| Topic: |
Science > Physics |
| User: |
"Lester Zick" |
| Date: |
08 May 2007 11:28:48 AM |
| Object: |
Infinitesimal Arithmetic |
Infinitesimal Arithmetic
~v~~
It's curious that for any finite r, infinitesimal i, and transfinite
I, the same people who understand I+r=I have such difficulty
understanding r+i=r.
~v~~
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| User: "Aatu Koskensilta" |
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| Title: Re: Infinitesimal Arithmetic |
04 Jun 2007 06:19:27 AM |
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On 2007-06-04, in sci.logic, Jonathan Hoyle wrote:
No argument there. Ultimately, all truths are those predicated upon
assumptions. One can never prove a statement "A". One can prove only
the conditional "If B then A".
So we can never prove a statement of the form "the algorithm A terminates on
every input"; all we can prove is "some bunch of statements implies the
algorithm A terminates on every input"? A curious idea in light of the fact
that we do act as if we knew that A terminates on every input given a
mathematical proof of A. Why should this be if all we have, in reality,
proved is that some bunch of statements implies A? Why should someone
interested in the algorithm A be interested in such seemingly random
implications?
--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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| User: "Wolf" |
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| Title: Re: Infinitesimal Arithmetic |
04 Jun 2007 08:23:13 AM |
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Aatu Koskensilta wrote:
On 2007-06-04, in sci.logic, Jonathan Hoyle wrote:
No argument there. Ultimately, all truths are those predicated upon
assumptions. One can never prove a statement "A". One can prove only
the conditional "If B then A".
So we can never prove a statement of the form "the algorithm A terminates on
every input"; all we can prove is "some bunch of statements implies the
algorithm A terminates on every input"? A curious idea in light of the fact
that we do act as if we knew that A terminates on every input given a
mathematical proof of A. Why should this be if all we have, in reality,
proved is that some bunch of statements implies A? Why should someone
interested in the algorithm A be interested in such seemingly random
implications?
If the proof is a true model of the algorithm, it will hold for the
algorithm. However, logical/mathematical proofs are always of the
general form
"If this premise P is true, then this conclusion Q is true."
So you have to show that the proof is a true model of the algorithm,
which may not be easy.
Note that logical/mathematical proof does not assert the truth of P. The
truth of P is something you have to justify outside the proof. There are
many ways of justifying the truth of a premise, but all of them are
ultimately empirical. That is, even people who claim not to be
empiricists rest their conviction that P is true on some experience.
I've found that few people are willing to admit this. That's why I like
mystics - they don't waffle about the grounds for their conviction, but
point to an Encounter with the Other, of which they, quite rightly IMO,
assert that it is uniquely their own. That's why they cannot communicate
it, but at best point to things that might bring you to recognise your
own encounter with the other. If you have one, that is.
It's also worth recalling that logical/mathematical proofs are
tautologies: P and Q must have the same truth value for the proof to
hold. But tautologies tempt people into believing they have found a
Truth about the World. It's easy to forget that the interpretation of
the abstract proof as a statement about the world rests on assumptions,
and that those assumptions cannot be proved. That's why Zick gets
furious when posters demand that he produce an instance of 'not not'
that applies top the world. He can't do it. So he trashes "empirical
mathematikers", who haven't experienced his insight into the Truth of
Science.
Me, I'm an empirical skeptic. I think. Maybe. Anyhow, my experience is
all I have to go on. And that includes my experience of my own behaviour
of 'thinking about things.' But I'm not sure of anything. Like
Descartes, I doubt everything. Unlike him, I doubt my doubting. Untangle
that logical knot, if you like. ;-)
--
Wolf
"Don't believe everything you think." (Maxine)
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| User: "Lester Zick" |
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| Title: Re: Infinitesimal Arithmetic |
04 Jun 2007 01:34:37 PM |
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On Mon, 04 Jun 2007 09:23:13 -0400, Wolf <ElLoboViejo@ruddy.moss>
wrote:
Aatu Koskensilta wrote:
On 2007-06-04, in sci.logic, Jonathan Hoyle wrote:
No argument there. Ultimately, all truths are those predicated upon
assumptions. One can never prove a statement "A". One can prove only
the conditional "If B then A".
So we can never prove a statement of the form "the algorithm A terminates on
every input"; all we can prove is "some bunch of statements implies the
algorithm A terminates on every input"? A curious idea in light of the fact
that we do act as if we knew that A terminates on every input given a
mathematical proof of A. Why should this be if all we have, in reality,
proved is that some bunch of statements implies A? Why should someone
interested in the algorithm A be interested in such seemingly random
implications?
If the proof is a true model of the algorithm, it will hold for the
algorithm. However, logical/mathematical proofs are always of the
general form
"If this premise P is true, then this conclusion Q is true."
Why are such demonstrations "always" of this form, Wolf? It's the
classic form of Aristotelean inference but doesn't demonstrate the
truth of P. Consequently the actual truth of Q remains problematic
despite its demonstration in terms of P.
So you have to show that the proof is a true model of the algorithm,
which may not be easy.
Note that logical/mathematical proof does not assert the truth of P. The
truth of P is something you have to justify outside the proof. There are
many ways of justifying the truth of a premise, but all of them are
ultimately empirical. That is, even people who claim not to be
empiricists rest their conviction that P is true on some experience.
Which may be true but strikes me as defective in scientific terms.
I've found that few people are willing to admit this. That's why I like
mystics - they don't waffle about the grounds for their conviction, but
point to an Encounter with the Other, of which they, quite rightly IMO,
assert that it is uniquely their own. That's why they cannot communicate
it, but at best point to things that might bring you to recognise your
own encounter with the other. If you have one, that is.
There is however a difference between the subjective or mystic origins
of convictions and demonstrations of truth for those convictions. True
mystics don't examine or allow examination of their basic convictions.
It's also worth recalling that logical/mathematical proofs are
tautologies: P and Q must have the same truth value for the proof to
hold. But tautologies tempt people into believing they have found a
Truth about the World. It's easy to forget that the interpretation of
the abstract proof as a statement about the world rests on assumptions,
and that those assumptions cannot be proved.
Is this a mystic "truth" not open to critical examination, Wolf, or
merely an assumption of truth whose truth is open to critical
examination and demonstration of truth? For if the former there is no
point to it since contrary assumptions of truth are equally valid.
That's why Zick gets
furious when posters demand that he produce an instance of 'not not'
that applies top the world. He can't do it.
Are we to assume the demonstration of boolean conjunctions in terms of
"not" is not a real world instance of the truth of alternatives to
"not not"? Curiouser and curiouser.
So he trashes "empirical
mathematikers", who haven't experienced his insight into the Truth of
Science.
I trash those too lazy or stupid to grasp critical demonstrations of
truth in universal terms not those who simply haven't experienced
insights. If you want revelations go to a seer.
My fury is not directed at those who don't understand but at those who
argue against demonstrations of truth in terms of their own
undemonstrated assumptions of truth and mystic revelations or at those
who consider themselves critical thinkers yet decline to argue
demonstrations of truth while enunciating their own revelations of
truth instead.
Me, I'm an empirical skeptic. I think. Maybe. Anyhow, my experience is
all I have to go on. And that includes my experience of my own behaviour
of 'thinking about things.' But I'm not sure of anything. Like
Descartes, I doubt everything. Unlike him, I doubt my doubting. Untangle
that logical knot, if you like. ;-)
In other words you're not much of anything.
~v~~
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| User: "Tony Orlow" |
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| Title: Re: Infinitesimal Arithmetic |
04 Jun 2007 12:58:58 PM |
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Wolf wrote:
Aatu Koskensilta wrote:
On 2007-06-04, in sci.logic, Jonathan Hoyle wrote:
No argument there. Ultimately, all truths are those predicated upon
assumptions. One can never prove a statement "A". One can prove only
the conditional "If B then A".
So we can never prove a statement of the form "the algorithm A
terminates on
every input"; all we can prove is "some bunch of statements implies the
algorithm A terminates on every input"? A curious idea in light of the
fact
that we do act as if we knew that A terminates on every input given a
mathematical proof of A. Why should this be if all we have, in reality,
proved is that some bunch of statements implies A? Why should someone
interested in the algorithm A be interested in such seemingly random
implications?
If the proof is a true model of the algorithm, it will hold for the
algorithm. However, logical/mathematical proofs are always of the
general form
"If this premise P is true, then this conclusion Q is true."
So you have to show that the proof is a true model of the algorithm,
which may not be easy.
Note that logical/mathematical proof does not assert the truth of P. The
truth of P is something you have to justify outside the proof. There are
many ways of justifying the truth of a premise, but all of them are
ultimately empirical. That is, even people who claim not to be
empiricists rest their conviction that P is true on some experience.
I've found that few people are willing to admit this. That's why I like
mystics - they don't waffle about the grounds for their conviction, but
point to an Encounter with the Other, of which they, quite rightly IMO,
assert that it is uniquely their own. That's why they cannot communicate
it, but at best point to things that might bring you to recognise your
own encounter with the other. If you have one, that is.
It's also worth recalling that logical/mathematical proofs are
tautologies: P and Q must have the same truth value for the proof to
hold. But tautologies tempt people into believing they have found a
Truth about the World. It's easy to forget that the interpretation of
the abstract proof as a statement about the world rests on assumptions,
and that those assumptions cannot be proved. That's why Zick gets
furious when posters demand that he produce an instance of 'not not'
that applies top the world. He can't do it. So he trashes "empirical
mathematikers", who haven't experienced his insight into the Truth of
Science.
Me, I'm an empirical skeptic. I think. Maybe. Anyhow, my experience is
all I have to go on. And that includes my experience of my own behaviour
of 'thinking about things.' But I'm not sure of anything. Like
Descartes, I doubt everything. Unlike him, I doubt my doubting. Untangle
that logical knot, if you like. ;-)
Very good, Wolf. I like that. We do go on assumptions, and like
Descartes, the only thing we're sure of is our own existence. The
assumptions we make and build on are often taught to us when young, and
often worth questioning later. But, unless we accept some assumptions,
at least functionally, there is nothing to build on logically. So, it
becomes a rather statistical exercise, the inductive side of logic,
determining the rules as best we can by what fits our experience. Life
is science, skepticism is healthy, faith is necessary, and thinking is
good. We sense, we think, we act, and then the world responds, and we
learn something through our senses. :)
Tony
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| User: "herbzet" |
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| Title: Re: Infinitesimal Arithmetic |
04 Jun 2007 11:05:26 AM |
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Wolf wrote:
[...]
That's why I like
mystics - they don't waffle about the grounds for their conviction, but
point to an Encounter with the Other, of which they, quite rightly IMO,
assert that it
I assume that "it" is the "Encounter", not the "Other".
is uniquely their own. That's why they cannot communicate
it, but at best point to things that might bring you to recognise your
own encounter with the other. If you have one, that is.
It is an illusion to suppose that there is an Other.
It's also worth recalling that logical/mathematical proofs are
tautologies: P and Q must have the same truth value for the proof to
hold.
False.
[...]
Me, I'm an empirical skeptic. I think. Maybe. Anyhow, my experience is
all I have to go on. And that includes my experience of my own behaviour
of 'thinking about things.' But I'm not sure of anything. Like
Descartes, I doubt everything. Unlike him, I doubt my doubting. Untangle
that logical knot, if you like. ;-)
It is an illusion to suppose there is an I.
Throw away your illusions.
--
hz
--
Posted via a free Usenet account from http://www.teranews.com
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| User: "Lester Zick" |
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| Title: Re: Infinitesimal Arithmetic |
06 Jun 2007 01:53:46 PM |
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On Mon, 04 Jun 2007 12:05:26 -0400, herbzet <herbzet@gmail.com> wrote:
Me, I'm an empirical skeptic. I think. Maybe. Anyhow, my experience is
all I have to go on. And that includes my experience of my own behaviour
of 'thinking about things.' But I'm not sure of anything. Like
Descartes, I doubt everything. Unlike him, I doubt my doubting. Untangle
that logical knot, if you like. ;-)
It is an illusion to suppose there is an I.
Throw away your illusions.
I assume you're joking, Herb, although maybe not. Certainly Bob Kolker
is a confirmed materialist. Can't tell what Wolf is. He talks a lot
but doesn't say much.
~v~~
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| User: "Wolf" |
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| Title: Re: Infinitesimal Arithmetic |
05 Jun 2007 09:41:13 PM |
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herbzet wrote:
Wolf wrote:
[...]
That's why I like
mystics - they don't waffle about the grounds for their conviction, but
point to an Encounter with the Other, of which they, quite rightly IMO,
assert that it
I assume that "it" is the "Encounter", not the "Other".
is uniquely their own. That's why they cannot communicate
it, but at best point to things that might bring you to recognise your
own encounter with the other. If you have one, that is.
It is an illusion to suppose that there is an Other.
It's also worth recalling that logical/mathematical proofs are
tautologies: P and Q must have the same truth value for the proof to
hold.
False.
[...]
Me, I'm an empirical skeptic. I think. Maybe. Anyhow, my experience is
all I have to go on. And that includes my experience of my own behaviour
of 'thinking about things.' But I'm not sure of anything. Like
Descartes, I doubt everything. Unlike him, I doubt my doubting. Untangle
that logical knot, if you like. ;-)
It is an illusion to suppose there is an I.
Throw away your illusions.
--
hz
Herbie, Herbie, you're suffering from terminal Presbyterian seriousness.
Lighten up.
--
Wolf
"Don't believe everything you think." (Maxine)
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| User: "herbzet" |
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| Title: Re: Infinitesimal Arithmetic |
05 Jun 2007 11:27:13 PM |
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Wolf wrote:
Herbie, Herbie, you're suffering from terminal Presbyterian seriousness.
Hmm, Presbyterians??? Maybe I've found my people.
Lighten up.
You don't know anything about logic or mysticism. Do you
know any jokes?
--
hz
--
Posted via a free Usenet account from http://www.teranews.com
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| User: "Lester Zick" |
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| Title: Re: Infinitesimal Arithmetic |
06 Jun 2007 01:48:37 PM |
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On Wed, 06 Jun 2007 00:27:13 -0400, herbzet <herbzet@gmail.com> wrote:
Wolf wrote:
Herbie, Herbie, you're suffering from terminal Presbyterian seriousness.
Hmm, Presbyterians??? Maybe I've found my people.
Lighten up.
You don't know anything about logic or mysticism. Do you
know any jokes?
I know several, Herb. Did you have anyone specific in mind?
~v~~
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| User: "Wolf" |
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| Title: Re: Infinitesimal Arithmetic |
06 Jun 2007 09:04:12 AM |
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herbzet wrote:
Wolf wrote:
Herbie, Herbie, you're suffering from terminal Presbyterian seriousness.
Hmm, Presbyterians??? Maybe I've found my people.
Lighten up.
You don't know anything about logic or mysticism. Do you
know any jokes?
--
hz
Er, I was joking. Maybe too subtly for a Presbyterian, even one who
doesn't know he is one. ;-)
I've read more books by and about mystics than I can recall. I did two
logic courses at university, B+ and A. I've obviously forgotten a lot.
So what do you take as signs that I don't know anything about logic?
--
Wolf
"Don't believe everything you think." (Maxine)
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| User: "Lester Zick" |
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| Title: Re: Infinitesimal Arithmetic |
06 Jun 2007 01:54:24 PM |
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On Wed, 06 Jun 2007 10:04:12 -0400, Wolf <ElLoboViejo@ruddy.moss>
wrote:
herbzet wrote:
Wolf wrote:
Herbie, Herbie, you're suffering from terminal Presbyterian seriousness.
Hmm, Presbyterians??? Maybe I've found my people.
Lighten up.
You don't know anything about logic or mysticism. Do you
know any jokes?
--
hz
Er, I was joking. Maybe too subtly for a Presbyterian, even one who
doesn't know he is one. ;-)
I've read more books by and about mystics than I can recall. I did two
logic courses at university, B+ and A. I've obviously forgotten a lot.
So what do you take as signs that I don't know anything about logic?
What logic?
~v~~
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| User: "herbzet" |
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| Title: Re: Infinitesimal Arithmetic |
06 Jun 2007 11:37:19 PM |
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Wolf wrote:
herbzet wrote:
Wolf wrote:
Herbie, Herbie, you're suffering from terminal Presbyterian seriousness.
Hmm, Presbyterians??? Maybe I've found my people.
Lighten up.
You don't know anything about logic or mysticism. Do you
know any jokes?
--
hz
Er, I was joking. Maybe too subtly for a Presbyterian, even one who
doesn't know he is one. ;-)
OK -- that's humorous. Good job.
I've read more books by and about mystics than I can recall. I did two
logic courses at university, B+ and A. I've obviously forgotten a lot.
So what do you take as signs that I don't know anything about logic?
Sometimes I meet people who profess to know something about
electronics. I mildly ask them what is Ohm's law, then sit
back and enjoy the spectacle.
The point is, Ohm's law is practically the first thing you learn
about electricity. It's elementary and indispensable.
When you say
"Logical/mathematical proofs are always of the general form
'If this premise P is true, then this conclusion Q is true'
... logical/mathematical proofs are tautologies: P and Q
must have the same truth value for the proof to hold",
it betrays a fundamental misunderstanding of what consitutes
a valid proof.
It is true that logical/mathematical proofs are of the form
"If P, then Q".
It's a terminological quibble as to whether logical/mathematical
proofs are always tautologies, but that's not what I'm objecting to.
When you say, however, that P and Q must have the same truth value
for the proof to hold, this is so _egregiously_ wrong, so basic,
so fundamentally pixelated ... it's like claiming to be an expert
in electronics and not knowing _Ohm's law_!
Do you think the following inference is valid or invalid?
All men have four legs, and Lassie is a man.
--------------------------------------------
.: Lassie has four legs.
--
dour herb
--
Posted via a free Usenet account from http://www.teranews.com
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| User: "Lester Zick" |
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| Title: Re: Infinitesimal Arithmetic |
07 Jun 2007 12:31:05 PM |
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On Thu, 07 Jun 2007 00:37:19 -0400, herbzet <herbzet@gmail.com> wrote:
Wolf wrote:
herbzet wrote:
Wolf wrote:
Herbie, Herbie, you're suffering from terminal Presbyterian seriousness.
Hmm, Presbyterians??? Maybe I've found my people.
Lighten up.
You don't know anything about logic or mysticism. Do you
know any jokes?
--
hz
Er, I was joking. Maybe too subtly for a Presbyterian, even one who
doesn't know he is one. ;-)
OK -- that's humorous. Good job.
I've read more books by and about mystics than I can recall. I did two
logic courses at university, B+ and A. I've obviously forgotten a lot.
So what do you take as signs that I don't know anything about logic?
Sometimes I meet people who profess to know something about
electronics. I mildly ask them what is Ohm's law, then sit
back and enjoy the spectacle.
The point is, Ohm's law is practically the first thing you learn
about electricity. It's elementary and indispensable.
The problem here, Herb, is that you are arguing EE by analogy with E.
I agree both use electricity.But I don't agree that you've established
the relevance of Ohm's law, or Mho's law for that matter, especially
with respect to EE logic. If you want to argue EE by all means do so.
Just don't expect anyone to accept the analogy without demonstration.
When you say
"Logical/mathematical proofs are always of the general form
'If this premise P is true, then this conclusion Q is true'
... logical/mathematical proofs are tautologies: P and Q
must have the same truth value for the proof to hold",
it betrays a fundamental misunderstanding of what consitutes
a valid proof.
It is true that logical/mathematical proofs are of the form
"If P, then Q".
It's a terminological quibble as to whether logical/mathematical
proofs are always tautologies, but that's not what I'm objecting to.
When you say, however, that P and Q must have the same truth value
for the proof to hold, this is so _egregiously_ wrong, so basic,
so fundamentally pixelated ... it's like claiming to be an expert
in electronics and not knowing _Ohm's law_!
Here again you claim "it's like claiming . . ." without demonstrating
why it's like claiming or the connection in necessary terms.
Do you think the following inference is valid or invalid?
All men have four legs, and Lassie is a man.
--------------------------------------------
.: Lassie has four legs.
Actually most men have three legs, right, left, and middle. The
problem with this kind of argument is that you can't much argue things
you don't already know. It's the difficulty with Aristotelean
syllogistic inference.
~v~~
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| User: "Wolf" |
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| Title: Re: Infinitesimal Arithmetic |
07 Jun 2007 08:10:56 AM |
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herbzet wrote:
Wolf wrote:
herbzet wrote:
Wolf wrote:
Herbie, Herbie, you're suffering from terminal Presbyterian seriousness.
Hmm, Presbyterians??? Maybe I've found my people.
Lighten up.
You don't know anything about logic or mysticism. Do you
know any jokes?
--
hz
Er, I was joking. Maybe too subtly for a Presbyterian, even one who
doesn't know he is one. ;-)
OK -- that's humorous. Good job.
I've read more books by and about mystics than I can recall. I did two
logic courses at university, B+ and A. I've obviously forgotten a lot.
So what do you take as signs that I don't know anything about logic?
Sometimes I meet people who profess to know something about
electronics. I mildly ask them what is Ohm's law, then sit
back and enjoy the spectacle.
The point is, Ohm's law is practically the first thing you learn
about electricity. It's elementary and indispensable.
When you say
"Logical/mathematical proofs are always of the general form
'If this premise P is true, then this conclusion Q is true'
... logical/mathematical proofs are tautologies: P and Q
must have the same truth value for the proof to hold",
it betrays a fundamental misunderstanding of what constitutes
a valid proof.
Yeah, yeah, I know. I was thinking of tautologies as logical identities,
"IFF P then Q". Sorry about not being explicit, and ignoring the
ambiguities of "tautology" as actually used by sloppy writers. ;-)
As far as I know, mathematical proofs are logical identities with P, Q
both true.
[...]
Do you think the following inference is valid or invalid?
All men have four legs, and Lassie is a man.
--------------------------------------------
.: Lassie has four legs.
--
dour herb
Sure. But as you well know, validity is not enough for a proof to hold.
So who's Lassie?
;-)
--
Wolf
"Don't believe everything you think." (Maxine)
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| User: "herbzet" |
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| Title: Re: Infinitesimal Arithmetic |
07 Jun 2007 10:38:24 PM |
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Wolf wrote:
herbzet wrote:
When you say
"Logical/mathematical proofs are always of the general form
'If this premise P is true, then this conclusion Q is true'
... logical/mathematical proofs are tautologies: P and Q
must have the same truth value for the proof to hold",
it betrays a fundamental misunderstanding of what constitutes
a valid proof.
Yeah, yeah, I know. I was thinking of tautologies as logical identities,
"IFF P then Q". Sorry about not being explicit, and ignoring the
ambiguities of "tautology" as actually used by sloppy writers. ;-)
Well, alright then.
As far as I know, mathematical proofs are logical identities
with P, Q both true.
There are lots of mathematical proofs that go "If the
Riemann hypothesis is true, then blah blah blah."
These are valid proofs, and will be no less valid if
the Riemann hypothesis turns out to be false. In some
cases it may turn out that the conclusion is true even
if the Riemann hypothesis is false.
[...]
Do you think the following inference is valid or invalid?
All men have four legs, and Lassie is a man.
--------------------------------------------
.: Lassie has four legs.
--
dour herb
Sure. But as you well know, validity is not enough for a proof to hold.
No, I don't know that. As far as I'm concerned, a proof holds if
it is a valid proof. This may or may not be different from what-
ever is common usage among mathematicians of the term "holds".
People use the term "proof" to mean that the truth of some
proposition has been made evident. Here in sci.logic it is
sometimes used in that sense, but more often it is used in
a more technical sense of showing that some proposition
logically follows from some premise or set of premises --
the truth or falsehood of the premises and conclusion is
not in question.
This sort of equivocation, as it happens, leads to some
quite non-trivial misunderstandings.
So who's Lassie?
See "Lassie Come-Home" by Eric Knight [1938].
Very worth it.
--
hz
--
Posted via a free Usenet account from http://www.teranews.com
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| User: "Wolf" |
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| Title: Re: Infinitesimal Arithmetic |
08 Jun 2007 07:59:29 AM |
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herbzet wrote:
Wolf wrote:
herbzet wrote:
When you say
"Logical/mathematical proofs are always of the general form
'If this premise P is true, then this conclusion Q is true'
... logical/mathematical proofs are tautologies: P and Q
must have the same truth value for the proof to hold",
it betrays a fundamental misunderstanding of what constitutes
a valid proof.
Yeah, yeah, I know. I was thinking of tautologies as logical identities,
"IFF P then Q". Sorry about not being explicit, and ignoring the
ambiguities of "tautology" as actually used by sloppy writers. ;-)
Well, alright then.
As far as I know, mathematical proofs are logical identities
with P, Q both true.
There are lots of mathematical proofs that go "If the
Riemann hypothesis is true, then blah blah blah."
These are valid proofs, and will be no less valid if
the Riemann hypothesis turns out to be false. In some
cases it may turn out that the conclusion is true even
if the Riemann hypothesis is false.
Granted. But then the conclusion must be true on other grounds than
Riemann, and that proof would be an identity, right?
[...]
Do you think the following inference is valid or invalid?
All men have four legs, and Lassie is a man.
--------------------------------------------
.: Lassie has four legs.
--
dour herb
Sure. But as you well know, validity is not enough for a proof to hold.
No, I don't know that. As far as I'm concerned, a proof holds if
it is a valid proof. This may or may not be different from what-
ever is common usage among mathematicians of the term "holds".
Usage differences, no problem.
People use the term "proof" to mean that the truth of some
proposition has been made evident. Here in sci.logic it is
sometimes used in that sense, but more often it is used in
a more technical sense of showing that some proposition
logically follows from some premise or set of premises --
the truth or falsehood of the premises and conclusion is
not in question.
This sort of equivocation, as it happens, leads to some
quite non-trivial misunderstandings.
Granted. I'll try to be more precise and explicit in future. At the cost
of tedium, even. ;-)
So who's Lassie?
See "Lassie Come-Home" by Eric Knight [1938].
Very worth it.
Yeah, I saw the movie way back when. Remember choking up.
Saw a remake some time later. They used one of those abominations that
the AKC thinks is a collie. Its coat was always perfectly groomed, too.
Awful.
See, I have a Presbyterian streak, too. Well, Lutheran, actually. ;-)
--
Wolf
"Don't believe everything you think." (Maxine)
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| User: "Lester Zick" |
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| Title: Re: Infinitesimal Arithmetic |
08 Jun 2007 11:59:33 AM |
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On Fri, 08 Jun 2007 08:59:29 -0400, Wolf <ElLoboViejo@ruddy.moss>
wrote:
See, I have a Presbyterian streak, too. Well, Lutheran, actually. ;-)
Not to mention Calvinist, Catholic, Jewish, Islamic, and polytheist I
suppose if one were willing to see ones life as a metaphor for every
bad idea handed down to us throughout history. Have another drink and
go back to sleep, Rip.
~v~~
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| User: "herbzet" |
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| Title: Re: Infinitesimal Arithmetic |
09 Jun 2007 03:08:49 AM |
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Wolf wrote:
herbzet wrote:
Wolf wrote:
As far as I know, mathematical proofs are logical identities
with P, Q both true.
There are lots of mathematical proofs that go "If the
Riemann hypothesis is true, then blah blah blah."
These are valid proofs, and will be no less valid if
the Riemann hypothesis turns out to be false. In some
cases it may turn out that the conclusion is true even
if the Riemann hypothesis is false.
Granted. But then the conclusion must be true on other grounds than
Riemann,
Right.
and that proof would be an identity, right?
At this point, I have to ask you what you mean by "an identity".
If from some mathematical premise(s) P we deduce Q, what do you
mean by saying that P and Q are "an identity"?
On the presumption that the mathematical premises are true, then
the conclusion will of course be true -- they will be identical
in truth-value.
But it will not always be the case that the premise(s) P will
be deducible _from_ Q -- the conclusion may not be identical
in deductive strength to the premise(s).
The situation is thus: If from premise(s) P we validly deduce Q,
then P -> Q is true because P -> Q is (broadly speaking) a tautology
-- even if the premise(s) P don't happen to be true.
If we assume, as usual, that our premise(s) P _are_ true, then Q will
also be true, and P <-> Q will be true, but P <-> Q will not necessarily
be a tautology -- the inference from Q to P may not be valid.
Does this make sense to you? If not, I ask again, what do you mean
by saying that a proof from P to Q is "an identity"?
[...]
I'll try to be more precise and explicit in future. At the cost
of tedium, even. ;-)
Practice makes perfect!
--
hz
--
Posted via a free Usenet account from http://www.teranews.com
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| User: "Lester Zick" |
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| Title: Re: Infinitesimal Arithmetic |
09 Jun 2007 05:10:35 PM |
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On Sat, 09 Jun 2007 04:08:49 -0400, herbzet <herbzet@gmail.com> wrote:
On the presumption that the mathematical premises are true, then
the conclusion will of course be true -- they will be identical
in truth-value.
.. . .
If we assume, as usual, that our premise(s) P _are_ true, then Q will
also be true,
And what if your premises P are not true? And what is the basis on
which you assume, as usual, that your premises P are true?
~v~~
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| User: "Tony Orlow" |
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| Title: Re: Infinitesimal Arithmetic |
10 Jun 2007 12:01:31 PM |
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Lester Zick wrote:
On Sat, 09 Jun 2007 04:08:49 -0400, herbzet <herbzet@gmail.com> wrote:
On the presumption that the mathematical premises are true, then
the conclusion will of course be true -- they will be identical
in truth-value.
. . .
If we assume, as usual, that our premise(s) P _are_ true, then Q will
also be true,
And what if your premises P are not true? And what is the basis on
which you assume, as usual, that your premises P are true?
~v~~
The measure of truth of premises, or of the validity of the logical
system, is in the conclusions derived, That's The Science of Science,
Lester. Feedback is the process of life and learning.
<3
Tony
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| User: "Lester Zick" |
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| Title: Re: Infinitesimal Arithmetic |
10 Jun 2007 05:13:44 PM |
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On Sun, 10 Jun 2007 13:01:31 -0400, Tony Orlow <tony@lightlink.com>
wrote:
Lester Zick wrote:
On Sat, 09 Jun 2007 04:08:49 -0400, herbzet <herbzet@gmail.com> wrote:
On the presumption that the mathematical premises are true, then
the conclusion will of course be true -- they will be identical
in truth-value.
. . .
If we assume, as usual, that our premise(s) P _are_ true, then Q will
also be true,
And what if your premises P are not true? And what is the basis on
which you assume, as usual, that your premises P are true?
~v~~
The measure of truth of premises, or of the validity of the logical
system, is in the conclusions derived, That's The Science of Science,
Lester. Feedback is the process of life and learning.
That's nice. Aphorisms in action. You shoulda been a priest.
~v~~
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| User: "herbzet" |
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| Title: Re: Infinitesimal Arithmetic |
10 Jun 2007 01:11:05 AM |
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Lester Zick wrote:
herbzet wrote:
On the presumption that the mathematical premises are true, then
the conclusion will of course be true -- they will be identical
in truth-value.
. . .
If we assume, as usual, that our premise(s) P _are_ true, then Q will
also be true,
And what if your premises P are not true?
Then the conclusion may be true or it may be false. That will depend
on the particular premises P and conclusion Q.
A valid inferential form applied to false premises will sometimes
lead to a true conclusion and sometimes not.
And what is the basis on
which you assume, as usual, that your premises P are true?
Oh, people just do. Probably because the standard math models
reality pretty well, allowing the design of things like bridges
and airplanes, calculate trajectories for spacecraft and military
ordnance, optimize factory production, etc., etc.
--
hz
--
Posted via a free Usenet account from http://www.teranews.com
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| User: "Lester Zick" |
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| Title: Re: Infinitesimal Arithmetic |
10 Jun 2007 11:42:59 AM |
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On Sun, 10 Jun 2007 02:11:05 -0400, herbzet <herbzet@gmail.com> wrote:
Lester Zick wrote:
herbzet wrote:
On the presumption that the mathematical premises are true, then
the conclusion will of course be true -- they will be identical
in truth-value.
. . .
If we assume, as usual, that our premise(s) P _are_ true, then Q will
also be true,
And what if your premises P are not true?
Then the conclusion may be true or it may be false. That will depend
on the particular premises P and conclusion Q.
A valid inferential form applied to false premises will sometimes
lead to a true conclusion and sometimes not.
And what is the basis on
which you assume, as usual, that your premises P are true?
Oh, people just do.
You mean they guess at whatever seems plausible to them and just hope
for the best?
Probably because the standard math models
reality pretty well,
Educated guesses often do and sometimes don't.
allowing the design of things like bridges
and airplanes, calculate trajectories for spacecraft and military
ordnance, optimize factory production, etc., etc.
Then apparently we can design all those things without knowing whether
premises are true or not. We can also cure cancer except for the times
we can't. All I'm interested in is whether premises are really true
and not what happens when we guess wrong. Somehow I rather doubt
Aristotelean syllogistic inference matters very much when it doesn't
shed any light on the truth of premises to begin with. That's probably
why syllogistic inference has a reputation as a sterile formalism. If
this, if that, if whatever. Right at present we can accommodate and
handle quantum relations quite effectively without knowing anything
mechanical about them. The same for relativity, angular mechanics, and
modern mathematical axioms. It would just be nice for a change if
science might concentrate what's actually true instead of saying "oh,
well it's just what people do because they don't really have a clue".
~v~~
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| User: "herbzet" |
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| Title: Re: Infinitesimal Arithmetic |
11 Jun 2007 01:20:19 AM |
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Lester Zick wrote:
herbzet wrote:
Lester Zick wrote:
And what is the basis on
which you assume, as usual, that your premises P are true?
Oh, people just do.
You mean they guess at whatever seems plausible to them and just hope
for the best?
Pretty much. Doesn't that agree with your observations?
Probably because the standard math models
reality pretty well,
Educated guesses often do and sometimes don't.
allowing the design of things like bridges
and airplanes, calculate trajectories for spacecraft and military
ordnance, optimize factory production, etc., etc.
Then apparently we can design all those things without knowing whether
premises are true or not. We can also cure cancer except for the times
we can't. All I'm interested in is whether premises are really true
and not what happens when we guess wrong. Somehow I rather doubt
Aristotelean syllogistic inference matters very much when it doesn't
shed any light on the truth of premises to begin with.
Well, logic is a methodological field. It is not concerned with
the truth of premises, just the validity of inferences.
As it happens, sometimes one _can_ establish the _falsehood_ of
premises by logic alone: when the premises validly imply something
known to be false, then at least one of the premises is also false.
When a premise can be shown false by this method, then its negation
is, of course, true.
That's probably
why syllogistic inference has a reputation as a sterile formalism. If
this, if that, if whatever. Right at present we can accommodate and
handle quantum relations quite effectively without knowing anything
mechanical about them. The same for relativity, angular mechanics, and
modern mathematical axioms. It would just be nice for a change if
science might concentrate what's actually true instead of saying "oh,
well it's just what people do because they don't really have a clue".
That _would_ be nice. It appears that science progresses by the
elimination of falsehoods, rather than the direct demonstration
of truth. It seems to be a refining process.
--
hz
--
Posted via a free Usenet account from http://www.teranews.com
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| User: "Lester Zick" |
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| Title: Re: Infinitesimal Arithmetic |
11 Jun 2007 01:47:27 PM |
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On Mon, 11 Jun 2007 02:20:19 -0400, herbzet <herbzet@gmail.com> wrote:
Lester Zick wrote:
herbzet wrote:
Lester Zick wrote:
And what is the basis on
which you assume, as usual, that your premises P are true?
Oh, people just do.
You mean they guess at whatever seems plausible to them and just hope
for the best?
Pretty much. Doesn't that agree with your observations?
Sure. But just try to tell that to empirics, mathematikers, and even
logicians - in other words those who consider themselves scientists -
and they'll jump all over you with protestations of innocence.
Nor do I think it necessarily has to be the case. In classic terms
science makes some guess and tries it out empirically but assuming
results are positive uses the guess as truth and proceeds to develop
consequential ideas on the basis of that assumption of truth.
But what if the guess turns out to be wrong despite appearances? After
all events can only be what they are but our explanations for them can
be totally erroneous. Then you become locked into a lie. And I can
think of at least four very specific and fundamental instances where
exactly that has happened with all kinds of pernicious consequences.
Probably because the standard math models
reality pretty well,
Educated guesses often do and sometimes don't.
allowing the design of things like bridges
and airplanes, calculate trajectories for spacecraft and military
ordnance, optimize factory production, etc., etc.
Then apparently we can design all those things without knowing whether
premises are true or not. We can also cure cancer except for the times
we can't. All I'm interested in is whether premises are really true
and not what happens when we guess wrong. Somehow I rather doubt
Aristotelean syllogistic inference matters very much when it doesn't
shed any light on the truth of premises to begin with.
Well, logic is a methodological field. It is not concerned with
the truth of premises, just the validity of inferences.
Yes but logic is only a methodological field unconcerned with the
truth of premises only because neither Aristotle nor any of his
successors was able to solve the problem of truth in universal terms
and had to settle for syllogistic inference despite their original
very real and ongoing search for truth and the demonstration of truth
and not just how to work with truth once found.
As it happens, sometimes one _can_ establish the _falsehood_ of
premises by logic alone: when the premises validly imply something
known to be false, then at least one of the premises is also false.
Except there is no "known to be false" in mechanically exhaustive,
demonstrable, universal terms. If there were we would know truth
itself and not have to rely on assumptions of truth. There is only
what we "take to be false" under various conditions, suppositions, and
assumptions of truth.
When a premise can be shown false by this method, then its negation
is, of course, true.
Yes but that process is problematic when assumptions of truth are
relied on to begin with such as mathematical axioms and Aristotle's
canons of logic. If something were shown to be universally false then
of course we could and would understand its alternative to be true in
universal terms. But that can never happen when what we rely on in
terms of truth against which to gauge falseness are mere assumptions
of truth rather than actual demonstrations of truth.
That's probably
why syllogistic inference has a reputation as a sterile formalism. If
this, if that, if whatever. Right at present we can accommodate and
handle quantum relations quite effectively without knowing anything
mechanical about them. The same for relativity, angular mechanics, and
modern mathematical axioms. It would just be nice for a change if
science might concentrate what's actually true instead of saying "oh,
well it's just what people do because they don't really have a clue".
That _would_ be nice. It appears that science progresses by the
elimination of falsehoods, rather than the direct demonstration
of truth. It seems to be a refining process.
Well sure I agree. Except that quite often the baby is thrown out with
the bathwater through association with errors and what is eliminated
aren't falsehoods at all just misapplications of explanations and
replaced with completely nonsensical jury rigged explanations.
Let me put it to you this way. Let's suppose we have some speculation
consisting of several different factors related and subordinated to
one another in various subtle yet specific ways which also happen to
conform to peoples' intuitions on such subjects. For example we might
consider such factors as space, time, dimensionality, and the
Michelson-Morley experiment.
Then we find Michelson-Morley doesn't work but instead of debugging
the experimental rationale and getting it to work we draft hodgepodges
of alternatives consisting of spacetime, hyperdimensionality, and the
necessary impotence of such experiments and justify all these results
as "counterintuitive" as if justification for the original factors was
intuition instead science. And the course of twentieth century science
is rife with precisely such speculative nonsense. In other words using
guesswork as the foundation for science is exactly like flying by the
seat of your pants, by guess and by golly, on a wing and a prayer.
~v~~
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| User: "Tony Orlow" |
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| Title: Re: Infinitesimal Arithmetic |
10 Jun 2007 11:58:14 AM |
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herbzet wrote:
Wolf wrote:
herbzet wrote:
Wolf wrote:
As far as I know, mathematical proofs are logical identities
with P, Q both true.
There are lots of mathematical proofs that go "If the
Riemann hypothesis is true, then blah blah blah."
These are valid proofs, and will be no less valid if
the Riemann hypothesis turns out to be false. In some
cases it may turn out that the conclusion is true even
if the Riemann hypothesis is false.
Granted. But then the conclusion must be true on other grounds than
Riemann,
Right.
and that proof would be an identity, right?
At this point, I have to ask you what you mean by "an identity".
If from some mathematical premise(s) P we deduce Q, what do you
mean by saying that P and Q are "an identity"?
On the presumption that the mathematical premises are true, then
the conclusion will of course be true -- they will be identical
in truth-value.
But it will not always be the case that the premise(s) P will
be deducible _from_ Q -- the conclusion may not be identical
in deductive strength to the premise(s).
The situation is thus: If from premise(s) P we validly deduce Q,
then P -> Q is true because P -> Q is (broadly speaking) a tautology
-- even if the premise(s) P don't happen to be true.
If we assume, as usual, that our premise(s) P _are_ true, then Q will
also be true, and P <-> Q will be true, but P <-> Q will not necessarily
be a tautology -- the inference from Q to P may not be valid.
Does this make sense to you? If not, I ask again, what do you mean
by saying that a proof from P to Q is "an identity"?
[...]
I'll try to be more precise and explicit in future. At the cost
of tedium, even. ;-)
Practice makes perfect!
--
hz
Q may be true under more constraints than the starting circumstances
specified in P. Is that what you mean?
aeo
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| User: "herbzet" |
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| Title: Re: Infinitesimal Arithmetic |
11 Jun 2007 01:23:39 AM |
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Tony Orlow wrote:
herbzet wrote:
On the presumption that the mathematical premises are true, then
the conclusion will of course be true -- they will be identical
in truth-value.
But it will not always be the case that the premise(s) P will
be deducible _from_ Q -- the conclusion may not be identical
in deductive strength to the premise(s).
The situation is thus: If from premise(s) P we validly deduce Q,
then P -> Q is true because P -> Q is (broadly speaking) a tautology
-- even if the premise(s) P don't happen to be true.
If we assume, as usual, that our premise(s) P _are_ true, then Q will
also be true, and P <-> Q will be true, but P <-> Q will not necessarily
be a tautology -- the inference from Q to P may not be valid.
Q may be true under more constraints than the starting circumstances
specified in P. Is that what you mean?
Q may be true under fewer constraints than are imposed by P.
So under the constraints imposed by P, Q is true. But Q may be
true under less restrictive constraints.
--
hz
--
Posted via a free Usenet account from http://www.teranews.com
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| User: "Tony Orlow" |
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| Title: Re: Infinitesimal Arithmetic |
11 Jun 2007 08:42:40 AM |
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herbzet wrote:
Tony Orlow wrote:
herbzet wrote:
On the presumption that the mathematical premises are true, then
the conclusion will of course be true -- they will be identical
in truth-value.
But it will not always be the case that the premise(s) P will
be deducible _from_ Q -- the conclusion may not be identical
in deductive strength to the premise(s).
The situation is thus: If from premise(s) P we validly deduce Q,
then P -> Q is true because P -> Q is (broadly speaking) a tautology
-- even if the premise(s) P don't happen to be true.
If we assume, as usual, that our premise(s) P _are_ true, then Q will
also be true, and P <-> Q will be true, but P <-> Q will not necessarily
be a tautology -- the inference from Q to P may not be valid.
Q may be true under more constraints than the starting circumstances
specified in P. Is that what you mean?
Q may be true under fewer constraints than are imposed by P.
I'm sorry. "Constraints" wasn't the right word. I meant "conditions",
meaning Q can be true in cases where some of the premises in P are false.
So under the constraints imposed by P, Q is true. But Q may be
true under less restrictive constraints.
--
hz
Right.
aeo
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| User: "herbzet" |
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| Title: Re: Infinitesimal Arithmetic |
12 Jun 2007 12:22:43 AM |
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Tony Orlow wrote:
herbzet wrote:
Tony Orlow wrote:
herbzet wrote:
On the presumption that the mathematical premises are true, then
the conclusion will of course be true -- they will be identical
in truth-value.
But it will not always be the case that the premise(s) P will
be deducible _from_ Q -- the conclusion may not be identical
in deductive strength to the premise(s).
The situation is thus: If from premise(s) P we validly deduce Q,
then P -> Q is true because P -> Q is (broadly speaking) a tautology
-- even if the premise(s) P don't happen to be true.
If we assume, as usual, that our premise(s) P _are_ true, then Q will
also be true, and P <-> Q will be true, but P <-> Q will not necessarily
be a tautology -- the inference from Q to P may not be valid.
Q may be true under more constraints than the starting circumstances
specified in P. Is that what you mean?
Q may be true under fewer constraints than are imposed by P.
I'm sorry. "Constraints" wasn't the right word. I meant "conditions",
meaning Q can be true in cases where some of the premises in P are false.
Yes, that is true. But:
We are discussing the case where the premises _are_ true (or at least
assumed true). What would it mean to say that, when the premises P
are true, the conclusion Q can be true in cases where the premises
are _not_ true?
What does it mean to say that, the premises P are true, and hence
the conclusion Q is true, and hence P <-> Q true, but that Q does
not imply P because Q "can be" true in "cases" where the premises
are false (although we know, or are assuming, that the premises
are actually true)?
You have to be careful to distinguish between a formula "Q -> P"
(which is the _form_ of a statement) and is neither true nor
false, and has various "cases", and the propositions which are
instances of that form, which are true or false, and do not
have cases -- they _are_ cases.
For a _formula_ Q -> P, we can say, Q will fail to validly
imply P when there are _cases_ of Q true and P false.
For a _proposition_ which is a _case_ (or _instance_) of the
formula Q -> P, what circumstance will enable us to say that
the antecedent proposition fails to imply the consequent proposition?
You have to tread carefully in how you state these things.
That's all I'm getting at.
So under the constraints imposed by P, Q is true. But Q may be
true under less restrictive constraints.
--
hz
Right.
aeo
--
hz
--
Posted via a free Usenet account from http://www.teranews.com
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| User: "Tony Orlow" |
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| Title: Re: Infinitesimal Arithmetic |
12 Jun 2007 10:35:59 AM |
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herbzet wrote:
Tony Orlow wrote:
herbzet wrote:
Tony Orlow wrote:
herbzet wrote:
On the presumption that the mathematical premises are true, then
the conclusion will of course be true -- they will be identical
in truth-value.
But it will not always be the case that the premise(s) P will
be deducible _from_ Q -- the conclusion may not be identical
in deductive strength to the premise(s).
The situation is thus: If from premise(s) P we validly deduce Q,
then P -> Q is true because P -> Q is (broadly speaking) a tautology
-- even if the premise(s) P don't happen to be true.
If we assume, as usual, that our premise(s) P _are_ true, then Q will
also be true, and P <-> Q will be true, but P <-> Q will not necessarily
be a tautology -- the inference from Q to P may not be valid.
Q may be true under more constraints than the starting circumstances
specified in P. Is that what you mean?
Q may be true under fewer constraints than are imposed by P.
I'm sorry. "Constraints" wasn't the right word. I meant "conditions",
meaning Q can be true in cases where some of the premises in P are false.
Yes, that is true. But:
We are discussing the case where the premises _are_ true (or at least
assumed true). What would it mean to say that, when the premises P
are true, the conclusion Q can be true in cases where the premises
are _not_ true?
You were saying that P -> Q does not imply Q -> P, because it may be
possible that Q ^ ~P.
What does it mean to say that, the premises P are true, and hence
the conclusion Q is true, and hence P <-> Q true, but that Q does
not imply P because Q "can be" true in "cases" where the premises
are false (although we know, or are assuming, that the premises
are actually true)?
Relax.
~(Q -> P) = ~(~Q v P) = Q ^ ~P
In words, "Q does not imply P" means "Q is true but not P"
You have to be careful to distinguish between a formula "Q -> P"
(which is the _form_ of a statement) and is neither true nor
false, and has various "cases", and the propositions which are
instances of that form, which are true or false, and do not
have cases -- they _are_ cases.
For a _formula_ Q -> P, we can say, Q will fail to validly
imply P when there are _cases_ of Q true and P false.
Riiiight. That's why I said Q can be true in cases where P is false,
hence Q doesn't imply P.
For a _proposition_ which is a _case_ (or _instance_) of the
formula Q -> P, what circumstance will enable us to say that
the antecedent proposition fails to imply the consequent proposition?
You have to tread carefully in how you state these things.
That's all I'm getting at.
Apparently.
So under the constraints imposed by P, Q is true. But Q may be
true under less restrictive constraints.
--
hz
Right.
aeo
--
hz
aeo
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