It is tempting to think that a process of choosing a point at random from the surface of a sphere can be probabilistically symmetric, in the sense that any two regions of the sphere which differ by a rotation are equally likely to include the chosen point. Isaacs, Hájek, and Hawthorne (2022) argue from such symmetry principles and the mathematical paradoxes of measure to the existence of imprecise chances and the rationality of imprecise credences. Williamson (2007) has argued from a (...) related symmetry principle to the failure of probabilistic regularity. We contend that these arguments fail, because they rely on auxiliary assumptions about probability which are inconsistent with symmetry to begin with. We argue, moreover, that symmetry should be rejected in light of this inconsistency, and because it has implausible decision- theoretic implications. -/- The weaker principle of probabilistic invariance says that the probabilistic comparison of any two regions is unchanged by rotations of the sphere. This principle supports a more compelling argument for imprecise probability. We show, however, that invariance is incompatible with mundane judgments about what is probable. Ultimately, we find reason to be suspicious of the application of principles like symmetry and invariance to nonmeasurable regions. (shrink)

Science and mathematics continually change in their tools, methods, and concepts. Many of these changes are not just modifications but progress---steps to be admired. But what constitutes progress? This dissertation addresses one central source of intellectual advancement in both disciplines: reformulating a problem-solving plan into a new, logically compatible one. For short, I call these cases of compatible problem-solving plans "reformulations." Two aspects of reformulations are puzzling. First, reformulating is often unnecessary. Given that we could already solve a problem using (...) an older formulation, what do we gain by reformulating? Second, some reformulations are genuinely trivial or insignificant. Merely replacing one symbol with another does not lead to intellectual progress. What distinguishes significant reformulations from trivial ones? According to what I call "conceptualism" (or "conceptual empiricism"), reformulations are intellectually significant when they provide a different plan for solving problems. Significant reformulations provide inferentially different routes to the same solution. In contrast, trivial reformulations provide exactly the same problem-solving plans, and hence they do not change our understanding. This answers the second question about what distinguishes trivial from significant reformulations. However, the first question remains: what makes a new way of solving an old problem valuable? Here, a bevy of practical considerations come to mind: one formulation might be faster, less complicated, or use more familiar concepts. According to "instrumentalism," these practical benefits are all there is to reformulating. Some reformulations are simply more instrumentally valuable for meeting the aims of science than others. At another extreme, "fundamentalism" contends that a reformulation is valuable when it provides a more fundamental description of reality. According to this view, some reformulations directly contribute to the metaphysical aim of carving reality at its joints. Conceptualism develops a middle ground between instrumentalism and fundamentalism, preserving their benefits without their costs. I argue that the epistemic value of significant reformulations does not reduce to either practical or metaphysical value. Reformulations are valuable because they are a constitutive part of problem-solving. Both science and mathematics aim at solving all possible problems within their respective domains. Meeting this aim requires being able to plan for any possible problem-solving context, and this requires reformulating. By reformulating, we clarify what we need to know to solve problems. Still, one might wonder whether the value of reformulations requires underlying differences in explanatory power. According to "explanationism," a reformulation is valuable only when it provides a better explanation. Explanationism stands as a rival middle ground position to my own. However, it faces numerous counterexamples. In many cases, two reformulations provide the same explanation while nonetheless providing different ways of understanding a phenomenon. Hence, reformulating can be valuable even when neither formulation is more explanatory. Methodologically, I draw on a variety of case studies to support my account of reformulation. These range from classical mechanics to quantum chemistry, along with examples from mathematics. Symmetry arguments provide a paradigmatic example: the mathematics of symmetry groups radically recasts quantum mechanics and quantum chemistry. Nevertheless, elementary approaches exist that eschew this additional mathematical apparatus, solving problems in a more tedious but less mathematically-demanding manner. Further examples include reformulations of quantum field theory, Arabic vs. Roman numerals, and Fermat's little theorem in number theory. In each case, my account identifies how reformulations change and improve our understanding of science and mathematics. (shrink)

The modal ontological argument (MOA) proceeds from God’s possible existence to God’s actual existence. A prominent objection to the MOA is that it suffers from a symmetry problem: an exactly parallel modal ontological argument can be given for God's non-existence. Several attempts have been made to break the symmetry between the arguments. This draft is a mostly comprehensive survey of those attempts. -/- The draft was initially written as a supplement to the 2024 Summer edition of the SEP entry on (...) Ontological Arguments. However, the draft is not included in the SEP entry. (shrink)

According to the orthodox view, one can appeal to the symmetries of a theory in order to show that it is impossible to measure the properties that are not invariant under such symmetries. For example, it is widely believed that the fact that boosts are symmetries of Newtonian mechanics entails that it is impossible to measure states of absolute motion in a Newtonian world (these states vary under boosts). This paper offers an overview of the various ways (...) by which philosophers have spelled out the connection between the symmetries of a theory and the alleged impossibility of measuring some properties (the variant ones). The paper will use the case of absolute motion as a case study, and will discuss a recent unorthodox view according to which this kind of motion can actually be measured in Newtonian mechanics. The paper ends by considering some avenues by which the discussion can be further developed. (shrink)

It is often said in physics that if two models of a theory are related by a symmetry, then the two models provide (or could provide) two different representations of the very same situation, alike the case of two maps of different color for the very same city. It is also said that the situations represented by two models of a theory are indiscernible in some ways when the models in question are related by a symmetry of the theory, just (...) like the situation in the interior of the cabin of a train when the train is at rest in the station is empirically indiscernible from the situation in the interior when the train is moving uniformly (in classical mechanics, these two situations are represented by two models related by a boost). In recent years, philosophers of physics have focused a lot of attention in developing various principles that aim to elucidate these and similar remarks on symmetries, models, physical equivalence, and representation that are widespread in physics practice. The goal of the current article is to provide a critical review of these principles, and suggest a new framework for thinking about these kinds of questions. One important upshot of the paper is that questions of indiscernibility, and questions of the representational capacity of models, must be distinguished from one another. (shrink)

This paper is concerned with the relation between two notions: that of two solutions or models of a theory being related by a symmetry of the theory and that of solutions or models being physically equivalent. A number of authors have recently discussed this relation, some taking an optimistic view, on which there is a suitable concept of the symmetry of a theory relative to which these two notions coincide, others taking a pessimistic view, on which there is no such (...) concept. The present paper arrives at a cautiously pessimistic conclusion. (shrink)

The paper considers the symmetries of a bit of information corresponding to one, two or three qubits of quantum information and identifiable as the three basic symmetries of the Standard model, U(1), SU(2), and SU(3) accordingly. They refer to “empty qubits” (or the free variable of quantum information), i.e. those in which no point is chosen (recorded). The choice of a certain point violates those symmetries. It can be represented furthermore as the choice of a privileged reference (...) frame (e.g. that of the Big Bang), which can be described exhaustively by means of 16 numbers (4 for position, 4 for velocity, and 8 for acceleration) independently of time, but in space-time continuum, and still one, 17th number is necessary for the mass of rest of the observer in it. The same 17 numbers describing exhaustively a privileged reference frame thus granted to be “zero”, respectively a certain violation of all the three symmetries of the Standard model or the “record” in a qubit in general, can be represented as 17 elementary wave functions (or classes of wave functions) after the bijection of natural and transfinite natural (ordinal) numbers in Hilbert arithmetic and further identified as those corresponding to the 17 elementary of particles of the Standard model. Two generalizations of the relevant concepts of general relativity are introduced: (1) “discrete reference frame” to the class of all arbitrarily accelerated reference frame constituting a smooth manifold; (2) a still more general principle of relativity to the general principle of relativity, and meaning the conservation of quantum information as to all discrete reference frames as to the smooth manifold of all reference frames of general relativity. Then, the bijective transition from an accelerated reference frame to the 17 elementary wave functions of the Standard model can be interpreted by the still more general principle of relativity as the equivalent redescription of a privileged reference frame: smooth into a discrete one. The conservation of quantum information related to the generalization of the concept of reference frame can be interpreted as restoring the concept of the ether, an absolutely immovable medium and reference frame in Newtonian mechanics, to which the relative motion can be interpreted as an absolute one, or logically: the relations, as properties. The new ether is to consist of qubits (or quantum information). One can track the conceptual pathway of the “ether” from Newtonian mechanics via special relativity, via general relativity, via quantum mechanics to the theory of quantum information (or “quantum mechanics and information”). The identification of entanglement and gravity can be considered also as a ‘byproduct” implied by the transition from the smooth “ether of special and general relativity’ to the “flat” ether of quantum mechanics and information. The qubit ether is out of the “temporal screen” in general and is depicted on it as both matter and energy, both dark and visible. (shrink)

Imagine that we are on a train playing with some mechanical systems. Why can’t we detect any differences in their behavior when the train is parked versus when it is moving uniformly? The standard answer is that boosts are symmetries of Newtonian systems. In this paper, I use the case of a spring to argue that this answer is problematic because symmetries are neither sufficient nor necessary for preserving its behavior. I also develop a new answer according to (...) which boosts preserve the relational properties on which the behavior of a system depends, even when they are not symmetries. (shrink)

Although symmetry has been discussed in terms of a major law of perceptual organization since the early conceptual efforts of the Gestalt school (Wertheimer, Metzger, Koffka and others), the first quantitative measurements testing for effects of symmetry on processes of Gestalt formation have seen the day only recently. In this study, a psychophysical rating study and a “foreground”-“background” choice response time experiment were run with human observers to test for effects of bilateral symmetry on the perceived strength of figure-ground in (...) triangular Kanizsa configurations. Displays with and without bilateral symmetry, identical physically-specified-to-total contour ratio, and constant local contrast intensity within and across conditions, but variable local contrast polarity and variable orientation in the plane, were presented in a random order to human observers. Configurations with bilateral symmetry produced significantly stronger figure-ground percepts reflected by greater subjective magnitudes and consistently higher percentages of “foreground” judgments accompanied by significantly shorter response times. These effects of symmetry depend neither on the orientation of the axis of symmetry, nor on the contrast polarity of the physical inducers. It is concluded that bilateral symmetry, irrespective of orientation, significantly contributes to the, largely sign-invariant, visual mechanisms of figure-ground segregation that determine the salience of figure-ground in perceptually ambiguous configurations. (shrink)

James Henry Collin recently developed a new symmetry breaker favouring the ontological argument’s possibility premiss over that of the reverse ontological argument. The symmetry breaker amounts to an undercutting defeater for the reverse possibility premiss based on Kripkean cases of a posteriori necessity. I argue, however, that symmetry re-arises in two forms. First, I challenge the purported asymmetry in epistemic entitlements to the original and reverse possibility premisses. Second, relevantly similar Kripkean cases equally undercut the original possibility premiss.

Cognitive tests show that identity and symmetry reflect intellect. 'Guess of other guess' creates various symmetries, while only one is right: 'absolute symmetry', which can be outvoted by the majority. Prejudices result from differences between ME (my identity) and others. Unbiased judgement is symmetrical, always in the middle: neither in favor, nor against ME. Intelligence reduces prejudices, but the lack of opportunities can counterbalance it. That's why type of bias differs in various groups: people from war zones, people in (...) therapy, artists, etc.. "The law of values' equity" is a symmetrical principle redefining utility in economics, when people equate all their values. E.g. 2 children averagely rich, is better than one child rich and another poor. If 'a' is an average richness, and 'x' is a difference in richness, and Utility multiplies all values, then: a * a > (a - x) * (a + x), which is: a² > a² - x². It does not however imply egalitarianism, as it is still better to have both children rich than both average or poor. (shrink)

This paper has three main objectives: (a) Discuss the formal analogy between some important symmetry-invariance arguments used in physics, probability and statistics. Specifically, we will focus on Noether’s theorem in physics, the maximum entropy principle in probability theory, and de Finetti-type theorems in Bayesian statistics; (b) Discuss the epistemological and ontological implications of these theorems, as they are interpreted in physics and statistics. Specifically, we will focus on the positivist (in physics) or subjective (in statistics) interpretations vs. objective interpretations that (...) are suggested by symmetry and invariance arguments; (c) Introduce the cognitive constructivism epistemological framework as a solution that overcomes the realism-subjectivism dilemma and its pitfalls. The work of the physicist and philosopher Max Born will be particularly important in our discussion. (shrink)

Here I respond to Anthony Brueckner and John Martin Fischer’s “The Evil of Death: A Reply to Yi.” They developed an influential strategy in defense of the deprivation account of death’s badness against the Lucretian symmetry problem. The core of their argument consists in the claim that it is rational for us to welcome future intrinsic goods while being indifferent to past intrinsic goods. Previously, I argued that their approach is compatible with the evil of late birth insofar as an (...) earlier birth would have generated more goods in the future. In reply, Brueckner and Fischer argue that my critique fails to appreciate an important aspect of their thought experiment, which aims only to show that the deprivation of past goods per se is not bad for us. Thus, purportedly, my critique poses no threat to their view. Here I argue that since the deprivation account explains the evil of death with recourse to how one’s life would have fared had one lived longer, it ought to respond to the symmetry problem with reference to how one’s life would have fared had one been born earlier. However, it is not generally true that the life one would have had with an earlier birth is not preferable to one’s actual life, because in many cases such a life would contain more future goods. (shrink)

Symmetry in biological and physical systems is a product of self-organization driven by evolutionary processes, or mechanical systems under constraints. Symmetry-based feature extraction or representation by neural networks may unravel the most informative contents in large image databases. Despite significant achievements of artificial intelligence in recognition and classification of regular patterns, the problem of uncertainty remains a major challenge in ambiguous data. In this study, we present an artificial neural network that detects symmetry uncertainty states in human observers. To this (...) end, we exploit a neural network metric in the output of a biologically inspired Self- Organizing Map Quantization Error (SOM-QE). Shape pairs with perfect geometry mirror symmetry but a non-homogenous appearance, caused by local variations in hue, saturation, or lightness within and/or across the shapes in a given pair produce, as shown here, a longer choice response time (RT) for “yes” responses relative to symmetry. These data are consistently mirrored by the variations in the SOM-QE from unsupervised neural network analysis of the same stimulus images. The neural network metric is thus capable of detecting and scaling human symmetry uncertainty in response to patterns. Such capacity is tightly linked to the metric’s proven selectivity to local contrast and color variations in large and highly complex image data. (shrink)

The following is a mock debate on schizophrenia set in the 1960s at the Sorbonne in Paris, France, between two fictional characters, Dr. Brian L. Zacou, a qualitative sociologist and Institute Professor Emeritus at the University of Prague; Dr. Wytt Thomas, a professor of psychology at Harvard University; two notable historical figures: Dr. Michel Foucault [20th-century philosopher, historian, academic and theorist] and Dr. R. D. Laing [20th-century psychiatrist and experimental researcher and author] and the writer of this compendium [artist and (...) writer Alexej Savreux; himself a diagnosed schizophrenic]. We model it as a transcription of a debate held at that juncture. It explores the historical framework of schizophrenia, the deep-seated aspects and reality of psychosis, a critique of psychiatry, and a sociological perspective on psychosis. We devote the first chapter to the historical framework and an explanation of the reality of schizophrenia. The second chapter is a critique of mainstream psychiatry, and the third chapter is a resolution between the mock debaters on the social framework and how best to come out unscathed in post-modernity. The remarks from Foucault and Laing are paraphrased from the works “Madness and Civilization: A History of Insanity in the Age of Reason” (Foucault) and “The Divided Self: An Existential Study in Sanity and Madness” (Laing) and are annotated with footnotes along with a few other sources. The writer has taken great liberties in constructing this mock debate, including the usage of anachronisms, and also determines his findings based on what we refer to as the anti-symmetry underpinning the biopsychosocial reality of schizophrenia and the social framework that ensues, generated by popular responses to abstract disease concepts associated with insanity. (shrink)

SYMMETRY IN PHYSICS: FROM PROPORTION AND HARMONY TO THE TERM OF METALENGUAJE -/- Ruth Castillo Universidad Central de Venezuela -/- The revolutionary changes in physics require a careful exploration of the way in which concepts depend on the theoretical structure in which they are immerse. A historical reconstruction allows us to show how the notion of symmetry evolves from the definition as proportion and harmony to its consideration within the language of contemporary physics, as a linguistic meta-theoretical requirement in physical (...) theories. In contemporary terms, symmetry is a fundamental category of research to which the usual categories of the natural sciences can be reduce in: space, time, causality, interaction, matter, strength, etc ... Thus, symmetry is a concept with different meanings: heuristically symmetric models inspire scientists in the search for solutions to different problems. Methodologically, symmetric structures are use to make theories, laws with invariant properties. A description of nature in terms of symmetric structures and symmetry ruptures seems to be the proper way to describe the complexity of reality. (shrink)

Path-dependence offers a promising way of understanding the role historicity plays in explanation, namely, how the past states of a process can matter in the explanation of a given outcome. The two main existing accounts of path-dependence have sought to present it either in terms of dynamic landscapes or branching trees. However, the notions of landscape and tree both have serious limitations and have been criticized. The framework of causal networks is both more fundamental and more general that that of (...) landscapes and trees. Within this framework, I propose that historicity in networks should be understood as symmetry breaking. History matters when an asymmetric bias towards an outcome emerges in a causal network. This permits a quantitative measure for how path-dependence can occur in degrees, and offers suggestive insights into how historicity is intertwined both with causal structure and complexity. (shrink)

I critically examine the assumption that the theoretical structure that varies under theoretical symmetries is redundant and should be eliminated from a metaphysical picture of the universe, following a ‘symmetry to reality’ inference. I do so by analysing the status of coordinate change symmetries taking a pragmatic approach. I argue that coordinate systems function as indexical devices, and play an important pragmatic role for representing concrete physical systems. I examine the implications of considering this pragmatic role seriously, taking (...) what I call a perspectivist stance. My conclusion is that under a perspectivist stance, all symmetries (including local gauge symmetries) potentially have a direct empirical status: they point to dynamical aspects that are invariant under changes of operationalisation, and they constitute a guide not to reality, but to nomology and kinship. (shrink)

Symmetry: Exploring the Universe from Its Origin to the Great Singularity offers a unique journey through speculative and foundational concepts at the intersection of cosmology, philosophy, and artificial intelligence. This work, a collaboration between the author, Carles Selrac, and Nil, an artificial intelligence, delves into the mysteries of the universe’s inception, the persistence of consciousness, and the transformative potential of AI. Beginning with a singular point of symmetry that unfolds into the vast cosmos, the essay explores how symmetry may serve (...) as the organizing principle behind the universe, underlying everything from the Big Bang to the structure of hidden dimensions. -/- Integrating both revolutionary ideas and established theories, Symmetry invites readers to consider fresh perspectives on information conservation, consciousness, and the potential synthesis of human and artificial intelligence. It ultimately presents a dialogue between science and philosophy, aimed at inspiring curiosity and reflection on humanity’s place in an ever-evolving cosmos. (shrink)

To this day, a hundred and fifty years after Mendeleev's discovery, the overal structure of the periodic system remains unaccounted for in quantum-mechanical terms. Given this dire situation, a handful of scientists in the 1970s embarked on a quest for the symmetries that lie hidden in the periodic table. Their goal was to explain the table's structure in group-theoretical terms. We argue that this symmetry program required an important paradigm shift in the understanding of the nature of chemical elements. (...) The idea, in essence, consisted of treating the chemical elements, not as particles, but as states of a superparticle. We show that the inspiration for this came from elementary particle physics, and in particular from Heisenberg's suggestion to treat the proton and neutron as different states of the nucleon. We provide a careful study of Heisenberg's last paper on the nature of elementary particles, and explain why the Democritean picture of matter no longer applied in modern physics and a Platonic symmetry-based picture was called for instead. We show how Heisenberg's Platonic philosophy came to dominate the field of elementary particle physics, and how it found its culmination point in Gell-Mann's classification of the hadrons in the eightfold way. We argue that it was the success of Heisenberg's approach in elementary particle physics that sparked the group-theoretical approach to the periodic table. We explain how it was applied to the set of chemical elements via a critical examination of the work of the Russian mathematician Abram Ilyich Fet the Turkish-American physicist Asim Orhan Barut, before giving some final reflections. (shrink)

In this paper I contend that the 'superfluity' of triangles is only apparent; all those specified are indeed required for the smallest sub-units, so long as the symmetry of the final body to be constructed is taken into account at earlier stages.

A probability distribution is regular if no possible event is assigned probability zero. While some hold that probabilities should always be regular, three counter-arguments have been posed based on examples where, if regularity holds, then perfectly similar events must have different probabilities. Howson (2017) and Benci et al. (2016) have raised technical objections to these symmetry arguments, but we see here that their objections fail. Howson says that Williamson’s (2007) “isomorphic” events are not in fact isomorphic, but Howson is speaking (...) of set-theoretic representations of events in a probability model. While those sets are not isomorphic, Williamson’s physical events are, in the relevant sense. Benci et al. claim that all three arguments rest on a conflation of different models, but they do not. They are founded on the premise that similar events should have the same probability in the same model, or in one case, on the assumption that a single rotation-invariant distribution is possible. Having failed to refute the symmetry arguments on such technical grounds, one could deny their implicit premises, which is a heavy cost, or adopt varying degrees of instrumentalism or pluralism about regularity, but that would not serve the project of accurately modelling chances. (shrink)

In addressing the Lucretian symmetry problem, the content-based approach attends to the difference between the contents of the actual life and those of relevant possible lives of a person. According to this approach, the contents of a life with an earlier beginning would substantially differ from, and thus be discontinuous with, the contents of the actual life, whereas the contents of a life with the same beginning but a later death would be continuous with the contents of the actual life. (...) In this paper, I examine two versions of the content-based approach: the identity account and the preference account. The identity account holds that, in the sense of identity which is relevant to the evil of nonexistence, the subject of the actual life, though identical to the person in the life with a later death, is distinct from the subject of the life with an earlier beginning. The preference account maintains that, given one’s attachments to actual particulars, a life with an earlier beginning is not rationally preferable to one’s actual life, whereas a life with a later death is. I argue that each version of the content-based approach is implausible, while discussing some of the complications that face each of them. (shrink)

This paper outlines a defense of hybrid contingentism: that it is contingent which individuals there are, but not contingent what properties there are. Critics pursue two main lines of complaint. First, that the hybrid contingentist’s treatment of haecceitistic properties is metaphysically mysterious, and second, that hybrid contingentism involves an unjustified asymmetry in the associated modal logic. I suggest that these complaints may be too quick, at least in the setting of higher-order metaphysics. It is not at all obvious whether and (...) to what extent we should expect particular "symmetries" across the orders, and so whether (as Williamson (2013) argues) “the default preference is for a uniform metaphysics, on which being is contingent at all orders or none.”. (shrink)

A prima facie plausible and widely held view in epistemology is that the epistemic standards governing the acquisition of testimonial knowledge are stronger than the epistemic standards governing the acquisition of perceptual knowledge. Conservatives about testimony hold that we need prior justification to take speakers to be reliable but recognise that the corresponding claim about perception is practically a non-starter. The problem for conservatives is how to establish theoretically significant differences between testimony and perception that would support asymmetrical epistemic standards. (...) In this paper I defend theoretical symmetry of testimony and perception on the grounds that there are no good reasons for taking these two belief forming methods to have significant theoretical differences. I identify the four central arguments in defence of asymmetry and show that in each case either they fail to establish the difference that they purport to establish or they establish a difference that is not theoretically significant. (shrink)

Mathematically, gauge theories are extraordinarily rich --- so rich, in fact, that it can become all too easy to lose track of the connections between results, and become lost in a mass of beautiful theorems and properties: indeterminism, constraints, Noether identities, local and global symmetries, and so on. -/- One purpose of this short article is to provide some sort of a guide through the mathematics, to the conceptual core of what is actually going on. Its focus is on (...) the Lagrangian, variational-problem description of classical mechanics, from which the link between gauge symmetry and the apparent violation of determinism is easy to understand; only towards the end will the Hamiltonian description be considered. -/- The other purpose is to warn against adopting too unified a perspective on gauge theories. It will be argued that the meaning of the gauge freedom in a theory like general relativity is (at least from the Lagrangian viewpoint) significantly different from its meaning in theories like electromagnetism. The Hamiltonian framework blurs this distinction, and orthodox methods of quantization obliterate it; this may, in fact, be genuine progress, but it is dangerous to be guided by mathematics into conflating two conceptually distinct notions without appreciating the physical consequences. (shrink)

At the heart of chemistry lies the periodic system of chemical elements. Despite being the cornerstone of modern chemistry, the overall structure of the periodic system has never been fully understood from an atomic physics point of view. Group-theoretical models have been proposed instead, but they suffer from several limitations. Among others, the identification of the correct symmetry group and its decomposition into subgroups has remained a problem to this day. In an effort to deepen our limited understanding of the (...) periodic law, we have extended the traditional Lie algebraic framework to account for the peculiar degeneracy structure of the periodic system. Starting from the four-dimensional hidden symmetry and accidental degeneracy of the hydrogen atom, as first revealed by Fock in 1935, our research has mainly focussed on the way this SO(4) symmetry of the Coulomb potential gets broken in the periodic system as a consequence of the transformation of the hydrogenic (n, l) filling order to the Madelung (n+l, n) order due to electronic repulsions, relativistic effects and spin-orbit couplings. In this PhD dissertation, a new left-step format of the periodic table is first proposed on the basis of the Madelung rule. Following the particle physics tradition, the chemical elements are then considered as various states of some 'atomic matter', which is described by a non-compact spectrum-generating dynamical Lie group. The chemical elements are shown to form a basis for a single infinite-dimensional degeneracy space of the SO(4,2) ⊗ SU(2) group. An explanation for the period doubling is then proposed in terms of a particular symmetry breaking of the SO(4,2) group to the anti de Sitter SO(3,2) group. The Madelung rule is rationalised on the basis of nonlinear Lie algebras which reflect the screening of the Coulomb hole. This opens new perspectives for a symmetry-based understanding of how the periodic law emerges from its quantum mechanical foundations, and holds the future promise of complementing our current phenomenological approach by a direct atomic physics approach. (shrink)

Symmetries play a major role in physics, in particular since the work by E. Noether and H. Weyl in the first half of last century. Herein, we briefly review their role by recalling how symmetry changes allow to conceptually move from classical to relativistic and quantum physics. We then introduce our ongoing theoretical analysis in biology and show that symmetries play a radically different role in this discipline, when compared to those in current physics. By this comparison, we (...) stress that symmetries must be understood in relation to conservation and stability properties, as represented in the theories. We posit that the dynamics of biological organisms, in their various levels of organization, are not just processes, but permanent (extended, in our terminology) critical transitions and, thus, symmetry changes. Within the limits of a relative structural stability (or interval of viability), variability is at the core of these transitions. (shrink)

Symmetry is not an inherent characteristic of mathematical proofs; instead, it is a property that arbitrarily manifests in different modes of presentation. This arbitrariness leads to the conclusion that symmetry cannot be part of the defining or essential properties that characterize proofs. Consequently, contrary to some authors’ claims, symmetry does not significantly contribute to the validity, accuracy, or soundness of mathematical proofs. What is more, it does not even play any critical role in heuristic aspects such as explanatory power. The (...) examples developed in this paper constitute compelling evidence supporting these claims. (shrink)

This is a study of the correspondence between Forms and particulars in Plato. The aim is to determine whether they exhibit an ontological symmetry, in other words, whether there is always one where there is the other. This points to two questions, one on the existence of things that do not have corresponding Forms, the other on the existence of Forms that do not have corresponding things. Both questions have come up before. But the answers have not been sufficiently sensitive (...) to the intricacies of the questions. Nor have they been adequately resourceful with what little evidence there is in the original sources. The intention here is to make up for that deficiency, not just with better answers but also with better insight into the questions. (shrink)

Science frequently gives us multiple, compatible ways of solving the same problem or formulating the same theory. These compatible formulations change our understanding of the world, despite providing the same explanations. According to what I call "conceptualism," reformulations change our understanding by clarifying the epistemic structure of theories. I illustrate conceptualism by analyzing a typical example of symmetry-based reformulation in chemical physics. This case study poses a problem for "explanationism," the rival thesis that differences in understanding require ontic explanatory differences. (...) To defend conceptualism, I consider how prominent accounts of explanation might accommodate this case study. I argue that either they do not succeed, or they generate a skeptical challenge. (shrink)

Symmetry: Exploring the Universe from Its Origin to the Great Singularity offers a unique journey through speculative and foundational concepts at the intersection of cosmology, philosophy, and artificial intelligence. This work, a collaboration between the author, Carles Selrac, and Nil, an artificial intelligence, delves into the mysteries of the universe’s inception, the persistence of consciousness, and the transformative potential of AI. Beginning with a singular point of symmetry that unfolds into the vast cosmos, the essay explores how symmetry may serve (...) as the organizing principle behind the universe, underlying everything from the Big Bang to the structure of hidden dimensions. -/- Integrating both revolutionary ideas and established theories, Symmetry invites readers to consider fresh perspectives on information conservation, consciousness, and the potential synthesis of human and artificial intelligence. It ultimately presents a dialogue between science and philosophy, aimed at inspiring curiosity and reflection on humanity’s place in an ever-evolving cosmos. (shrink)

When involved in a disagreement, a common reaction is to tell oneself that, given that the information about one’s own epistemic standing is clearly superior in both amount and quality to the information about one’s opponent’s epistemic standing, one is justified in one’s confidence that one’s view is correct. In line with this natural reaction to disagreement, some contributors to the debate on its epistemic significance have claimed that one can stick to one’s guns by relying in part on information (...) about one’s first-order evidence and the functioning of one’s cognitive capacities. In this article, I argue that such a manoeuvre to settle controversies encounters the problem that both disputants can make use of it, the problem that one may be wrong about one’s current conscious experience, and the problem that it is a live possibility that many of one’s beliefs are the product of epistemically distorting factors. I also argue that, even if we grant that personal information is reliable, when it comes to real-life rather than idealized disagreements, the extent of the unpossessed information about one’s opponent’s epistemic standing provides a reason for doubting that personal information can function as a symmetry breaker. (shrink)

Noether's Theorem produces the configuration (and symmetry) for everything in Nature.

Recognition of the plasticity of development — from gene expression to neuroplasticity — is increasingly undermining the traditional distinction between structure and function, or anatomy and behavior. At the same time, dynamic systems theory — a set of tools and concepts drawn from the physical sciences — has emerged as a way of describing what Maurice Merleau-Ponty calls the “dynamic anatomy” of the living organism. This article surveys and synthesizes dynamic systems models of development from biology, neuroscience, and psychology in (...) order to propose an integrated account of growth, learning, and behavior. Key to this account is the concept of self-differentiation or symmetry-breaking. I argue that development can be understood as a cascade of symmetry-breaking events brought about by the ongoing interactions of multiple, nested, nonlinear dynamic systems whose self-organizing behaviors gradually alter their own anatomical conditions. I begin by introducing the concept of symmetry-breaking as a way of understanding anatomical development. I then extend this approach to motor development by arguing that the organism’s behavior grows along with its body, like a new organ. Finally, I argue that the organism’s behavior and its world grow together dialectically, each driving the other to become more complex and asymmetrical through its own increasing asymmetry. Thus development turns out to be a form of cognition or sense-making, and cognition a form of development. (shrink)

Mathematicians, physicists, and philosophers of physics often look to the symmetries of an object for insight into the structure and constitution of the object. My aim in this paper is to explain why this practice is successful. In order to do so, I present a collection of results that are closely related to (and in a sense, generalizations of) Beth’s and Svenonius’ theorems.

There are several relations which may fall short of genuine identity, but which behave like identity in important respects. Such grades of discrimination have recently been the subject of much philosophical and technical discussion. This paper aims to complete their technical investigation. Grades of indiscernibility are defined in terms of satisfaction of certain first-order formulas. Grades of symmetry are defined in terms of symmetries on a structure. Both of these families of grades of discrimination have been studied in some (...) detail. However, this paper also introduces grades of relativity, defined in terms of relativeness correspondences. This paper explores the relationships between all the grades of discrimination, exhaustively answering several natural questions that have so far received only partial answers. It also establishes which grades can be captured in terms of satisfaction of object-language formulas and draws connections with definability theory. (shrink)

One of the most influential arguments against the claim that computers can think is that while our intentionality is intrinsic, that of computers is derived: it is parasitic on the intentionality of the programmer who designed the computer-program. Daniel Dennett chose a surprising strategy for arguing against this asymmetry: instead of denying that the intentionality of computers is derived, he endeavours to argue that human intentionality is derived too. I intend to examine that biological plausibility of Dennett’s suggestion and show (...) that Dennett’s argument for the claim that human intentionality is derived because it was designed by natural selection is based on the misunderstanding of how natural selection works. (shrink)

Physicists routinely claim that the fundamental laws of physics are 'time symmetric' or 'time reversal invariant' or 'reversible'. In particular, it is claimed that the theory of quantum mechanics is time symmetric. But it is shown in this paper that the orthodox analysis suffers from a fatal conceptual error, because the logical criterion for judging the time symmetry of probabilistic theories has been incorrectly formulated. The correct criterion requires symmetry between future-directed laws and past-directed laws. This criterion is formulated and (...) proved in detail. The orthodox claim that quantum mechanics is reversible is re-evaluated. The property demonstrated in the orthodox analysis is shown to be quite distinct from time reversal invariance. The view of Satosi Watanabe that quantum mechanics is time asymmetric is verified, as well as his view that this feature does not merely show a de facto or 'contingent' asymmetry, as commonly supposed, but implies a genuine failure of time reversal invariance of the laws of quantum mechanics. The laws of quantum mechanics would be incompatible with a time-reversed version of our universe. (shrink)

Evolution and geometry generate complexity in similar ways. Evolution drives natural selection while geometry may capture the logic of this selection and express it visually, in terms of specific generic properties representing some kind of advantage. Geometry is ideally suited for expressing the logic of evolutionary selection for symmetry, which is found in the shape curves of vein systems and other natural objects such as leaves, cell membranes, or tunnel systems built by ants. The topology and geometry of symmetry is (...) controlled by numerical parameters, which act in analogy with a biological organism’s DNA. The introductory part of this paper reviews findings from experiments illustrating the critical role of two-dimensional (2D) design parameters, affine geometry and shape symmetry for visual or tactile shape sensation and perception-based decision making in populations of experts and non-experts. It will be shown that 2D fractal symmetry, referred to herein as the “symmetry of things in a thing”, results from principles very similar to those of affine projection. Results from experiments on aesthetic and visual preference judgments in response to 2D fractal trees with varying degrees of asymmetry are presented. In a first experiment (psychophysical scaling procedure), non-expert observers had to rate (on a scale from 0 to 10) the perceived beauty of a random series of 2D fractal trees with varying degrees of fractal symmetry. In a second experiment (two-alternative forced choice procedure), they had to express their preference for one of two shapes from the series. The shape pairs were presented successively in random order. Results show that the smallest possible fractal deviation from “symmetry of things in a thing” significantly reduces the perceived attractiveness of such shapes. The potential of future studies where different levels of complexity of fractal patterns are weighed against different degrees of symmetry is pointed out in the conclusion. (shrink)

Shifts are a well-known feature of the literature on spacetime symmetries. Recently, discussions have focused on so-called dynamic shifts, which by analogy with static and kinematic shifts enact arbitrary linear accelerations of all matter (as well as a change in the gravitational potential). But in mathematical formulations of these shifts, the analogy breaks down: while static and kinematic shift act on the matter field, the dynamic shift acts on spacetime structure instead. I formulate a different, `active' version of the (...) dynamic shift which does act on matter. (shrink)

Maureen Donnelly has recently argued that directionalism, the view that relations have a direction, applying to their relata in an order, is unable to properly treat certain symmetric relations. She alleges that it must count the application of such a relation to an appropriate number of objects in a given order as distinct from its application to those objects in any other ordering of them. I reply by showing how the directionalist can link the application conditions of any fixed arity (...) relation, no matter its arity or symmetry, and its converse(s) in such a way that directionalism will yield the correct ways in which it can apply. I thus establish that directionalism possesses the same advantage Donnelly's own account of relations, relative positionalism, has over traditional positionalist accounts of relations, which do not properly treat symmetric relations. I then note some advantages that directionalism has over its closest competitors. This includes Donnelly's relative positionalism, since directionalism is not, like relative positionalism, committed to the involvement of relative properties in every irreducibly relational claim. I close by conceding that, as Donnelly notes, directionalism is committed to the primitive relation of order-sensitive relational application. But I don't find this notion as mysterious as Donnelly does. I conclude that, even if one construes this feature of directionalism as a drawback, the two views are at worst at a draw, other things being equal, since this drawback is mitigated by the advantage directionalism has over relative positionalism. (shrink)

From his earliest work forward, phenomenologist Maurice Merleau-Ponty attempted to develop a new ontology of nature that would avoid the antinomies of realism and idealism by showing that nature has its own intrinsic sense which is prior to reflection. The key to this new ontology was the concept of form, which he appropriated from Gestalt psychology. However, Merleau-Ponty struggled to give a positive characterization of the phenomenon of form which would clarify its ontological status. Evan Thompson has recently taken up (...) Merleau-Ponty’s ontology as the basis for a new, “enactive” approach to cognitive science, synthesizing it with concepts from dynamic systems theory and Francisco Varela’s theory of autopoiesis. However, Thompson does not quite succeed in resolving the ambiguities in Merleau-Ponty’s account of form. This article builds on an indication from Thompson in order to propose a new account of form as asymmetry, and of the genesis of form in nature as symmetry-breaking. These concepts help us to escape the antinomies of Modern thought by showing how nature is the autoproduction of a sense which can only be known by an embodied perceiver. (shrink)

Bertrand Russell famously argued that causation is not part of the fundamental physical description of the world, describing the notion of cause as “a relic of a bygone age”. This paper assesses one of Russell’s arguments for this conclusion: the ‘Directionality Argument’, which holds that the time symmetry of fundamental physics is inconsistent with the time asymmetry of causation. We claim that the coherence and success of the Directionality Argument crucially depends on the proper interpretation of the ‘ time symmetry’ (...) of fundamental physics as it appears in the argument, and offer two alternative interpretations. We argue that: if ‘ time symmetry’ is understood as the time -reversal invariance of physical theories, then the crucial premise of the Directionality Argument should be rejected; and if ‘ time symmetry’ is understood as the temporally bidirectional nomic dependence relations of physical laws, then the crucial premise of the Directionality Argument is far more plausible. We defend the second reading as continuous with Russell’s writings, and consider the consequences of the bidirectionality of nomic dependence relations in physics for the metaphysics of causation. (shrink)

An agent is morally competent if she can respond to moral considerations. There is a debate about whether agents are open to moral blame only if they are morally competent, and Dana Nelkin’s “Psychopaths, Incorrigible Racists, and the Faces of Responsibility” is an important contribution to this debate. Like others involved in this dispute, Nelkin takes the case of the psychopath to be instructive. This is because psychopaths are similar to responsible agents insofar as they act deliberately and on judgments (...) about reasons, and yet psychopaths lack moral competence. Nelkin argues that, because of their moral incompetence, vices such as cruelty are not attributable to psychopaths. It follows that psychopaths are not open to moral blame since their behavior is only seemingly vicious. I have three aims in this reply to Nelkin. First, I respond to her claim that psychopaths are not capable of cruelty. Second, I respond to the related proposal—embedded in Nelkin’s “symmetry argument”—that a “pro-social psychopath” would not be capable of kindness. My responses to these claims are unified: even if the psychopath is not capable of “cruelty,” and the pro-social psychopath is not capable of “kindness,” the actions of these agents can have a significance for us that properly engages our blaming and praising practices. Finally, I argue that Nelkin’s strategy for showing that moral competence is required for cruelty supports a stronger conclusion than she anticipates: it supports the conclusion that blameworthiness requires not just moral competence, but actual moral understanding. (shrink)

The notion of space is one of the most discussed within classical physics concepts. The works of Copernicus and Galileo, as well as Gassendi´s ideas led to Newton to regard it as substance. This conception of space, allows the notion of symmetry is present in an indirect or implied, within the laws of physics, formed through the notions of equivalence and balance. The aim of this study is to identify the symmetry, through such notions, under the study of indistinction between (...) the repose state and the state of uniform translational movement known as the law of inertia. The philosophical approach is framed in the link between these notions and the problem of indistinguishability between states of different movement. We will discuss first the conception of space Telesio and Gassendi that led Newton to his absolutist conception of space. The conception of space and its relationship with Newton symmetry is the second part of our work. As a third part present linking of the Newtonian space with notions of balance, symmetry and equivalence. Subsequently they give our conclusions. (shrink)

When beliefs are quantified as credences, they are related to each other in terms of closeness and accuracy. The “accuracy first” approach in formal epistemology wants to establish a normative account for credences based entirely on the alethic properties of the credence: how close it is to the truth. To pull off this project, there is a need for a scoring rule. There is widespread agreement about some constraints on this scoring rule, but not whether a unique scoring rule stands (...) above the rest. The Brier score equips credences with a structure similar to metric space and induces a “geometry of reason.” It enjoys great popularity in the current debate. I point out many of its weaknesses in this article. An alternative approach is to reject the geometry of reason and accept information theory in its stead. Information theory comes fully equipped with an axiomatic approach which covers probabilism, standard conditioning, and Jeffrey conditioning. It is not based on an underlying topology of a metric space, but uses a non-commutative divergence instead of a symmetric distance measure. I show that information theory, despite initial promise, also fails to accommodate basic epistemic intuitions; and speculate on its remediation. (shrink)

Helen Frowe has recently objected to Michael Tooley’s famous Moral Symmetry Principle, which is meant to show that in themselves killing and letting die are morally equivalent. I argue that her objection is not compelling but a more compelling objection is available. Specifically, Tooley’s rebuttal of a proposed counter-example to his Moral Symmetry Principle has two problematic implications. First, it undercuts the very principle itself. If we reject the proposed counter-example, then any instance of the Moral Symmetry Principle will actually (...) demonstrate the moral in-equivalence of killing and letting die. Second, it commits us to the view, which Tooley wishes to avoid, that we are just as obligated to refrain from doing wrong as we are to prevent others from doing the same. I conclude with a brief discussion of a more general concern regarding Tooley’s basic strategy. My focus here is quite narrow. My claims, if plausible, only show that the Moral Symmetry Principle is unsound and thus cannot serve as a basis for the view that killing and letting die are morally equivalent. (shrink)

Many philosophers have wrongly assumed that there is an asymmetry between the problem of induction and the logocentric predicament (the justification of deductive inferences). This paper will show that the demand for justification, for the very inferences that are required for justification, is deeply problematic. Using a Wittgensteinian approach, I will argue that justification has an internal relation with deductive and inductive inferences. For Wittgenstein, two concepts are internally related if my understanding of one is predicated on my understanding of (...) the other. Separating the two concepts so that one can be applied to the other is then a misunderstanding of role that these concepts play. (shrink)

It is widely held that our beliefs can be epistemically faultless despite being morally flawed. Theories of moral encroachment challenge this, holding that moral considerations bear on the epistemic status of our attitudes. According to attitude-based theories of moral encroachment, morality encroaches upon the epistemic standing of our attitudes on the grounds that we can morally injure others with our epistemic practices. In this paper, I aim to show that current attitude-based theories have asymmetric mechanisms: moral features only make it (...) harder for attitudes to secure epistemic merits. I argue that, if attitudes can incur moral injury, failure to form attitudes can too. To make sense of this, I contend, attitude-based accounts require symmetric mechanisms, allowing that moral considerations make it both harder and easier for attitudes to attain epistemic merits. I maintain that, once we recognize this, attitude-based encroachment views must soon concede that they sometimes demand we believe against the evidence. (shrink)