In chemistry the polyhedral skeletal electron pair theory (PSEPT) provides electron counting rules useful for predicting the structures of clusters such as borane and carborane clusters. The electron counting rules were originally formulated by Kenneth Wade, and were further developed by others including Michael Mingos; they are sometimes known as Wade's rules or the Wade–Mingos rules. The rules are based on a molecular orbital treatment of the bonding. These rules have been extended and unified in the form of the Jemmis mno rules.
Different rules (4n, 5n, or 6n) are invoked depending on the number of electrons per vertex.
The 4n rules are reasonably accurate in predicting the
structures of clusters having about 4 electrons per vertex, as is the
case for many boranes and carboranes. For such clusters, the structures are based on deltahedra, which are polyhedra in which every face is triangular. The 4n clusters are classified as closo-, nido-, arachno- or hypho-, based on whether they represent a complete (closo-) deltahedron, or a deltahedron that is missing one (nido-), two (arachno-) or three (hypho-) vertices.
However, hypho clusters are relatively uncommon due to the fact
that the electron count is high enough to start to fill antibonding
orbitals and destabilize the 4n structure. If the electron count
is close to 5 electrons per vertex, the structure often changes to one
governed by the 5n rules, which are based on 3-connected polyhedra.
As the electron count increases further, the structures of clusters with 5n electron counts become unstable, so the 6n rules can be implemented. The 6n clusters have structures that are based on rings.
A molecular orbital treatment can be used to rationalize the bonding of cluster compounds of the 4n, 5n, and 6n types.
The following polyhedra are closo polyhedra, and are the basis for the 4n rules; each of these have triangular faces. The number of vertices in the cluster determines what polyhedron the structure is based on.
Using the electron count, the predicted structure can be found. n is the number of vertices in the cluster. The 4n rules are enumerated in the following table.
Electron count
Name
Predicted structure
4n − 2
Bicapped closo
n − 2 vertex closo polyhedron with 2 capped (augmented) faces
4n
Capped closo
n − 1 vertex closo polyhedron with 1 face capped
4n + 2
closo
closo polyhedron with n vertices
4n + 4
nido
n + 1 vertex closo polyhedron with 1 missing vertex
4n + 6
arachno
n + 2 vertex closo polyhedron with 2 missing vertices
4n + 8
hypho
n + 3 vertex closo polyhedron with 3 missing vertices
4n + 10
klado
n + 4 vertex closo polyhedron with 4 missing vertices
Pb2− 10
When counting electrons for each cluster, the number of valence electrons is enumerated. For each transition metal present, 10 electrons are subtracted from the total electron count. For example, in Rh6(CO)16 the total number of electrons would be 6 × 9 + 16 × 2 − 6 × 10 = 86 – 60 = 26. Therefore, the cluster is a closo polyhedron because n = 6, with 4n + 2 = 26.
S2+ 4
Other rules may be considered when predicting the structure of clusters:
For clusters consisting mostly of transition metals, any main
group elements present are often best counted as ligands or interstitial
atoms, rather than vertices.
Larger and more electropositive atoms tend to occupy vertices of
high connectivity and smaller more electronegative atoms tend to occupy
vertices of low connectivity.
In the special case of boron hydride
clusters, each boron atom connected to 3 or more vertices has one
terminal hydride, while a boron atom connected to two other vertices has
two terminal hydrogen atoms. If more hydrogen atoms are present, they
are placed in open face positions to even out the coordination number of
the vertices.
For the special case of transition metal clusters, ligands are added to the metal centers to give the metals reasonable coordination numbers, and if any hydrogenatoms are present they are placed in bridging positions to even out the coordination numbers of the vertices.
In general, closo structures with n vertices are n-vertex polyhedra.
To predict the structure of a nido cluster, the closo cluster with n + 1
vertices is used as a starting point; if the cluster is composed of
small atoms a high connectivity vertex is removed, while if the cluster
is composed of large atoms a low connectivity vertex is removed.
To predict the structure of an arachno cluster, the closo polyhedron with n + 2 vertices is used as the starting point, and the n + 1 vertex nido
complex is generated by following the rule above; a second vertex
adjacent to the first is removed if the cluster is composed of mostly
small atoms, a second vertex not adjacent to the first is removed if the
cluster is composed mostly of large atoms.
Os6(CO)18, carbonyls omitted
Example: Pb2− 10
Electron count: 10 × Pb + 2 (for the negative charge) = 10 × 4 + 2 = 42 electrons.
Since n = 10, 4n + 2 = 42, so the cluster is a closo bicapped square antiprism.
Example: S2+ 4
Electron count: 4 × S – 2 (for the positive charge) = 4 × 6 – 2 = 22 electrons.
Since n = 4, 4n + 6 = 22, so the cluster is arachno.
Starting from an octahedron, a vertex of high connectivity is removed, and then a non-adjacent vertex is removed.
Example: Os6(CO)18
Electron count: 6 × Os + 18 × CO – 60 (for 6 osmium atoms) = 6 × 8 + 18 × 2 – 60 = 24
Since n = 6, 4n = 24, so the cluster is capped closo.
Starting from a trigonal bipyramid, a face is capped. The carbonyls have been omitted for clarity.
B 5H4− 5, hydrogen atoms omitted
Example: B 5H4− 5
Electron count: 5 × B + 5 × H + 4 (for the negative charge) = 5 × 3 + 5 × 1 + 4 = 24
Since n = 5, 4n + 4 = 24, so the cluster is nido.
Starting from an octahedron, one of the vertices is removed.
The rules are useful in also predicting the structure of carboranes.
Example: C2B7H13
Electron count = 2 × C + 7 × B + 13 × H = 2 × 4 + 7 × 3 + 13 × 1 = 42
Since n in this case is 9, 4n + 6 = 42, the cluster is arachno.
The bookkeeping for deltahedral clusters is sometimes carried out by
counting skeletal electrons instead of the total number of electrons.
The skeletal orbital (electron pair) and skeletal electron counts for
the four types of deltahedral clusters are:
The skeletal electron counts are determined by summing the total of the following number of electrons:
2 from each BH unit
3 from each CH unit
1 from each additional hydrogen atom (over and above the ones on the BH and CH units)
the anionic charge electrons
5n rules
As discussed previously, the 4n rule mainly deals with clusters with electron counts of 4n + k, in which approximately 4 electrons
are on each vertex. As more electrons are added per vertex, the number
of the electrons per vertex approaches 5. Rather than adopting
structures based on deltahedra, the 5n-type clusters have structures
based on a different series of polyhedra known as the 3-connected polyhedra, in which each vertex is connected to 3 other vertices. The 3-connected polyhedra are the duals of the deltahedra. The common types of 3-connected polyhedra are listed below.
n – 1 vertex 3-connected polyhedron with one vertex inserted into an edge
5n + 2
n – 2 vertex 3-connected polyhedron with two vertices inserted into edges
5n + k
n − k vertex 3-connected polyhedron with k vertices inserted into edges
Example: P4
Electron count: 4 × P = 4 × 5 = 20
It is a 5n structure with n = 4, so it is tetrahedral
Example: P4S3
Electron count 4 × P + 3 × S = 4 × 5 + 3 × 6 = 38
It is a 5n + 3 structure with n = 7. Three vertices are inserted into edges
Example: P4O6
Electron count 4 × P + 6 × O = 4 × 5 + 6 × 6 = 56
It is a 5n + 6 structure with n = 10. Six vertices are inserted into edges
6n rules
As more electrons are added to a 5n cluster, the number of electrons per vertex approaches 6. Instead of adopting structures based on 4n or 5n rules, the clusters tend to have structures governed by the 6n rules, which are based on rings. The rules for the 6n structures are as follows.
S8 crown
Total electron count
Predicted structure
6n – k
n-membered ring with k⁄2 transannular bonds
6n – 4
n-membered ring with 2 transannular bonds
6n – 2
n-membered ring with 1 transannular bond
6n
n-membered ring
6n + 2
n-membered chain (n-membered ring with 1 broken bond)
Example: S8
Electron count = 8 × S = 8 × 6 = 48 electrons.
Since n = 8, 6n = 48, so the cluster is an 8-membered ring.
6n + 2 cluster: hexane
Hexane (C6H14)
Electron count = 6 × C + 14 × H = 6 × 4 + 14 × 1 = 38
Since n = 6, 6n = 36 and 6n + 2 = 38, so the cluster is a 6-membered chain.
Isolobal vertex units
Provided a vertex unit is isolobal with BH then it can, in principle at least, be substituted for a BH unit, even though BH and CH are not isoelectronic. The CH+ unit is isolobal, hence the rules are applicable to carboranes. This can be explained due to a frontier orbital treatment. Additionally there are isolobal transition-metal units. For example, Fe(CO)3 provides 2 electrons. The derivation of this is briefly as follows:
Fe has 8 valence electrons.
Each carbonyl group is a net 2 electron donor after the internal σ- and π-bonding are taken into account making 14 electrons.
3 pairs are considered to be involved in Fe–CO σ-bonding and 3 pairs are involved in π-backbonding from Fe to CO reducing the 14 to 2.
Bonding in cluster compounds
closo-B 6H2− 6
MO diagram of B 6H2− 6
showing the orbitals responsible for forming the cluster. Pictorial
representations of the orbitals are shown; the MO sets of T and E
symmetry will each have two or one additional pictorial representation,
respectively, that are not shown here.
The boron atoms lie on each vertex of the octahedron and are sp hybridized.
One sp-hybrid radiates away from the structure forming the bond with
the hydrogen atom. The other sp-hybrid radiates into the center of the
structure forming a large bonding molecular orbital at the center of the
cluster. The remaining two unhybridized orbitals lie along the tangent
of the sphere like structure creating more bonding and antibonding
orbitals between the boron vertices. The orbital diagram breaks down as follows:
The 18 framework molecular orbitals, (MOs), derived from the 18 boron atomic orbitals are:
1 bonding MO at the center of the cluster and 5 antibonding MOs from the 6 sp-radial hybrid orbitals
6 bonding MOs and 6 antibonding MOs from the 12 tangential p-orbitals.
The total skeletal bonding orbitals is therefore 7, i.e. n + 1.
Transition metal clusters
Transition metal clusters use the d orbitals for bonding. Thus, they have up to nine bonding orbitals, instead of only the four present in boron and main group clusters. PSEPT also applies to metallaboranes
Clusters with interstitial atoms
Owing
their large radii, transition metals generally form clusters that are
larger than main group elements. One consequence of their increased
size, these clusters often contain atoms at their centers. A prominent
example is [Fe6C(CO)16]2-. In such
cases, the rules of electron counting assume that the interstitial atom
contributes all valence electrons to cluster bonding. In this way, [Fe6C(CO)16]2- is equivalent to [Fe6(CO)16]6- or [Fe6(CO)18]2-.
Grok (/ˈɡrɒk/) is a neologism coined by American writer Robert A. Heinlein for his 1961 science fiction novel Stranger in a Strange Land. While the Oxford English Dictionary summarizes the meaning of grok
as "to understand intuitively or by empathy, to establish rapport with"
and "to empathize or communicate sympathetically (with); also, to
experience enjoyment",
Heinlein's concept is far more nuanced, with critic Istvan
Csicsery-Ronay Jr. observing that "the book's major theme can be seen as
an extended definition of the term." The concept of grok
garnered significant critical scrutiny in the years after the book's
initial publication. The term and aspects of the underlying concept have
become part of communities such as computer science.
Descriptions in Stranger in a Strange Land
Critic David E. Wright Sr. points out that in the 1991 "uncut" edition of Stranger, the word grok "was used first without any explicit definition on page 22" and continued to be used without being explicitly defined until page 253 (emphasis in original). He notes that this first intensional definition is simply "to drink", but that this is only a metaphor "much as English 'I see' often means the same as 'I understand'". Critics have bridged this absence of explicit definition by citing passages from Stranger that illustrate the term. A selection of these passages follows:
Grok means "to understand",
of course, but Dr. Mahmoud, who might be termed the leading Terran
expert on Martians, explains that it also means, "to drink" and "a
hundred other English words, words which we think of as antithetical
concepts. 'Grok' means all of these. It means 'fear', it means
'love', it means 'hate' – proper hate, for by the Martian 'map' you
cannot hate anything unless you grok it, understand it so thoroughly
that you merge with it and it merges with you – then you can hate it. By
hating yourself. But this implies that you love it, too, and cherish it
and would not have it otherwise. Then you can hate – and (I think) Martian hate is an emotion so black that the nearest human equivalent could only be called mild distaste.
Grok means "identically
equal". The human cliché "This hurts me worse than it does you" has a
distinctly Martian flavor. The Martian seems to know instinctively what
we learned painfully from modern physics, that observer acts with
observed through the process of observation. Grok means to
understand so thoroughly that the observer becomes a part of the
observed – to merge, blend, intermarry, lose identity in group
experience. It means almost everything that we mean by religion,
philosophy, and science and it means as little to us as color does to a
blind man.
The Martian Race had encountered
the people of the fifth planet, grokked them completely, and had taken
action; asteroid ruins were all that remained, save that the Martians
continued to praise and cherish the people they had destroyed.
All that groks is God.
Etymology
Robert A. Heinlein originally coined the term grok in his 1961 novel Stranger in a Strange Land as a Martian
word that could not be defined in Earthling terms, but can be
associated with various literal meanings such as "water", "to drink",
"to relate", "life", or "to live", and had a much more profound
figurative meaning that is hard for terrestrial culture to understand
because of its assumption of a singular reality.
According to the book, drinking water is a central focus on Mars, where it is scarce.
Martians use the merging of their bodies with water as a simple example
or symbol of how two entities can combine to create a new reality
greater than the sum of its parts. The water becomes part of the
drinker, and the drinker part of the water. Both grok each other.
Things that once had separate realities become entangled in the same
experiences, goals, history, and purpose. Within the book, the statement
of divineimmanence verbalized among the main characters, "thou art God", is logically derived from the concept inherent in the term grok.
Heinlein describes Martian words as "guttural" and "jarring".
Martian speech is described as sounding "like a bullfrog fighting a
cat". Accordingly, grok is generally pronounced as a guttural gr terminated by a sharp k with very little or no vowel sound (a narrow IPA transcription might be [ɡɹ̩kʰ]). William Tenn suggests Heinlein in creating the word might have been influenced by Tenn's very similar concept of griggo, earlier introduced in Tenn's story Venus and the Seven Sexes (published in 1949). In his later afterword to the story, Tenn says Heinlein considered such influence "very possible".
Adoption and modern usage
In computer programmer culture
Uses of the word in the decades after the 1960s are more concentrated in computer culture, such as an InfoWorld
columnist in 1984 imagining a computer saying, "There isn't any
software! Only different internal states of hardware. It's all hardware!
It's a shame programmers don't grok that better."
The Jargon File, which describes itself as "The Hacker's Dictionary" and has been published under that name three times, puts grok in a programming context:
When you claim to "grok" some
knowledge or technique, you are asserting that you have not merely
learned it in a detached instrumental way but that it has become part of
you, part of your identity. For example, to say that you "know" Lisp
is simply to assert that you can code in it if necessary – but to say
you "grok" Lisp is to claim that you have deeply entered the world-view
and spirit of the language, with the implication that it has transformed
your view of programming. Contrast zen, which is a similar supernatural understanding experienced as a single brief flash.
The entry existed in the very earliest forms of the Jargon File in the early 1980s. A typical tech usage from the Linux Bible, 2005 characterizes the Unixsoftware development philosophy as "one that can make your life a lot simpler once you grok the idea".
The book Perl Best Practices defines grok as understanding a portion of computer code in a profound way. It goes on to suggest that to re-grok
code is to reload the intricacies of that portion of code into one's
memory after some time has passed and all the details of it are no
longer remembered. In that sense, to grok means to load everything into memory for immediate use. It is analogous to the way a processor caches
memory for short term use, but the only implication by this reference
was that it was something a human (or perhaps a Martian) would do.
The main web page for cURL, an open source tool and programming library, describes the function of cURL as "cURL groks URLs".
The book Cyberia covers its use in this subculture extensively:
This is all latter day usage, the
original derivation was from an early text processing utility from so
long ago that no one remembers but, grok was the output when it
understood the file. K&R would remember.
The keystroke logging software used by the NSA for its remote intelligence gathering operations is named GROK.
One of the most powerful parsing filters used in Elasticsearch software's logstash component is named grok.
A reference book by Carey Bunks on the use of the GNU Image Manipulation Program is titled Grokking the GIMP.
Tom Wolfe, in his book The Electric Kool-Aid Acid Test (1968), describes a character's thoughts during an acid trip: "He looks down, two bare legs, a torso rising up at him and like he is just noticing them for the first time... he has never seen any of this flesh before, this stranger. He groks over that..."
In his counterculture Volkswagen repair manual, How to Keep Your Volkswagen Alive: A Manual of Step-by-Step Procedures for the Compleat Idiot (1969), dropout aerospace engineer John Muir instructs prospective used VW buyers to "grok the car" before buying.
I caught the references to
Aristotle, the old man of the tribe with his unfortunate epistemological
paresis, and also to that feisty little lady I always imagine is really
the lost Anastasia, but I still didn’t grok. “What do you mean?” I
asked (...)
And in The Trick Top Hat, volume two of Schrödinger's Cat:
Williams went on. "You've got to think of time ripples, as well as space ripples, to grok the quantum world. ..."
A temporal paradox, time paradox, or time travel paradox, is a paradox, an apparent contradiction, or logical contradiction associated with the idea of time travel
or other foreknowledge of the future. While the notion of time travel
to the future complies with the current understanding of physics via
relativistic time dilation,
temporal paradoxes arise from circumstances involving hypothetical time
travel to the past – and are often used to demonstrate its
impossibility.
Types
"Causal loop" redirects here. For the cause and effect diagram, see causal loop diagram. For the plot device, see time loop.
Temporal paradoxes fall into three broad groups: bootstrap paradoxes, consistency paradoxes, and Newcomb's paradox. Bootstrap paradoxes violate causality by allowing future events to influence the past and cause themselves, or "bootstrapping", which derives from the idiom "pull oneself up by one's bootstraps."
Consistency paradoxes, on the other hand, are those where future events
influence the past to cause an apparent contradiction, exemplified by
the grandfather paradox,
where a person travels to the past to prevent the conception of one of
their ancestors, thus eliminating all the ancestor's descendants. Newcomb's paradox stems from the apparent contradictions that stem from the assumptions of both free will and foreknowledge of future events. All of these are sometimes referred to individually as "causal loops." The term "time loop" is sometimes referred to as a causal loop,
but although they appear similar, causal loops are unchanging and
self-originating, whereas time loops are constantly resetting.
Bootstrap paradox
A bootstrap paradox, also known as an information loop, an information paradox, an ontological paradox,
or a "predestination paradox" is a paradox of time travel that occurs
when any event, such as an action, information, an object, or a person,
ultimately causes itself, as a consequence of either retrocausality or time travel.
Backward time travel would allow information, people, or objects whose histories seem to "come from nowhere". Such causally looped events then exist in spacetime, but their origin cannot be determined. The notion of objects or information that are "self-existing" in this way is often viewed as paradoxical. Everett gives the movie Somewhere in Time
as an example involving an object with no origin: an old woman gives a
watch to a playwright who later travels back in time and meets the same
woman when she was young, and gives her the same watch that she will
later give to him. An example of information which "came from nowhere" is in the movie Star Trek IV: The Voyage Home, in which a 23rd-century engineer travels back in time, and gives the formula for transparent aluminum to the 20th-century engineer who supposedly invented it.
Predestination paradox
Smeenk
uses the term "predestination paradox" to refer specifically to
situations in which a time traveler goes back in time to try to prevent
some event in the past.
Grandfather paradox
The
consistency paradox or grandfather paradox occurs when the past is
changed in any way, thus creating a contradiction. A common example
given is traveling to the past and intervening with the conception of
one's ancestors (such as causing the death of the parent beforehand),
thus affecting the conception of oneself. If the time traveler were not
born, then it would not be possible for the traveler to undertake such
an act in the first place. Therefore, the ancestor lives to offspring
the time traveler's next-generation ancestor, and eventually the time
traveler. There is thus no predicted outcome to this. Consistency paradoxes occur whenever changing the past is possible. A possible resolution is that a time traveller can do anything that did happen, but cannot do anything that did not happen. Doing something that did not happen results in a contradiction. This is referred to as the Novikov self-consistency principle.
Variants
The grandfather paradox encompasses any change to the past, and it is presented in many variations, including killing one's past self. Both the "retro-suicide paradox" and the "grandfather paradox" appeared in letters written into Amazing Stories in the 1920s.
Another variant of the grandfather paradox is the "Hitler paradox" or
"Hitler's murder paradox", in which the protagonist travels back in time
to murder Adolf Hitler before he can instigate World War II and the Holocaust. Rather than necessarily physically preventing time travel, the action removes any reason for the travel, along with any knowledge that the reason ever existed.
Physicist John Garrison et al. give a variation of the paradox of
an electronic circuit that sends a signal through a time machine to
shut itself off, and receives the signal before it sends it.
Newcomb's paradox is a thought experiment showing an apparent contradiction between the expected utility principle and the strategic dominance principle. The thought experiment is often extended to explore causality
and free will by allowing for "perfect predictors": if perfect
predictors of the future exist, for example if time travel exists as a
mechanism for making perfect predictions
then perfect predictions appear to contradict free will because
decisions apparently made with free will are already known to the
perfect predictor. Predestination does not necessarily involve a supernatural power, and could be the result of other "infallible foreknowledge" mechanisms. Problems arising from infallibility and influencing the future are explored in Newcomb's paradox.
Proposed resolutions
Logical impossibility
Even without knowing whether time travel to the past is physically possible, it is possible to show using modal logic
that changing the past results in a logical contradiction. If it is
necessarily true that the past happened in a certain way, then it is
false and impossible for the past to have occurred in any other way. A
time traveler would not be able to change the past from the way it is, but would only act in a way that is already consistent with what necessarily happened.
Consideration of the grandfather paradox has led some to the idea
that time travel is by its very nature paradoxical and therefore
logically impossible. For example, the philosopher Bradley Dowden made this sort of argument in the textbook Logical Reasoning,
arguing that the possibility of creating a contradiction rules out time
travel to the past entirely. However, some philosophers and scientists
believe that time travel into the past need not be logically impossible
provided that there is no possibility of changing the past, as suggested, for example, by the Novikov self-consistency principle. Dowden revised his view after being convinced of this in an exchange with the philosopher Norman Swartz.
Illusory time
Consideration of the possibility of backward time travel in a hypothetical universe described by a Gödel metric led famed logician Kurt Gödel to assert that time might itself be a sort of illusion. He suggests something along the lines of the block time
view, in which time is just another dimension like space, with all
events at all times being fixed within this four-dimensional "block".
Physical impossibility
Sergey Krasnikov
writes that these bootstrap paradoxes – information or an object
looping through time – are the same; the primary apparent paradox is a
physical system evolving into a state in a way that is not governed by
its laws.
He does not find these paradoxical and attributes problems regarding
the validity of time travel to other factors in the interpretation of
general relativity.
Self-sufficient loops
A 1992 paper by physicists Andrei Lossev and Igor Novikov labeled such items without origin as Jinn, with the singular term Jinnee. This terminology was inspired by the Jinn of the Quran, which are described as leaving no trace when they disappear. Lossev and Novikov allowed the term "Jinn" to cover both objects and
information with the reflexive origin; they called the former "Jinn of
the first kind", and the latter "Jinn of the second kind".
They point out that an object making circular passage through time must
be identical whenever it is brought back to the past, otherwise it
would create an inconsistency; the second law of thermodynamics
seems to require that the object tends to a lower energy state
throughout its history, and such objects that are identical in repeating
points in their history seem to contradict this, but Lossev and Novikov
argued that since the second law only requires entropy to increase in closed systems, a Jinnee could interact with its environment in such a way as to regain "lost" entropy. They emphasize that there is no "strict difference" between Jinn of the first and second kind.
Krasnikov equivocates between "Jinn", "self-sufficient loops", and
"self-existing objects", calling them "lions" or "looping or intruding
objects", and asserts that they are no less physical than conventional
objects, "which, after all, also could appear only from either infinity
or a singularity."
The self-consistency principle developed by Igor Dmitriyevich Novikov expresses one view as to how backward time travel would be possible without the generation of paradoxes. According to this hypothesis, even though general relativity permits some exact solutions that allow for time travel that contain closed timelike curves that lead back to the same point in spacetime, physics in or near closed timelike curves
(time machines) can only be consistent with the universal laws of
physics, and thus only self-consistent events can occur. Anything a time
traveler does in the past must have been part of history all along, and
the time traveler can never do anything to prevent the trip back in
time from happening, since this would represent an inconsistency. The
authors concluded that time travel need not lead to unresolvable
paradoxes, regardless of what type of object was sent to the past.
Top:
original billiard ball trajectory. Middle: the billiard ball emerges
from the future, and delivers its past self a strike that averts the
past ball from entering the time machine. Bottom: The billiard ball
never enters the time machine, giving rise to the paradox, putting into
question how its older self could ever emerge from the time machine and
divert its course.
Physicist Joseph Polchinski considered a potentially paradoxical situation involving a billiard ball that is fired into a wormhole
at just the right angle such that it will be sent back in time and
collides with its earlier self, knocking it off course, which would stop
it from entering the wormhole in the first place. Kip Thorne referred to this problem as "Polchinski's paradox".
Thorne and two of his students at Caltech, Fernando Echeverria and
Gunnar Klinkhammer, went on to find a solution that avoided any
inconsistencies, and found that there was more than one self-consistent
solution, with slightly different angles for the glancing blow in each
case. Later analysis by Thorne and Robert Forward
showed that for certain initial trajectories of the billiard ball,
there could be an infinite number of self-consistent solutions.
It is plausible that there exist self-consistent extensions for every
possible initial trajectory, although this has not been proven. The lack of constraints on initial conditions only applies to spacetime outside of the chronology-violating region of spacetime; the constraints on the chronology-violating region might prove to be paradoxical, but this is not yet known.
Novikov's views are not widely accepted. Visser views causal loops and Novikov's self-consistency principle as an ad hoc solution, and supposes that there are far more damaging implications of time travel. Krasnikov similarly finds no inherent fault in causal loops but finds other problems with time travel in general relativity. Another conjecture, the cosmic censorship hypothesis, suggests that every closed timelike curve passes through an event horizon, which prevents such causal loops from being observed.
Parallel universes
The interacting-multiple-universes approach is a variation of the many-worlds interpretation
of quantum mechanics that involves time travelers arriving in a
different universe than the one from which they came; it has been argued
that, since travelers arrive in a different universe's history and not
their history, this is not "genuine" time travel. Stephen Hawking has argued for the chronology protection conjecture,
that even if the MWI is correct, we should expect each time traveler to
experience a single self-consistent history so that time travelers
remain within their world rather than traveling to a different one.
David Deutsch has proposed that quantum computation
with a negative delay—backward time travel—produces only
self-consistent solutions, and the chronology-violating region imposes
constraints that are not apparent through classical reasoning.
However Deutsch's self-consistency condition has been demonstrated as
capable of being fulfilled to arbitrary precision by any system subject
to the laws of classical statistical mechanics, even if it is not built up by quantum systems.
Allen Everett has also argued that even if Deutsch's approach is
correct, it would imply that any macroscopic object composed of multiple
particles would be split apart when traveling back in time, with
different particles emerging in different worlds.
In the philosophy of religion, a cosmological argument is an argument for the existence of God based upon observational and factual statements concerning the universe (or some general category of its natural contents) typically in the context of causation, change, contingency or finitude. In referring to reason and observation alone for its premises, and precluding revelation, this category of argument falls within the domain of natural theology. A cosmological argument can also sometimes be referred to as an argument from universal causation, an argument from first cause, the causal argument or the prime mover argument.
The concept of causation is a principal underpinning idea in all
cosmological arguments, particularly in affirming the necessity for a First Cause. The latter is typically determined in philosophical analysis to be God, as identified within classical conceptions of theism.
Plato (c. 427–347 BC) and Aristotle (c. 384–322 BC) both posited first cause arguments, though each had certain notable caveats. In The Laws (Book X), Plato posited that all movement in the world and the Cosmos was "imparted motion". This required a "self-originated motion" to set it in motion and to maintain it. In Timaeus, Plato posited a "demiurge" of supreme wisdom and intelligence as the creator of the Cosmos.
Aristotle argued against the idea of a first cause, often confused with the idea of a "prime mover" or "unmoved mover" (πρῶτον κινοῦν ἀκίνητον or primus motor) in his Physics and Metaphysics. Aristotle argued in favor of the idea of several unmoved movers, one powering each celestial sphere,
which he believed lived beyond the sphere of the fixed stars, and
explained why motion in the universe (which he believed was eternal) had
continued for an infinite period of time. Aristotle argued the atomist's assertion of a non-eternal universe would require a first uncaused cause – in his terminology, an efficient first cause – an idea he considered a nonsensical flaw in the reasoning of the atomists.
Like Plato, Aristotle believed in an eternal cosmos with no beginning and no end (which in turn follows Parmenides' famous statement that "nothing comes from nothing"). In what he called "first philosophy" or metaphysics, Aristotle did
intend a theological correspondence between the prime mover and a
deity; functionally, however, he provided an explanation for the
apparent motion of the "fixed stars"
(now understood as the daily rotation of the Earth). According to his
theses, immaterial unmoved movers are eternal unchangeable beings that
constantly think about thinking, but being immaterial, they are
incapable of interacting with the cosmos and have no knowledge of what
transpires therein. From an "aspiration or desire", the celestial spheres, imitate that purely intellectual activity as best they can, by uniform circular motion. The unmoved movers inspiring the planetary
spheres are no different in kind from the prime mover, they merely
suffer a dependency of relation to the prime mover. Correspondingly, the
motions of the planets are subordinate to the motion inspired by the
prime mover in the sphere of fixed stars. Aristotle's natural theology
admitted no creation or capriciousness from the immortal pantheon, but maintained a defense against dangerous charges of impiety.
Plotinus,
a third-century Platonist, taught that the One transcendent absolute
caused the universe to exist simply as a consequence of its existence (creatio ex deo). His disciple Proclus stated "The One is God". Centuries later, the IslamicphilosopherAvicenna (c. 980–1037) inquired into the question of being, in which he distinguished between essence (māhiyya) and existence (wuǧūd).
He argued that the fact of existence could not be inferred from or
accounted for by the essence of existing things, and that form and
matter by themselves could not originate and interact with the movement
of the Universe or the progressive actualization of existing things.
Thus, he reasoned that existence must be due to an agent cause
that necessitates, imparts, gives, or adds existence to an essence. To
do so, the cause must coexist with its effect and be an existing thing.
Early Christian theology
Steven Duncan writes that the cosmological argument "was first formulated by a Greek-speaking Syriac Christian neo-Platonist, John Philoponus,
who claims to find a contradiction between the Greek pagan insistence
on the eternity of the world and the Aristotelian rejection of the
existence of any actual infinite". Referring to the argument as the "'Kalam'
cosmological argument", Duncan asserts that it "received its fullest
articulation at the hands of [medieval] Muslim and Jewish exponents of Kalam ("the use of reason by believers to justify the basic metaphysical presuppositions of the faith").
Thomas Aquinas (c. 1225–1274) adapted and enhanced the argument he found in his reading of Aristotle, Avicenna (the Proof of the Truthful), and Maimonides to form one of the most influential versions of the cosmological argument.
His conception of first cause was the idea that the Universe must be
caused by something that is itself uncaused, which he claimed is that
which we call God:
The second way is from the nature
of the efficient cause. In the world of sense we find there is an order
of efficient causes. There is no case known (neither is it, indeed,
possible) in which a thing is found to be the efficient cause of itself;
for so it would be prior to itself, which is impossible. Now in
efficient causes it is not possible to go on to infinity, because in all
efficient causes following in order, the first is the cause of the
intermediate cause, and the intermediate is the cause of the ultimate
cause, whether the intermediate cause be several, or only one. Now to
take away the cause is to take away the effect. Therefore, if there be
no first cause among efficient causes, there will be no ultimate, nor
any intermediate cause. But if in efficient causes it is possible to go
on to infinity, there will be no first efficient cause, neither will
there be an ultimate effect, nor any intermediate efficient causes; all
of which is plainly false. Therefore it is necessary to admit a first
efficient cause, to which everyone gives the name of God.
Importantly, Aquinas' Five Ways, given the second question of his Summa Theologica,
are not the entirety of Aquinas' demonstration that the Christian God
exists. The Five Ways form only the beginning of Aquinas' Treatise on
the Divine Nature.
General principles
The infinite regress
A regress is a series
of related elements, arranged in some type of sequence of succession,
examined in backwards succession (regression) from a fixed point of
reference. Depending on the type of regress, this retrograde examination
may take the form of recursive
analysis, in which the elements in a series are studied as products of
prior, often simpler, elements. If there is no 'last member' in a
regress (ie. no 'first member' in the series) it becomes an infinite regress, continuing in perpetuity. In the context of the cosmological argument the term 'regress' usually refers to causal regress, in which the series is a chain of cause and effect,
with each element in the series arising from causal activity of the
prior member. Some variants of the argument may also refer to temporal regress, wherein the elements are past events (discrete units of time) arranged in a temporal sequence.
An infinite regress argument attempts to establish the falsity of a proposition by showing that it entails an infinite regress that is vicious. The cosmological argument is a type of positive infinite regress argument given that it defends a proposition (in this case, the existence of a first cause) by arguing that its negation would lead to a vicious regress. An infinite regress may be vicious due to various reasons:
Implausibility: The regress contradicts empirical evidence (eg. for the finitude of the past) or basic principles such as Occam's razor.
Explanatory failure: A failure of explanatory goals resulting in an
infinite regress of explanations. This may arise in the case of logical
fallacies such as begging the question or from an attempt to investigate causes concerning origins or fundamental principles.
Accidental and essential ordering of causes
Aquinas refers to the distinction found in Aristotle's Physics (8.5) that a series of causes may either be accidental or essential, though the designation of this terminology would follow later under John Duns Scotus at the turn of the 14th century.
In an accidentally ordered series of causes, earlier members need
not continue exerting causal activity (having done so to progress the
chain) for the series to continue. For example, in an ancestral lineage,
the ancestors need no longer exist in order for their descendents to
resume the bloodline. In an essential series, every prior member must
maintain causal interrelationship in order for the series to continue:
If a hand holds a stick that moves a rock along the ground, the rock
would stop motion as soon as the hand or stick ceases to exist.
Based upon this distinction Frederick Copleston (1907-1994) characterises two types of causation: Causes in fieri, which cause an effect's becoming, or coming into existence, and causes in esse, which causally sustain an effect, in being, once it exists.
Two specific properties of an essentially ordered series have significance in the context of the cosmological argument:
A first cause is essential: In the example illustrated above, the rock derives its causal power essentially from the stick, which derives its causal power essentially from the hand. Later members exercise no independent causal power in continuing the causal series.
It requires that all causes in the series exist simultaneously in time, or timelessly.
Thomistic philosopher, R. P. Phillips comments on the characteristics of essential ordering:
"Each member of the series of causes possesses being solely by
virtue of the actual present operation of a superior cause ... Life is
dependent inter alia on a certain atmospheric pressure, this
again on the continual operation of physical forces, whose being and
operation depends on the position of the earth in the solar system,
which itself must endure relatively unchanged, a state of being which
can only be continuously produced by a definite—if unknown—constitution
of the material universe. This constitution, however, cannot be its own
cause ... We are thus irresistibly led to posit a first efficient cause
which, while itself uncaused, shall impart causality to a whole series."
Versions of the argument
Aquinas's argument from contingency
In the scholastic era, Aquinas formulated the "argument from contingency", following Aristotle, in claiming that there must be something to explain the existence of the universe. Since the universe could, under different circumstances, conceivably not
exist (ie. it is contingent) its existence must have a cause. This
cause cannot be embodied in another contingent thing, but something that
exists by necessity (ie. that must exist in order for anything else to exist). It is a form of argument from universal causation,
therefore compatible with the conception of a universe that has no
beginning in time. In other words, according to Aquinas, even if the
universe has always existed, it still owes its continuing existence to
an uncaused cause, he states: "... and this we understand to be God."
Aquinas's argument from contingency is formulated as the Third Way (Q2, A3) in the Summa Theologica. It may be expressed as follows:
There exist contingent things, for which non-existence is possible.
It is impossible for contingent things to always exist, so at some time they do not exist.
Therefore, if all things are contingent, then nothing would exist now.
There exists something rather than nothing.
He concludes thereupon that contingent beings are an insufficient
explanation for the existence of other contingent beings. Furthermore,
that there must exist a necessary being, whose non-existence is impossible, to explain the origination of all contingent beings.
Therefore, there exists a necessary being.
It is possible that a necessary being has a cause of its necessity in another necessary being.
The derivation of necessity between beings cannot regress to infinity (being an essentially ordered causal series).
Therefore, there exists a being that is necessary of itself, from which all necessity derives.
That being is whom everyone calls God.
Leibnizian cosmological argument
In 1714, German philosopher Gottfried Leibniz presented a variation of the cosmological argument based upon the principle of sufficient reason.
He writes: "There can be found no fact that is true or existent, or any
true proposition, without there being a sufficient reason for its being
so and not otherwise, although we cannot know these reasons in most
cases." Stating his argument succinctly:
"Why is there something rather than nothing? The sufficient
reason ... is found in a substance which ... is a necessary being
bearing the reason for its existence within itself."
There is a contingent fact that includes all other contingent facts.
Therefore, there is an explanation of this fact.
This explanation must involve a necessary being.
This necessary being is God.
Premise 1 expresses the principle of sufficient reason (PSR). In premise 2, Leibniz proposes the existence of a logical conjunction
of all contingent facts. This may be regarded as the sum total of all
contingent reality, referred to in later literature as the Big
Conjunctive Contingent Fact (BCCF).
Premise 3 applies the PSR to the BCCF, given that it too, as a
contingency, has a sufficient explanation. It follows, in statement 4,
that the explanation of the BCCF must be necessary, not contingent,
given that the BCCF incorporates all contingent facts.
Statement 5 proposes that the necessary being explaining the
totality of contingent facts is God. Philosophers of religion, such as
Joshua Rasmussen and T. Ryan Byerly, have argued in defence of the
inference from 4 to 5.
Duns Scotus's argument
Inspired by Aquinas's argument of the unmoved mover, this metaphysical argument for the existence of God was formulated by influential Medieval Christian theologian Duns Scotus (1265/66–1308).
Like other philosophers and theologians, Scotus believed that his
statement for God's existence could be considered separate to that of
Aquinas. The form of the argument can be summarised as follows:
An effect cannot be produced by itself.
An effect cannot be produced by nothing.
A circle of causes is impossible.
Therefore, an effect must be produced by something else.
An accidentally ordered causal series cannot exist without an essentially ordered series.
Each member in an accidentally ordered series (except a possible first) exists via causal activity of a prior member.
That causal activity is exercised by virtue of a certain form.
Therefore, that form is required by each member to effect causation.
An essentially ordered causal series cannot regress to infinity.
Therefore [4,5,6], there exists a first agent.
Scotus affirms, in premise 5, that an accidentally ordered series of causes
is impossible without higher-order laws and processes that govern the
basic nature of all causal activity, which he characterises as
essentially ordered causes. Premise 6 continues, in accordance with Aquinas's discourses on the Second Way and Third Way, that an essentially ordered series of causes cannot be an infinite regress.
On this he posits that, if it is merely possible that a first agent exists, then it is necessarily
true that a first agent exists, given that the non-existence of a first
agent entails the impossibility of its own existence (by virtue of
being a first cause in the chain). Establishing this as basis, he argues that it is not impossible for a being to exist that is causeless by virtue of ontological perfection.
With the formulation of this argument, Scotus establishes the
first component of his 'triple primacy': The characterisation of a being
that is first in efficient causality, final causality and pre-eminence, or maximal excellence, which he ascribes to God.
A modern formulation of the cosmological argument that proposes, as its central thesis, the impossibility of an infinite temporal regress of events (or a past-eternal universe). Its premises defend the finitude of the past
through both philosophical and scientific arguments. Many of these
ideas originate in the writings of early Christian theologian John Philoponus (490–570 AD), developed within the proceedings of medieval Islamicscholasticism through the 9th to 12th centuries, eventually returning to Christian theological scholarship in the 13th century.
They were revitalised for modern academic discourse by philosopher and theologian William Lane Craig through publications such as The Kalām Cosmological Argument (1979) and the Blackwell Companion to Natural Theology (2009). The form of the argument popularised by Craig is expressed in two parts, as an initial deductivesyllogism followed by philosophical analysis of its conclusion.
Initial syllogism
Everything that begins to exist has a cause.
The universe began to exist.
Therefore, the universe has a cause.
Philosophical analysis of the conclusion
Craig argues that the cause of the universe necessarily embodies specific properties in creating the universe ex nihilo and in effecting creation from a timeless state (implying free agency). Based upon this analysis, he appends a further premise and conclusion:
If the universe has a cause, then an uncaused, personal Creator of the universe exists who sans (without) the universe is beginningless, changeless, immaterial, timeless, spaceless and enormously powerful.
Therefore, an uncaused, personal Creator of the universe exists, who sans the universe is beginningless, changeless, immaterial, timeless, spaceless and enormously powerful.
For scientific evidence of the finitude of the past, Craig appeals to the Borde-Guth-Vilenkin theorem, which posits a past boundary to cosmic inflation, and the general consensus on the standard model of cosmology, referring to the origin of the universe in the Big Bang.
For philosophical evidence, he cites the Hilbert's Hotelthought experiment and the tale of Tristram Shandy as proofs (respectively) of the impossibility of actual infinities
existing in reality and of forming an actual infinite by successive
addition. He concludes that past events, comprising a series of events
that are, (a) instantiated in reality, (b) formed by successive
addition, cannot be actually infinite.
He remarks upon the theological implications that follow from the final conclusion of this argument:
"... our whole universe was caused to exist by something beyond
it and greater than it. For it is no secret that one of the most
important conceptions of what theists mean by 'God' is Creator of heaven
and earth."
Criticism and discourse
"What caused the first cause?"
One
objection to the argument asks why a first cause is unique in that it
does not require any causes. Proponents argue that the first cause is
exempt from having a cause, as this is part of what it is to be the
first cause, while opponents argue that this is special pleading
or otherwise untrue. Critics often press that arguing for the first
cause's exemption raises the question of why the first cause is indeed
exempt,
whereas defenders maintain that this question has been answered by the
various arguments, emphasizing that none of the major cosmological
arguments rests on the premise that everything has a cause, and so the
question does not address the actual premises of an argument and rests
on a misunderstanding of them.
Andrew Loke states that, according to the Kalam cosmological argument,
only things which begin to exist require a cause. On the other hand,
something that is without beginning has always existed and therefore
does not require a cause. Loke and William Lane Craig
argue that an infinite regress of causes is impossible, therefore, that
there must be a first uncaused cause, even if one posits a plurality of
causes of the universe. Craig argues further that Occam's razor may be employed to remove unneeded further causes of the universe to leave a single uncaused cause.
"Why can't the universe be causeless?"
It is argued that the premise of causality has been arrived at via a posteriori (inductive) reasoning, which is dependent on experience. David Hume highlighted this problem of induction and argued that causal relations are not true a priori. However, as to whether inductive or deductive reasoning
is more valuable remains a matter of debate, with the general
conclusion being that neither is prominent.
Opponents of the cosmological argument argue that it is unwise to draw
conclusions from an extrapolation of causality beyond experience,
therefore, that the causal principle does not apply to the origin of the
universe.
Philosopher Robert Koons
argues that to deny causation is to deny all empirical ideas – for
example, if we know our own hand, we know it because of the chain of
causes including light being reflected upon one's eyes, stimulating the
retina and sending a message through the optic nerve into your brain. He
summarised the purpose of the argument as "that if you don't buy into
theistic metaphysics, you're undermining empirical science. The two grew
up together historically and are culturally and philosophically
inter-dependent ... If you say I just don't buy this causality principle
– that's going to be a big big problem for empirical science."
"Why should the cause be God?"
According
to this objection, the basic cosmological argument merely establishes
that a first cause exists, not that it has the attributes of a theistic god, such as omniscience, omnipotence, and omnibenevolence.
This is why the argument is often expanded to assert that at least some
of these attributes are necessarily true, for instance in the modern
Kalam argument given above.
Defenders of the cosmological arguments also reply that
theologians of note are aware of the need to additionally prove other
attributes of the first cause beyond that one exists. One notable
example of this is found in Aquinas' Summa Theologiae in which much of the first part (Prima Pars) is devoted to establishing the attributes of this first cause, such as its uniqueness, perfection, and intelligence.
Thus defenders of cosmological arguments would reply that while it is
true that the cosmological argument only establishes a first cause, this
is merely the first step which then allows for the demonstration of the
other theistic attributes.
Timeless origin of the universe
Some cosmologists and physicists, such as Carlo Rovelli, argue that a challenge to the cosmological argument is the nature of time: "One finds that time just disappears from the Wheeler–DeWitt equation." The Big Bang theory states that it is the point in which all dimensions came into existence, the start of both space and time.
Then, the question "What was there before the Universe?" makes no
sense; the concept of "before" becomes meaningless when considering a
situation without time.[61] This has been put forward by J. Richard Gott III, James E. Gunn, David N. Schramm, and Beatrice Tinsley, who said that asking what occurred before the Big Bang is like asking what is north of the North Pole.
However, some cosmologists and physicists attempt to investigate causes
for the Big Bang, using such scenarios as the collision of membranes. Philosopher Edward Feser
argues that most of the classical philosophers' cosmological arguments
for the existence of God do not depend on the Big Bang or whether the
universe had a beginning. The question is not about what got things
started, or how long they have been going, but rather what keeps them
going.
Avoiding an infinite regress
David Hume and later Paul Edwards have invoked a similar principle in their criticisms of the cosmological argument. William L. Rowe has called this the Hume-Edwards principle:
If the existence of every member of a set is explained, the existence of that set is thereby explained.
Nevertheless, David White argues that the notion of an infinite causal regress providing a proper explanation is fallacious. Furthermore, in Hume's Dialogues Concerning Natural Religion, the character Demea states that even if the succession of causes is infinite, the whole chain still requires a cause.
To explain this, suppose there exists a causal chain of infinite
contingent beings. If one asks the question, "Why are there any
contingent beings at all?", it does not help to be told that "There are
contingent beings because other contingent beings caused them." That
answer would just presuppose additional contingent beings. An adequate
explanation of why some contingent beings exist would invoke a different
sort of being, a necessary being that is not contingent.
A response might suppose each individual is contingent but the infinite
chain as a whole is not, or the whole infinite causal chain is its own
cause.
Edward Feser argues that an essentially ordered series of causes
cannot regress to infinity, even if it may be theoretically possible for
accidentally ordered causes to do so. Severinsen argues that there is an "infinite" and complex causal structure.
White tried to introduce an argument "without appeal to the principle
of sufficient reason and without denying the possibility of an infinite
causal regress".
A number of other arguments have been offered to demonstrate that an
actual infinite regress cannot exist, viz. the argument for the
impossibility of concrete actual infinities, the argument for the
impossibility of traversing an actual infinite, the argument from the
lack of capacity to begin to exist, and various arguments from
paradoxes.
Causal loop arguments
Some objections to the cosmological argument refer to the possibility of loops in the structure of cause and effect that would avoid the need for a First Cause. Gott and Li refer to the curvature of spacetime and closed timelike curves as possible mechanisms by which the universe may bring about its own existence. Richard Hanley
contends that causal loops are neither logically nor physically
impossible, remarking: "[In timed systems] the only possibly
objectionable feature that all causal loops share is that coincidence is
required to explain them."
However, Andrew Loke argues that there is insufficient evidence to
postulate a causal loop of the type that would avoid a First Cause. He
asserts that such a mechanism would suffer from the problem of vicious circularity, rendering it metaphysically impossible.
