Philosophy

– 4 – Journal of Siberian Federal University. Humanities & Social Sciences 1 (2014 7) 4-19 ~ ~ ~ УДК 113:530.145 Quantum Measure from a Philosophical Viewpoint Vasil Penchev* Bulgarian Academy of Sciences: Institute for the Study of Societies and Knowledge 31 Neofit Rilski, 1000 Sofia, Bulgaria Received 06.11.2013, received in revised form 29.11.2013, accepted 26.12.2013 The paper discusses the philosophical conclusions, which the interrelation between quantum mechanics and general relativity implies by quantum measure. Quantum measure is three-dimensional, both universal as the Borel measure and complete as the Lebesgue one. Its unit is a quantum bit (qubit) and can be considered as a generalization of the unit of classical information, a bit. It allows quantum mechanics to be interpreted in terms of quantum information, and all physical processes to be seen as informational in a generalized sense. This implies a fundamental connection between the physical and material, on the one hand, and the mathematical and ideal, on the other hand. Quantum measure unifies them by a common and joint informational unit. Quantum mechanics and general relativity can be understood correspondingly as the holistic and temporal aspect of one and the same, the state of a quantum system, e.g. that of the universe as a whole. Keywords: measurement, quantum mechanics, general relativity, quantum information, entanglement. © Siberian Federal University. All rights reserved * Corresponding author E-mail address: firstname.lastname@example.com The main question is how nothing (pure probability) can turn into something (physical quantity). The best idea is that both have a common measure. This problem is resolved implicitly in quantum mechanics introducing the formalism of wave functions, which are points in Hilbert space. Its approach can be equivalently represented explicitly in terms of quantum measure, i.e. by the notion of "qubit" defined and utilized in the theory of quantum information. The paper addresses the corresponding philosophical interpretation focused on the interrelation of quantum mechanics and general relativity, and thus on the problem of quantum gravity from a methodological viewpoint. 1. The Lebesgue (LM) and Borel (BM) measure They will be discussed in relation to the axiom of choice ("AC", "NAC" = no axiom of choice) and the continuum hypothesis ("CH", "NCH" = no continuum hypothesis): LM and BM coincide (AC, CH) Carathéodory's extension theorem as to Borel sets (BS) according to Carathéodory's extension – 5 – Vasil Penchev. Quantum Measure from a Philosophical Viewpoint theorem1, and either can be distinguished only unconstructively (AC; CH or NCH) as to nonBorel sets (NBS) or cannot be juxtaposed at all (NAC; NCH). Then one can ascribe whatever difference including no difference. That incomparability is a typical situation in quantum mechanics and it represents the proper content of "complementarity". Here the Lebesgue measure is implied to be one-dimensional since the real line is such. Dimensionality, LM, and BM: One must distinguish the dimensionality of the space being measured from the dimensionality of measure, by which the space is measured. The idea of probability as well as that of number in general is to be introduced a universal measure (quantity), by which all (pears, apples, distances, volumes and all the rest) can be measured as separately, item (quality) by item, as together. BM uses n-dimensional spheres, which it compares in radius independently of the number of dimensions. That radius is the Borel measure, and if it is finite, admits the Kolmogorov probability. One can suppose, though counterintuitively, the case, where the dimensionality of the space being measured is lesser than that of the measure, and that such a case may have a nonempty intersection with NCH. The conjecture would not make much sense while one does not point out a universal measure of the dimensionality greater than one. 2. Quantum measure (QM) and its construction QM is a three-dimensional universal one. A motivation may be for it to be introduced an as complete (like LM) as universal (like BM) measure. It should resolve the problem for completing BM in general (both AC & NAC): An alternative, but equivalent approach is to be measured empty intervals (without any points in them), i.e. discrete or quantum leaps, in the same way as complete intervals of continuum. In fact quantum mechanics is what forced the rising of such a measure (& probability). Moreover, quantum measure is more complete than LM in a sense or even is the most complete measure known to mankind since it can measure not only infinitely small empty, but any finite and even infinite leaps. However it postpones the question to complete them as no need to do it initially, on the one hand, and the general AC & NAC invariance even requires for the complete and incomplete case to be equated therefore rejecting the need of completion, on the other. That rather strange state of affairs is discussed in details below. Given BM, the construction of QM is the following: The objective is to be measured all the NBS as being reduced into some combination of the following three types partially complete: NBS complete in relative complement; NBS complete in countable union; NBS complete in countable intersection. A partial measure (or a partial probability as a finite measure) corresponds in each of the three cases above. If a NBS is incomplete in one or more, or even in all of the three relation above, its corresponding measure(s) [probability (-es)] is (are) accepted as zero. BM is the particular case where the three measures (probabilities) coincide. If a NBS is incomplete in any relation, it has a zero BM anyway. That backdoor is substantive for reconciling quantum theory based on QM and general relativity grounded on LM or BM in fact. That kind of construction will be called tricolor hereinafter. The tricolor has exact correspondences in set theory and logic. Let us now consider as an example of the case of tricolor or quantum probability compared – 6 – Vasil Penchev. Quantum Measure from a Philosophical Viewpoint with the classical one. One substitutes the unit ball for the interval of [0, 1]: The unit ball can be decomposed in a "spin" way into two orthogonal circles. The point of the unit ball generalizes that of [0, 1]. The point of the unit ball can be represented equivalently both as the two correlating complex numbers (the two projections on the orthogonal circles) and as three independent numbers (those of the tricolor above). As the interval of [0, 1] allows of introducing the unit of classical information, a bit, as the unit ball does the same for quantum information, a qubit: Since a bit can be thought as the alternative choice between two points: 0 or 1, a qubit might be thought as the choice of a point of the ball, i.e. as a choice among a continuum of alternatives in final analysis. The [0, 1] is the universal measuring unit of all what can be classically measured. It can be illustrated as a "tape measure" for anything which is something, but not nothing. However the unit ball is a more universal measuring unit since it can measure as anything which is something as nothing in a uniform way. In other words, it can measure as the continuous as the discrete without completing the latter with the continuum of a continuous medium of points, i.e. without transforming nothing in something. Consequently, the unit ball is the perfect measure for quantum mechanics since aids it in resolving its main question, namely: How can nothing (pure probability) become something (physical quantity)? Many philosophers reckon that the same kind of question, why there is something rather than nothing, is the beginning of philosophy, too. Quantum mechanics gives an answer, which is the single one that mankind has managed to reach and which, fortunately or unfortunately, is constructive besides. Given LM, the construction of QM is the following: One builds a tricolor measure as the BM for any dimension. One might consider a "vector" measure, which components are 3D balls. In fact, it is equivalent both to Minkowski and to Hilbert space. That unit-ball vector represents a unit covariant vector, i.e. just a measure. Any measure of the ball vector would be QM on LM. If the measure is the usual one of the vector length, the measured result would be a 3D ball rather than a 1D length. The axiom of choice does not use in that QM-on-LM construction. Using the axiom of choice, a ball is equivalent to any set of balls, which is known as the Banach and Tarski paradox2. So, one need not construct a ball-vector measure as above since it is directly equal to a ball (i.e. QM) according to the axiom of choice. The last two paragraphs show the original invariance of QM to the axiom of choice unlike LM and BM. As to BM that invariance is an undecidable statement. One might say that BM even possesses anyway a specific invariance or universality to the axiom of choice: the invariance of incompleteness: BM is incomplete as with the axiom of choice as without it. As to LM, it is complete without AC, but incomplete with AC: Indeed the construction of a Vitali set3, which is immeasurable by LM, requires necessarily AC. At the same time, the way of its construction shows that any Vitali set is a subset of a null set such as that of all the rational numbers within the interval [0, 1] since there is a one-to-one constructing mapping between the Vitali set and that set of the rational numbers. Consequently LM under the condition of AC is incomplete since there is a subset of a null set, which is immeasurable: the Vitali set. The consideration shows that LM occupies an intermediate position between the complete – 7 – Vasil Penchev. Quantum Measure from a Philosophical Viewpoint QM and the incomplete BM being partly complete (without AC) and partly incomplete (with AC). Thus LM can also demonstrate AC as the boundary between potential and actually infinity. LM under condition of AC can measure anything which is finite, but nothing which is infinite. QM unlike it can measure both even under AC. Thus the invariance of QM to the axiom of choice can be added to the motivation of QM since quantum mechanics needs such invariance: Really, quantum measuring requires the axiom of choice, and any quantum state by itself rejects it being due the "no hidden variables" theorems4,5. Consequently, the epistemological "equation", which equates any state "by itself" and the result of its measuring, needs that invariance in the case of quantum mechanics. A problem remains to be solved (as if): Is there a BM or LM, with which no QM corresponds after utilizing the aforesaid procedure? The finite or infinite discrete leaps are described by QM unlike LM and BM: Consequently QM can be accepted as more general. However, are there cases, too, which admit BM or LM, but not QM? Unfortunately that question is not one of abstract, purely mathematical interest since it is an interpretation of the quantum-gravity problem into the measure-theory language. General relativity uses LM, while quantum mechanics QM. If general relativity is true (as seems) and there is a LM (BM) which is not QM (LM-no-QM), then quantum gravity is an undecidable problem. Vice versa: Quantum gravity is resolvable if and only if QM is more general than (since it cannot be equivalent with) LM (BM). A try for a short answer might be as follows: The QM-on-LM construction excludes the LM-no-QM conjecture. However it cannot serve for refusing a nonconstructive proof of LM-noQM existence in general. Any pure proof of that kind, which requires necessarily the axiom of choice, can be neglected because of the QM invariance to AC/ NAC. No other proof of pure LM-no-QM existence can be omitted, but whether there are such ones, no one knows. That pure existence is not only a question of abstract and theoretical interest. It suggests that a more general measure than QM can be ever found on the base of LM-no-QM. One can suppose a new invariance to CH/ NCN similar to the QM invariance to AC/ NAC. In fact, it would be equivalent to the existence of a countable model for any mathematical structure of first order: This is a well-known direct corollary of the Löwenheim-Skolem theorem6. Thus that alleged as a new invariance would not expand out of QM, though. The reason is that CH implies AC. However one can continue the implication of AC from CH in the following way: AC implies Skolem's "paradox"7: The latter implies the impossibility to be compared infinite powers and that CH/ NCH is undecidable for the sake of that. That is: CH implies the undecidability of CH/ NCH, but NCH does not imply that undecidability since cannot imply AC. All this is another argument in favor of QM and against LM-no-QM. Anyway "QM & an undecidability of QM/ LM" satisfies almost all combinations of AC, CH, and their negations. Moreover it does not require LM-no-QM since LM and QM are complementary to each other where both AC & CH hold. As to the problem of "quantum gravity", this means the following: Quantum gravity as supposing QM is consistent as with NCH and the AC/ NAC invariance as with CH & AC. However it is not consistent with CH & NAC, in the domain of which general relativity is built, unfortunately. What about LM-no-QM in "CN & NAC"? Of course, one can construct QM on any LM – 8 – Vasil Penchev. Quantum Measure from a Philosophical Viewpoint there, too. That construction implies AC, and since NAC is valid there, the construction is forbidden, though. This is a very amazing state of affairs resembling the human rather than nature laws: QM is possible, but forbidden where general relativity is valid. After daring to construct QM in its territory, anyone turns out to be expulsed automatically in CH & AC where QM is admitted since it is complementary to LM and does not force LM to vanish. What implies all that? Quantum gravity is a question of choice. One can create the theory as of quantum gravity as of general relativity, however ought to choose preliminary which of them. They should be equivalent to each other in a sense and can be thought as one and the same. Consequently general relativity can be reckoned as the cherished quantum gravity. That is the case though it is very strange, even ridiculous. If and only if another and more general than QM measure be discovered so that the LM-no-QM be built constructively, then and only then general relativity and quantum gravity will be able to be distinguished effectively, i.e. experimentally. Vice versa: if an experimental refutation of general relativity be observed, a generalization of QM (GQM) will be implied: RIP both for Albert Einstein and for Niels Bohr since general relativity (LM) and quantum mechanics (QM) can be universal only together and reconciled. GQM will be able to resolve the dispute between them or will remove both when it comes. However we have not got any idea about GQM. Finally, the example of BS can be used to illustrate how the strange kind of as if undecidability of CH to AC, and hence the relation of general relativity and quantum mechanics in terms of measure: BS implies CH according to the Alexandroff – Hausdorff theorem8: Any uncountable BS has a perfect subset (and any perfect set has the power of continuum). However, CH implies AC in turn, and the latter does Skolem's "paradox", i.e. the incomparability (or more exactly, unorderability) of any two infinite powers. Consequently, BS can be consistent as with CH as with NCH since BS and CH are complementary in a sense. If the case is NCH, then AC is not implied and BS remains consistent as with CH as with NCH. Of course, this should be so as BM is a particular case of QM, and the latter is consistent with NCH (as well as CH & AC). All illustrate how it is possible for BS and BM to be consistent as with LM as with QM even where CH & NAC hold. That is the domain of general relativity, which should not exist if CH implies AC. Really CH implies AC only that AC implies the undecidabilty of CH or NCH, which allows of existing the area of general relativity. One can abstract the logical relation of general relativity and quantum mechanics by means of the same one of LM and QM. Roughly speaking, they are complementary because of a similar complementarity of CH/ NCH and AC/ NAC rooted in the amazing or even paradoxical properties of infinity: AC supposes a single infinity which ought to be countable. However, both CH and NCH suggest an infinite set of sets which can be countable (CH) in turn, too. That unordinary logical relation does not generate any contradictions. In fact, it contravenes only the prejudices. Anyway, we can attempt to explain and elucidate the reason of our confusion and misunderstanding: Anything in our experience can be either an indivisible whole (a much) or divided in parts (a many): No "much" can be a "many" in the same moment and vice versa. The above postulate is not valid as to infinity: It can be defined as that "much" which is a "many" or as that "many" which is "much". – 9 – Vasil Penchev. Quantum Measure from a Philosophical Viewpoint Consequently it can be equally seen as a single "much" consisting of a "many" of parts (after AC) or as a many of indivisible wholes ("much"-s) (after CH / NCH). To reconcile the two viewpoints onto infinity in a single illustration, one can utilize the image of cyclicality. While anything else consists of something else and is not self-referential or cyclic, infinity is what consists just of it self-referentially or cyclically: Its "much" is forced to return back into it as many units. AC suggests that cycle while CH or NCH unfold this cycle in a line. Consequently AC sees infinity as a well-ordering (line) bounded as a cycle while CH (or NCH) as many cycles wellordered in a line. Any contradictions between them do not arise since both are the same seen from opposite perspectives. 3. QM compared with LM and with BM QM can be compared as with BM as with the LM to stand out its essence and features: QM vs. BM: Similarities: – Both are supposed to be universal. – Both generate probabilities where they are bounded. – Both can be generated by BS. – There is a common viewpoint, according to which QM can be considered as a threedimensional or "tricolor" generalization of BM. Differences: – QM is complete, BM is not. – QM is three-dimensional, BM is onedimensional. – BM can be considered as the particular case where the three dimensions of QM coincides. QM vs. LM: Similarities: – Both are complete under NAC. – QM and LM correspond to each other "two-to-two", i.e. "± to ±" or in other symbols "square-to-square". – No one of QM and LM can be deduced from the other or represented as a particular case of the other. The differences from each other (see above) focus on a common 3D space where they vanish. One can utilize the metaphor of the two eyes or binocular sight for QM and LM. Differences: – QM is three-dimensional, while LM is of an arbitrary even infinite dimensionality. – The dimensionality of QM does not correspond to that of the space measured, in general. They can be interpreted differently even in the particular case, where they coincide (three dimensions). The dimensionality of LM always coincides with that. – QM is universal: It does not depend on the dimensionality of the space measured. LM is not universal: It does strictly correspond to the dimensionality of the space measured. If one uses the metaphor of binocular sight for QM and LM, then their "global focus" is always in the "plane" of QM, while LM can represent the "local development or change" dimension by dimension. 4. The origin of QM QM arose for quantum mechanics when Heisenberg's (1925) matrix mechanics9 and Schrödinger's (1926a)10 wave mechanics were united by the latter one11. Though Hilbert space guaranteed as a mathematical enough formalism, as John von Neumann showed12, for the quantum mechanics, the sense of that guarantee as well as its attitude toward the two initial components, matrix and wave mechanics accordingly, remained misunderstood: – 10 – Vasil Penchev. Quantum Measure from a Philosophical Viewpoint Heisenberg's matrix mechanics represented all quantum motions only as discrete rather than continuous or smooth. Schrödinger's wave mechanics represented all quantum motions only as smooth rather than discrete, though. Consequently the sense of quantum mechanics, which unites both by means of Hilbert space, is that, in fact, all quantum motions are invariant to the transition between the discrete and smooth. However wave mechanics had advantage that it could represent that invariance in terms of the continuous and smooth, which terms were dominating for classical mechanics, though they were only prejudices, a legacy of the past, needless or even harmful: The determent consisted in that the invariance of the discrete and smooth as to quantum motion remained tightly hidden in the mathematical apparatus of Hilbert space and accordingly misunderstood in physical interpretation. The real sense of QM is to suggest a common measure both for the discrete and for the continuous and smooth so that to offer a suitable language for their invariance required by quantum mechanics. The case of a (discrete) quantum leap measured by QM: Any quantum leap can be decomposed in harmonics by Fourier transform: Then any of those harmonics can be enumerated and considered as a QM for the nth dimension of Hilbert space. The nth dimension of Hilbert space can be interpreted as a frequency or consequently, as an energy corresponding one-to-one to it. The above construction shows the transition from real to complex Hilbert space and the transition from LM to QM as well. By the way the universality of QM is similar to that of complex numbers. The case of a continuous or smooth physical motion measured by QM: Since the continuous or smooth physical movement means a motion in Euclidean space, which is the usual three-dimensional one, it can be decomposed in successive 3D spheres or balls corresponding one-to-one as to all points of the trajectory in time as to the successive spheres or balls of the light cone in Minkowski space as well as to the successive dimensions of Hilbert space. Consequently those points of the trajectory can be enumerated and considered as a QM for the nth dimension of Hilbert space in an analogical way. Now the nth dimension of Hilbert space can be interpreted as a moment of time corresponding one-to-one to it. The two above constructions show why QM is universal as well as the sense of that universality. Since frequency (energy) and time are reciprocal (or complementary in terms of quantum mechanics), then they can be juxtaposed as the two dual spaces of Hilbert space connected and mapped one-to-one by Fourier transform. Max Born's probabilistic mechanics: Max Born suggested in 192613 that the square of the modulus of wave function represents a probability, namely that of the state corresponding to that wave function. However somehow it was called the "statistical interpretation" of quantum mechanics. The term of "interpretation" used by Max Born himself as an expression of scientific modesty and politeness should not mislead. Its utilizing shows a complete misunderstanding of Max Born's conjecture and a yearning for its understatement. In fact it was not and is not an interpretation, but another, third form of quantum mechanics among and with matrix and wave mechanics. This is the cause for one to call – 11 – Vasil Penchev. Quantum Measure from a Philosophical Viewpoint it probabilistic mechanics (after the expressions of "wave mechanics" and "matrix mechanics" are common) rather than an interpretation. Probabilistic mechanics shares Hilbert space with matrix and wave mechanics. However wave function (i.e. a point in Hilbert space) does not mean here a quantum leap decomposed in energies, neither a trajectory decomposed in time moments, but the characteristic function of a complex random quantity (or of two conjugate real quantity). One should say a few words on the Fourier transform of a complex random quantity and on its characteristic function: In fact their interrelation is quite symmetric and simple: The Fourier transform and the replacing of a complex random quantity by its conjugate swap the two dual Hilbert space. Consequently the characteristic function of the conjugate of the complex random quantity is just the complex random quantity itself. The interpretation of a complex random quantity and its conjugate is simple, too: Since a complex random quantity can be interpreted as two real conjugate (reciprocal) physical quantities such as e.g. time and frequency (energy), then the conjugate of the same random quantity must represent merely swapping between the corresponding physical quantities or the axes of the complex plane, or its rotation of π/2. Probabilistic vs. matrix mechanics: If one compares them, the differences would be only two: in interpretation and in choice between NAC and AC. However the wave function in both cases and despite the differences would be the same and the same point in Hilbert space. That sameness inspires invariance as to probabilistic vs. matrix "interpretation" as to NAC vs. AC. Since the wave function is a sum of the measured by QM, one can reduce completely that invariance in terms of QM: QM as quantum probability guarantees the former members, and it decomposed in dimensions (which are harmonics or energetic levels in the case) supplies the latter ones. A philosopher would emphasize the extraordinary universality both of Hilbert space and of QM contradicting common sense: Why and where exactly? QM is so universal that can measure both the unordered (and even unorderable in principle) and the well-ordered and thereof ordering it (them): In our case it can measure and order quantum probabilities (for the unorderable in principle) and quantum leaps (for the well-ordered in harmonics or energies), and therefore QM establishes a oneto-one mapping between quantum probabilities and quantum leaps: That one-to-one mapping is too shocking to the prejudices. It shows that a level of energy corresponds exactly to a quantum probability: That is a physical quantity (what is the former) can be equated with a real number being without any physical dimensionality (what is the latter): However this is what has been necessary for the objectives declared in the beginning of the paper: to demonstrate how QM allows of becoming nothing to something or vice versa and eo ipso creatio ex nihilo or reductio ad nihilum (i.e. true creation or true annihilation). Probabilistic vs. wave mechanics: All what has been said above about the links between probabilistic and matrix mechanics can be almost literally repeated again in that case. The immaterial differences are as follows: – The dual Hilbert space replaces its dual counterpart. – The well-ordering in time replaces that in frequency (energy). The one-to-one mapping based on QM establishes now a correspondence of a wave function as quantum probability with a continuous or smooth trajectory in time. – 12 – Vasil Penchev. Quantum Measure from a Philosophical Viewpoint A threefold (even fourfold) one-to-one mapping arises thereof: It states invariance or equivalence in a sense between the quantum leaps (for the discrete), the smooth trajectories in time (for the continuous) and the quantum probabilities (for the unorderable in principle). That threefold mapping shows how pure numbers even only the positive integers (for "nothing") can generate physical quantities in pairs of ("reciprocal") conjugates such as frequency (energy) and time. The stages of that generation are as follows: – Nothing. – The positive integers are given somehow, maybe by God as Leopold Kronecker reckoned: "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk"14: – Creation: Qubits (or QM) replaces each of the positive integers generating Hilbert space. – The Hilbert space generates that threefold mapping between quantum probability, energy and time and thereof the physical world arises already, too. Quantum mechanics seen as the unification of all three kinds of mechanics: probabilistic & matrix & wave mechanics: Quantum mechanics is better to be understood as the unification of all the three types of mechanics listed above instead only of the last two. The sense of that unification is the extraordinary invariance (or equivalence in a sense) of the discrete, continuous (smooth) and the probabilistic in the common form of quantum motion: Quantum motion can be already thought as a relation between two or more states despite whether each of them is considered as a discrete, continuous (smooth) or probabilistic one since it is always represented by one the same wave function in all the three cases: This calls for far-reaching philosophical conclusions, though: The difference not only between the discrete and continuous (smooth), but also that between both and the probabilistic is only seeming and accidental or even anthropomorphic in a sense. Quantum motion breaks down their barriers and allows any transitions between them. The probabilistic can be located "between" the discrete and continuous (smooth) and can be considered as something like a substance of that kind of transition. Accordingly the discrete and continuous can be supposed as the two extreme or particular cases of the probabilistic, which are opposite to each other, and that is not all: What is the physically existing according to common sense can be linked only to those two extremes. Physical reality ostensibly consists just of (and in) both since they are all the actual. According to the same common sense the probabilistic cannot be physically real since is not actual: However quantum mechanics shows that is the case, the probabilistic is physically real: "So much the worse for quantum mechanics because this means that it is incorrect or at least incomplete", declared the common sense then. Quantum mechanics rather than that "common sense" turns out to be the right again, though, experimentally verified15. If quantum mechanics is the right, what does it mean about the philosophical interrelation between reality and "virtuality"? "Virtuality" is a term coined here to denote just that new class required by quantum mechanics and involving both reality, i.e. the discrete and continuous (smooth), and "only" (ostensibly) the possible so that to allow of any transfer between them. Consequently virtuality is a term for the new constitution of being, according to which the barriers between the actual and the possible – 13 – Vasil Penchev. Quantum Measure from a Philosophical Viewpoint are broke down and all the kind of transitions between them are unrestrained. Thus virtuiality established by quantum mechanics can resolve our properly philosophical (and even theological) problem about creatio ex nihilo or reductio ad nihilum: The area of probability can describe very well both those creatio and reductio as states and processes: One can see the actual in creating or annihilating rigorously mathematically, i.e. in the process of creation or annihilation, as the change of probability. Not less striking is that the new "constitution" of virtuality suggests for mathematics to be more general than physics if the latter is defined and restricted only to the actual; or in other words, mathematics and a new and more general physics can and even should coincide. How to interpret the fermion and boson kind of spin statistics in the light of that unification? According to the so-called spin-statistics theorem16 all the quantum particles can be divided into two disjunctive classes after the second quantization: fermions and bosons: Since the second quantization maps the wave functions of the quantum particles "twoto-two", it admits two kinds of solving as to a swap of the space-time positions of two quantum particles: symmetric (++, −−) and antisymmetric (+−, −+). The bosons are supposed to be those of symmetric swap, and the fermions are those of the antisymmetric swap. Turns out the any number of bosons can share one the same state and wave function while if they are fermions, only two. The following can be easily spotted: Quantum probabilistic mechanics explains very well that property as to the bosons, and quantum matrix-wave mechanics explains it not less successfully as to the fermions: Indeed the opportunity of sharing a common state or wave function by the bosons is due to the sharing of a common probability by an arbitrary ensemble of quantum particles. That ensemble, which possibly consists of an infinite number of elements, is supposed not to be well-ordered. The same ensemble already well-ordered can be distinguished in two kinds of wellordering corresponding to the two fermions admitted in one the same state or wave function. The one is well-ordered to, and the other from infinity. If the ordering is in time and energy, the one fermion as if corresponds to the discrete "half" of wave-particle duality, and the other accordingly to its continuous (smooth) "half". Hence one can clearly see that the second quantization giving rise to spin statistics either is equivalent to, or is a particular case of that quantum mechanics, which includes as probabilistic as matrix and wave mechanics. Indeed, the sense of the second quantization is to be defined "quantum field". In fact, this is done by ascribing a wave function (i.e. a quantum state) to each space-time point. That quantum mechanics, which includes as probabilistic as matrix and wave mechanics, ascribes a spacetime point to each wave function. Then: If the quantum field is well-ordered, then the mapping between all the wave functions (Hilbert space) and all space-time points (Minkowski space) is one-to-one, and the second quantization is equivalent to that quantum mechanics, which involves probabilistic mechanics. Then any quantum particle must necessarily be either a boson or a fermion. If the quantum field is not well-ordered, it admits two opposite options as well as both together: Two or more space-time points to share one the same wave function, and thus the reverse mapping not to be well-defined: It will not be a standard function. – 14 – Vasil Penchev. Quantum Measure from a Philosophical Viewpoint Two or more wave functions to share one the same space-time point, and thus the straight mapping not to be well-defined: It will not be a standard function. However the most general case is two or more space-time points to share two or more wave functions (i.e. together both above). If that is the case, it can be equally described as some spacetime points, which share a part of some wave functions (entanglement) or as wave functions, which share a "part" of some space-time points ("quantum" gravity). The interrelation or equivalence of entanglement and gravity is being studied. If the quantum field is not well-ordered as above, it can be represented in a few ways (as well as in their combinations or mappings): – As a curving of Hilbert to Banach space. – As a curving of Minkowski to pseudoRiemainian space. – As quantum particles with an arbitrary spin: such which can be any real number. The transitions between the "probabilistic" wave function and "well-ordered" wave function in any of the above ways describe in essence the arising of "something from nothing and from time" ("time" is for the axiom of choice) as a continuous process as a quantum leap as well as a purely informational event. One can give examples of that arising in terms of classical (gravity) or quantum (entanglement) physics as a continuous process. Quantum mechanics, QM and AC: A question may have remained obscure: Does quantum mechanics need AC? Quantum mechanics is actually the only experimental science, which requires necessarily AC: Roughly speaking, the state before measuring has not to be well-ordered, but after that it has to. This means that measurement supposes the well-ordering theorem, which is equivalent to AC: The "no hidden variables" theorems do not allow of well-ordering before measuring. However even only the record of the measured results (which is after measuring, of course) forces they to be well-ordered. The basic epistemological postulates equate the states before and after measurement and thus imply AC. Though measuring requires the AC, it remains inapplicable before measuring. Consequently quantum mechanics is ought to be consistent both with AC and NAC in addition. The only possible conclusion is too extraordinary: Quantum mechanics is consistent as with AC as with NAC. However, quantum mechanics is not consistent with the absence as of AC as of NAC. We could see above that QM is linked to AC in the same extraordinary way. This means that quantum mechanics is consistent with QM as to AC, which should expect. Quantum mechanics, QM and the CH: That extraordinary interrelation between quantum mechanics, QM and AC goes on with CH: NCH is consistent as with AC as with NAC, thus quantum mechanics is consistent with NCH. Reversely, CH should (ostensibly) imply AC. However AC implies the undecidability of CH and NCH, then CH implies by means of AC the own undecidability. The only way out is then to admit the complementarity of CH and AC. Both quantum mechanics and QM share that extraordinary relation to CH by means of AC. Though the state of affairs is strange, it is not logically contradictory. It messes up only common sense. The cause of that ostensible muddle is the intervention of infinity, of which we try to think as of a finite entity. – 15 – Vasil Penchev. Quantum Measure from a Philosophical Viewpoint 5. Physical quantity measured by QM or by LM, or by BM The definition of physical quantity in quantum mechanics involves measuring by QM. It is a generalization of the corresponding notion in classical physics and exact science. One can see the quantum at all as that generalization from LM and BM to QM. The correspondence is the following: A few conclusions can be drawn from that correspondence: The sense of a point of dual Hilbert space is to be a "unit", or something like a reference frame, which can measure a point of Hilbert space. The measured value represents the distance between the "unit" point (or its conjugate point, too) and another point, which is for the measured quantity. This distance can be thought of also as a distance in the "reference frame" of that point. Both cases are "flat": They conserve the measure under translation and rotation. If the translation and rotation are understood as usual as translation and rotation in spacetime, then that "flatness" implies the classical laws of conservation complemented by Lorentz invariance. One can especially emphasize the time translation and energy conservation. The "flatness" in general can be equated with the axiom of choice. Indeed the well-ordering requires that flatness since otherwise a second dimension for ordering appears questioning the well-ordering made only in the first dimension. The above table (1) can be paraphrased in terms of the "crooked" as the following table 2 asking how both tables can refer to each other: The "crooked" case is that of general relativity and gravity. The question for the connection between the two tables addresses the problems of quantum gravity in terms of general relativity and measure theory. To be together in front of the eyes, one can combine the two tables as a new one (3):Table 3: The above table (3) shows that the problem of quantum gravity is a problem of measure: It concerns the ostensibly contradictory properties of infinity focused on how AC and CH should refer to each other. The gravity being as "crooked" as smooth in general relativity supposes the "classical" case of NAC and CH. However as CH implies AC, it should not exist. After all it arises anyway since AC in turn implies the undecidability between HC and NHC. In last analysis it concerns the property of infinity to be both cyclic and linear unlike anything in our usual experience. Table 1. From LM and BM to QM Quantity Unit Value The classical case Real quantity Unit Real number The quantum case Wave function Conjugate wave function Self-adjoint operator Table 2. Table 1 paraphrased Quantity Unit Value The classical case Real quantity Unit Real number The "crooked" case Contravariant vector Covariant vector Metric tensor – 16 – Vasil Penchev. Quantum Measure from a Philosophical Viewpoint One can think of the quantum and gravity case as complementary. In particular this means that the values of a quantity as a self-adjoint operator or as a metric tensor are complementary, too, as well as QM and the "crooked" LM therefore that QM and LM are even equivalent in the distinctive way of quantum mechanics Just this complementarity of QM and LM is taken in account as to NCH, which is consistent as with AC as with NAC, and for this, with the curious invariance of AC and NAC. For the above three as to the CH case, one can be free to suggest that it exactly repeats the NCH case in relation to AC and NAC because of the undecidability between CH and NCH after AC. This would means that any theory of quantum gravity is not necessary since the pair of quantum mechanics and general relativity can represent whatever the case is. The following should be highlighted in background of the just said: In the same extent, one can be free to admit the opposite: That is the CH case does not repeat the NCH one just because of the used undecidability between CH and NCH after AC. This will say that a theory of quantum gravity is possible though it will be never necessary. One can compare with the real state of affairs: Indeed many theories of quantum gravity appear constantly and supposedly some of them do not contradict the experiments just because they are possible. However they are not necessary in principle since general relativity does not contradict the experiments, too, and "Occam's razor" removes all of them remaining in hand only general relativity. The perspective Is there any measure more general and universal than QM? If there is, it could be called "generalized quantum measure" (GKM). According to current knowledge, we cannot even figure what might cause such a measure to be introduced or what would constitute. One can postulate the absolute universality of QM. This implies a series of philosophical conclusions and new interpretations of wellknown facts. The most of them have been already mentioned in a slightly different context above. What is worth to emphasize here is the following: A universal measure as QM suggests that all entities are not more than different forms of a substance shared by all of them as their fundament: It is quantum information and represents a general quantity, which is both mathematical and physical in its essence. The longstanding philosophical idea of a single and general substance can be already discussed in terms of exact science. Table 1 and Table 2 unified Quantity Unit Value The classical case Real quantity Unit Real number The quantum case Wave function Conjugate wave function Self-adjoint operator The gravity case Contravariant vector Covariant vector Metric tensor Quantum gravity ??? ??? ??? 1 Carathéodory, 1956, 149. 2 Banach and Tarski, 1924. 3 Vitali, 1905. 4 Neumann, 1932, 167-173. – 17 – Vasil Penchev. Quantum Measure from a Philosophical Viewpoint 5 Kochen and Specker 1968, 70. 6 Löwenheim, 1915; Skolem, 1919a; Skolem, 1919b. 7 Skolem, 1923. 8 Alexandroff, 1916; Hausdorff 1916; cf. Sierpiński 1924. 9 Heisenberg, 1925. 10 Schrödinger, 1926a. 11 Schrödinger, 1926b. 12 Neumann, 1932, 18-100. 13 Born 1926a; Born, 1926b; Born 1927a; Born 1927b; Born and Fock 1928; Born 1954. 14 Cited in: Weber, 1893, 19. 15 Bell 1964; Clauser and Horne 1974; Aspect et al, 1981; Aspect et al, 1982. 16 Fiertz, 1939; Pauli 1940. References Alexandroff, P. (1916). On the power of measurable (B) sets [Sur la puissance des ensembles 1. measurable (B)]. Comptes rendus hebdomadaires des séances de l'Académie des sciences, 162 (28.02.1916), 323-325. Aspect, A., Grangier, P., Roger, G. (1981). Experimental tests of realistic local theories via 2. Bell's theorem. Physical Review Letters, 47 (7), 460-463. doi:10.1103/PhysRevLett.47.460 Aspect, A., Grangier, P., Roger, G. (1982). Experimental Realization of Einstein-Podolsky-3. Rosen-Bohm Gedanken Experiment: A New Violation of Bell's Inequalities. Physical Review Letters, 49 (2), 91-94. doi:10.1103/PhysRevLett.49.91 Banach, S. and Tarski, A. (1924). On the decomposition of sets of points into parts respectively 4. congruent [Sur la decomposition des ensembles de points en parties respectivement congruentes]. Fundamenta Mathematicae, 6 (1), 244-277. Bell, J. (1964). On the Einstein – Podolsky – Rosen paradox. 5. Physics (New York), 1 (3), 195200. Born M. (1926a). On the quantum mechanics of shock events [Zur Quantenmechanik der 6. Stoβvorgänge]. Zeitschrift für Physik, 37,(12/ December), 863-867. Born, M. (1926b). The quantum mechanics of shock events [Quantenmechanik der 7. Stoβvorgänge]. Zeitschrift für Physik, 38 (11-12/ November), 803-827. Born, M. (1927a). The adiabatic principle in quantum mechanics [Das Adiabatenprinzip in 8. der Quantenmechanik]. Zeitschrift für Physik, 40, (3-4/ März), 167-192. doi: 10.1007/BF01400360 Born, M. (1927b). Physical aspects of quantum mechanics. 9. Nature, 119 (05.03.1927), 354-357. doi:10.1038/119354a0 Born, M. (1954). 10. The statistical interpretation of quantum mechanics (Nobel Lecture, December 11, 1954). Available at: http://nobelprize.org/nobel_prizes/physics/laureates/1954/bornlecture.pdf (accessed 23 November 2013). Born, M. and Fock, V. (1928). Proof of the adiabatic theorems [Beweis der Adiabatensatzes] . 11. Zeitschrift für Physik A: Hadrons and Nuclei, 51 (3-4), 165-180. doi: 10.1007/BF01343193 Carathéodori, C. Measure and integral and their algebraization [12. Mass und Integral und ihre Algebraisierung]. Basel, Birkhäuser, 1956. 337 p. Clauser, J. and Horne, M. (1974). Experimental consequences of objective local theories. 13. Physical Review D, 10 (2), 526-535. doi:10.1103/PhysRevD.10.526 Fiertz, M. (1939). On the relativistic theory of force-free particles with arbitrary spin [Über 14. die relativistische Theorie kräftefreier Teilchen mit beliebigem Spin]. Helvetica Physica Acta, 12 (1), 3-17. doi:10.5169/seals-110930 Vasil Penchev. Quantum Measure from a Philosophical Viewpoint Hausdorf, F. (1916). The power of Borel sets [Die Mächtigkeit der Borelschen Mengen]. 15. Mathematische Annalen, 77 (3), 430-437. doi:10.1007/BF01475871 Heisenberg, W. (1925). On the quantum-theoretical interpretation of kinematic and dynamic 16. relations [Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen]. Zeitschrift für Physik, 33 (1), 879-893. doi: 10.1007/BF01328377 Löwehheim, L. (1915). On possibilities in the calculus of relatives [Über Möglichkeiten im 17. Relativkalkül]. Mathematische Annalen, 76 (4), 447-470. doi: 10.1007/BF01458217 Kochen, S. and Specker, E. (1968). The problem of hidden variables in quantum mechanics. 18. Journal of Mathematics and Mechanics, 17 (1), 59-87. doi:10.1007/978-94-010-1795-4_17 Neumann, J. Mathematical foundation of quantum mechanics [19. Mathematische Grundlagen der Quantenmechanik]. Berlin, Verlag von Julius Springer, 1932. 262 p. Pauli, W. (1940). The Connection Between Spin and Statistics. 20. Physical Review, 58 (8/ October), 716-722. doi: 10.1103/PhysRev.58.716 Schrödinger, E. (1926a). An Undulatory Theory of the Mechanics of Atoms and Molecules. 21. Physical Review, 28 (6), 1049-1070. doi: 10.1103/PhysRev.28.1049 Schrödinger, E.22. (1926b). On the relation the Heisenber-Bohr-Jordan quantum mechanics to mine [Über das Verhältniss der Heisenberg-Born-Jordanischen Quantenmechanik zu der meinem]. Annalen der Physik, 384, (8), 734-756. Sierpiński, W. (1924). On the power of the measurable (B) sets [Sur la puissance des ensembles 23. measurable (B)]. Fundamenta Mathematicae, 5 (1), 166-171. Skolem, T. (1919a). Investigation on the axiom of the calculus of classes and on the problem 24. about the products and sums, to which some classes of propositions refer [Untersuchung über die Axiome des Klassenkalküls und über Produktationsund Summationsprobleme, welche gewisse Klassen von Aussagen betreffen]. Skrifter utgit av Videnskabsakademiet i Kristiania, I (3), 37-74. Skolem, T. (1919b). Logical and combinatoric investigations on the completeness and provability 25. of mathematical statements in relation to a theorem about dense sets [Logisch-kombinatorische Untersuchungen über die Erfüllbarkeit und Beweisbarkeit mathematischen Sätze nebst einem Theoreme über dichte Mengen]. Skrifter utgit av Videnskabsakademiet i Kristiania, I, (3), 1-36. Skolem, T. A few notes to the axiomatic foundation of set theory [Einige Bemerkungen zur 26. axiomatischen Begründung der Mengenlehre]. Matematikerkongressen i Helsingrofs den 4-7 Juli 1922, Den femte skandinaviska matematikerkongressen, Redogörelse, Helsinki: Akademiska-Bokhandeln, 1923, pp. 217-232. Weber27. , H. (1893). Leopold Kronecker [Leopold Kronecker]. Jahresbericht der Deutschen Mathematiker-Vereinigung, 2 (1891 – 1892), 5-31. Vitali, G. On the problem of the measure of the set of points of a straight line [28. Sul problema della misura dei gruppi di punti di una retta]. Bologna, Gamberini and Parmeggiani , 1905, pp. 231235. Vasil Penchev. Quantum Measure from a Philosophical Viewpoint Квантовое измерение с философской точки зрения Васил Пенчев Болгарская академия наук Институт исследований обществ и знаний Болгария, 1000 София, Неофит Рильски, 31 В данной статье представлены философские выводы о том, что квантовое измерение предполагает взаимосвязь между квантовой механикой и общей теорией относительности. Квантовое измерение является трехмерным, таким же универсальным, как мера Бореля, и таким же полным, как мера Лебега. Единица квантового измерения – бит (кубит) – может рассматриваться как генерализация единицы классической информации, бита. Квантовое измерение позволяет интерпретировать квантовую механику в рамках квантовой информации, и все физические процессы рассматриваются как информационные в обобщенном смысле, что предполагает фундаментальную взаимосвязь между физическим и материальным, с одной стороны, и математическим и идеальным – с другой. Квантовое измерение объединяет их посредством одной объединенной информационной единицы. Квантовая механика и теория общей относительности могут пониматься совместно как целостный и временный аспект одного и того же, состояние квантовой системы, т.е. состояние Вселенной в целом. Ключевые слова: измерение, квантовая механика, теория общей относительности, квантовая информация, перепутывание.


Review of Kirwan Beauty

Philosophy

Kirwan identifies three kinds of beauty theory within the Western tradition.  These are: âin the eye of the beholderâ theories; neoplatonic theories; and what he refers to as synaesthetic theories; which he discusses in chapters 2, 3 and 4 respectively.  He places himself within the synaesthetic tradition whose emphasis is apparently on the interaction between the beautiful object and the perceiver.  Kirwan, however, does not analyse this interaction.  Nor does he concern himself with what makes the experience of beauty possible, nor what characteristics of an object make it beautiful.  Instead, Kirwanâs theory of beauty amounts to a phenomenology of beauty.  Kirwan is interested in âthe structure of feeling involved in beauty â with respect to which the division between the perceived object and its ground is more important than any formal properties of either â (p.39).

BOOK REVIEW by Jennifer A. McMahon in The Journal of Aesthetics and Art Criticism 59:3 (2001): 334-336 Kirwan, James. Beauty. Manchester University Press, 1999, pp. x + 182, US$ (cloth), US$ (paper). Kirwan identifies three kinds of beauty theory within the Western tradition. These are: 'in the eye of the beholder' theories; neoplatonic theories; and what he refers to as synaesthetic theories; which he discusses in chapters 2, 3 and 4 respectively. He places himself within the synaesthetic tradition whose emphasis is apparently on the interaction between the beautiful object and the perceiver. Kirwan, however, does not analyse this interaction. Nor does he concern himself with what makes the experience of beauty possible, nor what characteristics of an object make it beautiful. Instead, Kirwan's theory of beauty amounts to a phenomenology of beauty. Kirwan is interested in "the structure of feeling involved in beauty – with respect to which the division between the perceived object and its ground is more important than any formal properties of either " (p.39). The feeling of beauty, according to Kirwan, is a "yearning without object, or yearning which suppresses its object" (p.39). Kirwan's theory of beauty consists in finding an analogue for this yearning in the realm of metaphysics (Ch.4). His is a theory of subjectivity; but a subjectivity in the tradition of Kant's 'universally communicable' subjectivity. In Kirwan's case, however, the basis of this communicability is not spelt out. Kirwan sets up his approach to the subject in chapter 1 when he writes: [I]t may be that beauty comes about only through a deliberately selective perception, a perception that is in some way aware of what it excludes, in so far as, for the sake of the beauty of an object, it can deliberately exclude the more complex object which is its ground. (It may even be that beauty exists only in this act of exclusion itself.) Even if this is not the case, to use the phrase 'beautiful object' to describe the true phenomenal ground of the instance of beauty may be misleading, for, though it is this ground which inspires the sensation of beauty in the subject, it is not, as an object, part of the subject's conscious experience of the sensation of beauty (p.12). At such times, Kirwan's analysis of his own experience of beauty promises to reveal useful distinctions. He does not, however, maintain this analytic vein, opting instead for broad sweeping claims such as that the experience of beauty is the effect of our desire for transcendence, or perfection (p.45). This idea of beauty vindicates the intuition underpinning the neo-platonic understanding of beauty. It is the idea that beauty's manifestations on earth are a mere reflection of 3 beauty proper, the latter a characteristic of the divine. Kirwan drops the framework of divinity, but it is not clear what he replaces it with. The grounding of his transcendental metaphysics is never made clear. It is perhaps easier to grasp Kirwan's idea of beauty by considering the kind of construal of beauty that it disallows. The idea that what is beautiful about an object is "quantifiable solely in terms of the properties of that object" (either to the exclusion or inclusion of the particular subject's interpretation of those properties) is incompatible with Kirwan's notion of beauty (p.47). Also the idea that "the pleasure of beauty is, somehow, identifiable with the pleasure attending the satisfaction of specific physiological goals; and that experiences of beauty are qualitatively different from one another ('high' or 'low')" is rejected by him (p.47). In effect then, Kirwan rejects, on the one hand, Clive Bell's 'formalism', and on the other, the idea that everything that pleases through the senses is beautiful. Neither does he entertain what has become a post-modern cliche that beauty is a cultural construct invented to impose mainstream tastes. Furthermore, his theory of beauty makes no assumptions about art (ch.8). Art may or may not be beautiful. Kirwan attempts to untangle the confusion which exists between the nature of art and the nature of beauty without making reference to specific examples of this confusion in the contemporary literature on beauty. Even so, this is a welcome and very useful distinction. Kirwan, then, recognizes the experience of beauty as an authentic part of what it is to be human but it must be said that what constitutes his 'theory' is left unhelpfully vague. He promises to examine the relation of his theory to three themes traditionally associated with beauty; knowledge, action and morality (in chapters 5,6 and 7 respectively), but we are left no better off regarding the grounds of his theory. One is left with a description of his attitude to beauty rather than an actual theory. For example, in chapter 9, Kirwan sums up: "I have described beauty, and told how the beauty of a thing is always in excess to anything we can hope from that thing, but there will still be beauty. For there is nothing beyond beauty. Everything is as it was. I have nothing to prescribe." (p.124) Such rhetoric, however, is interspersed with some revealing metaphors as when Kirwan reinterprets the Platonic relationship between the sensuous and formless beauty as like looking through the former to the latter with a telescope rather than climbing a ladder from one to the other (ch.7). But such revealing metaphors do not a theory make. Kirwan attempts to reveal rather than analyse. He attempts to keep the experience of beauty before us as a whole rather than dissect it into parts. The result is, however, that he asserts rather than argues. I am sympathetic to the dilemma of style regarding how best to treat the subject of beauty without reducing it beyond recognition. The problem is, however, that without analysis, what is acknowledged as constituting an experience of beauty needs to be taken as 5 a given. This will not do because the philosophical literature, not to mention the literature on beauty in the areas of art theory, sociology and psychology, is not consistent on what constitutes an experience of beauty. Kirwan identifies the locus of the nature of beauty as a subject's response to certain aspects of an object (and a certain construal of these aspects). However, while he explains what he thinks the experience of beauty is like, he presents no compelling argument as to why we should accept his account. At times, he seems to think all deeply moving feelings can be a part of the experience of beauty. According to Kirwan, the more the feeling exhibits a yearning kind of pleasure, a melancholy, the more cultivated the response to beauty is. (He had stated earlier that his theory was incompatible with the idea of high and low forms of beauty but obviously not with high and low forms of the experience of beauty!). Such feelings would preclude, say, the possibility of moral or mathematical beauty, even though Kirwan clearly wants to include moral beauty as a possibility (ch.7). Kirwan makes little of the ground of judgments of beauty. The fact is that without identifying some grounding for judgments of beauty, there is no way to differentiate between what constitutes an experience of beauty and what can sometimes accompany an experience of beauty. This is an important problem which causes many apparent disagreements about beauty and which Kirwan simply side steps. Furthermore, he does not address the classic problem of beauty; the problem concerning the basis for the intersubjectivity of experiences of beauty, when there are no stateable sufficient or necessary conditions for beauty. According to Kirwan, 'Beauty is immediately and (despite its felt subjectivity) irreducibly a sensation – it strikes me.'(p.6) But this belies the way judgments of beauty are defended. A configuration not previously perceived in an object by a particular perceiver can be pointed out to that perceiver so as to give rise to the experience of beauty in that perceiver. This suggests that beauty is reducible even though attempts to reduce beauty to principles seem doomed to failure. The way we defend judgments of beauty is hard to reconcile with the fact that we know something is beautiful by how it makes us feel, not by first judging whether it satisfies certain conditions. Kirwan makes a lot of the latter part of this point but rather than address the tension between this and the way judgments of beauty can be defended, he simply ignores one half of the equation. Kirwan's treatise reads like a poetic expression of what one person takes to be an experience of beauty. What his treatise can be said to contribute is yet another example of the difficulty of understanding beauty simply through a nonanalytical reflection on what is apparently a phenomenology of the experience of beauty. Jennifer A. McMahon Education University of Canberra


Philosophy

Randomization and Fair Judgment in Law and Science Julio Michael Stern∗, Marcos Antonio Simplicio†, Marcos Vinicius M. Silva‡, Roberto A. Castellanos Pfeiffer§ p.399-418 in José Acácio de Barros, Décio Krause, eds. (2020). A True Polymath: A Tribute to Francisco Antônio Dória. Rickmansworth, UK: College Publications. ISBN: 978-1-84890-351-7 Abstract Randomization procedures are used in legal and statistical applications, aiming to shield important decisions from spurious influences. This article gives an intuitive introduction to randomization and examines some intended consequences of its use related to truthful statistical inference and fair legal judgment. This article also presents an open-code Java implementation for a cryptographically secure, statistically reliable, transparent, traceable, and fully auditable randomization tool. Keywords: Randomization; Truthful inference; Fair judgment; Judicial autonomy and independence. ∗Institute of Mathematics and Statistics of the University of Sao Paulo firstname.lastname@example.com †Polytechnic School of the University of Sao Paulo email@example.com ‡Polytechnic School of the University of Sao Paulo firstname.lastname@example.com §Law School of the University of Sao Paulo, email@example.com 379 κληρω νυν πεπαλασθε διαμπερες oς κε λαχησιν : Let the lot be shaken for all of you, and see who is chosen. Iliad, VII, 171. י®פְר£ד! Mעְַצּומִי Nּובֵי הַּגֹור ל י®ׁש ּבִית MמִדŸי3נ¢י Casting the dice puts judgment quarrels to rest and keeps distinct essential powers duly separated. Proverbs 18:18. 1 Introduction Francisco Antonio Dória has had a consistent interest in randomness and chaos and, together with his collaborators, has investigated fundamental aspects of such phenomena. This article is our contribution to the Festschrift celebrating Doria's 75th birthday. This article analyses some pragmatical aspects of applying randomization in empirical science and law, considers some philosophical implications or premises justifying or motivating these applications, and offers some tools that promote good randomization practices. The Cardsharps (1594) marks the beginning of the independent career of the great Italian master Michelangelo Merisi da Caravaggio (1571-1610). This painting displays a wealthy but innocent looking boy playing cards with his opponent, a cardsharp, that cheats in two ways: On the one hand, the cardsharp hides in his belt spurious cards that he intends to use in illegitimate ways; on the other hand, a sinister looking and strategically positioned accomplice gives him access to privileged and undue information. Finally, the cardsharp carries a dagger, hinting at the dangers lurking in this environment of misrepresentation and deception. Caravaggio gives a beautiful depiction of some themes discussed in this article. First, the social importance of activities involving randomization, that is, the random setting of some variable of interest, like the drawing of dice or, in this painting, the distribution of playing cards. Second, it suggests the question – Why to randomize? 380 Figure 1: The Cardsharps (1594), by Michelangelo Caravaggio that is – Why should a rational agent abdicate the opportunity of making a deterministic choice introducing, on purpose, a random component in making a decision? If so – What is the role played by randomization? Finally – How to randomize? that is – What dangers could jeopardize a randomization process? and, if necessary – How to shield or immunize the process against these dangers? In order to answer these questions, we have to pay attention to some topics in Statistics, Computer Science and Cryptography; in addition, we have to examine some details concerning the design of empirical trials or the operation of legal systems. In this article, we investigate each one of the questions just raised, looking for an intuitive understanding of the role(s) played by randomization. In the final sections of the paper, we present an easy to use, open-code, traceable, auditable, secure, and statistically sound randomization toll that is ready for use in empirical trials and legal applications. This kind of secure randomization tool can prevent the possibility of misrepresentation and deception, as depicted in 381 the painting by Caravaggio. Moreover, even in situations where no misdeeds actually occur, the use of such a tool can be beneficial by fostering public confidence in the soundness of important decisions, by strengthening the resilience of public institutions, and by favoring the peaceful resolution of conflicts. 2 Social Importance of Randomization Gambling and lotteries exchange billions of Dollars every day worldwide. Hence, ensuring honesty and transparency in these activities should already be considered a meritory task. However, since ancient times, sortition (i.e., selection by lottery) is used for many other purposes. In the Iliad, one of the oldest texts of western culture (aprox. 1200BC), the Argonauts (crew of the ship Argo) selected a man to execute a dangerous task by sortition – see this paper's first opening quotation. In the same manner, modern societies often resort to sortition for drafting. Figure 2 displays some photographs related to compulsory enlistment for service in the USA, namely, military drafting during the Civil (left) and the Vietnam (center) wars, and selection for jury duty (right). In order to gain public trust, the sortitions for the Vietnam war were conducted in public view: Balls with calendar dates were placed in a transparent urn and some anniversary dates were then picked, giving the (un)lucky winners the opportunity to serve their country in the battlefield. A post hoc statistical analysis of these drawings revealed a significant bias favoring latter days of the calendar, corresponding to the last balls placed inside the urn, an unexpected effect of an ill-conceived randomization process that generated misunderstanding, frustration and conflict. Figure 2 (right) shows a letter calling a citizen for jury duty in the USA. In this process, an eligible citizen was chosen at random by running a computer program. Post hoc analyses of this randomization process revealed no significant bias or any other statistical anomaly. Nevertheless, the code of these computer programs were never made public, making the randomization process opaque and 382 Figure 2: Draft lotteries in war and peace non-verifiable, thus generating mistrust and resentment. Finally, in the world of science, good clinical trials are conducted by (double) blind and random attribution of patients to two or more distinct treatments. The objective of such a trial is to find out if a new or alternative treatment is significantly better than the old or standard one, according to well-established statistical criteria. In this situation, some frequently asked questions are: Why should a patient's treatment be selected at random? Why not give him or her the freedom to chose his or her proffered treatment? Why hide from a patient information about his or her own treatment? 3 Why to Randomize? Imagine a clinical trial where patients are free to choose a treatment according to their own will. Among patients, there will be rich and poor, people with different degrees of instruction, people with better or worst networks for support, etc. Obviously, rich, well educated, and well connected patients will have better access to good information and advice and, therefore, will be prone to make better decisions. Moreover, these same patients likely have better overall living conditions and, therefore, even with the same treatment, might have a better chance of recovery. Hence, this freedom of choice would automatically introduce confounding effects: After the trial is over, we would not be able to (completely) discern the beneficial and adverse consequences of distinct treatments from 383 consequences of preexisting conditions. Similar unwanted interference is generated by the placebo and nocebo confounding effects. If a patient knows to be receiving either a new, experimental and possibly wonderful drug, or else an old and possibly not very effective drug, his or her moral may be, respectively, lifted or depressed. That, in turn, may affect his or her overall health and chance of recovery. This is why, in a good clinical trial, treatment information is denied (blinding or censorship) to patients, and commonly also to their direct caretakers (double-blinding). There are in the medical literature plenty of examples of clinical trials that came to wrong conclusions in consequence of such confounding effects. The best known antidotes against these confounding effects rely in some form of randomization. The idea is to chose a patient's treatment based on a random variable that is independent of any potentially confounding variable. In so doing, the random element in the choice of treatment has the effect of breaking causal links that should not interfere with the experiment, allowing the trial to adequately focus on the causal links of interest – see Stern (2008) and Pearl (2009) for further details. Finally, let us consider the use of randomization in the legal system, like the selection of jurors or judge(s) for a given case. Figure 3 displays two pictures from ancient Egypt. On the left, a stone carving of approx. 2400BC shows two merchants using a two-pan balance to correctly measure amounts of goods for a fair commercial transaction. On the right, the Hunefer papyrus, of approx. 1275BC, shows the scale used by Maat, the goddess of justice, where the heart and the (de)merits of a man are measured. It should be clear that these two scales are essentially distinct – they belong to distinct contexts. The figure at the center suggests the possibility of "mixing" these two essences: Perhaps Maat could make a more benevolent assessment in the scale of justice if she, or her priests, received goods of commercial valuable... There we have, once more, a confounding effect, characterized by spurious influences between powers belonging to essentially distinct systems: 384 Figure 3: Essentially different powers: Economy and Justice in this case, the economic system and the justice system. How to avoid such confounding effects caused by spurious influences, fostering autonomous decisions in a strong and independent justice system? Surprisingly, the Hebrew bible already offers very good advice at this respect, as stated in the second opening quotation of this paper. Interestingly, the Hebrew root !Mעצ, etzem, whose literal meaning is bone, also generates words meaning essence (the etymological origin of the English word), strength, power and the modern Hebrew word for independence. Judges, even if perfectly honest, do not come to court as a blank slate, nor should they. Every judge has his or her own history of decisions and opinions. Hence, if the selection of judges could be influenced by the litigants or other interested parties, the richer, better informed, well connected, or otherwise more powerful parties would likely have an advantage in directing the case to a judge sympathetic to their arguments. For this reason, in many modern democracies, the distribution of a new judicial case must take the form of a random choice among the available judges or courts qualified to judge it. 4 How to Randomize? Previous sections discussed several applications of randomization and explanations of why to use it. This section describes some desirable characteristics of such a randomization process, including: 385 Statistical honesty: In a set of sortitions (random selections) in the system, the probability of any group of outcomes should be exactly as prescribed by the established rules. Cryptographic security: The outcome of the sortition should be unpredictable; moreover, no external agent should be able to influence the randomization process, even if the agent knows in detail the randomization mechanism being used and has state of the art knowledge of all relevant technologies involved. Transparency: All relevant information about the sortition process must be of public knowledge, including any pertinent detail about the randomization mechanisms being used. Auditability: All relevant occurrences of an actual randomization process must be traceable and auditable. Furthermore, it should not be possible to conceal any improper use of the randomization system. The first two requirements are of technical nature, stipulating that, in the randomization process, we should use "honest dice that cannot be tampered with" – or else a more convenient device, like a computational algorithm that adequately mimics all the relevant characteristics of "honest dice". For more technical discussions on theses characteristics, see Marcondes et al (2019), Saa and Stern (2019) and Silva et al. (2020). In order to emphasize the importance of the last two requirements, let us discuss a form of cheating known as rerandomization. In this kind of cheating, the agent responsible for a sortition has the privilege of using the randomization mechanism out of public scrutiny, examine the outcome, and chose at will either to make this process and its outcome public, or to hide this first try and randomize a second time, as if the first try never happened. Imagine for example the classic process of picking a ball from a transparent urn. However, instead of making a live presentation, the sortition ceremony is recorded for broadcasting at a later time. A dishonest agent could repeat and record the process twice, and only release 386 the recording that best fits his or her goals, as if it were the only recording ever made. It should be clear that the repeated use of this subterfuge gives the agent in charge some latitude to pick and choose, biasing the final outcome according to his or her convenience. Authority, Transparency and Understanding Why is transparency even required in a randomization process? Would it no be possible, or even easier, to anchor the credibility of the process on a principle of authority? If a given authority is responsible for a randomization process, doesn't the requirement of transparency imply an implicit doubt? If so, doesn't the requirement of transparency imply disrespect for the same authority? These are basic questions in philosophy of law, that can only be answered in a context that specifies the fundamental values and goals chosen by a given society. Niklas Luhmann (1985, 1989), a celebrated scholar in philosophy of law, postulates that the fundamental goal of the justice system is – "the congruent generalization of normative expectations". That is, the final objective of the legal system is the construction of a harmonious society, where citizens have a coherent view of what constitutes a good set of rules for social behavior (normative expectations). Moreover, a legal system should provide mechanisms that stimulate citizens to conform to normative expectations and inhibit their transgression. This conception of law requires from every citizen a well founded trust that the justice system is efficient and fair, preferably obtained by conscious understanding of laws and regulations and their forms of implementation. Moreover, a justice system conceived according to such principles is weak or fragile if sustained on blind faith on ad hoc authority, but strong and resilient if sustained by a conscious, engaged, and participative community. The articles of Silva et al. (2020) and Stern (2018) expand these ideas. 387 Figure 4: Modulo 12 or Clock Arithmetic 5 Modular Arithmetic and Trusted Roulettes In this section we discuss intuitive ideas for how to implement an honest randomization device that all interested parties can trust. In following sections of this article we offer a viable technological solution to the problem of randomization that satisfies all requirements stipulated in previous sections, following the general ideas hereby discussed. Modular arithmetic is an integer arithmetic system in which numbers "wrap around" after reaching a maximum value, m, called the modulus. A familiar example is the standard reading of a clock. After noon (12 o'clock), we restart counting from 1, 2, ... (p.m.). Notice that, in Figure 5, the position corresponding to noon (or midnight) is marked either by the modulus value, m = 12, or by the value zero – that is mathematically more convenient. In general, for positive integers, n and m > 1, we define n modm (read n modulo m) as the remainder of the division of n by m. For example, see Figure 5 (left): 13 mod 12 = 1, 14 mod 12 = 2, ... 23 mod 12 = 11, and 24 mod 12 = 12 mod 12 = 0. Figure 5 (right) illustrates the modular arithmetic operation (9 + 4) mod 12 = 1. Now imagine we have a game using a roulette or wheel of fortune, see Figure 5, with k participants, also known as the stakeholders, all of them wanting the privilege of spinning the roulette, and not trusting anyone else to do the job. How can we break this deadlock? 388 Figure 5: Rolling Dice and Spinning Roulettes We can solve the aforementioned impasse using the following protocol: 1. Provide each stakeholder with a well-balanced roulette, marked according to the numbers set {0, 1, 2 . . . (m− 1)}; 2. Ask each stakeholder to spin his roulette honestly, that is, with a not fully controlled and strong enough initial impulse so to produce any of the possible outcomes, {0, 1, 2 . . . (m−1)}, with the same probability, (1/m). Moreover ask each stakeholder to use his roulette independently, that is, to do so without sharing any information with other stakeholders or interested parties; 3. Collect and add, using modulo m arithmetic, the results produced by each one of the k stakeholders in order to produce the final result: nf = (n1 + n2, . . .+ nk) modm. We can guarantee that the final result, nf , produced by this protocol is equivalent to an "honest roulette", as long as at least one (any one) of the k stakeholders does his job as required. This guarantee is a corollary of the following theorem: Let x and y be independent random variables in {0, 1, 2 . . . (m− 1)}. Then, if any one of these random variables, x or y, is uniformly distributed, so is z = (x+ y) modm. Imagine, for example, that variable x is not random at all, but rather a known constant, c, namely, the initial state of the roulette. Furthermore, imagine that y is independent 389 of x and uniformly distributed in {0, 1, 2 . . . (m− 1)}. Under these conditions, the theorem states that the final state of the roulette, z = c + y is uniformly distributed, corresponding to the intuitive idea that, when using a well-constructed roulette, a strong enough impulse will produce a final outcome that "forgets" the initial state of the roulette. For further details and formal mathematical analyses, see Scozzafava (1993). In many applications in statistics, clinical trials, and complex sortitions, we need a random variable x uniformly distributed in the interval [0, 1[ of the real line. In computational procedures, this continuous variable can be approximated by a fraction n/m, where m in a large integer, and n ∈ {0, 1, 2, . . . (m − 1)}. This fraction can be translated to standard floating point notation, and then be further transformed into random variables with several probability distributions of interest in statistical modeling, see Hamersley (1964), Ripley (1987). Such uniform or non-uniform random variables can, in turn, be used in dynamic clinical trials, haphazard intentional sampling, adaptive sampling procedures, and other complex applications of interest in statistical modeling and decision science, see for example Fossaluza (2015) and Lauretto et al. (2012, 2017) and the bibliography therein. In the sequel, we describe a software implementing the protocol outlined in this section, including all necessary precautions in order to guarantee cryptographical security. In this software, every stakeholder is required to input a random number n between 0 and m = 9, 999, 999. Each stakeholder has the responsibility of producing his or hers random 7-digit decimal number in a way he or she finds appropriate. Random decimal digits can be easily produced, for example, by rolling 10-faced dice, available in several shapes with the required symmetry conditions, see Stern (2011). Figure 6 exhibits 10-faced dice shaped as a pentagonal trapezohedron (left) and a pentagonal antiprism (center), and also 20-faced dice shaped as an icosahedron (right); in order to produce a random decimal digit using icosahedral dice, the user should read only the last digit for outcomes ranging 390 Figure 6: Polyhedral dice for decimal digits from 10 to 20. 6 Intuitive Cryptology One crucial requirement of the random drawing approach described in Section 5 is that the the roulettes are run independently. Otherwise, a dishonest stakeholder Si could wait until the results of all roulettes are revealed, and then run his/her own roulette for manipulating the final outcome of the drawing: for example, suppose that the sum of the contributions from all stakeholders except Si is n = 3 for, say, m = 12; after learning this value of n, Si could force his/her own roulette to give ni = 2, thus obtaining nf = n+ ni modm = 5 as the final (manipulated) outcome of the drawing. To ensure this independence property, Silva et al. (2020) builds upon the properties of hash-based bit-commitment mechanisms. Intuitively, a hash function H is a cryptography construct analogous to fingerprinting for humans, as illustrated in Figure 7 – we refer the reader to Beutelspacher (1994), Bultel et al. (2017) and Fellows and Koblitz (1994) for intuitive introductions on key ideas of cryptology, an to Rogaway and Shrimpton (2004) for a concise but formal explanation of properties of cryptographic hash functions. Specifically, given the fingerprint for an unknown human being, it takes a lot of computational power to look all around the world for the owner of that fingerprint; on the other hand, given a fingerprint and the corresponding human, it is quite easy to check whether 391 or not they match, and it is hard (almost impossible) to find two different people with the same fingerprint (even considering identical twins). Similarly, suppose that someone computes the hash of a number n, i.e., a value h = H(n), which acts as a "fingerprint" for n; then, if only h is revealed, but n is kept secret, there is no simple mechanism for finding the value of n. Of course, one could test every possible value of n, checking if a guess ni is such that H(ni) = h, just like finding the owner of a fingerprint given only the fingerprint itself. However, the computational effort for performing such brute-force attack would be very large. Actually, in practice the cost for hash functions would be even larger than searching for a human who owns a fingerprint: while there are a few billions of humans in the world, a hash function can be used in such a manner that the number of tests would be as large as the number of atoms in the whole planet Earth! For this purpose, it suffices to combine the value of n with a large and unpredictable (e.g., random) mask r when computing the hash, i.e., to make h = H(r, n). As a result, even if there are only a few possible values of n to be tested, determining whether or not a guess ni is correct would require testing all possible values of r too. Therefore, it suffices to use a large-enough mask r (e.g., 256-bits) to ensure that any attempt of determining n via brute force would be computationally infeasible. When both n and h are revealed, on the other hand, it is easy to verify whether or not they match: it suffices to compute H(n) directly, and check if the output of this computation is identical to the provided value of h. However, like different humans should not have the same fingerprint, it is computationally hard to find two distinct values of n (say, n1 and n2) that have the same hash h. Hence, once h1 = H(n1) is revealed, one can say that the person who revealed it is "committed" to revealing n1, i.e., it would be hard to trick someone into believing that h1 was computed from any other input n2 6= n1. Such properties are used by Silva et al. (2020) to build a twophase procedure for ensuring the fairness of random draws: 392 n = 5 h = ef2d127de37b942baad06 145e54b0c619a1f22327b 2ebbcfbec78f5564afe39d fingerprinting hashing fingerprinting n ≠ 5 hashing Figure 7: Hash functions and their similarity with human fingerprinting. 1. Commitment phase: first, each stakeholder Si runs a roulette (honestly or not), getting a value ni as result. Then, Si computes the hash of ni, denoted hi, and reveals only hi to the other stakeholders, keeping ni itself secret. This prevents Si from learning the roulette results from his/her peers, and vice-versa. 2. Reveal phase: only after all hashes are received, every stakeholder Si reveals its own ni. The outcome of the drawing is then computed locally by Si by adding every ni together using modular arithmetic as explained in Section 5. In this case, even if Si is malicious and tries to delay the revelation of ni until it learns the partial outcome of the drawing from the values revealed by his/her peers, it would be already too late: after revealing hi in the commitment phase, Si has no choice but to reveal the already chosen ni, rather than some other value that might lead to a more desirable (but unfair) drawing outcome. 393 7 Java Implementation We developed a simple Java library for implementing the protocol described in Silva et. al (2020), and made it available under the MIT License at https://doi.org/10.24433/CO.6108166.v1 This library can, thus, be freely adapted for the needs of specific application scenario. To help in this task, we also provide a simple proof-of-concept graphical interface for testing purposes, which is depicted in Figure 8. More precisely, this figure shows: (a) A simple configuration interface for drawing a number among m = 10,000,000 candidates, i.e., from 0 to 9,999,999. The number of stakeholders participating in the drawing and additional metadata related to it can also be defined. (b) A snapshot of the Commitment phase, as seen by Stakeholder S0 in a drawing involving 5 stakeholders. In this snapshot, S0 is then free to choose a number n0 to commit, which is combined with a random mask for better security against brute force attacks. Meanwhile, S1, S3 and S4 have already sent the hashes h1, h3 and h4 of their own commitments, n1, n3 and n4, respectively; as a result, these stakeholders cannot modify the chosen values n1, n3 and n4 anymore. (c) A snapshot of the Reveal phase, as seen by Stakeholder S3. The figure shows that S3 is the only one who has not yet revealed the chosen value for n3, while all of his/her peers have already revealed n0, n1, n2 and n4. Nevertheless, S3 can only reveal the correct n3 (and corresponding mask), since the revealed value must match the committed value h3. (d) The completion of the protocol, when one of the eligible numbers (namely, 6,932,980) is picked with uniform probability based on all stakeholders' contributions n0, n1, n2, n3 and n4. 394 (a) Random drawing with 5 stakeholders (S0, S1, S2, S3 and S4) and modulus value m = 10, 000, 000. (b) S1, S3 and S4 after commitment, as seen by S0. Value committed by S4 is h4 = 4VtCmakPK1pBiAN4pDW7Pj . . . (c) S0, S1, S2 and S4 in reveal phase. Partial result as seen by S3. (d) Drawing result: 1,610,027 + 5,871,032 + 6,029,108 + 7,664,824 + 5,757,989 mod 10,000,000 = 6,932,980. Figure 8: Proof-of-concept Java implementation: screenshots 395 8 Final Remarks Previous articles of this research group have explored the need of randomization procedures in legal systems, like the random assignment (distribution) of legal cases to individual judges or courts, the sortition of jurors for a given case, etc., see Marcondes et al. (2019), Saa and Stern (2019), Silva et al. (2020). Moreover, these papers provide extensive discussions on how to build honest (statistically non-biased) and cryptographically secure procedures and protocols, on the sociological and political importance of using fully transparent and auditable procedures, and on the positive effects of using procedures fully compliant with the aforementioned desiderata in the constitution of strong and autonomous legal institutions. Finally, breaking away from vicious old habits can always be stimulated by respectful criticism, by firm encouragement, and by making available user friendly tools that facilitate the adoption of virtuous new habits without the imposition of additional difficulties beyond the already heavy load of overcoming corporate inertia. This paper provides such a tool, fully compliant with all technical desiderata, user friendly, written in freely available and open source code. The authors hope it will be soon put to use by Brazilian legal institutions and, if necessary, stand ready to help in this endeavor. Acknowledgments This work was supported by: Ripple's University Blockchain Research Initiative; CNPq (Brazilian National Council for Scientific and Technological Development – grants PQ 307648/2018-4 and 301198/2017-9); and FAPESP (São Paulo Research Foundation, grants CEPID-CeMEAI 2013/07375-0 and CEPID-Shell-RCGI 2014/50279-4). The authors are grateful for suggestions received from participants of the Interdisciplinary Colloquium on Probability Theory, held on October 10, 2019 at IEA-USP (Institute of Advanced Studies of the University of Sap Paulo), for early conversations with Julio Adolfo Zucon Trecenti from ABJ (Brazilian Jurimetrics Association), and for the mobile interface design con396 ceived by Giovanni A. dos Santos and Joao Paulo A. S. E. Lins. The authors are grateful for the invitation of Jean-Yves Beziau, from ABF (Brazilian Academy of Philosophy), and for the effort of José Acácio de Barros and Décio Krause, organizers of the Festschrift celebrating Doria's 75th birthday. References A. Beutelspacher (1994). Cryptology. Mathematical Association of America. X. Bultel, J. Dreier, P. Lafourcade, M. More (2017). How to explain modern security concepts to your children. Cryptologia, 41, 5, 422–447. 10.1080/01611194.2016.1238422 M. R. Fellows, N. Koblitz (1994). Kid Crypto. Congressus Numerantium, 99, 9–41. J.M. Hammersley and D.C. Handscomb (1964). Monte Carlo Methods. Methuen. V. Fossaluza, M.S. Lauretto, C.A.B. Pereira, and J.M. Stern (2015). Combining optimization and randomization approaches for the design of clinical trials. Springer Proceedings in Mathematics and Statistics, 118, 173–184. doi:10.1007/978-3-319-12454-4 14 M.S. Lauretto, F. Nakano, C.A.B. Pereira, J.M. Stern (2012). Intentional Sampling by Goal Optimization with Decoupling by Stochastic Perturbation. American Institute of Physics Conference Proceedings, 1490, 189-201. doi:10.1063/1.4759603 M.S. Lauretto, R.B. Stern, K.L. Morgam, M.H. Clark, J.M. Stern (2017). Haphazard Intentional Allocation and Rerandomization to Improve Covariate Balance in Experiments. American Institute of Physics Conference Proceedings, 1853, 050003, 1–8. doi:10.1063/1.4985356 N. Luhmann (1985). A Sociological Theory of Law. Routledge. N. Luhmann (1989). Ecological Communication. The Univ. of Chicago Press. M. Naor, M. Yung (1990). Public-key cryptosystems provably secure against chosen ciphertext attacks. Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, May 13-17, Baltimore, MD, p. 427-37. D. Marcondes, C. Peixoto, and J.M. Stern (2019). Assessing randomness in case assignment: The case study of the Brazilian Supreme Court. Law, 397 Probability and Risk, 18, 2/3, 97–114. doi:10.1093/lpr/mgz006 J. Pearl (2009). Causality: Models, Reasoning, and Inference. Cambridge Univ. Press. B. Ripley (1987). Stochastic Simulation. Wiley. P. Rogaway, T. Shrimpton (2004). Cryptographic hash-function basics: Definitions, implications, and separations for preimage resistance, secondpreimage resistance, and collision resistance. Lecture Notes in Computer Science, v.3017 p.371-88. O.T. Saa, J.M. Stern (2019). Auditable Blockchain Randomization Tool. Proceedings, 33, 1,, 17.1–17.6. doi:10.3390/proceedings2019033017 P. Scozzafava (1993). Uniform distribution and sum modulo m of independent random variables. Statistics and Probability Letters, 18, 4, 313-314. M.V.M. Silva, M.A. Simplicio, R.A.C. Pfeiffer, J.M. Stern (2020). A Fair, Traceable, Auditable and Participatory Randomization Tool for Legal Systems. arXiv:2006.02956 J.M. Stern (2008). Decoupling, Sparsity, Randomization, and Objective Bayesian Inference. Cybernetics and Human Knowing, 15, 2, 49-68. J.M. Stern (2011). Symmetry, Invariance and Ontology in Physics and Statistics. Symmetry, 3, 3, 611-635. doi:10.3390/sym3030611 J.M. Stern (2018). Verstehen (causal/interpretative understanding), Erklären (law-governed description/prediction), and Empirical Legal Studies. JITE Journal of Institutional and Theoretical Economics, 174, 1, 105-114. doi:10.1628/093245617X15120238641866 398 Cover of Doria's Festschrift book