Universe as Network

A simple model of spatiotemporal logic in terms of networks of interconnected events is constructed, and shown to give rise to discrete Clifford Algebra structure. Previous work by Tony Smith demonstrates that discrete Clifford Algebra structure can be used to derive a Feynman checkerboard version of the Standard Model plus gravity. Putting these two pieces together, one has a consistent and empirically accurate model of the universe as a discrete event network. Finally, the possibility of deriving the given rules of spatiotemporal event network dynamics from a yet simpler foundation of pattern maximization is discussed.

Contents

1. Introduction
2. Logical Event Networks
3. Temporal Event Networks
4. From Event Networks to Clifford Algebras
5. Evolving Event Networks

Two and a half decades ago, John Wheeler (1973) suggested that the laws of physics might ultimately be derivable from the statistics of large formulas of propositional logic. Thinking along related lines, David Finkelstein (1993), Tony Smith (1993) and others have sought to derive fundamental physics (the Standard Model of electromagnetic, strong and weak interactions; plus gravity) from variations of elementary set theory called "quantum set theory," "quantum nets," etc. However, these attempts have always fallen short of the goal.

The present investigation is in a similar spirit, but I believe the results are more successful. A simple, set-theoretic "event network" model of the universe is proposed, and it is shown how the structure of physical reality arises in this network through the emergence of simple rules for the interaction of "temporal events" (events with ordered links to other events). Specifically, event networks are shown to give rise to a discrete Clifford Algebra structure; and this algebra has been shown, in a series of papers by Tony Smith (1993, 1995, 1995a, 1997), to give rise to the Standard Model plus Gravity.

A sketch of Smith's development, called the D₄-D₅-E₆ model, may be useful background information. The real Clifford Algebras have periodicity 8, so the fundamental real Clifford Algebra is Cl(0,8)=Cl(8), whose vector, +half-spinor and -half-spinor representations are all isomorphic by triality to the octonions (Cayley algebra). The discrete octonions are naturally representable as E8 lattices, yielding an 8-dimensional model of spacetime. Reducing an E8 lattice to a 4-dimensional physical spacetime lattice produces a HyperDiamond lattice, and a Feynman checkerboard model on a HyperDiamond lattice represents the physics of the Standard Model plus Gravity. This model is related to the D₄, D₅ and E₆ Lie algebras; hence the name D₄-D₅-E₆. The model predicts values for the mass ratios of elementary particles that agree with experimental results in all cases except the truth quark, for which the D₄-D₅-E₆ model predicts a tree level constituent mass of about 130 GeV, whereas FermiLab reports a value of 180 GeV. However, a careful examination of FermiLab's experimental results shows that the actual data collected are consistent with the 130 GeV value (see Smith, 1997a).

Here we will not enter into the complexities of Feynman checkerboards and the like, but will rather take the D₄-D₅-E₆ model for granted, and content ourselves with a derivation of the basic Clifford Algebra structure used by the D₄-D₅-E₆ model from a simple underlying model of the universe as an event network. The present treatment of graphical logic and event networks is quite general and flexible, and could potentially be modified to harmonize with other versions of the Standard Model plus gravity, besides D₄-D₅-E₆. For example, the physics models given by Dixon (1994) also use Cl(8) in crucial and interesting ways. However, the match of event networks with the D₄-D₅-E₆ model is quite natural, more so than with any other current physics model; and the combination of D₄-D₅-E₆ with event networks provides the first solid implementation of Wheeler's concept of deriving physics from basic logic. The universe, insofar as it is described by known physical laws, is seen to emerge from simple logic-inspired dynamics in networks of discrete events.

Section 2 introduces the basic logical event network model, containing rules for the activation and interaction of atemporal events; and extends this model into a "spatial" event network model, involving a different interaction rule. Section 3 introduces the new concepts of chronons and contexts, necessary for specifying the interaction of temporal events. Section 4 proves the theorem that temporal event networks generate Discrete Clifford Algebras, and shows how spatiotemporal event networks can go one step further, and generate the discrete lattice versions of Clifford Algebra, as needed for the D₄-D₅-E₆ model. The final section speculatively explores a possible route to constructing an even simpler theory, by deriving the laws of spatiotemporal event interaction from a "principle of pattern maximization," stating that the universe's event networks evolve event transformation rules which produce structures of maximal emergent pattern.

The ideas described in this paper emerged during an ongoing research collaboration with Tony Smith, Onar Aam and Kent Palmer.

The basic idea of this investigation is to consider the universe as a network of interconnected events. "Interconnected" means that each event is linked to certain other events. Furthermore, since nothing has been introduced at this stage except for events and interconnections, each event is characterized by its interconnections, by its links to other events. each event is a collection of links.

In addition to the links that constitute its being, each event possesses two binary properties: it can be active or inactive; and it can be open or closed. Furthermore, pairs of events can also be either active or inactive: when two events are active, and there is a link between them, then the pair of events is active, and we say that the two events interact. The dynamics of the universe is determined largely by what happens when activity spreads through the event network. Both open and closed events are important, but in the following, unless otherwise specified, the default assumption will be that the word "event" refers to an open event.

At this level, there are four simple laws for event behavior:

• Identical events are collapsed into a single event

• When an open event becomes active, any identical links coming out of it disappear

• When two open events interact, a new event is created that has all the links of both members of the pair, and this new event becomes active

• When a closed event and another event interact, each event linked to by the first of the two events is allowed to interact with each event linked to by the second of t two events; then a new event is created, which links to all events created in this process

Some clarification of terminology may be useful. Two events are identical if neither one has a link pointing anywhere that some link pointing out of the other one does not also lead to. Two links are identical if they point to the same event. If an open event has two links pointing to the same event, then these two links are annihilated, but the event that they point to is not.

What happens when a pair of open events becomes active, according to these rules, is that first a new event is created which is the set-theoretic union of the two events: it has all the links of the first event, and all the links of the second event. Then, all links that occur more than once are annihilated, so that what is left is the collection of links that were in one of the original two events, but not both. This is what is called, in set theory, the symmetric difference.

The symmetric difference is formally identical to the XOR operator in logic; thus I call this event network model "graph-theoretical logic." Wheeler wanted to derive physics from propositional logic, and this is essentially what is being done here; the key point, however, is that the standard formulation of propositional logic is not workable for this purpose. The universe is a network of interconnected events, and so one needs a model in which logic emerges from event networks.

When two closed events interact, on the other hand, there is no cancellation. Instead, activation is passed through the network, and the "children" of the two events are allowed to interact, producing new offspring. Closed event interactions preserve structure rather than logically operating on it; they are intuitively

"spatial" in nature, representing a kind of primal space, and indeed are required to build up physical space out of event networks.

A digraph endowed with the rules for open events described in this section, I will call a logical event network. A digraph endowed with the rules for open and closed events described in this section, I will call a spatial event network.

So far, what we have is mainly a network representation of elementary set theory, or equivalently, propositional logic. The next stage is the emergence of time. Physical time is a higher-level construct, but it emerges from a lower-level temporal foundation that we call "primal time." Primal time is nothing more than the concept of order, together with a rule telling what happens when ordered entities become active. It is graph-theoretic temporal logic.

An event has order if the links coming out of it occur in a certain order, instead of being an unordered collection. An event with order may be called an "temporal event." The only temporal events that will be considered here are open temporal events; so I will omit the word "open" and consistently just speak of "temporal events." When a pair of temporal events becomes active, something different happens from what happens when a pair of events activates. The essence of the structure and dynamics of the physical universe is located precisely here: in the rule which tells what happens when two temporal events interact.

In order to articulate this rule we must introduce two new concepts: contexts and chronons. A "context," first, is an temporal event C, together with a collection of other temporal events that exist "within the context" of C, in the following sense: their links point only to themselves or to events that C also links to, and the order of their links is consistent with the order of C. Temporal events existing in the same context share a common ordering, and can interact as ordered entities.

A "chronon," next, is a fundamental unit of time, represented graphically as a link from an event to itself. Chronons, self-links, obey the three basic event network rules just like links spanning two different events. But they also obey an additional rule:

• If an open event has links to two events, which differ only in that one of the two has a chronon and the other does not, then both of the two links are annihilated.

Note that this rule provides for annihilation of links even without activity. Pairs of identical links coming out of active events are annihilated; but pairs of links to events that are identical except for the presence of a chronon are always annihilated, whether the event they come out of is active or not.

Now suppose that temporal event A and temporal event B exist within a common context, and that temporal event A links to temporal event B. What happens when both events become active? Because it is A that points to B, and not vice versa, we may say that A acts on B. When A acts on B, a new temporal event is created, containing all the links belonging to either A or B. This is as would occur if A and B were simply events, without an intrinsic concept of order, rather than temporal events. But something else also happens. What happens is that, each time one of B's links encounters one of A's links that comes after it in the common contextual ordering, a chronon is created, attached to the new event that is produced by the interaction of A and B. In other words:

• When two linked temporal events in a common context are activated, a new temporal event is created, containing all the links possessed by either of the original temporal events. each of the target temporal event's links creates chronons attached to the new temporal event, one for each of the source temporal event's links that comes after it in the contextual ordering. Then, the new event becomes active.

The implications of this rule are interesting. When the new temporal event created by the interaction becomes active, then pairs of identical links coming out of this temporal event are annihilated -- and this applied to chronons as well. Therefore, if the number of chronons created was even, the annihilation process will remove all chronons. On the other hand, if the number of chronons created was odd, the annihilation process will remove all but one chronon. The result is that temporal events, upon interaction, produce temporal events with one chronon or no chronons.

A logical event network, augmented with the rules described in this section, I will call a temporal event network. A spatial event network, augmented with the rules described in this section, I will call a spatiotemporal event network.

4. From Event Networks to Clifford Algebra

Smith (1995, 1997) shows that the discrete lattice subset DCL(8) of the Clifford algebra Cl(0,8)=Cl(8) can be used to derive all the structures of physical reality. However, as Barry Simon (1996) has shown, the Clifford algebra Cl(N) is the group algebra of a special finite group, which I call the Discrete Clifford Group, DCLG(N). From an event network perspective, once the DCLG has emerged, the discrete Clifford Algebra DCL(N) can be easily derived using interacting closed events. The main work is in deriving the DCLG itself from temporal event interactions.

Simon defines the DCLG in terms of a basis of elements {e₁, ..., e_N}, and the subsets of {1,...,N}.

Where A is a subset of {1,...,N} and ei is a basis vector, the elements of DCLG(N) are of the form +e_A or -e_A, where e_A is the Clifford element e_j(1)e_j(2)...e_j(k), and j(i), j(2),...,j(k) are the elements of the subset A. Since we have used + and -, this yields 2^N+1 elements in DCLG(N).

The multiplication rule for DCLG(N) is given by

(x_Ae_a) (x_Be_b) = x_A,Be_C

where C is the symmetric difference of A and B, and x_A, x_B, and x_A,B are +1 or -1. The sign x_A,B of the product is determined by an algorithm using the rules.

e_ie_i = +1

e_ie_j = - e_je_i for i =/= j

The algorithm works as follows. First, one creates a sequence of elements e_X by concatenating the sequence representations e_Aand e_B (placing e_B at the end of e). Then one performs a series of operations on e_X, using the rule e_ie_j = - e_je_i to move each of the B-elements to the left until it either:

• meets a similar element, in which case it is cancelled with the similar element using e_ie_i = +1, or

• finds a place in between two A-elements in the proper order.

According to this algorithm, each time a B-element is moved to the left, the sign of e_X changes. If the total number of moves is even, then the sign of e_X is unchanged by the algorithm, and x_{A, B}= x_Ax_B. If the total number of moves is odd, then the sign of e_X is reverse by the algorithm, and x_{A, B} = - x_Ax_B. Elegantly, the algorithm also performs the symmetric difference operation: the only elements left in e_X after the transformation process is complete, are the ones that are present in both e_Aand e_B.

Some simple eexamples may be helpful here. Suppose we have

A = e₁ e₂ e₃

B = e₂ e₃ e₄

nd we wish to compute the product

A * B = e₁ e₂ e₃ * e₂ e₃ e₄

According to the DCLG algorithm, we move the elements of B over, one by one, until they meet their

respective elements of A. each time we move an element of B past an element of A, we switch the sign of the product from its initial positive value. So to compute A*B, we have

eX = e₁ e₂ e₃ e₂ e₃ e₄ =

- e₁ e₂ e₂ e₃ e₃ e₄ =

- e₁ e₄

The sign is negative. And what is left over after cancellation, e₁ e₄, is eexactly the symmetric difference of the sets A and B.

Or, consider:

e₁ e₂ e₅ * e₁ e₂ e₄ =

- e₁ e₂ e₁ e₅ e₂ e₄ =

e₁ e₁ e₂ e₅ e₂ e₄ =

- e₂ e₂ e₅ e₄ =

e₄ e₅

Again, each switch does a sign change, and the resultant is the signed symmetric difference of the multiplicands.

The essential meaning of this peculiar sign algorithm is best understood by reformulating the eexpansions in terms of elements as bit strings, sequences of zeroes and ones. The length of the bit strings is N, the number of basis elements. In an algebra with four basis elements, for eexample, the element e₁ is represented 1000; the element e₃ is represented 0010; the element e₂e₃ is represented 0110; the element e₁e₂e₃e₄ is represented 1111.

Consider again the first eexample above, e₁ e₂ e₃ * e₂ e₃ e₄ = -e₁ e₄. In bit string notation, this result is represented

1110 * 0111 = - 1001

To compute the sign using only bit strings, one reasons as follows. First, one lines up the two multiplicands, one under the other:

1110 *

0111

Then, for each 1 in the second multiplicand (B), one adds up the number of 1's in the first multiplicant (A) that come strictly after it (moving from left to right). (For instance, for the 1 in the second place in B in this example, we have the one 1 in the third place in A. For the 1 in the third place in B in this example, we have no ones coming after it in A; and likewise for the 1 in the fourth place in B in this example.) Then, one takes the sum one has gotten by this procedure, and uses it to determine the sign of the product. If this sum is odd, the sign of the product is negative. If the sum is even, the sign of the product is positive.

In this example, we have moved through the 1 bits of the representation of the second multiplicand B, summing for each one the number of 1 bits of A that come after it. Equivalently, one can sum over all the 1's in the first multiplicand, A, summing for each one of these, the number of 1's in B that occur strictly before it. The two procedures always give the same result.

Similarly, the second example above, e₁ e₂ e₅ * e₁ e₂ e₄, translates to

11001 *

11010

Here, for the 1 in the first place of B, we have two 1's in A coming after it; for the 1 in the second place of B, we have one 1 in A coming after it; for the 1 in the fourth place of B, we have one 1 in A coming after it. So, the sum is four, and the sign of the product is plus.

This formulation of the DCLG makes the connection with event networks relatively clear. First, the symmetric difference, as already pointed out, is executed by the activation of a pair of connected events in an event network. Suppose one has a set of N events, called S(N)={e₁,...,e_N}, and then considers two events A and B, each one linking to some subset of S(N). If the pair (A,B) becomes active, then according to the rules of event networks they create a new node, which, after annihilation of identicals, contains links to precisely the symmetric difference of the subsets of S(N) linked to by A and B individually.

And what about the sign? Quite simply, chronons in temporal event networks correspond to plus and minus signs in DCLG. Continuing the example of the previous paragraph, we may posit an event C which links to all elements of S(N), and consider this as a context. The two events A and B then operate in a common context. When they become simultaneously activated, they create a new node by the laws of event networks -- and they also create chronons attached to this new node. each time a link coming out of B encounters a link coming out of A that comes after it in the ordering, a chronon is created. If the number of chronons created is even, then no chronon remains after annihilation; if the number of chronons created is odd, then one chronon remains after annihilation. The result is the same as in the bit string formulation of Simon's DCLG sign algorithm. We thus have the following result:

Theorem: The interactions of pairs of temporal events operating within a common context C, according to the laws of temporal event networks, are isomorphic to Discrete Clifford Algebra on the set of events linked to by C

The isomorphism is the mapping that assigns a basis element of DCLG(N) to each element linked to by the context C. each temporal event is mapped to the combination of basis elements corresponding to its ordered sequence of links to other events; and the sign of the combination corresponding to an temporal event is determined by whether the temporal event has a chronon or not.

The step from DCLG(N) to DCL(N), finally, is not a large one. Clifford algebra is obtained by considering the M=2 ^N+1 elements {V₁,...,V_M} of the DCLG to combine, not only by DCLG multiplication, but also by summation and scalar multiplication, according to standard vector space laws. Clifford algebra, as normally studied, is a continuous construction, involving either the reals or the complex numbers as a scalar field. But for the purposes of building the D₄-D₅-E₆ theory of the Standard Model plus gravity, or other Feynman checkerboard physics models, one does not require the full continuous Clifford Algebra Cl(N), only the discrete lattice subset DCL(N) obtained by considering {V₁,...,V_M} as a vector space with scalars drawn from the integers. In particular, for building Smith's model, one requires DCL(8), which is the maximal Clifford algebra with any original structure, as shown by the Clifford algebra periodicity theorem.

To obtain DCL from event networks, one uses the notion, introduced above, of closed events. Each element of DCL(N) is of the form a₁ V₁ + ... + a_M V_M where the ai are integers. This is merely an event which has a₁ links pointing to V₁, a₂ links pointing to V₂, etc. What is required to generate DCL(N) from events is to posit that such events interact with each other in a closed rather than open way. This means that, when a pair (A,B) of these events becomes active, what happens is that each event linked to by A interacts with each event linked to by B, and a new node is created, bearing links to all the new nodes created by these interactions. The interactions of the events linked to by A and the events linked to by B take place according to the rules laid out already for the interactions of temporal events.

In this way one gets all the algebraic structures needed for physics out of event networks. The DCLG derives from temporal event interaction, whereas the DCL comes out of closed event interactions on top of the DCLG. Roughly, closed event interactions correspond to primal space, whereas temporal event interactions correspond to primal time. The combination of the two would seem to be the minimal amount of structure required to get event networks to give rise to the particular structures of the physical universe.

The model presented here is by far the simplest foundation for the laws of physical reality yet identified. However, it is still not entirely satisfactory, because the most crucial part -- the rule for combination of temporal events -- is more complicated than one would like. Also, the arrangement of closed and open events required to get DCL out of DCLG, while simple, has still a whiff of arbitrariness about it. These things are what permit event networks to give rise to the structure of physical reality, as opposed to arbitrary self-organizing network structure. But if one could replace these rules with something equivalent yet simpler and even more intuitive, this would obviously be desirable. In this section I will describe an approach that I believe will eventually lead to the replacement of the spatiotemporal event interaction rules with something simpler. All the steps in this programme have not been carried out yet, but the basic idea is clear.

The key, I believe, is yet another of John Wheeler's notions, "law without law" (Wheeler, 1993). Wheeler has proposed that physical laws be viewed as emergent rather than rigid entities, and that at the beginning of the universe, laws emerged from a primordial "law soup." Of course, this notion leads to an infinite regress: if laws evolve, then according to what laws do laws evolve? If these "second-order laws" of law evolution evolve, then according to what laws do they evolve? Etc. Obviously, the imposition of laws on the universe cannot be avoided altogether, in physical theory. What one can do, however, is to follow the regress until it leads to laws that seem insusceptible to further simplification.

Toward this end, I propose, the spatiotemporal event interactions proposed here may ultimately be derivable from a simpler "pattern maximization principle" which states, quite simply, that

• the rules of the universe's event networks are the ones that give rise to the maximum amount of emergent structure

The maximization of emergent pattern is a dynamical principle which I have shown, in previous publications, to apply broadly across evolving biological and psychological systems (Goertzel, 1993, 1997). Here it is proposed to apply to the "evolution" of physical law as well. The concept of emergent pattern can be formalized in various ways, using algorithmic information theory (Chaitin, 1988) and related notions from theoretical computer science; for the present purposes, however, the details are not required, and it is sufficient to know that such formalizations exist.

The pattern-maximization principle represents a reversal of the development of the previous sections of this paper. There I introduced open, temporal and closed event interaction, and used these to derive Clifford algebra structure. The D₄-D₅-E₆ model picks up where this leaves off, and exploits the numerous symmetries of Clifford algebra to obtain the Standard Model plus gravity. The pattern maximization principle talks about the same things, but it works in an opposite direction: it takes the numerous symmetries of Clifford algebras, as displayed in the D₄-D₅-E₆ model, as the reason for the emergence of Clifford algebras from the universe's event networks. The rules of spatiotemporal event interaction are said to emerge because they give rise to Clifford algebras, which are the maximally patterned algebraic structures.

To explore this notion in more detail, we must step back from the particular rules proposed here, and probe a little deeper into the notion of a "rule" itself. A rule, in the present context, is an "event transformation rule" -- a way of transforming individual events, or pairs of events, into other events. And all event transformation rules are ultimately expressible in terms of the basic operations of creating, copying or deleting links. In particular, the rules considered here are all special cases of one simple meta-rule. To state this meta-rule, define the notion of "perimeter" as follows: each event is its own 0-perimeter; the set of events linked to by a given event the 1-perimeter of that event; and the set of events linked to by the events in the n-perimeter of an event is the (n+1)-perimeter of that event. And define the N-links of an event as the set of links coming out of events in the n-perimeter of the event, for n no greater than N. In this language, the meta-rule as follows:

• There is some N and some M so that, when a collection of events is activated, a collection of new events is created, each link of which is either drawn from the N-links of the interacting events, or drawn from a fixed set of M additional links. Some, all or none of the original events may be deleted.

Any rule falling under this general template, I call an event transformation rule; and any collection of linked events obeying some collection of event transformation rules, I call an event network. In a general event network, each event may carry an integer label, telling what type of event it is. Different event transformation rules may apply to different types of events. A general event network is thus a finite collection of labeled events, together with a finite collection of event transformation rules, each one corresponding to at least one possible combination of event labels. The logical and temporal event networks introduced above are specialized event networks, involving three types of events (open, closed and temporal) and a few particular event transformation rules, with interaction restricted to single events and pairs of events.

An event transformation rule is a mapping from collections of events (n-perimeters of the events involved in an interaction) to collections of events (the events created as a result of interaction). In order to implement the pattern-maximization approach to deriving physics from event networks, one must show that the logical and temporal event transformation rules, as presented here, are are optimal in the space of event transformation rules, in the sense that they lead to networks with more emergent patterns per event and per link than any other collection of event transformation rules.

At this stage, however, another twist suggests itself. Because event transformation rules map finite sets into finite sets, any event transformation rule can be expressed in terms of propositional logic; and propositional logic can, in turn, be expressed in terms of logical event networks, without reference to closed or temporal events. All that one needs to get propositional logic are simple interactions between open events, because these give the symmetric difference, which is equivalent to the XOR operator, which generates the AND, OR and NOT operators of logic. Thus, one can view general event transformation rules as particular combinations of events, which operate internally according to the rules of logical event networks.

One arrives in this way at a hierarchical vision of the universe. On the first level, there is the basic event network, with events interacting according to transformation rules. On the next level up, there are the event networks that constitute the transformation rules for the first-level event network. Collapse of the two levels occurs if the networks in the first-level event network are used within the second-level network, as transformation rules. The pattern-maximization principle states that the second-level event network evolves in such a way as to cause the first-level event network to manifest the maximum amount of emergent pattern per event and per link.

This part of the current theory is still hypothetical, and not yet fully developed. What remains to be done is, essentially, to show that spatiotemporal event networks defined here do indeed manifest maximal emergent pattern in some natural sense. The important thing for the present, however, is that by focusing attention on the level of discrete event interaction, the notion of "law without law" becomes discrete and workable rather than entirely nebulous and abstract.

Chaitin, G. (1988). Algorithmic Information Theory. New York: Addison-Wesley.

Dixon, G. (1994). Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics

Finkelstein, David (1993). "Thinking Quantum," Cybernetics and Systems 24,, pp. 139-149
Goertzel, Ben (1993). The Evolving Mind. New York: Gordon and Breach.

Goertzel, Ben (1997). From Complexity to Creativity. New York: Plenum.

Simon, Barry (1996). Representations of Finite and Compact Groups, AMS Graduate Studies in Mathematics, vol. 10, 1996

Smith, Tony (1997). "From Sets to Quarks," preprint

Smith, Tony (1997a). "Truth Ceng Zi: Mt. 130 GeV," preprint, http://galaxy.cau.edu/tsmith/TCZ.html

Smith, Tony(1995). "Gravity and the Standard Model with 130 GeV Truth Quark from D₄-D₅-E₆ Model using 3x3 Octonion Matrices," preprint, hep-ph/9501252

Smith, Tony (1995a). "HyperDiamond Feynman Checkerboard in 4-Dimensional Spacetime," preprint, quant-ph/9503015

Wheeler, John A., in Misner, Thorne and Wheeler (1973). Gravitation. San Francisco: W.H. Freeman

Wheeler, John A. (1993). "On recognizing 'law without law," American Journal of Physics 51 (5), 398-404