From Wikipedia, the free encyclopedia
Infinity (symbolically represented by
∞) is a concept in mathematics and philosophy that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity. In mathematics, infinity is defined in the context of
set theory. The word comes from the
Latin infinitas or "unboundedness."
.^ (As I use the phrase "infinite interval", a temporal interval is infinite if and only if it contains at least an aleph-zero number of equal-lengthed finite temporal intervals [e.g., an aleph-zero number of hours]).- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ Moreover, the real/Cambridge distinction is both vague and equivocal and is often used to mark several different distinctions.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ The same holds for T', such that there are an infinite number of intervals with the order type alpha that are earlier than T (and an infinite number of intervals with the order type alpha that are later than T).- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
The German mathematician
Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries.
.^ There is no difference in kind between Smith (1993a) and Einstein's book Relativity ; one difference in degree is that there is a greater number of mathematical equations in Einstein's book.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
For example, the set of
integers is
countably infinite. However, the set of
real numbers is
uncountably infinite.
.^ One can define changes (i.e., all changes that exist) in terms of acquiring or losing an n -adic property by virtue of a causal event or process .- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ There are an infinite number of temporal intervals with the order type alpha-one and the interval T2 composed of all these intervals has the order type alpha-two.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^{[1]} The paradoxical nature of infinity is illustrated by the idea of a
grand hotel, with infinitely many rooms—all of which are occupied by guests—but can nevertheless manage to accommodate a new guest by moving each existing guest over, one by one, to other rooms.
History
Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks, unable to codify infinity in terms of a formalized mathematical system approached infinity as a philosophical concept.
Early Indian
The
Isha Upanishad of the
Yajurveda (c. 4th to 3rd century BC) states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity".
The
Indian mathematical text
Surya Prajnapti (c.
.^ There exists a set of numbers with this order type, and there is no logical or metaphysical difficulty with supposing that the set of all hours is isomorphic to this set of numbers.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ There are an infinite number of temporal intervals with the order type alpha-one and the interval T2 composed of all these intervals has the order type alpha-two.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
Each of these was further subdivided into three orders:
- Enumerable: lowest, intermediate, and highest
- Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
- Infinite: nearly infinite, truly infinite, infinitely infinite
.^ There exists a set of numbers with this order type, and there is no logical or metaphysical difficulty with supposing that the set of all hours is isomorphic to this set of numbers.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ The same holds for T', such that there are an infinite number of intervals with the order type alpha that are earlier than T (and an infinite number of intervals with the order type alpha that are later than T).- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ This line of reasoning proceeds infinitely, leading to the hypothesis that there are an infinite number of temporal series of order type omega-star plus omega, both before the series {.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
On both physical and
ontological grounds, a distinction was made between
asaṃkhyāta ("countless, innumerable") and
ananta ("endless, unlimited"), between rigidly bounded and loosely bounded infinities.
Buddhism
In some
Buddhist imagery including
Tibetan Buddhist
thangka &
vajrayana meditation deities such as
Chenrezig the deity is often pictured holding a
mala twisted in the middle to form a figure of 8. This represents the endless (infinite) cycle of existence, of birth, death & rebirth, i.e. the [infinity] of
samsara.
Early Greek
.^ Nor is there any instantaneous event that causes the event at instant I, since if there were such a causal relation, the instantaneous event E would cause the instantaneous event E', with this causal relation "bypassing" an infinite number of subsequent instantaneous events that separate E from E'.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ There is a more general argument that shows that any object, concrete or abstract, exists in time if time exists.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ (As I use the phrase "infinite interval", a temporal interval is infinite if and only if it contains at least an aleph-zero number of equal-lengthed finite temporal intervals [e.g., an aleph-zero number of hours]).- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
.^ However, if we are not talking merely about finite intervals, but about all intervals, finite and infinite, then time does "begin" in the sense that it has a first infinite interval, namely, the maximal interval with the order type w*.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
Mathematics
Calculus
.^ There are an infinite number of temporal intervals with the order type alpha-one and the interval T2 composed of all these intervals has the order type alpha-two.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ There is no difference in kind between Smith (1993a) and Einstein's book Relativity ; one difference in degree is that there is a greater number of mathematical equations in Einstein's book.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ (As I use the phrase "infinite interval", a temporal interval is infinite if and only if it contains at least an aleph-zero number of equal-lengthed finite temporal intervals [e.g., an aleph-zero number of hours]).- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
To Leibniz, both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties.
^{[2]}^{[3]}
Real analysis
In
real analysis, the symbol
, called "infinity", denotes an unbounded
limit.
means that
x grows without bound, and
means the value of x is decreasing without bound. If
f(
t) ≥ 0 for every
t, then
- means that f(t) does not bound a finite area from a to b
- means that the area under f(t) is infinite.
- means that the total area under f(t) is finite, and equals n
- means that the sum of the infinite series converges to some real value a.
- means that the sum of the infinite series diverges in the specific sense that the partial sums grow without bound.
.^ (As I use the phrase "infinite interval", a temporal interval is infinite if and only if it contains at least an aleph-zero number of equal-lengthed finite temporal intervals [e.g., an aleph-zero number of hours]).- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ Moreover, the real/Cambridge distinction is both vague and equivocal and is often used to mark several different distinctions.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
Points labeled
and
can be added to the
topological space of the real numbers, producing the
two-point compactification of the real numbers. Adding algebraic properties to this gives us the
extended real numbers.
.^ This line of reasoning proceeds infinitely, leading to the hypothesis that there are an infinite number of temporal series of order type omega-star plus omega, both before the series {.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
Projective geometry also introduces a
line at infinity in plane geometry, and so forth for higher dimensions.
Complex analysis
As in real analysis, in
complex analysis the symbol
, called "infinity", denotes an unsigned infinite
limit.
means that the magnitude
| x | of
x grows beyond any assigned value. A
point labeled can be added to the complex plane as a
topological space giving the one-point
compactification of the complex plane. When this is done, the resulting space is a one-dimensional
complex manifold, or
Riemann surface, called the extended complex plane or the
Riemann sphere.
.^ Similar arguments hold for other real n -adic properties in terms of which temporal existence may (allegedly) be defined, such as light-connectability, movement and the like.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ One instantaneous event is not caused by any event that occupies the immediately preceding instant, since there is no immediately preceding instant.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ There exists a set of numbers with this order type, and there is no logical or metaphysical difficulty with supposing that the set of all hours is isomorphic to this set of numbers.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
On the other hand, this kind of infinity enables division by zero, namely
for any complex number
z except for zero. In this context it is often useful to consider
meromorphic functions as maps into the Riemann sphere taking the value of
at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of
Möbius transformations.
Nonstandard analysis
The original formulation of
infinitesimal calculus by Newton and Leibniz used
infinitesimal quantities. In the twentieth century, it was shown that this treatment could be put on a rigorous footing through various
logical systems, including
smooth infinitesimal analysis and
nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers.
.^ Since there is no spatial midpoint between S and p and no spatial place relative to which the correspondence relation obtains, the present event or state of p's standing in the correspondence relation to S is absolutely present.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ Each present, abstract state A cannot be present relatively to a reference frame, or be simultaneous with the other abstract states relatively to a reference frame, since there is no spatial midpoint between these abstract states; these abstract states are instead present and simultaneous absolutely.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ There is no difference in kind between Smith (1993a) and Einstein's book Relativity ; one difference in degree is that there is a greater number of mathematical equations in Einstein's book.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
For example, if H is an infinite number, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to
non-standard calculus is fully developed in
H. Jerome Keisler's book (see below).
Set theory
A different form of "infinity" are the
ordinal and
cardinal infinities of set theory.
.^ But Cantor has shown us that there are "counterintuitive" truths about transfinite cardinals, ordinals and other order types.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
This modern mathematical conception of the quantitative infinite developed in the late nineteenth century from work by Cantor,
Gottlob Frege,
Richard Dedekind and others, using the idea of collections, or sets.
.^ One event consists in Socrates' exemplification of dying and the other event consists in this first event's exemplifying corresponding to a part of the proposition q .- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ This implies that p acquires and loses the property of being a part of a conjunctive proposition one of whose conjuncts is transiently corresponding to a state of affairs.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
.^ In STR, the laws have the same form in each frame (are covariant), but the infinite multiplicity of different contents makes its postulation of laws infinitely less simple than the Lorentzian postulate of one content.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
.^ But Cantor has shown us that there are "counterintuitive" truths about transfinite cardinals, ordinals and other order types.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ There are an infinite number of temporal intervals with the order type alpha-one and the interval T2 composed of all these intervals has the order type alpha-two.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
.^ In fact, the principle of discrete temporal infinitude implies that time's order type cannot be any ordinal number (recall that not all order types are ordinals).- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ There exists a set of numbers with this order type, and there is no logical or metaphysical difficulty with supposing that the set of all hours is isomorphic to this set of numbers.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ The same holds for T', such that there are an infinite number of intervals with the order type alpha that are earlier than T (and an infinite number of intervals with the order type alpha that are later than T).- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
.^ (As I use the phrase "infinite interval", a temporal interval is infinite if and only if it contains at least an aleph-zero number of equal-lengthed finite temporal intervals [e.g., an aleph-zero number of hours]).- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ There is no difference between a sequence of finite equal-lengthed intervals (e.g., hours) beginning or ending arbitrarily at some interval and a sequence of infinite intervals beginning or ending arbitrarily at some interval, apart from the finite/infinite difference .- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ The principle of nonarbitrary duration, applied to time, is properly generalized to intervals of any length, finite or infinite.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is
countably infinite. If a set is too large to be put in one to one correspondence with the positive integers, it is called
uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity.
.^ Nor is there any instantaneous event that causes the event at instant I, since if there were such a causal relation, the instantaneous event E would cause the instantaneous event E', with this causal relation "bypassing" an infinite number of subsequent instantaneous events that separate E from E'.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ It may be objected that there is a relevant difference between the finite/infinite cases.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ (As I use the phrase "infinite interval", a temporal interval is infinite if and only if it contains at least an aleph-zero number of equal-lengthed finite temporal intervals [e.g., an aleph-zero number of hours]).- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
Cardinality of the continuum
.^ There is no difference in kind between Smith (1993a) and Einstein's book Relativity ; one difference in degree is that there is a greater number of mathematical equations in Einstein's book.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ The same holds for T', such that there are an infinite number of intervals with the order type alpha that are earlier than T (and an infinite number of intervals with the order type alpha that are later than T).- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ There are an infinite number of temporal intervals with the order type alpha-one and the interval T2 composed of all these intervals has the order type alpha-two.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
Namely, Cantor showed that
(see
Cantor's diagonal argument).
.^ One instantaneous event is not caused by any event that occupies the immediately preceding instant, since there is no immediately preceding instant.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ There exists a set of numbers with this order type, and there is no logical or metaphysical difficulty with supposing that the set of all hours is isomorphic to this set of numbers.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ Each present, abstract state A cannot be present relatively to a reference frame, or be simultaneous with the other abstract states relatively to a reference frame, since there is no spatial midpoint between these abstract states; these abstract states are instead present and simultaneous absolutely.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
However, this hypothesis can neither be proved nor disproved within the widely accepted
Zermelo-Fraenkel set theory, even assuming the
Axiom of Choice.
.^ (As I use the phrase "infinite interval", a temporal interval is infinite if and only if it contains at least an aleph-zero number of equal-lengthed finite temporal intervals [e.g., an aleph-zero number of hours]).- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
The first three steps of a fractal construction whose limit is a
space-filling curve, showing that there are as many points in a one-dimensional line segment as in a two-dimensional square.
.^ Two distant events E 1 and E 2 emit light signals that are observed to arrive simultaneously at the apparent spatial midpoint between E 1 and E 2 , relative to the reference frame R, but the signal from E 1 is observed to arrive first at the apparent midpoint relative to the reference frame R'.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ One event consists in Socrates' exemplification of dying and the other event consists in this first event's exemplifying corresponding to a part of the proposition q .- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ Consider one argument for the nonexistence of these properties: .- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when
Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or
hypercube, or finite-dimensional space.
.^ "Time" is reductively definable in terms of an observable spatial relationship between one physical thing (the hands of a clock) and another physical thing (points on the clock dial).- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ In the present case, no relevant difference between numbers and times has ever been established, i.e., a difference that would show times cannot correspond to these numbers.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
Geometry and topology
Fractals
The structure of a
fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have an infinite perimeter resp. an infinite surface area. An example for a fractal curve of infinite length is the
Koch snowflake.
Mathematics without infinity
Physics
In
physics, approximations of
real numbers are used for
continuous measurements and
natural numbers are used for
discrete measurements (i.e. counting). It is therefore assumed by physicists that no
measurable quantity could have an infinite value
^{[citation needed]} , for instance by taking an infinite value in an
extended real number system (see also:
hyperreal number), or by requiring the counting of an infinite number of events. It is for example presumed impossible for any body to have infinite mass or infinite energy. Concepts of infinite things such as an infinite
plane wave exist, but there are no experimental means to generate them.
Theoretical applications of physical infinity
It should be pointed out that this practice of refusing infinite values for measurable quantities does not come from
a priori or ideological motivations, but rather from more methodological and pragmatic motivations
^{[citation needed]}.
.^ I shall briefly present one argument for absolute simultaneity that is based solely on physicalism, with its sets, after I have presented the detailed argument in terms of platonic realism.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ But we need not rely on Lorentz's theory to show this; any world in which time has a topology and metric that cannot be known by observations of physical clocks (but in which Lorentz's laws also do not obtain) is a world that shows (7) is false.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ (Anti-platonism includes Aristotelian realism, conceptualism, physicalism, trope theory and the many varieties of nominalism.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
.^ If time stopped at noon, August 22, 1999, this would be arbitrary since the current boundary conditions of the universe (the current amount and arrangement of mass-energy, etc.,), in conjunction with the laws of nature, imply that it is probable that the universe will continue to exist after this time.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ Each present, abstract state A cannot be present relatively to a reference frame, or be simultaneous with the other abstract states relatively to a reference frame, since there is no spatial midpoint between these abstract states; these abstract states are instead present and simultaneous absolutely.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ Nor is there any instantaneous event that causes the event at instant I, since if there were such a causal relation, the instantaneous event E would cause the instantaneous event E', with this causal relation "bypassing" an infinite number of subsequent instantaneous events that separate E from E'.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
.^ It may be objected that there is a relevant difference between the finite/infinite cases.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
.^ It may be objected that there is a relevant difference between the finite/infinite cases.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ But we need not think that it is logically or metaphysical necessary that concrete objects or real changes exist at some time in order for there to be time.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ Reality would contain fewer entities if we supposed that time ended then, rather than lasted longer or had an infinite future.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
.^ If we adopt a substantival theory of time or spacetime, as many philosophers of physics have done since the 1970s, then (it may be alleged) my above criticisms of these theories do not hold.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
This may help to indicate the limitations of a theory.
This point of view does not mean that infinity cannot be used in physics. For convenience's sake, calculations, equations, theories and approximations often use
infinite series, unbounded
functions, etc., and may involve infinite quantities. Physicists however require that the end result be physically meaningful.
.^ The verificationism of these theories has sometimes been noted in a general way in the philosophical literature, [1] but the verificationist arguments need to be pinpointed precisely and their unsoundness made plain to view.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ But we need not rely on Lorentz's theory to show this; any world in which time has a topology and metric that cannot be known by observations of physical clocks (but in which Lorentz's laws also do not obtain) is a world that shows (7) is false.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
.^ There is no difference between a sequence of finite equal-lengthed intervals (e.g., hours) beginning or ending arbitrarily at some interval and a sequence of infinite intervals beginning or ending arbitrarily at some interval, apart from the finite/infinite difference .- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
One example is the singularity in the description of
black holes.
.^ (As I use the phrase "infinite interval", a temporal interval is infinite if and only if it contains at least an aleph-zero number of equal-lengthed finite temporal intervals [e.g., an aleph-zero number of hours]).- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ For example, he writes in his seminal 1916 paper "The Foundation of the General Theory of Relativity": .- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ Thus, we reject the "received wisdom" that the following conditionals are true: (a) "if STR is false, a neo-Lorentzian theory is true", (b) "if time appears frame-relative but is really absolute, a neo-Lorentzian theory is true".- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
This is an example of what is called a
mathematical singularity, or a point where a physical theory breaks down.
.^ If we adopt a substantival theory of time or spacetime, as many philosophers of physics have done since the 1970s, then (it may be alleged) my above criticisms of these theories do not hold.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ Perhaps his argument may be more charitably construed as the argument that some objects exist in time, but that "an object O's exemplification of an n-adic property F is an event simultaneous with other events" does not imply "the object O exists in time".- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
Two other examples occur in inverse-square force laws of the gravitational force equation of
Newtonian gravity and
Coulomb's Law of electrostatics. At r=0 these equations evaluate to infinities.
Cosmology
.^ There exists a set of numbers with this order type, and there is no logical or metaphysical difficulty with supposing that the set of all hours is isomorphic to this set of numbers.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ Nor is there any instantaneous event that causes the event at instant I, since if there were such a causal relation, the instantaneous event E would cause the instantaneous event E', with this causal relation "bypassing" an infinite number of subsequent instantaneous events that separate E from E'.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ Given that the possibility of there being time before the big bang can be derived from our preceding arguments, the important question to address in this section is whether it is probable or improbable that there is time before the big bang.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
Does the universe have infinite volume? Does space
"go on forever"? This is an important open question of
cosmology. Note that the question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar
topology; if one travelled in a straight line through the universe perhaps one would eventually revisit one's starting point.
If, on the other hand, the universe were not curved like a sphere but had a flat topology, it could be both unbounded and infinite. The curvature of the universe can be measured through
multipole moments in the spectrum of the
Cosmic Background Radiation. As to date, analysis of the radiation patterns recorded by the
WMAP spacecraft hints that the universe has a flat topology.
.^ For example, if our local temporal series has the order type w* + w, and consists of a universe that contracts for an infinite amount of time (corresponding to the {.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
.^ "Considering now the measurement of time, we give an indication that one cannot measure a time t for which ct is less than the Planck length [the Planck length is 10 -33 centimeter.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ For example, if our local temporal series has the order type w* + w, and consists of a universe that contracts for an infinite amount of time (corresponding to the {.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ I shall not here address the question of whether it is possible for P1 and P2 to be satisfied by a time with a different structure.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
Logic
In
logic an
infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."
^{[5]}
Computing
.^ The number zero corresponds to the present hour, the negative numbers correspond to past hours, and the positive numbers to future hours.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
Some
programming languages (for example,
J and
UNITY) specify
greatest and least elements, i.e. values that compare (respectively) greater than or less than all other values. These may also be termed
top and
bottom, or
plus infinity and
minus infinity; they are useful as
sentinel values in
algorithms involving
sorting,
searching or
windowing.
.^ If the (possible) causal relations between events is relative to a frame of reference, but there is nonetheless an absolute temporal ordering of events, then relative to one of these frames of reference, some event can causally influence another that is absolutely earlier.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
^ There is no spatial midpoint between Socrates' death and the set of which his death is the sole member that could make a relative B-relation possible.- ABSOLUTE SIMULTANEITY AND THE INFINITY OF TIME 12 January 2010 7:53 UTC www.qsmithwmu.com [Source type: FILTERED WITH BAYES]
The arts and cognitive sciences
Perspective artwork utilizes the concept of imaginary
vanishing points, or
points at infinity, located at an infinite distance from the observer. This allows artists to create paintings that 'realistically' depict distance and foreshortening of objects. Artist
M. C. Escher is specifically known for employing the concept of infinity in his work in this and other ways.
From the perspective of cognitive scientists
George Lakoff and Nunez, concepts of infinity in mathematics and the sciences are metaphors, based on what they term the Basic Metaphor of Infinity (BMI), namely the ever-increasing sequence <1,2,3,...>.
The infinity symbol
John Wallis introduced the infinity symbol to mathematical literature.
The precise origin of the infinity symbol,
, is unclear. One possibility is suggested by the name it is sometimes called—the
lemniscate, from the Latin
lemniscus, meaning "ribbon".
John Wallis is usually credited with introducing
as a symbol for infinity in 1655 in his
De sectionibus conicis. One conjecture about why he chose this symbol is that he derived it from a
Roman numeral for 1000 that was in turn derived from the
Etruscan numeral for 1000, which looked somewhat like
CIƆ and was sometimes used to mean "many." Another conjecture is that he derived it from the Greek letter ω (
omega), the last letter in the
Greek alphabet.
^{[6]} Also, before typesetting machines were invented, ∞ was easily made in printing by typesetting an 8 type on its side.
The infinity symbol is available in standard
HTML as
∞
and in
LaTeX as
\infty
. In
Unicode, it is the character at code point U+221E (∞), or 8734 in
decimal notation.
See also
References
Other references
- Amir D. Aczel (2001). The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity. New York: Pocket Books. ISBN 0-7434-2299-6.
- D. P. Agrawal (2000). Ancient Jaina Mathematics: an Introduction, Infinity Foundation.
- Bell, J. L.: Continuity and infinitesimals. Stanford Encyclopedia of philosophy. Revised 2009.
- L. C. Jain (1982). Exact Sciences from Jaina Sources.
- L. C. Jain (1973). "Set theory in the Jaina school of mathematics", Indian Journal of History of Science.
- George G. Joseph (2000). The Crest of the Peacock: Non-European Roots of Mathematics (2nd edition ed.). Penguin Books. ISBN 0-14-027778-1.
- H. Jerome Keisler: Elementary Calculus: An Approach Using Infinitesimals. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html
- Eli Maor (1991). To Infinity and Beyond. Princeton University Press. ISBN 0-691-02511-8.
- John J. O'Connor and Edmund F. Robertson (1998). 'Georg Ferdinand Ludwig Philipp Cantor', MacTutor History of Mathematics archive.
- John J. O'Connor and Edmund F. Robertson (2000). 'Jaina mathematics', MacTutor History of Mathematics archive.
- Ian Pearce (2002). 'Jainism', MacTutor History of Mathematics archive.
- Rudy Rucker (1995). Infinity and the Mind: The Science and Philosophy of the Infinite. Princeton University Press. ISBN 0-691-00172-3.
- N. Singh (1988). 'Jaina Theory of Actual Infinity and Transfinite Numbers', Journal of Asiatic Society, Vol. 30.
- David Foster Wallace (2004). Everything and More: A Compact History of Infinity. Norton, W. W. & Company, Inc.. ISBN 0-393-32629-2.
External links
- A Crash Course in the Mathematics of Infinite Sets, by Peter Suber. From the St. John's Review, XLIV, 2 (1998) 1-59. The stand-alone appendix to Infinite Reflections, below. A concise introduction to Cantor's mathematics of infinite sets.
- Infinite Reflections, by Peter Suber. How Cantor's mathematics of the infinite solves a handful of ancient philosophical problems of the infinite. From the St. John's Review, XLIV, 2 (1998) 1-59.
- Infinity, Principia Cybernetica
- Hotel Infinity
- Source page on medieval and modern writing on Infinity
- The Mystery Of The Aleph: Mathematics, the Kabbalah, and the Search for Infinity
- Dictionary of the Infinite (compilation of articles about infinity in physics, mathematics, and philosophy)