Re: existence of numbers in mathematics #629 Correcting Math
FredJeffries
> Jeffries:
> "I await the outcry about the double standard of Mr Plutonium's being
> allowed to invoke fabulous entities like a walking virus but
> forbidding others from talking about finite numbers larger than
> 10^500"
>
> From context, common sense implies
Common sense? COMMON SENSE?! Examine the context?!
What happened to your PRINCIPLES, man?!
> that AP really intended to say
> that the _ratio_ of the circumference of the universe to the length
> of a virus is less than 10^500, _not_ that viruses can walk. Hence
> no double standard.
>
Seems to me that what he said was that the existence of a walking
virus would make a finite number greater than 10^500 absurd, therefore
it is not the case that there is a finite number greater than 10^500.
Using that logical pattern, why can't we say: the existence of a
finite number greater than 10^500 would make the idea that "all finite
numbers are smaller than 10^500" absurd, therefore it is not the case
that all finite numbers are smaller than 10^500.
Archimedes Plutonium
> "*Prime* *Simplicity*", *Mathematical
> Intelligencer<https://mail.google.com/wiki/Mathematical_Intelligencer>
I was contacted that the above has Archimedes Plutonium's ideas
published in Mathematical Intelligencer.
Whether it lists my name is questionable?
Whether it refers to my work posted to sci.math?
Can someone check on this? thanks
Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
David R Tribble
> > [0] Michael *Hardy* and Catherine Woodgold,
> > "*Prime* *Simplicity*", *Mathematical
> > Intelligencer<https://mail.google.com/wiki/Mathematical_Intelligencer>
>
> I was contacted that the above has Archimedes Plutonium's ideas
> published in Mathematical Intelligencer.
>
> Whether it lists my name is questionable?
> Whether it refers to my work posted to sci.math?
> Can someone check on this? thanks
Search online:
http://www.springerlink.com/content/121223/?k=archimedes+plutonium
David Bernier
>> [0] Michael *Hardy* and Catherine Woodgold,
>> "*Prime* *Simplicity*", *Mathematical
>> Intelligencer<https://mail.google.com/wiki/Mathematical_Intelligencer>
>
>
> I was contacted that the above has Archimedes Plutonium's ideas
> published in Mathematical Intelligencer.
The authors write that Euclid's proof that there is no largest
prime number is a constructive one. They mention several
forms of distortions that misrepresent the proof, for example
those who say that Euclid's proof was a proof by contradiction.
Harold Edwards wrote a letter to the Editors showing his
enthusiasm at having the proof presented as it appears
in the Elements.
Cf.:
< http://www.springerlink.com/content/m0t8727288823ug5/ >
[first page only to non-subscribers].
David Bernier
Archimedes Plutonium
David Bernier wrote:
> Archimedes Plutonium wrote:
> >> [0] Michael *Hardy* and Catherine Woodgold,
> >> "*Prime* *Simplicity*", *Mathematical
> >> Intelligencer<https://mail.google.com/wiki/Mathematical_Intelligencer>
> >
> >
> > I was contacted that the above has Archimedes Plutonium's ideas
> > published in Mathematical Intelligencer.
>
>
> The authors write that Euclid's proof that there is no largest
> prime number is a constructive one. They mention several
> forms of distortions that misrepresent the proof, for example
> those who say that Euclid's proof was a proof by contradiction.
>
> Harold Edwards wrote a letter to the Editors showing his
> enthusiasm at having the proof presented as it appears
> in the Elements.
>
> Cf.:
> < http://www.springerlink.com/content/m0t8727288823ug5/ >
>
>
> [first page only to non-subscribers].
>
> David Bernier
>
Yes, thanks for the link and able to read the first page. It is
obvious to me
that the article is about correcting alot of the mistakes and
misconceptions of the
proof of Euclid's Infinitude of Primes.
I have been posting many of the mistakes since about 1993 to 1994 and
which finally
became an Internet book. So I doubt that the authors of Hardy and
Woodgold had
a correction of Euclid's Infinitude of Primes proof in 1993.
I see this as a thievery of my ideas posted to the Internet from 1993
onwards.
I would not mind at all, if I had been mentioned or listed in the
references of that
Mathematical Intelligencer article. But my name and my work on
Euclid's IP is
nowhere to be seen in that article.
So I am writing Mathematical Intelligencer to have my name inserted as
reference to
that article.
It is not intellectually honest of people in academia who often read
the internet and sci
newsgroups. To see someone building a case and then to pick those
ideas and publish
it in some magazine, years later, and never give credit to the first
origins of the body
of those ideas.
I am writing Mathematical Intelligencer a letter over this issue.
And it is not the first time that "others" have lifted ideas off of my
sci postings and never
bothered to cite reference. I can recall several others:
(1) the Dalai Lama with his book titled "universe in an atom" or words
to that effect.
(2) E.E. Escultura (pardon the spelling) who published a
counterargument over the Wiles's
Fermat's Last Theorem using p-adics for which I had started years
earlier.
(3) Some professor emeritus in California school system published
about human evolution
saying "homo sapiens evolved with stonethrowing where throwing
increased the advantages
of getting food, more mates, and warding off rivals" or words to that
effect. This was published but I had almost the identical words and
ideas published on the sci newsgroups by me many months earlier, I
forgotten now but it may have been a full year earlier.
Anyway, what I am saying is that the science publications are not
taking the sci newsgroups
seriously enough, and thinking that anything appearing in the sci
newsgroups is fair game to
lift and steal without any attribution to its original author.
Of course, I am never going to go chasing down the Dalai Lama, and if
I did, only to say to
him "thank you, and wish him luck on all that he strives for in life."
But I am concerned about these other forms of thievery of intellectual
property of others
who post in the sci newsgroups building up cases and theories and
organized books.
It may have been that Hardy and Woodgold never read sci.math from 1993
to 2010. Or it
maybe that they avidly read sci.math and especially posts by
Archimedes Plutonium on
Euclid's Infinitude of Primes. But the thing is, that the SCI.MATH
newsgroup have a public
date time group of its messages, and that much of that article by
Hardy and Woodgold trespasses all over my program of correcting
Euclid's IP proof.
Mathematical Intelligencer can expect a letter from me.
It is time that these publishers recognize that the Sci newsgroups are
not a free place to
steal the works of others, and that they should be referenced, if
others have written those
ideas and posted them in the newsgroups.
P.S. Escultura politely apologized to me in the sci.math newsgroup,
and I thanked him, or
I hope I did, and all is forgiven. But the Californian on human
evolution, never so much as
opened his mouth about his belated-published- idea.
It appears that the Mathematical Intelligencer article is 2009 and my
internet sci.math
book on Euclid's IP was much earlier than 2009, having started it in
1993.
I do not think enough people and enough attention has been given to
this problem of
old time publishers "stealing from the newsgroups" and not giving
proper credit to the
newsgroup author of those ideas.
And maybe it is one of the reasons that so many of the old time
publishers are going out of
business, not only because they find it hard to make money from their
"business model" but because they are too slow to get out information
and they are often "lifting or stealing information" from others
without giving the true discoverer of those ideas the proper credit.
A huge flaw of old time science publications was that the publishers
could often "steal"
the works of others and publish it under their own names. This is a
feature not often
discussed or looked into, but there was some corruption in the past,
in science where
the publishing houses of science did bad things. By and large, the
science publishers
were honest people, but in a few cases, the thievery of ideas and the
credit to original
authors is enough to have ruined those authors life in science.
And I personally have firsthand knowledge of a case of publishing
thievery, in that a
friend of mine from Stanford had sent a submission to a physics
journal and his submission
was rejected, only to see some of his submission content printed in a
later issue having
the same ideas and math as his own submission, but accredited to a
different author.
So do not tell me, or naively think that science journals are a
bedrock of good honest
behaviour, but rather a place where alot of politics and dishonesty
occurs, along with
waiting for ages to see your article in print.
Boner J. Skidmark
"Archimedes Plutonium" anagram: "Tiresome unlaid chump"
Archimedes Plutonium
publication.
The good is that I was right or correct all along and that Euclid's IP
proof was a
"direct method" not an indirect. And this article is perhaps the first
"in print article"
to point out to muddle headed mathematics professors that they never
really understood
Euclid's IP proof well enough. So that this published article starts a
new momentum
forward to cleanse the math community of their lousy misunderstanding
of Euclid's IP
proof.
The bad news is that Hardy & Moongold and the editors of Mathematical
Intelligencer
chose to lift Archimedes Plutonium's work without reference or
attribution on this subject
and present it into Mathematical Intelligencer as their own original
work without so much as
even a recognition of all the work done by AP on this subject, for the
issue of attribution
is lacking in this article. Maybe the magazine feels that sci
newsgroup posts are unworthy
of referencing and that ideas and posts are free to lift from the sci
newsgroups.
They probably think they can get away with it because all my work on
this subject was posted
to the sci.math newsgroup from 1993 to present day, and many editors
have the sneeky suspicion that whatever appears on the newsgroups is
free to steal as their own thoughts and
research.
I do not know the code of ethics or whether it is a codified law for
science publishers to make
attribution. Whether Hardy and Moongold broke some sort of copyright
law or whether Mathematical Intelligencer broke some legal law of
citing reference. Michael Hardy does have
posts to the sci newsgroups where he is involved with my post and
especially Euclid's Infinitude of Primes proof, so that Michael Hardy
cannot say he "does not know AP and
his posts about Euclid's proof."
Also, in my sci.math posts, I listed many math book authors who got
Euclid's proof all wrong
as to whether it was direct or indirect method. To Hardy and Moongold,
this would have been
a question of constructive versus contradiction whereas I prefered the
terms direct and indirect method, and they list many more
mathematicians who
had Euclid's proof all wrong than I listed. I listed about 30
mathematicians who got
Euclid's IP all wrong, whereas Hardy and Moongold list twice the
number.
And funny how Hardy and Moongold seem to think that the Euclid IP in
"A Mathematician's
Apology, G.H.Hardy, that Moongold & Hardy seem to think his proof was
correct, when in fact it was wrong. This is a fault of Hardy &
Moongold by not listing their
own direct and indirect proofs at the start, rather than the
philosophical spiel.
So Mathematical Intelligencer with Hardy & Moongold have committed a
injustice to
Archimedes Plutonium by writing a article that contains very much the
same ideas
that I had written a decade earlier, and never attributing any credit
to Archimedes Plutonium.
Now many in the science community should be concerned about this, not
because I was
treated unfairly, but because it makes the science newsgroups as a
open arena for thievery
and stealing of ideas by someone with access to a publisher, and for
them to thence claim
original research.
I admit there is alot of trash posts in the newsgroups, but there are
also many gems in the
newsgroups that should not be stolen just because someone with easier
access to a publisher
choses to steal from another.
And I also want to correct Hardy & Moongold, because they did not even
get a correct Euclid
IP proof out with this article of theirs in Mathematical
Intelligencer. Because if Arthur Rubin is
reading the Hardy & Moongold article and since Arthur Rubin is the
watchdog over the Wikipedia article on Euclid's IP and wherein that
article is in grave error, that Arthur Rubin
still does not learn anything to correct his foggy notions of a true
and valid Euclid proof
of Infinitude of Primes. In other words, Arthur Rubin, after reading
Hardy & Moongold, would
not understand his misconceptions of Euclid's proof. And Wikipedia
would still list a invalid
proof of the Infinitude of Primes.
By the way, Wikipedia, to their credit has archived the discussion as
to whether Euclid's proof was direct or indirect.
Hardy and Moongold, hem and haw around with philosophy spiel,
philosophy chitter chatter,
when they should have been focused logically. A pure slab of logic.
Show the two methods in
detail, one next to the other, up front and out in front. Show why
many mix the two and end up
with a invalid argument.
One would think that if you are going to correct the Euclid IP proof,
that you would set aside
a space in the article and give both methods, one after the other, so
that people can see clearly what the two methods are and why they
differ, both the constructive and contradiction
proof. Moongold and Hardy never did this.
Someone well equiped to correct Euclid's IP proof is someone who in
the first paragraph gives
the Direct or Constructive proof, then in the second paragraph gives
the Indirect or Contradiction proof. Then the rest of the article
would be to show the differences between the
two. This is what I have done with my book in sci.math of my book
"Correcting Math".
I do not see the clarity of mind the clarity of logic of Hardy and
Moongold to give both methods of proof. I see only chitter chatter,
philosophical meandering by Hardy and Moongold.
To their credit, Hardy and Moongold do recognize that Euclid's IP is a
direct or constructive proof. But to their discredit, they lifted
these ideas from my work in sci.math and feel they
have no obligation to cite me or list me as reference. To their
discredit, their article is obscure and obfuscates the argument. To
their discredit, they lack a pure slab of logic to know that
the first paragraph is the Constructive proof and the second is the
Contradiction proof and the
discussion from there on out is simply to point out the differences.
Hardy and Moongold did not have that slab of logic to know to give
both proof methods. It would be like buying a book
on rules on all the football games expecting to understand the rules
of both American football and soccer football, yet it never a mentions
soccer.
I doubt that Hardy and Moongold have enough logic to even do both a
contructive and
contradiction Euclid IP correctly,
and I pose this as a challenge to both Hardy and Moongold to post in
sci.math their
two methods of Euclid's IP since they omitted that clarity in their
Mathematical Intelligencer
article.
Please post your attempts of Euclid's IP here.
But I still will write a letter to the editors of Mathematical
Intelligencer, saying basically that
they owe an obligation to the newsgroups that their "writers of
articles" must attribute
work of ideas that are lifted from the sci.math and other sci
newsgroups when lifted. That
just because ideas and work is posted to sci.math, does not mean those
ideas are free
to steal and publish as if they were your own original research.
Cassidy Furlong
my Dick Tracy watch only dysplays one letter
of the book at a time. so, I decided to use it
as a pseudorandom-number generator --
that no-one will ever be able to crack.
how many of us have ever understood a proof of this theorem?... well,
if not, we'll never get p-adic numbers, or AP-didactical ones, either.
thus&so:
well, there's phi of me to one o'you; go figure!
> outnumber the intelligent so, odds are that the first replier to a post is not even dot.dot
--the duke of oil!
Rationale. In addition to political, economic, and mechanical
feasibility, one must consider the environmental consequences of
choosing ethanol over gasoline. In par- ticular, the amount of air
pollution released in the form of CO2 and other green house gases
(GHGs) is a crucial point of interest. In order to model the
difference in ethanol and gasoline emissions, it is necessary to
calculate the final mass of GHGs (in the case where 10% of the
gasoline energy supply has been replaced by ethanol) minus the ini-
tial mass (before the 10% replacement was implemented). If the result
is negative, the 10% ethanol scenario gives off fewer GHGs; if it is
positive, it gives off more.
Assumptions and calculations. Our model is based on the following
assump- tions:
1.
Itisassumedthatnearlyallofthegasolinerequiredfortheproductionofethanol
is used in the farming and harvesting stage, while other energy
sources (i.e., coal)
http://www.maa.org/pubs/cmj47.pdf
http://tarpley.net/online-books/george-bush-the-unauthorized-biography/chapter-8-the-permian-basin-gang/
Bill Dubuque
>
> The bad news is that Hardy & Moongold [7] and the editors of Mathematical
> Intelligencer chose to lift Archimedes Plutonium's work without reference
> or attribution on this subject and present it into Mathematical
> Intelligencer as their own original work without so much as even a
> recognition of all the work done by AP on this subject, for the
> issue of attribution is lacking in this article. Maybe the magazine
> feels that sci newsgroup posts are unworthy of referencing and that
> ideas and posts are free to lift from the sci newsgroups.
In fact it's quite common that authors fail to acknowledge discussions
on electronic forums. This has happened to me many times, e.g. see my
emails below for a striking case regarding my old Wronskian-based proof
of Mason's ABC theorem. Certainly your huge number of posts here on
Euclid's proof have helped make it widely known that Euclid's proof
was not a proof by contradiction and, moreover, that many authors
were not aware of such. A quick google search shows that Michael Hardy
participated in one of these early threads in 1994 [4], so one would
presume that he knew of such discussions here, esp. since he seems
to be a fairly active member since then (using many different email
addresses - which makes it a bit difficult to locate all of his posts).
So it seems a bit strange when Michael Hardy claims in the article
that he first learned Euclid's proof was not by contradiction only in
2007, from Jitse Niesen on the Citizendium web site (presumably [5]).
Also, it seems like a strange coincidence that one of his "students"
thought he discovered a way to turn Euclid's proof into a proof of the
twin-prime conjecture - as you often claimed here. Perhaps Michael's
memory for sources is fuzzy. Or perhaps his editor's did not like
references to newsgroups. In any case I just wanted to let you know
that you are not alone - it's happened to me and probably to many
other frequent posters. Below are said emails on one of my examples:
-------- excerpt of email from 14 Mar 2005 --------
Hi, I just noticed a reference to Noah Snyder's proof of Mason's ABC
theorem in your lecture [0]. I wonder if this is really any different
than the Wronskian viewpoint that I've pointed out since the mid 80's,
which has been mentioned in passing in various places online since at
least '96 e.g. [2] (and [1], one of many MathWorld pages based wholly
on one of my math-fun posts - some uncredited). Do you know how
I might obtain a copy of Snyder's article? (later: see [6])
[...]
Thanks so much for making Synder's paper available. As I suspected
Snyder's proof is essentially the same proof I gave over 20 years ago.
I've mentioned this online in many places, e.g. I wrote in 1996 [2]
to math-fun & sci.math
Mason's abc theorem may be viewed as a very special instance of a
Wronskian estimate: in Lang's proof the corresponding Wronskian
identity is c^3 W(a,b,c) = W(W(a,c),W(b,c)), thus if a,b,c are
linearly dependent then so are W(a,c),W(b,c); the sought bounds
follow upon multiplying the latter dependence relation through by
N0 = r(a) r(b) r(c), where r(x) = x/gcd(x,x').
This is *precisely* what Snyder does in his proof. The first hit on
Googling "Mason's theorem" is the MathWorld page [1], which refers to
my 1996 post [2] which mentions this proof. Thus I'm surprised that
Snyder and his mentors/editors didn't know this.
There are actually much deeper things one can do from this Wronskian
viewpoint - it is a fundamental approximation tool in differential
algebra. In the mid eighties I was working for the Macsyma group on
effective approaches for special functions using tools from differential
algebra, so I was quite familiar with Wronskian tools. Hence it was a
nice confluence of events that Mason's work happened around the same
time, since I immediately recognized the relationship. If I'm lucky
enough to re-obtain my math library I should dig up some of my notes on
this and write a letter to the editor since there are still some things
worthy of mention.
--Bill Dubuque
[0] http://www.fen.bilkent.edu.tr/~franz/ag05/ag-02.pdf
[1] http://mathworld.wolfram.com/MasonsTheorem.html
see below for a snapshot
[2] http://groups.google.com/group/sci.math/msg/4a53c1e94f1705ed
http://google.com/groups?selm=WGD.96Ju...@berne.ai.mit.edu
[4] http://groups.google.com/group/sci.math/msg/2c73f0dab34a188d
[5] http://locke.citizendium.org/wiki/Talk:Prime_number/Archive_2
[6] Noah Snyder, An Alternate Proof of Mason's Theorem,
Elemente der Mathematik, Vol. 55, Issue 3, 2000, pp. 93-94
http://dx.doi.org/10.1007/s000170050074
[7] Michael Hardy and Catherine Woodgold
Prime Simplicity, Math. Intelligencer, v.31 , 4, Dec. 2009, 44-52
http://dx.doi.org/10.1007/s00283-009-9064-8
-------- Mathworld page [1] excerpted from my math-fun post --------
Mason's theorem may be viewed as a very special case of a Wronskian estimate
(Chudnovsky and Chudnovsky 1984). The corresponding Wronskian identity in the
proof by Lang (1993) is c^3 W(a,b,c) = W(W(a,c),W(b,c)) so if a, b, and c
are linearly dependent, then so are W(a,c) and W(b,c). More powerful Wronskian
estimates with applications toward Diophantine approximation of solutions of
linear differential equations may be found in Chudnovsky and Chudnovsky (1984)
and Osgood (1985).
REFERENCES:
Chudnovsky, D. V. and Chudnovsky, G. V. "The Wronskian Formalism for Linear
Differential Equations and Pade Approximations." Adv. Math. 53, 28-54, 1984.
Dubuque, W. "poly FLT, abc theorem, Wronskian formalism [was: Entire
solutions of f^2+g^2=1]." math...@cs.arizona.edu posting, Jul 17, 1996.
Lang, S. "Old and New Conjectured Diophantine Inequalities."
Bull. Amer. Math. Soc. 23, 37-75, 1990.
Lang, S. Algebra, 3rd ed. Reading, MA: Addison-Wesley, 1993.
Mason, R. C. Diophantine Equations over Functions Fields. Cambridge, England:
Cambridge University Press, 1984.
Osgood, C. F. "Sometimes Effective Thue-Siegel-Roth-Schmidt-Nevanlinna Bounds,
or Better." J. Number Th. 21, 347-389, 1985.
Stothers, W. W. "Polynomial Identities and Hauptmodulen."
Quart. J. Math. Oxford Ser. II 32, 349-370, 1981.
Chudnovsky, D. V. and Chudnovsky, G. V. "The Wronskian Formalism for
Linear Differential Equations and Pade Approximations."
Adv. Math. 53, 28-54, 1984
Cassidy Furlong
that words typed aroundhereinat could have been a trigger, but
what will show a neccesity, that any one grokked a theorem
of Arivaderci Petroleum?
> Mason's abc theorem may be viewed as a very special instance of a
> Wronskian estimate: in Lang's proof the corresponding Wronskian
> identity is c^3 W(a,b,c) = W(W(a,c),W(b,c)), thus if a,b,c are
> linearly dependent then so are W(a,c),W(b,c); the sought bounds
> follow upon multiplying the latter dependence relation through by
> N0 = r(a) r(b) r(c), where r(x) = x/gcd(x,x').
thus&so:
are not dilation of time and length (in the direction
of time-travellin' (sik), directly porportional?
thus&so:
how many of us'd ever understood a proof of the unfinity of the
primes?... well,
if not, we'll never get p-adic numbers, or AP-didactical ones,
either. anyway,
p-adics are cool, when subsumed in Galois theory (or vise-versa .-)
Archimedes Plutonium
Thanks for the information Bill. Can you talk about the obligations of
a journal
of science or math, as to checking the references or literature so
that a new submission is
not a trespass over earlier works by other people?
Bill, what I want to know is that a magazine like Nature or Science
has a review
process of new incoming submissions to check to see if the information
has already
been given credit for to others in an earlier time and so these
magazines usually will reject
a submission if the information was in large part covered by another
author in an earlier time, and Bill, would math journals like
Mathematical Intelligencer
be bound to such same credit to author rules?
When I was in an active talk with a science magazine to publish a
feature of the Atom
Totality, I was told that my work went through this scrutiny process
of checking the older
literature to see if mine was original, and was obviously original and
that a check through the science
channels revealed no authors, ever, having dealt with Atom Totality
and Superdeterminism.
So I suspect that such rules of submissions to physics journals being
vetted for earlier work must be a part of math journals vetting rules
procedure? I cannot see that math would have a looser vetting of
submissions.
I can see how editors would say "no electronic references". But in
this case of where
someone takes a huge chunk of my work and publishes it as his/her own
ideas, is rather
a injustice.
Bill, can you discuss what procedures math journals have to protect
earlier authors of the
same ideas, such as your Wronskian?
Can you say something as to whether there are legal issues when a
publisher publishes the same ideas without reference to the original
author?
All that Michael Hardy and Catherine Moongold had to do was to include
Archimedes Plutonium, sci.math posts on Euclid's Infinitude of Primes
1993 to present.
Simply the recognition that much of what Hardy & Moongold are saying
was covered earlier
by Archimedes Plutonium.
And the saddest part of this story, is that the Mathematical
Intelligencer (MI) article does not
have a valid proof of Euclid's Infinitude of Primes showing. That
neither Hardy nor Moongold
could furnish a valid Euclid IP by contradiction. And their offering
of a proof by construction
based on ystein Ore is overabundantly messy, where Stillwell gives a
more lucid and easier
construction proof.
So I doubt that Hardy or Moongold knows what a valid Euclid IP by
contraction is. And they
should be ashamed of themselves and the editors of MI by calling out
so many mathematicians claiming their Euclid proof was in error, yet
Hardy and Moongold never deliver
a valid Euclid proof by contradiction themselves.
In one of the earliest paragraphs of this article, it says words to
the effect "that the Euclid
number, all the primes that exist multiplied together and add 1, is
not necessarly a prime number". By that statement alone from Hardy and
Moongold, it is obvious that neither Hardy nor Moongold can do a valid
Euclid Infinitude of Primes by Contradiction. Because to do a valid
proof by Contradiction, then the Euclid number of P+1 is necessarily
prime. This is something that Hardy and Moongold have yet to learn.
And this is what makes their article
in MI a travesty of Euclid's proof. The only thing Hardy and Moongold
got correct is that
Euclid's proof was a constructive proof.
But for Hardy and Moongold, unable to do a valid Euclid contradiction
proof and then to go
and lambast numerous mathematicians for errors, yet Hardy and Moongold
not even able to
do a valid Euclid contradiction proof is really sad that math
magazines are of such poor quality. Perhaps that is why they picked a
math dropout and an electrical engineering student
to write a article on Euclid.
So, editors of Mathematical Intelligencer, keep up the poor and shoddy
work of publishing
shoddy math.
Archimedes Plutonium
>
> So I doubt that Hardy or Moongold knows what a valid Euclid IP by
> contraction is. And they
Typing too fast, obviously for that should read "contradiction" not
contraction
and my mind is between physics and math so I can see how that error
slip came
about. I corrected in the original with a (sic) sign.
But let me just briefly comment on how a logical article to
Mathematical Intelligencer MI
would have read over this issue.
It would have given the construction proof in the first paragraph,
noting how it increases
set cardinality.
In the second paragraph it would have displayed the proof by
contradiction, and the key
lines in the proof that makes alot of math professors go wrong and
most every one in the
general public go into error is the line that says that the Euclid
Number-- all the primes
in existence multiplied together and add 1 is necessarily prime
itself. In the Contradiction
proof of infinitude of primes, what makes it so difficult is that P+ 1
is necessarily prime
due to the formalism of the logical setup.
Both Hardy and Moongold and even GH Hardy were unable to recognize
that this was the
foul up or the stumbling block of nearly everyone who ventures into a
Euclidean IP by
contradiction. They make the error and then they try to defend
themselves with the silly example of 1+2x3x5x7x11x13 = 59x509. In the
proof by Contradiction that number is necessarily prime because you
assumed the primes were finite and so 59 x 509 is
necessarily prime given your assumption. This is what trips up so many
people, even
Hardy and Moongold and a long list of professors of mathematics.
That in the proof by contradiction the number P+1 is necessarily
prime. And neither Hardy
nor Moongold made that clear. And since they failed to make that
clear, they had no right
to lambast their long list of math professors who made mistakes. It is
true that Euclid's proof
was a constructive proof or direct proof.
But the Mathematical Intelligencer article was far to shoddy and
lacking in logic to make
clear the errors of Euclid's Infinitude of Primes by Contradiction.
And I do not appreciate Mathematical Intelligencer stealing the bulk
of my ideas on Euclid's
Infinitude of Primes proof. Actually, I am glad that alot of magazines
of science and especially
math are going out of business due to people getting information over
the Internet instead. The
old magazine peer review was mostly a clubhouse society that was
biased and prejudicial and very unfair to alot of authors.
The worst case I have seen is where my friend from Stanford submitted
a article only to be
rejected and only to see his ideas and math being included in another
article of a different
author. In other words, the journal stole his work. And my friend
never really recovered from
that incident.
Archimedes Plutonium
Infinitude of Primes proof
(IP) since about 1993 to present, and wrote an entire book on this,
and pointed out the
mistakes that math professors were making on Euclid's IP proof. Now I
had every right
to do this because I gave both methods of Euclid's IP proof. But in a
recent article
to Mathematical Intelligencer (MI) by Michael Hardy and Catherine
Moongold, they do not provide both proof methods, nor do they even
provide their own but refer to others such as
Ore. They lambast many math professors. Now I feel they have no right
to do that because
they never provided their own renditions of both methods. I had the
right to lambast because
I furnished valid proofs both constructive and by contradiction (I
called them direct and indirect).
From reading the MI article, I doubt that Hardy and Moongold could
even do a valid indirect method, since in their article they say that
Euclid's number is not necessarily prime. That tells
me they do not know the valid proof of Euclid's IP indirect. And
obviously the editor of MI
does not know a valid IP indirect proof, or he would have stopped the
publishing of the article.
So I think that the editors of MI, and Hardy and Moongold, have no
right in lambasting their long list of mathematicians who they claim
made errors. I agree with this article that Euclid's IP was a
constructive (direct method) proof.
And I feel that this article is a lifting of my work on this subject
from my posts to sci.math
from 1994 to present. I am politely saying "lifting" but others can
call it a stealing of my work.
But then again, Hardy and Moongold never provide their own renditions
of the proof, either
constructive or contradiction.
And as I remarked from the fact that they say Euclid's number P + 1 is
not necessarily prime
indicates that Hardy, Moongold and the Mathematical Intelligencer
editors are too inept at
producing a valid Euclid IP by contradiction.
Here is my versions of both methods, in long form and short form.
DIRECT Method (constructive method), long-form; Infinitude of Primes
Proof
(1) Definition of prime as a positive integer divisible
only by itself and 1.
(2) Statement: Given any finite collection of primes
2,3,5,7,11, ..,p_n possessing a cardinality n Reason: given
(3) Statement: we find another prime by considering W+1 =(2x3x...xpn)
+1 Reason: can always operate on given numbers
(4) Statement: Either W+1 itself is a prime Reason: numbers are either
unit, composite or prime
(5) Statement: Or else it has a prime factor not equal to any of the
2,3,...,pn
Reason: numbers are either unit, composite or prime
(6) Statement: If W+1 is not prime, we find that prime factor Reason:
We take the square root of W+1 and
we do a prime search through all the primes from 2 to
square-root of W+1 until we find that prime factor which
evenly divides W+1
(7) Statement: Thus the cardinality of every finite set can be
increased. Reason: from steps (3) through (6)
(8) Statement: Since all/any finite cardinality set can be increased
by one more prime, therefore the set of primes is an infinite set.
Reason:
going from the existential logical quantifier to the universal
quantification
INDIRECT (contradiction) Method, Long-form; Infinitude of Primes Proof
(1) Definition of prime as a positive integer divisible
only by itself and 1.
(2) The prime numbers are the numbers 2,3,5,7,11, ..,pn,... of set S
Reason: definition of primes
(3.0) Suppose finite, then 2,3,5, ..,p_n is the complete series set
with p_n the largest prime Reason: this is the supposition step
(3.1) Set S are the only primes that exist Reason: from step (3.0)
(3.2) Form W+1 = (2x3x5x, ..,xpn) + 1. Reason: can always operate and
form a new number
(3.3) Divide W+1 successively by each prime of
2,3,5,7,11,..pn and they all leave a remainder of 1.
Reason: can always operate
(3.4) W+1 is necessarily prime. Reason: definition of prime, step (1).
(3.5) Contradiction Reason: pn was supposed the largest prime yet we
constructed a new prime, W+1, larger than pn
(3.6) Reverse supposition step. Reason (3.5) coupled with (3.0)
(4) Set of primes are infinite Reason: steps (1) through (3.6)
So in words, the Euclid Infinitude of Primes proof, Indirect in short-
form goes like this:
1) Definition of prime
2) Hypothetical assumption, suppose set of primes 2,3,5,7,.. is
finite with P_k the last and final prime
3) Multiply the lot and add 1 (Euclid's number) which I call W+1
4) W+1 is necessarily prime
5) contradiction to P_k as the last and largest prime
6) set of primes is infinite.
And Euclid's IP, Direct or constructive in short-form goes like this:
1) Definition of prime
2) Given any finite set of primes
3) Multiply the lot and add 1 (Euclid's number) which I call W+1
4) Either W+1 is prime or we conduct a prime factor search
5) this new prime increases the set cardinality by one more prime
6) since this operation of increasing set cardinality occurs for any
given
finite set we start with, means the primes are infinite set.
I am still writing MI a letter, telling them to respectfully add a
correction in
a upcoming issue where the Correction lists Archimedes Plutonium as a
reference
to my sci.math postings over this work on Euclid IP proofs for that
Hardy/Moongold article. I have intellectual
property rights of the ideas I posted on Euclid's IP proof and my
ideas that Euclid
was a constructive proof and why mathematicians got it wrong was
posted many
years earlier than ever did Michael Hardy and Catherine Moongold and
the editors
of MI write their article.
Archimedes Plutonium
> "*Prime* *Simplicity*", *Mathematical
> Intelligencer<https://mail.google.com/wiki/
Mathematical_Intelligencer>
First off, let me apologize to Catherine Woodgold, for I mistakenly
somehow
typed in Moongold and that word seemed to have electronically
increased,
and where that came from, I have no idea other than in the
heat of anger of seeing part of my work printed without any
attribution towards
my long hours of work.
In that Mathematical Intelligencer (MI) article, I see now that Hardy/
Woodgold
and MI editors are complaining only about whether mathematicians saw
Euclid's IP
as either direct-constructive or as a indirect-contradiction proof and
were lambasting
those that figured Euclid's IP was indirect. I too, was lambasting all
those who
thought Euclid's IP was indirect.
But the major difference between me and the MI/Hardy/Woodgold, is that
I posted
from 1993 to present that few if any mathematicians had a valid proof
argument for
the indirect Euclid IP. So I lambasted the mathematics community for
two things--
(a) Euclid's IP was direct-constructive and not indirect-contradiction
(b) few if any
mathematician ever did a valid indirect-contradiction method proof.
Whereas MI/Hardy
/Woodgold is lambasting for one thing-- that Euclid's IP is direct
contructive.
So it is on the (b) feature that MI/Hardy/Woodgold are silent about in
their MI article
and for which in that article, one can see that the editors of MI and
Hardy and Woodgold
would not be able to do a valid Euclid Infinitude of Primes proof
indirect-contradiction
based on the fact that in their opening paragraphs MI/Hardy/Woodgold
write this:
--- quoting from Mathematical Intelligencer their article ---
Then he said, ‘‘By our assumption, no
other primes than those exist. This number is therefore not
divisible by any primes. Since it is not divisible by any
primes, it must itself be prime. But that contradicts our initial
assumption that no other primes than p exist.’’
We shall see that this account of what Euclid did in his
famous proof of the infinitude of primes is commonplace
among some (not all) of the best number-theorists and
among a broad cross-section of mathematicians and others,
and that it is historically wrong.
--- end quoting ---
--- quoting MI ---
Only the premise that a set contains all prime numbers could
make one conclude that if a number is not divisible by any
primes in that set, then it is not divisible by any primes.
Only the statement that p is not divisible by any
primes makes anyone conclude that that number ‘‘is therefore
itself prime’’, to quote no less a number-theorist than G. H.
Hardy
--- end quoting MI ---
MI/Hardy/Woodgold are wrong on those ideas quoted. And those quotes
evinces me, at least me, that MI/Hardy/Woodgold could not do a valid
Euclid Infinitude of Primes proof because they could never stomach a
step in which it says "Euclid's number is necessarily a prime number".
You see, the editors of MI, and Hardy and Woodgold have that all
wrong.
The above is actually the valid proof of Euclid IP indirect-
contradiction. Provided
the definition was the first step in the proof which would force
Euclid's number
to be necessarily prime.
That Euclid's number is indeed a necessarily prime number given the
assumption
space ordered up by the reductio ad absurdum method. Under that
assumption,
if 3 and 5 were the only primes in existence, then Euclid's number 16
would be a
prime number in that assumptive space. What the editors of MI and
Hardy and
Woodgold fail to understand is that the only valid contradiction proof
has Euclid's
number a "necessarily prime number".
It is due to this confusion, that many thought that Euclid's IP could
be either constructive
or contradiction. But when you realize that Euclid's number is
necessarily prime in the
contradiction method, then you realize that Euclid's IP proof could
only have been
constructive increasing set cardinality, for Euclid never states that P
+1 was necessarily
prime.
So I should include this MI article with the editors of MI and Hardy/
Woodgold as unable
to deliver a valid contradiction proof of Euclid's Infinitude of
Primes proof. And although
this article does point out one truth-- Euclid's proof was direct
construction. This article
only contributes far more confusion to the issue than what it is worth
to have in
print.
One of the main reasons that there is so much confusion and mistakes
over the
Euclid infinitude of primes proof is because of lack of logic. Lack of
logic to list
both methods, up front in the first paragraph. Why chitter chatter
around with
a metaphorical classroom teacher.
For anyone to clear the confusion would have to be able to submit both
the methods of
proof to contrast one another and that is where MI/Hardy/Woodgold have
failed miserably
and have only added to the confusion.
In as few of words as possible the mistakes made are these when doing
both methods
simultaneously:
(a) most people forget that the definition of primes is the first
statement
(b) set cardinality increase to any finite set is the constructive
method and it
may have a lemma in there about fetching the new prime, but the fact
that you
are increasing set cardinality makes the method, unmistakeably
constructive.
And in fact, no lemmas are needed, but when a confused lad or lassy
enter
a lemma, well, its excess baggage
(c) the most often occurring error in Euclid IP and which has tripped
up most
mathematicians doing it, is that in the contradiction method, Euclid's
number
is necessarily a new prime. Most people trip up on this because they
forget that the
reductio ad absurdum structure demands P+1 to be prime, and in their
tripping up,
they cite goofball examples such a 2x3x5x7x11x13 (+1) has prime
factors. This is
goofball logic because under the assumptive space of reductio ad
absurdum that
those primes were all the primes that existed and hence P+1 is
necessarily prime
no matter what examples you pull out of your hat, for when you assume
that 3 and
5 are the only primes that exist in contradiction method, then 16 is a
new prime in that
space. This is very hard for even top mathematicians, let alone the
general public
or MI editors or Hardy and Woodgold to understand.
The formal LOGIC demands that Euclid's number P+1 is necessarily prime
in
the contradiction method.
But I still will send a letter to MI editors, scolding them for their
lack of attribution
for my 16 years of hard work and postings to sci.math for which that
article lifted
much of their conclusions, although not their mistakes, from my
postings.
I will ask MI editors to include my name with attributions to that
article of Hardy and
Woodgold in a future issue correction page, citing Archimedes
Plutonium's sci.math
postings on the subject of Euclid's Infinitude of Primes proof from
1993 to 2009.
It is about time that magazines and news outlets stop stealing from
the electronic
media without attribution. Sci.math is not a place to freely steal
ideas and pretend as
if they are your own original ideas, and if you see someone making the
same arguments
over whether Euclid IP was direct or indirect and has an earlier date
of claim, you should
not publish an article without attribution.
sttsc...@tesco.net
> no matter what examples you pull out of your hat, for when you assume
> that 3 and
> 5 are the only primes that exist in contradiction method, then 16 is
> a prime
Perhaps you will find a simple argument with strings easier to follow.
If say a,b,c are string "primes" then you can form an infinite number
of strings of the form a, b, c, aaa, ab, aab, aaaacc etc.
But can you order all of them so that no two consecutive strings
share a common "prime".
a,b,c,ab,cc,aab,ccc, bb, etc.
Obviously not. The string "abc" must occur somewhere in the
ordering and any string that follows must share some "prime"
with "abc".
The analogy with natural numbers is obvious.
No two consecutive naturals share a prime divisor.
If you assume there is a finite number
of primes w amd w+1 must share a prime divisor. A contradiction.
Whatever "comes immediately after" abc shares a prime
factor with it whether it is prime or not a,b,c, aaabbc, etc.
In the case of the naturals all primes assumed to exust
precede w, every number after the largest prime is therefore
composite.
porky_...@my-deja.com
wrote:
> I see you are still having problems with numbers and simple logic.
>
Archimedes Plutonium
Archimedes Plutonium wrote:
> > [0] Michael *Hardy* and Catherine Woodgold,
> > "*Prime* *Simplicity*", *Mathematical
> > Intelligencer<https://mail.google.com/wiki/
> Mathematical_Intelligencer>
>
There is a paragraph in that article that riles me. And shows how out
of place are the
authors of the article. There basic overall premiss is true that
Euclid wrote a Constructive
proof, what I called direct method. And an author trying to convince
the world of mathematics
that Euclid wrote a constructive proof, should not be running to Ore
for his book example, but
should be writing out a valid direct method and valid indirect method
and then compare and
eliminate the obviously wrong method.
For this is an exercise in logic, afterall, so the writeup, unlike
what Hardy/Woodgold/MI did
is not a logical writeup, rather a scatterbrained writeup.
The constructive proof in modern terminology which Euclid did not have
was that of set theory
increasing set cardinality by one more prime given any finite set of
primes. So someone with
set theory proof of Euclid IP, not Ore, should have been chosen, but
then again, neither Hardy,
Woodgold, or the MI editors are logicians. When Euclid did his proof,
he was talking about
"more than any assigned multitude" (if I remember the wording
correctly). Anyway, Euclid's
wording is what we recognize as modern day set theory of increasing
set cardinality.
But anyway, if the authors had been capable of writing their own two
proofs of Euclid IP, one
in construction and the other in contradiction. And if they were valid
proofs, they would notice
that the Euclid Number in contradiction method must be necessarily
prime in order for the proof to be valid. Now, nowhere in Euclid's
writing and no translator of ancient Greek has ever
found where Euclid says the P+1 number is necessarily prime, but
rather where Euclid does a
prime factor search. A valid Infinitude of Primes proof requires a
prime factor search in the Constructive method, never in the
Contradiction method because P+1 is automatically prime due to the
structure of the reductio ad absurdum with the definition of prime in
step one and the assumptive step in step two.
So, if Mathematical Intelligencer, had a superb write up of Euclid's
IP and why it is constructive proof, would have had the first
paragraph of a valid direct method proof and the
second paragraph of a valid indirect method proof. And the third
paragraph would simply say
that Euclid wrote a direct method because he was increasing set
cardinality, and impossible
for Euclid to have written a valid proof as reductio ad absurdum since
he never noticed
that P+1 was necessarily prime in that method (provided, of course we
insist that Euclid
gave a valid proof).
So what I have asked all interested people, whether mathematicians or
general public, is
to give both methods, and see for yourself whether you have the
logical mind needed to
deliver two valid proofs.
But what riles me about that article, other than no attribution to all
my work in sci.math
on this topic, is this paragraph by the authors:
--- quoting from Mathematical Intelligencer ---
Any proof that is not by contradiction can be rewritten
as a proof by contradiction in a way that superficially seems
trivial, but that can have quite unexpected consequences:
just prepend to the proof the assumption that the theorem
is false, then. . .
--- end quoting ---
Now I know that Michael Hardy is a statistician, not a logician, and
that Catherine
Woodgold is an electrical engineer and not a logician. But what riles
me is why
the editors of Mathematical Intelligencer did not contact a logician
such as Thomason
of Yale whose book I learned Symbolic Logic whilst in College. My
guess that a
statistician and electrical engineer was given the opportunity to fill
a number of
pages in MI, is because of the sensitive nature of this article. Keep
in mind that
many mathematicians, even living ones, are named as committing an
error of
logic by thinking the Euclid IP was indirect method. So any author is
going to
upset alot of living mathematicians and tarnishing their reputation as
not knowing
better that Euclid's IP was direct and not indirect. So not too many
takers for an
offer to fill pages, lambasting many a mathematician who erred in
Euclid's IP proof.
In that circumstance, one can thus see why a statistician and
electrical engineer
were allowed to write the article.
But let me address that riling paragraph about prepending and any
proof turned into a
proof by contradiction.
Proof by contradiction is vastly more complicated than merely
prepending.
And I know of several proofs in mathematics that have only a direct
method
and never a indirect was found. And I know of several math proofs that
have
only a indirect method but never a direct method found.
The trouble here and what riles me is that Hardy and Woodgold are way
out of their
realm of expertise. By gosh, they could not even do this article with
their own
direct and indirect Euclid IP. And for them to pontificate on "any
direct proof can
be prepended and turned into a indirect proof" is so misleading and so
far out in
left field, that the editors of Mathematical Intelligencer should be
ashamed of themselves.
The Indirect method is very complex and complicated procedure and is
not a simple
prepending. What the authors have in mind is that you can negate a
proof. We establish
as true A, then we negate it as not A, and then the proof ends up
again as A. But that is
not a proof by contradiction. See Thomason, Symbolic Logic, for a
proof by contradiction
is far more involved than any simple prepend.
And it is obvious that in mathematics and physics, that you can never
have both a constructive and contradiction method of proof for every
statement. Proof: the majority
of geometry statements have only a direct method proof. You cannot
prepend in geometry.
So I think that one paragraph is horribly misleading to many young
readers, who will
go away with the idea that every proof in mathematics has both a
contradiction and constructive proof. This subject of whether a proof
has both methods is an open area
of research in mathematics, and is not for some ribald authors in a
ribald article in
a ribald magazine to pontificate over.
Archimedes Plutonium
David Bernier wrote:
> Archimedes Plutonium wrote:
> >> [0] Michael *Hardy* and Catherine Woodgold,
> >> "*Prime* *Simplicity*", *Mathematical
> >> Intelligencer<https://mail.google.com/wiki/Mathematical_Intelligencer>
> >
> >
> > I was contacted that the above has Archimedes Plutonium's ideas
> > published in Mathematical Intelligencer.
>
>
> The authors write that Euclid's proof that there is no largest
> prime number is a constructive one. They mention several
> forms of distortions that misrepresent the proof, for example
> those who say that Euclid's proof was a proof by contradiction.
>
> Harold Edwards wrote a letter to the Editors showing his
> enthusiasm at having the proof presented as it appears
> in the Elements.
>
> Cf.:
> < http://www.springerlink.com/content/m0t8727288823ug5/ >
>
>
> [first page only to non-subscribers].
>
> David Bernier
Funny how they say get your information faster on the Internet. But
when it comes to basics such as a mailing address for Mathematical
Intelligencer MI, one has to run to an actual
hardcopy.
Hard for me to understand Harold Edwards enthusiasm for finally
exposing the truth about
whether Euclid's Infinitude of Primes (IP) proof was direct or
indirect, when according to Wikipedia that Mr. Edwards had
enthusiasm for the fake proof of Fermat's Last Theorem of Wiles,
considering it is a fake proof because Wiles never defines what is the
difference between a finite number and an
infinite number. You see, in mathematics, if you leave the question
unanswered as to when
numbers become infinite numbers and no longer are finite numbers, then
you end up with a
whole gaggle of statements that can never be proven true or false. I
am not talking about
undecidability, but am talking about, simply precision of definition
which is the main job of
mathematics in the first place.
Fermat's Last Theorem has boatloads of numbers that satisfy
a^n + b^n = c^n to any exponent when a number is ambiguous as to
whether it is finite or infinite. The Peano Axioms never well-defines
the difference between a finite number and
an infinite number. Ask anyone what finite number means and then ask
them whether
....33333 is finite. According to Peano axioms and Fermat's Last
Theorem the number
....33333 is as finite as the number 3 because the Peano axioms never
tells us where
finite ends and infinite begins. And this is the reason Goldbach
Conjecture, Fermat's Last Theorem, Twin Primes, Riemann Hypothesis
have no proof, nor will they ever be proven so long as noone
in math gives a **precision definition of finite versus infinite
number**. Some people may
think, oh well AP is just complaining, but then look at geometry where
they do well-define
a finite line as a line segment and a infinite line as a line ray or a
ray infinite in both directions.
If Geometry is wise enough to well define finite line from infinite
line, then why is number
theorists too dumb and too stupid to well define finite number from
infinite number? Is it because, well, too many mathematicians will
have the shame of pie in their face?
Now I did give a precision
definition of finite number versus infinite number and it is the only
way to go on this chore.
I used the king of science-- physics, because physics contains all of
mathematics as a tiny
subset of itself. In physics, numbers give out at 10^500 for integers.
That number is so huge,
that their is nothing in physics of the Planck Units that makes any
physical sense. But mathematicians are really not very bright and not
very smart of a class of people. Mathematicians in large part are
lemmings and parrots who follow fashion trends of any
given century rather than follow Physics and Logic. If you define
Finite as all the numbers
smaller than 10^500, you instantly clear out all of those Number
theory problems unsolved
and unsolvable.
Now it sounds as though I am pretty harsh on mathematicians and on
Harold Edwards,
which according to Wikipedia Mr. Edwards is a founder of the magazine
Mathematical Intelligencer.
But I am not harsh enough, because mathematics has progressed so far,
so fast, but
it cannot even address and correct something as old as Euclid's
Infinitude of Primes proof
when such is under scrutiny. Mathematics is too much of a old man's
clubhouse that
entrenches fake math. And Mr. Edwards is part of that problem itself.
Did Edwards ever write out a Euclid Infinitude of Primes proof in one
of his books? I would
guess he did. And I would further guess that Edwards was never bright
enough to do both the
direct and indirect method of proof. No, I would guess that Edwards,
as a founder of MI,
was not even bright enough to ask some writer to expose both a valid
direct and valid indirect
proof of IP in a MI issue. About the only brightness of Edwards
concerning Euclid's IP is to
get some authors to talk about a statistics of how many thought
Euclid's IP was constructive
or contradiction. A statistical expose of how many mathematicians
voted constructive rather
than contradiction. That is not really much progress but it is snail's
pace progress.
A bright editor, on the other hand would have summoned someone to
write a article exposing what the valid Euclid IP constructive versus
the valid Euclid IP contradiction methods looked like. Edwards was not
bright enough for such a project, because, probably, Edwards was never
able to give a valid Euclid contradiction proof himself.
To give a valid Euclid contradiction method proof means you must say
in the proof that the
Euclid Number is necessarily prime, otherwise your attempt is an
invalid proof. Edwards probably never could see that, and so we have
at best from MI a statistical article, or a roll-call
of how many mathematicians think Euclid IP is constructive "please
raise your hands".
We don't have an Edwards in control of the situation. An Edwards who
wants the "real and whole truth" about Euclid's Infinitude of Primes,
who is not a scaredy-cat about printing a valid direct alongside a
valid indirect.
Why do we not have that? Because obviously, about 90% of all those
mathematicians who
ever wrote down a Euclid IP in book or print form, have a mangled and
garbled mess and invalid proof. Imagine that, 90%
flunking in a Euclid IP proof. That is worse, by far than a freshman
Calculus class on a surprize quiz.
The Internet and newsgroups such as sci.math is helping mathematics by
training our focus
away from magazines and publications, which is a good thing. Because
the major reason that
math is so slow in correcting itself and why math has fakeries such as
Cantor infinities and
Godel nonsense and Wiles FLT, is that fakeries can hide behind
entrenched math publications. The Internet is "open to all" and
admittedly most of the Internet is worthless nonsense, but the small
percentage of the best of the Internet quickens the pace of fake
math being exposed. Fake math that hides behind entrenched journal
writings.
We now have the question of whether Euclid IP is direct or indirect by
this article of
Hardy/Woodgold/MI. But the deeper question as to "could any
mathematician prior to
1993, ever give a valid Euclid IP indirect?" I say noone prior to
1993, realized that
Euclid's Number must necessarily be prime for the proof to be valid.
So, does Edwards have the audacity, the integrity to have a writeup in
Mathematical
Intelligencer exposing the fact, the idea that only when Euclid's
Number is necessarily
prime that you have a valid reductio ad absurdum proof of Infinitude
of Primes? Does Edwards
measure up to the contest of valid indirect method?
Can someone please post the mailing address of Mathematical
Intelligencer.
sttsc...@tesco.net
<plutonium.archime...@gmail.com> wrote:
> David Bernier wrote:
> > Archimedes Plutonium wrote:
> > >> [0] Michael *Hardy* and Catherine Woodgold,
> > >> "*Prime* *Simplicity*", *Mathematical
> To give a valid Euclid contradiction method proof means you must say
> in the proof that the
> Euclid Number is necessarily prime, otherwise your attempt is an
> invalid proof.
I see you still have not grasped a few simple ideas.
1) Every natural greater than 1 has a prime divisor
2) GCD(n,n+1) = 1
3) If there is a last prime then GCD(w,w+1) <> 1
Contradiction.
Therefore, the number of primes is infinite
Presumably even you can see that 1) and 2) are true statements
and that the assumption that primes are finite in number leads
to the contradiction GCD(w,w+1) <> 1
The necessity of "necessary primes" is simply
a delusion on your part
Archimedes Plutonium
> "*Prime* *Simplicity*", *Mathematical
> Intelligencer<https://mail.google.com/wiki/Mathematical_Intelligencer>
I raise a challenge to the editors of Mathematical Intelligencer (MI).
As I said earlier,
I am upset that they lifted 1/2 of my work of thousands of posts to
sci.math from 1993 to present on this topic of Euclid's Infinitude of
Primes
proof in that it was a Direct/Constructive proof and not a Indirect/
Contradiction proof
with no reference attribution.
But the heart of the issue or problem is really not whether Euclid did
a Constructive
or a Contradiction proof. Is it? The real important matter is whether
someone, anyone
can do a valid Constructive alongside a valid Contradiction proof. So
that the demonstrator
does two proofs. And let the bystanders, the audience, the students
and pupils see
what the difference is.
Why does Hardy/Woodgold yakkity yak about direct method, never
displaying a valid
direct alongside a valid indirect? If this were a court of law and a
defense and prosecution
case, why not allow the evidence for both the Direct along with the
Indirect. The trial
would be horribly unfair and slanted, if the Direct showed evidence
but the Indirect
was not allowed to show any evidence.
Reserve an entire issue of MI of some future upcoming issue just
showing the Euclid Infinitude of Primes proof of having at least 6 or
more different people doing two proofs, one direct, the
other indirect of Euclid's IP.
That is not too much to ask, seeing that the above article implies
that scores of mathematicians cannot even tell whether a proof is
constructive or contradiction.
Someone posted that Harold Edwards was enthusiastic about this 2009
article above.
But is Harold Edwards enthusiastic enough to be reasonable and logical
to have
his two versions, direct and indirect printed in MI?
Is Hardy and Woodgold up for the challenge of having in their own
words
publish both direct and indirect method alongside each other of Euclid
IP?
To demonstrate to anyone that a Toyota has different mechanics than a
GM, we need to
have both vehicles in hand and peering at their engines in close
scrutiny.
My challenge is reasonable; my challenge is logical and forthright.
In that article the authors praise Ore for doing a direct method, but
they never
show where Ore does a valid indirect method. Is it that Ore (hopefully
still alive to
take the challenge) cannot do the indirect method validly or is it
that noone asked
Ore to do both methods with a valid proof to compare?
In that article, Hardy/Woodgold excoriate Devlin for claims that P+1
was necessarily prime.
Is Devlin still alive to take the challenge of having both direct and
indirect printed in
a future issue of MI? Because the excoriation of Devlin would turn
into praise, because
in the valid indirect method, P+1 can only be necessarily prime.
So can MI take the challenge? Reserve an entire future edition of
their magazine to
the printing of a select number of mathematicians, (if mine are wanted
would be happy to
oblige). But to reserve space in this future issue that is devoted to
having both a valid
direct and valid indirect Euclid IP.
When we ever run into a mechanics challenge, when people want to know
the difference
of a Toyota versus a GM, or the difference of a direct IP or indirect
IP, we do not bring
in just the Toyota and just talk about the GM.
If we are to have a Mechanics challenge as to whether Toyota prius or
the GM volt
are this and that, then we act rational and logical and commonsense,
in that we get both vehicles and do a thorough analysis of the engines
and body of both. We do not just get the
prius and blabber about the volt. After we examined both, then we can
talk about their
differences.
It is rather funny and ridiculuous that mathematics, the science of
precision, and yet
a article on Euclid's Infinitude of Primes proof lambasting tens even
hundreds even thousands
of mathematicians and others for failing to recognize Euclid did a
"direct method", yet
not once did it occur to the authors or editors of that article, that
they should have
the direct and indirect up front and in plain view, before any
accusations of error
are hurled.
So, a huge can of worms has been opened here with this article,
accusing many a
mathematician that they failed and flunked with error of not knowing
direct from indirect.
But that is only the start of this can of worms. The larger issue is
whether any mathematician
ever understood what a valid Euclid Infinitude of Primes indirect
method looks like? Does
Michael Hardy, or Catherine Woodgold, or Harold Edwards know what a
valid Euclid Indirect
method proof of Infinitude of Primes looks like? We will never know
unless they can
summon themselves into displaying both direct and indirect proofs, one
alongside the other.
So, take the challenge, and see if you really do know the Infinitude
of Primes proof.
Or whether you want to do a lousy job of presenting only the Direct
and then go accusing
others of not knowing the difference between direct and indirect.
spudnik
it take, to collate & interpret your gargantuan proofology?
dood, take heed:
proof requires the use of the two words,
neccesity & sufficiency in ordinary language;
if you can't set that bar & shimmy under or
jump over it, how can any one else be expected to parse your ****?
> So, take the challenge, and see if you really do know the Infinitude
> of Primes proof.
> Or whether you want to do a lousy job of presenting only the Direct
> and then go accusing
> others of not knowing the difference between direct and indirect.
thus&so:
waht if the same guy/ette who was the source d'Eaugate
for Bernward, was also the Vice President,
who purposely set his mattress on fire in the first tower
(second was hit by a 757 filled with fuel for most
of a transcontinental flight, minus the steering loop);
and, so, how many mattresses'd he have'd to set,
to make for a controlled demolition?
well, some of us believe that
he was not just the acting president --
especially since the impeachment of Bill C..
also, what in Heck is a one-ball centrifuge --
doesn't one need two, at the least, for balance?
thus&so:
Hensel's lemma -- yeah, team!...
anyway, what is it called, that one can use "mod p"
on either side of the equation or inequality?
> 2^(p-2) mod p is just 2^-1 mod p == (p+1)/2
--BP's cap&trade plus free beer points on your CO2 creds at ARCO!
http://wlym.com
Archimedes Plutonium
Archimedes Plutonium wrote:
> > [0] Michael *Hardy* and Catherine Woodgold,
> > "*Prime* *Simplicity*", *Mathematical
> > Intelligencer<https://mail.google.com/wiki/Mathematical_Intelligencer>
--- quoting from Mathematical Intelligencer MI of Ore's direct proof
---
page 65 of
Øystein Ore’s book Number Theory and Its History [104]:
Euclid’s proof runs as follows. Let a;b;c;...;k be any
family of prime numbers. Take their product P ¼ ab...k
and add 1. Then P + 1 is either a prime or not a prime.
If it is, we have added another prime to those given. If it
is not, it must be divisible by some prime p. But p cannot
be identical with any of the given prime numbers
a;b;...;k because then it would divide P and also P + 1;
hence it would divide their difference, which is 1, and
this is impossible. Therefore a new prime can always be
found to any given (finite) set of primes.
Within this argument, we do find one small lemma
proved by contradiction.
--- end quoting ---
Now I was thinking that we never really needed any reductio ad
absurdum
anywhere and anyplace in the full Direct Euclid Infinitude of Primes.
Can I
make that so? I think so, for mine own direct method proofs have no
contradiction. And as I wrote them in the early or mid 1990s I used
the
unique prime factorization theorem(UPFAT). I am sure that Euclid was
privy to
UPFAT and that every Direct method uses UPFAT, only we glide past
saying so.
One of the reasons that I like to give "reasons" alongside the
statements is to
keep me on track. Here are my most recent Direct Method in both
shortform and
longform:
Euclid's IP, Direct or constructive in short-form goes like this:
1) Definition of prime
2) Given any finite set of primes
3) Multiply the lot and add 1 (Euclid's number) which I call W+1
4) Either W+1 is prime or we conduct a prime factor search
5) this new prime increases the set cardinality by one more prime
6) since this operation of increasing set cardinality occurs for any
given
finite set we start with, means the primes are infinite set.
Notice in step 4 of shortform, we know that either W+1 or a factor of
W+1 has a prime which is not in the finite list at the start and we
know
this from UPFAT. So UPFAT eliminates the need to do what Ore or even
Euclid did, since when we invoke UPFAT, and we cannot escape from
invoking UPFAT as Euclid invokes UPFAT and Ore invokes UPFAT.
The Unique Prime Factorization theorem tells us that either W+1 is
prime
or has a prime factor. So the only thing remaining is to mechanically
fetch
the new prime and we do that by taking the square root of W+1 and
going through
all the primes less than or equal to the square root of W+1.
Now I can improve the above in that statements 4,5 have the reasons of
Unique
Prime Factorization theorem.
Some where along the long road of writing those proofs I deleted the
UPFAT, only to
see that I now need it to streamline the Direct method.
In the Direct method, no lemma is needed and no lemma having a
argument
of contradiction is needed. In the Direct method, it is all a
constructive method. The so
called lemma of Ore as stipulated by the authors Hardy/Woodgold, was
excess baggage
and that the either P+1 was prime or had a prime factor comes not from
having a reductio
ad absurdum but comes from UPFAT.
I am convinced, only there can never be a proof of this, but convinced
that in Ancient Greek times, and fairly common today in math circles,
is that the expression "suppose false.." is a
expression commonly blurted out and with no bearing to the proof at
hand. Just as is
the expression "absolutely so" or the expression "without a doubt". So
I think what happened
is that Euclid or the translators of Euclid, filled in spaces in the
proof with expressions such
"suppose false.." Expressions that fill in between sentences or
phrases. And that the expression "suppose false.." was not a method
that Euclid was engaged in, but merely some
words that leads him into further steps of the proof.
An expression in modern times of math proofs is the expression "this
further implies..."
Now when seeing the word "implies" in a proof, one can mistakenly
think it is the logical
implication, and it may well be, but often it is just the
mathematician using common words
to push along his argument.
Now any one of us can take any easy test. We can pick up a proof and
see how easy it is
to insert the phrase "suppose not true.." Almost with any math proof
we are reading, we can
insert that phrase and although it does not launch a reductio ad
absurdum lemma at that
point of the proof write-up, it still looks like it fits in that spot
of the proof.
So what I am saying is that in Euclid's proof or in Ore's proof, both
needed to say that
either P+1 is prime or has a prime factor and the reason is UPFAT, but
if you do not say
UPFAT, you can mislead yourself by saying "suppose false, then the 1
between P and P+1
is divisable.
Summary: I am convinced that no lemma of reductio ad absurdum is ever
needed in the
Direct/Constructive proof. I suspect that in Ancient Greek times the
expression
"suppose false.." was an expression of common usage and did not mean
that the
mathematician was launching a reductio ad absurdum.
Archimedes Plutonium
Archimedes Plutonium wrote:
(snipped)
Now here is this proof after it is corrected with the Unique Prime
Factorization
theorem:
>
> DIRECT Method (constructive method), long-form; Infinitude of Primes
> Proof
>
>
> (1) Definition of prime as a positive integer divisible
> only by itself and 1.
>
>
> (2) Statement: Given any finite collection of primes
> 2,3,5,7,11, ..,p_n possessing a cardinality n Reason: given
>
>
> (3) Statement: we find another prime by considering W+1 =(2x3x...xpn)
> +1 Reason: can always operate on given numbers
>
>
> (4) Statement: Either W+1 itself is a prime Reason: Unique Prime
Factorization theorem
>
>
> (5) Statement: Or else it has a prime factor not equal to any of the
> 2,3,...,pn
Reason: Unique Prime Factorization theorem
>
> (6) Statement: If W+1 is not prime, we find that prime factor Reason:
> We take the square root of W+1 and
> we do a prime search through all the primes from 2 to
> square-root of W+1 until we find that prime factor which
> evenly divides W+1
>
>
> (7) Statement: Thus the cardinality of every finite set can be
> increased. Reason: from steps (3) through (6)
>
>
> (8) Statement: Since all/any finite cardinality set can be increased
> by one more prime, therefore the set of primes is an infinite set.
> Reason:
> going from the existential logical quantifier to the universal
> quantification
>
I want to say something further that I noticed and is probably a Lemma
disease
of reducto ad absurdum lemmas.
Notice that Euclid's translated proof appears to have a lemma of
contradiction,
and that Ore seems to have retained that lemma of contradiction.
But I said that the Euclid IP direct method needs no lemma of
contradiction at all
if you plugg in the Unique Prime Factorization Theorem UPFAT.
So what I am wondering is if the world of math has a proliferation or
reproduction of
lemmas of contradiction by all those who forget that there is some
theorem they should
be applying and not be applying a argument of contradiction. So that
if Ore had
realized he was using UPFAT, only he did not use UPFAT, and instead
argued there is
a prime factor with a lemma contradiction.
So that if Ore had realized or recognized he was using UPFAT, just
needed to state that
there exists a prime factor, not because P then P+1 has 1 divisible,
not because of that,
but because UPFAT was invoked and that Ore had not realized he was
using UPFAT. Not
realizing that UPFAT was used, then Ore launched a lemma of
contradiction.
So I am wondering whether a huge number of lemmas by contradictions in
other proofs
are used because the author invoked another theorem but did not
realize it and then launched
a needless lemma. I know alot of math proofs that seem to have strings
and strings of
lemmas. And for every lemma by contradiction, I would hazard to guess
the author invoked an
already established theorem, and did not realize he was using the
theorem and thus created excess baggage of a lemma.
So lemmas by contradiction are needless and heedless contraptions for
which the author should have listed the theorem invoked and kept the
proof as streamlined direct method.
Now if memory serves me, there are some mathematicians who when faced
with a lemma by
contradiction, will stop at that point in the proof and search around
and if they do not find
an existing theorem, will pause in the proof and actually state a
theorem and prove it there,
then picking up the original proof to continue. They do this because
they abhor most proofs, even lemmas
by contradiction.
sttsc...@tesco.net
<plutonium.archime...@gmail.com> wrote:
> Archimedes Plutonium wrote:
> > > [0] Michael *Hardy* and Catherine Woodgold,
> > > "*Prime* *Simplicity*", *Mathematical
> > >
> needed to say that
> either P+1 is prime or has a prime factor and the reason is UPFAT, but
> if you do not say
> UPFAT, you can mislead yourself by saying "suppose false, then the 1
> between P and P+1
> is divisable.
No proof whether direct or indirect will go through unless
you can exclude the possibility that the integer following
w= "some product of primes" or the "product of all primes assumed to
exist" etc. is a unit i.e. has no prime divisors.
The essence of this proof is that no two consecutive integers
share a prime divisor. Whatever w+1 might be, you know it must
have at least one prime divisor that is not a prime divisor of w
otherwise GCD(w,w+1) <> 1.
If w is the product of all primes you have a contradiction, there are
no other primes that could divide w+1 without gcd(w,w+1)<> 1. If w is
some product of primes then w+1 must be divisible by some other prime
or primes.
I am surprised it has taken you 20 years to realize this.
Archimedes Plutonium
never have a
proof of Twin Primes, Perfect Numbers, Goldbach Conjecture, Fermat's
Last Theorem,
Riemann Hypothesis and thousands of Number theory conjectures. Before
I discuss
the reason why, let me just post my latest current up to date
Infinitude of Primes Proof
both direct and indirect methods.
And Euclid's IP, Direct or constructive in short-form goes like this:
1) Definition of prime
2) Given any finite set of primes
3) Multiply the lot and add 1 (Euclid's number) which I call W+1
4) Either W+1 is prime or we conduct a prime factor search
5) this new prime increases the set cardinality by one more prime
6) since this operation of increasing set cardinality occurs for any
given finite set we start with, means the primes are infinite set.
So in words, the Euclid Infinitude of Primes proof, Indirect in
short-
form goes like this:
1) Definition of prime
2) Hypothetical assumption, suppose set of primes 2,3,5,7,.. is
finite with P_k the last and final prime
3) Multiply the lot and add 1 (Euclid's number) which I call W+1
4) W+1 is necessarily prime
5) contradiction to P_k as the last and largest prime
6) set of primes is infinite.
INDIRECT (contradiction) Method, Long-form; Infinitude of Primes Proof
and
the numbering is different to show the reductio ad absurdum structure
as
given by Thomason and Fitch in Symbolic Logic book.
(1) Definition of prime as a positive integer divisible
only by itself and 1.
(2) The prime numbers are the numbers 2,3,5,7,11, ..,pn,... of set S
Reason: definition of primes
(3.0) Suppose finite, then 2,3,5, ..,p_n is the complete series set
with p_n the largest prime Reason: this is the supposition step
(3.1) Set S are the only primes that exist Reason: from step (3.0)
(3.2) Form W+1 = (2x3x5x, ..,xpn) + 1. Reason: can always operate and
form a new number
(3.3) Divide W+1 successively by each prime of
2,3,5,7,11,..pn and they all leave a remainder of 1.
Reason: can always operate
(3.4) W+1 is necessarily prime. Reason: definition of prime, step
(1).
(3.5) Contradiction Reason: pn was supposed the largest prime yet we
constructed a new prime, W+1, larger than pn
(3.6) Reverse supposition step. Reason (3.5) coupled with (3.0)
(4) Set of primes are infinite Reason: steps (1) through (3.6)
Now let me begin a discussion as to why there is a proof of the
Infinitude of Primes
but never going to be a proof of Twin Primes, its immediate cousin.
Nor will there be
a proof of Goldbach, Riemann H., Fermat's Last Theorem, and thousands
of other
math conjectures especially in Number theory.
The reason is very simple and clear. That mathematics community
refuses to well define
with precision what a **finite number is, versus a infinite number**.
So let me try to
explain and possibly prove that we can have a proof of infinitude of
primes but not Twin Primes when mathematics does not well-define
finite versus infinite numbers.
First off, note that the proposition of Infinitude of Primes is
different from the propositions
of Twin Primes, Goldbach, Riemann, FLT, and a difference not of just
content but a difference
of dimension. With Infinitude of Primes we simply seek "more primes
than any assigned multitude". But in Twin Primes, we seek that
condition plus the added condition that P_1
and P_2 is separated by a length of 2 units. In the Goldbach, FLT,
Riemann Hypothesis,
Perfect Numbers, we ask far more than a question of "more than any
assigned multitude."
So the Infinitude of Primes proof is a infinitude proof of what one
can call one dimensional
but the Twin Primes proof is two dimensional.
Now when Mathematics well defines the finite number versus the
infinite number, as what I
did, earlier in this book Correcting Math, I defined finite number as
all those numbers less than
the largest Planck Unit number in Physics, which is 10^500. There are
no more meaningful
and measureable items in physics beyond 10^500. So, if any
mathematician accepts that as
a well-defined finite number versus infinite number, then immediately,
before the ink is dry,
immediately the Twin Primes, Goldbach, Riemann, FLT and every other
Number theory question open but unsolved is solved. The reason that
mathematics does not have a clean house, a polished gleaming floor, a
pristine and beautiful infrastructure and interior that is
spic and span clean throughout, is because mathematicians have never
realized nor bothered
to define Finite Number versus Infinite Number, and as long as they
refuse to do that, they have a dirty and messy house.
Now, let me try to show and explain and possibly prove that you can
prove Infinitude of Primes
with or without a well-defined Finite versus Infinite Number. But the
moment you try to prove
Twin Primes or Goldbach or FLT or Riemann, that there never is or will
be a proof under the
ill-defined Finite Number (or more technically the non-defined finite
number).
To prove that Primes are infinite, all I needed to do under the 10^500
definition is show that
there is one number larger than 10^500 that is prime relative to all
the numbers below it. To
prove that Twin Primes are infinite, here again, all I need do is show
two numbers Q1 and Q2
that are relatively prime to all numbers below them and which are
separated by a length of
2 units.
But in the old system of mathematics where never anyone steps forward
and well defines
Finite Number versus Infinite Number then the question of the
infinitude of Twin Primes becomes a question of whether 9999....99913
versus 9999....99911 are primes.
So is 13, 11 twin primes? Yes of course. Is 913, 911 twin primes? Well
we have to do some
checking. Is 9913, and 9911 twin primes? Again more checking. The
point here is that when
you never well define Finite Number versus Infinite Number and then
pose as a Conjecture
of mathematics that of Twin Primes, then you have a conjecture that is
unprovable, since
you never am able to make clear what numbers your conjecture is
supposed to answer.
The numbers .....9999913 and ......9999911 are sometimes prime and
sometimes not. But
if you well define Finite versus Infinite as 10^500, then you have a
well-defined proposition
of Twin Primes and you can prove it within a hour.
Now getting back to the question as to why Euclid Infinitude of Primes
is able to muster or
manage to glide past the fact that there is a proof, yet
mathematicians never well defined
Finite Number. What allows Euclid IP to be provable under the ill-
defined old math, yet its
cousin conjecture of Twin Primes is never provable?
The answer I believe, although I need to check it out is that of what
I called one dimensional
versus those other conjectures as two or higher dimensions and by
dimensions I mean that
the place-value is what makes Euclid's IP one dimensional whereas Twin
Primes is two dimensional. Earlier in this book I often said, how do I
fix the place value of the number
99999...99997 to tell me if it is prime or composite? If the 9s were
even numbered then
the whole number is divisible by a prime but not if the 9s are an odd
numbered lot.
If you notice in the proofs of Infinitude of Primes, either direct or
indirect, the place-value
of the numbers in question of the proof are never in question. For all
other conjectures that
are unproven such as Twin Primes, Goldbach, FLT, Riemann Hypothesis
and thousands of
Number theory conjectures, all of them, when closely examined require
a place-value checking
in their proof.
This is why, under the old stained and blemished mathematics where
finite-number versus
infinite-number is never defined properly can still yield Euclid's
Infinitude of Primes proof,
but can never enter into a proof of Twin Primes or other conjectures.
Now I think I am able to prove these ideas, mathematically by using
Geometry where
geometry has well-defined a finite-line versus a infinite-line. And we
can see why the
place-value for Algebra would be the analog of infinite line rays or
infinite lines in geometry.
Archimedes Plutonium
Archimedes Plutonium wrote:
(snipped all else)
>
>
> INDIRECT (contradiction) Method, Long-form; Infinitude of Primes Proof
> and
> the numbering is different to show the reductio ad absurdum structure
> as
> given by Thomason and Fitch in Symbolic Logic book.
>
>
> (1) Definition of prime as a positive integer divisible
> only by itself and 1.
>
> (2) The prime numbers are the numbers 2,3,5,7,11, ..,pn,... of set S
> Reason: definition of primes
>
> (3.0) Suppose finite, then 2,3,5, ..,p_n is the complete series set
> with p_n the largest prime Reason: this is the supposition step
>
> (3.1) Set S are the only primes that exist Reason: from step (3.0)
>
> (3.2) Form W+1 = (2x3x5x, ..,xpn) + 1. Reason: can always operate and
> form a new number
>
> (3.3) Divide W+1 successively by each prime of
> 2,3,5,7,11,..pn and they all leave a remainder of 1.
> Reason: can always operate
No, the reason for that is the Unique Prime Factorization theorem
>
> (3.4) W+1 is necessarily prime. Reason: definition of prime, step
> (1).
>
> (3.5) Contradiction Reason: pn was supposed the largest prime yet we
> constructed a new prime, W+1, larger than pn
>
> (3.6) Reverse supposition step. Reason (3.5) coupled with (3.0)
>
> (4) Set of primes are infinite Reason: steps (1) through (3.6)
>
Archimedes Plutonium
sttsc...@tesco.net
<plutonium.archime...@gmail.com> wrote:
> And Euclid's IP, Direct or constructive in short-form goes like this:
> 1) Definition of prime
> 2) Given any finite set of primes
> 3) Multiply the lot and add 1 (Euclid's number) which I call W+1
> 4) Either W+1 is prime or we conduct a prime factor search
How do you know w+1 is prime ? If w+1 has no prime fcators
then a prime factor search will find no primes.
> 5) this new prime increases the set cardinality by one more prime
What new prime ?
> 1) Definition of prime
> 2) Hypothetical assumption, suppose set of primes 2,3,5,7,.. is
> finite with P_k the last and final prime
> 3) Multiply the lot and add 1 (Euclid's number) which I call W+1
> 4) W+1 is necessarily prime
W+1 is not divisible by any prime 2, 3, 6,..pk so must be a unit.
So W+1 is not necessarily prime.
Archimedes Plutonium
Now in the above I eluded to geometry as a way of proving that the
reason
Twin Primes, Goldbach, FLT, Riemann H in the old math could never be
proven
because they forgot to well-define Finite versus Infinite Number, or
simply were
not bright enough to recognize that not all numbers are just finite
numbers but that
some are infinite numbers.
The way geometry is going to help me prove that Euclid IP is provable
yet Twin
Primes is not provable is because line rays in geometry are infinite
along with
infinite-lines, but these two types of lines are far different from
finite-lines which are
line segments. And the Place-Value on numbers for algebra is why Twin
Primes, Goldbach,
FLT and Riemann Hypothesis are unprovable, whilst Euclid's Infinitude
of Primes has a easy proof, for it escapes the Place Value checking.
Archimedes Plutonium
Let me recap what this adventure is plunging into. We notice the old
math system never
defined with precision the difference between a finite number and an
infinite number.
So that 3 is a Peano axiom number as well as 3333....33333. So the
next question was, since
the old math is so poor in definitions, cannot even distinguish
between a finite number and
allows for a proof of
Infinitude of Primes but never any proof of Twin Primes? In other
words, the old math is
messy and ill-defined, so should that have prevented even a Euclid
Infinitude of Primes proof?
The answer is obviously not, in that even though it was messy, it
still allowed for a IP
proof. It even allows for a proof that the counting numbers are
infinite, since you add one more
to the highest number of any finite set.
But the boundary line for the old math of having a ill defined finite-
number versus infinite-number is that of Infinitude of Primes and Twin
Primes. If we define finite number as all
numbers less than 10^500, we have instantly a proof of Twin Primes,
since we find a pair
of twin primes larger than 10^500, and (10^500)+1 and (10^500) +3 are
the first two
candidates.
Now I said the reason that Twin Primes, FLT, Goldbach, Riemann
Hypothesis were different from Infinitude of Primes is that those were
two dimensional asking more than just whether the
primes were infinite. RH asks alot of complex questions of the state
of infinity. Goldbach and
FLT and Twin Primes deal with infinity but also deal with operations
of infinity such as adding in infinity. Whereas Euclid's Infinitude of
Primes is a one dimensional conjecture. And I said that this
complexity deals with the place-value of infinite-numbers.
But I maybe able to give a geometry reason for why Twin primes is
unprovable when finite versus infinite numbers are never defined
properly. In geometry we have finite lines as line segments. Now can
finite lines form to make a infinite-line? It seems that noone has
ever
asked that before. Since all finite lines are line segments, only an
infinite-number of finite
line segments can make an infinite line or infinite line ray. But old
math never defined "infinite
number". So in geometry we can never build a infinite line from that
of finite line segments
unless of course we define infinite-number. If we define the finite
number as less than 10^500
then if we had 10^500 one unit line segments we can put them together
to form an infinite line
or infinite line ray. Likewise, we can demonstrate that the addition
of all line segments of 1 + 2
+ 3 + 4 + . . + 10^500 forms a infinite line ray. We can also prove
that there is an infinitude of
prime segments since we add a prime that is larger than 10^500. We
also can prove an infinitude of Twin Prime segments by just adding a
twin prime larger than 10^500.
So how does Geometry in fact escape the dilemma mess of old math
Number theory that
never well defined finite-number versus infinite-number? How did
Geometry sneek past
Number theory with a well defined finite-line versus infinite-line?
Well most human minds
sense right and wrong with geometry far easier than they do with
algebra and quantities.
We know a line is finite since it has two endpoints. We know a line is
infinite if it has at least
one arrow rather than two endpoints. And no matter how many finite
line segments we add together, we never can turn those line segments
into a infinite line ray, unless we know what
an infinite-number is. If we define infinite number as greater than
10^500, then we can build
a infinite line ray out of finite line segments.
Sorry, this post is too long already, and I am not able to explain why
Twin primes has no proof
yet regular primes has a proof when the old system of math never
defined finite-number from
infinite-number. The geometry talk does not explain the cutoff from
Regular primes to Twin Primes unprovability. I am searching for a
geometry reason why Twin Primes is never provable, rather than the
Place-Value explanation. It may come down that I cannot explain it
without Place Value.
Archimedes Plutonium
Archimedes Plutonium wrote:
(snipped)
>
> Sorry, this post is too long already, and I am not able to explain why
> Twin primes has no proof
> yet regular primes has a proof when the old system of math never
> defined finite-number from
> infinite-number. The geometry talk does not explain the cutoff from
> Regular primes to Twin Primes unprovability. I am searching for a
> geometry reason why Twin Primes is never provable, rather than the
> Place-Value explanation. It may come down that I cannot explain it
> without Place Value.
>
In my last post I noted an interesting theorem that noone in goemetry
or mathematics
proposed or proved.
Theorem: in geometry there never can be constructed a infinite-line
from any number
of finite line segments.
Proof: Since old math does not recognize infinite-numbers, that no
matter how many
finite number of line segments we put together, they still will never
summon into an
infinite-line-ray. However, if a precision definition is given in
geometry or algebra saying
that finite-number means all numbers less than 10^500 and over that is
infinite-numbers.
Well, with that definition we can say that an infinite number of
finite line segments builds
a infinite-line-ray.
Now the problem of Place Value is also solved and I can stop typing in
numbers
like 9999.....99999 or 99999....99997.
Because when finite-number is defined as less than 10^500 and all
numbers above that
are infinite-numbers. Then we can type in infinite numbers with place
values such as
10^505 is an infinite number and has a place value of significance
tied into all the numbers
below it.
So in a sense, I believe, the only way to solve this crisis in
mathematics of well-defining
finite number to infinite-number is to place this boundary (I prefer
the largest number in
Physics) and then this boundary thus gives meaning to Place-Value of
infinity.
What it all does, in the end, is make us understand that infinity
means merely, beyond our
ability to count or measure physical things. Physics stops for us at
infinity. The infinity
in the old math meant someone on a spiritual order or a religious
nonsense order of something eternal, everlasting.
So that when we ask, are the primes infinite, what we really are
asking is whether there are
10^500 set of primes. And when we ask are the Twin Primes infinite, we
are asking if there is
a set of twin primes that has a cardinality equal to 10^500.
And when we ask how many unit line segments does it take to form a
infinite-line-ray, the
answer is 10^500 such unit line segments put together forms a infinite-
line-ray.
P.S. I made a gross error in previous posts by saying the primes and
twin primes are infinite
if just one such is found that is larger than 10^500. Corrected, that
should have read, a set is
infinite if its cardinality is greater than 10^500, for it does no
good to just find one sample beyond the boundary of 10^500. One must
find an entire set containing 10^500 such types
of numbers to call them an infinite set.
Archimedes Plutonium
Archimedes Plutonium wrote:
(snipped)
>
> Theorem: in geometry there never can be constructed a infinite-line
> from any number
> of finite line segments.
Let me add to the above "in old math"
> Proof: Since old math does not recognize infinite-numbers, that no
> matter how many
> finite number of line segments we put together, they still will never
> summon into an
> infinite-line-ray. However, if a precision definition is given in
> geometry or algebra saying
> that finite-number means all numbers less than 10^500 and over that is
> infinite-numbers.
> Well, with that definition we can say that an infinite number of
> finite line segments builds
> a infinite-line-ray.
Let me summarize so far. Old Math never well defined what it means to
be a finite-number
versus an infinite-number. Due to this lack of precision, we get a
whole slew of unsolved and
unsolvable problems such as Twin Primes, Goldbach, FLT, Riemann
Hypothesis and thousands of number theory problems. The question then
arises, why can we still prove Infinitude of Primes with such a system
that lacks precision definitions, yet unable to prove
more complicated statements such as Twin Primes? And the answer seems
to be that at a
raw primitive level of complexity such as Infinitude of Regular Primes
is do-able with no precision on infinite-numbers, but with a tiny bit
more complexity such as Twin Primes conjecture or Perfect Numbers or
Goldbach, that the lack of precision of what is a finite-number versus
an infinite number, the lack of precision catches up and prevents
there to
ever be a proof.
So what is the added complexity thrown into the Twin Primes that just
prevents there ever
being a proof? With regular primes we have a contruction mechanism of
"multiply the lot and
add 1". Now with Twin Primes we can have that mechanism by saying
multiply the lot and
consecutively add 1 and subtract 1 giving two numbers separated by an
interval of 2 units.
So now, why does that not work? It does not work because we can never
separate out the
twin primes from the regular primes. And because, well, the Twin
Primes conjecture maybe
false in that as the twin primes runs out, the quad primes the six-
primes, the 8 primes still keep going and a new category of 2N primes
arises to take the place of the vanished twin
primes. So in turn, as the twin primes then the quad primes then the
six primes vanish, a new
category of 2N primes emerges then the 2N+2 primes then the 2N+4
primes and etc etc.
Now if the Twin Primes is that much more complex of a conjecture, just
imagine for a moment
how much more exceedingly complex is the Riemann Hypothesis versus the
tiny bit more complexity of the Twin Primes over the Euclid Infinitude
of Primes.
However, with this disparity in complexity there is a simple solution
that makes all these unsolved problems disappear. It is to well-define
Finite Number versus Infinite Number
and the only way that can be achieved is to pick a boundary line.
There is no better boundary than to use the largest number in Physics
which is 10^500 as the Coulomb Interactions in an
atom of element 109. That number is so huge that there are no more
physical measurement
at that number, nor its inverse 10^-500.
So now, how does that well-defined precision definition of Finite
Number as less than
10^500 and any number over is infinite, how does that render these
unsolved problems
solveable? Easy, for we have a mechanical proof means of unsolvable
problems. If we can
start tabulating twin-primes and can begin to list a set with
cardinality approaching 10^500
of twin-primes then we conclude twin-primes are infinite. For Goldbach
the same story
of showing that the first 10^500 even numbers are the sums of two
primes. For FLT,
to show that there are no pythagorean triples in the first 10^500
numbers. For Riemann
H. to show that all the first 10^500 primes lie in the 1/2 Real strip.
Now some of those tasks or chores are going to take longer than any
computer could
reach. Because 10^500 is so huge that we need shortcut algorithms to
help us. But as the
computers crunch through say 10^20 of these numbers we get the feeling
that there really
is no stopping the verification up to 10^500.
The trouble with the old math is that it viewed the Universe as some
idealistic Newtonian
absolutist Universe, a sort of Platonic ideals running about. Whereas
the Universe is more of a finite arena of measurement and ability to
count and measure and where Physics is
in control over math and where logic is not Aristotelian straight line
logic but rather dualistic
circular logic.
Archimedes Plutonium
infinite-line but
Algebra or Number theory was ill-defined with its finite-number versus
infinite-number
and that is why mathematics could never prove Twin Primes, Perfect
Numbers,
Goldbach C. , Fermat's Last Theorem, Riemann Hypothesis and thousands
of
number theory conjectures. In this theorem, we show there never can be
constructed a infinite-line in geometry since the other half of
mathematics, the old-math never well defined
infinite-number versus finite-number.
Proof: Since old math does not recognize infinite-numbers, that no
matter how many finite number of line segments we put together, they
still will never
summon into an infinite-line-ray. However, if a precision definition
is given in
all numbers less than 10^500 and 10^500 and beyond are infinite-
numbers.
Well, with that definition we can build an infinite-line-ray in
geometry by
adding together 10^500 units of line-segments of finite line segments
building
an infinite-line-ray. QED
So mathematics at this moment is in an awfully messy and precarious
position. The
Geometry side of mathematics is far more perfect in its definitions
and has its house
in a superb order. The Number or Algebra side of mathematics is
horribly messy, stained
and dirty with never any precision definition of what it means to be a
finite-number
versus a infinite-number. It is the reason why Mathematics has never
been able to prove
the oldest conjecture on record-- Perfect Numbers Conjecture and the
second oldest
conjecture-- Twin Primes. Math, the old math will never prove these
two conjectures nor the
thousands of others such as Riemann Hypothesis, so long as mathematics
is lethargic and
ignoring the definition of finite-number versus infinite-number.
Now everyone is going to carp and complain that, whoa, 10^500 does not
feel like infinity.
But then everyone never thought about Physics all that much. That
there is no physical
existence of anything beyond 10^500 or below 10^-500. There is no time
available of 10^500 seconds. Noone and nothing can count to 10^500.
Our best computers can never deliver a set
of the first 10^500 primes. Our best computers can never verify
Goldbach or FLT or Riemann
Hypothesis out to 10^500.
Beyond 10^500 has no Physical Meaning, and yet, when I say that this
number is excellent
pick as the boundary between finite and infinite, we hear the
screaming and shrieking of
nattering nutter ivory towered professors of mathematics. Those
professors never understood
or learned Physics and all they did was live in a idealistic Platonic
world. They were more
philosophers and religionists rather than being scientists where math
is but the science of
precision. Those professors of mathematics truly belong more in the
psychology department
of the Universities rather than in the science-math departments.
Transfer Principle
<plutonium.archime...@gmail.com> wrote:
> Theorem: In old-math, geometry had well-defined finite-line versus
> infinite-line but Algebra or Number theory was ill-defined with
> its finite-number versus infinite-number and that is why
> mathematics could never prove Twin Primes, Perfect Numbers,
> Goldbach C. , Fermat's Last Theorem, Riemann Hypothesis and thousands
> of number theory conjectures.
I haven't posted in the AP threads in a while, since I usually
avoid the Atom Totality threads. But now that AP has returned
to Correcting Math, I will return to participating.
> However, if a precision definition is given in
> mathematics for geometry or algebra saying that finite-number means
> all numbers less than 10^500 and 10^500 and beyond are infinite-
> numbers.
So obviously, AP has returned to his 10^500-infinity idea.
> It is the reason why Mathematics has never been able to prove
> the oldest conjecture on record-- Perfect Numbers Conjecture and the
> second oldest conjecture-- Twin Primes.
The "Perfect Numbers Conjecture"? Is this the conjecture that
all perfect numbers are even, or the conjecture that there
exist infinitely many even perfect numbers? I would guess the
latter, since I fail to see how the former has anything to do
with the 10^500-infinity theory, and the latter is analogous
to the Twin Primes Conjecture.
Actually, I take that back. I suppose that one could include
10^500 by stating, "if no odd perfect number less than 10^500
exists, then no odd perfect number exists." As of now, it's
proved that no odd perfect number less than 10^300 exists, so
we still have 200 orders of magnitude left to go before we
reach AP's limit.
> Our best computers can never verify Goldbach or FLT or Riemann
> Hypothesis out to 10^500.
AP appears to be saying here that if a conjecture states that
infinitely many natural numbers satisfy some property, then
we only need to check to see whether 10^500 naturals satisfy it
before declaring the conjecture true, and if the conjecture is
that no natural numbers satisfy some property, then we only
need to check to up 10^500.
Obviously, this claim fails in standard theory ("Old Math.") Let
me chew on this for a while...
Archimedes Plutonium
Transfer Principle wrote:
> On Jul 1, 12:56 pm, Archimedes Plutonium
> <plutonium.archime...@gmail.com> wrote:
> > Theorem: In old-math, geometry had well-defined finite-line versus
> > infinite-line but Algebra or Number theory was ill-defined with
> > its finite-number versus infinite-number and that is why
> > mathematics could never prove Twin Primes, Perfect Numbers,
> > Goldbach C. , Fermat's Last Theorem, Riemann Hypothesis and thousands
> > of number theory conjectures.
>
> I haven't posted in the AP threads in a while, since I usually
> avoid the Atom Totality threads. But now that AP has returned
> to Correcting Math, I will return to participating.
>
> > However, if a precision definition is given in
> > mathematics for geometry or algebra saying that finite-number means
> > all numbers less than 10^500 and 10^500 and beyond are infinite-
> > numbers.
>
> So obviously, AP has returned to his 10^500-infinity idea.
>
So now, have you ever questioned yourself as to why you would need any
number larger than 10^500 to frame infinity? A virus, if it could
walk, and walk
across the entire circumference of the Universe would be far less than
10^200
virus steps. Our current Cosmos of best estimate is less than 10^100
meters
circumference. Our smallest lengths are measured in 10^-15 as the
diameter of
the proton, so why would anyone need 10^-500?
What I am saying is, why step into any mathematics class and waffle on
about
everlasting distance as infinity, or nonstop as infinity. When Physics
already gave
out at about 10^200, much less than 10^500.
Trouble with most people that end up studying mathematics, is that
they bury their
heads in sand of complexity, whilst all around them they lose the
commonsense
that a 10 year old has more commonsense.
The axiom in mathematics that one and only one line is parallel to a
given line, drawn
from a point not on the line, is seen by every mathematician as a line
that goes to infinity
and is parallel and never meets the given line. But why in the world
do they all imagine
this parallel line as something that is beyond where Physics ends? Why
do they
think it must be larger than a metric of 10^500? Is it because people
stepped out of
church, before they learned Euclid's Geometry and that Infinity to
them is something of a God's distance, as a parallel line is eternal
and everlasting? When they should have stepped
out of Physics class knowing that there is no physical meaning beyond
10^500.
I am the first mathematician that recognized full well that Physics is
above Math, and that
Math is a tiny part of Physics. That is the biggest problem as to why
mathematicians
still think that infinity is something special, when it is nothing
special. Infinity is where
Physics no longer has measuring or where Physics no longer needs
numbers. Physics
is good up to 10^500 because that is the number of Coulomb
interactions inside element
109. That is infinity for mathematics.
For mathematicians to waffle on about numbers larger than 10^500, when
those numbers
have no physical meaning, is no different than writing a dime fiction
novel and thinking the
story was true.
> > It is the reason why Mathematics has never been able to prove
> > the oldest conjecture on record-- Perfect Numbers Conjecture and the
> > second oldest conjecture-- Twin Primes.
>
> The "Perfect Numbers Conjecture"? Is this the conjecture that
> all perfect numbers are even, or the conjecture that there
> exist infinitely many even perfect numbers? I would guess the
> latter, since I fail to see how the former has anything to do
> with the 10^500-infinity theory, and the latter is analogous
> to the Twin Primes Conjecture.
>
I meant both of those Perfect Number conjectures, and I differ with
you on the
description of the first, for I take it that 1 is the only odd perfect
number conjecture.
So I have to go through all the numbers from 2 to 10^500 to see if any
other odd number
is a perfect number.
> Actually, I take that back. I suppose that one could include
> 10^500 by stating, "if no odd perfect number less than 10^500
> exists, then no odd perfect number exists." As of now, it's
> proved that no odd perfect number less than 10^300 exists, so
> we still have 200 orders of magnitude left to go before we
> reach AP's limit.
>
How far has the Riemann Hypothesis been checked out for? Is it 10^20?
How far has FLT been checked out? Is it 10^10?
How far has Goldbach been checked out? Is it 10^15?
> > Our best computers can never verify Goldbach or FLT or Riemann
> > Hypothesis out to 10^500.
>
> AP appears to be saying here that if a conjecture states that
> infinitely many natural numbers satisfy some property, then
> we only need to check to see whether 10^500 naturals satisfy it
> before declaring the conjecture true, and if the conjecture is
> that no natural numbers satisfy some property, then we only
> need to check to up 10^500.
>
That is correct. Another way of saying it is that the largest
meaningful
number in Physics is the number that is infinity for mathematics. That
the
infinity in mathematics is no larger than the largest number in
Physics.
> Obviously, this claim fails in standard theory ("Old Math.") Let
> me chew on this for a while...
Say, LWalk, can you help me out on public relations? I have used the
term
"old-math" but such a moniker has been used throughout history. So I
need a new
moniker. Should I call it the "ill-defined math", or how about "sloth-
math" since the
community knows they never defined finite number versus infinite
number? Or
how about "rumdummy math" which has that poetic ring and someone can
use it
in a song. Can you help me out on this, because math is supposed to be
the
science of precision but noone seems to care to do their jobs of
defining finite-number
versus infinite-number. Should I call it rumdummy-math? I like the
ring of that.
Transfer Principle
<plutonium.archime...@gmail.com> wrote:
> Transfer Principle wrote:
> > Actually, I take that back. I suppose that one could include
> > 10^500 by stating, "if no odd perfect number less than 10^500
> > exists, then no odd perfect number exists." As of now, it's
> > proved that no odd perfect number less than 10^300 exists, so
> > we still have 200 orders of magnitude left to go before we
> > reach AP's limit.
> How far has the Riemann Hypothesis been checked out for? Is it 10^20?
So far, about 10^13 nontrivial zeros of zeta have been confirmed
to have real part 1/2. The imaginary part of the last known
zero is also within an order of magnitude of 10^13.
> How far has FLT been checked out? Is it 10^10?
FLT has been completely proved. Therefore, it works for all
natural exponents, 10^500 and beyond.
> How far has Goldbach been checked out? Is it 10^15?
Thereabouts.
> Say, LWalk, can you help me out on public relations? I have used the
> term "old-math" but such a moniker has been used throughout history.
> So I need a new moniker. Should I call it the "ill-defined math", or
> how about "sloth-math" since the community knows they never defined
> finite number versus infinite number? Or how about "rumdummy math"
> which has that poetic ring and someone can use it in a song. Can you
> help me out on this, because math is supposed to be the science of
> precision but noone seems to care to do their jobs of defining
> finite-number versus infinite-number. Should I call it rumdummy-
> math? I like the ring of that.
For some reason, I doubt that the the users of "rundummy-math" would
like the ring of that name.
I just prefer to call it "standard math," since, as of today, it is
the standard theory.
Archimedes Plutonium
Transfer Principle wrote:
> On Jul 1, 10:50 pm, Archimedes Plutonium
> <plutonium.archime...@gmail.com> wrote:
> > Transfer Principle wrote:
> > > Actually, I take that back. I suppose that one could include
> > > 10^500 by stating, "if no odd perfect number less than 10^500
> > > exists, then no odd perfect number exists." As of now, it's
> > > proved that no odd perfect number less than 10^300 exists, so
> > > we still have 200 orders of magnitude left to go before we
> > > reach AP's limit.
> > How far has the Riemann Hypothesis been checked out for? Is it 10^20?
>
> So far, about 10^13 nontrivial zeros of zeta have been confirmed
> to have real part 1/2. The imaginary part of the last known
> zero is also within an order of magnitude of 10^13.
>
> > How far has FLT been checked out? Is it 10^10?
>
> FLT has been completely proved. Therefore, it works for all
> natural exponents, 10^500 and beyond.
Disagree. The Wiles allegedry never distinguishes between a finite-
number and
a infinite-number, as a geometer could never get away with geometry
without
precision defining finite-line versus infinite-line. So Wiles never
proved anything.
FLT is still open to proof and the only proof is to show that it is
true for the first
10^500. There are boatloads of counterexamples in p-adics. I doubt
there is any
counterexample in the first 10^500, but I can be surprized real
easily. Peano
in his axioms never distinguished an infinite-number from a finite-
number and
so all the math proofs prior to a precision definition of infinite-
number versus
finite-number are suspect as not a proof at all.
>
> > How far has Goldbach been checked out? Is it 10^15?
>
> Thereabouts.
>
> > Say, LWalk, can you help me out on public relations? I have used the
> > term "old-math" but such a moniker has been used throughout history.
> > So I need a new moniker. Should I call it the "ill-defined math", or
> > how about "sloth-math" since the community knows they never defined
> > finite number versus infinite number? Or how about "rumdummy math"
> > which has that poetic ring and someone can use it in a song. Can you
> > help me out on this, because math is supposed to be the science of
> > precision but noone seems to care to do their jobs of defining
> > finite-number versus infinite-number. Should I call it rumdummy-
> > math? I like the ring of that.
>
> For some reason, I doubt that the the users of "rundummy-math" would
> like the ring of that name.
>
> I just prefer to call it "standard math," since, as of today, it is
> the standard theory.
I do not know which came first, the physicists calling their particle
physics as the
Standard Model or the mathematicians calling their accepted axioms and
theorems
as the "standard theory". I think a better name would be Accepted
theory. Because
in a hundred years from now, what was standard today will be
trashcanned or modified by the
next century.
The Standard theory of mathematics is missing a precision definition
of finite-number
versus infinite-number. That is a huge gaping hole in all of
mathematics and affects even
geometry. Because you cannot build a infinite line ray out of line-
segments.
You need a infinite-number of line-segments to build a infinite line-
ray.
So the Standard theory of mathematics is a flawed system and this is
very evident in that
we have conjectures dating over 2,000 years old-- perfect numbers and
twin primes. What is
blocking their proofs is the lack of defining finite-number versus
infinite-number. The flaw
of lack of a precision definition is what is backlogging most of those
unproven conjectures.
So to call our present day theory of mathematics the Standard theory
should have a subtraction in it.
Standard Theory of Mathematics minus a precision definition of finite-
number versus infinite number.
That is a whole sentence for a moniker and I need something shorter.
Something like Rumdummy-Math.
Or how about Imprecise Math rather than Standard Math.
Would you accept Imprecise-Math as what you call Standard Math, LWalk?
Imprecise-Math does not offend anyone, LWalk, so I much prefer
Rumdummy Math,
because it gets a better reaction by the listener, to understand that
they never defined
finite-number versus infinite-number.
Perhaps it should have been added into the Peano Axioms of a precision
definition of
finite-number? A new axiom stating that finite-number are all those
numbers less than
10^500.
Of course, if that were tacked on, then all of Cantors nonsense would
be recognized as
nonsense. And that even the proof of Godel's undecidables is further
nonsense since
his proof never recognized finite-number versus infinite-number.
So a precision definition of finite-number versus infinite-number
would sweep clean much
of mathematics as we know it. The Calculus would remain the same but
all that nonsense
about continuity would be thrown out, since the world has no
continuity as imagined by
mathematicians. This is another one of those philosophy or religion
ideas that crept into
mathematics. So there would be a little cleaning up in Calculus as for
the limit concept,
but most of Calculus remains untouched as to its usefulness.
So when mathematicians refuse to recognize their mistake of never
precision defining finite-number versus infinite-number, it is not
proper to call theirs the Standard theory but rather the
Imprecise theory.
Jesse F. Hughes
>> Say, LWalk, can you help me out on public relations? I have used the
>> term "old-math" but such a moniker has been used throughout history.
>> So I need a new moniker. Should I call it the "ill-defined math", or
>> how about "sloth-math" since the community knows they never defined
>> finite number versus infinite number? Or how about "rumdummy math"
>> which has that poetic ring and someone can use it in a song. Can you
>> help me out on this, because math is supposed to be the science of
>> precision but noone seems to care to do their jobs of defining
>> finite-number versus infinite-number. Should I call it rumdummy-
>> math? I like the ring of that.
>
> For some reason, I doubt that the the users of "rundummy-math" would
> like the ring of that name.
Yeah, but AP is right: rumdummy math is so lyrical that someone is just
*bound* to use it in a song.
It can be the new math analog to "We Shall Overcome".
--
Jesse F. Hughes
'If you're not making mistakes you're not doing extreme mathematics."
-- James S. Harris, extreme mathematician par excellence
Marshall
> On Jul 1, 12:56 pm, Archimedes Plutonium
>
> <plutonium.archime...@gmail.com> wrote:
> > Theorem: In old-math, geometry had well-defined finite-line versus
> > infinite-line but Algebra or Number theory was ill-defined with
> > its finite-number versus infinite-number and that is why
> > mathematics could never prove Twin Primes, Perfect Numbers,
> > Goldbach C. , Fermat's Last Theorem, Riemann Hypothesis and thousands
> > of number theory conjectures.
>
> I haven't posted in the AP threads in a while, since I usually
> avoid the Atom Totality threads. But now that AP has returned
> to Correcting Math, I will return to participating.
It's a match made in heaven!
Marshall
Archimedes Plutonium
So let me try going from geometry's precision well-defining of finite-
line versus infinite-line
to see if I can well-define Algebra's numbers. I do not expect to make
progress on this
because geometry has three different types of geometry with its
NonEuclidean elliptic
and hyperbolic.
One thing is sure, that if we posted a list of unsolved problems, over
90% come from
Number theory Algebra and less than 10% come from Geometry. This is
witness to the
fact that the concepts over in geometry are pretty much well defined,
especially
finite line and infinite line.
For starters, I defined AP-adics as for example 9999....9997 with
frontview and backview
and infinity in the middle. And Hensel P-adics has .....999999 = -1.
Now in Euclidean plane geometry we have a line-segment, or finite-line
such as
__________ and we can consider it as say 0 to 9 where we have
0_________9
where the 0 and 9 are endpoints, or we can say it is from -3 to 6
such as -3___________6
And in geometry we have a infinite line-ray as ---------> where the
endpoint
could be any number, finite-number since in old-math that was the only
type of
number given in Peano Axioms. And the arrow end of that line-ray is
what?
Well the arrow end cannot be the frontview nor the backview in
9999....9997
of AP-adics, but the arrow can be the dots of the Hensel p-adic ......
99999
but that is sort of messy since that number is -1 afterall.
So none of this really is working out, is it.
So now let me inject the definition that 10^500 is the boundary
between finite
number and infinite number. Now given the infinite line ray of --------
>, does
it make any better sense? Perhaps so, if we stipulate that the
endpoint is
0 and that the arrow is beyond 10^500. Or, the endpoint is 10^-500 and
not
zero. And then what is a infinite line such as <---------->, is it
from (-)10^500
arrow to (+)10^500 arrow? So there is alot of complications and
mischief
here. The translation of infinite line ray and infinite line to a
precision definition
of infinite-number for Algebra is not working smoothly, is it. There
are some
bumps in the road of this adventure.
So in cases like this, we should defer to the king of sciences--
physics. What
would physics say is a precision well defined infinite number? Well
physics
needs negative numbers, especially for electricity and magnetism, so
then
the infinite number should have an arrow if it exceeds either the
positive
10^500 or the negative 10^500. So in this fashion, the Hensel p-adics
do not
qualify as infinite-numbers because ....99999 is -1 and we easily
measure
or count to -1.
The AP-adics fair even worse than the Hensel p-adics.
Again, what the 10^500 boundary does, is show us that it is the only
way for
Number theory to define finite number versus infinite-number and still
tie back
or connect with geometry's definition of finite line versus infinite
line. Any other
system of infinite number such as AP-adics or Hensel p-adics are
artificial
and flawed. For years now I have been harping that the Hensel p-adics
are merely
a narrowed disguised set of restricted Reals, or a subset of Reals
with a
fancy operation. That the Hensel p-adics were not an independent set
of new numbers.
But the AP-adics, although independent new set of numbers, are they
realistic or fiction?
Apparently they are fiction and have no
reality. The number 10^600 is an infinite number when 10^500 is the
boundary
and both numbers have place values, but the AP-adic number
99999.....99997 or 333....4444
has no place value, and never will since 10^500 means infinity and so
the
dots in 99999.....99997 mean 10^500 dots, thus absurdity.
So Hensel p-adics cannot be infinite numbers, nor can AP-adics. Both
those
systems are either fictional sets or as in the case of Hensel p-adics,
a disguised
subset of the Reals.
When we have the boundary between finite and infinite as a number
exceeding
10^500, we have to throw out or discard both the Hensel P-adics and
the AP-adics.
Both those number systems were devised with a misunderstanding of
infinity.
Infinity means going beyond a certain point. Going beyond where
Physics can no
longer measure or count or do anything. If physics is not in the
picture, then it is
infinite. So the arrows in an infinite line ray or an infinite line
mean that there is no
longer any physics going on, or we cannot reach that physics. Infinity
means more of "unknown physics" rather than "endless". The concept of
"endless" is like the concept
of the "present bald king of France" or the "fire breathing dragon".
Endlessness and continuity
were never realities in physics and so are fictions.
Enough said for one post. And note, in this post I rejected mine own
AP-adics, which I
rejected a long time ago when I went with the 10^500. But I never
publically declared the
rejection of AP-adics. Here I declare I have thrown them out as
fiction. And it goes to show
we keep things or ideas, only when there is nothing better to replace
them. And it looks
as though the only infinite-numbers definition that makes sense is to
call numbers a little
larger than 10^500 such as 10^600, but that any other system such as
6666....88888
or .......11111 is just folly.
Archimedes Plutonium
here. The translation of infinite line ray and infinite line to a
precision definition
reality. The number 10^600 is an infinite number when 10^500 is the
boundary
Archimedes Plutonium
Archimedes Plutonium
Archimedes Plutonium wrote:
(big snip)
Well the story above gets very complicated. I was going under the
impression of
Wikipedia about the Unique Prime Factorization Theorem (UPFAT) which
Wikipedia calls
the Fundamental Theorem of Arithmetic.
--- quoting Wikipedia on UPFAT ---
The theorem was practically proved by Euclid (in book 7 of Euclid's
elements, propositions 30 and 32), but the first full and correct
proof is found in the Disquisitiones Arithmeticae by Carl Friedrich
Gauss.
--- end quoting ---
But I was struck by this quote of Weil's book "Number theory", 1984,
page 5: "Even in Euclid,
we fail to find a general statement about the uniqueness of the
factorization of an integer into primes; surely he may have been aware
of it, but all he has is a statement (Eucl.IX.14) about the l.c.m. of
any number of given primes. Finally, the proof for the existence of
infinitely many
primes (Eucl.IX.20).. "
Which brings up an interesting question. That if Euclid really
understood the UPFAT which was
referred to in IX.14, but which may not have been realized its
"uniqueness property", that Euclid could have avoided the lemma of
reductio ad absurdum in his proof of IX.20.
Euclid and Ore, do not need a lemma of contradiction if they simply
said that either P+1
is prime or has a prime factor not on the list and simply justified
that with UPFAT.
So was Euclid fully cognizant of UPFAT? Apparently not, and that
Weil's evaluation seems
to be accurate in that Euclid did not have UPFAT.
So here is probably another squabble about the history of mathematics.
Whether UPFAT
was not in existence until Gauss fully proved it, and only a notion
before Gauss?
If Euclid had been fully aware of Unique Prime Factorization, he would
not have needed that
lemma in Infinitude of Primes proof.
And my other point is also relevant, that it seems as though a direct
proof, if it has
any lemmas of reductio ad absurdum contained within that Direct proof,
is a sign of
weakness of the proof, in that it should be all direct method
throughout. That if there
appears a lemma of contradiction method, means that the author of the
proof is unaware
of a existing theorem that covers the issue at hand. Or in the case of
Euclid, to set
aside the proof and prove the Uniqueness of the Prime Factors.
So it is likely that a lemma of contradiction is most often a sign of
weakness in a proof or a mask for a theorem already in existence or a
theorem that needs to be proven and thus eliminate that lemma.
Archimedes Plutonium
> "*Prime* *Simplicity*", *Mathematical
The above magazine article uses Ore's proof as what Euclid did as a
Direct/Constructive
proof of Infinitude of primes. I went and looked up this book by Ore.
--- quoting from Number Theory and Its History, Oystein Ore, 1948,
page 65 ---
Euclid's proof runs as follows: let a, b, c, . . ., k be any family of
prime numbers. Take their
product P = ab x . . x k and add 1. The P+1 is either a prime or not a
prime. If it is,
we have added another prime to those given. If it is not, it must be
divisible by some prime
p. But p cannot be identical with any of the given prime numbers a,
b, . . ., k because then it
would divide P and also P+1; hence it would divide their difference,
which is 1 and this is impossible. Therefore a new prime can always be
found to any given (finite) set of primes.
--- end quoting Ore ----
I agree that Euclid did a Direct/Constructive proof of Infinitude of
Primes and I agree
the above is a valid proof.
But I have some minor issues with the above. The direct method is
increasing set cardinality
of any given finite set. So Ore begins by calling it a "family" of
prime numbers yet ends
with set theory of "given (finite) set of primes." So why not be
consistent and remove "family".
I understand Ore was trying to be as exacting to Euclid's own proof,
only given modern
day math language, and it is this lemma by contradiction "would divide
P and also P+1"
that concerns me.
I had always thought that Euclid had proved the Unique Prime
Factorization Theorem
(UPFAT), but reading Weil's book page 5, Euclid fell short of UPFAT.
So when Ore
goes into the sentence saying "it must be divisible by some prime p."
And the only
justification for that claim, as far as I can see is UPFAT.
So I guess Weil was correct, in that Euclid was unaware that the
factorization of any
number beyond 1 is a list of unique primes. For if Euclid had UPFAT,
then Euclid and
Ore could have skipped or eliminated this portion of the above: ". .
because then it
would divide P and also P+1; hence it would divide their difference,
which is 1 and this is impossible."
So if Euclid and then Ore writing a translation of the proof, had had
the UPFAT in
Ancient Greek times, then there would not be a need for a lemma of
contradiction.
Euclid and Ore could have said after forming P+1 that either P+1 was
prime or that
P+1 had a prime factor not on the original list, all due to UPFAT.
But I think it becomes more serious than this. I think when Euclid
wrote and Ore writes
"it must be divisible by some prime p." there is no justification for
that claim other than
UPFAT. So I think UPFAT is a necessary theorem in order to do the
Direct/Constructive proof.
So I take my words back, Euclid may have had a flaw that was a flaw
that cannot be
covered up. And if that is true, then Euclid did not have a valid
proof but came exceedingly
close to a valid proof with a missing theorem of unique prime
factorization required to do the
proof.
So I think that UPFAT is necessary to have in both the Direct and
Indirect methods and a valid
end result cannot accrue unless UPFAT is used. There seems to be no
justification for
"it must be divisible by some prime p." unless you know UPFAT and
invoke it at that juncture.
Maybe Weil is wrong and that Euclid really had UPFAT, because Eucl.IX.
14 preceded Eucl.IX.20
quote of Weil's book "Number theory", 1984,
page 5: "Even in Euclid,
we fail to find a general statement about the uniqueness of the
factorization of an integer into primes; surely he may have been
aware
of it, but all he has is a statement (Eucl.IX.14) about the l.c.m.
of
any number of given primes. Finally, the proof for the existence of
infinitely many
primes (Eucl.IX.20).. "
Should we ask the question, can you have done Euclid's book containing
the infinitude
of primes and not know that the numbers have a unique prime
factorization?
Archimedes Plutonium
mistakes of logic
in Euclid's IP proof.
So we have the geometry side of mathematics with a precision
definition of finite-line
versus infinite-lines. Those definitions involve the fact that a
finite-line is a line segment
with two endpoints. The infinite-line involves two types called a
infinite-line-ray which has
a endpoint and a arrow to infinity at the other end. The infinite-line
has no endpoint, but
only two arrows.
So can we actually use the Geometry definition, since it is a
precision definitions, and use
them to formulate what the finite-number versus infinite-number
definitions should be? There
is nothing to say that numbers must be similar to lines. But there
must be some sort of
consistency or coherence between the definitions.
In a prior post I remarked that the AP-adics and the Hensel p-adics
could not be the infinite-number definition.
So what I must do is analyze how the Reals behave once we inject the
notion that
the boundary between finite number and infinite number is 10^500 due
to Physics.
Now is there any hint from geometry alone that there is a boundary
between finite-line and
infinite-line? Not that I can discern. Perhaps the idea that all
finite-lines have two endpoints, and these two endpoints somehow force
a boundary between finite lines and infinite lines.
But that does not look to be true. Then, the only other thing I can
inspect is the idea that
Geometry can never build a infinite-line from finite-lines unless
there exists an infinite-number.
That was the theorem I proved a few posts back.
So here we have a glimpse that we cannot use geometry line definitions
as a template
to defining finite number versus infinite-number since the infinite-
number is essential to
geometry, and geometry dependent on having infinite-numbers.
So what do the Reals and the Cartesian coordinate system look like
once a axiom
is injected into math, into the Peano axioms stating that there is a
boundary between
finite-number versus infinite-number and it is exactly 10^500.
Well, the Reals will no longer have continuity as a concept because
anything smaller
than 10^-500 leaves holes. And the graphs no longer need anything
beyond 10^500.
So starting at zero as a endpoint and then placing an arrow to this
line at 10^500
is that an infinite-line-ray? And having an arrow at (-)10^500 and
another arrow at
(+)10^500 is that an infinite-line?
There are vexing questions such as whether a infinite line ray is
formed by having
0 as end point and employing all the micronumbers between 0 and 1 such
as
10^-500, so do we have micronumbers as infinite? In modern math we say
call this
the limit for calculus but we do not call it a micro infinity. I am of
the opinion that
infinity means merely beyond the ability to do physics measurements so
that a
micro infinity makes just as much sense as a large scale infinity.
Like anything new, there are vexing new issues and questions. But look
at the benefit
of this new program. We no longer have unsolved problems anywhere in
mathematics,
nor do we ever have to search for a new proving techniques, because
there is now one
standard way of proving all Algebra or Number theory problems, run
through 10^500 of
those numbers.
And, unless I am mistaken, this new program vastly improves Calculus,
since in calculus,
the experts of that field spent most of their time on issues of
continuity. With this program,
continuity was never a mathematical concept, just as fire breathing
dragons are not a part
of biology.
FredJeffries
<plutonium.archime...@gmail.com> wrote:
> Transfer Principle wrote:
>
> > So obviously, AP has returned to his 10^500-infinity idea.
>
> So now, have you ever questioned yourself as to why you would need any
> number larger than 10^500 to frame infinity? A virus, if it could
> walk, and walk
> across the entire circumference of the Universe would be far less than
> 10^200
> virus steps.
I await the outcry about the double standard of Mr Plutonium's being
allowed to invoke fabulous entities like a walking virus but
forbidding others from talking about finite numbers larger than 10^500
Archimedes Plutonium
Geometry has precision definitions of finite-line versus infinite-line
and they have
two types of infinite-line, the line-ray with one arrow and the line
with two arrows.
But Geometry does not imply or hint or suggest that there must be a
boundary
between finite line and infinite line, unless we realize that you
cannot build a
infinite-line ray without using infinite-number. And you cannot define
infinite-number
unless you select from physics a number where there is no longer any
Physics
meaning in the Universe. That involves the Planck Units and 10^500 has
no more
physics measuring allowed.
So does Geometry offer a template for Peano axioms and Algebra to
define infinite-number?
Yes it does. For if you examine the Peano Axioms and insert an
additional new axiom
of this:
t t0 are finite
and any Natural Number of 10^500 and larger is an infinite-number.
So now, let us look at how Geometry lines are template, that means
they are similar
in meaning and in building, are the template of Numbers in Algebra.
The two numbers 0 and 1 as endpoints, in Peano axioms would be a line
segment 0 to 1. The two numbers 1 and 10^500 as endpoints forms a line-
segment also, a finite-line, since it is less than 10^500. However the
two numbers as endpoints 0 and 10^500 forms not a line-segment but an
infinite-line-ray since it has a length equal to or greater than
10^500. Now the
two numbers of 5 and ((10^500) +3) forms a line segment because the
length is less than
10^500.
So we see here that Geometry serves as a template of how to precision
define finite-number
versus infinite-number.
Now in the organic building of the Peano Natural Numbers to the
Rationals then to the
Reals, in that building process we included the negative numbers and
with their inclusion
we use the Geometry template to correspond with the infinite-line with
its two arrows. So that
an infinite-line corresponds with (-)10^500 to that of (+)10^500.
I am not familar with the actual history of mathematics and geometry
in particular and this
information is probably lost in ancient times. The information as to
whether we first
discovered, and obviously true the finite-line as a line segment. And
then the next line
discovered would have been, according to this post, the next line
discovered would have
been a infinite-line-ray with its single arrow in a direction. The
discovery of a infinite-line
with two arrows would have been the last discovery of lines in
geometry. History probably
lost that information, but that is what makes common sense. To think
that the progression
of discovery of different lines in geometry went from finite line to
that of infinite-line of two
arrows suggests that we would have known 0 exists as a number and that
the negative numbers existed in the very early development of
mathematics, yet we know as a fact that
0 took a long time to be recognized and understood and even longer for
the negative numbers.
So that not until we had infinite-line-ray long time established would
we ever have the
double arrowed infinite-line discovered. This would have been
centuries before the
Euclid parallel postulate where a infinite-line double arrowed was
required. Even then, the
infinite-line-ray would have been sufficient.
So far, I have only been able to ascertain that a boundary between
finite and infinite
is necessary in Algebra and Number theory in order to build a infinite-
line in geometry
from that of finite-lines. So far I needed both Algebra and Geometry
together to
assert that a boundary must exist.
But can I find other areas of mathematics that require this boundary
between finite and infinite?
Another example that could also demand that Algebra and Geometry must
have this
boundary between finite and infinite. It is the old paradox of the
turtle and rabbit race, called
Zeno's paradox, that the turtle is given a head start lead and
according to the paradox the turtle wins the race. The explanation for
it is of course Physics that the rabbit wins due
to the concept of speed. But when mathematicians try explaining this
paradox with the infinities of small distances, it is never a
satisfactory explanation.
But now, let us inject into that
explanation in the turtle rabbit race there are only four distances of
1, 2, 3 and 4, where the
turtle starts the race on 2 and only has to reach 4, and where the
rabbit is on 1 and has to
reach 4 to win. So that by the time that the turtle reaches 3, the
rabbit is already on 4. Here
we begin to see that if mathematics has no boundaries between finite
and infinite such as
the boundary of 10^-500 where there are nothing but holes and gaps
between numbers this small or smaller, then we begin to realize that
Zeno's paradox is a reflection of the fact that
mathematics has no absolute continuity.
What I am trying to say is that Geometry really does have a boundary
between finite and
infinite otherwise the Zeno rabbit and turtle race would not be a
paradox. Here I am talking
of micro infinity 10^-500, not the macro infinity of 10^500. So is not
the Zeno paradox really
about the idea that there must exist a boundary between finite and
infinite.
Summary: in the macro world, we must have a boundary between finite
and infinite in order
for geometry to be able to construct a infinite line from finite
lines, otherwise no amount of
finite lines could ever be a infinite line. In the micro world there
must also be a boundary
between finite and infinite otherwise the turtle truly wins all the
Zeno races.
Archimedes Plutonium
Geometry has precision definitions of finite-line versus infinite-line
and they have
two types of infinite-line, the line-ray with one arrow and the line
with two arrows.
But Geometry does not imply or hint or suggest that there must be a
boundary
between finite line and infinite line, unless we realize that you
cannot build a
infinite-line ray without using infinite-number. And you cannot define
infinite-number
unless you select from physics a number where there is no longer any
Physics
meaning in the Universe. That involves the Planck Units and 10^500 has
no more
physics measuring allowed.
So does Geometry offer a template for Peano axioms and Algebra to
define infinite-number?
Yes it does. For if you examine the Peano Axioms and insert an
additional new axiom
of this:
Axiom: numbers are finite if less than 10^500
So far, I have only been able to ascertain that a boundary between
finite and infinite
What I am trying to say is that Geometry really does have a boundary
between finite and
infinite otherwise the Zeno rabbit and turtle race would not be a
paradox. Here I am talking
of micro infinity 10^-500, not the macro infinity of 10^500. So is not
the Zeno paradox really
about the idea that there must exist a boundary between finite and
infinite.
Summary: in the macro world, we must have a boundary between finite
and infinite in order
for geometry to be able to construct a infinite line from finite
lines, otherwise no amount of
finite lines could ever be a infinite line. In the micro world there
must also be a boundary
between finite and infinite otherwise the turtle truly wins all the
Zeno races.
Archimedes Plutonium
sttsc...@tesco.net
<plutonium.archime...@gmail.com> wrote:
> > [0] Michael *Hardy* and Catherine Woodgold,
> > "*Prime* *Simplicity*", *Mathematical
> Should we ask the question, can you have done Euclid's book containing
> the infinitude
> of primes and not know that the numbers have a unique prime
> factorization?
This must have been pointed out to you hundreds of times ovet the
years.
If there are naturals >1 that have no prime divisors, you
cannot conclude that w+1 has prime divisors other than those of
w and so the prime divisors of w might be all the prime
divisors there are.
You don't even need uniqueness.
The fact that every N>1 has at least one prime
divisor is sufficient.
Maybe you have not realized that w and w+1 are consecutive integrs
with a greatest common measure of 1.
1 < w <w+1, if any prime p divides w then p does not divide w+1
If every prime that exists divides w then no prime divides w+1. This
contradicts the fact that every n>1 has a prime divisor.
Let 2 ,3, 5 are the only primes that exist
2, 3 and 5 divide 120. None of these primes divide
120+1 = 121 = 11*11 (or 120-1 =119=17*7). This a contradiction.
As every n>1 has a prime divisor and the only prime divors
assumed to exist are 2,3,5 at least one of them must divide
121 or 119. None does.
Archimedes Plutonium
numbers, yet with
defining infinity as starting with 10^500 really does not stir up the
mathematicians like
a hornet's nest. Sure a few will come out of their hive or clam shell
and take a peek at
what is going on, but then right back to the ivory tower sipping more
coffee.
But then when they hear that 10^500 as the boundary between finite and
infinite
causes there to be no more absolute continuity of Reals less than
10^-500, well, that
is a hornet's nest of mathematicians buzzing around all over the
place.
Funny how talk of large infinities is easily ignored as idle talk, but
once you talk about
small infinities, then you are treading over limit concept and
convergence and even the
definition of Reals as Dedekind cuts.
Now the proof I gave a few days ago that you must have the boundary
between finite
and infinite-number and 10^500 is a reasonable choice, the proof that
you cannot build
a infinite line in geometry from any amount of finite lines unless
there was a boundary
to denote infinite-numbers. So you are stuck in geometry when Algebra
never defines
infinite-number versus finite-number. Stuck in not being able to
construct an infinite-line.
But much earlier, I forgotten if 2009, I showed where absolute
continuity in the small scale
of between two consecutive integers, that if absolute continuity
existed, then one can
construct a triangle that has two angles of 90 degrees, and thus more
than 180 degrees.
Another supporting evidence that you cannot have absolute continuity
comes from Quantum
Mechanics of physics were energy, distance and time are quantized,
meaning whole multiples
with emptiness in between. Quantum Mechanics means gaps or holes in
between. So the pursuit of absolute continuity in mathematics for
centuries or milleniums, or for how long exactly has mathematics been
concerned over continuity, seems rather funny and misplaced
effort, because Quantum Mechanics is all about discreteness and thus
holes and gas
in between.
So it looks as though I have now two proofs that a boundary exists
between finite versus infinite lines and a boundary exists between
finite versus infinite numbers.
Proof One is that you cannot build a infinite-line-ray without there
being a boundary in numbers where the next number and all larger are
infinite numbers, such as 10^500.
Proof Two is that if you have no boundary between finite versus
infinite lines, or finite versus
infinite numbers in the small scale of mathematics, then you can
construct a Euclidean
triangle which has two right-angles and the sum of angles is greater
than 180 degrees.
Proof Two thus implies that if you have a boundary between finite and
infinite, whether lines
or numbers, implies that you cannot have absolute continuity in
mathematics.
So, there is some really heavy mathematics here with this issue of
precision defining finite
versus infinite.
Archimedes Plutonium
Now, curiously enough, this new axiom appended to the existing Peano
Axioms strengthens
the Natural Numbers as defined by those axioms.
New Axiom: define finite-Natural Number as all less than 10^500 (the
largest meaningful
number in physics) and define infinite Natural Number as all those
equal to or larger than
10^500.
So curiously, we find that this new axiom when appended to the old
Peano axioms does
not upset or disturb the existing lot. However, in the program of
building the Rationals and
then the Reals, well, we get into a huge fight because many of the
concepts in the Rationals
and Reals are untenable.
Perhaps the best news of this new axiom is that of our methods of
proof in mathematics
hinges more on brute calculations rather than some clever trick such
as "multiply the lot
and add 1" in infinitude of primes. That is where Perfect Numbers,
Twin Primes, Goldbach,
FLT, Riemann Hypothesis will be proven in going through all the
numbers to 10^500.
Beyond 10^500 is meaningless in physics. You cannot measure beyond it;
you cannot count
to it; you cannot experiment beyond it. So if physics can never do
anything with numbers larger than 10^500, it is rather absurd to think
that some mathematician sipping coffee in
ivory towers has meaning for numbers larger than 10^500.
Archimedes Plutonium
imagine a
number larger than 10^500 that it immediately exists just as 1 or 2
exists. But existence
is not a mathematical provence. Existence comes from physics and
existence
in physics comes from the duality of particle versus wave. Another way
of stating
this, is that existence comes from the interaction between one
material matter versus
a second material matter. The largest number in physics is 10^500 or
thereabouts as the
number of Coulomb interactions inside a atom of element 109. So you
have every
coulomb interaction represented in 10^500.
But to a mathematician who thinks about 10^900, and because he can
think it or write
it down or talk about it, does not constitute the existence of 10^900,
just as when someone
thinks about fire breathing dragons or witches flying on brooms does
not mean they are
real and have existence.
Existence of math and for math numbers comes from Physics in that
numbers represent
physical interactions of matter.
In my other book, I discuss how the numbers pi and "e" come into
existence as the fact that
in physics, plutonium of the Plutonium Atom Totality has 22 subshells
in 7 shells and for with
19 subshells are occupied, which translates in rational number form of
22/7 and 19/7. So here
we see directly how Physics creates numbers as a reflection of
physical interaction of matter.
So numbers have reality and this reality is a reflection of physical
interaction of matter.
Mathematics is not a thought excercise of just mere imagination and
math is not something
that exists independent of the physical universe. Math is a result of
the interactions of the
matter in the physical universe.
We will never see a electron individually, nor see a 0 kelvin, nor see
a north pole, nor a negative charge. Just as we will never see a
number 1 or 3 or 85 come knocking on our
door. But all of those exist as interactions between matter.
A fire breathing dragon does not exist because it is not in biology
but only in an errant
imagination of the mind. Likewise the number 10^500 exists because it
is a physics
measurable amount in element 109, but the number 10^600 does not exist
except in
an errant imagination of the mind.
Archimedes Plutonium
with alot of subplots and
channels. It is a better "spy story" than any spy mystery novel. Its
ending is finally coming
to front and center stage. Its ending is not happy but sad, for not
only did most of the mathematics community unable to deliver a valid
proof of Infinitude of Primes but that
Euclid himself could not deliver a valid proof since he lacked the
Fundamental theorem
of Arithmetic-- unique prime factorization.
Here are the two proof of Infinitude of Primes in both methods of
direct/constructive proof
and of the indirect/contradiction method.
Reason: unique prime factorization theorem
(3.4) W+1 is necessarily prime. Reason: definition of prime, step
(1).
(3.5) Contradiction Reason: pn was supposed the largest prime yet we
constructed a new prime, W+1, larger than pn
(3.6) Reverse supposition step. Reason (3.5) coupled with (3.0)
(4) Set of primes are infinite Reason: steps (1) through (3.6)
Now if we take Ore's example of Euclid's proof of IP as a faithful
representation and **translation** of Euclid's ancient proof and we
further take the words and evaluation of Weil on page 5 of his book
"Number theory" as true, and no reason not to accept his evaluation.
Then we come to the conclusion that unless you know the Unique
Prime Factorization theorem (UPFAT) of numbers before doing the
Infinitude
of Primes proof, that it is required in critical parts of the proof no
matter
if the direct method or indirect method is used. Both methods require
UPFAT.
--- quoting from Number Theory and Its History, Oystein Ore, 1948,
page 65 ---
Euclid's proof runs as follows: let a, b, c, . . ., k be any family
of
prime numbers. Take their
product P = ab x . . x k and add 1. The P+1 is either a prime or not
a
prime. If it is,
we have added another prime to those given. If it is not, it must be
divisible by some prime
p. But p cannot be identical with any of the given prime numbers a,
b, . . ., k because then it
would divide P and also P+1; hence it would divide their difference,
which is 1 and this is impossible. Therefore a new prime can always
be
found to any given (finite) set of primes.
--- end quoting Ore ----
quote of Weil's book "Number theory", 1984,
page 5: "Even in Euclid,
we fail to find a general statement about the uniqueness of the
factorization of an integer into primes; surely he may have been
aware
of it, but all he has is a statement (Eucl.IX.14) about the l.c.m.
of
any number of given primes. Finally, the proof for the existence of
infinitely many
primes (Eucl.IX.20).. "
Now we should not be alarmed that Euclid failed to deliver a valid
proof, or a nearly
valid proof with just a minor flaw in it, because the key essential
ingredient is forming
the number P+1, and in my estimate the forming of that number means
the person is
on the right track.
And we should not be surprized because modern day mathematicians made
many mistakes
such as not knowing the difference between IP direct or indirect, let
alone invoking
UPFAT.
And we note also, that Hilbert and others spent considerable time in
cleaning up the geometry
proofs of Euclid where many holes and gaps in reasoning took place. We
should not expect
Euclid's book and works to have been 100% true without any flaws or
gaps. Likewise, we should expect a gap in Euclid's Infinitude of
Primes proof.
Not only was Euclid's proof of Infinitude of Primes a direct or
contructive proof method, but
it omitted or was flawed without the use of Unique Prime
Factorization. Weil points this out.
And we can see that Euclid did not have UPFAT because he refers to a
lemma of contradiction with Ore's talk of P and P+1 and division of 1
by a prime.
Now the question is, can we prove the Unique Prime Factorization
theorem from the idea
in Euclid's/Ore's lemma of contradiction with P and P+1 as a
foundation? Is UPFAT equivalent
to that section of the proof by Euclid and Ore where they apply the
lemma on P and P+1?
I think the answer is no. That the lemma used by Euclid and Ore are
not sufficient to be
turned into the Unique Prime Factorization theorem. I say this because
before the lemma is
reached in Ore's above proof, he says this "If it is not, it must be
divisible by some prime p." That statement by Ore (Euclid?) requires
UPFAT as justification
for making the statement. So that UPFAT enters the picture before the
lemma is applied.
So I conclude this by saying, that in order to be able to do a valid
Infinitude of Primes proof
that one needs to have Unique Prime Factorization Theorem present
before IP is started.
And I think that Weil is correct that UPFAT may have been
subconsciously known in Ancient
Greek times by Euclid, it was not consciously known and thus not
proven.
Maybe the Arabian mathematicians centuries after Euclid proved UPFAT,
for I doubt that it
required until the 1800s for Gauss to have proved UPFAT. Perhaps
Fibonacci proved UPFAT,
or Tartaglia? Probably those Arabic mathematicians proved it, a few
centuries after Euclid.
Archimedes Plutonium
Well here is what Wikipedia has to say about UPFAT (Fundamental
theorem of
Arithmetic) and its history:
--- quoting from Wikipedia ---
The theorem was practically proved by Euclid (in book 7 of Euclid's
elements, propositions 30 and 32), but the first full and correct
proof is found in the Disquisitiones Arithmeticae by Carl Friedrich
Gauss. It may be important to note that Egyptians like Ahmes used
earlier practical aspects of the factoring, such as the lowest common
multiple, allowing a long tradition to develop, as formalized by
Euclid, and rigorously proven by Gauss.
--- end quoting ---
From the way that Euclid handled his Infinitude of Primes proof, is
evidence that UPFAT was
not known to ancient Greece for then it would have no need of the
lemma in Ore's/ Euclid's
proof.
And I doubt that mathematics history was without a proof of UPFAT
until Gauss in the 1800s
performed. I suspect someone gave a proof within a hundred or 200
years circa Euclid.
Archimedes Plutonium
Archimedes Plutonium wrote:
(snipped)
>
>
> quote of Weil's book "Number theory", 1984,
> page 5: "Even in Euclid,
> we fail to find a general statement about the uniqueness of the
> factorization of an integer into primes; surely he may have been
> aware
> of it, but all he has is a statement (Eucl.IX.14) about the l.c.m.
> of
> any number of given primes. Finally, the proof for the existence of
> infinitely many
> primes (Eucl.IX.20).. "
>
Maybe Weil was just being too exaggerating. Maybe all we need for the
historical record
is for an ancient text to show a sequence such as this:
1 = 1
2 = 2
3 = 3
4 = 2x2
5 = 5
6 = 2x3
7 = 7
8 = 2x2x2
9 = 3x3
10 = 2x5
11 = 11
12 = 2x2x3
13 = 13
14 = 2x7
etc etc
So that if in Euclid's writings we see some sequence like that then we
can say Euclid was
aware of UPFAT and that it was proven in his time. And that Gauss
would only later refine
the proof.
Maybe Weil was just being overly harsh.
Archimedes Plutonium
--- quoting Wikipedia on the proof of uniqueness for Fundamental
theorem of Arithmetic ---
A proof of the uniqueness of the prime factorization of a given
integer proceeds as follows. Let s be the smallest natural number that
can be written as (at least) two different products of prime numbers.
Denote these two factorizations of s as p1···pm and q 1···qn, such
that s = p1p2···pm = q 1q2···qn. By Euclid's proposition either p1
divides q1, or p1 divides q 2···qn. Both q1 and q 2···qn must have
unique prime factorizations (since both are smaller than s), and thus
p1 = qj (for some j). But by removing p1 and qj from the initial
equivalence we have a smaller integer factorizable in two ways,
contradicting our initial assumption. Therefore there can be no such
s, and all natural numbers have a unique prime factorization.
--- end quoting ---
That is satisfying as a proof of UPFAT, to me. So I fail to see why
Weil says what he
says on page 5 of "Number theory"? Is Weil one to make spurious
complaints?
Archimedes Plutonium
Well, maybe this is more of a correcting of Andre Weil and his 1984
book
"Number Theory" then a correcting of Euclid. So why would Weil say
what
he says on page 5? Does he have some personal gripe on the Ancients of
Euclid?
Archimedes Plutonium
principle states that if is a prime and , then or (where means
divides). A corollary is that (Conway and Guy 1996). The
fundamental theorem of arithmetic is another corollary (Hardy and
Wright 1979).
Better yet, all we need is a concept in Ancient Greek times for which
that concept
is dependent on knowing full well the Unique Prime Factorization of a
given number.
Now I think the concept of Perfect Numbers can not go anywhere without
the understanding
of Unique Prime Factorization.
What is the word for the concept of "unique" in ancient Greek? Did
Weil ever comb through
the Euclid Elements for the Greek concept of "unique"?
Wikipedia calls it Euclid's Lemma and Wolfram's calls it Euclid's
Theorem
for which Unique Prime Factorization is obtained. and Wolfram writes:"
The fundamental theorem of arithmetic is another corollary (Hardy
and Wright 1979)."
I do not know for what reason that Weil made that remark in his book,
and rather than
cast aspersion on Euclid's work, it casts aspersion on Andre Weil as
to whether
he was a competent in mathematics,
and casts doubt that any of Weil's mathematics is reliable. I dare
say, it is possible to
throw out all of Andre Weil's work in mathematics as "not true" and
not affect any of
mathematics as we know it, and all because of a flippant errant remark
about past
mathematics. Perhaps, Weil should have joined with his sister in
philosophy rather
than have entered in mathematics.
Archimedes Plutonium
Archimedes Plutonium wrote:
> A theorem sometimes called "Euclid's first theorem" or Euclid's
> principle states that if is a prime and , then or (where means
> divides). A corollary is that (Conway and Guy 1996). The
> fundamental theorem of arithmetic is another corollary (Hardy and
> Wright 1979).
>
Sorry, that is not my writing, for it is a transmission error. I
wanted to quote Wolfram
on Unique Prime Factorization with Euclid's theorem but it failed to
transmit.
I apologize to Wolfram and have deleted the above in my original post
with a (sic) sign.
I did quote Wolfram in that post with part of that paragraph.
Sorry again for the transmission error.
But let me not spoil a whole post on admitting typing errors. Let me
continue with
the train of thought.
I unfortunately encountered Weil's comment that disparages Euclid, and
ran with it
on the side of Weil, but come to find out that it was Weil that was
clearly wrong.
I guess Weil expected the ancient Greeks to go wiling on page after
page about
the unique prime factor concept, whereas all the Greeks did was prove
it and no
mention thereafter.
What Weil failed to appreciate is that much of the ideas and concepts
of Number
theory depend on this Unique Prime Factorization theorem (UPFAT) and
thus called the Fundamental
Theorem of Arithmetic. So if Weil had stopped for a moment and
thought, "hey, all
we know about Perfect Numbers depends on UPFAT and Euclid spent
considerable time
on perfect numbers.
My concern is the Infinitude of Primes proof and UPFAT is critical for
that proof. Here again,
if Weil had stopped and thought about it, that Euclid could not have
done a valid IP proof
absent of an understanding of UPFAT.
So here we have a case of a famous mathematician Andre Weil, making a
spurious remark
and deprecating remark of ancient mathematicians, which only comes
back to reflect on
Mr Weil himself, that he was not a sharp and first rate mathematician
and that it is likely that
all of his contributions to mathematics were error prone. It is likely
that none of Weil's math
is true.
Archimedes Plutonium
with "missing mass".
I was interrupted from that physics book by this:
2009 Mathematical Intelligencer magazine article:
> [0] Michael *Hardy* and Catherine Woodgold,
> "*Prime* *Simplicity*", *Mathematical
> Intelligencer<https://mail.google.com/wiki/
Mathematical_Intelligencer>
Which when compared to my postings on this subject to sci.math from
1993
to 2009 was a "lifting of my postings" without attribute by the
Mathematical
Intelligencer (MI) article. One poster said that magazine editors are
"afraid" of
referencing the electronic sci. newsgroups. Well, they have to get
used to
it for the newsgroups are going to be a larger body of referencing
than most
individual books or magazines or periodicals.
In that article by Hardy/Woodgold, they do get across the true message
that
Euclid's proof was direct/constructive and not indirect/contradiction.
But the
article fails to show where most mistakes are made on the indirect
method,
and the article even suggests that Hardy/Woodgold and editors of MI
could
not do a valid proper Infinitude of Primes proof indirect method based
on their
inability to recognize that P+1 is necessarily prime in the
contradiction method.
It is a wonder that whenever supposedly logic persons are doing a
discussion
over the logic of Infinitude of Primes proof and then fail to give
both methods a
showing, side by side one another, and then lambast others for
committing errors.
Seems to me, if you are going to talk about Euclid's IP proof of
direct versus
indirect, the most logical article would show the two methods, but
here in
MI , Hardy and Woodgold and editors could only muster a showing of the
direct
method and then lambasting hundreds of mathematicians in that they did
a
indirect method.
So on the Internet of the science newsgroups, of sci.math and
sci.logic, I have
come up with a challenge that whenever anyone does Euclid's Infinitude
of Primes
Proof, that they do two proofs, one of the direct method and the other
of the
indirect method. I guess Hardy and Woodgold did not want to do that
ultra logical
exposition because, perhaps, maybe they felt they would be stealing
too much
of my sci.math postings without proper attribute in that Mathematical
Intelligencer
issue.
Without further delay, here are the two methods of proof of Euclid's
Infinitude of Primes.
And the major stumbling block is in the indirect that P+1 or I used W
+1 for Euclid's
number is ** necessarily prime **. Most authors, especially
mathematicians in books
make that mistake of thinking that P+1 in contradiction method is not
necessarily prime
for they cite some silly irrelevant example of 1+(2x3x5x7x11x13) =
59x509. That example
is actually part of the direct method proof where you have the list of
finite primes as
2,3,5,7,11,13 and where the constructive proof ends up fetching the 59
and 509 increasing
the set cardinality. But in the Indirect Method, we have to fetch a
new prime not on the list
of the supposed hypothetical list of all the primes in existence. In
the Indirect, we cannot go
scrambling around looking for a prime factor in P+1. Our only chance
of a new prime is P+1
itself. And it is the structure, the logical structure of the indirect
method (reductio ad absurdum) that the structure of logic allows you
to refer to step 1 where you defined a prime
number as divisible only by itself and 1, it is this definition in the
Indirect that permits you
to boldly claim that P+1 is necessarily prime. So when mathematics
professors writing books
on the Euclid Infinitude of primes and not recognizing that P+1 in
contradiction method is
necessarily prime, have failed to deliver a valid proof. For those
that cannot understand their
silly example of 59 x 509 is no example of the indirect method, well,
here is an example
to show them they are wrong. Start with definition of primes. Suppose
the set of all primes is finite and that 3 and 5 are the only primes
in existence. Thus we have 5 as the last and largest
prime. Form P+1 which is (3 x 5) +1 = 16. Now, 16 is necessarily prime
in this hypothetical
supposition space. Yet we all know that outside this supposition space
that 16 is not prime. But that makes no difference because Logic is
structure, and in this proof method, it is all about logical
structure. So that 16 is necessarily prime given the definition of
prime and the
supposition hypothesis and the contradiction follows from the fact
that 16 as prime is larger
than the largest supposed prime of 5. So it is no wonder that
hundreds, thousands of mathematicians themselves messed this up and
mixed the two methods. They forgot that it
is the logical structure that renders the proof and not irrelevant
examples.
Archimedes Plutonium
Archimedes Plutonium wrote:
> I need to get back to my physics book, where I am in the middle of it
> with "missing mass".
>
> I was interrupted from that physics book by this:
>
> 2009 Mathematical Intelligencer magazine article:
> > [0] Michael *Hardy* and Catherine Woodgold,
> > "*Prime* *Simplicity*", *Mathematical
> > Intelligencer<https://mail.google.com/wiki/
> Mathematical_Intelligencer>
>
Now from that Mathematical Intelligencer MI article Ore's
constructive proof was taken to be the same as Euclid's only
in modern day language.
--- quoting from Number Theory and Its History, Oystein Ore, 1948,
page 65 ---
Euclid's proof runs as follows: let a, b, c, . . ., k be any family
of
prime numbers. Take their
product P = ab x . . x k and add 1. The P+1 is either a prime or
not
a
prime. If it is,
we have added another prime to those given. If it is not, it must
be
divisible by some prime
p. But p cannot be identical with any of the given prime numbers a,
b, . . ., k because then it
would divide P and also P+1; hence it would divide their
difference,
which is 1 and this is impossible. Therefore a new prime can always
be
found to any given (finite) set of primes.
--- end quoting Ore ----
As mentioned before, I have some complaints about the above Ore/Euclid
rendition in that set theory should have been spoken more of, such as
increasing
set cardinality.
I find the proof valid, but I disagree that the lemma is needed, for I
think the lemma
is excess baggage-- "hence it would divide their difference". In both
Ore and Euclid's
rendition, all that need to be stated was the Unique Prime
Factorization theorem which
would have rendered prime factors if P+1 was not prime itself and thus
no lemma of
contradiction need be posed.
Now there was a bit of fuss in this thread, when I read this passage
by Weil after looking in the
library for the Ore book:
quote of Weil's book "Number theory", 1984,
page 5: "Even in Euclid,
we fail to find a general statement about the uniqueness of the
factorization of an integer into primes; surely he may have been
aware
of it, but all he has is a statement (Eucl.IX.14) about the l.c.m.
of
any number of given primes. Finally, the proof for the existence
of
infinitely many
primes (Eucl.IX.20).. "
I have no idea as to why Weil felt he had to be deprecatory of Euclid.
For
all that Andre Weil had to do was put on his thinking cap and realize
that
many of the number theory concepts of ancient greek time could not
have
progressed the distance they did without knowing the Fundamental
theorem
of Arithematic--unique prime factorization theorem (UPFAT). The
concept of
perfect-numbers would not have occurred without UPFAT. And one must
keep
in mind that in Ancient Greek, they did not have the decimal number
notation
and that is why we see so often numbers in Euclid represented as
lengths of
line. And so the reason that Weil probaby never sees, nor Euclid ever
write
about unique prime factorization, is the difficulty of even writing
numbers
not in a decimal notation.
So this diversion caused by Weil flippant remark caused me to explore
whether the Greeks had UPFAT or whether Weil was in error. And as it
turns
out, Weil was mistaken. I do not know if there is any moral theme to
the Weil
diversion, perhaps when you call into question other mathematicians,
that your
own work is then in question also. For I have the feeling that once
mathematics
has defined the boundary between finite and infinite as 10^500, that
much or
maybe all of Weil's work in mathematics becomes untruthful and
irrelevant,
just as his remark of Greeks over unique prime factorization.
So I think the Euclid and Ore proofs are valid, albeit with excess
unneeded baggage
of a lemma. When all that was needed was to invoke UPFAT for the prime
factor search.
P.S. I need to remark about an item in that MI article saying words to
the effect that
any direct proof can be turned into an indirect method proof. For I
feel that is a
mistake and ironic that the MI article is to clear up errors and
errors of myth, and here
with this idea of turning any direct method into an indirect may
create a brand new
erroneous myth that future mathematicians have to straighten out.
Archimedes Plutonium
edition, but also
I see in the news, now, that Paabo and Noonan have released the
Neanderthal Genome
project and for me to find out if the HACNS1 gene is more in keeping
with the chimp
than with the homo-sapiens HACNS1. In other words, I look forward to
HACNS1 proving
the Rockthrowing theory of Anthropology. What made us humans was a
throwing ape
after about 8 to 10 million years of evolution. So I must not dilly
dally when I need not
on math.
Let me point out that in the Mathematical Intelligencer article of
Euclid's infinitude of primes
proof by authors Hardy/Woodgold, that they make a mistake in my
opinion by spreading
a myth that you can convert a direct method proof into a proof by
contradiction. The trouble
with that idea is that we see many cases in mathematics where a direct
proof but never
a indirect and vice versa. Much of geometry proofs are not amenable to
a "suppose false".
What does physics have to say about direct versus indirect methods of
proof? Plenty. And
I feel the most pertinent issue of physics for direct and indirect
proof methods is that Physics
is based on Quantum Mechanics where the foundation is duality. Duality
is not straight-line
linear logic but is rather circular logic, or commonly called
nonlinear. Quantum Mechanics
is not either or, but rather probability.
To say that every direct method proof of mathematics has an indirect
also, would be saying
that mathematics is linear logic or Aristotelian logic throughout. And
if math is but a tiny
subset of physics and where physics is duality and circular nonlinear
logic, then we expect
asymmetry between proofs by direct method versus indirect method. Some
proofs can only
be direct method whereas others only indirect according to Quantum
Mechanics.
Also we should distinguish between actual proof by contradiction or
reductio ad absurdum
(indirect) as a method of proof from the negation operator. "Suppose
not" for a statement
does not mean automatically we initiated a reductio ad absurdum for we
may have ignited just
a negation of a statement. The difference between Reductio Ad Absurdum
and Negation, is what G. H. Hardy in his book A Mathematician's
Apology referred to as a total gambit where
you place the entire axioms and all the theorems of math and pitt them
against a supposition statement. The Negation of a statement is not
the pitting of all of math against one statement.
So we have to be careful about distinguishing a Negation of a
statement versus the Reductio
ad Absurdum method.
The evidence in math itself suggests that some proofs can only be
either direct or indirect
but not both.
Physics suggests from duality, that math should not have both direct
and indirect for
every proof.
Maybe I will have some time opening in the future to further develop
these ideas, but
now I do not have that time.
sttsc...@tesco.net
> > "*Prime* *Simplicity*", *Mathematical
> I had always thought that Euclid had proved the Unique Prime
> Factorization Theorem
> (UPFAT), but reading Weil's book page 5, Euclid fell short of UPFAT.
> So when Ore
> goes into the sentence saying "it must be divisible by some prime p."
> And the only
> justification for that claim, as far as I can see is UPFAT.
The uniqueness part is superfluous.
The smallest divisor d >1 of M> 1 must be a prime.
If d >1 is composite, d has a divisor d', d'<d
This means every m>1 has a prime divisor, the smallest divisor
of m that exceeds 1.
1) Every n>1 has at least one prime diviosr
2) GCD(n, n+1) = 1
3) Assume the primes are finite in number
Let L= LCM of these primes
4) GCD(L,L+1) <>1
3) => 4), 4) is false, therefore 3) is false.
Archimedes Plutonium
separate discussion
for infinity on the small scale. Here is another example of Quantum
Duality in mathematics
in that infinity, which we usually think of as beyond large, that it
must be reconciled
with microscale and infinity there.
In July 2010, this month: I gave this as proof that large scale
infinity requires a boundary
such as 10^500 which says that finite is any integer smaller than
10^500, and 10^500 itself
is an infinite number as well as any number greater than 10^500.
I wrote in July 2010:
Theorem: In old-math, geometry had well-defined finite-line versus
infinite-line but
Algebra or Number theory was ill-defined with its finite-number
versus
infinite-number
and that is why mathematics could never prove Twin Primes, Perfect
Numbers,
Goldbach C. , Fermat's Last Theorem, Riemann Hypothesis and
thousands
of
number theory conjectures. In this theorem, we show there never can
be
constructed a infinite-line in geometry since the other half of
mathematics, the old-math never well defined
infinite-number versus finite-number.
Proof: Since old math does not recognize infinite-numbers, that no
matter how many finite number of line segments we put together, they
still will never
summon into an infinite-line-ray. However, if a precision definition
is given in
mathematics for geometry or algebra saying that finite-number means
all numbers less than 10^500 and 10^500 and beyond are infinite-
numbers.
Well, with that definition we can build an infinite-line-ray in
geometry by
adding together 10^500 units of line-segments of finite line
segments
building
an infinite-line-ray. QED
I referred to another proof of mine in 2009 where I said that
betweeness axiom
no longer held. So I need to refurbish this betweenness proof to prove
that in
small scale that you also need a boundary between finite and infinite
number,
and here it is 10^-500
--- quoting old post in part ---
Newsgroups: sci.logic, sci.math, sci.physics
From: Archimedes Plutonium <plutonium.archime...@gmail.com>
Date: Thu, 18 Jun 2009 21:51:09 -0700 (PDT)
Local: Thurs, Jun 18 2009 11:51 pm
Subject: proving the Betweenness Axiom contradicts the Parallel Axiom
Betweenness Axiom:
If A and B are any two points, then (1) there is a point C such
that A-B-C, and (2) there is a point D such that A-D-B.
Parallel Axiom:
Given a line and a point not on the line there exists one and only
one line parallel from that point to the given line.
Now I need to prove that those two Euclidean Geometry axioms
are contradictory. I did this earlier in this book by setting up a
triangle that becomes smaller and smaller and which would thence
have two 90 degree angles.
But let me try it using the same scheme a bit differently.
here I have a given line with a parallel line from that point not on
the
given line called A:
--------------------------A----------------------------
--------------------------------------------------------
Now what I do is form a right triangle using A
and two points on the given line called B and C
like this:
--------------------------A-----------------------------
--------------------------B------------------------C---
Now here is the interesting feature of the Old
Euclidean geometry axioms in that they are
contradictory. As I do the infinite-downward
-regression of Betweenness the C point
approaches infinitely close to the B point
such as this picture
--------------------------A-----------------------------
--------------------------BC---------------------------
Since the axiom of Betweenness never ends
means that C becomes B and the triangle
is merely a line segment AB and no longer
a triangle of ABC and before it becomes
a mere line segment it becomes a triangle
with two 90 degree angles.
Now if I went the other way of B approaching
C as such:
--------------------------A-----------------------------
--------------------------------------------------BC---
In this direction what ends up is a line segment
AC and where the B becomes C.
In the first case I have a triangle which has two
90 degree angles, and in the second
case I have a triangle whose angle sum is equal to
zero since side AB and BC vanished into becoming AC.
--- end quoting old post in part ---
Some minor adjustments to the above, that if infinity is without end
and
no boundary between finite and infinite number, then what happens in
the Microworld or small scale is that you end up with either a
triangle
that has angle sum greater than 180 degrees and two right-angles, or
you have a very slender scalene triangle whose tiny side, like in
Calculus,
making the side as tiny as we want to make it, so that finally, we end
up with
a triangle that has two sides of a scalene triangle as parallel since
they never
meet.
SUMMARY: in mathematics when we do not admit that there must be a
boundary
between finite number versus infinite number such as 10^500 large
scale and 10^-500 small
scale, that we end up not able to build a infinite line ray in
geometry simply because
infinite-number is not available. And on the small scale, there also
needs a boundary
between finite number versus infinite number, or else we have a
triangle sum greater
than 180 degrees and a triangle with two parallel sides.
Archimedes Plutonium
You see, what the above two proofs do, is not only tell us the
difference between
infinity at large and infinity for the small, is that it tells us the
meaning of the concept
of infinity, for which none of us have ever realized we are using a
false idea for infinity.
We were thinking infinity means endless, whereas it only really means
the end
of physical measure or physical experimentation. So if you cannot have
a physics,
then mathematics is merely just "ideas of fiction". We can pretend
that a woman becomes
a witch and rides on brooms or we can pretend that lizards become
large and shot fire out
of their mouth. Likewise we can pretend that a line goes on forever or
that a number always
exists between any two numbers no matter how small.
What those two proofs show is that mathematics is no longer math as
the science of precision, so long as finite number versus infinite
number is never well-defined.
Infinity simply means that the end of Physics has arrived, where
Physics is no longer
meaningful and where Physics has duality as logic and not Aristotelian
linear logic. Logic
is nonlinear in Quantum Mechanics. So when we reach 10^500 and go
beyond, we have
reached infinity for the large. And when we reach 10^-500 and try
going smaller, we cannot
because there is no more physics at either end of these numbers.
The reason Twin Primes, Perfect Numbers, FLT, Goldbach, Riemann
Hypothesis were never
proved and never provable is because 10^500 and 10^-500 in the case of
Poincare Conjecture
are the end of the line for infinity. Infinity means the end-of-
physics. Physics is a bigger subject than all of mathematics and, all
of mathematics fits inside of Physics as a subset. The bedroom in your
house is a subset of your house, and mathematics is a bedroom or
closet space of your house.
Notice the difference in those two proofs of infinity at large and
infinity in the small. Infinity at
large when not well defined as 10^500, then noone in geometry can ever
build or construct
or erect a infinite line from finite lines. Since we have no
definition of infinite-number, then
no matter how many finite lines are joined together, they can never
build an infinite line.
Now in the infinite small case, what happens when that is not
precisely defined such as
10^-500, what happens here is that you have triangles that can be
built where the angle
sum in Euclidean geometry is greater than 180 degrees because
triangles can possess two
right-angles. Or, you have the odd situation where, since infinity in
old math means never
ending, even of physics, you have the odd situation of a triangle
between 0 and 1 that never
meets in a third point but has only two points of intersection of its
three sides.
I am afraid that math of old was too much philosophical imagination
run amok rather than
doing their job of precision definition of terms and concepts. Even
the Peano Axioms of the
late 1800s never precision defined finite-number versus infinite-
number. And that is why
Mathematics is so much in the weeds and is the least progressive of
the sciences. At least
when someone like Pons and Fleischmann announce cold fusion in a test
tube, the physics
community can clear that up in a matter of 3 years. In the mathematics
community, time
moves slothlike and something as important as precision definition of
finite-number takes
more than 3 milleniums because we have already endured 3 milleniums of
math without the
community so much as taking notice, and precision defining finite-
number versus infinite-number.
Archimedes Plutonium
Someone wrote:
>>
>> 1) Every n>1 has at least one prime diviosr
>> 2) GCD(n, n+1) = 1
>> 3) Assume the primes are finite in number
>> Let L= LCM of these primes
>> 4) GCD(L,L+1) <>1
>>
>> 3) => 4), 4) is false, therefore 3) is false.
>>
Someone called asking me to critique the above garbled mess.
I would bet that for every 1000 people that do a Infinitude of Primes
proof, only 1 is
going to see it clearly from start to finish and the other 999 are
going to add excess baggage
or even invalid steps.
The corollary of every number larger than 1 has at least one prime
divisor comes immediately
from the definition of prime.
So we have this.
Corollary: Every number larger than 1 has at least one prime divisor.
Proof: Definition of prime is a number divisible only by itself and
one. Hence
every number has at least one prime divisor.
So the above gnarled mess is but one step excess baggage more than
what any proof of Infinitude of Primes needs.
Now I could have added that Corollary into my direct or indirect proof
methods, but why bother, when the definition of prime says the same
thing, but says it much more directly and
clearly. I could add five alternative definitions of prime numbers to
any of my proofs and would not have affected the validity of the
outcome, but it sure would have made a mess.
So there are going to be thousands of people who add excess nonsense
into a Infinitude of Primes proof, rather than that one in a thousand
clear mind that can do IP without nonsense
and excess baggage.
You still have to go through all these steps to reach a valid proof
whether you add the Corollary or not:
Hope that helps.
Archimedes Plutonium
Archimedes Plutonium wrote:
> Someone wrote:
>
> >>
> >> 1) Every n>1 has at least one prime diviosr
> >> 2) GCD(n, n+1) = 1
> >> 3) Assume the primes are finite in number
> >> Let L= LCM of these primes
> >> 4) GCD(L,L+1) <>1
> >>
> >> 3) => 4), 4) is false, therefore 3) is false.
> >>
>
> Someone called asking me to critique the above garbled mess.
>
The above is not a proof, for it is an invalid attempt. I can even
shorten the above
to expose its flaws:
Another invalid attempt:
(1) Every n > 1 has at least one prime divisor
(2) Since the set N is infinite thus the prime divisors are infinite,
thus primes infinite
QED
Whoever wrote the above mess wanting to use prime divisors, rather
than writing out
a clear proof in long form, either could not write it out in long form
or did not want to
for it would expose the flaws.
The flaws of the above is that you have to actually produce a new
prime, whether
direct or indirect. So when you crank-dwell on every n>1 has at least
one prime
divisor, you never are able to actually produce a new prime not on the
starting list
and thus can never claim a contradiction.
The contradiction in the Indirect is that P+1 is larger than the
largest supposed prime.
So unless you produce a new prime, then all you have is just crank-
talk.
sttsc...@tesco.net
<plutonium.archime...@gmail.com> wrote:
> Someone wrote:
>
> >> 1) Every n>1 has at least one prime diviosr
> >> 2) GCD(n, n+1) = 1
> >> 3) Assume the primes are finite in number
> >> Let L= LCM of these primes
> >> 4) GCD(L,L+1) <>1
>
> >> 3) => 4), 4) is false, therefore 3) is false.
>
>
> divisor comes immediately
> from the definition of prime.
>
> So we have this.
>
> Corollary: Every number larger than 1 has at least one prime divisor.
> Proof: Definition of prime is a number divisible only by itself and
> one. Hence
> every number has at least one prime divisor.
1 is divisible by itself (1) and 1 is divisible by 1.
Are you saying that 1 is prime or not ?
If it is why do you always leave it off your list of
primes 2,3,,....
(see below : (2) Statement: Given any finite collection of primes
2,3,5,7,11, ..,p_n possessing a cardinality ...)
If 1 is prime then unique factorization does not apply.
6 = 1x2x3 = 1x1x2x3 etc.
Obviously if 1 is prime then the claim that w+1 is not
divisible by any of the primes that divide w is false.
1 divides w and 1 divides w+1. you cannot conclide there
is a new prime.
Def: n >1 is prime if n =ab => a=1 or b =1
The definition is not empty as 2 is prime by the definition.
There is nothing in the definition that says
every n >1 has a prime divisor.
There are three alternatives for every n:
n is prime, n is composite or n is a unit.
At least you now seem to undestand that
"Every n> 1 has a prime divisor" is a true statement
every if you can't prove it or simply don't understand there is
a need to prove it.
A) Every n>1 has a prime divisor (theorem)
B) GCD(n,n+1) =1 (theorem)
C) Assume 2 is the only prime.
Don't you even dimly see that A) and C) together
imply that every n>1 is an even number ?
By A) every n>1 is divisible by a prime, 2 is the only prine
therfore every n>1 is divisible by 2 and so even
In other words
For m, m' >1
m = 2n
m+1 = 2n'
GCD(m,m+1) >= 2
But B) tells us that GCD(m,m+1) =1
If GCD(m,m+1) is true for all m
then GCD(m,m+1) cannot be >= 2.
This is a contradiction.
If A) and B) are true then C) must be false
Therefore, 2 is not the only prime.
It really does not matter whether you choose 2 or
n primes the same principle applies.
A) Every n>1 has a prime divisor (theorem)
B) GCD(n,n+1) =1 (theorem)
C) Assume there are n pimes a,b,c,...,z
Some m is divisible by all n primes
m+1 is divisble by some prime say p
GCD(m,m+1) >=p
Archimedes Plutonium
the largest number
in physics is closer to 10^631 or maybe in the opposite direction of
10^498. The number
of Coulomb interactions inside an atom of plutonium is approaching
that of 10^400. The number Coulomb interactions inside of Element 109
is approximately 10^500. I think we
are near to manufacturing element 114 and whose Coulomb Interactions
would be larger
than 10^500. But this is an assignment for the Physicists to work out
as to their largest
number, and to their smallest number. Whatever their largest number
is, we can take the
inverse as the smallest number in physics.
And of course, talk like this, where Physics dominates over
mathematics, where physics
actually takes over the control of the subject of mathematics is a
first time occurrence in
the history of both subjects. Before, math dominated over physics. But
with the Atom Totality theory, we have a recognition that all of
mathematical thought and math proof arises out of
what atoms are and what atoms do. Atoms have shape and size and thus
gives rise to the
math subject of geometry. And atoms are numerous and have quantity of
them, and thus giving rise to algebra and number theory.
The idea of infinity in mathematics, before the Atom Totality, was
very philosophical and overly idealistic. Thinking that a concept of
infinity meant "endlessness". With the Atom Totality,
the concept of Infinity means merely an "end of Physics". So whenever
mathematicians
end a sentence with "out to infinity." What they really mean is "out
to where there can be no
more physics counting or measuring or experimentation."
So what does anyone care about a number larger than 10^500 when no
possible physics nor
biology could ever do or use that number as size or quantity? What
good is it to talk about
or ruminate about 10^-700, when no physical act can even deal with
10^-70? Jokingly, one can
say that as the mathematicians developed math from Pythagoras to the
year 1990, they seemed to have lost most of their commonsense along
the way.
It was recognized with the advent of Quantum Mechanics in the 1900 to
1930s that Newtonian's absolute time and absolute space was never
more. But the mathematicians
have never undergone such a cleaning out revolution. They still think
that "endless or forever"
have this Newtonian absolutist quality. They cannot seem to understand
that endless, ends
where Physics ends. If Physics cannot go beyond a large number, then
math cannot go beyond there either. And if Physics cannot reach a tiny
number, then mathematics is also
detoured.
The logic for both math and physics is not just one logic of Aristotle
of linear straight line
logic. The logic that pervades physics is duality logic which is
nonlinear. So that Physics
has both linear logic and nonlinear logic, but math never recognized
this. Math somewhat
saw duality and symmetry in things such as the 5 regular polyhedra or
the duality of
calculus integral with calculus derivative, but math never saw that
their entire subject is
a duality where geometry is dual to algebra.
So it is about time for mathematics to have wholescale sweeping
changes. For mathematics
to undergo a revolution to clean out its old decayed house. And there
is no better place to
start the cleaning out, then to get back to math's fundamental job.
The Fundamental Job
of mathematics is precision defining and we need a precision
definition of finite-number
versus infinite-number.
sttsc...@tesco.net
<plutonium.archime...@gmail.com> wrote:
> Archimedes Plutonium wrote:
> > Someone wrote:
>
> > >> 1) Every n>1 has at least one prime diviosr
> > >> 2) GCD(n, n+1) = 1
> > >> 3) Assume the primes are finite in number
> > >> Let L= LCM of these primes
> > >> 4) GCD(L,L+1) <>1
>
> > >> 3) => 4), 4) is false, therefore 3) is false.
>
> > Someone called asking me to critique the above garbled mess.
>
> The above is not a proof, for it is an invalid attempt. I can even
> shorten the above
> to expose its flaws:
I can see you do not understand what is involved in a proof
by contradiction.
Obviously, the idea that every n>1 has at least one
prime divisor is too commplex for you.
On the other hand, you seem to concede sometimes
that every N>1 has a unique prime factorization
A) Every N>1 has a unique prime factorization.
B) No pair of consecutive naturals has a common factor
other than 1
A) and B) are theorems and so true.
H) Assume that p,q,r are the only primes
If H is true then by A) every natural must be some product of
p,q,r. The only primes that exist.
In sequence, the naturals could have prime factorizations
1, p, q, pp, r, pq, rr, ppq, ........., pppqqqqrrrr, .....
p= 1+1, q = p+1, pp= q+1, r = pp+1, etc
As no two consecutive naturals have a common factor other than 1
GCD(p,q) =1, GCD(pq,r) = 1 etc.
pqr, pqr+1 are consecutive naturals and so have
no common factor greater than 1
pqr+1 by unique factorization must be some product of p,q,r
But whatever unique product of primes pqr+1 equals
p,q,r,pq,pr,qr,pp, .., ppppppqqqrrrrrr, etc
pqr and pqr+1 must share a common factor other than 1.
This is a contradiction as a pair of consecutive naturals
has no common divisor other than 1
The assumption that the primes are finite in number is therefore
false.
The primes are infinite in number.
If the reasoning is false say which statement
or deduction is false or invalid.
If you cannot indicate which particular statement or deduction is
false or invalid, the proof must be valid
Aatu Koskensilta
> I can see you do not understand what is involved in a proof by
> contradiction.
That's entirely possible, but it's utterly pointless to formulate the
standard proof of the infinitude of primes as an indirect proof.
> If you cannot indicate which particular statement or deduction is
> false or invalid, the proof must be valid
The validity of the proof in no way depends on Archimedes Plutonium's
abilities.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechan kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
sttsc...@tesco.net
> "sttscitr...@tesco.net" <sttscitr...@tesco.net> writes:
> > I can see you do not understand what is involved in a proof by
> > contradiction.
>
> That's entirely possible, but it's utterly pointless to formulate the
> standard proof of the infinitude of primes as an indirect proof.
Which proof is the standard proof ?
Who is trying to formulate the "standard proof" as an "indirect
proof ?
> > If you cannot indicate which particular statement or deduction is
> > false or invalid, the proof must be valid
>
> The validity of the proof in no way depends on
Archimedes Plutonium's
> abilities.
"you" = "one"
"If you don't train, you'll never be a top athlete"
David R Tribble
> Of course I used 10^500 because of its well rounded off. But perhaps
> the largest number
> in physics is closer to 10^631 or maybe in the opposite direction of
> 10^498. The number
> of Coulomb interactions inside an atom of plutonium is approaching
> that of 10^400. The number Coulomb interactions inside of Element 109
> is approximately 10^500. I think we
> are near to manufacturing element 114 and whose Coulomb Interactions
> would be larger
> than 10^500. But this is an assignment for the Physicists to work out
> as to their largest
> number, and to their smallest number. Whatever their largest number
> is, we can take the
> inverse as the smallest number in physics.
Given as few as 500 elementary particles (or molecules,
or asteroids), there are 500! total interactions among them
(gravitational, electromagnetic, whatever, take your pick).
And 500! = 1.220x10^1134, which well exceeds 10^500.
For 1,000 particles or bodies, it's 1000! = 4.023x10^2567.
Now imagine the celestial bodies in the Oort cloud, which number
in the millions, and all of which have a gravitational influence on
one another.
Archimedes Plutonium
What does your calculator get for 231! =
And what does your calculator get for 10^500 = what N!
Now you can blame me for your misunderstanding of "physics
interaction"
as much as blame yourself for not understanding "interaction". For I
have
often just written "Coulomb interactions". So for a bar magnet of its
North and
South poles, I see it as 1 magnetic field, whereas you are seeing this
as
the result of a large number of protons and electrons involved and the
entire
protons and electrons throughout the Cosmos making that magnetic
field.
So let me apologize for adding to your misunderstanding of "physics
interaction."
The Coulomb Interaction is the mediating of a photon between a proton
and electron.
So if the number of protons in the Cosmos was 10^60 and the number of
protons and
electrons and photons was 10^61, then we are never going to get
anything far beyond
10^61 or 61!.
The field of study of Combinations and Permutations is a vast field,
and the
factorial -- number of ways of arranging N things into a sequence,
becomes a
large number very quickly, but very much misunderstood in physics.
Now I do not have time to get into this subject with any depth, but
only to start the
conversation. I am not in the position to precision define Coulomb
Interaction, but
as time goes by, I can get at the root of it, and by starting here.
I have the suspicion that N! is the volume of the Cosmos and that the
volume
is related to Time. So that if the Cosmos is a 231Pu Atom Totality, it
has a volume
of 231!. So what are those units of Volume? They are not units of some
distance,
but rather they are units of time. Distance and time are interrelated
and can be converted
from one to the other, much as I have done in the derivation of speed
of light
where I convert 5,300 km of the Earth's sphere into 5,300 seconds.
I remarked often before to the Internet sci newsgroups that time is
the arrangement
of atoms. If all atoms that exist were frozen into place, where no
movement or motion
exists, then time would itself stop. To have zero time means there is
no change in position
of all the atoms in the Universe. Now if there is a change in position
of just one atom whereas
all the other atoms remained still, then time still exists.
Now the reason that "time travel" is science fiction and always will
be science fiction is because to travel back in time or forward in
time means you have to rearrange all the atoms
to be in those former or future positions. So a program like the
"Doctor Who" will always
be science fiction.
So now the 231! is a huge number and let us say it is the volume of
the Cosmos in units
of time so that each cubic unit is a cube of time and there are 231!
of these cubic time units.
In some of those cubic time units sits a particle with mass, in others
sit almost a vaccuum.
So that when astronomers claim the Cosmos is 10^14 billion years old,
another way of saying that, is that the Cosmos is 231! time units old,
in which the previous Atom Totality may have
been 228! old.
Now as for the misconception of Coulomb Interactions or interactions
in general, Inside a atom of plutonium also has 231 elementary
particles of protons and electrons, creating a
microworld volume of 231!. So we have big space with a volume of 231!
and a microspace
of inside the atom with volume 231!. So the concept of a physics
interaction is far different
from the idea of everything doing something to everything else.
The concept of Physics Interaction deals more with the idea that a
Magnetic Field is in place
as 1 magnetic field, not as 1,000 electrons doing 1000! interactions.
So I am sorry that all my past posts have only increased the
misconception and misunderstanding of Coulomb Interaction. And that I
never took a time-out to try to
reign in a understanding of ** Physics Interaction**. It will take an
entire big book
to cover the subject, decently, and I never had that time. But can
start it.
Archimedes Plutonium
What does your calculator get for 231! =
Archimedes Plutonium
Archimedes Plutonium wrote:
(snipped for brevity)
>
> The concept of Physics Interaction deals more with the idea that a
> Magnetic Field is in place
> as 1 magnetic field, not as 1,000 electrons doing 1000! interactions.
>
> So I am sorry that all my past posts have only increased the
> misconception and misunderstanding of Coulomb Interaction. And that I
> never took a time-out to try to
> reign in a understanding of ** Physics Interaction**. It will take an
> entire big book
> to cover the subject, decently, and I never had that time. But can
> start it.
>
>
Also, I am not going to be posting the same post for two different
books.
I generally have used Atkins book "Quanta", 1991 as a definitional
book of
quantum mechanics terms. On page 39 and 40 he talks about the
definition
of "boson" and that is the closest definition to Interaction for
physics. So the idea of
a force in physics is that a atom of plutonium of 231 and iron of 56
has photon exchanges
for the Coulomb force, not that 231! photons are exchanged in any one
given instant of time,
but rather that 231/2 photons are exchanged and for iron that 56/2
photons are exchanged but that the electromagnetic field of the
plutonium atom is 231! interactions at any instant of time and the EM
field of iron is 56! interactions at any instant of time.
So that it is wrong to think that there are 1,000 galaxies all
containing 1,000 atoms at least
and thus 1,000,000! is the force interactions. The force interactions
of those galaxies would be
1,000,000 photons at any given instant of time.
Now I want to expand on the idea that the largest number in physics is
231! approx equal to
10^500.
As was mentioned in the prior post that N! is permutations and meaning
that N! ways to
arrange N distinct items in a sequence. So what I have done is
modified that idea of
arranging N distinct items into N! sequences. Think of each of those
sequences as a
individual cell of Space-Time. So that for 231! there are 231! cells
that occupy the
Atom Totality Universe. Some of those cells have mass matter inside
them, so are nearly
a vaccuum. And consider all of these cells as the volume of the
Universe and also the
Time of the Universe. In a previous Atom Totality of say Iron there
were 56! cells and the time
of that Cosmos was 56! time units.
Now it is well known in Quantum Mechanics that position and momentum
are canonical conjugate dualities, as well as time and energy. So that
when you do physics of the very small numbers and the very large
numbers, unlike Mathematics, the logic is duality. All of mathematics
to date is Aristotelian straight line or linear logic, but physics has
not only linear logic but also has duality-logic, especially when it
comes to the very tiny and the very large.
QM is mysterious to everyone, even physicists and the source of this
mystery is that the human mind cannot think in terms of duality logic,
and can only pick our way through this
logic. The human mind can easily deal with straightline logic. So when
we deal with millions
and billions and trillions or less and when we deal with 1/1000 of a
meter or nanometers we are
dealing with what the human mind is comfortable with and the dualities
are not in view, although they are present. But it is not until we get
into large numbers like 10^60 or 10^40, or the small world of 10^-40,
not to speak of 10^500 or 10^-500, that we no longer can find truths
with Aristotelian logic or linear logic.
So that a number such as 231! can be either a Space volume in terms of
cells, or can be
a time units as cells. So that we can imagine that the Cosmos is an
Atom Totality and is
built of 231! cells of the duality of position versus momentum.
Alternatively, we can picture
those 231! multiplications as representing cells of time, as in the
time versus energy duality.
Now normally, we picture multiplication of 1x2x...230x231, we picture
that geometrically as
a 231 dimensional object in space. But remember that is old time math
of linear-logic. Here
we have the new math of duality logic. The 231! of ways of arranging
231 distinct items becomes 231! cells that compose all of the
Universe. Each cell is a unit of time, or a unit
of position in the Universe.
Archimedes Plutonium
Archimedes Plutonium wrote:
(snipped)
>
> QM is mysterious to everyone, even physicists and the source of this
> mystery is that the human mind cannot think in terms of duality logic,
> and can only pick our way through this
> logic. The human mind can easily deal with straightline logic. So when
> we deal with millions
> and billions and trillions or less and when we deal with 1/1000 of a
> meter or nanometers we are
> dealing with what the human mind is comfortable with and the dualities
> are not in view, although they are present. But it is not until we get
> into large numbers like 10^60 or 10^40, or the small world of 10^-40,
> not to speak of 10^500 or 10^-500, that we no longer can find truths
> with Aristotelian logic or linear logic.
>
You see, our knowledge of Quantum Mechanics is the poking around with
Aristotelian straightline logic and we run into something like the
Double Slit
Experiment which our linear minds is unable, at first to understand.
We keep
poking around with it and slowly we piece together the explanation
because
the explanation is Quantum Mechanics of nonlinear logic or duality
logic.
Our minds do not understand it directly because we cannot think
duality logic
but we can come to the truthful conclusions. This has been the way of
understanding
physics all along, in that we see the magnetic field, a mysterious
force that
pervades space and we slowly put it together into the Maxwell
Equations. But the
mysteries and misunderstandings always remain because our minds were
never
at home in duality logic, our minds were built to function as linear
logic.
Now this example out of mathematics that N! is the total number of
ways of arranging
N distinct items into a sequence, called permutations in old math. But
let us use that
example of N! and examine it in new-math where Physics is supreme and
math but a
tiny subset. Let us examine N! as duality logic in Quantum Mechanics.
Let us take 254! which is close enough to 10^500. The number 254! is
element 100 and
is where the StrongNuclear Force no longer exists. In a sense, physics
no longer exists
if there is no StrongNuclear Force available.
Now in linear logic we think that we can always multiply and that the
rules of Old-math
apply no matter how large the numbers we are multiplying. But what if
the Universe was
a sphere and we multiplied 10^600 x 10^700? In Old math, that would be
very acceptable.
In New Math, duality logic steps in and says the Universe is spherical
and beyond a certain
multiplication, that we no longer get "good answers", for our numbers
are so large that we
physically have no place to put those numbers since we circumnavigated
all the way back around to our starting point. In Old Math they would
say that a airplane that flew 50,000 km
x 3 times in a straight line is "alright math", whereas in fact it is
bad math because Earth
is only 40,000 circumference, so the multiplication breaks down.
So what New Math does is recognize that Physics is supreme and math
but the backseat
passenger.
So there is a new meaning to the N! of Old Math. Old Math saw N! as
only a linear logic
of multiplication in Euclidean Geometry. But in Elliptic or Hyperbolic
Geometry N! now takes
on an entirely new meaning. N! means more like volume building where
254! is equal to
10^500 and it builds 10^500 cells. Each of those possible sequences of
that 254! is a cell
with volume. So instead of a distance travelling around and around a
sphere by a large number such as 254!, the new meaning of 254! is that
it yields a construction of the sphere
itself in volume with cells. Now in Old Math, this cell building was
recognized when we did
2x3 or 4x5x6 to create volume, but it only allowed volume with
multiplication of three or less
multipliers because of orthogonality was only three orthogonals of
dimension. In Old math
it was never thought that 254! had 254 orthogonals for 254 dimensions.
And Quantum Mechanics has only 3 physical dimensions so orthogonality
is not an issue here. What is
an issue is that in duality logic, multiplication breaks down because
it no longer makes sense
on Elliptical geometry. In Elliptic Geometry, the number 254! now
becomes not a straight line
enlarged by multiplication, but becomes a multiplication machine that
spews out sequences, and each sequence is a volume-cell of Elliptic
geometry. So when asked to do 254! on a
sphere surface, we are not travelling around and around that sphere,
but we are in fact
cutting up that sphere into tiny compartments or cells that are 10^500
cells inside.
You see, when physics is in control, the answers always come back to
where there is
commonsense physical reality, something with hands on experimenting
you can apply. When math is in control, you often get nothing but
imagination run amok and seldom any physical connection to anything,
where old math
is on par with religion and philosophy.
Archimedes Plutonium
Note the sheer beauty of this authorship writing where I am weaving in
a thread
into two books, one the Correcting Math and the other the Atom
Totality. The posts
are relevant to two books and this is perhaps the first time in human
history where
a author can weave a single post into two books. So long as I can keep
the numbering
track correct.
Anyway, let me not discourage people from learning Quantum Mechanics
by telling them
it is Duality logic where our minds were only built for linear logic.
We cannot picture or understand how a entity can be two things at the
same time. We cannot understand how
a human if quantum physics were applied that a human being would be
simultaneously
at one location and also be scattered into millions of fragmented
pieces over a vast space.
This is why QM is so difficult, because our minds were not built to
see or understand it
easily. We have to poke around with our linear logic to get at the
truth of QM.
But there is one help that may be a major help. Duality Logic is
represented by geometry
in the fact that NonEuclidean geometry of Elliptic with Hyperbolic is
a duality to Euclidean
Geometry. So if we can criss cross back and forth between NonEuclidean
Geometry and
Euclidean Geometry, knowing the features of all three geometries, that
you cannot have
lines to infinity in Elliptic geometry but can have them in Euclidean
geometry, so as we
criss cross back and forth between these three, we are in a sense
navigating in Quantum
Mechanics. NonEuclidean geometry is dual logic compared to Euclidean
Geometry. So our
minds are not completely helpless in navigating Quantum Mechanics.
But let me point out a pretty detail that this N! has brought up. A
long time ago when doing
the math book AP-adics, I started that program with "All Possible
Digit Arrangements" as the
foundation of numbers. That was a substitute for axioms. But as we
look at N! and know it
to be all the possible ways of arranging N distinct items into a
sequence, we realize that
N! is a representation of "All Possible Digit Arrangements". For 1,2,3
we have
123
132
231
213
312
321
Now think of each of those arrangements as a cell, so that for a
Sphere we have the
number 254! would be the number 10^500 but not a number represented by
a line
on the sphere that goes around and around many times. Rather the
number 254! for
the sphere in Elliptic geometry is a complete building of that sphere
from the inside
out of 10^500 individual cells. If the sphere were represented by 3!
it would have
just 6 cells.
So this confluence of ideas, of my earlier work in axiomatizing
numbers as All Possible
Digit Arrangements has now blossomed over into the idea that N! is All
Possible
Digit Arrangements and that they are cells that build up the entire
Elliptic geometry.
Now the cells can be all on the surface of the sphere and the cells
can be ordered so the
cells can represent TIME. So that a atom of plutonium would have more
time-cells than
an atom of iron or bismuth.
And we begin to see how multiplication breaks down from that of
Euclidean geometry
where we have no problem in 254! as a line that goes very long and
far. But in Elliptic
geometry where the long line bends back around to the start and then
it looses meaning.
So that multiplication in Elliptic geometry is not the same as
multiplying in Euclidean.
Transfer Principle
<plutonium.archime...@gmail.com> wrote:
> In my other book, I discuss how the numbers piand "e" come into
> existence as the fact that in physics, plutonium of the Plutonium
> Atom Totality has 22 subshells in 7 shells and for with 19
> subshells are occupied, which translates in rational number form
> of 22/7 and 19/7. So here we see directly how Physics creates
> numbers as a reflection of physical interaction of matter.
I usually don't like to bump up threads that are this old, but I
just remembered that today is Pi Approximation Day, and so I bump
up one of the few recent threads that mentions the particular
approximation of pi, namely 22/7, that inspired this day.
I notice that AP tries to find a way to connect the approximation
22/7 to his Plutonium Atom Totality theory. Remember that I'm not
an Atom Totalitarian, and so I don't agree that the promixity of
22/7 to pi proves that the universe is a Plutonium atom.
But Happy Pi Approximation Day anyway, AP. Meanwhile, since I'm
here in this thread anyway, let me respond to some old discussions.
Jeffries:
"I await the outcry about the double standard of Mr Plutonium's being
allowed to invoke fabulous entities like a walking virus but
forbidding others from talking about finite numbers larger than
10^500"
From context, common sense implies that AP really intended to say
that the _ratio_ of the circumference of the universe to the length
of a virus is less than 10^500, _not_ that viruses can walk. Hence
no double standard.
Jesse Hughes:
"Yeah, but AP is right: rumdummy math is so lyrical that someone is
just
*bound* to use it in a song."
Speaking of names of theories, the poster Inverse19 Mathematics has
given the name "Fermatists" to those who believe that Wiles's proof
of FLT is valid. Notice that AP agrees with I19 in rejecting the
validity of the proof. So I wonder whether "Fermatism" is a better
name for the mainstream theory, with "Fermatists" being the name
for the adherents of the mainstream theory.
Jesse F. Hughes
> On Jul 4, 12:24 am, Archimedes Plutonium
> <plutonium.archime...@gmail.com> wrote:
>> In my other book, I discuss how the numbers piand "e" come into
>> existence as the fact that in physics, plutonium of the Plutonium
>> Atom Totality has 22 subshells in 7 shells and for with 19
>> subshells are occupied, which translates in rational number form
>> of 22/7 and 19/7. So here we see directly how Physics creates
>> numbers as a reflection of physical interaction of matter.
>
> I usually don't like to bump up threads that are this old, but I
> just remembered that today is Pi Approximation Day, and so I bump
> up one of the few recent threads that mentions the particular
> approximation of pi, namely 22/7, that inspired this day.
>
> I notice that AP tries to find a way to connect the approximation
> 22/7 to his Plutonium Atom Totality theory. Remember that I'm not
> an Atom Totalitarian, and so I don't agree that the promixity of
> 22/7 to pi proves that the universe is a Plutonium atom.
I'm not positive, but I *think* that AP is claiming that pi is exactly
22/7, not approximately.
--
"To solve this problem, we define a security flag, known as the 'evil'
bit, in the IPv4 [RFC791] header. Benign packets have this bit set to
0; those that are used for an attack will have the bit set to 1."
-- RFC 3514
Transfer Principle
> Transfer Principle <lwal...@lausd.net> writes:
> > I notice that AP tries to find a way to connect the approximation
> > 22/7 to his Plutonium Atom Totality theory. Remember that I'm not
> > an Atom Totalitarian, and so I don't agree that the promixity of
> > 22/7 to pi proves that the universe is a Plutonium atom.
> I'm not positive, but I *think* that AP is claiming that pi is exactly
> 22/7, not approximately.
Is he? I decided to start a Google search for old AP posts to find
out the answer. I found one interesting post all the way from 1993,
where the OP claimed that he had found a proof of Fermat's Last
Theorem (oh, and BTW, the OP's name was Andrew _Wiles_), and AP
(still posting as Ludwig Plutonium) challenged him by giving the
10-adics (AP-adics) as a counterexample. Then he started discussing
Atom Totality.
Anyway, I found a relevant discussion from 24th August, 2007. I'm
loath to bring up old posts since that leads to arguments, but this
was the most recent that I could find:
Archimedes Plutonium wrote:
>> Chapter: (pi) and (e) explained
>> A theory of physics that explains everything has to explain why pi has
>> a value of 3.14159.... [...]
>> And since the Atom Totality has a Riemannian geometry of a shape like
>> a sphere the Atom Totality has a circumference and an diameter. Since it is
>> one big atom or 231Plutonium which is the 5f6 and has
>> 22 subshells inside of 7 shells. Now that is the *Rational
>> Approximation of the subshells and shells*. The actual number of subshells and
>> shells yields the number (pi) precisely.
Proginoskes wrote:
> It looks like AP is saying that pi is exactly 22/7 here. Either that,
> or the number of shells or subshells is an irrational number. AP
> should know better, with him having been a math major a long time ago.
Tribble wrote:
He seems to be saying that 22/7 is a rational approximation
to the "actual number [ratio] of subshells and shells", which
is "pi precisely". So it seems that he's saying that pi is the
ratio between the actual number of subshells and shells,
which means that pi is a rational number.
Either that or he's saying that the number of shells and
subshells are not integers. It's hard to tell.
In other words, inconclusive.
Maybe I'll go and ask AP directly, in one of his more recent
threads so that he'll notice.
Archimedes Plutonium
With my newfound wisdom about 10^500 as boundary between finite and
infinite. Let me
close the debate over approx or exact 22/7 as 22 subshells in 7
shells.
Quantum Physics is duality logic, and things come in duals.
Mathematics is Aristotelian logic where everything is either yes or no
and nothing in between. In physics, you can have 22 subshells/ 7
shells as being both exactly and approximately. In other words,
Physics has a more comprehensive idea or theory of what it means to be
transcendental. If the wavefunction of physics is uncollapsed then the
electron is not a single ball but a bunch of dots scattered in space.
If the electron is collapsed and moving in the wire, it is a ball
object.
So the problem when mathematicians read AP's talk, is that they come
from a background of linear Aristotelian logic that either pi is exact
or is approx. A physicist coming to this discussion with a foundation
of quantum mechanics of duality logic, knows that a plutonium atom of
its 22subshells in 7 shells is both rational and irrational/
transcendental. It does not bother the physicist at all because he
knows that the electron is both particle and wave in
the Double Slit Experiment. It surely bugs the heck out of the
mathematician who expects everything to be either exact or approx.
Physics can have both, and does have both.
And this is another reason why Physics is above mathematics, in that
math is a subset of physics. For physics has both duality logic and
the Aristotelian logic for a confined range.
When we make a boundary of finite-number with infinite number at
10^500, we also are doing
mathematics a huge favor, by saying that Aristotelian Logic is only in
this range and beyond,
we cannot be certain that multiplication, addition or any other
mathematical operation is valid
or consistent. So when we mark the boundary at 10^500 we are saying
that mathematics can be that linear logic, that Aristotelian logic
confined to that range.
Does that clear things up? Physics is broader than math and in physics
pi and "e" are both
exactly 22/7 and 19/7 and also approx 22/7 and 19/7. Physics can have
both for it has, always, both particle and wave. But mathematics is
more narrow and a subset of physics and
can only have a yes or no.
Now this brings up the interesting question as to what physics deems
as the meaning of
"transcendental number" in mathematics. Apparently it has something to
do with the collapsed wavefunction or the uncollapsed wavefunction.
Whether the electron is a single ball, or is a dot-cloud. A
transcendental number would thus correspond geometrically to a
electron-dot-cloud. A algebraic number would thus correspond to the
collapsed wavefunction and thus a single ball. So out of this, we can
start seeing a better understanding of what it means to be a
transcendental number versus a algebraic number.