Comparing theories: the dynamics of changing vocabulary. A case-study in relativity theory

There are several first-order logic (FOL) axiomatizations of special relativity theory in the literature, all looking essentially different but claiming to axiomatize the same physical theory. In this paper, we elaborate a comparison, in the framework of mathematical logic, between these FOL theories for special relativity. For this comparison, we use a version of mathematical definability theory in which new entities can also be defined besides new relations over already available entities. In particular, we build an interpretation of the reference-frame oriented theory SpecRel into the observationally oriented Signalling theory of James Ax. This interpretation provides SpecRel with an operational/experimental semantics. Then we make precise, "quantitative" comparisons between these two theories via using the notion of definitional equivalence. This is an application of logic to the philosophy of science and physics in the spirit of Johan van Benthem's work.Comment: 27 pages, 8 figures. To appear in Springer Book series Trends in Logi

Twin Paradox and the logical foundation of relativity theory

We study the foundation of space-time theory in the framework of first-order logic (FOL). Since the foundation of mathematics has been successfully carried through (via set theory) in FOL, it is not entirely impossible to do the same for space-time theory (or relativity). First we recall a simple and streamlined FOL-axiomatization SpecRel of special relativity from the literature. SpecRel is complete with respect to questions about inertial motion. Then we ask ourselves whether we can prove usual relativistic properties of accelerated motion (e.g., clocks in acceleration) in SpecRel. As it turns out, this is practically equivalent to asking whether SpecRel is strong enough to "handle" (or treat) accelerated observers. We show that there is a mathematical principle called induction (IND) coming from real analysis which needs to be added to SpecRel in order to handle situations involving relativistic acceleration. We present an extended version AccRel of SpecRel which is strong enough to handle accelerated motion, in particular, accelerated observers. Among others, we show that the Twin Paradox becomes provable in AccRel, but it is not provable without IND.Comment: 24 pages, 6 figure

Divergent mathematical treatments in utility theory

In this paper I study how divergent mathematical treatments affect mathematical modelling, with a special focus on utility theory. In particular I examine recent work on the ranking of information states and the discounting of future utilities, in order to show how, by replacing the standard analytical treatment of the models involved with one based on the framework of Nonstandard Analysis, diametrically opposite results are obtained. In both cases, the choice between the standard and nonstandard treatment amounts to a selection of set-theoretical parameters that cannot be made on purely empirical grounds. The analysis of this phenomenon gives rise to a simple logical account of the relativity of impossibility theorems in economic theory, which concludes the paper

Topological reversibility and causality in feed-forward networks

Systems whose organization displays causal asymmetry constraints, from evolutionary trees to river basins or transport networks, can be often described in terms of directed paths (causal flows) on a discrete state space. Such a set of paths defines a feed-forward, acyclic network. A key problem associated with these systems involves characterizing their intrinsic degree of path reversibility: given an end node in the graph, what is the uncertainty of recovering the process backwards until the origin? Here we propose a novel concept, \textit{topological reversibility}, which rigorously weigths such uncertainty in path dependency quantified as the minimum amount of information required to successfully revert a causal path. Within the proposed framework we also analytically characterize limit cases for both topologically reversible and maximally entropic structures. The relevance of these measures within the context of evolutionary dynamics is highlighted.Comment: 9 pages, 3 figure

Proportions of the aesthetic African-Caribbean face : idealized ratios, comparison with the golden proportion and perceptions of attractiveness

Abstract Background In the absence of clear guidelines for facial aesthetic surgery, most surgeons rely on expert intuitive judgement when planning aesthetic and reconstructive surgery. One of the most famous theories regarding “ideal” facial proportions is that of the golden proportion. However, there are conflicting opinions as to whether it can be used to assess facial attractiveness. The aim of this investigation was to assess facial ratios of professional black models and to compare the ratios with the golden proportion. Methods Forty photographs of male and female professional black models were collected. Observers were asked to assign a score from 1 to 10 (1 = not very attractive, 10 = very attractive). A total of 287 responses were analysed for grading behaviour according to various demographic factors by two groups of observers. The best graded photographs were compared with the least well-graded photographs to identify any differences in their facial ratios. The models’ facial ratios were calculated and compared with the golden proportion. Results Differences in grading behaviour were observed amongst the two assessment groups. Only one out of the 12 facial ratios was not significantly different from the golden proportion. Conclusions Only one facial ratio was observed to be similar to the golden proportion in professional model facial photographs. No correlation was found between facial ratios in professional black models with the golden proportion. It is proposed that an individualistic treatment for each ratio is a rather better method to guide future practice

The Mathematical Universe

I explore physics implications of the External Reality Hypothesis (ERH) that there exists an external physical reality completely independent of us humans. I argue that with a sufficiently broad definition of mathematics, it implies the Mathematical Universe Hypothesis (MUH) that our physical world is an abstract mathematical structure. I discuss various implications of the ERH and MUH, ranging from standard physics topics like symmetries, irreducible representations, units, free parameters, randomness and initial conditions to broader issues like consciousness, parallel universes and Godel incompleteness. I hypothesize that only computable and decidable (in Godel's sense) structures exist, which alleviates the cosmological measure problem and help explain why our physical laws appear so simple. I also comment on the intimate relation between mathematical structures, computations, simulations and physical systems.Comment: Replaced to match accepted Found. Phys. version, 31 pages, 5 figs; more details at http://space.mit.edu/home/tegmark/toe.htm

Treatment of bipolar disorder: a complex treatment for a multi-faceted disorder

Background: Manic-depression or bipolar disorder (BD) is a multi-faceted illness with an inevitably complex treatment. Methods: This article summarizes the current status of our knowledge and practice of its treatment. Results: It is widely accepted that lithium is moderately useful during all phases of bipolar illness and it might possess a specific effectiveness on suicidal prevention. Both first and second generation antipsychotics are widely used and the FDA has approved olanzapine, risperidone, quetiapine, ziprasidone and aripiprazole for the treatment of acute mania. These could also be useful in the treatment of bipolar depression, but only limited data exists so far to support the use of quetiapine monotherapy or the olanzapine-fluoxetine combination. Some, but not all, anticonvulsants possess a broad spectrum of effectiveness, including mixed dysphoric and rapid-cycling forms. Lamotrigine may be effective in the treatment of depression but not mania. Antidepressant use is controversial. Guidelines suggest their cautious use in combination with an antimanic agent, because they are supposed to induce switching to mania or hypomania, mixed episodes and rapid cycling. Conclusion: The first-line psychosocial intervention in BD is psychoeducation, followed by cognitive-behavioral therapy. Other treatment options include Electroconvulsive therapy and transcranial magnetic stimulation. There is a gap between the evidence base, which comes mostly from monotherapy trials, and clinical practice, where complex treatment regimens are the rule

Towards the clinical implementation of pharmacogenetics in bipolar disorder.

BackgroundBipolar disorder (BD) is a psychiatric illness defined by pathological alterations between the mood states of mania and depression, causing disability, imposing healthcare costs and elevating the risk of suicide. Although effective treatments for BD exist, variability in outcomes leads to a large number of treatment failures, typically followed by a trial and error process of medication switches that can take years. Pharmacogenetic testing (PGT), by tailoring drug choice to an individual, may personalize and expedite treatment so as to identify more rapidly medications well suited to individual BD patients.DiscussionA number of associations have been made in BD between medication response phenotypes and specific genetic markers. However, to date clinical adoption of PGT has been limited, often citing questions that must be answered before it can be widely utilized. These include: What are the requirements of supporting evidence? How large is a clinically relevant effect? What degree of specificity and sensitivity are required? Does a given marker influence decision making and have clinical utility? In many cases, the answers to these questions remain unknown, and ultimately, the question of whether PGT is valid and useful must be determined empirically. Towards this aim, we have reviewed the literature and selected drug-genotype associations with the strongest evidence for utility in BD.SummaryBased upon these findings, we propose a preliminary panel for use in PGT, and a method by which the results of a PGT panel can be integrated for clinical interpretation. Finally, we argue that based on the sufficiency of accumulated evidence, PGT implementation studies are now warranted. We propose and discuss the design for a randomized clinical trial to test the use of PGT in the treatment of BD