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Science Forum Index » Logic Forum » Rucker on Infinitesimals (was '0.999... = 1')
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| Author |
Message |
| Paul Holbach |
 Posted: Sun Jan 04, 2004 5:19 pm |
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Guest
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"In our ordinary real number system, we say that the number K with
decimal expansion .99999... is the samas 1. An informal argument for
this is sketched below:
10K = 9.999...
- K = .9999...
____________
9K = 9
K = 1
But maybe this argument is misleading. What if there is some number,
call it 1 - 1/omega, that is greater than any finite string .9...9 of
nines, yet less than 1? If K were actually equal to 1 - 1/omega, the
informal argument used in the last paragraph would not work, for this
argument overlooks the fact that the difference between 10K and 10 is
ten times as great as the difference between K and 1. There is a
residual infinitesimal quantity below that does not get canceled out:
10K = 10 - 10/omega
- K = 1 - 1/omega
_________________
9K = 9 - 9/omega
K = 1 - 1/omega
Intuitively, nothing could be more natural than to go ahead and talk
about 1/omega, 1/Aleph-1, and so on. Just as we move from the natural
numbers to the fractions and then on to the reals, should we not be
able to move from the whole ordinal numbers to some richer number
field?
Curiously, Cantor himself was very much opposed to this step. When a
fellow mathematician attempted to use Cantorīs transfinite numbers to
develop a theory of infinitely small quantities, Cantor accused him of
trying to 'infect mathematics with the Cholera-Bacillus of
infinitesimals'. Cantor even constructed a proof that no number can be
infinitesimal. This proof, however, is just as circular and worthless
as finitist attempts to prove that no number can be infinite. In both
cases, the desired conclusion is smuggled in as part of the definition
of 'number'.
Why was Cantor so vehemently opposed to infinitesimals? In his
valuable essay, 'The Metaphysics of the Calculus', Abraham Robinson
suggests that Cantor already had enough problems trying to defend
transfinite numbers. It seems likely that, consciously or otherwise,
Cantor deemed it politically wise to go along with othodox
mathematicians on the question of infinitesimals. Cantorīs stance
might be compared to that of a pro-marijuana Congressional candidate
who advocates harsh penalties for the sale or use of heroin. Yet, as
we shall see, there is almost as much justification for infinitesimals
as there is for Cantorīs transfinite ordinals.
Formally speaking, it is as consistent to say that there is a number
between all of .9, .99, .999, ... and 1 as it is to say that there is
a number greater than all of 1, 2, 3, ... . And just as we go on to
find more and more ordinals piled atop one another, we can go on to
find more and more infinitesimals squeezed beneath each other.
[...]
But so great is the average personīs fear of infinity that to this day
calculus all over the world is being taught as a study of *limit
processes* instead of what it really is: *infinitesimal analysis*.
As someone who has spent a good portion of his adult life teaching
calculus courses for a living, I can tell you how weary one gets of
trying to explain the complex and fiddling theory of limits to wave
after wave of uncomprehending freshman.
I often think of C. H. Hintonīs words from a similar context:
'How pleasant it would be to let pass away some of the verbiage I
learnt at school--learnt because teachers must live, I suppose. The
apeing and prolonged caw called grammar, the cackling of the human hen
over the egg of language--I should like to unlearn grammar.'
But there is hope for a brighter future. Robinsonīs investigations of
the hyperreal numbers have put infinitesimals on a logically
unimpeachable basis, and here and there calculus texts based on
infinitesimals have appeared [*]."
[*: - Keisler, H. J. (1976). /Elementary Calculus/. Boston: Prindle,
Weber & Schmidt.
- Henle & Kleinberg (1978). /Infinitesimal Calculus/. Cambridge,
Mass.: MIT Press.]
[Rucker, Rudy (1995). /Infinity and the Mind/. Princeton, NJ:
Princeton University Press. (pp. 79/80 + 87)]
PH |
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| Jerry Comisar |
| Posted: Sun Jan 04, 2004 7:20 pm |
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Guest
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Or, 1/3 = .333333...
3 * 1/3 = 1 = .99999...
Paul Holbach wrote:
Quote: "In our ordinary real number system, we say that the number K with
decimal expansion .99999... is the samas 1. An informal argument for
this is sketched below:
10K = 9.999...
- K = .9999...
____________
9K = 9
K = 1
But maybe this argument is misleading. What if there is some number,
call it 1 - 1/omega, that is greater than any finite string .9...9 of
nines, yet less than 1? If K were actually equal to 1 - 1/omega, the
informal argument used in the last paragraph would not work, for this
argument overlooks the fact that the difference between 10K and 10 is
ten times as great as the difference between K and 1. There is a
residual infinitesimal quantity below that does not get canceled out:
10K = 10 - 10/omega
- K = 1 - 1/omega
_________________
9K = 9 - 9/omega
K = 1 - 1/omega
Intuitively, nothing could be more natural than to go ahead and talk
about 1/omega, 1/Aleph-1, and so on. Just as we move from the natural
numbers to the fractions and then on to the reals, should we not be
able to move from the whole ordinal numbers to some richer number
field?
Curiously, Cantor himself was very much opposed to this step. When a
fellow mathematician attempted to use Cantorīs transfinite numbers to
develop a theory of infinitely small quantities, Cantor accused him of
trying to 'infect mathematics with the Cholera-Bacillus of
infinitesimals'. Cantor even constructed a proof that no number can be
infinitesimal. This proof, however, is just as circular and worthless
as finitist attempts to prove that no number can be infinite. In both
cases, the desired conclusion is smuggled in as part of the definition
of 'number'.
Why was Cantor so vehemently opposed to infinitesimals? In his
valuable essay, 'The Metaphysics of the Calculus', Abraham Robinson
suggests that Cantor already had enough problems trying to defend
transfinite numbers. It seems likely that, consciously or otherwise,
Cantor deemed it politically wise to go along with othodox
mathematicians on the question of infinitesimals. Cantorīs stance
might be compared to that of a pro-marijuana Congressional candidate
who advocates harsh penalties for the sale or use of heroin. Yet, as
we shall see, there is almost as much justification for infinitesimals
as there is for Cantorīs transfinite ordinals.
Formally speaking, it is as consistent to say that there is a number
between all of .9, .99, .999, ... and 1 as it is to say that there is
a number greater than all of 1, 2, 3, ... . And just as we go on to
find more and more ordinals piled atop one another, we can go on to
find more and more infinitesimals squeezed beneath each other.
[...]
But so great is the average personīs fear of infinity that to this day
calculus all over the world is being taught as a study of *limit
processes* instead of what it really is: *infinitesimal analysis*.
As someone who has spent a good portion of his adult life teaching
calculus courses for a living, I can tell you how weary one gets of
trying to explain the complex and fiddling theory of limits to wave
after wave of uncomprehending freshman.
I often think of C. H. Hintonīs words from a similar context:
'How pleasant it would be to let pass away some of the verbiage I
learnt at school--learnt because teachers must live, I suppose. The
apeing and prolonged caw called grammar, the cackling of the human hen
over the egg of language--I should like to unlearn grammar.'
But there is hope for a brighter future. Robinsonīs investigations of
the hyperreal numbers have put infinitesimals on a logically
unimpeachable basis, and here and there calculus texts based on
infinitesimals have appeared [*]."
[*: - Keisler, H. J. (1976). /Elementary Calculus/. Boston: Prindle,
Weber & Schmidt.
- Henle & Kleinberg (1978). /Infinitesimal Calculus/. Cambridge,
Mass.: MIT Press.]
[Rucker, Rudy (1995). /Infinity and the Mind/. Princeton, NJ:
Princeton University Press. (pp. 79/80 + 87)]
PH |
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| Jerry Comisar |
| Posted: Sun Jan 04, 2004 7:20 pm |
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Guest
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Or, 1/3 = .333333...
3 * 1/3 = 1 = .99999...
Paul Holbach wrote:
Quote: "In our ordinary real number system, we say that the number K with
decimal expansion .99999... is the samas 1. An informal argument for
this is sketched below:
10K = 9.999...
- K = .9999...
____________
9K = 9
K = 1
But maybe this argument is misleading. What if there is some number,
call it 1 - 1/omega, that is greater than any finite string .9...9 of
nines, yet less than 1? If K were actually equal to 1 - 1/omega, the
informal argument used in the last paragraph would not work, for this
argument overlooks the fact that the difference between 10K and 10 is
ten times as great as the difference between K and 1. There is a
residual infinitesimal quantity below that does not get canceled out:
10K = 10 - 10/omega
- K = 1 - 1/omega
_________________
9K = 9 - 9/omega
K = 1 - 1/omega
Intuitively, nothing could be more natural than to go ahead and talk
about 1/omega, 1/Aleph-1, and so on. Just as we move from the natural
numbers to the fractions and then on to the reals, should we not be
able to move from the whole ordinal numbers to some richer number
field?
Curiously, Cantor himself was very much opposed to this step. When a
fellow mathematician attempted to use Cantorīs transfinite numbers to
develop a theory of infinitely small quantities, Cantor accused him of
trying to 'infect mathematics with the Cholera-Bacillus of
infinitesimals'. Cantor even constructed a proof that no number can be
infinitesimal. This proof, however, is just as circular and worthless
as finitist attempts to prove that no number can be infinite. In both
cases, the desired conclusion is smuggled in as part of the definition
of 'number'.
Why was Cantor so vehemently opposed to infinitesimals? In his
valuable essay, 'The Metaphysics of the Calculus', Abraham Robinson
suggests that Cantor already had enough problems trying to defend
transfinite numbers. It seems likely that, consciously or otherwise,
Cantor deemed it politically wise to go along with othodox
mathematicians on the question of infinitesimals. Cantorīs stance
might be compared to that of a pro-marijuana Congressional candidate
who advocates harsh penalties for the sale or use of heroin. Yet, as
we shall see, there is almost as much justification for infinitesimals
as there is for Cantorīs transfinite ordinals.
Formally speaking, it is as consistent to say that there is a number
between all of .9, .99, .999, ... and 1 as it is to say that there is
a number greater than all of 1, 2, 3, ... . And just as we go on to
find more and more ordinals piled atop one another, we can go on to
find more and more infinitesimals squeezed beneath each other.
[...]
But so great is the average personīs fear of infinity that to this day
calculus all over the world is being taught as a study of *limit
processes* instead of what it really is: *infinitesimal analysis*.
As someone who has spent a good portion of his adult life teaching
calculus courses for a living, I can tell you how weary one gets of
trying to explain the complex and fiddling theory of limits to wave
after wave of uncomprehending freshman.
I often think of C. H. Hintonīs words from a similar context:
'How pleasant it would be to let pass away some of the verbiage I
learnt at school--learnt because teachers must live, I suppose. The
apeing and prolonged caw called grammar, the cackling of the human hen
over the egg of language--I should like to unlearn grammar.'
But there is hope for a brighter future. Robinsonīs investigations of
the hyperreal numbers have put infinitesimals on a logically
unimpeachable basis, and here and there calculus texts based on
infinitesimals have appeared [*]."
[*: - Keisler, H. J. (1976). /Elementary Calculus/. Boston: Prindle,
Weber & Schmidt.
- Henle & Kleinberg (1978). /Infinitesimal Calculus/. Cambridge,
Mass.: MIT Press.]
[Rucker, Rudy (1995). /Infinity and the Mind/. Princeton, NJ:
Princeton University Press. (pp. 79/80 + 87)]
PH |
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| The Ghost In The Machine |
| Posted: Sun Jan 04, 2004 8:01 pm |
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Guest
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In sci.logic, Paul Holbach
<paulholbachSPAMBAN@freenet.de>
wrote
on 4 Jan 2004 14:19:31 -0800
<881c8779.0401041419.1097372c@posting.google.com>:
Quote: "In our ordinary real number system, we say that the number K with
decimal expansion .99999... is the samas 1. An informal argument for
this is sketched below:
10K = 9.999...
- K = .9999...
____________
9K = 9
K = 1
But maybe this argument is misleading.
That argument suffers from a major problem, as one can readily see
by considering the finite case.
Kn = .999...99
10*Kn = 9.999...90
10*Kn - Kn = 8.999...91
Quote: What if there is some number,
call it 1 - 1/omega, that is greater than any finite string .9...9 of
nines, yet less than 1? If K were actually equal to 1 - 1/omega, the
informal argument used in the last paragraph would not work, for this
argument overlooks the fact that the difference between 10K and 10 is
ten times as great as the difference between K and 1. There is a
residual infinitesimal quantity below that does not get canceled out:
10K = 10 - 10/omega
- K = 1 - 1/omega
_________________
9K = 9 - 9/omega
K = 1 - 1/omega
Intuitively, nothing could be more natural than to go ahead and talk
about 1/omega, 1/Aleph-1, and so on. Just as we move from the natural
numbers to the fractions and then on to the reals, should we not be
able to move from the whole ordinal numbers to some richer number
field?
K = .9999... = 1 - 1/\infty
K/10 = .09999... = 0.1 - 1/(10*\infty)
0.9 + K/10 = 0.9999... = 1 - 1/(10*\infty)
Might work but then one would have to, as you state below,
be very careful as to where the infintesimals and transfinites go.
Some possible questions, for example, from college freshman (well,
OK, one ex-college freshman  ).
[1] What is 1/\infty + 1/\infty?
[a] 2/\infty
[b] 1/\infty
[c] 0
[d] indeterminate
[e] the expression cannot be simplified, but must be left in
this form
[2] What is 1/\infty - 1/\infty?
[a] 0
[b] 1/\infty
[c] -1/\infty
[d] indeterminate
[e] the expression cannot be simplified
[3] What is (1/\infty)/(1/\infty)?
[a] 1
[b] \infty
[c] any number you want
[d] undefined
[e] the expression cannot be simplified
[4] What is 1/\infty^2?
[a] 1/\infty
[b] 0
[c] any number you want
[d] undefined
[e] the expression cannot be simplified
[5] What is 1/sqrt(\infty)?
[a] 1/\infty
[b] 0
[c] any number you want
[d] undefined
[e] the expression cannot be simplified
[6] What is 0.999... + 1/\infty?
[a] 1
[b] 0.999....
[c] any number you want
[d] undefined
[e] the expression cannot be simplified
[7] What is the relationship between 1/\infty and 0?
[a] 1/\infty > 0
[b] 1/\infty = 0
[c] indeterminate
[8] What does 1/3 equal?
[a] 0.333....
[b] 0.333.... + 1/\infty
[c] 0.333.... + 1/(3*\infty)
[d] indeterminate
[e] the expression cannot be expanded into an infinite decimal
[9] What does 3*(1/3) equal?
[a] 1
[b] 0.999...
[c] 1 - 1/\infty
[d] 1 - 3/\infty
[e] indeterminate
[f] the expression cannot be simplified
[10] What does sqrt(1 - 1/\infty) equal?
[a] 1 - 1/(2*\infty) + 3/(8*\infty^2) - 5/(16*\infty^3) + ...
[b] 1 - 1/(2*\infty)
[c] 1 - 1/\infty
[d] 1
[e] indeterminate
[f] the expression cannot be simplified
[11] What does lim(h->0+) (1+h)^(1/h) equal?
[a] Euler's number, 'e'
[b] (1+1/\infty)^\infty
[c] the expression cannot be simplified
[12] What does 1/(1 - 1/\infty) equal?
[a] 1 + 1/\infty + 1/\infty^2 + 1/\infty^3 + ...
[b] 1 + 1/\infty
[c] 1
[d] indeterminate
[e] the expression cannot be simplified
[13] What is lim(x->0-) (x^2/x)?
[a] 0
[b] -1/\infty
[c] 1/\infty
[d] \infty/\infty^2
[e] indeterminate
[f] the expression cannot be simplified
[14] What is sum(i=1,+\infty) (1/\infty)?
[a] 0
[b] 1
[c] e
[d] indeterminate
[e] the expression cannot be simplified
[15] Does the trichotomy principle always hold?
[a] yes
[b] yes, but it may be tricky to determine when \infty is in there
[c] only when \infty is not involved
[d] no
[e] unknown
All of these will have to be dealt with in some fashion.
Standard mathematics uses the sequence
1c2a3d4b5b6a7b8a9a10d11a12c13a14d15a, which implicitly sets 1/\infty=0
pretty much everywhere.
Quote: Curiously, Cantor himself was very much opposed to this step. When a
fellow mathematician attempted to use Cantorīs transfinite numbers to
develop a theory of infinitely small quantities, Cantor accused him of
trying to 'infect mathematics with the Cholera-Bacillus of
infinitesimals'. Cantor even constructed a proof that no number can be
infinitesimal. This proof, however, is just as circular and worthless
as finitist attempts to prove that no number can be infinite. In both
cases, the desired conclusion is smuggled in as part of the definition
of 'number'.
It depends on how one defines "number". If one uses Cauchy
sequences one has problems, for example, as Cauchy sequences
lead to limits.
0.999... is such a sequence.
f_1 = 0.9
f_2 = 0.99
f_3 = 0.999
etc.
The classical definition of limit for this case is:
f = lim(s->+\infty) f_s if, for any epsilon > 0 I pick, one
can show an N such that for all s > N, abs(f_s - f) < epsilon.
If f = 1 - 1/\infty, I pick epsilon = 1/2*\infty, and then wonder
what N satisfies this definition, bearing in mind 0 < 1/\infty < 1/10^N
for any integer N (although the nonpositive integers aren't all that
interesting )
You may also recall Douglas Adams' number 2^(\infty - 1).
Quote: Why was Cantor so vehemently opposed to infinitesimals? In his
valuable essay, 'The Metaphysics of the Calculus', Abraham Robinson
suggests that Cantor already had enough problems trying to defend
transfinite numbers. It seems likely that, consciously or otherwise,
Cantor deemed it politically wise to go along with othodox
mathematicians on the question of infinitesimals. Cantorīs stance
might be compared to that of a pro-marijuana Congressional candidate
who advocates harsh penalties for the sale or use of heroin. Yet, as
we shall see, there is almost as much justification for infinitesimals
as there is for Cantorīs transfinite ordinals.
Formally speaking, it is as consistent to say that there is a number
between all of .9, .99, .999, ... and 1 as it is to say that there is
a number greater than all of 1, 2, 3, ... . And just as we go on to
find more and more ordinals piled atop one another, we can go on to
find more and more infinitesimals squeezed beneath each other.
[...]
But so great is the average personīs fear of infinity that to this day
calculus all over the world is being taught as a study of *limit
processes* instead of what it really is: *infinitesimal analysis*.
As someone who has spent a good portion of his adult life teaching
calculus courses for a living, I can tell you how weary one gets of
trying to explain the complex and fiddling theory of limits to wave
after wave of uncomprehending freshman.
I often think of C. H. Hintonīs words from a similar context:
'How pleasant it would be to let pass away some of the verbiage I
learnt at school--learnt because teachers must live, I suppose. The
apeing and prolonged caw called grammar, the cackling of the human hen
over the egg of language--I should like to unlearn grammar.'
But there is hope for a brighter future. Robinsonīs investigations of
the hyperreal numbers have put infinitesimals on a logically
unimpeachable basis, and here and there calculus texts based on
infinitesimals have appeared [*]."
Let us hope Robinson has some answers for my questions.
I've not read his works, though.
The Leibnitz notation [dy/dx] is currently a limit,
even if it does look like the division of two infinitesimals.
If y(x) = x^2, then dy/dx = lim(h->0) (y(x+h) - y(x))/h
= lim(h->0) (x^2 + 2hx + h^2 - x^2) / h
= lim(h->0) (2hx + h^2)/h
= lim(h->0) (2x + h) = 2x, for example.
Integration is similar. This notation can be abused nastily, and is.
Quote:
[*: - Keisler, H. J. (1976). /Elementary Calculus/. Boston: Prindle,
Weber & Schmidt.
- Henle & Kleinberg (1978). /Infinitesimal Calculus/. Cambridge,
Mass.: MIT Press.]
[Rucker, Rudy (1995). /Infinity and the Mind/. Princeton, NJ:
Princeton University Press. (pp. 79/80 + 87)]
PH
--
#191, ewill3@earthlink.net
It's still legal to go .sigless. |
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| The Ghost In The Machine |
| Posted: Sun Jan 04, 2004 8:01 pm |
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Guest
|
In sci.logic, Paul Holbach
<paulholbachSPAMBAN@freenet.de>
wrote
on 4 Jan 2004 14:19:31 -0800
<881c8779.0401041419.1097372c@posting.google.com>:
Quote: "In our ordinary real number system, we say that the number K with
decimal expansion .99999... is the samas 1. An informal argument for
this is sketched below:
10K = 9.999...
- K = .9999...
____________
9K = 9
K = 1
But maybe this argument is misleading.
That argument suffers from a major problem, as one can readily see
by considering the finite case.
Kn = .999...99
10*Kn = 9.999...90
10*Kn - Kn = 8.999...91
Quote: What if there is some number,
call it 1 - 1/omega, that is greater than any finite string .9...9 of
nines, yet less than 1? If K were actually equal to 1 - 1/omega, the
informal argument used in the last paragraph would not work, for this
argument overlooks the fact that the difference between 10K and 10 is
ten times as great as the difference between K and 1. There is a
residual infinitesimal quantity below that does not get canceled out:
10K = 10 - 10/omega
- K = 1 - 1/omega
_________________
9K = 9 - 9/omega
K = 1 - 1/omega
Intuitively, nothing could be more natural than to go ahead and talk
about 1/omega, 1/Aleph-1, and so on. Just as we move from the natural
numbers to the fractions and then on to the reals, should we not be
able to move from the whole ordinal numbers to some richer number
field?
K = .9999... = 1 - 1/\infty
K/10 = .09999... = 0.1 - 1/(10*\infty)
0.9 + K/10 = 0.9999... = 1 - 1/(10*\infty)
Might work but then one would have to, as you state below,
be very careful as to where the infintesimals and transfinites go.
Some possible questions, for example, from college freshman (well,
OK, one ex-college freshman ).
[1] What is 1/\infty + 1/\infty?
[a] 2/\infty
[b] 1/\infty
[c] 0
[d] indeterminate
[e] the expression cannot be simplified, but must be left in
this form
[2] What is 1/\infty - 1/\infty?
[a] 0
[b] 1/\infty
[c] -1/\infty
[d] indeterminate
[e] the expression cannot be simplified
[3] What is (1/\infty)/(1/\infty)?
[a] 1
[b] \infty
[c] any number you want
[d] undefined
[e] the expression cannot be simplified
[4] What is 1/\infty^2?
[a] 1/\infty
[b] 0
[c] any number you want
[d] undefined
[e] the expression cannot be simplified
[5] What is 1/sqrt(\infty)?
[a] 1/\infty
[b] 0
[c] any number you want
[d] undefined
[e] the expression cannot be simplified
[6] What is 0.999... + 1/\infty?
[a] 1
[b] 0.999....
[c] any number you want
[d] undefined
[e] the expression cannot be simplified
[7] What is the relationship between 1/\infty and 0?
[a] 1/\infty > 0
[b] 1/\infty = 0
[c] indeterminate
[8] What does 1/3 equal?
[a] 0.333....
[b] 0.333.... + 1/\infty
[c] 0.333.... + 1/(3*\infty)
[d] indeterminate
[e] the expression cannot be expanded into an infinite decimal
[9] What does 3*(1/3) equal?
[a] 1
[b] 0.999...
[c] 1 - 1/\infty
[d] 1 - 3/\infty
[e] indeterminate
[f] the expression cannot be simplified
[10] What does sqrt(1 - 1/\infty) equal?
[a] 1 - 1/(2*\infty) + 3/(8*\infty^2) - 5/(16*\infty^3) + ...
[b] 1 - 1/(2*\infty)
[c] 1 - 1/\infty
[d] 1
[e] indeterminate
[f] the expression cannot be simplified
[11] What does lim(h->0+) (1+h)^(1/h) equal?
[a] Euler's number, 'e'
[b] (1+1/\infty)^\infty
[c] the expression cannot be simplified
[12] What does 1/(1 - 1/\infty) equal?
[a] 1 + 1/\infty + 1/\infty^2 + 1/\infty^3 + ...
[b] 1 + 1/\infty
[c] 1
[d] indeterminate
[e] the expression cannot be simplified
[13] What is lim(x->0-) (x^2/x)?
[a] 0
[b] -1/\infty
[c] 1/\infty
[d] \infty/\infty^2
[e] indeterminate
[f] the expression cannot be simplified
[14] What is sum(i=1,+\infty) (1/\infty)?
[a] 0
[b] 1
[c] e
[d] indeterminate
[e] the expression cannot be simplified
[15] Does the trichotomy principle always hold?
[a] yes
[b] yes, but it may be tricky to determine when \infty is in there
[c] only when \infty is not involved
[d] no
[e] unknown
All of these will have to be dealt with in some fashion.
Standard mathematics uses the sequence
1c2a3d4b5b6a7b8a9a10d11a12c13a14d15a, which implicitly sets 1/\infty=0
pretty much everywhere.
Quote: Curiously, Cantor himself was very much opposed to this step. When a
fellow mathematician attempted to use Cantorīs transfinite numbers to
develop a theory of infinitely small quantities, Cantor accused him of
trying to 'infect mathematics with the Cholera-Bacillus of
infinitesimals'. Cantor even constructed a proof that no number can be
infinitesimal. This proof, however, is just as circular and worthless
as finitist attempts to prove that no number can be infinite. In both
cases, the desired conclusion is smuggled in as part of the definition
of 'number'.
It depends on how one defines "number". If one uses Cauchy
sequences one has problems, for example, as Cauchy sequences
lead to limits.
0.999... is such a sequence.
f_1 = 0.9
f_2 = 0.99
f_3 = 0.999
etc.
The classical definition of limit for this case is:
f = lim(s->+\infty) f_s if, for any epsilon > 0 I pick, one
can show an N such that for all s > N, abs(f_s - f) < epsilon.
If f = 1 - 1/\infty, I pick epsilon = 1/2*\infty, and then wonder
what N satisfies this definition, bearing in mind 0 < 1/\infty < 1/10^N
for any integer N (although the nonpositive integers aren't all that
interesting )
You may also recall Douglas Adams' number 2^(\infty - 1).
Quote: Why was Cantor so vehemently opposed to infinitesimals? In his
valuable essay, 'The Metaphysics of the Calculus', Abraham Robinson
suggests that Cantor already had enough problems trying to defend
transfinite numbers. It seems likely that, consciously or otherwise,
Cantor deemed it politically wise to go along with othodox
mathematicians on the question of infinitesimals. Cantorīs stance
might be compared to that of a pro-marijuana Congressional candidate
who advocates harsh penalties for the sale or use of heroin. Yet, as
we shall see, there is almost as much justification for infinitesimals
as there is for Cantorīs transfinite ordinals.
Formally speaking, it is as consistent to say that there is a number
between all of .9, .99, .999, ... and 1 as it is to say that there is
a number greater than all of 1, 2, 3, ... . And just as we go on to
find more and more ordinals piled atop one another, we can go on to
find more and more infinitesimals squeezed beneath each other.
[...]
But so great is the average personīs fear of infinity that to this day
calculus all over the world is being taught as a study of *limit
processes* instead of what it really is: *infinitesimal analysis*.
As someone who has spent a good portion of his adult life teaching
calculus courses for a living, I can tell you how weary one gets of
trying to explain the complex and fiddling theory of limits to wave
after wave of uncomprehending freshman.
I often think of C. H. Hintonīs words from a similar context:
'How pleasant it would be to let pass away some of the verbiage I
learnt at school--learnt because teachers must live, I suppose. The
apeing and prolonged caw called grammar, the cackling of the human hen
over the egg of language--I should like to unlearn grammar.'
But there is hope for a brighter future. Robinsonīs investigations of
the hyperreal numbers have put infinitesimals on a logically
unimpeachable basis, and here and there calculus texts based on
infinitesimals have appeared [*]."
Let us hope Robinson has some answers for my questions.
I've not read his works, though.
The Leibnitz notation [dy/dx] is currently a limit,
even if it does look like the division of two infinitesimals.
If y(x) = x^2, then dy/dx = lim(h->0) (y(x+h) - y(x))/h
= lim(h->0) (x^2 + 2hx + h^2 - x^2) / h
= lim(h->0) (2hx + h^2)/h
= lim(h->0) (2x + h) = 2x, for example.
Integration is similar. This notation can be abused nastily, and is.
Quote:
[*: - Keisler, H. J. (1976). /Elementary Calculus/. Boston: Prindle,
Weber & Schmidt.
- Henle & Kleinberg (1978). /Infinitesimal Calculus/. Cambridge,
Mass.: MIT Press.]
[Rucker, Rudy (1995). /Infinity and the Mind/. Princeton, NJ:
Princeton University Press. (pp. 79/80 + 87)]
PH
--
#191, ewill3@earthlink.net
It's still legal to go .sigless. |
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| Paul Holbach |
| Posted: Sun Jan 04, 2004 9:13 pm |
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Guest
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Quote: paulholbachSPAMBAN@freenet.de (Paul Holbach) wrote in message news:
881c8779.0401041419.1097372c@posting.google.com>...
[...]
But there is hope for a brighter future. Robinsonīs investigations of
the hyperreal numbers have put infinitesimals on a logically
unimpeachable basis, and here and there calculus texts based on
infinitesimals have appeared [*]."
[*: - Keisler, H. J. (1976). /Elementary Calculus/. Boston: Prindle,
Weber & Schmidt.
- Henle & Kleinberg (1978). /Infinitesimal Calculus/. Cambridge,
Mass.: MIT Press.]
[Rucker, Rudy (1995). /Infinity and the Mind/. Princeton, NJ:
Princeton University Press. (pp. 79/80 + 87)]
I have just discovered that the entire text of Keislerīs book, to
which Rucker refers, can be downloaded for free at the following link:
http://www.math.wisc.edu/~keisler/calc.html
PH |
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| Paul Holbach |
| Posted: Sun Jan 04, 2004 9:13 pm |
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Guest
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Quote: paulholbachSPAMBAN@freenet.de (Paul Holbach) wrote in message news:
881c8779.0401041419.1097372c@posting.google.com>...
[...]
But there is hope for a brighter future. Robinsonīs investigations of
the hyperreal numbers have put infinitesimals on a logically
unimpeachable basis, and here and there calculus texts based on
infinitesimals have appeared [*]."
[*: - Keisler, H. J. (1976). /Elementary Calculus/. Boston: Prindle,
Weber & Schmidt.
- Henle & Kleinberg (1978). /Infinitesimal Calculus/. Cambridge,
Mass.: MIT Press.]
[Rucker, Rudy (1995). /Infinity and the Mind/. Princeton, NJ:
Princeton University Press. (pp. 79/80 + 87)]
I have just discovered that the entire text of Keislerīs book, to
which Rucker refers, can be downloaded for free at the following link:
http://www.math.wisc.edu/~keisler/calc.html
PH |
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| Tim Smith |
| Posted: Mon Jan 05, 2004 12:12 am |
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Guest
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In article <881c8779.0401041419.1097372c@posting.google.com>, Paul Holbach
wrote:
Quote: But there is hope for a brighter future. Robinsonīs investigations of the
hyperreal numbers have put infinitesimals on a logically unimpeachable
basis, and here and there calculus texts based on infinitesimals have
appeared [*]."
What is the current status of this? I read somewhere, but don't recall
where that it was proven that Robinson's approach and the standard approach
are exactly equivalent in what they can do, so that while Robinson's
approach is more intuitive, it cannot lead to any more or less than the
standard approach. Thus, while the non-standard approach would be better if
everyone was starting from scratch, there is so much already in standard
terms, that learning the standard approach is necessary, and once you've
done that, there is no need to learn the non-standard approach, since you
can do everything using the standard approach.
Basically, a mathematical society gets to choose one way or the other, and
then they are stuck with it. We hit on the limit approach rather than the
rigorization of infenitesimals, and now are stuck.
So, is that correct?
--
--Tim Smith |
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| Back to top |
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| Tim Smith |
| Posted: Mon Jan 05, 2004 12:12 am |
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Guest
|
In article <881c8779.0401041419.1097372c@posting.google.com>, Paul Holbach
wrote:
Quote: But there is hope for a brighter future. Robinsonīs investigations of the
hyperreal numbers have put infinitesimals on a logically unimpeachable
basis, and here and there calculus texts based on infinitesimals have
appeared [*]."
What is the current status of this? I read somewhere, but don't recall
where that it was proven that Robinson's approach and the standard approach
are exactly equivalent in what they can do, so that while Robinson's
approach is more intuitive, it cannot lead to any more or less than the
standard approach. Thus, while the non-standard approach would be better if
everyone was starting from scratch, there is so much already in standard
terms, that learning the standard approach is necessary, and once you've
done that, there is no need to learn the non-standard approach, since you
can do everything using the standard approach.
Basically, a mathematical society gets to choose one way or the other, and
then they are stuck with it. We hit on the limit approach rather than the
rigorization of infenitesimals, and now are stuck.
So, is that correct?
--
--Tim Smith |
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| Lewis Mammel |
| Posted: Mon Jan 05, 2004 2:28 am |
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Guest
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Tim Smith wrote:
Quote:
Basically, a mathematical society gets to choose one way or the other, and
then they are stuck with it. We hit on the limit approach rather than the
rigorization of infenitesimals, and now are stuck.
So, is that correct?
No, because the original idea is the infinitesimal - "Through P let there
be drawn to this [ spherical ] surface two lines HK, IL, intercepting
very small arcs HI, KL ; ... " - Prop. LXX Theor. XXX of Newtons' Principia.
This is his proof that there is no gravitational attraction within
a hollow uniform sphere.
His reasoning is very intuitive, and the presentation is quite sketchy.
I think nonstandard analysis is a more direct justification of this
type of reasoning, as opposed to the limit justification which casts
it in somewhat different terms.
In physics, one never really makes the distinction between "delta x"
and "dx", and I think if one understands "delta x" as "first order in
x" the equation can be made rigorous, as witness the nonstandard
construction of infinitesimals in terms of sequences.
Lew Mammel, Jr. |
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| Lewis Mammel |
| Posted: Mon Jan 05, 2004 2:28 am |
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Guest
|
Tim Smith wrote:
Quote:
Basically, a mathematical society gets to choose one way or the other, and
then they are stuck with it. We hit on the limit approach rather than the
rigorization of infenitesimals, and now are stuck.
So, is that correct?
No, because the original idea is the infinitesimal - "Through P let there
be drawn to this [ spherical ] surface two lines HK, IL, intercepting
very small arcs HI, KL ; ... " - Prop. LXX Theor. XXX of Newtons' Principia.
This is his proof that there is no gravitational attraction within
a hollow uniform sphere.
His reasoning is very intuitive, and the presentation is quite sketchy.
I think nonstandard analysis is a more direct justification of this
type of reasoning, as opposed to the limit justification which casts
it in somewhat different terms.
In physics, one never really makes the distinction between "delta x"
and "dx", and I think if one understands "delta x" as "first order in
x" the equation can be made rigorous, as witness the nonstandard
construction of infinitesimals in terms of sequences.
Lew Mammel, Jr. |
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| Tony Thomas |
| Posted: Mon Jan 05, 2004 4:25 am |
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Guest
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Paul
I have been pushing this barrow for some time but the fruit is regarded as
rotten or forbidden by the cogniscenti.
I do appreciate that the 'doctrine' of real numbers has been forged by some
of the world's finest intellects and is not to be taken lightly, however,
your argumant for a bit left over in convergent infinite series may be
flawed.
Consider:
Sn = (9/10 + 9/100 + ...+9/10^n) = 1 - 9/10^n
which is true for all values of n.
The mysterious zen bull's tail never disappears altogether but seems to
converge to your omega rather than to zero. But the high priests say that as
n tends to infinity the tail tends to zero.
The trick is that this expression and others like it can be said to never
exceed its limit, which seems to be true.
When we say that 2x is the differential coefficient of x^2 we are are
including such a limit in the expression, but this works OK. The point is
that the use of such limits is guaranteed to be harmless, provided that we
do not admit any pesky infinitessimals.
Some time ago (1950s), mathemeticians invented things like supernatiural
numbers and integrated them into analysis, so the whole question is water
under the bridge. The annoying residual is all that Cantorian set theoretic
doctrine that doesn't seem to be consistent with the newly
respectable infinite and infinitessimal constructions. (but I don't know
because I am loath to divert my meagre intellectual resources to climbing
these arcane heights.)
My preference is for a 'number' system which includes both infinitessimal
and infinite 'numbers'
which can be operated on by familiar arithmetical rules. This requires a
shift in perspective which
regards the finite domain as a special case within an infinte context.
This can be illustrated by the following (crude) formulation.
Ur = b^rw
Where Ur is a cardinal, b is a finite base, r is a finite integral parameter
and w is an infinite number.
When:
r = 0, Ur = 1
r = 1, Ur = 1* (the first infinite cardinal relative to (b,w)
r = -1, Ur = *1 (the first infinitesimal 1/1*)
The main conclusion, which Cantor would have seen right away is that this
leads to an infinite hierarchy of infinities and thrusts beyond all hope the
primitive idea of absolute infinity.
Being a religious nut, he couldn't stand the face of God when he saw it.
In the doctrine of real numbers, the set of the irrational numbers is
defined negatively as all other numbers on the real line than the rationals.
A detailed classification of these is impossible but their general type is
defined by the infinite convergent sequences used in Cantor's diagonal
argument. This leaves open the possibility of claiming that the real line is
absolutely continuous. This seems fair enough if we allow that Ao stands for
absolute infinity. But as the diagonal argument shows, this Ao is far too
small a base to ensure absolute continuity. Here the mystification about the
diagonal begins. Suffice to say, the argument is a peitio pricipii and
therfore flawed.
The genii has been out of the bottle for some time but a lot of
mathematicians don't seem to know or can't face the prospect of extending
the kingdom of real numbers to include all those demons from hell.
Tony Thomas
"Paul Holbach" <paulholbachSPAMBAN@freenet.de> wrote in message
news:881c8779.0401041419.1097372c@posting.google.com...
Quote: "In our ordinary real number system, we say that the number K with
decimal expansion .99999... is the samas 1. An informal argument for
this is sketched below:
10K = 9.999...
- K = .9999...
____________
9K = 9
K = 1
But maybe this argument is misleading. What if there is some number,
call it 1 - 1/omega, that is greater than any finite string .9...9 of
nines, yet less than 1? If K were actually equal to 1 - 1/omega, the
informal argument used in the last paragraph would not work, for this
argument overlooks the fact that the difference between 10K and 10 is
ten times as great as the difference between K and 1. There is a
residual infinitesimal quantity below that does not get canceled out:
10K = 10 - 10/omega
- K = 1 - 1/omega
_________________
9K = 9 - 9/omega
K = 1 - 1/omega
Intuitively, nothing could be more natural than to go ahead and talk
about 1/omega, 1/Aleph-1, and so on. Just as we move from the natural
numbers to the fractions and then on to the reals, should we not be
able to move from the whole ordinal numbers to some richer number
field?
Curiously, Cantor himself was very much opposed to this step. When a
fellow mathematician attempted to use Cantorīs transfinite numbers to
develop a theory of infinitely small quantities, Cantor accused him of
trying to 'infect mathematics with the Cholera-Bacillus of
infinitesimals'. Cantor even constructed a proof that no number can be
infinitesimal. This proof, however, is just as circular and worthless
as finitist attempts to prove that no number can be infinite. In both
cases, the desired conclusion is smuggled in as part of the definition
of 'number'.
Why was Cantor so vehemently opposed to infinitesimals? In his
valuable essay, 'The Metaphysics of the Calculus', Abraham Robinson
suggests that Cantor already had enough problems trying to defend
transfinite numbers. It seems likely that, consciously or otherwise,
Cantor deemed it politically wise to go along with othodox
mathematicians on the question of infinitesimals. Cantorīs stance
might be compared to that of a pro-marijuana Congressional candidate
who advocates harsh penalties for the sale or use of heroin. Yet, as
we shall see, there is almost as much justification for infinitesimals
as there is for Cantorīs transfinite ordinals.
Formally speaking, it is as consistent to say that there is a number
between all of .9, .99, .999, ... and 1 as it is to say that there is
a number greater than all of 1, 2, 3, ... . And just as we go on to
find more and more ordinals piled atop one another, we can go on to
find more and more infinitesimals squeezed beneath each other.
[...]
But so great is the average personīs fear of infinity that to this day
calculus all over the world is being taught as a study of *limit
processes* instead of what it really is: *infinitesimal analysis*.
As someone who has spent a good portion of his adult life teaching
calculus courses for a living, I can tell you how weary one gets of
trying to explain the complex and fiddling theory of limits to wave
after wave of uncomprehending freshman.
I often think of C. H. Hintonīs words from a similar context:
'How pleasant it would be to let pass away some of the verbiage I
learnt at school--learnt because teachers must live, I suppose. The
apeing and prolonged caw called grammar, the cackling of the human hen
over the egg of language--I should like to unlearn grammar.'
But there is hope for a brighter future. Robinsonīs investigations of
the hyperreal numbers have put infinitesimals on a logically
unimpeachable basis, and here and there calculus texts based on
infinitesimals have appeared [*]."
[*: - Keisler, H. J. (1976). /Elementary Calculus/. Boston: Prindle,
Weber & Schmidt.
- Henle & Kleinberg (1978). /Infinitesimal Calculus/. Cambridge,
Mass.: MIT Press.]
[Rucker, Rudy (1995). /Infinity and the Mind/. Princeton, NJ:
Princeton University Press. (pp. 79/80 + 87)]
PH |
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| Back to top |
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| Tony Thomas |
| Posted: Mon Jan 05, 2004 4:25 am |
|
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Guest
|
Paul
I have been pushing this barrow for some time but the fruit is regarded as
rotten or forbidden by the cogniscenti.
I do appreciate that the 'doctrine' of real numbers has been forged by some
of the world's finest intellects and is not to be taken lightly, however,
your argumant for a bit left over in convergent infinite series may be
flawed.
Consider:
Sn = (9/10 + 9/100 + ...+9/10^n) = 1 - 9/10^n
which is true for all values of n.
The mysterious zen bull's tail never disappears altogether but seems to
converge to your omega rather than to zero. But the high priests say that as
n tends to infinity the tail tends to zero.
The trick is that this expression and others like it can be said to never
exceed its limit, which seems to be true.
When we say that 2x is the differential coefficient of x^2 we are are
including such a limit in the expression, but this works OK. The point is
that the use of such limits is guaranteed to be harmless, provided that we
do not admit any pesky infinitessimals.
Some time ago (1950s), mathemeticians invented things like supernatiural
numbers and integrated them into analysis, so the whole question is water
under the bridge. The annoying residual is all that Cantorian set theoretic
doctrine that doesn't seem to be consistent with the newly
respectable infinite and infinitessimal constructions. (but I don't know
because I am loath to divert my meagre intellectual resources to climbing
these arcane heights.)
My preference is for a 'number' system which includes both infinitessimal
and infinite 'numbers'
which can be operated on by familiar arithmetical rules. This requires a
shift in perspective which
regards the finite domain as a special case within an infinte context.
This can be illustrated by the following (crude) formulation.
Ur = b^rw
Where Ur is a cardinal, b is a finite base, r is a finite integral parameter
and w is an infinite number.
When:
r = 0, Ur = 1
r = 1, Ur = 1* (the first infinite cardinal relative to (b,w)
r = -1, Ur = *1 (the first infinitesimal 1/1*)
The main conclusion, which Cantor would have seen right away is that this
leads to an infinite hierarchy of infinities and thrusts beyond all hope the
primitive idea of absolute infinity.
Being a religious nut, he couldn't stand the face of God when he saw it.
In the doctrine of real numbers, the set of the irrational numbers is
defined negatively as all other numbers on the real line than the rationals.
A detailed classification of these is impossible but their general type is
defined by the infinite convergent sequences used in Cantor's diagonal
argument. This leaves open the possibility of claiming that the real line is
absolutely continuous. This seems fair enough if we allow that Ao stands for
absolute infinity. But as the diagonal argument shows, this Ao is far too
small a base to ensure absolute continuity. Here the mystification about the
diagonal begins. Suffice to say, the argument is a peitio pricipii and
therfore flawed.
The genii has been out of the bottle for some time but a lot of
mathematicians don't seem to know or can't face the prospect of extending
the kingdom of real numbers to include all those demons from hell.
Tony Thomas
"Paul Holbach" <paulholbachSPAMBAN@freenet.de> wrote in message
news:881c8779.0401041419.1097372c@posting.google.com...
Quote: "In our ordinary real number system, we say that the number K with
decimal expansion .99999... is the samas 1. An informal argument for
this is sketched below:
10K = 9.999...
- K = .9999...
____________
9K = 9
K = 1
But maybe this argument is misleading. What if there is some number,
call it 1 - 1/omega, that is greater than any finite string .9...9 of
nines, yet less than 1? If K were actually equal to 1 - 1/omega, the
informal argument used in the last paragraph would not work, for this
argument overlooks the fact that the difference between 10K and 10 is
ten times as great as the difference between K and 1. There is a
residual infinitesimal quantity below that does not get canceled out:
10K = 10 - 10/omega
- K = 1 - 1/omega
_________________
9K = 9 - 9/omega
K = 1 - 1/omega
Intuitively, nothing could be more natural than to go ahead and talk
about 1/omega, 1/Aleph-1, and so on. Just as we move from the natural
numbers to the fractions and then on to the reals, should we not be
able to move from the whole ordinal numbers to some richer number
field?
Curiously, Cantor himself was very much opposed to this step. When a
fellow mathematician attempted to use Cantorīs transfinite numbers to
develop a theory of infinitely small quantities, Cantor accused him of
trying to 'infect mathematics with the Cholera-Bacillus of
infinitesimals'. Cantor even constructed a proof that no number can be
infinitesimal. This proof, however, is just as circular and worthless
as finitist attempts to prove that no number can be infinite. In both
cases, the desired conclusion is smuggled in as part of the definition
of 'number'.
Why was Cantor so vehemently opposed to infinitesimals? In his
valuable essay, 'The Metaphysics of the Calculus', Abraham Robinson
suggests that Cantor already had enough problems trying to defend
transfinite numbers. It seems likely that, consciously or otherwise,
Cantor deemed it politically wise to go along with othodox
mathematicians on the question of infinitesimals. Cantorīs stance
might be compared to that of a pro-marijuana Congressional candidate
who advocates harsh penalties for the sale or use of heroin. Yet, as
we shall see, there is almost as much justification for infinitesimals
as there is for Cantorīs transfinite ordinals.
Formally speaking, it is as consistent to say that there is a number
between all of .9, .99, .999, ... and 1 as it is to say that there is
a number greater than all of 1, 2, 3, ... . And just as we go on to
find more and more ordinals piled atop one another, we can go on to
find more and more infinitesimals squeezed beneath each other.
[...]
But so great is the average personīs fear of infinity that to this day
calculus all over the world is being taught as a study of *limit
processes* instead of what it really is: *infinitesimal analysis*.
As someone who has spent a good portion of his adult life teaching
calculus courses for a living, I can tell you how weary one gets of
trying to explain the complex and fiddling theory of limits to wave
after wave of uncomprehending freshman.
I often think of C. H. Hintonīs words from a similar context:
'How pleasant it would be to let pass away some of the verbiage I
learnt at school--learnt because teachers must live, I suppose. The
apeing and prolonged caw called grammar, the cackling of the human hen
over the egg of language--I should like to unlearn grammar.'
But there is hope for a brighter future. Robinsonīs investigations of
the hyperreal numbers have put infinitesimals on a logically
unimpeachable basis, and here and there calculus texts based on
infinitesimals have appeared [*]."
[*: - Keisler, H. J. (1976). /Elementary Calculus/. Boston: Prindle,
Weber & Schmidt.
- Henle & Kleinberg (1978). /Infinitesimal Calculus/. Cambridge,
Mass.: MIT Press.]
[Rucker, Rudy (1995). /Infinity and the Mind/. Princeton, NJ:
Princeton University Press. (pp. 79/80 + 87)]
PH |
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| Back to top |
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| G. A. Edgar |
| Posted: Mon Jan 05, 2004 1:28 pm |
|
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|
Guest
|
In article <kn6Kb.37906$Pg1.14619@newsread1.news.pas.earthlink.net>,
Tim Smith <reply_in_group@mouse-potato.com> wrote:
Quote: In article <881c8779.0401041419.1097372c@posting.google.com>, Paul Holbach
wrote:
But there is hope for a brighter future. Robinsonīs investigations of the
hyperreal numbers have put infinitesimals on a logically unimpeachable
basis, and here and there calculus texts based on infinitesimals have
appeared [*]."
What is the current status of this? I read somewhere, but don't recall
where that it was proven that Robinson's approach and the standard approach
are exactly equivalent in what they can do,
correct
Quote: so that while Robinson's
approach is more intuitive,
not clear
Quote: it cannot lead to any more or less than the
standard approach.
There are no new theorems in standard mathematics that can be proved by
non-standard methods.[*] In some cases non-standard proofs may be
shorter or easier in some other sense. In other cases longer or
harder.
Quote: Thus, while the non-standard approach would be better if
everyone was starting from scratch,
a wild assumption
Quote: there is so much already in standard
terms, that learning the standard approach is necessary, and once you've
done that, there is no need to learn the non-standard approach, since you
can do everything using the standard approach.
Basically, a mathematical society gets to choose one way or the other, and
then they are stuck with it. We hit on the limit approach rather than the
rigorization of infinitesimals, and now are stuck.
a rather extreme way to say it
Quote:
So, is that correct?
Except for specialists, one need only learn one system. And that might
as well be the commonly-used system, so one can talk to other people.
Specialists may learn, and use, two or more systems.
.. . .
[*] This equivalence does depend on the Axiom of Choice. Proof of the
existence of non-standard models relies on the Compactness Theorem of
model theory. Non-standard analysis can prove the Boolean Algebra
Prime Ideal Theorem without any further use of AC.
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/ |
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| Back to top |
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| G. A. Edgar |
| Posted: Mon Jan 05, 2004 1:28 pm |
|
|
|
Guest
|
In article <kn6Kb.37906$Pg1.14619@newsread1.news.pas.earthlink.net>,
Tim Smith <reply_in_group@mouse-potato.com> wrote:
Quote: In article <881c8779.0401041419.1097372c@posting.google.com>, Paul Holbach
wrote:
But there is hope for a brighter future. Robinsonīs investigations of the
hyperreal numbers have put infinitesimals on a logically unimpeachable
basis, and here and there calculus texts based on infinitesimals have
appeared [*]."
What is the current status of this? I read somewhere, but don't recall
where that it was proven that Robinson's approach and the standard approach
are exactly equivalent in what they can do,
correct
Quote: so that while Robinson's
approach is more intuitive,
not clear
Quote: it cannot lead to any more or less than the
standard approach.
There are no new theorems in standard mathematics that can be proved by
non-standard methods.[*] In some cases non-standard proofs may be
shorter or easier in some other sense. In other cases longer or
harder.
Quote: Thus, while the non-standard approach would be better if
everyone was starting from scratch,
a wild assumption
Quote: there is so much already in standard
terms, that learning the standard approach is necessary, and once you've
done that, there is no need to learn the non-standard approach, since you
can do everything using the standard approach.
Basically, a mathematical society gets to choose one way or the other, and
then they are stuck with it. We hit on the limit approach rather than the
rigorization of infinitesimals, and now are stuck.
a rather extreme way to say it
Quote:
So, is that correct?
Except for specialists, one need only learn one system. And that might
as well be the commonly-used system, so one can talk to other people.
Specialists may learn, and use, two or more systems.
.. . .
[*] This equivalence does depend on the Axiom of Choice. Proof of the
existence of non-standard models relies on the Compactness Theorem of
model theory. Non-standard analysis can prove the Boolean Algebra
Prime Ideal Theorem without any further use of AC.
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/ |
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