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JSH: My research, a roadmap
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David Moran
science forum Guru Wannabe


Joined: 13 May 2005
Posts: 252

PostPosted: Sat Jul 15, 2006 2:35 am    Post subject: Re: My research, a roadmap Reply with quote

<jstevh@msn.com> wrote in message
news:1152929075.517531.182010@h48g2000cwc.googlegroups.com...
Quote:
My research speaks for itself.

I have given a definition of mathematical proof:

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

Figured out the key properties that define rings that are like the ring
of integers:

http://mymath.blogspot.com/2005/03/object-ring.html

Found my own prime counting function, which unlike any other known
relies on summing a partial difference equation, which is also why it
finds primes on its own, unlike any other known:

http://mymath.blogspot.com/2005/06/counting-primes.html

Fighting mathematicians who have done their best to ignore my research
I wrote the first prime counting function article for the Wikipedia,
where my latest version is now found in the history of the current
page:

http://en.wikipedia.org/w/index.php?title=Prime_counting_function&old...

There readers can see my prime counting function in its fully
mathematicized "pure" form, and see how it is a summation, so they can
make the leap to understanding how it relates to a partial differential
equation and an integration.

I had a paper published in a formally peer reviewed mathematical
journal--and then the editors withdrew it after sci.math pressure
against it:

http://www.emis.de/journals/SWJPAM/vol2-03.html

Link is to a site mirror as the electronic journal DIED a few months
later.

That paper covered some pioneering research advancing modular algebra
or the algebra of congruences, extending on the work started by Gauss:

http://mymath.blogspot.com/2005/07/tautological-spaces-factoring.html

Which is a line of attack I used to find a short proof of Fermat's Last
Theorem:

http://mymath.blogspot.com/2006/03/proof-of-fermats-last-theorem.html

But I've even considered problems in logic and set theory, handling
supposed contradictions:

http://mymath.blogspot.com/2005/06/three-valued-logic.html

and

http://mymath.blogspot.com/2005/05/logical-formedness-axioms.html

and

http://mymath.blogspot.com/2005/06/3-logic-more-basics.html

Even some of my minor research is significant, as I talked about a
simple way to find primes using quadratic residues:

http://mymath.blogspot.com/2006/04/method-for-quickly-finding-primes....

The only explanation given the breadth of my research, and dramatic
events like a math journal imploding after publishing then retracting a
paper of mine is that it is so huge that mathematicians who are living
in a political society today--where their word is more important than
their research--are fighting a war to deny acceptance of any of it.

If any piece of my research is acknowledged as important from my
definition of mathematical proof to my ideas about finding primes then
they have to fear that the world will realize what they are doing, so
the math wars as I call them are political ones.

It is a fight of group power against mathematical truth.


James Harris


There is nothing that says the mathematical community has to acknowledge
anyone's proof AFAIK. Nobody cares about your "research" for many obvious
reasons, one of those being that you're a notorious crank.

Dave
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David Moran
science forum Guru Wannabe


Joined: 13 May 2005
Posts: 252

PostPosted: Sat Jul 15, 2006 2:50 am    Post subject: Re: My research, a roadmap Reply with quote

"David Moran" <dmoran21@cox.net> wrote in message
news:5wYtg.68250$fG3.52730@dukeread09...
Quote:

jstevh@msn.com> wrote in message
news:1152929075.517531.182010@h48g2000cwc.googlegroups.com...
My research speaks for itself.

I have given a definition of mathematical proof:

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

Figured out the key properties that define rings that are like the ring
of integers:

http://mymath.blogspot.com/2005/03/object-ring.html

Found my own prime counting function, which unlike any other known
relies on summing a partial difference equation, which is also why it
finds primes on its own, unlike any other known:

http://mymath.blogspot.com/2005/06/counting-primes.html

Fighting mathematicians who have done their best to ignore my research
I wrote the first prime counting function article for the Wikipedia,
where my latest version is now found in the history of the current
page:

http://en.wikipedia.org/w/index.php?title=Prime_counting_function&old...

There readers can see my prime counting function in its fully
mathematicized "pure" form, and see how it is a summation, so they can
make the leap to understanding how it relates to a partial differential
equation and an integration.

I had a paper published in a formally peer reviewed mathematical
journal--and then the editors withdrew it after sci.math pressure
against it:

http://www.emis.de/journals/SWJPAM/vol2-03.html

Link is to a site mirror as the electronic journal DIED a few months
later.

That paper covered some pioneering research advancing modular algebra
or the algebra of congruences, extending on the work started by Gauss:

http://mymath.blogspot.com/2005/07/tautological-spaces-factoring.html

Which is a line of attack I used to find a short proof of Fermat's Last
Theorem:

http://mymath.blogspot.com/2006/03/proof-of-fermats-last-theorem.html

But I've even considered problems in logic and set theory, handling
supposed contradictions:

http://mymath.blogspot.com/2005/06/three-valued-logic.html

and

http://mymath.blogspot.com/2005/05/logical-formedness-axioms.html

and

http://mymath.blogspot.com/2005/06/3-logic-more-basics.html

Even some of my minor research is significant, as I talked about a
simple way to find primes using quadratic residues:

http://mymath.blogspot.com/2006/04/method-for-quickly-finding-primes....

The only explanation given the breadth of my research, and dramatic
events like a math journal imploding after publishing then retracting a
paper of mine is that it is so huge that mathematicians who are living
in a political society today--where their word is more important than
their research--are fighting a war to deny acceptance of any of it.

If any piece of my research is acknowledged as important from my
definition of mathematical proof to my ideas about finding primes then
they have to fear that the world will realize what they are doing, so
the math wars as I call them are political ones.

It is a fight of group power against mathematical truth.


James Harris


There is nothing that says the mathematical community has to acknowledge
anyone's proof AFAIK. Nobody cares about your "research" for many obvious
reasons, one of those being that you're a notorious crank.

Dave


I still wonder if you'd object to anything in my paper for my degree in math
over Partial Differential Equations. Surely you'd find something wrong since
I am a dirty lying mathematician, right?

Dave
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Proginoskes
science forum Guru


Joined: 29 Apr 2005
Posts: 2593

PostPosted: Sat Jul 15, 2006 3:07 am    Post subject: Re: JSH: My research, a roadmap Reply with quote

jstevh@msn.com wrote:
Quote:
My research speaks for itself.

I have given a definition of mathematical proof:

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

Too bad you don't know what constitutes a "logical step".

Quote:
Figured out the key properties that define rings that are like the ring
of integers:

http://mymath.blogspot.com/2005/03/object-ring.html

This object ring has two properties:

1. 1 and -1 are the only rationals that are units in the ring.

2. Given a member m of the ring there must exist a non-zero member n
such that mn is an integer, and if mn is not a factor of m, then n
cannot be a unit in the ring.

There is only one ring that satisfies both conditions: the ring Z of
integers.

What follows is an actual proof. JSH may want to take notes.

The first part of property 2. just states that the object ring R must
be a subring of Q (the set of rational numbers). The second part
states:

(2b) If mn is not a factor of m, then n cannot be a unit in R.

Since P implies Q is the same as Not Q implies not P, (2b) is
equivalent to:

(2b') If n is a unit in R, then mn is a factor of m.

and (2b') is seen to be true in any case. [Proof: If n is a unit in R,
there exists an a in R such that an = 1, by definition of a unit. Then
(mn)(a) = m(na) = m,
so mn is a factor of m.]

Thus, we have reduced the definition of the "object ring" to (A) 1 and
-1 are the only rationals that are units in R, and (B) R is a subset of
Q.

Now Z (the set of integers) satisfies this definition, so suppose R is
not Z. Since any ring that contains 1 also contains Z, and R is not Z,
R contains an element a/b, where a,b are integers, b is nonzero, b is
not 1 or -1, and
gcd(a,b) = 1.

Since gcd(a,b) = 1, there exist integers r and s such that
a r + b s = 1, so
(a/b) r + s = (1/b).
Now, a/b is in R, r is in R, and s is in R, and the result of sums and
products of elements of R is in R, so the left-hand side is in R. Hence
1/b is also in R; however, (1/b)*b = 1, and b is in R, which makes
(1/b) a unit. Since b is not 1 or -1, (1/b) can't be 1 or -1. This
contradicts (A).

Hence the only candidate for the "object ring" is Z. End of proof.

Quote:
Found my own prime counting function,

FORMULA

Quote:
which unlike any other known
relies on summing a partial difference equation, which is also why it
finds primes on its own, unlike any other known:

None of the other prime functions I know of require human intervention.

Quote:
http://mymath.blogspot.com/2005/06/counting-primes.html

Fighting mathematicians who have done their best to ignore my research
I wrote the first prime counting function article for the Wikipedia,
where my latest version is now found in the history of the current
page:

http://en.wikipedia.org/w/index.php?title=Prime_counting_function&old...

There readers can see my prime counting function in its fully
mathematicized "pure" form, and see how it is a summation, so they can
make the leap to understanding how it relates to a partial differential
equation and an integration.

I had a paper published in a formally peer reviewed mathematical
journal--and then the editors withdrew it after sci.math pressure
against it:

http://www.emis.de/journals/SWJPAM/vol2-03.html

They withdrew it after they realized it had a mistake. Another paper in
the same issue of that journal (by Plotnikov) also contained a mistake.

BTW, this link shows that the "proof" you wrote doesn't match your
definition of a proof.

Quote:
Link is to a site mirror as the electronic journal DIED a few months
later.

That paper covered some pioneering research advancing modular algebra
or the algebra of congruences, extending on the work started by Gauss:

http://mymath.blogspot.com/2005/07/tautological-spaces-factoring.html

Which is a line of attack I used to find a short proof of Fermat's Last
Theorem:

http://mymath.blogspot.com/2006/03/proof-of-fermats-last-theorem.html

This short proof relies on the existence of the "object ring", whose
existence is doubtable. Thus, if the "object ring" falls, so does the
FLT proof.

Quote:
But I've even considered problems in logic and set theory, handling
supposed contradictions:

http://mymath.blogspot.com/2005/06/three-valued-logic.html

and

http://mymath.blogspot.com/2005/05/logical-formedness-axioms.html

and

http://mymath.blogspot.com/2005/06/3-logic-more-basics.html

Even some of my minor research is significant, as I talked about a
simple way to find primes using quadratic residues:

http://mymath.blogspot.com/2006/04/method-for-quickly-finding-primes....

The only explanation given the breadth of my research,

Just because your research is broad doesn't mean you're a crank. Check
out Archimedes Plutonium's page, http://www.iw.net/~a_plutonium

Quote:
and dramatic
events like a math journal imploding after publishing then retracting a
paper of mine is that it is so huge that mathematicians who are living
in a political society today--where their word is more important than
their research--are fighting a war to deny acceptance of any of it.

If any piece of my research is acknowledged as important from my
definition of mathematical proof to my ideas about finding primes then
they have to fear that the world will realize what they are doing, so
the math wars as I call them are political ones.

And if not, JSH will remain an anonymous crank. The odds are
10^(10^100) to 1 in favor of the crank outcome.

Quote:
It is a fight of group power against mathematical truth.

Yeah, right.

--- Christopher Heckman
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Arturo Magidin
science forum Guru


Joined: 25 Mar 2005
Posts: 1838

PostPosted: Sat Jul 15, 2006 3:29 am    Post subject: Re: JSH: My research, a roadmap Reply with quote

In article <1152932838.606239.104660@m79g2000cwm.googlegroups.com>,
Proginoskes <CCHeckman@gmail.com> wrote:

[.snip.]

Quote:

This object ring has two properties:

1. 1 and -1 are the only rationals that are units in the ring.

2. Given a member m of the ring there must exist a non-zero member n
such that mn is an integer, and if mn is not a factor of m, then n
cannot be a unit in the ring.

There is only one ring that satisfies both conditions: the ring Z of
integers.

What follows is an actual proof.

No, it's not. You goofed on this in sci.math as well. There is no
warrant for asserting that property 2 implies the ring must be a
subring of Q; the ring of algebraic integers satisfies both
conditions, among many rings that do.

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
magidin@math.berkeley.edu
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G.E. Ivey
science forum Guru


Joined: 29 Apr 2005
Posts: 308

PostPosted: Sat Jul 15, 2006 11:34 am    Post subject: Re: JSH: My research, a roadmap Reply with quote

You very first two statements in the first website:

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

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

are wrong. A mathematical proof doesn not "begin with a truth", it begins with an hypothesis. Mathematics does not speak about "truth", it speaks about "validity".
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Frederick Williams
science forum addict


Joined: 19 Nov 2005
Posts: 97

PostPosted: Sat Jul 15, 2006 2:13 pm    Post subject: Re: JSH: My research, a roadmap Reply with quote

jstevh@msn.com wrote:
Quote:

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?

--
Remove "antispam" and ".invalid" for e-mail address.
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jstevh@msn.com
science forum Guru


Joined: 21 Jan 2006
Posts: 951

PostPosted: Sat Jul 15, 2006 10:32 pm    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?


My suggestion is that you go look at any proof, other than a proof by
contradiction, and if you consider its start with the question of
whether or not it is a truth, you will find that it is.

Or, if that sounds too vague, look at any mathematical proof you can
find, and see if it starts with something that is NOT true.

Proofs by contradiction are a special case as they involve showing
there is a proof by assuming something that you think is not true, and
finding a contradiction with that assumption, so a proof by
contradiction is the taking of a conclusion of a proof, and negating it
to find some contradiction.

You still need an underlying proof.

Oh, for those who wonder, it is significant that I gave a nice,
succinct definition of mathematical proof because no one else has.

And if you don't believe me, just go look. Check your dictionaries and
encyclopedias or whatever, and you'll find this bizarre thing that no
one else has a clear, unambiguous or non-contradictory definition of
mathematical proof.

Those disagreeing should feel free to reply in this thread with other
definitions as they see fit.


James Harris
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Bob Terwilliger
science forum addict


Joined: 22 Feb 2006
Posts: 56

PostPosted: Sun Jul 16, 2006 12:44 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 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?



My suggestion is that you go look at any proof, other than a proof by
contradiction, and if you consider its start with the question of
whether or not it is a truth, you will find that it is.

What about proof by contrapositive? This is first year shite dude, check
it out: Instead of directly proving the statement "A implies B", you can
instead prove the logical equivalent "not B implies not A" (this is the
contrapositive of the original statement).

Quote:

Or, if that sounds too vague, look at any mathematical proof you can
find, and see if it starts with something that is NOT true.

See above.

Quote:

Proofs by contradiction are a special case as they involve showing
there is a proof by assuming something that you think is not true, and
finding a contradiction with that assumption, so a proof by
contradiction is the taking of a conclusion of a proof, and negating it
to find some contradiction.

You still need an underlying proof.

You're yammering here!

Quote:

Oh, for those who wonder, it is significant that I gave a nice,
succinct definition of mathematical proof because no one else has.


Ya, if you say so!

Quote:
And if you don't believe me, just go look. Check your dictionaries and
encyclopedias or whatever, and you'll find this bizarre thing that no
one else has a clear, unambiguous or non-contradictory definition of
mathematical proof.

Those disagreeing should feel free to reply in this thread with other
definitions as they see fit.

Sure, I'll reply. Just to recap your "definition" of mathematical proof:

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

It is obvious from this statement that you haven't a clue.

First, who's qualified to say what a "truth" is? For example, compare
Euclidean and non-Euclidean geometry. In Euclidean geometry, the
"parallel postulate" is assumed true, whereas in non-Euclidean geometry,
the "parallel postulate" is assumed false.

Both these geometries are valid mathematical systems. They both prove
theorems starting from their perspective axioms, and in many cases the
theorems between the geometries contradict one another.

For example, in Euclidean geometry, one consequence of the "parallel
postulate" is that the angle sum of any triangle is always 180 degrees.
Yet, in hyperbolic geometry, the angle sum of any triangle is always
less than 180 degrees. Even more, in elliptic geometry, the angle sum of
any triangle is always greater than 180 degrees (I may have mixed up
hyperbolic and elliptic... forgive me).

Who are you to decide which axioms are "truth"? Maybe in your negligible
region of the universe, Euclidean geometry reigns supreme, yet in other
regions, one of the non-Euclidean geometries rules. How can you say for
sure?

So, as is easily seen, your "definition" of mathematical proof is at
best naive (and at worst, pure shite).

The best one can say about "direct" mathematical proof is that it starts
with a set of axioms and proceeds by logical steps to a conclusion. Then
one can claim that the conclusion follows from the axioms, but can't
really claim that the conclusion is true (after all maybe your axioms
aren't true, whatever "true" means).

Can you follow this Jimmy?

Quote:


James Harris
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Frederick Williams
science forum addict


Joined: 19 Nov 2005
Posts: 97

PostPosted: Sun Jul 16, 2006 1:49 am    Post subject: Re: JSH: My research, a roadmap Reply with quote

jstevh@msn.com wrote:
Quote:

Frederick Williams wrote:
jstevh@msn.com wrote:

"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?


My suggestion is that you go look at any proof, other than a proof by
contradiction, and if you consider its start with the question of
whether or not it is a truth, you will find that it is.

Or, if that sounds too vague, look at any mathematical proof you can
find, and see if it starts with something that is NOT true.

My point was this: a proof of Pythagoras's theorem begins with (some of)
the axioms of Euclidean geometry. If we consider the conjunction, C, of
those axioms then it begins with C. In what sense is C _true_? If the
axioms of Euclidean geometry _are_ true in your opinion, why do you
think they are true, and what is the status of hyperbolic geometry?

Quote:
encyclopedias or whatever, and you'll find this bizarre thing that no
one else has a clear, unambiguous or non-contradictory definition of
mathematical proof.

Any book on logic will contain a definition of "proof" in (at least) one
sense. There are of course other senses. As for telling us to look
things up in books, shouldn't you do that yourself first?

I take it that your reference to proof by contradiction is a hint that
you know you're wrong. Whether you know precisely how you're wrong I
doubt.

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Proginoskes
science forum Guru


Joined: 29 Apr 2005
Posts: 2593

PostPosted: Sun Jul 16, 2006 4:21 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 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?


My suggestion is that you go look at any proof, other than a proof by
contradiction, and if you consider its start with the question of
whether or not it is a truth, you will find that it is.

So a proof by contradiction is not a proof by your definition then,
right?

Quote:
Or, if that sounds too vague, look at any mathematical proof you can
find, and see if it starts with something that is NOT true.

Proofs by contradiction are a special case as they involve showing
there is a proof by assuming something that you think is not true, and
finding a contradiction with that assumption, so a proof by
contradiction is the taking of a conclusion of a proof, and negating it
to find some contradiction.

You still need an underlying proof.

Oh, for those who wonder, it is significant that I gave a nice,
succinct definition of mathematical proof because no one else has.

And if you don't believe me, just go look. Check your dictionaries and
encyclopedias or whatever, and you'll find this bizarre thing that no
one else has a clear, unambiguous or non-contradictory definition of
mathematical proof.

You really need to read some books about logic. Douglas Hofstader's
_Godel, Escher, and Bach_ gives such a definition, and is fun to boot.

--- Christopher Heckman
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jstevh@msn.com
science forum Guru


Joined: 21 Jan 2006
Posts: 951

PostPosted: Sun Jul 16, 2006 4:40 am    Post subject: Re: JSH: My research, a roadmap Reply with quote

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

Frederick Williams wrote:
jstevh@msn.com wrote:

"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?


My suggestion is that you go look at any proof, other than a proof by
contradiction, and if you consider its start with the question of
whether or not it is a truth, you will find that it is.

Or, if that sounds too vague, look at any mathematical proof you can
find, and see if it starts with something that is NOT true.

My point was this: a proof of Pythagoras's theorem begins with (some of)
the axioms of Euclidean geometry. If we consider the conjunction, C, of
those axioms then it begins with C. In what sense is C _true_? If the
axioms of Euclidean geometry _are_ true in your opinion, why do you
think they are true, and what is the status of hyperbolic geometry?


Axioms are believed to be truths.

If they are truths then arguments based on them that follow logical
steps to a conclusion have true conclusion and are proofs.

If they are not true then those arguments are not proofs.

Do you get it yet?

Your knowledge of truth or falsehood is irrelevant to truth or
falsehood.

So even if you think something is true that is not, it still is not
true.

So what is the status of hyperbolic geometry?

If arguments in the field begin with truths and proceed by logical
steps then they give conclusions which are true.

The definition is tautological, and is therefore, perfect.

Your belief about whether or not something is true is irrelevant to
whether or not it is actually a proof, as, after all, within a finite
timeframe, you will be dead, will proofs vanish into thin air without
you around to know they are true?

Of course not.

Now then, how do you determine truth?

Quote:
encyclopedias or whatever, and you'll find this bizarre thing that no
one else has a clear, unambiguous or non-contradictory definition of
mathematical proof.

Any book on logic will contain a definition of "proof" in (at least) one
sense. There are of course other senses. As for telling us to look
things up in books, shouldn't you do that yourself first?


Been there, done that, which is why I felt the need to give the
definition.

I noticed that no one copied in a definition of mathematical proof from
some other source.

I suggest someone try.

Go out, look over what's out there, and see if you can find a better
definition.

Quote:
I take it that your reference to proof by contradiction is a hint that
you know you're wrong. Whether you know precisely how you're wrong I
doubt.


I've gone back and forth with myself about proofs by contradiction.

Writing definitions isn't easy (try it if you think I'm wrong).

It took me a while to come up with the definition of mathematical
proof. It's real work.

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

Assuming that the statement that follows is false, then using it as a
truth will lead to a contradiction, proving that it is false.

So then, proofs by contradiction also begin with a truth, where there
is something understood and not always stated that begins every such
proof.

James Harris
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Frederick Williams
science forum addict


Joined: 19 Nov 2005
Posts: 97

PostPosted: Sun Jul 16, 2006 2:19 pm    Post subject: Re: JSH: My research, a roadmap Reply with quote

jstevh@msn.com wrote:

Quote:

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".

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.

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Frederick Williams
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PostPosted: Sun Jul 16, 2006 2:39 pm    Post subject: Re: JSH: My research, a roadmap Reply with quote

jstevh@msn.com wrote:

Quote:
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.

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.

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.

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PostPosted: Sun Jul 16, 2006 2:50 pm    Post subject: Re: JSH: My research, a roadmap Reply with quote

Frederick Williams wrote:
Quote:
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.

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.

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.

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 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?

Quote:
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.

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.


James Harris
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Frederick Williams
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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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