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JSH: My research, a roadmap
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Frederick Williams
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PostPosted: Tue Jul 18, 2006 10:14 am    Post subject: Re: JSH: My research, a roadmap Reply with quote

jstevh@msn.com wrote:
Quote:

Frederick Williams wrote:

Note how the conclusion not-A no longer depends on the assumption A, it
only depends on other assumptions (if any) made in the rest of the proof
marked with the ellipsis. The question of whether A is true or not
doesn't arise.


Nope. That makes no sense.

Here's another example.

One can see that truth is not presupposed when something is being proved
by considering the proof of a conditional

p --> q.

One may(*) assume p and derive q from it, then one concludes p --> q
_and_ the assumption p is discharged. Note that what is proved is

p --> q

_not_

assuming p is true: p --> q.

Note also that p may be false. If it is, p --> q is true.

Note (*) I make no claim that this is the only way to prove a
conditional.

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Frederick Williams
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PostPosted: Mon Jul 17, 2006 2:59 pm    Post subject: Re: JSH: My research, a roadmap Reply with quote

jstevh@msn.com wrote:

Quote:

The axioms are consistent within an appropriate space, so Euclidean
geometry works fine in a flat space, while hyperbolic geometery works
well in one that is hyperbolic.

Hyperbolic space is a model of the axioms of hyperbolic geometry
(similarly with Euclidean geometry). No question of "working well"--you
do use strange language.

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Bob Terwilliger
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PostPosted: Mon Jul 17, 2006 3:45 am    Post subject: Re: JSH: My research, a roadmap Reply with quote

jstevh@msn.com wrote:

Quote:
Bob Terwilliger wrote:

jstevh@msn.com wrote:

Frederick Williams wrote:


jstevh@msn.com wrote:



So what is the status of hyperbolic geometry?

That's what I'm wondering. More specifically, what do you think the
status of hyperbolic geometry and Euclidean geometry are? (I think
"status" is the plural of "status".) It can't be that all the theorems
of hg are true and all the theorems of Eg are true, because some of them
contradict one another (see Bob Terwilliger's remarks on the angle sum
of a triangle). So let's go back to the axioms that those theorems are
derived from: which are true and which are false? Note that if you
insist that proofs begin with truths then you haven't defined "proof"
until you've defined "truth".



I've defined proof relative to truth, which is where I ended in my
reply when I explained the tautological properties of my definition.

Ya, and I define wizengool relative to schmizenshmoff. If I don't define
schmizenshmoff for you, will you really know what a wizengool is? Hell
no! So as has been pointed out, your definition of proof, the one that
uses the undefined concept of truth, is garbage.



But anyone reading what you said sees nonsense in the wording, but
there is already in most people a sense of what truth is.

Even the word "proof" in other contexts besides mathematics already has
a connection with truth, so truth is something that is bigger.

By defining mathematical proof in terms of truth, I get a succinct
definition that covers all the important points. and besides, the
definition I give IS correct, with the criticism now being that I don't
define truth.

So? I'll take the definition as it is and acknowledge that I don't
define truth.

But the definition of mathematical proof stands, as it is perfect
anyway.


I have not defined truth.

Your knowledge of whether or not a proof is a proof is irrelevant to
whether or not it's a proof.


Right, and the whole point is that nobody, even the great James Hashish,


Useless dig noted. My name is not "Hashish".


cannot say what truth is when it comes to mathematical proof. Again,
when one proves something mathematically, one can only claim that the
conclusion follows from the axioms. Any truth attributed to said theorem
is subjective, not absolute. Now go back to your definition of
mathematical proof and see just how flawed it really is. I'll help you
here... if you don't know what truth is, then how can you apply your
definition?



You stated a falsehood when you claimed above that "truth attributed to
said theorem
is subjective".

You may make an argument based on what you think are truths, which are
not, but that does not make the truth value of the argument subjective.

Your beliefs are irrelevant to the truth.

But what is truth?

A mathematical proof begins with a truth and proceeds by logical steps
to a conclusion which then must be true--the functional definition of
mathematical proof that I have given.

How do I apply that definition?

I note that a mathematical proof begins with a truth and proceeds by
logical steps to a conclusion which then must be true.

But what if someone challenges an argument claiming it's not a proof?

Well, if it proceeds by logical steps, then their challenge is on the
beginning.

Like, if someone wants to attack axioms or something.


To simplify it further for you James, one could take false (again,
subjective) statements as their axioms and by logical steps derive
results that follow from those axioms. If one didn't make any mistakes
in the derivations, then one would have a CONSISTENT system (meaning you
cannot derive two contradictory results) of mathematics. The axioms used
may not correspond to your immediate reality, but that doesn't negate
the consistency of the system (and perhaps in a different part of the
universe, the assumed false axioms correspond to reality, thus rendering
them true).



Truths are not relative, so your statement above contains falsehoods,
and in this case also contains a contradictory statement.

By definition axioms are true: "false axioms" is a direct
contradiction.

It's like saying true equals false.

The problem I think you have here is your BELIEF about truths versus
whether or not something is actually true.

The truth value of any proposition is independent of your beliefs as
you may simply have your wires crossed or be dsyfunctional in some way
where your brain just doesn't work right to determine truth.


To make the point clear, Euclidean and non-Euclidean geometry assume
identical axioms except for one, the parallel postulate. Many results
are derived from the axioms in each geometry, and many of these results
contradict results from the other system. Because of your environment,
your intuition may tell you that Euclidean geometry is true, and that
non-Euclidean geometries are false. Yet, it has been proven that if
non-Euclidean geometry is INCONSISTENT, then so is Euclidean geometry.
Bottom line: your definition of mathematical proof fails here.



It's about context. I gave the analogy in another reply in this thread
of a mechanic who usually works on car engines who does work on boat
engines, and finds that some of the basic axioms of working on car
engines don't apply.

But he's not working on car engines, he's working on a boat engine.

The difference with geometries is space.

I like to think there are underlying axioms which determine the choice
dependent on the space.

Note that by defining mathematical proof in terms of truth I've created
a perfect definition--for mathematical proof--leaving the definition of
truth outside of it.

To attack my definition you have to prove that there is no definition
of truth, or prove that a mathematical proof can behave differently
than I state, given a definition of truth.


James Harris


Seeing as how the above is getting cluttered, I'll simply recap here.

Your purported definition of mathematical proof depends on an undefined
notion... the notion of truth. You use the notion of truth in your
definiton in the absolute sense; i.e., truth is independent of a being's
consciousness... it exists whether a being is there to perceive it or
not. To be concrete, you've made the following statement, and others
similar to it:

"The truth value of any proposition is independent of your beliefs".

Yet, when it comes to the contradictions between Euclidean and
non-Euclidean geometry, which are consistent mathematical systems, you
say that truth depends on context (flat space or curved space)!

You're too stupid to realise that you're contradicting yourself and that
your definition of mathematical proof is meaningless.

There's much more to your idiocy than what I've pointed out here, but I
don't feel like educating you! You're way more entertaining as an
uninformed wannabe with no discernable mathematical skills.
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mensanator@aol.compost
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PostPosted: Mon Jul 17, 2006 2:18 am    Post subject: Re: JSH: My research, a roadmap Reply with quote

jstevh@msn.com wrote:
Quote:
Bob Terwilliger wrote:
jstevh@msn.com wrote:
Frederick Williams wrote:

jstevh@msn.com wrote:


So what is the status of hyperbolic geometry?

That's what I'm wondering. More specifically, what do you think the
status of hyperbolic geometry and Euclidean geometry are? (I think
"status" is the plural of "status".) It can't be that all the theorems
of hg are true and all the theorems of Eg are true, because some of them
contradict one another (see Bob Terwilliger's remarks on the angle sum
of a triangle). So let's go back to the axioms that those theorems are
derived from: which are true and which are false? Note that if you
insist that proofs begin with truths then you haven't defined "proof"
until you've defined "truth".



I've defined proof relative to truth, which is where I ended in my
reply when I explained the tautological properties of my definition.

Ya, and I define wizengool relative to schmizenshmoff. If I don't define
schmizenshmoff for you, will you really know what a wizengool is? Hell
no! So as has been pointed out, your definition of proof, the one that
uses the undefined concept of truth, is garbage.


But anyone reading what you said sees nonsense in the wording, but
there is already in most people a sense of what truth is.

Even the word "proof" in other contexts besides mathematics already has
a connection with truth, so truth is something that is bigger.

By defining mathematical proof in terms of truth, I get a succinct
definition that covers all the important points. and besides, the
definition I give IS correct, with the criticism now being that I don't
define truth.

So? I'll take the definition as it is and acknowledge that I don't
define truth.

But the definition of mathematical proof stands, as it is perfect
anyway.


I have not defined truth.

Your knowledge of whether or not a proof is a proof is irrelevant to
whether or not it's a proof.


Right, and the whole point is that nobody, even the great James Hashish,

Useless dig noted. My name is not "Hashish".

cannot say what truth is when it comes to mathematical proof. Again,
when one proves something mathematically, one can only claim that the
conclusion follows from the axioms. Any truth attributed to said theorem
is subjective, not absolute. Now go back to your definition of
mathematical proof and see just how flawed it really is. I'll help you
here... if you don't know what truth is, then how can you apply your
definition?


You stated a falsehood when you claimed above that "truth attributed to
said theorem
is subjective".

You may make an argument based on what you think are truths, which are
not, but that does not make the truth value of the argument subjective.

Your beliefs are irrelevant to the truth.

But what is truth?

A mathematical proof begins with a truth and proceeds by logical steps
to a conclusion which then must be true--the functional definition of
mathematical proof that I have given.

How do I apply that definition?

I note that a mathematical proof begins with a truth and proceeds by
logical steps to a conclusion which then must be true.

But what if someone challenges an argument claiming it's not a proof?

Well, if it proceeds by logical steps, then their challenge is on the
beginning.

Like, if someone wants to attack axioms or something.

To simplify it further for you James, one could take false (again,
subjective) statements as their axioms and by logical steps derive
results that follow from those axioms. If one didn't make any mistakes
in the derivations, then one would have a CONSISTENT system (meaning you
cannot derive two contradictory results) of mathematics. The axioms used
may not correspond to your immediate reality, but that doesn't negate
the consistency of the system (and perhaps in a different part of the
universe, the assumed false axioms correspond to reality, thus rendering
them true).


Truths are not relative, so your statement above contains falsehoods,
and in this case also contains a contradictory statement.

By definition axioms are true: "false axioms" is a direct
contradiction.

It's like saying true equals false.

The problem I think you have here is your BELIEF about truths versus
whether or not something is actually true.

The truth value of any proposition is independent of your beliefs as
you may simply have your wires crossed or be dsyfunctional in some way
where your brain just doesn't work right to determine truth.

To make the point clear, Euclidean and non-Euclidean geometry assume
identical axioms except for one, the parallel postulate. Many results
are derived from the axioms in each geometry, and many of these results
contradict results from the other system. Because of your environment,
your intuition may tell you that Euclidean geometry is true, and that
non-Euclidean geometries are false. Yet, it has been proven that if
non-Euclidean geometry is INCONSISTENT, then so is Euclidean geometry.
Bottom line: your definition of mathematical proof fails here.


It's about context. I gave the analogy in another reply in this thread
of a mechanic who usually works on car engines who does work on boat
engines, and finds that some of the basic axioms of working on car
engines don't apply.

But he's not working on car engines, he's working on a boat engine.

The difference with geometries is space.

I like to think there are underlying axioms which determine the choice
dependent on the space.

Note that by defining mathematical proof in terms of truth I've created
a perfect definition--for mathematical proof--leaving the definition of
truth outside of it.

To attack my definition you have to prove that there is no definition
of truth, or prove that a mathematical proof can behave differently
than I state, given a definition of truth.


This one needs to be preserved.

Quote:

James Harris
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Frederick Williams
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PostPosted: Mon Jul 17, 2006 1:36 am    Post subject: Re: JSH: My research, a roadmap Reply with quote

jstevh@msn.com wrote:

Quote:

Logical arguments connect truths to truths.

No. Consider this logical argument:

(1) Assume: A and not-A
(2) Hence: Contradiction!
(3) Proof by contradiction: not-(A and not-A)

Now the assumption, (A and not-A) is false, right? But the conclusion
not-(A and not-A) is true, right? So there is a logical argument from a
falsehood to a truth, and you are refuted.

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Frederick Williams
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PostPosted: Mon Jul 17, 2006 1:34 am    Post subject: Re: JSH: My research, a roadmap Reply with quote

Frederick Williams wrote:
Quote:

jstevh@msn.com wrote:

My research speaks for itself.

I have given a definition of mathematical proof:

http://mymath.blogspot.com/2005/07/definition-of-mathematical-proof.html



In which we read:

"A mathematical proof begins with a truth and proceeds by logical steps
to a conclusion which then must be true."

In what sense does a proof of, say, Pythagoras's Theorem begin with a
truth?

And now to the next obvious question: what are "logical steps"?

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Frederick Williams
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PostPosted: Mon Jul 17, 2006 1:33 am    Post subject: Re: JSH: My research, a roadmap Reply with quote

jstevh@msn.com wrote:

Quote:

But anyone reading what you said sees nonsense in the wording, but
there is already in most people a sense of what truth is.

Even the word "proof" in other contexts besides mathematics already has
a connection with truth, so truth is something that is bigger.

By defining mathematical proof in terms of truth, I get a succinct
definition that covers all the important points. and besides, the
definition I give IS correct, with the criticism now being that I don't
define truth.

So? I'll take the definition as it is and acknowledge that I don't
define truth.

Why not define it anyway? After all you write above that there is
already in most people a sense of what truth is. So it shouldn't be too
hard for you.

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Frederick Williams
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PostPosted: Mon Jul 17, 2006 1:29 am    Post subject: Re: JSH: My research, a roadmap Reply with quote

jstevh@msn.com wrote:
Quote:

Frederick Williams wrote:
jstevh@msn.com wrote:

My position now--having just thought about it earlier after writing
that post--is that proofs by contradiction automatically begin with the
following truth:

This is how it works:

_assume_ A
:
:
contradiction
therefore, conclude not-A and _discharge_ the assumption A.


Hmmm...but you're going in a circle, right?

No.

Quote:

That is, you say, assume something is true,

No. A is not assumed to be true. Here's an example that uses proof by
contradiction (here not-B plays the role of A) and conditional proof:

Assumptions --------------------> B, not B
---------
Contradiction ------------------> B & not B
--------- (proof by
Conclude not-(2nd assumption) --> not not B contradiction)
---------------
Discharge the first assumption -> B --> not not B (cond. proof)

So we have proved B --> not not B . Now notice something: that is true
whether B is true or not. Here's the truth table;

B --> not not B
---------------
t t t f t
f t f t f

So, no "not B", the assumption for the proof by contradiction, is _not_
assumed to be true, rather the whole, that's "B --> not not B" is true
whether "not B" is true or not.

Quote:
then you find a
contradiction, circle back to say, ok, this thing is NOT true.

Logical arguments connect truths to truths.

Therefore, it is logical that something that is not true, when you
attempt to connect it to a truth, you will get a contradiction, so
there is an underlying proof that starts your proof by contradiction,
as otherwise, why bother with the exercise?

Note how the conclusion not-A no longer depends on the assumption A, it
only depends on other assumptions (if any) made in the rest of the proof
marked with the ellipsis. The question of whether A is true or not
doesn't arise.


Nope. That makes no sense.

See my example above to show that A (= not-B in that case) is not
assumed to be true or false, rather it can be either.

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Frederick Williams
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PostPosted: Mon Jul 17, 2006 1:11 am    Post subject: Re: JSH: My research, a roadmap Reply with quote

Frederick Williams wrote:
Quote:

jstevh@msn.com wrote:

My research speaks for itself.

I have given a definition of mathematical proof:

http://mymath.blogspot.com/2005/07/definition-of-mathematical-proof.html



In which we read:

"A mathematical proof begins with a truth and proceeds by logical steps
to a conclusion which then must be true."

In what sense does a proof of, say, Pythagoras's Theorem begin with a
truth?

Since you tried to introduce the notion of truth into that of proof (and
confused yourself in the process) it may be that you are mixing up the
syntactic notion of proof with the semantic notion of logical
consequence.

A sentence phi is a logical consequence of a set of sentences S iff
every model of S is a model of phi. Let's symbolize that by

S |= phi

then the question arises; is it the case that

S |= phi iff S |- phi

(where S |- phi is defined as in my previous post)?

It is the case that

S |= phi iff S |- phi

for elementary logic such as fol=. But in number theory T (less than PA
is required), assuming that it is consistent, there will be sentences
gamma such that

T |= gamma but neither T |- gamma nor T |- not-gamma.

Nor will it help to strengthen T (so long as it remains consistent), see
G\"odel[**]. Note that mathematicians routinely use a theory of sets
that contains sufficient number theory, so that |= and |- in mathematics
cannot be equated.

"Model of a set of sentences" is defined by Tarski[*] as follows:

Let L be any class of sentences. We replace all extra-logical
constants which occur in the sentences belonging to L by
corresponding variables, like constants being replaced by
like variables, and unlike by unlike. In this way we obtain a
class L' of sentential functions. An arbitrary sequence of
objects which satisfies every sentential function
of the class L' will be called a model or realization of the
class L of sentences. If, in particular, the class L consists
of a single sentence X, we shall also refer to the model of the
class L the model of the sentence X. In terms of these
concepts we can define the concept of logical consequence as
follows:

The sentence X follows logically from the sentences
of the class K if and only if every model of the
class K is also a model of the sentence X.

Should the jargon be unclear:

sentence means formula with no free variables;
sentential function means formula possibly with free variables;
logical constant means truth-functional connectives,
quantifiers, identity sign;
like constants... means if constant c is replaced by x at
one occurrence, then it is replaced
by x at all occurrences, and no
constant d not equal to c is
replaced by x;
sequence... satisfies means the sequence c_1, c_2, ... is said
to satisfies phi(x_1, x_2, ...) if
phi(c1, c2, ...) where c1, c2, ...
are the names of the objects c_1, c_2,
..., and x_1, x_2, ... are variables.

[*] Tarski "On the Concept of Logical Consequence" in "Logic, Semantics,
Metamathematics" Oxford UP 1956, Hackett 1983.

As Tarski himself notes this definition was framed by Bolzano 100 years
earlier:

Bolzano "Wissenschaftslehre..." translated as "Theory of Science", ed
George, U California P.

[**] G\"odel "On Formally Undecidable Propositions of _Principia
Mathematica_ and Related Systems" in:

Heijenoort, Jean van, ed, "From Frege to G\"odel A Source Book in
Mathematical Logic, 1879--1931" Harvard U P.

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Frederick Williams
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PostPosted: Mon Jul 17, 2006 1:07 am    Post subject: Re: JSH: My research, a roadmap Reply with quote

jstevh@msn.com wrote:
Quote:

Frederick Williams wrote:
jstevh@msn.com wrote:

I have not defined truth.

I noticed.

Yup. Instead I defined mathematical proof RELATIVE to truth.

I'll put you out of your misery and tell you what a proof is. You don't
need to take my word for it: you can look it up in a book or ask the
professional logicians who post here from time to time.

We presuppose a background logic: first order logic with identity is
popular, so let's say it's that. Let's shorten "first order logic with
identity" to "fol=". We presuppose some formulation of fol= with a set
of axioms, and a set of rules.

Given a set S of formulae and a formula phi a proof of phi from S is a
finite sequence of formulae

phi_1, phi_2, phi_3, ..., phi_n

such that:

(1) phi_n is phi

and every formula in the sequence is either:

(2) an element of S, or

(3) an axiom of fol=, or

(3) a consequence of earlier formulae in the sequence by an application
of a rule of fol=.

If there is a proof of phi from S we write

S |- phi.

If S is the empty set we write

|- phi

and say that phi is proved.

Note that the word "true" isn't used and there is no need to use it.
Note that, if S is a set of axioms for Euclidean geometry[*] and S' is a
set of axioms for hyperbolic geometry[**] it is quite possible that

S |- phi and S' |- not-phi,

for example let phi be "the angle sum of a triangle is 180 degrees"
(expressed in the appropriate language). Now, if you insist that proofs
proceed from truths, are the sentences in S or S' or both true? Is phi
or not-phi true?

-----------------------------------------------------------------------

If you want to read about fol= I would recommend

Church, A "Introduction to Mathematical Logic" Princeton UP.

Don't miss section 55 which discusses the formulation of (among other
things) geometry in the language of fol=.

Note the difference between S |- phi and |- phi (i.e. S = the empty
set). Not long ago a sci.log'er got them mixed up and took quite a lot
of persuading that he'd done so.

So that's what proof is "officially". Almost always when a
mathematician claims to be proving something what he means is that he
hopes to convince his audience that he or they could formulate the proof
in the official manner (probably with the axioms of a formal theory of
sets in S as well as the particular axioms of his subject) if only they
had the time and the patience. Whether you believe them or think that it
is a wicked deception is up to you: I take the charitable view, your
writings suggest that you may not.

A final point. Church is relentlessly axiomatic with only a passing
mention of the natural deduction techniques of Ja\'skowski and Gentzen
which encapsulate something more like the usual mathematical reasoning.
If you want a recommendation of a text using natural deduction you'll
have to seek one from someone else. If you have access to an academic
library (or if the library you do have access to has something like what
in Britain is called interlibrary loans) then you might like to look at
Gentzen's papers in the American Philosophical Quarterly vol 1 pp
288--306 and vol 2 pp 204--218. For an entirely introductory book on
logic I would recommend Tarski "An Introduction to Logic and the
Methodology of the Deductive Sciences" Oxford UP (and Dover?). Terrible
title, great book.

[* I have in mind a first order formulation, such as Tarski's.]
[** Similar, Schwabh\"auser.]


Quote:

It makes no difference. Hyperbolic geometry is an abstraction quite
independent of humanity. So, I'll ask again: are the axioms (and thus
the theorems) oh hyperbolic geometry true or false?


The axioms are consistent within an appropriate space, so Euclidean
geometry works fine in a flat space, while hyperbolic geometery works
well in one that is hyperbolic.

That's a bit circular, isn't it?

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PostPosted: Mon Jul 17, 2006 12:04 am    Post subject: Re: JSH: My research, a roadmap Reply with quote

Bob Terwilliger wrote:
Quote:
jstevh@msn.com wrote:
Frederick Williams wrote:

jstevh@msn.com wrote:


So what is the status of hyperbolic geometry?

That's what I'm wondering. More specifically, what do you think the
status of hyperbolic geometry and Euclidean geometry are? (I think
"status" is the plural of "status".) It can't be that all the theorems
of hg are true and all the theorems of Eg are true, because some of them
contradict one another (see Bob Terwilliger's remarks on the angle sum
of a triangle). So let's go back to the axioms that those theorems are
derived from: which are true and which are false? Note that if you
insist that proofs begin with truths then you haven't defined "proof"
until you've defined "truth".



I've defined proof relative to truth, which is where I ended in my
reply when I explained the tautological properties of my definition.

Ya, and I define wizengool relative to schmizenshmoff. If I don't define
schmizenshmoff for you, will you really know what a wizengool is? Hell
no! So as has been pointed out, your definition of proof, the one that
uses the undefined concept of truth, is garbage.


But anyone reading what you said sees nonsense in the wording, but
there is already in most people a sense of what truth is.

Even the word "proof" in other contexts besides mathematics already has
a connection with truth, so truth is something that is bigger.

By defining mathematical proof in terms of truth, I get a succinct
definition that covers all the important points. and besides, the
definition I give IS correct, with the criticism now being that I don't
define truth.

So? I'll take the definition as it is and acknowledge that I don't
define truth.

But the definition of mathematical proof stands, as it is perfect
anyway.

Quote:

I have not defined truth.

Your knowledge of whether or not a proof is a proof is irrelevant to
whether or not it's a proof.


Right, and the whole point is that nobody, even the great James Hashish,

Useless dig noted. My name is not "Hashish".

Quote:
cannot say what truth is when it comes to mathematical proof. Again,
when one proves something mathematically, one can only claim that the
conclusion follows from the axioms. Any truth attributed to said theorem
is subjective, not absolute. Now go back to your definition of
mathematical proof and see just how flawed it really is. I'll help you
here... if you don't know what truth is, then how can you apply your
definition?


You stated a falsehood when you claimed above that "truth attributed to
said theorem
is subjective".

You may make an argument based on what you think are truths, which are
not, but that does not make the truth value of the argument subjective.

Your beliefs are irrelevant to the truth.

But what is truth?

A mathematical proof begins with a truth and proceeds by logical steps
to a conclusion which then must be true--the functional definition of
mathematical proof that I have given.

How do I apply that definition?

I note that a mathematical proof begins with a truth and proceeds by
logical steps to a conclusion which then must be true.

But what if someone challenges an argument claiming it's not a proof?

Well, if it proceeds by logical steps, then their challenge is on the
beginning.

Like, if someone wants to attack axioms or something.

Quote:
To simplify it further for you James, one could take false (again,
subjective) statements as their axioms and by logical steps derive
results that follow from those axioms. If one didn't make any mistakes
in the derivations, then one would have a CONSISTENT system (meaning you
cannot derive two contradictory results) of mathematics. The axioms used
may not correspond to your immediate reality, but that doesn't negate
the consistency of the system (and perhaps in a different part of the
universe, the assumed false axioms correspond to reality, thus rendering
them true).


Truths are not relative, so your statement above contains falsehoods,
and in this case also contains a contradictory statement.

By definition axioms are true: "false axioms" is a direct
contradiction.

It's like saying true equals false.

The problem I think you have here is your BELIEF about truths versus
whether or not something is actually true.

The truth value of any proposition is independent of your beliefs as
you may simply have your wires crossed or be dsyfunctional in some way
where your brain just doesn't work right to determine truth.

Quote:
To make the point clear, Euclidean and non-Euclidean geometry assume
identical axioms except for one, the parallel postulate. Many results
are derived from the axioms in each geometry, and many of these results
contradict results from the other system. Because of your environment,
your intuition may tell you that Euclidean geometry is true, and that
non-Euclidean geometries are false. Yet, it has been proven that if
non-Euclidean geometry is INCONSISTENT, then so is Euclidean geometry.
Bottom line: your definition of mathematical proof fails here.


It's about context. I gave the analogy in another reply in this thread
of a mechanic who usually works on car engines who does work on boat
engines, and finds that some of the basic axioms of working on car
engines don't apply.

But he's not working on car engines, he's working on a boat engine.

The difference with geometries is space.

I like to think there are underlying axioms which determine the choice
dependent on the space.

Note that by defining mathematical proof in terms of truth I've created
a perfect definition--for mathematical proof--leaving the definition of
truth outside of it.

To attack my definition you have to prove that there is no definition
of truth, or prove that a mathematical proof can behave differently
than I state, given a definition of truth.


James Harris
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Bob Terwilliger
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Posts: 56

PostPosted: Sun Jul 16, 2006 4:35 pm    Post subject: Re: JSH: My research, a roadmap Reply with quote

jstevh@msn.com wrote:
Quote:
Frederick Williams wrote:

jstevh@msn.com wrote:


So what is the status of hyperbolic geometry?

That's what I'm wondering. More specifically, what do you think the
status of hyperbolic geometry and Euclidean geometry are? (I think
"status" is the plural of "status".) It can't be that all the theorems
of hg are true and all the theorems of Eg are true, because some of them
contradict one another (see Bob Terwilliger's remarks on the angle sum
of a triangle). So let's go back to the axioms that those theorems are
derived from: which are true and which are false? Note that if you
insist that proofs begin with truths then you haven't defined "proof"
until you've defined "truth".



I've defined proof relative to truth, which is where I ended in my
reply when I explained the tautological properties of my definition.

Ya, and I define wizengool relative to schmizenshmoff. If I don't define
schmizenshmoff for you, will you really know what a wizengool is? Hell
no! So as has been pointed out, your definition of proof, the one that
uses the undefined concept of truth, is garbage.

Quote:

I have not defined truth.

Your knowledge of whether or not a proof is a proof is irrelevant to
whether or not it's a proof.


Right, and the whole point is that nobody, even the great James Hashish,
cannot say what truth is when it comes to mathematical proof. Again,
when one proves something mathematically, one can only claim that the
conclusion follows from the axioms. Any truth attributed to said theorem
is subjective, not absolute. Now go back to your definition of
mathematical proof and see just how flawed it really is. I'll help you
here... if you don't know what truth is, then how can you apply your
definition?

To simplify it further for you James, one could take false (again,
subjective) statements as their axioms and by logical steps derive
results that follow from those axioms. If one didn't make any mistakes
in the derivations, then one would have a CONSISTENT system (meaning you
cannot derive two contradictory results) of mathematics. The axioms used
may not correspond to your immediate reality, but that doesn't negate
the consistency of the system (and perhaps in a different part of the
universe, the assumed false axioms correspond to reality, thus rendering
them true).

To make the point clear, Euclidean and non-Euclidean geometry assume
identical axioms except for one, the parallel postulate. Many results
are derived from the axioms in each geometry, and many of these results
contradict results from the other system. Because of your environment,
your intuition may tell you that Euclidean geometry is true, and that
non-Euclidean geometries are false. Yet, it has been proven that if
non-Euclidean geometry is INCONSISTENT, then so is Euclidean geometry.
Bottom line: your definition of mathematical proof fails here.

Quote:
As for hyperbolic geometry versus Euclidean, there is the issue of the
space.

Oh ya, that makes your position clear... NOT! Be a man dude... take a
stand! Which system of geometry is valid and which is invalid? Your
definiton uses truth in the absolute sense, so by your definition, one
of these systems cannot be valid.

Quote:

I think the concept that is escaping you is that YOUR KNOWLEDGE of
truth is irrelevant to whether or not something actually is true.


Correct! But who are you to say what's actually true, and if you can't
say what's true, then how the hell do you apply your definiton? Perhaps
you think your definition doesn't need to be applied? Well, a definition
that can't be applied is useless!

Quote:
Consider, a mad demon on a rampage steps on earth and kills every
living thing above bacteria, wiping out all of humanity in the process,
will hyperbolic geometry be valid or not?

If your answer is that it no longer exists

Who ever said this? To the best of my knowledge, these words are yours
and yours alone... nobody has ever said such a thing.

Quote:
without human beings, well,
what if there is another sentient race in spaceships which has been
watching human beings for millennia while on a brief vacation, and they
know about all of humanity's activities, so they know of it.

They watch with some amusement when the mad demon squashes most of life
on this planet and then wander off as now it's boring and they still
have vacation time.

Does hyperbolic geometry have validity or not?


On truth and proof, you might like to read an article of Tarski's, it's
not technical:

Truth and Proof, Scientific American, June 1969.



I might. Might have already read it though as I read a lot and widely.

Insert joke here!

Quote:

Why don't you consider that JUST MAYBE you are not the exalted one that
you imagine yourself to be and that all of humanity is not either so
that MAN IS NOT THE MEASURE OF ALL THINGS, so your beliefs about truth
are irrelevant to what is actually true.

I suggest to you that when humanity goes its own way off into oblivion
the universe will still continue just fine, and mathematics will still
be here.

I think most agree with this? The question is whether or not your
"definition" of mathematical proof is valid. Your definition uses the
notion of "absolute truth" (truth independent of the existence of any
conscious being). You, nor anybody else, can say what truth is (witness
Euclidean and non-Euclidean geometry). Thus, your definition is shite!

Quote:

James Harris
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jstevh@msn.com
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Joined: 21 Jan 2006
Posts: 951

PostPosted: Sun Jul 16, 2006 3:46 pm    Post subject: Re: JSH: My research, a roadmap Reply with quote

Frederick Williams wrote:
Quote:
jstevh@msn.com wrote:

My position now--having just thought about it earlier after writing
that post--is that proofs by contradiction automatically begin with the
following truth:

This is how it works:

_assume_ A
:
:
contradiction
therefore, conclude not-A and _discharge_ the assumption A.


Hmmm...but you're going in a circle, right?

That is, you say, assume something is true, then you find a
contradiction, circle back to say, ok, this thing is NOT true.

Logical arguments connect truths to truths.

Therefore, it is logical that something that is not true, when you
attempt to connect it to a truth, you will get a contradiction, so
there is an underlying proof that starts your proof by contradiction,
as otherwise, why bother with the exercise?

Quote:
Note how the conclusion not-A no longer depends on the assumption A, it
only depends on other assumptions (if any) made in the rest of the proof
marked with the ellipsis. The question of whether A is true or not
doesn't arise.


Nope. That makes no sense.

Consider my research on factoring where I say that I have shown a route
to neat factoring method, and someone is considering the truth or
falsehood of that position.

They may test it out by assuming it to be true, and trying the method,
knowing that if it does not work, then they may have proven that my
position is not true.

Now you can consider someone who assumes it is correct without trying
to prove it's not, but either way the proof by contradiction that
envelopes their behavior tells you that if it's not true, then acting
as if it is true, will lead to a contradiction.

So you're encompassed by a larger proof, whether you know it or not.

Quote:
Suppose the set of those other assumptions is Sigma, then we have proved
that A follows from Sigma. If Sigma is the set of axioms of, say,
Euclidean geometry then we say that A is a theorem of Euclidean
geometry.

If Sigma is the empty set then we say that A is a theorem of our
underlying logic. If that logic is sound then we say that A is true.
But note that soundness needs proving, in the case of first order logic
that's not difficult, in the case of higher order logic or set theory
G\"odel's second incompleteness theorem shows that it's more of a
problem. The best one can say is that no contradictions have turned up
yet.


And that fits in well with my analogy to factoring as someone using my
latest idea who fails has not necessarily proven that the idea fails.

However, there are a finite number of ways to check, so you can remove
uncertainty within the finite system, as then Goedel's theorem doesn't
apply.

A key point in my ideas then on defining mathematical proof, and I
think a great thing as I like the simple definition, is that given ANY
proposition it must be true that if that proposition is false, then it
will lead to a contradiction because a proof begins with a truth and
proceeds by logical steps to a conclusion which then must be true.

Get it?

By the definition, you have that something that leads to a
contradiction is not true, so proof by contradiction relies on the
definition, and the requirement that something not true lead to a
contradiction is the true start.


James Harris
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jstevh@msn.com
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Joined: 21 Jan 2006
Posts: 951

PostPosted: Sun Jul 16, 2006 3:35 pm    Post subject: Re: JSH: My research, a roadmap Reply with quote

Frederick Williams wrote:
Quote:
jstevh@msn.com wrote:

I have not defined truth.

I noticed.

Yup. Instead I defined mathematical proof RELATIVE to truth.

Quote:
Your knowledge of whether or not a proof is a proof is irrelevant to
whether or not it's a proof.

Proofs are checkable by machine.


Machines like human beings assume the start with a truth, and check the
steps to make sure each is logical.

Quote:
As for hyperbolic geometry versus Euclidean, there is the issue of the
space.

I think the concept that is escaping you is that YOUR KNOWLEDGE of
truth is irrelevant to whether or not something actually is true.

I've never claimed that that is so. Just once you're telling me
something I already believe.

Consider, a mad demon on a rampage steps on earth and kills every
living thing above bacteria, wiping out all of humanity in the process,
will hyperbolic geometry be valid or not?

It makes no difference. Hyperbolic geometry is an abstraction quite
independent of humanity. So, I'll ask again: are the axioms (and thus
the theorems) oh hyperbolic geometry true or false?


The axioms are consistent within an appropriate space, so Euclidean
geometry works fine in a flat space, while hyperbolic geometery works
well in one that is hyperbolic.

The issue is context. So, consider a car mechanic who has basic
rules--axioms--about working a car engine and keeping it running, but
he wanders over to a motorboat, where a lot of the same rules apply,
but some don't, and ends up scratching his head wondering if his basic
axioms are wrong.

I think what's neat about such questions is the necessity of underlying
axioms that cover both contexts and give you conditionals, so that one
set of rules apply in flat space, while another applies when space is
curved.


James Harris
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Frederick Williams
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Joined: 19 Nov 2005
Posts: 97

PostPosted: Sun Jul 16, 2006 2:58 pm    Post subject: Re: JSH: My research, a roadmap Reply with quote

jstevh@msn.com wrote:
Quote:

I have not defined truth.

I noticed.

Quote:
Your knowledge of whether or not a proof is a proof is irrelevant to
whether or not it's a proof.

Proofs are checkable by machine.

Quote:
As for hyperbolic geometry versus Euclidean, there is the issue of the
space.

I think the concept that is escaping you is that YOUR KNOWLEDGE of
truth is irrelevant to whether or not something actually is true.

I've never claimed that that is so. Just once you're telling me
something I already believe.

Quote:
Consider, a mad demon on a rampage steps on earth and kills every
living thing above bacteria, wiping out all of humanity in the process,
will hyperbolic geometry be valid or not?

It makes no difference. Hyperbolic geometry is an abstraction quite
independent of humanity. So, I'll ask again: are the axioms (and thus
the theorems) oh hyperbolic geometry true or false?

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