viernes, 11 de agosto de 2017

On the Universe as a Process of Unique Events



On the Universe as a Process of Unique Events

Notes from (and on) the article by Marina Cortês and Lee Smolin, "The Universe as a Process of Unique Events" (Physical Review, 2014). With a few corrections and observations.

In this important article, Cortês and Smolin (herafter C&S) describe a new class of models of quantum space-time based on energetic causal sets and show that under natural conditions space-time emerges from them. These are causal sets whose causal links are labeled by energy and momentum and conservation laws are applied at events. The models are motivated by principles they propose govern microscopic physics which posit a fundamental irreversibility of time. One consequence is that each event in the history of the universe has a distinct causal relationship to the rest. This requires a novel form of dynamics which can be applied to uniquely distinctive events. They hence introduce a new kind of deterministic dynamics for a causal set in which new events are generated from pairs of progenitor events by a rule which is based on extremizing the distinctions between causal past sets of events. This dynamics is asymmetric in time, but they find evidence from numerical simulations of a 1 + 1 dimensional model, that an effective dynamics emerges which restores approximate time reversal symmetry. Finally they also present a natural twistorial representation of energetic causal sets.

Thus far the abstract. Only mathematicians and first-class theoretical physicists are advised to read the original paper in Physics Review, linked below. This is a mere attempt at popularization (a modest one, as the paper has already won an award in cosmology). Although I have detected several typos, the mathematics of the paper is beyond my ken, so I will "skip the equations" in one fell swoop and focus on C&S's account of the import and significance of the operations. One must point out, though, that:

- the theory in this paper is complemented by two companion papers by C&S in which the mathematical model for fundamental irreversibility in cosmology is further specified and developed, and some of its implications for quantum foundations are drawn out.

- the mathematical treatment of this issue should be placed in the context of Lee Smolin's evolutionary cosmology, as exposed in Time Reborn and (with Roberto Mangabeira Unger) in The Singular Universe and the Reality of Time. A lively and accessible summary is available (in English and Spanish) in Smolin's interview with Robert Sawyer linked in the works cited.  This theory includes not just a new conception of time and cosmology, but also a "downgrading" of the status of mathematics, which is demoted from its transcendental position dictating the underlying structure of reality, beyond time, to a more modest instrumental role in a human world. Mathematical models are tools which, though they do not uncover any fundamental realm transcending the universe (such as the mind of God, etc.), are useful in modelling for their manipulation a selection of aspects of physical reality, or allowing operations within the much more localized realm of human communication and research in scientific disciplines. That is, in spite of the phrasing used sometimes in the paper, the mathematical model presented here is not the formula which generates reality, the way E=mc2 is sometimes conceived as being inscribed in some constellation. It is a mathematical set of formulas which are meant to account what other theories leave out, i. e. the singularity of events in the universe whose description is necessitated by the cosmological scope of the theory.

The paper helps to provide a crucial theoretical foundation which allows the (consilient)  embedding of classical physics, and its mathematical formulation of time-reversible dynamics, within the novel paradigm of evolutionary cosmology based on the irreversibility and the inclusive reality of time. A fuller exposition of this paradigm, which amounts to a substantial revision of the current paradigm of physical science and mathematics, is available in the volume The Singular Universe and the Reality of Time, by Roberto Mangabeira Unger and Lee Smolin.



neutrinos


The current scientific consensus about time, the one they reject, is formulated thus by C&S:

"Time is an illusion. It emerges as we coarse grain, zooming out from timeless fundamental physics. It's not a fundamental quantity of nature. In searches for quantum gravity we are to seek an underlying Plack length description, expressed in the form of timeless equations, where time is not to play any role. When time emerges it is to parametrize equations that are symmetric under time reversal. The evident time-asymmetry of nature is then held to be an accident, due to an improbable choice of initial conditions."

C&S complain that this view "twice diminishes our basic experience of the world as an unfolding series of moments, to the realm of accident and illusion (i) because the asymmetry of time is held to be an accident, and (ii) because time itself is held to be emergent".

[Note, by the way, that the appeal to "our basic experience of the world" may be misleading here, because the whole issue of the experience of time in its perceptual-animal sense, not to mention the peculiar dimensions of human time, is outside the scope of this paper, which focuses altogether on the basic processes giving rise to events in nature, quite apart from the issue of how these events are construed by sentient beings.]

"In this paper we will propose the diametrically opposite view. We develop the hypothesis that time is both fundamental and irreversible, as opposed to reversible and emergent. We will argue that the irreversible passage of time must be incorporated in fundamental physics to enable progress in our current understanding. The true laws of physics may evolve in time and depend on a distinction between the past, present and future, a distinction which is absent in the standard block universe perspective.

The present instant is a primitive and part of fundamental processes, and the laws of physics may refer to it preferentially. There is a process continually acting in the present bringing into existence the next moment. The present may code aspects of past states but the past is no longer accessible, as could be argued to be the case in the block universe picture. Along the same lines the future has yet to happen, and aspects of it may even be open, i.e. not computable from a complete knowledge of the present.

We'll argue that, in contrast to the time reversal symmetry which is standard practice, time is fundamentally asymmetric and irreversible. The future is different from the past: the process by which the present becomes the past and gives rise to the future cannot be inverted to allow the perfect reconstruction of the past.

Based on this view we propose four principles. The first will concern the nature of time.

(i) Principle A  Time is a fundamental quantity; the agency of time is the most elementary process in physics, by which new events are created out of present events. Causality results directly from the irreversible agency of time.

(ii) Principle B   Time has a fundamental directionality. The future develops out of the present constantly; there are no causal loops and no regions or phenomena where time "evolves backwards." This implies that the fundamental laws that evolve the future from the past are irreversible in the sense that they have no inverse by which the past state can be reconstructed from the present state.

A second pair frame the way that the dynamics of the world may be expressed.

(i) Principle C  We choose a relational point of view, according to which the space-time properties of an object or event arise from its relationship with other objects or events. All space-time properties have a dynamical origin.

(ii) Principle D  Energy is fundamental. Energy and momentum are not emergent from space-tim, rather the opposite is the case, space-time is emergent from a more fundamental causal and dynamical regime in which energy and momentum are primitives.

We chose to model these assumptions within a discrete framework. This means that we envision the history of the universe as a set of events, endowed with both a causal structure and intrincis energy-momentum variables.

Before presenting this model, we not that the assumption that events are described relationally has a strong consequence, which is the

(i) Uniqueness of events  The relational properties of each event in cosmological evolution make it unique and distinguishable from all others.

This is a consequence of the demand that each event be distinguishable by its relational properties. Within a discrete causal structure an event can only be distinguished by its causal past. Furthermore the events that make up the causal past of a given event cannot have any absolute labels, for those are only labeled by their causal pasts. But those are part of the causal past of the given event. This implies that any two events must have nonisomorphic causal pasts in that there is no map that takes one to the other preserving their causal pasts. This can be considered to be a consequence of Leibniz's principle of the identity of the indiscernible.

Uniqueness need not contradict our standard scientific method which we base on statistical inference from repeated experiments. Repeated systems do occur—to a sufficient degree of approximation—if we are considering subsets of the whole universe. Whenever we apply a boundary to define the system under study in the laboratory, we are severing the relations it has with the remainder of the system. Truncating these connections makes it possible for the subsystem to appear approximately similar to other subsets as well as to itself when subject to repetitions in time of the initial conditions. Similarity is local, and a result of truncation, and repetition is never exact when the whole system is considered.

Uniqueness of events is a strong requirement as it demands each event has a causal past that is complex enough to be district from the causal pasts of all the other events. As a result, fundamental laws acting on unique events need to take their complexity into consideration. This would seem to imply large informational inputs and outputs, which appears to contradict the idea that elementary events should be simple. We will show below that this query has a simple answer, which is that a space-time must emerge within which the network of complex historical relations embeds. When an embedding of the history of events in a space-time exists one can use coordinates on the space-time to uniquely identify each event. this is a highly nontrivial requirement, but in the next section we describe a model in which it is satisfied.

Having set out the physical picture which follows from our assertions, we present in Sec.II a simple model which is useful to study them. In this model a causal set [1] is generated by an event generator which acts according to a rule satisfying the principles we laid out. The events are endowed with energy and momenta which are propagated from old to new events subject to conservation and energy-momentum relations. We propose to call causal sets so endowed energetic causal sets. We find some remarkable results:

(1) Space-time is not part of the fundamental description but emerges as an arena for a statistical description of the fundamental process.

(2) The emergent equations of motion, which describe the embedding of the causal processes in space-time, cannot always be solved consistently but when they can there is a classical limit which is a theory of interacting relativistic particles.

(3) Numerical studies of the model in 1+1 dimensions show that a reversible effective dynamics can emerge from irreversible evolution rules.

(4) In 3 + 1 dimensions the model can be reformulated in the language of twistor theory."

(...)

[In this model,] "the history of the universe is described as a set of N events, EI, I = 1, ..., N.

Principle A is incorporated here, in that each event is created as the result of a process acting on the prior set of events. The process is the activity of time. In the model we call it the events generator.

The event generator must make two decisions each time it creates an event: First, which of the prior set of events are to be progenitors of the new event. Second, how are the properties of the new event determined from the properties of its progenitors.

Each event is endowed with energy and momenta, which are held to be primitive properties. These are conveyed to an event from its progenitors. This realizes Principle D."

(...)

"We can gain more insight into the physics of this model by introducing a statistical formulation. Since the dynnamics which generates the causal set is deterministic, but non-local, we have to define carefully what ensemble the statistical averaging corresponds to. We also have to respect the assumption that each fundamental event is unique when all the information about its causal past is known. There is also a single, unique causal history for the whole universe, Cuniverse. A statistical ensemble can then only arise from the consideration of subsystemss of intermediate scale, neither cosmological nor universal. We propose then to introduce statistics by studyin ensembles composed of subsystems which are characterized by incomplete information about their causal pasts."

(...)

"The emergence of the zaI's as coordinates of the embeddings of the events in a space-time neatly solves the puzzle of how to give dynamics to unique events. As we posed the question in the Introduction, the question was how to conceive of a law or a rule for generating unique events when that rula had to be simple, but what distinguishes the events in a big universe are the intricacies of their histories. The answer is to invent space-time so that each event has a unique embedding in that space-time."

(...)

Twistor formulation

"We can solve the energy-momentum constraints for massless particles (3) in terms of two component spinors. (...) So we elegantly reconstruct the embeddings of the causal processes in Minkowski space-time."


Description of the model

 
"We will consider events that have two incoming photons and two outgoing photons, in which one pair is left-movin, the other is right moving" (...)  "The spatial compactness is also helpful to get an interesting model; without it the worldliness disappears as nothing happens after a finite number of interactions" [Garbled text; my hypothetical reconstruction].

(...)

"Each event is distinguished by its past, as well as its position and time labels, and is stored in the past of the parent event that contributed the left incoming ray. Hence all events belong to one and only one past, and the intersection of all pasts is null."

However[,] even a moderately complex measure of each [past] quickly becomes computationally unfeasible because the number of differences of pasts to compute in the generation of each new [event] increases exponentially with the number of intervening events. Since our sole requirement for each past is that it be unique, the need for complexity can be alleviated by taking a simple measure of each as the average of the space-intervals of events in that past. This we know to be unique since we have continuum space-time available. So the emergence of space-time from the interaction momenta is at the same time ensuring [the] uniqueness of each past. (...) Each event is distinguished by its past, as well as its position and time labels, and is stored in the past of the parent event that contributed the left incoming ray. Hence all events belong to one and only one past, and the intersection of all pasts is null.

This model is very simple[:] the momenta play a small role, which is just to propagate themselves acoording to the conservation rules. But it can be used to study the question of how time reversible approximate laws might emerge out of time irreversible fundamental laws, and hence address our central assumption, principle A."

(...)

Description of results

"We saw evidence for a simple characterization of the time evolution in which the systems pass in time thorugh two distinct phases. The systems begin in a disordered phase, followed by an ordered phase we call the locked in phase. In the disordered phase, the time asymmetry of the event generation rule manifests itself in a visible time asymmetry of the pattern of space-time positions of the events. In this initial phase, the events form a roughly random pattern in space-time, characterized by a large variety of spatial positions, within an envelope that is asymmetric under time reversal. In the [locked-in] phase an approximate time symmetry emerges. This second phase is dominated by persistent repeating patterns we can call quasiparticles."

[Note the similarity to the account of the origin of repetition, equilibration and partially stable systems out of an original undifferentiated force in Herbert Spencer's theory of evolutionary cosmology, as expounded in his First Principles.]

"Each run began with a period of seemingly chaotic behavior, manifested by a disordered ensemble of events in which the time asymmetry of the algorithm is evident. In this disordered phase all the past-sets intervene with similar weights in the generation of new events. Variety is maximal. Any two past-sets interact only very briefly—two times—just enough for both paris of photons of the parent to be used and discarded, and interaction moves on to another pair." (...) "In Fig. 1 we see a very different behavior emerging after a sufficient number of events has been generated. As time progresses, we gradually begin to see two past-sets interlocking, ever so briefly for a short succession of events, and we start to see momentarily glimpses of stable trajectories in the plotted evolution. The interlocking between pairs of past-sets lasts longer each time it occurs. These give rise to recognizable trajectories of quasiparticles." (...) "Quasiparticles move in straight lines in space-time ,and as such have a dynamics which appears to be unchanged under reversal of the time coordinate. We call this the lock-in phase." (... ) "given enough time models eventually evolve towards the lock-in and regular phase".

"The general behavior is thus an evolution from initial disordered behavior, from which a few quasiparticles eventually set off and interact only amongst themselves. The other pasts are left behind and stop interacting. These results support the hypothesis that approximately time reversal symmetric behavior can emerge from time asymmetric evolution rules."

"The emergence of regularity always seems to be related to the loss of novelty in the system. In our case the novelty element is the random number of cycles added at each event generation. In the beginning all pasts intervene in similar amounts, any pair interacts only twice and interaction moves on to the next pair. The information in each past is minimal and novelty, in the form of the random number of cycles added, plays a more determining role in the generation of new events than the interaction rule. However, with each event generation the information in each past-set builds up, and begins to weigh increasingly more in the selection of pairs. Correspondingly novelty plays progressively a minor role, until there are no new positions created and the number of possible space positions is constant. This is represented by the emergence of quasiparticles, with regular trajectories, which we observe in all four figures. This is the lock-in phase which is identifiable by the regularity of structures in the network. We observed that, quite surprisingly, this lock-in regular phase exists [for each] of the interaction rules and initial conditions we considered."

[Note that the 'novelty' C&S refer to here is exactly the reverse of 'the novelty of emergent phenomena', as the novelty of emergent phenomena, such as quasiparticles, is enabled precisely by the build-up of information from the past, from the increasing complexity of locked-in phenomena, and from the loss of the short-term randomness of the initial phase as described here. This process has also been put in verse by Abraham Cowley. See 'Chaos and the Emergence of Order in Evolutionary Cosmology. Two evolutionary accounts'.]

"So there is an interplay between the random input and the regularity of the rule we apply. The observed tendency is for the regularity to eventually overcome the amount of diversity. Irreversible dynamics seems to have the tendency to evolve toward predictable, reversible evolution."

"The general system is irreversible because each state may not have a unique predecessor, even if determinism requires each state to have a unique successor. But once a limit cycle is entered, each state has both a unique predecessor and a unique successor. Hence, restricted to the limit cycles, the system may be said to be governed by an effective reversible dynamics. This will be discussed in more detail in [Fundamental Irreversibility in Cosmology]."

Summary and conclusions

"We began this work as we began the paper, by a search for principles that could guide the search for a physics that could be applied to the whole universe. This must be a novel form of dynamics that can be applied to a unique system—the universe as a whole, that is neither quantum mechanics nor general relativity, from which quantum physics and space-time emerge as approximate descriptions of systems that may be regarded to a sufficient degree of approximation as isolated subsystems."

A. Summary of ideas

"We formulated four principles, which we presented as Principles A to D in Sec. I. The first two have to do with the nature of time and assert that time, taken as the agency that produces new events from present events, is truly fundamental and irreversible. This focuses attention on an agent which creates novel events—what we call the event generator[.] The third principle asserts the philosophy of relationalism—that space-time emerges from the action of that agent as a reflection of the network of causal relatins that agent creates. The fourth principle limits the reach of relationalism to assert that in addition to their relational, spatial-temporal properties, events are endowed by intrinsic properties, and these include energy and momenta.

From these principles we draw several conclusions.

(1) We work within an ontology of events, and the causal processes that create new events out of old events.

(2) Each event should be uniquely distinct from the rest. This is a consequence of Leibniz's principle of the identity of the indiscernible applied to an ontology of events and causal processes.

(3) The fundamental dynamics resides in the choices made by the events generator: which pairs (or small sets of) present events will give rise to a new event and how are the properties o those progenitor events—both relational and intrinsic—transferred to the newly created event. We should emphasize that no existing paradigm addresses the question of why particular events are created. The existing causal set of dynamics are either stochastic or based on a sum of stories approach to a quantum physics. Another paradigm based on an events ontology—the Feynman diagram approach to QFT—is also based on a sum over stories.

(4) The event generator must make use of and create tha data that uniquely identifies each event.

This emphasis on the uniqueness of events, and its role in the fundamental dynamics, stands in  strong contrast to the existing paradigms for dynamics, which assume that microscopic systems are simple and come in vast ensembles of identical copies. This puts the emphasis on symmetry principles applied to identical elementary particles. In contrast to this we assert that at the level of the elementary events and the dynamics that generates them every event is unique so there are no ensembles of identical events and no fundamental symmetries.

Consequently we assert that the kind of physics we have developed to this point, based on identical properties classified by symmetries, must emerge at an intermediate scale. We can say more specifically that this emergence has to do with truncating the description within which each event is unique, so as to give an effective description of subsystems of the universe with minimal information.

It is also natural to conjecture that this truncation is responsible for the statistical nature of quantum physics. The stochastic description arises by neglecting data that renders each event unique which results in a description of emergent quasiparticles that fall into large classes of nearly identical copies. It is at this emergent level that symmetry and identity arise as aspects of a statistical and approximate description. We can express this by saying that there are hidden variables which reside in the data that makes each event unique and determines which events take place.

Time reversal invariance is amongst the symmetries that must emerge from the coarse graining that neglects the unique identity of each event. From our perspective fundamental physics is time asymmetric and the apparent time symmetry of the laws of nature is approximate and emergent.

We explored the implications of this new approach to fundamental physics by inventing a new kind of model of space-time physics: an energetic causal set, which differs from the usual causal sets by having events endowed with intrinsic energy and momenta, which are transmitted by causal processes. We discovered that under natural assumptions—a small enough number of progenitors—space-time naturally emerges as an approximate description based on stationary phase approximation that appears in a statistical description of these systems. This addresses a long standing difficulty of causal set models to generate emergent [low-dimensional] space-times.

We can offer a suggestion of why space-time emerges, which is that it resolves a problem faced by the event generator, which is how to uniquely label each event in a manner that requires a small amount of information and so is computationally efficient. At first, each event is distinguished by its causal past, but to specify these in a large universe to the point they render each event unique would take a vast amount of information. The emergent [spacetime] summarizes this information in a small set of numbers."

[Note that the phrasing and presuppositions here are curiously formalistic, and ascribe to the universe processes which are actually relative to the mathematical theory used to model it. This is particularly striking in view of Smolin's instrumental view of mathematics; this may be attributed to the mathematical views of Cortês, or perhaps it is a case of resolutely adopting for the sake of the reasoning the mathematicians' view of mathematics as literally generating the structure of reality. I propose that this be considered a rhetorical licence.]

(...)

B. Future prospects

"A key goal of future work will be to identify the limit where the science of the subsystems breaks down and uniqueness of cosmological events must be taken into account. Locally the structures in the universal network appear identical, and we achieve the limit of repeatable experiments by truncating the description to subsystems. There is however a scale where this frequentist science breaks down, subsystems are no longer similar, and we need to extend a subsystem to include its unique history in order to obtain a consistent cosmological dynamics."

[This might be the way to address such phenomena as quantum entanglement]

"In the model of unique events, the present is defined as the set of open events, instants that are in the process of completing. In the model we described here, these are those events or instants that still have unused photons. So the present is not a single instant, but a set of instants. This has a parallel in the discussion of whether the present moment is thin or thick: in this view the present is thick, made of those events that are in the process of realizing themselves."

[Note once again how the formal needs of the model lead C&S to use a language that seems to contradict their assumption of time as an irreducible primitive; mathematics necessarily maps time and events into a simultaneous atemporal graph in which some of the operations represent 'the present'.]

"One important question to study is the nature of the set of present events. In the model we studied in the last section, the present is continually generated from interaction of open events. Even in the case of stationary quasiparticles, that have static space-time trajectories, they [are in fact interacting:] their trajectories are created mutually by interacting with each other. There is no absolute stillness, the agency of time is continual. Both in the emergence of quasiparticles and in the irreversible phase, at each moment, one event or two events are transpoted into the past, stop being part of the present, and a new event is created, taking their place, and is brought into the present.

The amount of information in the current live events set characterizes the potential for novelty and diversity of the future network. There are multiple possibilities for how to [quantify] the amount of latent novelty in a system. In our model, novelty arises with the random number of cycles introduced at each event generation. As argued above, when this novelty is exhausted lock-in occurs and the regular phase is reached. In the regular phase the diversity in the system is constant and limited. It will be an interesting [experiment] to make the injection of novelty vary in time and observe how the [system] reacts with regards to its reversible or irreversible dynamics."

[This sounds like a promising avenue to further theorize the "locked-in" nature of such phenomena as strong atomic force, the binding of quarks, etc.as subsystems which partially escape the action of time, or dimensional scales of nature where "time has stopped" at a local level]

"Our results suggest that the irreversibility persists so long as diversity in the system is abundant. If there is a fixed quantity of novelty, the tendency is for it to be used up in the generation of irreversibility. When novelty is exhausted, the amount of diversity (however we choose to quantify it) becomes limited, and the dynamics regularizes, becoming patterned and reversible."

[One might argue that, at a convenient emergent distance, this is the matter with my blog of late; the influx of novelty is null, time has stopped or a locked-in phase has been reached, and the dynamics has become patterned and reversible]

"It is notable to observe that the lock-in and consequent emergence of reversible dynamics is robust and occurs whether the interaction rule is local or nonlocal i.e. whether the progenitors of now events are chosen to have past-sets which are maximally similar or maximally diverse.

A goal for future work will be to understand in more detail how fundamental irreversibility gives rise to standard reversible laws; what is the quantity that represents novelty; and how does the transition to the reversible regime occur."









Works cited

Cortês, Marina, and Lee Smolin, "The Universe as a Process of Unique Events." Physical Review D: Particles, Fields, Gravitation and Cosmology 90.8 (6 Oct. 2014).
http://www.research.ed.ac.uk/portal/en/publications/the-universe-as-a-process-of-unique-events(3c29b15e-2aa7-4660-a81d-e4a31ffaff19).html
http://dx.doi.org/10.1103/PhysRevD.90.084007
2017
_____. "The Universe as a Process of Unique Events." ArXiv (25 Nov. 2015).*
https://arxiv.org/abs/1307.6167
2017
_____. "Quantum Energy Causal Sets." Physical Review D: Particles, Fields, Gravitation and Cosmology 90, 044035 (14 August 2014)
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.90.044035
https://doi.org/10.1103/PhysRevD.90.044035
2017
_____. "Fundamental Irreversibility in Cosmology." Forthcoming (2014).

García Landa, José Angel.  "Chaos and the Emergence of Order in Evolutionary Cosmology: Two Classical Accounts." Ibercampus (Vanity Fea) 29 Aug. 2015.
http://www.ibercampus.eu/chaos-and-the-emergence-of-order-in-evolutionary-cosmology-two-classical-3358.htm
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_____. "Primeros Principios, Resumen y Conclusión." ResearchGate 5 May 2016.
https://www.researchgate.net/publication/301891379
2016
_____. "El paradigma evolucionista en Físíca y en Cosmología (The Evolutionary Paradigm in Physics and in Cosmology)." Social Science Research Network 16 Jan. 2015.
http://papers.ssrn.com/abstract=2550659
2015
_____."La realidad inclusiva del tiempo." Social Science Research Network 19 July 2017.
https://ssrn.com/abstract=3002081
2017

Sawyer, Robert. "Lee Smolin / April 23, 2013 / Appel Salon." Video interview. YouTube (Toronto Public library) 17 May 2013.
http://youtu.be/3y5IQ3kqMt8
2013
_______. "Lee Smolin habla sobre El renacer del tiempo." Academia 27 Dec. 2014.
https://www.academia.edu/9914789/
2014

Smolin, Lee. Time Reborn: From the Crisis in Physics to the Future of the Universe. Boston and New York: Houghton Mifflin, 2013.
_____. "Lee Smolin public lecture: Time Reborn." YouTube (Perimeter Institute for Theoretical Physics) 12 Dec. 2014.
https://youtu.be/ATxi0_-7HqQ
2016

Spencer, Herbert. First Principles. 6th ed. (The Thinker's Library). London: Watts, 1937.

"Time Comes First: Cortês and Smolin Win Cosmology Prize." Perimeter Institute 6 Jan. 2015.*
https://www.perimeterinstitute.ca/news/time-comes-first-cort-s-and-smolin-win-cosmology-prize
2017

Unger, Roberto Mangabeira, and Lee Smolin. The Singular Universe and the Reality of Time. Cambridge: Cambridge UP, 2015.



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