should produce accurate predictions may inadvertently lead to instrumentalism. Some of Green and Shapiro's critics (...) hint at a better approach, which would eschew predictive testing in favor of testing the applicability of the theory to particular cases. (shrink)

Fukuyama's argument that we have recently reached ?The End of History?; is defended against writers who fail to appreciate the Hegelian meaning of Fukuyama's ?Endism,?; but is criticized for using simplistic dichotomies that evade the economic and ideological convergence of East and West. Against Fukuyama, the economic critique of socialism, revisionist scholarship on early Soviet economic history, and the history of the libertarian ideas of Rousseau, Kant, Hegel and Marx are deployed to show that history ?ended?; years ago: the creeds (...) of the First and Second Worlds sprang from common assumptions; and even before Eastern European reform movements, both sides of the Iron Curtain had moved to economies that are neither capitalist nor socialist. (shrink)

Hayek developed two contradictory epistemologies. The epistemology for which he is famous attributed dispersed knowledge to economic actors and credited the price system for aggregating and communicating this knowledge. The other epistemology attributed to human and non-human organisms alike the error-prone interpretation of stimuli, which could never truly be said to be “knowledge.” Several of the paradoxes of Hayek's economic and political thought that are explored in this symposium can be explained by the triumph of the first epistemology over the (...) second, including his historical interpretation of socialism as a planning mentality; his tendentious definitions of “liberty” and “justice”; and his opposition to economic redistribution even as he endorsed all manner of economic and social regulations. (shrink)

The volume number is a lagging indicator, 1 but Critical Review was founded 20 years ago. As we enter our third decade, I am pleased to report that Routledge, storied publisher of the two most impo...