Aristotle on the Infinite,
Space, and Time:
Infinite
Covered
in Physics book 3
chapters 4-8
Two
types of investigators from Aristotle’s time
· Some
held infinite a principle part of things
· Aristotle
held the infinite only as a characteristic
5 reasons where someone
could believe an infinite exists
1) Time
is seemingly unlimited
2) Unlimited divisions within a magnitude
3) If
a cyclical process does not end, then there is an unlimited source
of generation and destruction
4) The
regress argument, in that if things have a limit limited by
another thing, which is limited by another thing, etc., then
ultimately there is no limit
5) Numbers
and magnitudes may continue to play out forever in thoughts
Before
considering these 5 reasons Aristotle wants to evaluate two types
of quantity
2
types of classifications on the infinite
-mathematical
-metaphysical
Mathematical
2
types of quantity from Metaphysics
5.13
1)
Multitude
· Numerable
and can be divided into discrete units that are not continuous
with one another
· Each
unit can always have another unit added to it
o Like
counting
· No
upper limit to this but a definite lower limit where the series
began
· Argument
for an infinite magnitude is of this kind
o Any
magnitude could continue to increase in size because another unit
of measurement may be added to it infinitely
· REASONS
WHY ARISTOTLE DOES NOT BELIEVE IN AN INFINITE MAGNITUDE:
2)
Magnitude
· Divisible
into parts that are
continuous with one another
o These
thoughts are probably better expressed later within the topic of
space
o Infinite
divisions of magnitude
§ Ex.
Linear magnitude may be divided in half, and those halves divided
into halves, etc towards infinity
o No
lower limit to this but a definite upper limit
§ Converse
of the addition of multitudes
· Something
already conceived of in Aristotle’s time:
o Zeno’s
Dichotomy in Physics 6.2
o Take
a specified portion of a finite magnitude
add that same ratio of the removal taken from the whole and places
it onto the portion
§ Repeating
ad infinitum will never
allow for the original magnitude to be realized thought it appears
one is always adding more onto the new whole at every turn
§ Geometric
series
· Each
unit is the same proportion of the preceding unit
Metaphysical
· Physics book 3
· Aristotle
knows he cannot dismiss the infinite because number sequences
could not continue nor could magnitudes be divided infinitely
· But
he doesn’t believe that infinites ever actually come about, only
their potential is in existence
o Defines
it as not something of which there is nothing outside of, but
“that which there is always something outside.”
o An
infinite’s potentiality is something from which things may tend in
the direction of infinite but are never realized
§ In
the process, another step may always be added and no completion
can come about
· Two
types of investigators from Aristotle’s time
o Some
held infinite a participle part of things
o Aristotle
held the infinite only as a characteristic
A
return to the 5 reasons
why an “actual” infinite is believed
1) Time
is not considered an infinite by Aristotle but this is discussed
later
2) A
division of magnitudes is only a potential infinite and never
realized into actuality
3) (pic)
Cyclical processes like generations of people or the rotations of
the sun are dismissed as sequences that are numerable and not
continuous. Each may
be divided with a beginning and end and, therefore, possess
limits. A system of
limits is a limited system and no infinite has both an upper and a
lower limit
a. This
is also discussed later and is a very important aspect of the
division of time
4) (pic)
The regress argument. Being
acted upon, or touched upon by another thing is not a limit. There are already limits
imposed on the separate units connected through the interpretation
of “touch”. He
affectively brings the argument back to the conclusion of the 3rd
reason by limiting the units involved
5) Thoughts
are not direct movers of reality or material meaning that a
conception of infinite is not its creation. Nor is it possible to
ever create an infinite is someone’s thoughts, they must stop
eventually
Difference
of
eternal and infinite
· 1st
of 5 reasons
· Motion
and time are labeled as eternals
o No
beginning and no end
o No
infinites
· Distinction
o Same
as 3rd of 5 reasons
§ Time
and motion may be divided into discrete units that exist with an
enumerated sequence
§ May
continue forever but is a limited system
§ An
infinite system may not possess an upper and a lower limit
§ Therefore,
time and motion is not an infinite
Space
The
concept of space is
either
1) A
reservoir for physical possibilities as made up of a “prime”
material
2) A
framework for relative locations
3) A
space in which things ARE, AND
move within. That
space is the container of the body and one may move around within
it
Dismissal
of concepts
1) Aristotle
favors
his idea of matter over space as a “prime” material that is the
substance of the body
3) Aristotle
believes that the body/form are separate from space
Before
continuing
with space, must understand an IMPORTANT BASIC STRUCTURAL
PROPORTY that holds strongly in all three sections
· Continuity
Categories listed lines,
surfaces, bodies, time, and place as continuous discrete
quantities
· Discussed
at length in Physics
· Metaphysics 5.13 noted that
a magnitude is divisible into continuous parts
· Concerning
magnitude
o Parts
of a magnitude have prior parts
§ Points
to lines
§ Lines
to planes
§ Planes
to bodies
o Where
two parts meet, there is a prior part
§ Ex.
When two 1-D lines meet there is a 0-D point
o A
part can NOT be composed of its prior parts
§ Physics 5.3, a line is
not composed of points (its prior parts) because it would take an
infinite amount of 0-D points to create a 1-D line.
· Collective density
§ Infinites
cannot exist
§ Therefore,
a magnitude is not made up of prior parts
o A
magnitude is continuous because the potential of its division into
an infinite number of smaller units
§ Distributive density
· A
magnitude in a given dimension is not reducible to magnitudes of
lesser dimensions, lesser magnitudes are dependent on the 3-D body
and are not separable
Physics book 4
introduces the word “place”
· Space
is the framework for relative locations
· Place
is that relative location
It
is assumed
· All
things exist somewhere
· Understanding
of motion is in respect to place
o This
is called local motion
· Physics 4.1 Space, that
houses “places” includes all the 3-D directions
o Up,
down, left, right, forwards, backwards
Physics 4.4
Aristotle’s
requirements for space
· It
is where a thing is but “place” is not part of the thing
o As
mentioned and reasoned for his dismissal of the 3rd
option for concepts of space
· Place
is neither smaller nor larger than the thing occupying it, it’s
the exact same size
· As
not part of the occupying body, space may be left behind once the
body vacates from the place
Aristotle’s
4 candidates for what
satisfy this criterion
1) Form
of the body
2) Matter
3) An
extension between the outside of the body (its 2-D surface planes)
that displace the space into other space
4) The
outside of the body without an extension in addition to the
magnitude
Dismissal
of the first 3 candidates, acceptance of the 4th
· 1
and 2 are dismissed because form and matter belong to the thing in
the place as expressed in his dismissal of the 1st and
3rd concept of space
· The
3rd candidate is dismissed because it creates an
infinite
o If
space was displaced by the body, then that space would have to
displace other space ad
infinitum
· The
4th candidate is accepted by Aristotle
o Described
as an unmoved boundary around the body
o Form
is the boundary of the thing
o They
do not interfere with one another’s role
These
descriptions
do not relate place to positions in space
· An
interpretation of the translation by H. Lang
o Describes
place as not the first innermost boundary but, “first as the whole
heaven is what first surrounds everything that is contained within
the heavens”
o This
places a limit on space creating individual units of place the
same way as a limit can be imposed on a line creating new lines
o This
translation grants orientation to the six directions with a place
in which to compare to other places, to allow for local motion
Time
Physics 4.11
Mathematical
analysis
of time
· Time
is listed as among the examples of a continuous quantity
· Time
can be divided into smaller metered units (described in Categories)
· Each
unit is defined by the points that separate them and are the
limits of each unit
The
NOW
· Now
is understood only through change (and change is motion)
· Change
is a measure, through time, so that each defines the other
o Differences
of time and motion
§ Time
had no location
§ Time
does not change its measures
· The
Now is understood in reference to a magnitude
Ex.
A
line
o As
lines are not composed of points, time is not composed of Nows
§ Nows
are the limits of time in the same way as a line bisecting another
will create a point that is a new limit of the line
o The
Now shares no part of time any more than the point does with the
line
Physics 4.12
o The
transient nature of Nows keeps them from being indivisible units
of time and allows them to mark out, as limits, intervals with
which can be used to compare to others in measurement
Physics 4.13
The
last piece of the complete schema Physics 4.12
· The
cycle of the stars around the earth is a continuous quantity that
can be divided, like time
perhaps it is time?
·
Because its motion is cyclical, it can be divided into discrete
units and enumerated
· Time
is (also?) linear and not cyclical, is divided arbitrarily without
a reference
· Using
the framework of a cyclical motion, time may be divided into
discrete units measured in relation to the cycles
· Time
is then measured by a repeating motion as a unit of that motion,
with respect to a beginning and end created from the demarcation
of the Nows
· Time
may then measure other motions using the same temporal cadence if
it remains the same for a meaningful measurement of change as a
product of difference
The
overall relationship between infinite, space, and time
· Continuity
is conceived because an infinite cannot be actualized
· This
creates a difference in an eternal and infinite
· Continuity
also relegates the disposition of all magnitudes to its 3-D
identity as does time to its nows
· Meanwhile,
place provides a location in space in which one can measure motion
· Motion
can divide time in between the nows using the continuity of time
· Time
can then be used to measure motion in turn