Epistemology 201: The Science of Science
Lester Zick
Epistemology 201: The Science of Science
---------
(Scientific Reduction)
If asked what science is, most people would reply that science is
empirical in nature and conducts experiments to falsify unsound
hypotheses. Some might also vouchsafe that mathematics is a
scientific discipline to the extent that unsound hypotheses can be
falsified through contradictions with foundational axioms. But why
do we need science at all and how is it used?
Let's look at reality in general through unscientific eyes. What we
see is a collation of events in historical terms. We see them succeed
one another and opine that various events cause one another. There is
no way to determine whether this is true in any fundamental sense,
only that history documents that various sequences of events have in
fact succeeded one another.
So, what is science is expected to do? What is apparent to everyone is
that we have one historical tapestry of events and science is expected
to make sense of that tapestry. But how to do this? Basically science
can only make sense of the tapestry by reducing the number and
complexity of causes evident for events. Science must take the run of
events evident to everyone and show which characteristics and
properties govern the emergence of certain events as manifested in the
characteristics and properties of other events.
In so doing science regresses consideration of events to properties of
events and shows how the emergence of one event is implied in the
emergence of other events. And thereby science reduces the panoply
of history to manifold considerations evident in all or most events.
Methodology
---------
Initially at least empirical sciences approach this reduction in the
same way mathematics does through finite tautological regression.
However, whereas mathematics regresses its observations through to
consistency with foundational axioms, through the vagaries of history
empirical sciences are left only with contradiction between empirical
observations as the basis for its regressive foundation.
Consequently empirical science has been left with no understanding of
its own intellectual mechanics. It pretends to be different from logic
and mathematics and claims no finite tautological regressions limit
its empiricism. However, this is only partly true just as it is only
partly true for mathematics and logic. None of the three have finite
ending points that limit application of the respective disciplines.
But all three have finite tautological regressions which define and
limit their starting points, what I refer to as ur regressions.
Ur Regressions for True, False, and Not
--------------
True, False, and Not are defined in reciprocal terms in the following
way. For any empirical observation [subject] the proposition
p:[subject][not subject] is always true. And the proposition
p:[subject not subject] is always false. And the empirical observation
P:[not] is always true because the proposition
P:[not not] is always false.
--------------
These seem to be the only reducible definitions for true, false, and
not. The problem is analogous to the definition of factorability in
mathematics where given i=j*k we have for any number, i, two factors,
j and k but only one equation, which means there is no general
solution possible for factors of i lying between 1 and i.
--------------
In other words every empirical observation is regressable through
tautologies or it cannot be true because tautologies in the formal
sense are always true. Tautologies are not perfect, however, because
even though they account for everything true they do not account for
everything. In order to do that they would also have to account not
only for everything true but everything false as well. And we find
that perfecting ordinary tautologies requires the addition of some
component which is always false. For example, for
t:[subject][not subject]
t:[subject][not subject][subject not subject]
wherein the self contradictory alternative [subject not subject] is
appended to an ordinary tautology to form a comprehensive or perfect
tautology inclusive of all possibilities.
However such a regression through to self contradiction is not
possible in the case of one empirical observation [not] which forms
an irreducible regression directly in tautuological terms:
T:[not][not not]
inclusive of all possibilities. From which we conclude [not] or
contradiction forms the basis of all tautological regressions in
general, which in fact is exactly consistent with the form of the
tautology itself.
Regards - Lester
tadchem
Lester Zick wrote:
> If asked what science is, most people would reply that science is
> empirical in nature and conducts experiments to falsify unsound
> hypotheses.
Science is most decidedly NOT a democracy. It matters squat what "most
people" think.
> Some might also vouchsafe that mathematics is a
> scientific discipline to the extent that unsound hypotheses can be
> falsified through contradictions with foundational axioms.
Certainly no empirical scientist would make such an assertion. Science
relies on falsifiability primarily through empirical testing;
mathematical falsifiability comes through rigorous logic.
> But why
> do we need science at all and how is it used?
We need science because sometimes our best logic is inadequate, and our
axioms are not alwaysd explicitly stated and thus available for logical
analysis. Empirical validation/falsification works without detailed
analysis of the logic, but can only apply itself to that which can be
measured - repeatibly and independently.
> Let's look at reality in general through unscientific eyes.
If you are smart, you will soon recognize the need to develop a
perception of science.
> What we
> see is a collation of events in historical terms.
We could also choose to organize our libraries alphabetically by height
(in Imperial units), but there *ARE* more effective ways to accomplish
the same task.
> We see them succeed
> one another and opine that various events cause one another. There is
> no way to determine whether this is true in any fundamental sense,
> only that history documents that various sequences of events have in
> fact succeeded one another.
Analysis by exception - amazingly inefficient...
> So, what is science is expected to do?
Science is expected to do what every *osophy, *ology, and *ism is
trying to do - help us understand how the universe works so that we can
better insure the survival of ourselves and our descendants.
The distinguishing feature of science is that it is *methodical* rather
than doctrinal. Science works using the scientific method - a cyclic
interaction between interpretation and experimentation. Science
unhesitatingly discards ideas that are shown to be inaccurate
descriptions of how the universe works, and requires that new ideas
provide some *testable statements* about how the universe works. By
throwing out all the straw men, science eventually closes in on the
golden needles.
> What is apparent to everyone is
> that we have one historical tapestry of events and science is
expected
> to make sense of that tapestry.
It is apparent to many of us scientists that the most important
elements of the 'historical tapestry' are those that tell us what
*doesn't* work, so we can avoid repeating those mistakes. The chief
value of the historical tapestry is mainly as an error log.
<Gotta Go - boss is coming>
Tom Davidson
Richmond, VA
tadchem
Lester Zick wrote:
> If asked what science is, most people would reply that science is
> empirical in nature and conducts experiments to falsify unsound
> hypotheses.
Science is most decidedly NOT a democracy. It matters squat what "most
people" think.
> Some might also vouchsafe that mathematics is a
> scientific discipline to the extent that unsound hypotheses can be
> falsified through contradictions with foundational axioms.
Certainly no empirical scientist would make such an assertion. Science
relies on falsifiability primarily through empirical testing;
mathematical falsifiability comes through rigorous logic.
> But why
> do we need science at all and how is it used?
We need science because sometimes our best logic is inadequate, and our
axioms are not alwaysd explicitly stated and thus available for logical
analysis. Empirical validation/falsification works without detailed
analysis of the logic, but can only apply itself to that which can be
measured - repeatibly and independently.
> Let's look at reality in general through unscientific eyes.
If you are smart, you will soon recognize the need to develop a
perception of science.
> What we
> see is a collation of events in historical terms.
We could also choose to organize our libraries alphabetically by height
(in Imperial units), but there *ARE* more effective ways to accomplish
the same task.
> We see them succeed
> one another and opine that various events cause one another. There is
> no way to determine whether this is true in any fundamental sense,
> only that history documents that various sequences of events have in
> fact succeeded one another.
Analysis by exception - amazingly inefficient...
> So, what is science is expected to do?
Science is expected to do what every *osophy, *ology, and *ism is
trying to do - help us understand how the universe works so that we can
better insure the survival of ourselves and our descendants.
The distinguishing feature of science is that it is *methodical* rather
than doctrinal. Science works using the scientific method - a cyclic
interaction between interpretation and experimentation. Science
unhesitatingly discards ideas that are shown to be inaccurate
descriptions of how the universe works, and requires that new ideas
provide some *testable statements* about how the universe works. By
throwing out all the straw men, science eventually closes in on the
golden needles.
> What is apparent to everyone is
> that we have one historical tapestry of events and science is
expected
> to make sense of that tapestry.
It is apparent to many of us scientists that the most important
elements of the 'historical tapestry' are those that tell us what
*doesn't* work, so we can avoid repeating those mistakes. The chief
value of tha historical tapestry is mainly as an error log.
Lester Zick
comp.ai.philosophy wrote:
>
>Lester Zick wrote:
>
>> If asked what science is, most people would reply that science is
>> empirical in nature and conducts experiments to falsify unsound
>> hypotheses.
>
>Science is most decidedly NOT a democracy. It matters squat what "most
>people" think.
It matters squat what people who pay scientists' bills think? Elitist
bullshit.
>> Some might also vouchsafe that mathematics is a
>> scientific discipline to the extent that unsound hypotheses can be
>> falsified through contradictions with foundational axioms.
>
>Certainly no empirical scientist would make such an assertion. Science
>relies on falsifiability primarily through empirical testing;
>mathematical falsifiability comes through rigorous logic.
Yes, well, one day perhaps scientists can learn to put their minds in
gear sometime before, during, or after the reading process.
>> But why
>> do we need science at all and how is it used?
>
>We need science because sometimes our best logic is inadequate, and our
Please don't confuse your logic with our best logic.
>axioms are not alwaysd explicitly stated and thus available for logical
>analysis. Empirical validation/falsification works without detailed
>analysis of the logic, but can only apply itself to that which can be
>measured - repeatibly and independently.
Empirical contradiction doesn't seem to have done much for your logic.
>> Let's look at reality in general through unscientific eyes.
>
>If you are smart, you will soon recognize the need to develop a
>perception of science.
Unlike yourself? What do you think I've just done, dufus?
>> What we
>> see is a collation of events in historical terms.
>
>We could also choose to organize our libraries alphabetically by height
>(in Imperial units), but there *ARE* more effective ways to accomplish
>the same task.
Not in your case.
>> We see them succeed
>> one another and opine that various events cause one another. There is
>> no way to determine whether this is true in any fundamental sense,
>> only that history documents that various sequences of events have in
>> fact succeeded one another.
>
>Analysis by exception - amazingly inefficient...
Your prose certainly is.
>> So, what is science is expected to do?
>
>Science is expected to do what every *osophy, *ology, and *ism is
>trying to do - help us understand how the universe works so that we can
>better insure the survival of ourselves and our descendants.
No lie?
>The distinguishing feature of science is that it is *methodical* rather
>than doctrinal.
Thanks for the tutorial.
> Science works using the scientific method - a cyclic
>interaction between interpretation and experimentation.
A cyclic interaction between empirical observation and ignorance.
> Science
>unhesitatingly discards ideas that are shown to be inaccurate
>descriptions of how the universe works, and requires that new ideas
>provide some *testable statements* about how the universe works. By
>throwing out all the straw men, science eventually closes in on the
>golden needles.
Quite an anal analogy. How did science every get along without you?
>> What is apparent to everyone is
>> that we have one historical tapestry of events and science is
>expected
>> to make sense of that tapestry.
>
>It is apparent to many of us scientists that the most important
>elements of the 'historical tapestry' are those that tell us what
>*doesn't* work, so we can avoid repeating those mistakes. The chief
>value of tha historical tapestry is mainly as an error log.
><Gotta Go - boss is coming>
I'd be fascinated to know exactly which elements of the historical
tapestry didn't work? Better ask your boss to avoid repeating the same
mistakes as those elements of the historical tapestry which didn't
work.
Regards - Lester
Neil W Rickert
> Epistemology 201: The Science of Science
> ---------
> (Scientific Reduction)
>If asked what science is, most people would reply that science is
>empirical in nature and conducts experiments to falsify unsound
>hypotheses.
And if asked what scientist are, they might say that scientists are
men who wear white lab coats.
Not useful.
> Some might also vouchsafe that mathematics is a
>scientific discipline to the extent that unsound hypotheses can be
>falsified through contradictions with foundational axioms.
Many mathematicians would disagree.
> But why
>do we need science at all and how is it used?
A solipsist such as yourself does not need science. You can just
make it up as you go along. The rest of us find it useful to deal
with the world that we find ourselves occupying.
>Let's look at reality in general through unscientific eyes. What we
>see is a collation of events in historical terms. We see them succeed
>one another and opine that various events cause one another. There is
>no way to determine whether this is true in any fundamental sense,
>only that history documents that various sequences of events have in
>fact succeeded one another.
Hume say that causation is constant conjunction.
You appear to be repeating Hume's most serious mistake.
robert j. kolker
Lester Zick wrote:
>
> Axioms, mathematical and otherwise, are subject to empirical
> contradiction just as in common, ordinary, mundane science.
Axioms are formal entities. Their truth or falsity is a function of how
they are mapped into statements about the world. Mathematics per se has
no empirical content.
Bob Kolker
Lester Zick
<mitchs...@yahoo.com> in comp.ai.philosophy wrote:
>Lester Zick wrote:
>
>[snip]
>
>Why so hostile? Seems ol' Tadchem took some time out of his workday
>to consider some ideas with you. Didn't you want discussion?
I certainly welcome the discussion but maybe I missed the part where
he considers what I said.
Regards - Lester
Lester Zick
in comp.ai.philosophy wrote:
>Zick listens to his voices. External input is not tolerated.
Sure it is, Uncle Ox. I listen to you, don't I? I just don't listen to
your handwaving. Tell me, if a hand waves in the forest and no one is
around to hear it does it make a sound? You should know.
Regards - Lester
Jason
> > scientific discipline
>
> not a science.
Hey, who holds the heavenly taxonomy here? Catholic rank please.
Maybe you are right, that maths isn't science, whatever "science" means. Maybe
it is art, whatever that is. Maybe maths is just maths... but then again maybe
it isn't. Let the people Know the answers cast the first insult.
Some people say maths isn't a science because it is not empirical, as if
empiricism is the necessary condition to wear the reverent name "science". If
this is the case, Maths can be let into the faith. They proved the four colour
map problem by empirical means didn't they? How are primes found? Chaitin
suggests that maths needs to be quasi-empirical because they've run into the
limits of prediction.
Mitch Perkins
[snip]
Why so hostile? Seems ol' Tadchem took some time out of his workday
to consider some ideas with you. Didn't you want discussion?
Mitch
Lester Zick
Axioms, mathematical and otherwise, are subject to empirical
contradiction just as in common, ordinary, mundane science.
Regards - Lester
Lester Zick
<ricke...@cs.niu.edu> in comp.ai.philosophy wrote:
>lester...@worldnet.att.net (Lester Zick) writes:
>
>> Epistemology 201: The Science of Science
>> ---------
>> (Scientific Reduction)
>
>>If asked what science is, most people would reply that science is
>>empirical in nature and conducts experiments to falsify unsound
>>hypotheses.
>
>And if asked what scientist are, they might say that scientists are
>men who wear white lab coats.
So do shrinks.
>Not useful.
Especially to those trying to practice what they don't understand.
>> Some might also vouchsafe that mathematics is a
>>scientific discipline to the extent that unsound hypotheses can be
>>falsified through contradictions with foundational axioms.
>
>Many mathematicians would disagree.
Because mathematikers have special mystical insight when it comes to
the validation of mathematical axioms that doesn't require empirical
contradiction for validation.
>> But why
>>do we need science at all and how is it used?
>
>A solipsist such as yourself does not need science. You can just
>make it up as you go along. The rest of us find it useful to deal
>with the world that we find ourselves occupying.
A solipsist such as myself only requires the mechanics of finite
tautological regression to explain your need for empirical
contradiction and finite mystical handwaving you make up as you go
along to support the mathematical axioms you find difficult to justify
in terms of finite tautological regression while complaining about the
mystic handwaving employed by behaviorists to justify their nonsense.
>>Let's look at reality in general through unscientific eyes. What we
>>see is a collation of events in historical terms. We see them succeed
>>one another and opine that various events cause one another. There is
>>no way to determine whether this is true in any fundamental sense,
>>only that history documents that various sequences of events have in
>>fact succeeded one another.
>
>Hume say that causation is constant conjunction.
And I'm sure Hume's opinion really matters to science.
>You appear to be repeating Hume's most serious mistake.
Whereas you just repeat everybody's mistakes and call it science.
Regards - Lester
Uncle Al
>
> Epistemology 201: The Science of Science
> ---------
> (Scientific Reduction)
>
> If asked what science is, most people would reply that science is
> empirical in nature and conducts experiments to falsify unsound
> hypotheses.
Galilei, Galileo. "Discorsi e Dimostrazioni Matematiche Intorno a Due
Nuove Scienze" (Appresso gli Elsevirii, Leida: 1638)
> Some might also vouchsafe that mathematics is a
> scientific discipline
Bullshit. Mathematics has no empirical falsification. Mathematics is
not a science.
> to the extent that unsound hypotheses can be
> falsified through contradictions with foundational axioms. But why
> do we need science at all and how is it used?
Idiot. Do you read what you write? Aristotle deduced grasshoppers
have four legs. Zick deduces he is competent. The value of empirical
falsification is obvious.
[snip 100 lines of crap]
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/qz.pdf
Uncle Al
Zick listens to his voices. External input is not tolerated.
--
Lester Zick
in comp.ai.philosophy wrote:
>Lester Zick wrote:
>>
>> Epistemology 201: The Science of Science
>> ---------
>> (Scientific Reduction)
>>
>> If asked what science is, most people would reply that science is
>> empirical in nature and conducts experiments to falsify unsound
>> hypotheses.
>
>Galilei, Galileo. "Discorsi e Dimostrazioni Matematiche Intorno a Due
>Nuove Scienze" (Appresso gli Elsevirii, Leida: 1638)
Gee, Uncle Ox, I'd have thought Galileo would have published in
Latin. But that's okay because you apparently don't understand Latin,
Italian, or English.
>> Some might also vouchsafe that mathematics is a
>> scientific discipline
>
>Bullshit. Mathematics has no empirical falsification. Mathematics is
>not a science.
Well, Uncle Ox, it occurs to me that you are factually remiss with
respect to axioms. However, it would scarcely be the first time.
>> to the extent that unsound hypotheses can be
>> falsified through contradictions with foundational axioms. But why
>> do we need science at all and how is it used?
>
>Idiot. Do you read what you write? Aristotle deduced grasshoppers
>have four legs. Zick deduces he is competent. The value of empirical
>falsification is obvious.
Unfortunately not in your case. I deduce that you have four legs and
no brain.
Regards - Lester
Lester Zick
Axioms are formal entities because they are used formally and not
because they have no empirical basis. Mathematics per se has no
empirical content because it doesn't and can't justify its axioms
mathematically. But mathematical axioms are still subject to empirical
contradiction just as any scientific insight.
Regards - Lester
Jason
I repeat, some mathematical proofs are by empirical means... where the proof of
formal statements are being tested and studied by experiment. Amongst other
things, it has become an artefact of the social world, studied by empirical
means.
Jason
> >> not a science.
> >
> >Hey, who holds the heavenly taxonomy here? Catholic rank please.
> >
> >Maybe you are right, that maths isn't science, whatever "science" means.
Maybe
> >it is art, whatever that is. Maybe maths is just maths... but then again
maybe
> >it isn't. Let the people Know the answers cast the first insult.
> >
> >Some people say maths isn't a science because it is not empirical, as if
> >empiricism is the necessary condition to wear the reverent name "science".
If
> >this is the case, Maths can be let into the faith. They proved the four
colour
> >map problem by empirical means didn't they? How are primes found? Chaitin
> >suggests that maths needs to be quasi-empirical because they've run into the
> >limits of prediction.
>
> Axioms, mathematical and otherwise, are subject to empirical
> contradiction just as in common, ordinary, mundane science.
This is just maths as a tool versus maths as a formal system. But why not talk
about maths as a demand for bubble-gum?
I don't think there is any disagreement if people read your "some might vouch"
qualifications. They seem more interested in getting in touch with their PMS
and having a knowledge-off about maths.
robert j. kolker
Lester Zick wrote:
>
>
> Gee, Uncle Ox, I'd have thought Galileo would have published in
> Latin. But that's okay because you apparently don't understand Latin,
> Italian, or English.
Galileo was particular about publishing his major works in the
vernacular. He wanted to be read -and understood- by as many people as
possible. It turns out he is a very gifted and witty writer. His
-Dialogs Concerning Two New World Systems- were extremely witty and his
rather satirical representation of Pope Urban's (and Aristotle's) views
by way of Simplicio got hin into a lot of trouble.
Galileo was an ueber smart-ass which his talent gave him the right to
be. In terms of the politics in which the Church was embroiled at the
time he was too smart by a half. It cost him.
Bob Kolker
robert j. kolker
Lester Zick wrote:
>
> Axioms are formal entities because they are used formally and not
> because they have no empirical basis.
A distinction without a difference.
What is the empirical content of "two points determine a line"? Is it
that two points determine a straight line (like a string stretch taut)
or is it that two points determine a shortest path on a surface
containing the points? And what is a point. In projective geometry a
point could be a vector in a certain direction and the straight line the
plane determined by a pair of vectors each interpreted as a point.
The fact of ambiguity shows that this geometric axiom has multiple
meanings (i.e. it is context dependent) and its form does not determine
a particular empirical or specific meaning.
Bob Kolker
robert j. kolker
Jason wrote:
>
> I repeat, some mathematical proofs are by empirical means...
A proof is a proof (in mathematics) solely through the logical
mechanisms that enables one step of the proof to be infered from an
earlier step or steps. A rigorous proof is strictly formal, even if non
formal means were used to discover it. Distinguish between discovery and
justification. Discovery can be very empirical and heuristic, but
justification (actual proof) is formal.
Bob Kolker
Jason
Maths as a formal system is incomplete, so some statements cannot be proven as
derivations from the axioms. Some of these statements are true or false under
the standard interpretation of the language of mathematics. In these cases,
discovery IS justification. Proof is empirical:
The four colour map problem was finally 'proved' by computer. That is, every
possible combination of neighbouring map shapes were tried and tested. This is
empirical. There is (or at least was at the time) no know formal method to
prove it.
Neil W Rickert
>I repeat, some mathematical proofs are by empirical means... where the proof of
Repeating it does not make it so.
robert j. kolker
> The four colour map problem was finally 'proved' by computer. That is, every
> possible combination of neighbouring map shapes were tried and tested. This is
> empirical. There is (or at least was at the time) no know formal method to
> prove it.
The discovery was empirical. The justification was formal. It was still
necessary to show that the cases enumerated by the computer program were
exhaustive and mutually exclusive.
Proofs are almost always found by heuristic means, but they must be
presented in a formal or near formal manner to show that they are
correct proofs.
Bob Kolker
Neil W Rickert
>The four colour map problem was finally 'proved' by computer. That is, every
>possible combination of neighbouring map shapes were tried and tested. This is
>empirical. There is (or at least was at the time) no know formal method to
>prove it.
This is a misunderstanding.
If you are able to prove that a problem reduces to a finite number of
cases, and if you are able to verify those cases, that amounts to
formal proof. Empirical "proof" would involve actually coloring maps
and counting how many colors. It would not be accepted as proof by
mathematicians.
mme...@cars3.uchicago.edu
combinations is infinite. It was only *after* you had a proof that
any of these combinations is equivalent to one from a smaller, finite
set, that you could go and check the combinations in this finite set,
one by one. Absent the proof, and yes, it was a formal proof, you
could run all the computers in the world for eternity, getting
nowhere.
So, yes, there most certainly *was* a formal method to prove it,
consisting of reducing the problem to a finite number of cases.
Mati Meron | "When you argue with a fool,
me...@cars.uchicago.edu | chances are he is doing just the same"
The Sophist
> Maths as a formal system is incomplete, so some statements cannot be proven as
> derivations from the axioms. Some of these statements are true or false under
> the standard interpretation of the language of mathematics. In these cases,
> discovery IS justification. Proof is empirical:
I fail to see how the fact that some statements we accept as true don't
follow from our basic axioms, or even the fact that we know that no
matter what axioms we pick there will be some statements which are true
which don't follow from those axioms, shows that those truths are
empirical. After all, any given truth follows from the right choice of
axioms (trivially, the axiom set including that truth as an axiom). And
it is not by experience that we know which claims are false, either;
false claims in mathematics lead to formal contradictions. There is no
need for experience to contradict them.
> The four colour map problem was finally 'proved' by computer. That is, every
> possible combination of neighbouring map shapes were tried and tested. This is
> empirical. There is (or at least was at the time) no know formal method to
> prove it.
Exhaustion is a formal method. It's just not in this case a formal
method which is within the physical capacities of humans to apply (the
four colour problem is just a little too exhausting).
--
Aaron Boyden
The main division between the so-called Continental and Analytic
traditions has been disputes over whether the task of being unclear
should be carried out in natural language or in a formal system.
Jason
Formal proof is derivation from the system's axioms, while formal justification
is what exactly? It seems you're trading on an ambiguity in the word 'formal'.
Jason
of
>
> Repeating it does not make it so.
Ignoring me doesn't make it not so.
Jason
Well, they verified the cases by computer. But how do you prove an algorithm is
correct? If there is an effective method to prove that an algorithm is correct
then what can prove this effective method is correct? The 'proof' of the four
colour problem is partially inductive. Empiricism has leaked into mathematics
via computers.
The computer literally did the colouring of maps and counting of the colours.
It was accepted after they tried it on other computers with different
programmes. But again, this is inductive evidence. That it is a legitimate
proof is controversial.
Jason
They key to the proof being empirical is that a computer is used. So if the
algorithm can be deductively proven to be correct, then I'll concede. Otherwise
all that can be done is to test it on different computers with different
programmes until we're satisfied, which is inductive.
My contention is not that the four colour problem is not formally provable. It
may well be by someone with a lot of time on their hands. But unless the
referees are prepared to spend the same amount of time, then they can only
appeal to a computer proof.
BTW, this is not to say that it is not admissible as a proof. I think it should
be. It is just another argument for quasi-empirical mathematics.
Jason
as
> > derivations from the axioms. Some of these statements are true or false
under
> > the standard interpretation of the language of mathematics. In these cases,
> > discovery IS justification. Proof is empirical:
>
> I fail to see how the fact that some statements we accept as true don't
> follow from our basic axioms, or even the fact that we know that no
> matter what axioms we pick there will be some statements which are true
> which don't follow from those axioms, shows that those truths are
> empirical. After all, any given truth follows from the right choice of
> axioms (trivially, the axiom set including that truth as an axiom). And
> it is not by experience that we know which claims are false, either;
> false claims in mathematics lead to formal contradictions. There is no
> need for experience to contradict them.
If maths develops according to the way we see the world working, then the notion
of mathematical 'truth' can be extended from the limited formal system notion of
'truth', to our every day understanding. I believe there are mathematical
statements that are true in our world, but are not provable by the current
axioms of maths. These statements are the ones I am referring to. They need to
be empirically verified, which might then provide impetus for the axioms to
change.
> > The four colour map problem was finally 'proved' by computer. That is,
every
> > possible combination of neighbouring map shapes were tried and tested. This
is
> > empirical. There is (or at least was at the time) no know formal method to
> > prove it.
>
> Exhaustion is a formal method. It's just not in this case a formal
> method which is within the physical capacities of humans to apply (the
> four colour problem is just a little too exhausting).
Exhaustion and testing by algorithm is not so formal. The computer has to be
trusted to be doing the right task. I don't know about yours, but my computer
is not to be trusted at the best of times. Let alone the software I write...
Mitch Harris
>
>>So, yes, there most certainly *was* a formal method to prove it,
>>consisting of reducing the problem to a finite number of cases.
>
> They key to the proof being empirical is that a computer is used. So if the
> algorithm can be deductively proven to be correct, then I'll concede. Otherwise
> all that can be done is to test it on different computers with different
> programmes until we're satisfied, which is inductive.
How are noncomputer proofs tested? By having lots of people test them
out. That process (of checking the proof) is just as inductive.
> My contention is not that the four colour problem is not formally provable. It
> may well be by someone with a lot of time on their hands. But unless the
> referees are prepared to spend the same amount of time, then they can only
> appeal to a computer proof.
>
> BTW, this is not to say that it is not admissible as a proof. I think it should
> be. It is just another argument for quasi-empirical mathematics.
For reference, see:
http://www.math.gatech.edu/~thomas/FC/fourcolor.html
The proof of 4CT is not quasi-empirical. No random sampling was made.
The cases are exhaustive. Yes, you need to prove that the algorithm
checking the cases is correct, and that the set of cases is
exhaustive. But that's not more empirical than any noncomputer proof.
--
Mitch Harris
(remove q to reply)
Albert
Indeed. It is, as you pointed out, only evidence. Proof is only
possible in mathematics.
--
"Don't you see that the whole aim of Newspeak is to narrow the
range of thought? In the end we shall make thoughtcrime literally
impossible, because there will be no words in which to express it."
-- George Orwell as Syme in "1984"
tadchem
Lester Zick wrote:
> >Science is most decidedly NOT a democracy. It matters squat what
"most
> >people" think.
>
> It matters squat what people who pay scientists' bills think? Elitist
> bullshit.
<Straw man, ad himinem attack>
> >Certainly no empirical scientist would make such an assertion.
Science
> >relies on falsifiability primarily through empirical testing;
> >mathematical falsifiability comes through rigorous logic.
>
> Yes, well, one day perhaps scientists can learn to put their minds in
> gear sometime before, during, or after the reading process.
<veiled ad hominen attack>
> >We need science because sometimes our best logic is inadequate, and
our
>
> Please don't confuse your logic with our best logic.
<veiled ad hominen attack>
> >axioms are not alwaysd explicitly stated and thus available for
logical
> >analysis. Empirical validation/falsification works without detailed
> >analysis of the logic, but can only apply itself to that which can
be
> >measured - repeatibly and independently.
>
> Empirical contradiction doesn't seem to have done much for your
logic.
<ad hominen attack>
> >If you are smart, you will soon recognize the need to develop a
> >perception of science.
>
> Unlike yourself? What do you think I've just done, dufus?
<ad hominen attack>
> >We could also choose to organize our libraries alphabetically by
height
> >(in Imperial units), but there *ARE* more effective ways to
accomplish
> >the same task.
>
> Not in your case.
<ad hominen attack>
> >Analysis by exception - amazingly inefficient...
>
> Your prose certainly is.
<veiled ad hominen attack>
> > Science works using the scientific method -
a cyclic
> >interaction between interpretation and experimentation.
>
> A cyclic interaction between empirical observation and ignorance.
<metonymy>
> >Science
> >unhesitatingly discards ideas that are shown to be inaccurate
> >descriptions of how the universe works, and requires that new ideas
> >provide some *testable statements* about how the universe works. By
> >throwing out all the straw men, science eventually closes in on the
> >golden needles.
>
> Quite an anal analogy. How did science every get along without you?
Actually a mixed metaphor, injected for illustrative effect...
Your ungrammatical sarcasm is gratuitous.
> >It is apparent to many of us scientists that the most important
> >elements of the 'historical tapestry' are those that tell us what
> >*doesn't* work, so we can avoid repeating those mistakes. The chief
> >value of tha historical tapestry is mainly as an error log.
> ><Gotta Go - boss is coming>
>
> I'd be fascinated to know exactly which elements of the historical
> tapestry didn't work? Better ask your boss to avoid repeating the
same
> mistakes as those elements of the historical tapestry which didn't
> work.
'Elements of the historical tapestry' that didn't work include
Aristotelian gravity, geocentric astronomy, alchemy, phlogiston,
Lysenkoism, 'cold fusion', polywater, and many, many more half-baked
ideas that were sooner or later invalidated by empiricism. You should
already be aware of these.
> Regards - Lester
It is apparent from your responses that you are not really interested
in establishing a dialectic exchange, but rather in luring people to
post comments addressed to you that allow you to freely abuse them.
You use cheap rhetorical devices instead of addressing the *substance*
of my comments.
If you are interested in continuing in this vein, you will receive
enlightenment by carefully studying all the key words linked on the
following page:
http://en.wikipedia.org/wiki/Rhetoric
You can do so without any further attention from me.
Tom Davidson
Richmond, VA
robert j. kolker
Jason wrote:
>
> Formal proof is derivation from the system's axioms, while formal justification
> is what exactly? It seems you're trading on an ambiguity in the word 'formal'.
In the real world, "proofs" given in the math journals are in reality
outlines of how one would go about constructing a real honest to God
formal proof. The reasons why complete formal proofs are not giveen are:
1. The detail is boring to the point of being soporific.
2. The size of a typical monthly publication would expand to several
Manhattan Yellow Page Directory sized publication. A sad waste of both
wood and time.
Bob Kolker
>
>
robert j. kolker
Jason wrote:
>
> The computer literally did the colouring of maps and counting of the colours.
> It was accepted after they tried it on other computers with different
> programmes. But again, this is inductive evidence. That it is a legitimate
> proof is controversial.
Any indicative proof (and that is all that is ever printed in the
journals) has a empirical aspect to it. Consider Wile's proof of FLT. It
took a committee of experts to go over it with a fine tooth comb (or the
mental equivalent of one) and the first attempt was flaws. In the second
attempt by Wile's the no error was found (which does not prove
conclusively that it is error free). And that is why it is generally
accepted by the community of working mathematicians. It was declared a
kosher proof by a committee of rabbis in a manner of speaking.
Bob Kolker
robert j. kolker
Jason wrote:
>
> They key to the proof being empirical is that a computer is used. So if the
> algorithm can be deductively proven to be correct, then I'll concede. Otherwise
> all that can be done is to test it on different computers with different
> programmes until we're satisfied, which is inductive.
Why is a human using a rule book any less empirical than a computer
driven by its program?
Bob Kolker
robert j. kolker
Jason wrote:
>
> Exhaustion and testing by algorithm is not so formal. The computer has to be
> trusted to be doing the right task. I don't know about yours, but my computer
> is not to be trusted at the best of times. Let alone the software I write...
Von Neuman proved that an arbitrarily reliable automaton could be
constructed from unreliable parts, as long as the errors were
statistically independent.
If you want high reliability go to an engineer. If you want certainty,
go to church.
Bob Kolker
robert j. kolker
Albert wrote:
> Indeed. It is, as you pointed out, only evidence. Proof is only
> possible in mathematics.
In the ideal instance. In the real world, proofs have become so long and
involved that the question of their correctness is an empirical issue.
Only in Platon's realm would a mathematical construct be pure and right.
That fact that proofs are carried out by error prone humans introduces a
genuine empirical issue as to whether a "proof" is really a proof.
There is an irreducible empirical aspect to the question of whether a
particular -claimed- proof is indeed a proof. See the history of Wile's
two proofs of FLT. The first one was wrong (an error was actually
exhibited) and the second was -declared to be right- by a committee,
since they could not find any errors.
Bob Kolker
Neil W Rickert
[context is discussion of 4 color problem]
>Well, they verified the cases by computer.
Lots of mathematics is verified with pencil and paper. Does the use
of pencil and paper make it empirical.
> But how do you prove an algorithm is
>correct?
The same way that you demonstrate that a mathematical proof is
correct. Oh, by the way, there are many incorrect proofs in the
published literature.
> The 'proof' of the four
>colour problem is partially inductive.
If that is correct, then all logic is partially inductive.
>The computer literally did the colouring of maps and counting of the colours.
The computer was used as a book keeping tool, to keep track of
details to numerous for ordinary human attention. This is not an
empirical investigation, except in the strange meanings you seem to
be giving to "empirical".
>It was accepted after they tried it on other computers with different
>programmes.
Traditional proofs are accepted only after people have worked through
the details of the proof. Recoding the program and running on
different computers is just part of the normal working through a
proof.
> But again, this is inductive evidence. That it is a legitimate
>proof is controversial.
You are misusing "inductive", much as you have been misusing "empirical".
Although there was some initial controversy over the idea of using a
computer in the proof, I have not seen much evidence that it is
currently considered controversial.
Lester Zick
The only mystery here is that you consider your remarks substantive
much less responsive to what I wrote. In case you have any lingering
thoughts on the subject, my rejoinders to you were in kind. None of
your remarks represented a serious critique of what I posted. And most
either duplicated what I said or were completely extraneous. I think
you'll find that when your remarks approach serious criticism, my
replies to your remarks will be comparably responsive.
On 25 Jan 2005 06:43:40 -0800, "tadchem" <thomas....@dla.mil> in
comp.ai.philosophy wrote:
Regards - Lester
Lester Zick
<now...@nowhere.net> in comp.ai.philosophy wrote:
>
>
>Lester Zick wrote:
>>
>> Axioms are formal entities because they are used formally and not
>> because they have no empirical basis.
>
>A distinction without a difference.
In which case your empirical observation regarding the formality of
axioms is also nugatory.
>What is the empirical content of "two points determine a line"?
The observation is the empirical content.
> Is it
>that two points determine a straight line (like a string stretch taut)
>or is it that two points determine a shortest path on a surface
>containing the points?
This is a complementary empirical justification for the original
empirical observation.
> And what is a point. In projective geometry a
>point could be a vector in a certain direction and the straight line the
>plane determined by a pair of vectors each interpreted as a point.
In geometries developed beyond the level of axioms, points are already
assumed
>The fact of ambiguity shows that this geometric axiom has multiple
>meanings (i.e. it is context dependent) and its form does not determine
>a particular empirical or specific meaning.
Which geometric axiom are you referring to? Definition of a straight
line or of a point? Axioms all have multiple supplementary empirical
justifications which is why they are axioms to begin with rather than
propositions. Problematic empirical observations are just the positive
part of tautologies.
Regards - Lester
The Sophist
> If maths develops according to the way we see the world working, then the notion
> of mathematical 'truth' can be extended from the limited formal system notion of
> 'truth', to our every day understanding. I believe there are mathematical
> statements that are true in our world, but are not provable by the current
> axioms of maths. These statements are the ones I am referring to. They need to
> be empirically verified, which might then provide impetus for the axioms to
> change.
Could you provide an example of such statements? The only truths not
provable in the current axioms of math that come to my mind are
artificial examples like Goedel sentences, which we believe to be true
for theoretical reasons (they'd better be true, or arithmetic is
inconsistent), not on the basis of anything in our experience.
> Exhaustion and testing by algorithm is not so formal. The computer has to be
> trusted to be doing the right task. I don't know about yours, but my computer
> is not to be trusted at the best of times. Let alone the software I write...
So a method is only formal if it is applied by an infallible reasoner?
Well, that's an easy way to rule out anything from being formally
proven, but it's quite a non-standard use of the term.
RFHall
(Lester Zick) wrote:
>On 24 Jan 2005 10:21:56 -0800, "tadchem" <thomas....@dla.mil> in
>comp.ai.philosophy wrote:
>
>>
>>Lester Zick wrote:
>>
>>> If asked what science is, most people would reply that science is
>>> empirical in nature and conducts experiments to falsify unsound
>>> hypotheses.
>>
>>Science is most decidedly NOT a democracy. It matters squat what "most
>>people" think.
>
>It matters squat what people who pay scientists' bills think? Elitist
>bullshit.
>
>>> Some might also vouchsafe that mathematics is a
>>> scientific discipline to the extent that unsound hypotheses can be
>>> falsified through contradictions with foundational axioms.
>>
>>Certainly no empirical scientist would make such an assertion. Science
>>relies on falsifiability primarily through empirical testing;
>>mathematical falsifiability comes through rigorous logic.
>
>Yes, well, one day perhaps scientists can learn to put their minds in
>gear sometime before, during, or after the reading process.
>
>>> But why
>>> do we need science at all and how is it used?
>>
>>We need science because sometimes our best logic is inadequate, and our
>
>Please don't confuse your logic with our best logic.
>
>>axioms are not alwaysd explicitly stated and thus available for logical
>>analysis. Empirical validation/falsification works without detailed
>>analysis of the logic, but can only apply itself to that which can be
>>measured - repeatibly and independently.
>
>Empirical contradiction doesn't seem to have done much for your logic.
>
>>> Let's look at reality in general through unscientific eyes.
>>
>>If you are smart, you will soon recognize the need to develop a
>>perception of science.
>
>Unlike yourself? What do you think I've just done, dufus?
>
I guess I may have touched on this in an essay some time ago, but it
seems to me that a great many participants of usenet use it as their
outlet of frustration. They write inserts that obviously show they
haven't read the entire piece, and continue on with insults and
denigrations that appear to be blatant efforts to scare the poster
away. It's always nice to see people reap what they sow, I suppose,
even if there is no progress made. Once in a while there is something
worthwhile, even more original than you would find in any publication.
RFHall
Realistic Idealism
Don't shoot me, I'm just the messenger.
Lester Zick
Surely also in logic, Albert, through finite tautological regression
on which mathematics would have to be based if proof is possible.
Regards - Lester
Lester Zick
<har...@tcs.inf.tu-dresden.de> in comp.ai.philosophy wrote:
>Jason wrote:
> >
>>>So, yes, there most certainly *was* a formal method to prove it,
>>>consisting of reducing the problem to a finite number of cases.
>>
>> They key to the proof being empirical is that a computer is used. So if the
>> algorithm can be deductively proven to be correct, then I'll concede. Otherwise
>> all that can be done is to test it on different computers with different
>> programmes until we're satisfied, which is inductive.
>
>How are noncomputer proofs tested? By having lots of people test them
>out. That process (of checking the proof) is just as inductive.
Good observation. So the proofing of proofs is lack of contradiction,
exactly the same criterion as empirical contradiction used in science.
>> My contention is not that the four colour problem is not formally provable. It
>> may well be by someone with a lot of time on their hands. But unless the
>> referees are prepared to spend the same amount of time, then they can only
>> appeal to a computer proof.
>>
>> BTW, this is not to say that it is not admissible as a proof. I think it should
>> be. It is just another argument for quasi-empirical mathematics.
>
>For reference, see:
>
> http://www.math.gatech.edu/~thomas/FC/fourcolor.html
>
>The proof of 4CT is not quasi-empirical. No random sampling was made.
>The cases are exhaustive. Yes, you need to prove that the algorithm
>checking the cases is correct, and that the set of cases is
>exhaustive. But that's not more empirical than any noncomputer proof.
>
>--
>Mitch Harris
>(remove q to reply)
>
>
Regards - Lester
Lester Zick
It isn't. Just less reliable.
Regards - Lester
Lester Zick
comp.ai.philosophy wrote:
>Jason wrote:
>
>> Maths as a formal system is incomplete, so some statements cannot be proven as
>> derivations from the axioms. Some of these statements are true or false under
>> the standard interpretation of the language of mathematics. In these cases,
>> discovery IS justification. Proof is empirical:
>
>I fail to see how the fact that some statements we accept as true don't
>follow from our basic axioms, or even the fact that we know that no
>matter what axioms we pick there will be some statements which are true
>which don't follow from those axioms, shows that those truths are
>empirical.
Observations whose only justification as true relies on the lack of
contradiction with other observations also considered true makes the
observations empirical in the sense I use the term through finite
tautological regression, or they cannot be true at all if tautologies
are always true.
> After all, any given truth follows from the right choice of
>axioms (trivially, the axiom set including that truth as an axiom). And
>it is not by experience that we know which claims are false, either;
>false claims in mathematics lead to formal contradictions. There is no
>need for experience to contradict them.
>
>> The four colour map problem was finally 'proved' by computer. That is, every
>> possible combination of neighbouring map shapes were tried and tested. This is
>> empirical. There is (or at least was at the time) no know formal method to
>> prove it.
>
>Exhaustion is a formal method. It's just not in this case a formal
>method which is within the physical capacities of humans to apply (the
>four colour problem is just a little too exhausting).
>
>--
>Aaron Boyden
>
>The main division between the so-called Continental and Analytic
>traditions has been disputes over whether the task of being unclear
>should be carried out in natural language or in a formal system.
Regards - Lester
Lester Zick
<jasonstev...@free.net.nz> in comp.ai.philosophy wrote:
>> >> Bullshit. Mathematics has no empirical falsification. Mathematics is
>> >> not a science.
>> >
>> >Hey, who holds the heavenly taxonomy here? Catholic rank please.
>> >
>> >Maybe you are right, that maths isn't science, whatever "science" means.
>Maybe
>> >it is art, whatever that is. Maybe maths is just maths... but then again
>maybe
>> >it isn't. Let the people Know the answers cast the first insult.
>> >
>> >Some people say maths isn't a science because it is not empirical, as if
>> >empiricism is the necessary condition to wear the reverent name "science".
>If
>> >this is the case, Maths can be let into the faith. They proved the four
>colour
>> >map problem by empirical means didn't they? How are primes found? Chaitin
>> >suggests that maths needs to be quasi-empirical because they've run into the
>> >limits of prediction.
>>
>> Axioms, mathematical and otherwise, are subject to empirical
>> contradiction just as in common, ordinary, mundane science.
>
>
>This is just maths as a tool versus maths as a formal system. But why not talk
>about maths as a demand for bubble-gum?
>
>I don't think there is any disagreement if people read your "some might vouch"
>qualifications. They seem more interested in getting in touch with their PMS
>and having a knowledge-off about maths.
The really interesting thing in all this is that people are focusing
on the wrong issue. The idea that math is a tautological extrapolation
on science in general, although true, is hardly as relevant or
interesting in my estimation as the reduction of science in general
through finite tautological regression.
Regards - Lester
Lester Zick
<now...@nowhere.net> in comp.ai.philosophy wrote:
>
>
>Lester Zick wrote:
>
>>
>>
>> Gee, Uncle Ox, I'd have thought Galileo would have published in
>> Latin. But that's okay because you apparently don't understand Latin,
>> Italian, or English.
>
>Galileo was particular about publishing his major works in the
>vernacular. He wanted to be read -and understood- by as many people as
>possible. It turns out he is a very gifted and witty writer. His
>-Dialogs Concerning Two New World Systems- were extremely witty and his
>rather satirical representation of Pope Urban's (and Aristotle's) views
>by way of Simplicio got hin into a lot of trouble.
>
>Galileo was an ueber smart-ass which his talent gave him the right to
>be. In terms of the politics in which the Church was embroiled at the
>time he was too smart by a half. It cost him.
My apologies. Thanks for the correction. It would be nice if Uncle Ox
could be corrected so easily.
Regards - Lester
Albert
'Also in logic' implies mathematics is something other than
logic. Axioms and postulates are statements that are *accepted*
as true in order to study the consequences that follow from them.
Are you sure that you are not confusing 'proof' with 'truth'?
Lester Zick
For sure. What's impressive is how much frustration there is and how
easily it is liberated.
> They write inserts that obviously show they
>haven't read the entire piece, and continue on with insults and
>denigrations that appear to be blatant efforts to scare the poster
>away. It's always nice to see people reap what they sow, I suppose,
>even if there is no progress made. Once in a while there is something
>worthwhile, even more original than you would find in any publication.
Thanks, Rich. Very well put. It would be nice if people actually
considered what they write about so cavalierly. Everyone seems to
have an opinion to go along with the other orifice. The price of easy
access on the usenet, I suppose. With easy entree comes the peanut
gallery. Sometimes you get through though. One mind at a time.
Regards - Lester
Lester Zick
comp.ai.philosophy wrote:
Point taken. I wasn't sure where your reference to mathematics was
intended to lead.
> Axioms and postulates are statements that are *accepted*
>as true in order to study the consequences that follow from them.
>Are you sure that you are not confusing 'proof' with 'truth'?
Well, there is a fine distinction I would draw in connection with
demonstration as a process versus truth as a definable criterion for
the process. Axioms and postulates are empirical in nature whereas
demonstration or proof is tautological in nature. Finite tautological
regression renders empirical observations not drawn in terms of the
regression pleonastic. In other words, what I'm suggesting is that
axioms should be founded on finite tautological regression thus
eliminating the ambiguous empirical character of axioms.
Regards - Lester
mme...@cars3.uchicago.edu
>> >> A proof is a proof (in mathematics) solely through the logical
>> >> mechanisms that enables one step of the proof to be infered from an
>> >> earlier step or steps. A rigorous proof is strictly formal, even if non
>> >> formal means were used to discover it. Distinguish between discovery and
>> >> justification. Discovery can be very empirical and heuristic, but
>> >> justification (actual proof) is formal.
>> >
>as
>> >derivations from the axioms. Some of these statements are true or false
>under
>> >the standard interpretation of the language of mathematics. In these cases,
>> >discovery IS justification. Proof is empirical:
>> >
>> >possible combination of neighbouring map shapes were tried and tested. This
>is
>> >empirical. There is (or at least was at the time) no know formal method to
>> >prove it.
>> >
>> combinations is infinite. It was only *after* you had a proof that
>> any of these combinations is equivalent to one from a smaller, finite
>> set, that you could go and check the combinations in this finite set,
>> one by one. Absent the proof, and yes, it was a formal proof, you
>> could run all the computers in the world for eternity, getting
>> nowhere.
>>
>
>They key to the proof being empirical is that a computer is used.
Nope. The key to the proof above is the step where you reduce the
apriori infinite set of possibilities to a finite one, at which point
the individual cases can be enumerated and checked one by one. This is a
standard technique used in math for centuries. Now, would the finite
set consist of just few or, at most, few dozen cases, you could've
checked them by hand. Since the number, though finite, is still
rather large, you use a computer. The computer here is not a "key",
just a time saving device. That's all.
>
>My contention is not that the four colour problem is not formally provable. It
>may well be by someone with a lot of time on their hands. But unless the
>referees are prepared to spend the same amount of time, then they can only
>appeal to a computer proof.
>
And if you take a proof aof a problem which reduces to few hundred
cases, with somebody checking them manually one by one, then you can
either believe that he made no error in checking or spend the same
amount of time repeating the verification. And even if, after
spending all the time needed, you got the same result, there is still
a non-zero (though very small) probability that the original checker
made an error in some place and you just happened to repeat same
error. This doesn't make it "empirical".
Mati Meron | "When you argue with a fool,
me...@cars.uchicago.edu | chances are he is doing just the same"
mme...@cars3.uchicago.edu
>
>
>Jason wrote:
>>
>> Exhaustion and testing by algorithm is not so formal. The computer has to be
>> trusted to be doing the right task. I don't know about yours, but my computer
>> is not to be trusted at the best of times. Let alone the software I write...
>
>constructed from unreliable parts, as long as the errors were
>statistically independent.
>
>If you want high reliability go to an engineer. If you want certainty,
>go to church.
>
Jason
Absolutely. This idea can be stretched like glad-wrap over humans as well. It
shows how thin the idea of proof can be at times. Rather than proof being this
mathematical ideal, in reality it is just what is socially accepted for the
moment. Maths has a long history of proofs that were not proofs.
Jason
>
> >Well, they verified the cases by computer.
>
> Lots of mathematics is verified with pencil and paper. Does the use
> of pencil and paper make it empirical.
Yes, the idea can cover pencil and paper as well. It doesn't make their use
empirical, but it does show how fragile the concept can become.
> > But how do you prove an algorithm
is
> >correct?
>
> The same way that you demonstrate that a mathematical proof is
> correct. Oh, by the way, there are many incorrect proofs in the
> published literature.
Yes, I think you're right. There is no algorithm that can check algorithms, but
we could do it on a case-by-case basis I suppose. I didn't do this when they
accepted the proof though, by the sounds.
> > The 'proof' of the
four
> >colour problem is partially inductive.
>
> If that is correct, then all logic is partially inductive.
I'm not quite sure I follow your reasons here, but I don't think there is
disagreement for the most part.
> >The computer literally did the colouring of maps and counting of the colours.
>
> The computer was used as a book keeping tool, to keep track of
> details to numerous for ordinary human attention. This is not an
> empirical investigation, except in the strange meanings you seem to
> be giving to "empirical".
While I'd like to take credit for the strangeness, but it was Chaitin (and
Tymoczko for the four colour problem). The strange meaning of "empirical" here
is the lack of correctness proof in algorithms, trust in their implementation,
trust in computers. You wouldn't run the proof on one computer once, it would
be run on different computers using different algorithms several times before
accepting the proof.
If it turns out that something is not provable due to incompleteness, then why
shouldn't it be proved empirically and a new axiom added?
> >It was accepted after they tried it on other computers with different
> >programmes.
>
> Traditional proofs are accepted only after people have worked through
> the details of the proof. Recoding the program and running on
> different computers is just part of the normal working through a
> proof.
>
> > But again, this is inductive evidence. That it is a legitimate
> >proof is controversial.
>
> You are misusing "inductive", much as you have been misusing "empirical".
I don't think so. These concepts are not so easily nailed down and defined.
What inductive reasoning doesn't use deductive reasoning and vice-versa?
>
> Although there was some initial controversy over the idea of using a
> computer in the proof, I have not seen much evidence that it is
> currently considered controversial.
Compsci and maths are merging in discrete maths. I'm sure mathematicians are
more willing to use computers as they learn more about algorithms and
vice-versa.
Jason
> >>consisting of reducing the problem to a finite number of cases.
> >
> > algorithm can be deductively proven to be correct, then I'll concede.
Otherwise
> > all that can be done is to test it on different computers with different
> > programmes until we're satisfied, which is inductive.
>
> out. That process (of checking the proof) is just as inductive.
Yep :)
>
> > My contention is not that the four colour problem is not formally provable.
It
> > may well be by someone with a lot of time on their hands. But unless the
> > referees are prepared to spend the same amount of time, then they can only
> > appeal to a computer proof.
> >
> > BTW, this is not to say that it is not admissible as a proof. I think it
should
> > be. It is just another argument for quasi-empirical mathematics.
>
> For reference, see:
>
> http://www.math.gatech.edu/~thomas/FC/fourcolor.html
>
> The proof of 4CT is not quasi-empirical. No random sampling was made.
> The cases are exhaustive. Yes, you need to prove that the algorithm
> checking the cases is correct, and that the set of cases is
> exhaustive. But that's not more empirical than any noncomputer proof.
The sampling was in the algorithms, their implementations and the instances they
were run.
Jason
If something cannot be proven formally by incompleteness, but it can be proven
empirically since maths does seem to overlap the world, then it is reasonable
that proof by empirical means should be used and a new axiom added.
Jason
How to you prove that an algoritm is correct and that it has been implemented
and carried out correctly?
Because it has been used for centuries, it should be used today is an empirical
reason.
> >
> >My contention is not that the four colour problem is not formally provable.
It
> >may well be by someone with a lot of time on their hands. But unless the
> >referees are prepared to spend the same amount of time, then they can only
> >appeal to a computer proof.
> >
> And if you take a proof aof a problem which reduces to few hundred
> cases, with somebody checking them manually one by one, then you can
> either believe that he made no error in checking or spend the same
> amount of time repeating the verification. And even if, after
> spending all the time needed, you got the same result, there is still
> a non-zero (though very small) probability that the original checker
> made an error in some place and you just happened to repeat same
> error. This doesn't make it "empirical".
It does, or at least did, to some mathematicians. We hold on to this
mathematical ideal as if it actually happens. The history of maths is full of
bogus proofs.
Jason
:)
Perhaps they're not so different. Engineers did invent "Plug and Pray"
technology.
Jason
notion
> > of mathematical 'truth' can be extended from the limited formal system
notion of
> > 'truth', to our every day understanding. I believe there are mathematical
> > statements that are true in our world, but are not provable by the current
> > axioms of maths. These statements are the ones I am referring to. They
need to
> > be empirically verified, which might then provide impetus for the axioms to
> > change.
>
> Could you provide an example of such statements? The only truths not
> provable in the current axioms of math that come to my mind are
> artificial examples like Goedel sentences, which we believe to be true
> for theoretical reasons (they'd better be true, or arithmetic is
> inconsistent), not on the basis of anything in our experience.
Unprovable hypotheses about Prime numbers spring to mind. The Riemann
Hypothesis has a lot of empirical weight, but has never been proven. I don't
think this one has been shown to be outside of proof though. The Continuum
Hypothesis has been shown to lie outside of mathematical proof. But
mathematicians don't wait around for proofs, they use these hypotheses and
qualify that their results are dependent. They use them on empirical grounds
because by all counts they look correct.
> > Exhaustion and testing by algorithm is not so formal. The computer has to
be
> > trusted to be doing the right task. I don't know about yours, but my
computer
> > is not to be trusted at the best of times. Let alone the software I
write...
>
> So a method is only formal if it is applied by an infallible reasoner?
> Well, that's an easy way to rule out anything from being formally
> proven, but it's quite a non-standard use of the term.
It is an ideal. And yes, a non-standard use of the term. But then this is a
philosophy channel so we're allowed to explore the limits of concepts.
Jason
"Lester Zick" <lester...@worldnet.att.net> wrote in message
news:41f6797c...@netnews.att.net...
We don't engage in dialogue, the dialogue engages us :)
Mitch Perkins
> The only mystery here is that you consider your remarks substantive
> much less responsive to what I wrote. In case you have any lingering
> thoughts on the subject, my rejoinders to you were in kind. None of
> your remarks represented a serious critique of what I posted. And
most
> either duplicated what I said or were completely extraneous. I think
> you'll find that when your remarks approach serious criticism, my
> replies to your remarks will be comparably responsive.
>
Does not compute - why didn't you respond with the above paragraph in
the first place, instead of the nastiness?
I found Tom's remarks both substantive and responsive to what you
wrote.
Mitch
mme...@cars3.uchicago.edu
>and carried out correctly?
>
has been implemented correctly?
If you're asking for absolute certainty, regarding anything, you're
bound to be disappointed and, beyond some point, the quest for
certainty may become counterproductive. Engineers have a saying to
the effect that "'Perfect' is the enemy of 'good'" and it applies
here.
But, again, the computer issue is a red herring, here. There is no
qualitative difference between computer checked proofs and human
checked proofs. In both cases a non-zero probability of error exists.
>Because it has been used for centuries, it should be used today is an empirical
>reason.
>
If you want to use "empirical" as a synonym for "not absolutely
certain" then, first, this is a highly unorthodox use of the term and,
second, it is totally useless. Since, as I said, nothing is
absolutely certain, the word "empirical" as you use it applies to
everything and any term that applies to everything serves no useful
purpose.
>
>> >My contention is not that the four colour problem is not formally provable.
>It
>> >may well be by someone with a lot of time on their hands. But unless the
>> >referees are prepared to spend the same amount of time, then they can only
>> >appeal to a computer proof.
>> >
>> And if you take a proof aof a problem which reduces to few hundred
>> cases, with somebody checking them manually one by one, then you can
>> either believe that he made no error in checking or spend the same
>> amount of time repeating the verification. And even if, after
>> spending all the time needed, you got the same result, there is still
>> a non-zero (though very small) probability that the original checker
>> made an error in some place and you just happened to repeat same
>> error. This doesn't make it "empirical".
>
>It does, or at least did, to some mathematicians. We hold on to this
>mathematical ideal as if it actually happens. The history of maths is full of
>bogus proofs.
>
Indeed. And Santa Claus doesn't really exist. So?
Jason
If this where sci.engineering then I would conceed. But here we see that
empiricism is a sliding-scale concept. At what point we decide something is
empirical is arbitrary, but the impression is that there is a hard-line. Like
people saying that maths is not empirical, therefore not science. The question
becomes, 'what is empirical?' and we see that the non-empirical element of maths
is not so certain.
> >Because it has been used for centuries, it should be used today is an
empirical
> >reason.
> >
> If you want to use "empirical" as a synonym for "not absolutely
> certain" then, first, this is a highly unorthodox use of the term and,
> second, it is totally useless. Since, as I said, nothing is
> absolutely certain, the word "empirical" as you use it applies to
> everything and any term that applies to everything serves no useful
> purpose.
Philosophical debates aren't won by appealing to orthodoxy or usefulness.
> >> >My contention is not that the four colour problem is not formally
provable.
> >It
> >> >may well be by someone with a lot of time on their hands. But unless the
> >> >referees are prepared to spend the same amount of time, then they can only
> >> >appeal to a computer proof.
> >> >
> >> And if you take a proof aof a problem which reduces to few hundred
> >> cases, with somebody checking them manually one by one, then you can
> >> either believe that he made no error in checking or spend the same
> >> amount of time repeating the verification. And even if, after
> >> spending all the time needed, you got the same result, there is still
> >> a non-zero (though very small) probability that the original checker
> >> made an error in some place and you just happened to repeat same
> >> error. This doesn't make it "empirical".
> >
> >It does, or at least did, to some mathematicians. We hold on to this
> >mathematical ideal as if it actually happens. The history of maths is full
of
> >bogus proofs.
> >
> Indeed. And Santa Claus doesn't really exist. So?
So the heavenly ideal of formal mathematical proof is bogus in practice. Proof
is social acceptance, nothing more. Empirical methods saturate everything we
do, and yet people are ready to blow trumpets in the name of non-empirical
mathematics. My aim is not to convince people that maths is empirical, although
this is the argument point, it is to get them to put down their trumpets.
robert j. kolker
Jason wrote:
> So the heavenly ideal of formal mathematical proof is bogus in practice. Proof
> is social acceptance, nothing more. Empirical methods saturate everything we
> do, and yet people are ready to blow trumpets in the name of non-empirical
> mathematics. My aim is not to convince people that maths is empirical, although
> this is the argument point, it is to get them to put down their trumpets.
The objects of mathematics (generally) have no physical existence so the
term emprical does not apply to them. However the question whether a
sequence of steps is a proof or an outline of a proof is an empirical
matter. Social acceptance is not just a matter of good fellowship and
cordiality. When a committee of mathematicians examins a proof they are
realy looking at its structure. The social accpetance comes with an
agreement that no error has been found (which does not mean there is no
error, necessarily).
Bob Kolker
>
>
>
mme...@cars3.uchicago.edu
>[snip]
>empiricism is a sliding-scale concept. At what point we decide something is
>empirical is arbitrary, but the impression is that there is a hard-line. Like
>people saying that maths is not empirical, therefore not science. The question
>becomes, 'what is empirical?' and we see that the non-empirical element of maths
>is not so certain.
>
the "computer issue". As for "emirical", I already commented on your
use of this term, no point in repeating myself.
>
>> >Because it has been used for centuries, it should be used today is an
>empirical
>> >reason.
>> >
>> If you want to use "empirical" as a synonym for "not absolutely
>> certain" then, first, this is a highly unorthodox use of the term and,
>> second, it is totally useless. Since, as I said, nothing is
>> absolutely certain, the word "empirical" as you use it applies to
>> everything and any term that applies to everything serves no useful
>> purpose.
>
>Philosophical debates aren't won by appealing to orthodoxy or usefulness.
>
I'll try to remember this if I'll ever engage in a philosophical
debate:-)
>
>> >> >My contention is not that the four colour problem is not formally
>provable.
>> >It
>> >> >may well be by someone with a lot of time on their hands. But unless the
>> >> >referees are prepared to spend the same amount of time, then they can only
>> >> >appeal to a computer proof.
>> >> >
>> >> And if you take a proof aof a problem which reduces to few hundred
>> >> cases, with somebody checking them manually one by one, then you can
>> >> either believe that he made no error in checking or spend the same
>> >> amount of time repeating the verification. And even if, after
>> >> spending all the time needed, you got the same result, there is still
>> >> a non-zero (though very small) probability that the original checker
>> >> made an error in some place and you just happened to repeat same
>> >> error. This doesn't make it "empirical".
>> >
>> >It does, or at least did, to some mathematicians. We hold on to this
>> >mathematical ideal as if it actually happens. The history of maths is full
>of
>> >bogus proofs.
>> >
>> Indeed. And Santa Claus doesn't really exist. So?
>
>So the heavenly ideal of formal mathematical proof is bogus in practice.
:-))) The purist's battle cry, "if it ain't perfect, it is bogus."
Count me mildly amused.
Mitch Harris
>>> It is just another argument for quasi-empirical mathematics.
>>
>>For reference, see:
>>
>> http://www.math.gatech.edu/~thomas/FC/fourcolor.html
>>
>>The proof of 4CT is not quasi-empirical. No random sampling was made.
>>The cases are exhaustive. Yes, you need to prove that the algorithm
>>checking the cases is correct, and that the set of cases is
>>exhaustive. But that's not more empirical than any noncomputer proof.
>
> The sampling was in the algorithms, their implementations and the instances they
> were run.
But that is not "empirical" sampling.
The sampling made for Pons Asinorum (Euclid I.5) is every school child
ever required to read it. (which I suppose means that it hasn't been
proved very well :) )
All I'm saying is that your use of "sampling" makes computer proofs
just as empirical as human ones.
robert j. kolker
mme...@cars3.uchicago.edu wrote:
>
> :-))) The purist's battle cry, "if it ain't perfect, it is bogus."
> Count me mildly amused.
Once again proving that the Best is the Enemy of the Good.
Bob Kolker
Lester Zick
<mitchs...@yahoo.com> in comp.ai.philosophy wrote:
I didn't respond with the above paragraph first because I considered
Tom's remarks silly and directed at comparatively minor points in the
post. If you'd seriously like to examine why, I'll be happy to oblige.
Regards - Lester
Lester Zick
<jasonstev...@free.net.nz> in comp.ai.philosophy wrote:
So I've noticed. I'm beginning to wonder if mathematikers and
physicists lack critical reading comprehension skills for non numeric
text.
>> The idea that math is a tautological extrapolation
>> on science in general, although true, is hardly as relevant or
>> interesting in my estimation as the reduction of science in general
>> through finite tautological regression.
>>
>> Regards - Lester
>
>
>
Regards - Lester
Lester Zick
comp.ai.philosophy wrote:
>In article <QAzJd.11563$mo2.8...@news.xtra.co.nz>, "Jason" <jasonstev...@free.net.nz> writes:
[. . .}
>If you want to use "empirical" as a synonym for "not absolutely
>certain" then, first, this is a highly unorthodox use of the term and,
>second, it is totally useless. Since, as I said, nothing is
>absolutely certain, the word "empirical" as you use it applies to
>everything and any term that applies to everything serves no useful
>purpose.
The term empirical only means problematic and the term certainty only
means finite tautological regression through to self contradictory
alternatives as should be intuitvely obvious to any casual observer
who bothered to read the original post through to its conclusion which
it seems mathematikers have some considerable difficulty doing as they
prefer to concentrate instead on the true but relatively minor
contention that mathematics in general rests on problematic empirical
problematic foundations.
Regards - Lester
Lester Zick
<jasonstev...@free.net.nz> in comp.ai.philosophy wrote:
[. . .]
>> But, again, the computer issue is a red herring, here. There is no
>> qualitative difference between computer checked proofs and human
>> checked proofs. In both cases a non-zero probability of error exists.
>
>If this where sci.engineering then I would conceed. But here we see that
>empiricism is a sliding-scale concept. At what point we decide something is
>empirical is arbitrary, but the impression is that there is a hard-line. Like
>people saying that maths is not empirical, therefore not science. The question
>becomes, 'what is empirical?' and we see that the non-empirical element of maths
>is not so certain.
Mathematics is the tautological elaboration of axioms. There are two
empirical aspects to mathematics: verification of tautological proofs
and the axiomatic foundations on which mathematics rests. Proofs as
such are simply representations of the tautologies involved in proofs.
These are only accidentally empirical whereas the axioms on which
mathematics are inherently empirical because they aren't demonstrated
by finite tautological regression to self contradictory alternatives.
Regards - Lester
robert j. kolker
Lester Zick wrote:
>
> Mathematics is the tautological elaboration of axioms.
Poicare' demonstrated in -Science et Hypohtesis- that Peano induction is
not tautological.
Bob Kolker
Lester Zick
Not sure what Peano induction means. However there are no true
empirical observations which are not tautologically reducible. If that
doesn't include Peano induction, I'm really not sure what it can be
nor what definition for tautological Poincare could have been using.
Regards - Lester
robert j. kolker
>
> Not sure what Peano induction means.
See induction postulate among the Peano postulates for arithmetic. I
assume you are familiar with proof by induction which is not to be
confused with empirical induction.
Bob Kolker
mme...@cars3.uchicago.edu
>
>
>mme...@cars3.uchicago.edu wrote:
>
>>
>> :-))) The purist's battle cry, "if it ain't perfect, it is bogus."
>> Count me mildly amused.
>
>
Lester Zick
Not really sure. The problem is there are a great many empirical
observations concerning mathematical and tautological proofs.
Tautologies are relatively clear and succinct. Induction is one of
those ambiguous extra tautological areas.
Regards - Lester
robert j. kolker
>
> Not really sure. The problem is there are a great many empirical
> observations concerning mathematical and tautological proofs.
> Tautologies are relatively clear and succinct. Induction is one of
> those ambiguous extra tautological areas.
Let S be a subset of the integers Z. if 0 in S and for all n whenever
n in S then n+1 in S, then S = Z.
Alternately let P stand for a predicate. if P(0) and (n)[P(n) -> P(n+1)]
then (n)P(n).
I am surprise that you do not know induction. It is taught in the ninth
or tenth grade of high school.
Bob Kolker
Paul Bramscher
> Jason wrote:
>
>> Maths as a formal system is incomplete, so some statements cannot be
>> proven as
>> derivations from the axioms. Some of these statements are true or
>> false under
>> the standard interpretation of the language of mathematics. In these
>> cases,
>> discovery IS justification. Proof is empirical:
>
>
> follow from our basic axioms, or even the fact that we know that no
> matter what axioms we pick there will be some statements which are true
> which don't follow from those axioms, shows that those truths are
> empirical. After all, any given truth follows from the right choice of
> axioms (trivially, the axiom set including that truth as an axiom). And
> it is not by experience that we know which claims are false, either;
> false claims in mathematics lead to formal contradictions. There is no
> need for experience to contradict them.
Different scopes to truth, which is the essence of Goedel. A system
with missing axioms, perhaps others missing due to "explanatory gap"
might produce a truth which is internally consistent -- given the
limited number of available axioms -- but perhaps which might not hold
true if another axiom is added.
Given modern computers, the exercise of solving truth tables is now
exposed for what it is: it does not yield any surprises, can be
perfectly performed by a machine, and it produces results which are
necessarily true or necessarily false. The challenge in science, as
opposed to math/logic, is to bring the axioms to the table in the first
place.
>
>> The four colour map problem was finally 'proved' by computer. That
>> is, every
>> possible combination of neighbouring map shapes were tried and
>> tested. This is
>> empirical. There is (or at least was at the time) no know formal
>> method to
>> prove it.
>
>
> Exhaustion is a formal method. It's just not in this case a formal
> method which is within the physical capacities of humans to apply (the
> four colour problem is just a little too exhausting).
>
The Sophist
> Different scopes to truth, which is the essence of Goedel. A system
> with missing axioms, perhaps others missing due to "explanatory gap"
> might produce a truth which is internally consistent -- given the
> limited number of available axioms -- but perhaps which might not hold
> true if another axiom is added.
What do you mean by it "producing a truth"? If you mean that there is a
statement which follows from the system with "missing axioms" which
would become untrue if more axioms were added, this is of course
impossible; adding axioms can't make a system weaker. If you mean that
there is a statement which in Goedel fashion must be true for the system
to be consistent, then again adding new axioms cannot possibly show it
to be false. If you mean there is a statement which is independent of
the system with "missing axioms", I am curious as to why you refer to it
as a truth; what about independence gives us any reason to even suspect
truth?
--
Aaron Boyden
The main division between the so-called Continental and Analytic
traditions has been disputes over whether the task of being unclear
should be carried out in natural language or in a formal system.
Lester Zick
I haven't been in the ninth or tenth grades for some decades now. It
turns out I couldn't wait around for people to make up their minds
about things like induction and deduction.
The problem is that things like induction and deduction are properties
of combinations of tautologies which are either speculative or false.
And if speculative they are subject to finite tautological regression
like every other empirical observation. I see nothing in the
application of tautologies that invalidates tautological regression.
Regards - Lester
robert j. kolker
Lester Zick wrote:
>
> The problem is that things like induction and deduction are properties
> of combinations of tautologies which are either speculative or false.
> And if speculative they are subject to finite tautological regression
> like every other empirical observation. I see nothing in the
> application of tautologies that invalidates tautological regression.
The induction principle cannot be derived from tautologies such as the
law of the excluded middle or the law of non-contradiction. The
induction principle cannot be derived from logic at all.
Bob Kolker
Paul Bramscher
(2) This Cortland is a variety of apple.
Necessary conclusion #1: This Cortland is red.
That's a necessary truth.
Let's add a couple more axioms:
(3) Decayed apples turn brown.
(4) This Cortland is decayed.
New (necessary) conclusion #2: This Cortland is brown.
In either case, the abstract rules of the game are the same -- but the
end result changes based on the addition of more axioms. Let's say that
you're holding a brown apple, know nothing about the process of decay
(explanatory gap). Are are compelled to either think that what you hold
must not be an apple (it contradicts axiom #1 above) or that there is
some missing axiom which must be accounted for the obvious wrong
conclusion #1 if the apple you hold is brown.
Ponder this very carefully before you respond with sophistry.
robert j. kolker
Paul Bramscher wrote:
> (3) Decayed apples turn brown.
> (4) This Cortland is decayed.
>
> New (necessary) conclusion #2: This Cortland is brown.
Only if a Cortland is an apple.
Bob Kolker
The Sophist
> (1) All apples are red.
> (2) This Cortland is a variety of apple.
>
> Necessary conclusion #1: This Cortland is red.
>
> That's a necessary truth.
Sure, it is a necessary truth that this conclusion follows from those
premisses.
> Let's add a couple more axioms:
>
> (3) Decayed apples turn brown.
> (4) This Cortland is decayed.
>
> New (necessary) conclusion #2: This Cortland is brown.
Well, of course this follows, but now the axiom set is inconsistent, so
anything at all follows from it. The original conclusion also still
follows, for instance.
Lester Zick
Then it can't be true. The laws of logic you cite aren't mechanical.
They don't represent reducible mechanisms. The tautology is reducible
in the sense that it represents a mechanism finitely regressable to
self contradictory alternatives which is always true for that reason.
Consequently if induction cannot be regressed through something that
is always true (i.e. tautologies) it cannot be true.
However I assume for the sake of argument that induction can be true.
Therefore I also have to assume it is regressable through tautologies
despite your contention to the contrary. Induction is just an instance
of empirical observation in this regard whatever its ostensible
justification may be in math.
Regards - Lester
Paul Bramscher
> Paul Bramscher wrote:
>
>> (1) All apples are red.
>> (2) This Cortland is a variety of apple.
>>
>> Necessary conclusion #1: This Cortland is red.
>>
>> That's a necessary truth.
>
>
> Sure, it is a necessary truth that this conclusion follows from those
> premisses.
>
>> Let's add a couple more axioms:
>>
>> (3) Decayed apples turn brown.
>> (4) This Cortland is decayed.
>>
>> New (necessary) conclusion #2: This Cortland is brown.
>
>
> Well, of course this follows, but now the axiom set is inconsistent, so
> anything at all follows from it. The original conclusion also still
> follows, for instance.
I mean that #3 and #4 are added to #1 and #2 (not floating out there on
their own):
This:
(1) All apples are red.
(2) This Cortland is a variety of apple.
Color = red
Versus this:
(1) All apples are red.
(2) This Cortland is a variety of apple.
(3) Decayed apples turn brown.
(4) This Cortland is decayed.
Color = brown
So the addition of more axiomatic statements (to a previously existing
system) clearly changes what must be (necessarily) true. This game, I'm
sure could produce an infinite flip-flop between colors of the apple.
(1) All apples are red.
(2) This Cortland is a variety of apple.
(3) Decayed apples turn brown.
(4) This Cortland is decayed.
(5) Red paint will make things red.
(6) This apple is painted with red paint.
Color = red
And so on -- new information changes what the truth table (necessarily)
concludes. Truths are scoped situations, only in relation to their
axioms. So I humbly submit that since even a mere machine can perform
truth table logic flawlessly (probably better than humans), that the
real problem facing logic is in the assembly of relevant axioms. I see
this as largely the domain of science, human sensibilities and biases
(experimental, procedural, social, financial limitation, etc.)
Unfortunately, that's where things get icky.
Gregory L. Hansen
Paul Bramscher <brams00...@tc.umn.edu> wrote:
>The Sophist wrote:
>This:
>
>(1) All apples are red.
>(2) This Cortland is a variety of apple.
>
>Color = red
>
>Versus this:
>
>(1) All apples are red.
>(2) This Cortland is a variety of apple.
>(3) Decayed apples turn brown.
>(4) This Cortland is decayed.
>
>Color = brown
Color = brown and red. It may be decayed, but it's still an apple, and
postulate (1) explicitly said that all apples (not just undecayed apples)
are red.
>And so on -- new information changes what the truth table (necessarily)
>concludes. Truths are scoped situations, only in relation to their
>axioms. So I humbly submit that since even a mere machine can perform
>truth table logic flawlessly (probably better than humans), that the
>real problem facing logic is in the assembly of relevant axioms. I see
>this as largely the domain of science, human sensibilities and biases
>(experimental, procedural, social, financial limitation, etc.)
--
"No one need be surprised that the subject of contagion was not clear to
our ancestors."-- Heironymus Fracastorius, 1546
Lester Zick
Only because axioms are empirical observations conventionally
viewed as not finitely regressable through to self contradictory
alternatives. Same problem with science in conventional terms.
Regards - Lester
Jason
Well it is random sampling I suppose out of a number of possible algorithms, but
yeah, I agree mostly.
Jason
There is no mechanical method to prove something, so in this sense it is a
creative process for us. This tautological element is idealistic once again.
Jason
Possibly. People in general, it seems, are more interested in saying than
listening. We want to tell people what to do rather than vice-versa. Will to
power maybe :) Not really. I for one are not a strong reader and with little
time and so many posts, skim reading is necessary some times. I try not to
critique things I haven't read, but I miss-read things at times as Albert never
tires of pointing out.
Paul Bramscher
> In article <ctbckb$ak2$1...@lenny.tc.umn.edu>,
> Paul Bramscher <brams00...@tc.umn.edu> wrote:
>
>>The Sophist wrote:
>
>
>>This:
>>
>>(1) All apples are red.
>>(2) This Cortland is a variety of apple.
>>
>>Color = red
>>
>>Versus this:
>>
>>(1) All apples are red.
>>(2) This Cortland is a variety of apple.
>>(3) Decayed apples turn brown.
>>(4) This Cortland is decayed.
>>
>>Color = brown
>
>
> Color = brown and red. It may be decayed, but it's still an apple, and
> postulate (1) explicitly said that all apples (not just undecayed apples)
> are red.
Good point. It can be rectified in a number of ways, probably involving
a rephrasing of conflicting axioms.
This makes the problem more complex. Adding new axioms may not only
change the conclusion, but it may also require redefinition or
elimination of previous axioms -- lest the problem become non-computable.
In fact, if I state that a decayed apple turns brown, and a red-painted
(and yet decayed) apple is red, then we have another non-computable
situation.
But it can be resolved if it introduce time sensitivity into the laying
down of the axioms, or process them in order sensitive means (such as
reverse polish notation, etc.) In that case, computability is still
possible.
Gregory L. Hansen
It was a model system anyway, not much sense in putting too much effort
into it. But some rephrasing is appropriate like
(1) All ripe apples are red.
(2) All decayed apples are brown.
(3) An apple cannot be both ripe and decayed.
etc.
--
"Don't try to teach a pig how to sing. You'll waste your time and annoy
the pig."
Jesse F. Hughes
> (1) All apples are red.
> (2) This Cortland is a variety of apple.
>
> Necessary conclusion #1: This Cortland is red.
>
> That's a necessary truth.
That is not the typical sense of "necessary truth". That the Cortland
is red is necessary *given* (1) and (2), but it is not called a
necessary truth simpliciter. It is contingent. It could have been
otherwise (say if not all apples are red).
>
> Let's add a couple more axioms:
>
> (3) Decayed apples turn brown.
> (4) This Cortland is decayed.
>
> New (necessary) conclusion #2: This Cortland is brown.
>
> In either case, the abstract rules of the game are the same -- but the
> end result changes based on the addition of more axioms.
This is nonsense. You can't assert that both (1) and (3) are true
under any reasonable interpretation of (1). I guess you're trying to
sketch default (non-monotonic) reasoning, but it doesn't work if you
say that *all* apples are red.
Perhaps it is more plausible if you say "Apples are red," provided
that the reader knows you mean that *normally* apples are red.
--
Jesse F. Hughes
"Of course, my ability to admit my mistakes and correct them is a
trait that many of you seem to never have properly appreciated."
-- JSH, discussing his 1463rd "proof" of Fermat's Last Theorem.
Lester Zick
<jasonstev...@free.net.nz> in comp.ai.philosophy wrote:
>"Lester Zick" <lester...@worldnet.att.net> wrote in message
>news:41f7b7d4...@netnews.att.net...
[. . .]
>> >> The really interesting thing in all this is that people are focusing
>> >> on the wrong issue.
>> >
>> >We don't engage in dialogue, the dialogue engages us :)
>>
>> So I've noticed. I'm beginning to wonder if mathematikers and
>> physicists lack critical reading comprehension skills for non numeric
>> text.
>
>Possibly. People in general, it seems, are more interested in saying than
>listening. We want to tell people what to do rather than vice-versa. Will to
>power maybe :) Not really. I for one are not a strong reader and with little
>time and so many posts, skim reading is necessary some times. I try not to
>critique things I haven't read, but I miss-read things at times as Albert never
>tires of pointing out.
Well we all have a will to power or at least influence and we all mis
construe things as I have with Albert on occasion. If you find a post
worth responding to I suggest you take some care reading it. I find
that many if not most mathematikers and physikers are in the fields
they're in more to avoid critical reading and thinking comprehension
than any lack of intelligence on their part.
Regards - Lester
The Sophist
> The Sophist wrote:
>
>> Paul Bramscher wrote:
>>
>>> (1) All apples are red.
>>> (2) This Cortland is a variety of apple.
>>>
>>> Necessary conclusion #1: This Cortland is red.
>>>
>>> That's a necessary truth.
>>
>>
>>
>> Sure, it is a necessary truth that this conclusion follows from those
>> premisses.
>>
>>> Let's add a couple more axioms:
>>>
>>> (3) Decayed apples turn brown.
>>> (4) This Cortland is decayed.
>>>
>>> New (necessary) conclusion #2: This Cortland is brown.
>>
>>
>>
>> Well, of course this follows, but now the axiom set is inconsistent,
>> so anything at all follows from it. The original conclusion also
>> still follows, for instance.
>
>
> I mean that #3 and #4 are added to #1 and #2 (not floating out there on
> their own):
Obviously, that's why I noted that the new set is inconsistent. 1 and 3
can't both be true.
If we were talking about the real world, rather than introducing axioms,
the solution to the contradiction would be obvious. 1 is just false of
real world apples. But that doesn't seem relevant to the theoretical
points we're discussing here.