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Matheology § 110
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WM
Oct 5, 2012, 4:32:53 AM10/5/12
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Matheology § 110
Belief in the existence of the infinite comes mainly from five
considerations:
(1) From the nature of time - for it is infinite.
(2) From the division of magnitudes - for the mathematicians also use
the notion of the infinite.
Further, how can the infinite be itself any thing, unless both number
and magnitude, of which it is an essential attribute, exist in that
way? If they are not substances, a fortiori the infinite is not. It is
plain, too, that the infinite cannot be an actual thing and a
substance and principle.
This discussion, however, involves the more general question whether
the infinite can be present in mathematical objects and things which
are intelligible and do not have extension, as well as among sensible
objects. Our inquiry (as physicists) is limited to its special
subject-
matter, the objects of sense, and we have to ask whether there is or
is not among them a body which is infinite in the direction of
increase.
We may begin with a dialectical argument and show as follows that
there is no such thing. If 'bounded by a surface' is the definition of
body there cannot be an infinite body either intelligible or sensible.
Nor can number taken in abstraction be infinite, for number or that
which has number is numerable. If then the numerable can be numbered,
it would also be possible to go through the infinite.
It is plain from these arguments that there is no body which is
actually infinite.
But on the other hand to suppose that the infinite does not exist in
any way leads obviously to many impossible consequences: there will be
a beginning and an end of time, a magnitude will not be divisible into
magnitudes, number will not be infinite. If, then, in view of the
above considerations, neither alternative seems possible, an arbiter
must be called in; and clearly there is a sense in which the infinite
exists and another in which it does not. We must keep in mind that the
word 'is' means either what potentially is or what fully is. Further,
a thing is infinite either by addition or by division.
Now, as we have seen, magnitude is not actually infinite. But by
division it is infinite. (There is no difficulty in refuting the
theory of indivisible lines.) The alternative then remains that the
infinite has a potential existence.
The infinite exhibits itself in different ways-in time, in the
generations of man, and in the division of magnitudes. For generally
the infinite has this mode of existence: one thing is always being
taken after another, and each thing that is taken is always finite,
but always different.
But in the direction of largeness it is always possible to think of a
larger number: for the number of times a magnitude can be bisected is
infinite. Hence this infinite is potential, never actual: the number
of parts that can be taken always surpasses any assigned number. But
this number is not separable from the process of bisection, and its
infinity is not a permanent actuality but consists in a process of
coming to be, like time and the number of time.
With magnitudes the contrary holds. What is continuous is divided
ad infinitum, but there is no infinite in the direction of increase.
For the size which it can potentially be, it can also actually be.
Hence since no sensible magnitude is infinite, it is impossible to
exceed every assigned magnitude; for if it were possible there would
be something bigger than the heavens.
Our account does not rob the mathematicians of their science, by
disproving the actual existence of the infinite in the direction of
increase, in the sense of the untraversable. In point of fact they do
not need the infinite and do not use it. They postulate only that the
finite straight line may be produced as far as they wish. It is
possible to have divided in the same ratio as the largest quantity
another magnitude of any size you like. Hence, for the purposes of
proof, it will make no difference to them to have such an infinite
instead, while its existence will be in the sphere of real
magnitudes.
It remains to dispose of the arguments which are supposed to support
the view that the infinite exists not only potentially but as a
separate thing. Some have no cogency; others can be met by fresh
objections that are valid.
[Aristoteles: Physics (350 v. Chr.)]
http://www.greektexts.com/library/Aristotle/Physics/eng/index.html
Regards, WM
Belief in the existence of the infinite comes mainly from five
considerations:
(1) From the nature of time - for it is infinite.
(2) From the division of magnitudes - for the mathematicians also use
the notion of the infinite.
Further, how can the infinite be itself any thing, unless both number
and magnitude, of which it is an essential attribute, exist in that
way? If they are not substances, a fortiori the infinite is not. It is
plain, too, that the infinite cannot be an actual thing and a
substance and principle.
This discussion, however, involves the more general question whether
the infinite can be present in mathematical objects and things which
are intelligible and do not have extension, as well as among sensible
objects. Our inquiry (as physicists) is limited to its special
subject-
matter, the objects of sense, and we have to ask whether there is or
is not among them a body which is infinite in the direction of
increase.
We may begin with a dialectical argument and show as follows that
there is no such thing. If 'bounded by a surface' is the definition of
body there cannot be an infinite body either intelligible or sensible.
Nor can number taken in abstraction be infinite, for number or that
which has number is numerable. If then the numerable can be numbered,
it would also be possible to go through the infinite.
It is plain from these arguments that there is no body which is
actually infinite.
But on the other hand to suppose that the infinite does not exist in
any way leads obviously to many impossible consequences: there will be
a beginning and an end of time, a magnitude will not be divisible into
magnitudes, number will not be infinite. If, then, in view of the
above considerations, neither alternative seems possible, an arbiter
must be called in; and clearly there is a sense in which the infinite
exists and another in which it does not. We must keep in mind that the
word 'is' means either what potentially is or what fully is. Further,
a thing is infinite either by addition or by division.
Now, as we have seen, magnitude is not actually infinite. But by
division it is infinite. (There is no difficulty in refuting the
theory of indivisible lines.) The alternative then remains that the
infinite has a potential existence.
The infinite exhibits itself in different ways-in time, in the
generations of man, and in the division of magnitudes. For generally
the infinite has this mode of existence: one thing is always being
taken after another, and each thing that is taken is always finite,
but always different.
But in the direction of largeness it is always possible to think of a
larger number: for the number of times a magnitude can be bisected is
infinite. Hence this infinite is potential, never actual: the number
of parts that can be taken always surpasses any assigned number. But
this number is not separable from the process of bisection, and its
infinity is not a permanent actuality but consists in a process of
coming to be, like time and the number of time.
With magnitudes the contrary holds. What is continuous is divided
ad infinitum, but there is no infinite in the direction of increase.
For the size which it can potentially be, it can also actually be.
Hence since no sensible magnitude is infinite, it is impossible to
exceed every assigned magnitude; for if it were possible there would
be something bigger than the heavens.
Our account does not rob the mathematicians of their science, by
disproving the actual existence of the infinite in the direction of
increase, in the sense of the untraversable. In point of fact they do
not need the infinite and do not use it. They postulate only that the
finite straight line may be produced as far as they wish. It is
possible to have divided in the same ratio as the largest quantity
another magnitude of any size you like. Hence, for the purposes of
proof, it will make no difference to them to have such an infinite
instead, while its existence will be in the sphere of real
magnitudes.
It remains to dispose of the arguments which are supposed to support
the view that the infinite exists not only potentially but as a
separate thing. Some have no cogency; others can be met by fresh
objections that are valid.
[Aristoteles: Physics (350 v. Chr.)]
http://www.greektexts.com/library/Aristotle/Physics/eng/index.html
Regards, WM
Michael Stemper
Oct 5, 2012, 12:53:34 PM10/5/12
to
In article <24060a9a-4475-45a0...@o30g2000vbu.googlegroups.com>, WM <muec...@rz.fh-augsburg.de> writes:
>Matheology =A7 110
those who cannot.
"Three, Sire!"
--
Michael F. Stemper
#include <Standard_Disclaimer>
Life's too important to take seriously.
>Matheology =A7 110
>
>Belief in the existence of the infinite comes mainly from five
>considerations:
>(1) From the nature of time - for it is infinite.
>(2) From the division of magnitudes - for the mathematicians also use
>the notion of the infinite.
There are five kinds of mathematicians -- those who can count and
>Belief in the existence of the infinite comes mainly from five
>considerations:
>(1) From the nature of time - for it is infinite.
>(2) From the division of magnitudes - for the mathematicians also use
>the notion of the infinite.
those who cannot.
"Three, Sire!"
--
Michael F. Stemper
#include <Standard_Disclaimer>
Life's too important to take seriously.
FredJeffries
Oct 5, 2012, 1:04:50 PM10/5/12
to
On Oct 5, 9:53 am, mstem...@walkabout.empros.com (Michael Stemper)
wrote:
surprise...surprise and fear...fear and surprise.... Our two weapons
are fear and surprise...and ruthless efficiency.... Our *three*
weapons are fear, surprise, and ruthless efficiency...and an almost
fanatical devotion to the Pope.... Our *four*...no... *Amongst* our
weapons.... Amongst our weaponry...are such elements as fear,
surprise....
wrote:
> In article <24060a9a-4475-45a0-a101-906717cea...@o30g2000vbu.googlegroups.com>, WM <mueck...@rz.fh-augsburg.de> writes:
>
> >Matheology =A7 110
>
> >Belief in the existence of the infinite comes mainly from five
> >considerations:
> >(1) From the nature of time - for it is infinite.
> >(2) From the division of magnitudes - for the mathematicians also use
> >the notion of the infinite.
>
> There are five kinds of mathematicians -- those who can count and
> those who cannot.
>
> "Three, Sire!"
>
NOBODY expects the Spanish Inquisition! Our chief weapon is
>
> >Matheology =A7 110
>
> >Belief in the existence of the infinite comes mainly from five
> >considerations:
> >(1) From the nature of time - for it is infinite.
> >(2) From the division of magnitudes - for the mathematicians also use
> >the notion of the infinite.
>
> There are five kinds of mathematicians -- those who can count and
> those who cannot.
>
> "Three, Sire!"
>
surprise...surprise and fear...fear and surprise.... Our two weapons
are fear and surprise...and ruthless efficiency.... Our *three*
weapons are fear, surprise, and ruthless efficiency...and an almost
fanatical devotion to the Pope.... Our *four*...no... *Amongst* our
weapons.... Amongst our weaponry...are such elements as fear,
surprise....
Michael Stemper
Oct 5, 2012, 1:13:09 PM10/5/12
to
In article <cfdba9cb-d2c8-413a...@wz4g2000pbc.googlegroups.com>, FredJeffries <fredje...@gmail.com> writes:
>On Oct 5, 9:53=A0am, mstem...@walkabout.empros.com (Michael Stemper) wrote:
>> In article <24060a9a-4475-45a0-a101-906717cea...@o30g2000vbu.googlegroups.com>, WM <mueck...@rz.fh-augsburg.de> preaches on:
>> >Matheology 110
>On Oct 5, 9:53=A0am, mstem...@walkabout.empros.com (Michael Stemper) wrote:
>> In article <24060a9a-4475-45a0-a101-906717cea...@o30g2000vbu.googlegroups.com>, WM <mueck...@rz.fh-augsburg.de> preaches on:
>> >Matheology 110
>>
>> >Belief in the existence of the infinite comes mainly from five
>> >considerations:
>> >(1) From the nature of time - for it is infinite.
>> >(2) From the division of magnitudes - for the mathematicians also use
>> >the notion of the infinite.
>>
>> There are five kinds of mathematicians -- those who can count and
>> those who cannot.
>>
>> "Three, Sire!"
>
>NOBODY expects the Spanish Inquisition! Our chief weapon is
>surprise...surprise and fear...fear and surprise.... Our two weapons
>are fear and surprise...and ruthless efficiency.... Our *three*
>weapons are fear, surprise, and ruthless efficiency...and an almost
>fanatical devotion to the Pope.... Our *four*...no... *Amongst* our
>weapons.... Amongst our weaponry...are such elements as fear,
>surprise....
Why don't you come in again?
>> >Belief in the existence of the infinite comes mainly from five
>> >considerations:
>> >(1) From the nature of time - for it is infinite.
>> >(2) From the division of magnitudes - for the mathematicians also use
>> >the notion of the infinite.
>>
>> There are five kinds of mathematicians -- those who can count and
>> those who cannot.
>>
>> "Three, Sire!"
>
>NOBODY expects the Spanish Inquisition! Our chief weapon is
>surprise...surprise and fear...fear and surprise.... Our two weapons
>are fear and surprise...and ruthless efficiency.... Our *three*
>weapons are fear, surprise, and ruthless efficiency...and an almost
>fanatical devotion to the Pope.... Our *four*...no... *Amongst* our
>weapons.... Amongst our weaponry...are such elements as fear,
>surprise....
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