Constructivist or Crank?
Math1723
always been a bit cranky in his Mathematics. You know the usual: has
problems with infinity, doesn't like Georg Cantor, doesn't like the
Axiom of Choice, etc. Recently, he told me that I was being unfair in
judging him to be a "crank" but was rather a "Constructivist" (or
"Intuitionist", he seems to use those terms interchangeably). I know
a little about it, as it was a Philosophy of Mathematics popularized
by Brouwer and others about a century ago. My friend argues that it
is still a legitimate Mathematical perspective, and I am doing him
(and other Constructivists) a disservice when I speak of Cantorian
infinities as fact.
I do not feel my friend is truly a Constructivist, as he accepts the
Law of the Excluded Middle and some existence proofs (which
Constructivists do not). However, this discussion has made me wonder:
Is Constructivism still considered a legitimate part of Mathematics?
What is the appropriate way to respond to this?
Aatu Koskensilta
> However, this discussion has made me wonder: Is Constructivism still
> considered a legitimate part of Mathematics?
Sure.
> What is the appropriate way to respond to this?
By pointing out your friends ramblings are not in fact in accord with
any school of thought we meet in constructivist mathematics. There are
many brands of constructivism but in each and every of them it is a
central tenet that the meaning of mathematical statements is to be
explained in terms of proofs or constructions. The law of the excluded
middle amounts, on any constructivist reading, to the assertion we can
decide every mathematical problem, an obviously problematic stance.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
FredJeffries
>
> Is Constructivism still considered a legitimate part of Mathematics?
It's even relevant to one of your other interests: see
Wattenberg, Frank (1988). "Nonstandard Analysis and Constructivism?"
Studia Logica 47 (3).
http://www.jstor.org/stable/20015382
http://www.springerlink.com/content/t4nm4314r8526070/
Math1723
>
> It's even relevant to one of your other interests: see
> Wattenberg, Frank (1988). "Nonstandard Analysis and Constructivism?"
> Studia Logica 47 (3).
>
> http://www.jstor.org/stable/20015382
> http://www.springerlink.com/content/t4nm4314r8526070/
Wow, very interesting. I find Nonstandard Analysis very interesting,
but I hadn't considered it to be remotely related to anything
Constructivist. Admittedly, I'm surprised, since both the analytic
and set-theoretic constructions of *R seem to involve things that
would be unacceptable to Constructivists (or so I would have
thought). Particularly with the embracing of infinite and
infinitesimal values.
Thanks for the link!
Helmut Richter
> A buddy of mine (thank God he he has not yet discovered sci.math) has
> always been a bit cranky in his Mathematics. You know the usual: has
> problems with infinity, doesn't like Georg Cantor, doesn't like the
> Axiom of Choice, etc. Recently, he told me that I was being unfair in
> judging him to be a "crank" but was rather a "Constructivist" (or
> "Intuitionist", he seems to use those terms interchangeably).
The difference between a mathematician and a crank is not so much what he
does but how he does it. If someone finds some mathematical notion or some
axiom dubious, he may do so if he is able
- to understand what he criticises to an extent that his criticism does
not only hit a straw man built by himself
- to explain to other mathematicians what he finds dubious (they need not
share his qualms but they should understand why some people have them)
- either to develop a consistent alternative theory or to admit that he
has none.
I think, people like Brouwer meet these criteria. In contrast to that, a
typical crank will present no or wrong ideas and proofs and, when they are
not accepted, complain that everybody is so brainwashed that he cannot see
the truth. (Of course, I do now know which category your buddy is in.)
--
Helmut Richter
Math1723
>
> By pointing out your friends ramblings are not in fact in accord with
> any school of thought we meet in constructivist mathematics. There are
> many brands of constructivism but in each and every of them it is a
> central tenet that the meaning of mathematical statements is to be
> explained in terms of proofs or constructions. The law of the excluded
> middle amounts, on any constructivist reading, to the assertion we can
> decide every mathematical problem, an obviously problematic stance.
From a Formalist perspective, I essentially see Constructivism as just
another system with certain axioms missing (so that the Law of the
Excluded Middle is undecidable, etc.) I frankly don't find it
terribly interesting, as so much of Mathematics gets lost, but I
certainly can accept that you can build a consitent system by
axiomitizing only the Constructivist postulates.
But taking off my Formalist hat, I must say I find it strange to have
a religious belief in Constructivism. What can be more obvious than
the L.E.M. Perhaps my Platonism is showing here, but I believe (for
example) that the Googolplex-th digit of pi is fixed, and the truth
value of the Reimann Hypothesis is constant, whether or not we ever
determine it.
I accept that believing that Fermat's Last Theorem was true even back
when the dinosaurs roamed the Earth is no less religious than what
Constructivists hold dear. Perhaps it's merely a difference of
religions. But if you start with a Formalist perspective, it doesn't
matter what you personally believe, everyone can agree that: beginning
with these Constructivist assumptions, you get these Constructivist
theorems, and beginning with these other assumptions, you get other
theorems. What I think distinguishes the cranks is their insistence
that certain theorems are wrong, period, and their unwillingness to
reason based upon assumptions, and go where logic leads.
My opinion. Sorry, returning my soapbox.
Math1723
>
> (Of course, I do now know which category your buddy is in.)
Oh he's definitely a crank. He's my buddy, and I love him to death,
but he's still just a crank behind Constructivist clothing.
Speaking of which, have you ever read "Mathematical Cranks" by
Underwood Dudley? [ http://www.amazon.com/Mathematical-Cranks-Spectrum-Underwood-Dudley/dp/0883855070
] It's actually quite entertaining.
Tim Little
> From a Formalist perspective, I essentially see Constructivism as
> just another system with certain axioms missing (so that the Law of
> the Excluded Middle is undecidable, etc.)
That's missing almost the entirety of Constructivism.
> But taking off my Formalist hat, I must say I find it strange to have
> a religious belief in Constructivism.
I'd say it is no stranger than any other belief, and much less strange
than most. In fact I'd find it difficult to call any belief in
constructivism "religious". If anything, in your implied analogy it's
a *lack* of belief in entities that cannot be directly proven to
exist.
> What can be more obvious than the L.E.M. Perhaps my Platonism is
> showing here
Mathematical Platonism, on the other hand, appears rather strange to
me.
> but I believe (for example) that the Googolplex-th digit of pi is
> fixed
Given a fairly standard set of definitions, it is easily provable that
all models will agree on the Googleplex-th digit of pi. So I think it
is reasonable to call this "true".
I am not at all sure that all models will agree on the Googleplex-th
digit of Chaitin's Omega. I expect its definition includes some
programs for which halting behaviour is formally undecidable in a
great many systems. As a Platonist, what would you say about Omega's
googleplexth digit?
> and the truth value of the Reimann Hypothesis is constant, whether
> or not we ever determine it.
It has not been proven (to my knowledge), that all reasonable models
of arithmetic will agree on the truth value of the Riemann Hypothesis.
I would be much more reluctant to accept that it has constant truth
value.
What is your Platonic opinion on the truth value of the Continuum
Hypothesis?
--
Tim
Math1723
> On 2011-01-04, Math1723 <anonym1...@aol.com> wrote:
>
> > From a Formalist perspective, I essentially see Constructivism as
> > just another system with certain axioms missing (so that the Law of
> > the Excluded Middle is undecidable, etc.)
>
> That's missing almost the entirety of Constructivism.
Okay then. I apparently do not understand Constructivism. Feel free
to fill me in on what I am missing.
> I am not at all sure that all models will agree on the Googleplex-th
> digit of Chaitin's Omega. I expect its definition includes some
> programs for which halting behaviour is formally undecidable in a
> great many systems. As a Platonist, what would you say about Omega's
> googleplexth digit?
I am not familiar enough with Chaitin's constant to give you an
informed answer to that at this time.
> What is your Platonic opinion on the truth value of the Continuum
> Hypothesis?
Similar to that of Euclid's Parallel Postulate (and other Undecidable
Propositions): if you wish it to be true, simply add that axiom to
your list; if not, you can add one of its negations. Being
undecidable, you are free to choose either way, and there are models
in which it is true, and model in which it is false (just as there is
Euclidean Geometry and at least two Non-Euclidean Geometries). If you
don't care either way, you can leave the axiom out altogether, and
deal with theorems which do not touch upon the matter. (Leaving the
Parallel Postulate undecided will give you only theorems which
correspond to true statements in both types of Geometries.)
David Bernier
> On 2011-01-04, Math1723<anony...@aol.com> wrote:
>> From a Formalist perspective, I essentially see Constructivism as
>> just another system with certain axioms missing (so that the Law of
>> the Excluded Middle is undecidable, etc.)
>
> That's missing almost the entirety of Constructivism.
>
Suppose we have a constructive proof that if p is a prime and
k >= 1, then there is a field with p^k elements.
Does the proof contain a method to construct such a field
for a given p and n?
I'm assuming that all results used in the proof also have
constructive proofs.
As to what a constructive proof is, at the minimum the
Law of the Excluded Middle couldn't be used.
-- David
Transfer Principle
> On Tue, 4 Jan 2011, Math1723 wrote:
> > A buddy of mine (thank God he he has not yet discovered sci.math) has
> > always been a bit cranky in his Mathematics. You know the usual: has
> > problems with infinity, doesn't like Georg Cantor, doesn't like the
> > Axiom of Choice, etc. Recently, he told me that I was being unfair in
> > judging him to be a "crank" but was rather a "Constructivist" (or
> > "Intuitionist", he seems to use those terms interchangeably).
> The difference between a mathematician and a crank is not so much what he
> does but how he does it. If someone finds some mathematical notion or some
> axiom dubious, he may do so if he is able
I've been criticized more than almost every other sci.math
poster for failing to understand this distinction. So let
me try your suggested definition (which I take to be the
definition of a regular mathematician, a NON-"crank"):
> - to understand what he criticises to an extent that his criticism does
> not only hit a straw man built by himself
OK. But that's the hard part -- is determining whether the
poster truly understands what he criticizes or not. This
is especially true when posters on both sides start
throwing insults around.
> - to explain to other mathematicians what he finds dubious (they need not
> share his qualms but they should understand why some people have them)
Most posters have no trouble saying what they find dubious,
whether it's the Axiom of Infinity, uncountable sets, the
existence of two distinct infinite expansions for certain
rationals, abstract algebra, the lack of a largest finite
number, and so on.
> - either to develop a consistent alternative theory or to admit that he
> has none.
But of course, developing a consistent alternate theory is
rather difficult. I've seen many threads in which posters
attempt to do so, but fail to meet the high standards of
mathematical rigor. And many other posters become tired of
seeing these failed attempts.
Thanks for the suggestions.
Tim Little
> Okay then. I apparently do not understand Constructivism. Feel free
> to fill me in on what I am missing.
Everything about *why* one might choose to study systems with such
axioms. In particular, the concept that a mathematical entity can
only be proven to exist by constructing it.
The Law of the Excluded Middle is held (at least partially) invalid
not necessarily as a primary principle, but because it leads to
non-constructive existence proofs.
> I am not familiar enough with Chaitin's constant to give you an
> informed answer to that at this time.
Ah, okay. The idea is that for a certain computer language, every
program is a string of length n, and Chaitin's Omega can be defined as
the sum of 2^-n for all programs that halt. The sum is bounded below
by 0 and bounded above by 1.
Presumably, a Platonist would believe that every program either halts
or it does not, and so the constant is well-defined. I am not so sure
that it could be considered well-defined, since it (provably!) cannot
be defined in any formal system.
>> What is your Platonic opinion on the truth value of the Continuum
>> Hypothesis?
>
> Similar to that of Euclid's Parallel Postulate (and other
> Undecidable Propositions): if you wish it to be true, simply add
> that axiom to your list; if not, you can add one of its negations.
> Being undecidable, you are free to choose either way
Undecidability is only relative to a formal system. *Every*
nontrivial proposition is undecidable in some formal system. What
makes CH more "undecidable" than, say, the googleplexth digit of pi?
(I realize this is a bit of a loaded question. I am nonetheless
interested in the reasoning behind any answer)
--
Tim
Math1723
>
> >> What is your Platonic opinion on the truth value of the Continuum
> >> Hypothesis?
>
> > Similar to that of Euclid's Parallel Postulate (and other
> > Undecidable Propositions): if you wish it to be true, simply add
> > that axiom to your list; if not, you can add one of its negations.
> > Being undecidable, you are free to choose either way
>
> Undecidability is only relative to a formal system. *Every*
> nontrivial proposition is undecidable in some formal system. What
> makes CH more "undecidable" than, say, the googleplexth digit of pi?
Maybe I didn't explain it well. The Parallel Postulate (at least to
me) is not inherently true or false. There are models in which it is
true and models in which it is false. You decide which model by
choosing whether to postulate PP or ~PP. CH is no different. There
are models in which it is true and models in which it is false. It's
undecidable, so you decide.
The Googolplexth digit of pi, on the other hand, is not undecidable
(at least not in the same way). There is only one answer to the
question. Just because I don't know that answer (and probably never
will) does not change this. It is as certain as 2+2=4.
Hundreds of millions of years ago, no one knew the first decimal digit
of pi. No dinosaur knew or cared. But I contend that the answer was
still 1 ... as it shall ever be long after humanity has died out.
Han de Bruijn
> On 2011-01-05, Math1723 <anonym1...@aol.com> wrote:
>
> > Okay then. I apparently do not understand Constructivism. Feel free
> > to fill me in on what I am missing.
>
> Everything about *why* one might choose to study systems with such
> axioms. In particular, the concept that a mathematical entity can
> only be proven to exist by constructing it.
In Constructivism, axioms are NOT the stuff mathematics is built upon.
It's the other way around. We construct (the rest of) mathematics from
e.g. the Naturals (one HAS to start somewhere, not ?) and then axioms,
and logic as well, come up as entities accompanying that process. This
at least is constructivism ("intuitionism") as originally conceived by
L.E.J. Brouwer.
> The Law of the Excluded Middle is held (at least partially) invalid
> not necessarily as a primary principle, but because it leads to
> non-constructive existence proofs.
>
> > I am not familiar enough with Chaitin's constant to give you an
> > informed answer to that at this time.
>
> Ah, okay. The idea is that for a certain computer language, every
> program is a string of length n, and Chaitin's Omega can be defined as
> the sum of 2^-n for all programs that halt. The sum is bounded below
> by 0 and bounded above by 1.
Constructivism is the primary paradigm in Computer Science.
> Presumably, a Platonist would believe that every program either halts
> or it does not, and so the constant is well-defined. I am not so sure
> that it could be considered well-defined, since it (provably!) cannot
> be defined in any formal system.
>
> >> What is your Platonic opinion on the truth value of the Continuum
> >> Hypothesis?
>
> > Similar to that of Euclid's Parallel Postulate (and other
> > Undecidable Propositions): if you wish it to be true, simply add
> > that axiom to your list; if not, you can add one of its negations.
> > Being undecidable, you are free to choose either way
>
> Undecidability is only relative to a formal system. *Every*
> nontrivial proposition is undecidable in some formal system. What
> makes CH more "undecidable" than, say, the googleplexth digit of pi?
>
> (I realize this is a bit of a loaded question. I am nonetheless
> interested in the reasoning behind any answer)
>
> --
> Tim
Han de Bruijn
Han de Bruijn
> On Jan 4, 11:53 am, Helmut Richter <hh...@web.de> wrote:
>
>
>
> > (Of course, I do now know which category your buddy is in.)
>
> Oh he's definitely a crank. He's my buddy, and I love him to death,
> but he's still just a crank behind Constructivist clothing.
Be kind to him ! Maybe better call him a "pink constructivist", a term
coined up quite a while ago by Torkel Franzen(+). (Pink constructivism
is constructivism without all the trouble needed to make it rigourous.
It's like not really eating the pudding, just smelling the flavour.)
I am a pink constructivist myself. As a physicist by education, I'm
rather focussed on applications, not so much on foundations - exept
when the latter represent a problem for the applications.
> Speaking of which, have you ever read "Mathematical Cranks" by
> Underwood Dudley? [http://www.amazon.com/Mathematical-Cranks-Spectrum-Underwood-Dudley/d...
> ] It's actually quite entertaining.
Han de Bruijn
Math1723
>
> In Constructivism, axioms are NOT the stuff mathematics is built upon.
> It's the other way around. We construct (the rest of) mathematics from
> e.g. the Naturals (one HAS to start somewhere, not ?) and then axioms,
> and logic as well, come up as entities accompanying that process. This
> at least is constructivism ("intuitionism") as originally conceived by
> L.E.J. Brouwer.
The "where you start" *ARE* your axioms, whether you call them that or
not. Okay, so you begin with the Naturals. Well then, your first
axiom(s) involve postulating their existence. Axioms are nothing more
than your starting assumptions. (These include the basic axioms of
logic as well, which you also appear to be starting with.)
> > The Law of the Excluded Middle is held (at least partially) invalid
> > not necessarily as a primary principle, but because it leads to
> > non-constructive existence proofs.
Okay, so your stating axioms do not involve L.E.M. Fine. (Although
frankly I don't see what's wrong with existence proofs, but hey, these
are your axioms, not mine, so start however you'd like.)
My point is: whether you formally define your starting assumptions as
"axioms" or not, they are indeed your axioms. You take them as true
and then proceed. Formalists have no problem with this, and can
easily follow what you are doing. They just need to know what the
rules of the game are.
The main distinction (I see) between Formalists and Constructivists is
that Formalists are agnostic about which axioms to start with. They
are happy to begin wherever you want and see what theorems result.
Constructivists, on the other hand, have very specific axioms they
wish to start with, with an emphasis on certain axioms they are
specifically excluding. From that perspective, Constructivism appears
to be little more than Formalism with some "hard coded" axioms.
(I suspect that this will be objected to, but there you have it.)
Marshall
>
> Speaking of which, have you ever read "Mathematical Cranks" by
> ] It's actually quite entertaining.
I love that book. I found it a fascinating glimpse into some obscure
failure modes of human consciousness. In this way it was somewhat
akin to The Man Who Mistook His Wife for a Hat.
Marshall
Math1723
I'd love to see one on Newsgroup Cranks. Imagine a place where
classic sci.math cranks James Harris, Nathan the Great, Archimedes
Plutonium, Pertti Lounesto and others can have their day in the sun.
It seems that there has been some thought in this direction for such a
sequel: http://ns.comap.com/wwwdev.comap.com/pdf/999/On-Jargon-How-Call-Crank-Crank.pdf
.
Tim Golden BandTech.com
> On Jan 4, 8:53 am, Helmut Richter <hh...@web.de> wrote:
>
> > On Tue, 4 Jan 2011, Math1723 wrote:
> > > A buddy of mine (thank God he he has not yet discovered sci.math) has
> > > always been a bit cranky in his Mathematics. You know the usual: has
> > > problems with infinity, doesn't like Georg Cantor, doesn't like the
> > > Axiom of Choice, etc. Recently, he told me that I was being unfair in
> > > judging him to be a "crank" but was rather a "Constructivist" (or
> > > "Intuitionist", he seems to use those terms interchangeably).
> > The difference between a mathematician and a crank is not so much what he
> > does but how he does it. If someone finds some mathematical notion or some
> > axiom dubious, he may do so if he is able
This is some nice writing Math1723 is putting out.
To me constructivism means that we have freedom to construct, but that
what we construct from is critical, and axioms can support this means,
but the constructivist may take more interest in modifying those
axioms, rather than merely extending off of a set of accepted axioms.
I understand that my own opinion may not be consistent with the
historical usage of the term, but it is a word with serious meaning.
'Utilitarian' within philosophy got misappropriated into pleasure
seekers by some interpretations, and the freedom with which involved
individuals could sway what could have been a strong stance into a
weak one cannot be overlooked. Just one strong anticonstructivist
(probably a highly established person) could sway it. I am not
strongly versed on it, so I'm just able to put my own opinion here.
Anyway, when a newcomer goes ahead and overrides old axioms that
others treat as bedrock the potential for such abuse arises.
The believability of a construction is an individual judgement, or at
least the image of the ideal mathematician should include this level
of scrutiny. This is deeply lacking within the academic environment,
where mimicry is the mode of productivity. As to whether a modern
student is even capable of scrutinizing a construction, well, lets
just admit that this quality is not in their training. Along with this
level of scrutiny comes the ability to treat problems as open, and
this harks back to constructive freedom.
I suppose that it is true that an intuitionist might just have a
higher bar on the complexity of axioms that are acceptable, simply
because he accepts that level. I suspect all axioms are ultimately
intuitionist, or should be, since otherwise we will have a problem
under scrutiny. Mimicry is a fast way around this problem. These
problems must be left open beyond the axiomatic level. Physics is not
so far away from math here, since ordinary perception is relied upon
at some level. Also then the human being itself deserves tremendous
scrutiny within this system, and the mathematician writes himself a
ticket here which may not be valid, especially within an academic
environment so reliant upon mimicry.
He who mimics best is best rewarded in the current system, and it is
possible that more high quality constructions have not risen up for
this simple reason.
I predict that the cartesian product may fail under the constructivist
paradigm. By cartesian product I mean simply the construction of a two
dimensional space from a one dimensional space. We see that this is
not simply done, as for instance we may exist in a three dimensional
space, but cannot freely cross it with itself to construct a six
dimensional space. The means of dimensional construction is no longer
convincing. I only say this through a new lens called polysign
numbers, but the abuses that arise within abstract algebra are fine
indicators. What has been regarded as dimensional increase via
cartesian product can be had in another way; a way which actually can
generate support for spacetime.
- Tim
Math1723
wrote:
>
> This is some nice writing Math1723 is putting out.
Thank you. (I hope you were not being sarcastic.) :-)
> To me constructivism means that we have freedom to construct, but that
> what we construct from is critical, and axioms can support this means,
> but the constructivist may take more interest in modifying those
> axioms, rather than merely extending off of a set of accepted axioms.
I have always been uncomfortable with understanding what exactly
Constructivists believe. I am also a bit uncomfortable with the
vehemence of their writings (granted, this was a century ago). Most
of my Mathematical training in college and grad school never even
intersected with this philosophical issue, and it was (and still is)
easy to simply ignore this Mathematical sect.
I started by stepping back and considering things from a Formalist
perspective. I may be unclear what exactly Constructivists believe,
but whatever it is, it should be able to be formalized into an
axiomatic schema. When I suggested this, someone (presumably a
Constructivist) rejected this, saying I was missing the point of it.
Perhaps so, but it seems to me you can't get away from saying you are
starting with certain assumptions and rules of inference, and why not
just lay them out and move forward that way? Formalists are fine with
that.
I was also taken to task about my referring to Constructivism as being
"religious". Well, sorry, but I think it is. To be fair, I have no
problem calling my Platonism religious as well. But the difference
is, at least I can discuss these questions without getting into an
Inquisition about it.
Alright, for example, let's take the Axiom of Choice. I believe in
the Axiom of Choice. It seems very intuitive to me to be able to take
one item out of each of an arbitrary collection of sets. I never
understood why there were people who objected to it. Pretty straight
forward to me. But even though my belief/religion/Platonism/whatever
has me believe in A.C., I can certainly study other axiomatic systems
in which Choice is false. There is a richness of Mathematical
theorems and interesting objects (such as infinite but Dedekind finite
sets) to study when doing so.
My belief/religion/Platonism/whatever also has me believe in the Law
of the Excluded Middle. This is why I don't "believe" in
Constructivism. To deny LEM seems completely bizarre to me. Worse
still, it becomes exceedingly frustrating when a basic logical tool is
removed from consideration (which is why I have found Smooth
Infinitesimal Analysis so unfulfilling). Still, I certainly accept
the results that come from Constructivist Mathematics, since it is
interesting to see what different theorems can come from different
starting presumptions.
> I understand that my own opinion may not be consistent with the
> historical usage of the term, but it is a word with serious meaning.
> 'Utilitarian' within philosophy got misappropriated into pleasure
> seekers by some interpretations, and the freedom with which involved
> individuals could sway what could have been a strong stance into a
> weak one cannot be overlooked. Just one strong anticonstructivist
> (probably a highly established person) could sway it. I am not
> strongly versed on it, so I'm just able to put my own opinion here.
> Anyway, when a newcomer goes ahead and overrides old axioms that
> others treat as bedrock the potential for such abuse arises.
I think what can change some "cranks" into a "non-cranks" would be the
simple process of organizing their assumptions into a clean and
rigorous axiomatic schema. This process requires a person to be
willing to let go of emotional attachments to what he wants the
theorems to be, but instead to let the flow of logic go where it
will. It doesn't matter, for example, if you emotionally desire
0.999... to be less than 1. It matters only what your assumptions and
the laws of logic dictate.
As for "overriding" old axioms, there really is no such thing. It is
merely a matter of defining what you mean, if you decide to use terms
of assumptions that are different from the norm.
Consider Archimedes Plutonium. Here's a crank who once "proved" FLT
by substituting the natural numbers with p-adic numbers. A.P.
believes that his p-adics *should be* the "true" natural numbers
(whatever that means). However, he could stop being a crank tomorrow
by simply acknowledging that the definition of his numbers (which he
wants to call "natural numbers") are not the same as the traditional
Mathematical definition. Then he can show how FLT applies (or
doesn't) to these other numbers, and no one would be in disagreement.
But it's A.P.'s religious fervor which prevents him from becoming
legit.
So, just define what you mean, and for God's sakes, don't take a
common mathematical term (like "natural number") and change its
meaning behind your back. You will quickly be dispatched to crankhood
once you make statements which would be absurd using the common
meanings.
As for Constructivism, apparently I am not the right person to talk
with. I sincerely desire to give an accurate accounting of this
philosophy (even though I don't share it), but other presumed
Constructivists have me I have "missed their point", so there you have
it.
Good Luck.
Math1723
>
> Be kind to him ! Maybe better call him a "pink constructivist", a term
> coined up quite a while ago by Torkel Franzen(+). (Pink constructivism
> is constructivism without all the trouble needed to make it rigourous.
> It's like not really eating the pudding, just smelling the flavour.)
>
> I am a pink constructivist myself. As a physicist by education, I'm
> rather focussed on applications, not so much on foundations - exept
> when the latter represent a problem for the applications.
LOL, "Pink Constructivist", I love it! In any case, I don't believe
the issue with him is a level of pinkness as it is his refusal to let
go of some attachments to ideas that are absurd (and I suspect that he
knows is the case, but has too much of an emotional investment to
actually let go).
Jesse F. Hughes
> I was also taken to task about my referring to Constructivism as being
> "religious". Well, sorry, but I think it is. To be fair, I have no
> problem calling my Platonism religious as well. But the difference
> is, at least I can discuss these questions without getting into an
> Inquisition about it.
It is probably less controversial if you simply call these philosophical
differences rather than religious beliefs.
--
Conservative, n:
A statesman who is enamored of existing evils, as distinguished
from the Liberal who wishes to replace them with others.
-- Ambrose Bierce
Math1723
> Math1723 <anonym1...@aol.com> writes:
> > I was also taken to task about my referring to Constructivism as being
> > "religious". Well, sorry, but I think it is. To be fair, I have no
> > problem calling my Platonism religious as well. But the difference
> > is, at least I can discuss these questions without getting into an
> > Inquisition about it.
>
> It is probably less controversial if you simply call these philosophical
> differences rather than religious beliefs.
Fair point.
Tim Little
> attachments to what he wants the theorems to be, but instead to let
> the flow of logic go where it will.
The lack of such willingness is already one feature that distinguishes
essentially all mathematical cranks from non-cranks. So it boils down
to "cranks could become non-cranks if they changed those traits that
make them cranks".
--
Tim
mike3
I'm curious about this "crank" phenomenon. What is it about infinity
and all that that makes it so awful that they want to reject it?
Han de Bruijn
Sigh ! How I recognize that .. But people get older, and - sometimes -
become wiser as well.
Han de Bruijn
Han de Bruijn
The reason why we want to reject infinity is not so cranky. As far as
mathematics is a _science_, it should meet scientific standards. There
is simply NO scientific evidence for "infinity". Completed infinity is
a theological / religious element in mathematics, and hence it should
not be there at all.
Note. In fact there are two kind of infinities playing a role. We have
the good old concept of a LIMIT in the first place, which is perfectly
acceptable (see my Integral questions in this group). But on the other
hand we have infinite Cardinals and Ordinals, quite different stuff ..
I find mathematics a bit schizophrenic in this respect. Why not all of
infinity based upon limits ? Cannot imagine anything else is needed.
Han de Bruijn
Tim Little
> I'm curious about this "crank" phenomenon. What is it about infinity
> and all that that makes it so awful that they want to reject it?
I don't think many crank reject infinity altogether. Most seem to
have very specific ideas they feel *must* hold about infinities,
regardless of how they are defined. Of course, another property of
most math cranks is that they don't know what a definition is and
refuse to learn, so perhaps that is not too surprising.
--
Tim
Aatu Koskensilta
> As far as mathematics is a _science_, it should meet scientific
> standards. There is simply NO scientific evidence for "infinity".
What sort of "scientific evidence" is there for the number 3 or the
number 2^321^44426523?
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Tim Little
> As far as mathematics is a _science_, it should meet scientific
> standards.
It is not a science at all. It is an activity that may be *used* in
the sciences.
--
Tim
Han de Bruijn
> Han de Bruijn <umum...@gmail.com> writes:
>
> > As far as mathematics is a _science_, it should meet scientific
> > standards. There is simply NO scientific evidence for "infinity".
>
> What sort of "scientific evidence" is there for the number 3 or the
> number 2^321^44426523?
Oh well, admittedly that argument of mine is: too concise for comfort.
Han de Bruijn
Han de Bruijn
> On 2011-01-06, Han de Bruijn <umum...@gmail.com> wrote:
>
> > As far as mathematics is a _science_, it should meet scientific
> > standards.
>
> It is not a science at all. It is an activity that may be *used* in
> the sciences.
Only if that activity meets scientific standards.
Han de Bruijn
Marshall
> On Jan 6, 7:47 am, mike3 <mike4...@yahoo.com> wrote:
>
> > I'm curious about this "crank" phenomenon. What is it about infinity
> > and all that that makes it so awful that they want to reject it?
>
> The reason why we want to reject infinity is not so cranky. As far as
> mathematics is a _science_, it should meet scientific standards. There
> is simply NO scientific evidence for "infinity".
There is no scientific evidence for "zero" either.
Marshall
Han de Bruijn
As far as the _real_ (read: "floating point") "zero" is concerned, you
have a point.
Han de Bruijn
JT
I always thought that math was a construct of our mind to deal with
existence noneexistence of elements, i can not see the idea of math
being developed in a vaccua without objects, groups, fractioned
objects.
And it has developed from there, as far as i see it all mathematical
operands steem from 1 weither you count, summation of a group of
elements or divide an element into fractions.
It all steem from one, i can not for my mind understand why someone
would think of a noneexistent element as part of this group of
operands called numbers. I can see why some want to call it 0 zero and
be the result from an operation, but the use of it beyond that seem
like shadowplay to me.
Regardless if there can be an infinite granularity of one nathematical
element,object is hard to say but probably.
But i feel that nonequatified mathematical operands should not be used
in evaluation of mathematical expressions, i have a very limited
mathematical knowledge, but i am a good problem solver. And from that
i know that it is the quality of the tools that may gain you success
or failure, so one most chose one tools careful when trying to solve a
problem, and if there are no suitable tools you may have to invent
some to solve the problem.
I hardly remember what a limit is but someone maybe can shed me some
light upon the subject with a few words.
JT
Tim Golden BandTech.com
> On Jan 5, 10:22 am, "Tim Golden BandTech.com" <tttppp...@yahoo.com>
> wrote:
>
>
>
> > This is some nice writing Math1723 is putting out.
>
> Thank you. (I hope you were not being sarcastic.) :-)
No, I am sincere, and I know that usenet posters are frail on
encouragement. It is great to have stong and open thinkers here, and I
am happy to place you in that category based on early impressions.
Still, I mean no alliance and am happy to point out below that all of
mathematics is constructed, and it is the root of this word that I
would prefer we use, rather than some bouffant interpretation. Each
individual deserves credit for their own ability to provide an
interpretation, or a variation of past interpretations. As men are
born as children and as math is built by men then as the child is a
blank slate so must mathematics be at the outset.
>
> > To me constructivism means that we have freedom to construct, but that
> > what we construct from is critical, and axioms can support this means,
> > but the constructivist may take more interest in modifying those
> > axioms, rather than merely extending off of a set of accepted axioms.
>
> I have always been uncomfortable with understanding what exactly
> Constructivists believe. I am also a bit uncomfortable with the
> vehemence of their writings (granted, this was a century ago). Most
> of my Mathematical training in college and grad school never even
> intersected with this philosophical issue, and it was (and still is)
> easy to simply ignore this Mathematical sect.
>
> I started by stepping back and considering things from a Formalist
> perspective. I may be unclear what exactly Constructivists believe,
> but whatever it is, it should be able to be formalized into an
> axiomatic schema. When I suggested this, someone (presumably a
> Constructivist) rejected this, saying I was missing the point of it.
> Perhaps so, but it seems to me you can't get away from saying you are
> starting with certain assumptions and rules of inference, and why not
> just lay them out and move forward that way? Formalists are fine with
> that.
Yes, but by denying any freedom at the axiomatic level of construction
the chance at discovering new fundamentals withers. This is perhaps
the most exciting part, which few in this day will appreciate, because
it has been schooled out of them. The idea that we are not done
building out the fundamentals; that no matter how long existing
mathematics stands that the correct approach will always be an open
one which includes scrutiny of its axioms and toying with their
variations; this is a powerful stance that I believe the
constructivist of this day should adhere to, though we should as well
admit that the probability of finding such new fundamental material is
only slight, and that as we travel down that path we may be merely
verifying existing mathematics, but this makes one a true
mathematician rather than a mime.
Here I see you've contradicted yourself. If a variation on old axioms
exist and that variation has extensions then the old axioms become
more dubious. For instance the symbology of the real number has often
been taken as fundamental, yet it can be broken down as
s x
where s is sign and x is unsigned magnitude. The s is a discrete form
and has a modulo two behavior. This form is extensible and while we
will not generally see the real number constructed with the axiomatic
form
- 1 + 1 = 0
it is this form which allows for the generalization of sign such that
a three-signed form can take on the property
- 1 + 1 * 1 = 0 .
These ones can be replaced with a constant or even a single variable,
or even an n-signed value. This three-signed form recovers the complex
numbers from the same ruleset that constructs the real numbers. It
builds a family of number systems.
You see, whether you personally are willing to grant the
generalization of sign which has previously been restricted to a
modulo two property could be a fair instance by which a conservative
will deny, but a progressive will entertain. Regardless the fact
remains that this is a personal choice, but to deny that this is a
construction would be foolhardy. The real numbers are merely a
construction, and that they are named 'real' suggests that we ought to
consider physics as well as philosophy when we get to the level that
you are discussing. You seem to be willing to stay within mathematics,
but at the blank state level of a child who would construct the first
math then this stance does not work, for there would be little to work
with. As I see you accept the 'natural' number as fundamental, then
let's include nature into this bridge, which I would say physics
covered, but to some this may be more acceptable. To isolate
mathematics will never be meaningful. It's source lays in the pursuit
of truth, and this is a more adequate principle than LEM. If LEM comes
to be observed then we have a stronger candidate than if LEM is
denied.
>
> Consider Archimedes Plutonium. Here's a crank who once "proved" FLT
> by substituting the natural numbers with p-adic numbers. A.P.
> believes that his p-adics *should be* the "true" natural numbers
> (whatever that means). However, he could stop being a crank tomorrow
> by simply acknowledging that the definition of his numbers (which he
> wants to call "natural numbers") are not the same as the traditional
> Mathematical definition. Then he can show how FLT applies (or
> doesn't) to these other numbers, and no one would be in disagreement.
> But it's A.P.'s religious fervor which prevents him from becoming
> legit.
>
> So, just define what you mean, and for God's sakes, don't take a
> common mathematical term (like "natural number") and change its
> meaning behind your back. You will quickly be dispatched to crankhood
> once you make statements which would be absurd using the common
> meanings.
There are easier ways to attain crankhood here, such as merely
thinking freely.
Beware, for you may be next. Really I am willing to entertain the
human race as a bunch of cranks. The evidence is everywhere, and the
social aspects of the human mind indicate that mimicry is the
strongest response, and that language follows directly from mimicry,
and in that mathematics is such language then there are serious
problems with humans approaching mathematics, especially when they
write themselves a blank ticket. Usenet paints an adequate picture of
real human interactions; unfiltered and accountably so.
>
> As for Constructivism, apparently I am not the right person to talk
> with. I sincerely desire to give an accurate accounting of this
> philosophy (even though I don't share it), but other presumed
> Constructivists have me I have "missed their point", so there you have
> it.
Well, you seem to have failed to reflect a notion of flexibility at
the axiomatic level which to me is very important to the notion of
construction. Still I admit that I am askance to the 'Constructivist'
paradigm that you seem to be railing against. None the less
construction itself should be maintained without letting it become a
bouffant term. All mathematicians practice construction, but the ones
who merely mimic the ones who came before are lacking in their
abilities. As we pursue construction we should be more open to failure
and its detection, and this is generally lacking in modern academics,
where the best mime will take the cake, and the variants will be
demoted.
>
> Good Luck.
Good luck to you too. I'll be looking for your posts.
- Tim
Math1723
wrote:
>
> Yes, but by denying any freedom at the axiomatic level of construction
> the chance at discovering new fundamentals withers.
Formalists do not deny "any freedom at the axiomatic level of
construction" since you are free to begin with whatever axioms you'd
like. You may find that some axioms you take will result in an
inconsistent formal system. Other times, you will end up with
something which is equivalent to, or a subset of, a current system.
Others still yield something different but not very interesting.
Every now and again, you end up with something new and interesting.
This is essentially how Non-Euclidean Geometry became investigated.
It began with people trying to prove Euclid's Parallel Postulate, by
starting with its negation, using that as an axiom, and hoping to
yield and inconsistent system. To everyone's surprise, rather than an
inconsistent system, new forms of geometries were discovered, provably
shown to be no less consistent than Euclidean Geometry.
> Here I see you've contradicted yourself. If a variation on old axioms
> exist and that variation has extensions then the old axioms become
> more dubious.
I don't see how. Just because there are consistent Geometries with
differing axioms with respect to the Parallel Postulate, how does that
make the axiom (or its negation) dubious? By definition, an axiom is
a statement that is taken as true within the corresponding model. As
long as the set of axioms you start with are not inconsistent (for
example, taking both PP and ~PP as axioms), the resulting theorems
will represents truths of any model to which it applies.
> You see, whether you personally are willing to grant the
> generalization of sign which has previously been restricted to a
> modulo two property could be a fair instance by which a conservative
> will deny, but a progressive will entertain.
Again, as a formalist, I don't see this as a conservative vs.
progressive issue. Although I am not familiar with these "polysign"
numbers (I have never heard of them prior to recent posts on
sci.math), but as long as you define them using a consistent set of
axioms, feel free to see where they take you. If you end up with an
inconsistency, then you merely need to reexamine your beginning set of
axioms.
> There are easier ways to attain crankhood here, such as merely
> thinking freely.
I disagree. Crankhood is marked by stubbornness and refusal to accept
the results of beginning hypotheses (ironically, the very thing they
think non-cranks are doing). Typically, it is found by an emotional
embracing (or repulsion) of certain ideas, irrespective of their
mathematical merit. There are cranks who refuse to accept, for
example, the consistency of transfinite ordinals and cardinals, and
will refer to Mathematicians as "Cantorians" (as if they are
"followers" of Georg Cantor). But they are missing the point. You
don't *have* to adopt the Axiom of Infinity (or any other ZFC axiom)
if you don't want, for whatever private system you wish. But what you
do have to do is accept the Mathematical proofs that say "If you start
from these assumptions, the following will result." Such cranks have
such an emotional attachment to the denial of the axioms (due to some
very naive intuityions about infinity), they cannot bring themselves
accept relative Mathematical consistency. They treat these as
religious doctrine, rather than Mathematics.
As stated earlier, I believe in the Axiom of Choice. What could be
simpler than taking one item from each of a collection of sets? Yet,
I am happy to investigate systems using the Axiom of Determinacy (an
Axiom which is inconsistent with A.C.). So what? I can leave my
religion at home when I do Mathematics. I know many Mathematicians
who object to the concept of Infinitesimals, and simply don't believe
they "exist". Yet they all acknowledge the relative consistency of
Non-Standard Analysis (whether they find it interesting or not).
Cranks, on the other hand, do not dispassionately accept the results
of whatever system you are in. They refuse to accept the logic of the
matter, but would rather hold onto some emotional intuition which, in
the end, cannot be consistently maintained.
> > As for Constructivism, apparently I am not the right person to talk
> > with. I sincerely desire to give an accurate accounting of this
> > philosophy (even though I don't share it), but other presumed
> > Constructivists have me I have "missed their point", so there you have
> > it.
>
> Well, you seem to have failed to reflect a notion of flexibility at
> the axiomatic level which to me is very important to the notion of
> construction.
I am not sure what you mean by this. I would argue that Formalists
are the most flexible with respect to the axiomatic level. Choose
your axioms, be my guest!
With respect to Constructivism, I certainly accept that beginning with
their assumptions, you get Constructivist theorems. Personally, I
find it less interesting, since so much is lost within such a
framework. It is essentially a subset of what I already know. You
get the same basic theorems of most things, but lose some of the most
interestingly unusual objects (non-measurable sets, etc). Worse
still, the removal of the Law of the Excluded Middle is particularly
annoying, robbing me of a basic tool in Mathematics. But hey, you can
change your logical axioms just as you can change you mathematical
ones. If that's your thing, have at it.
If you are truly interested in Constructivist Mathematics, one area of
study you might enjoy (related to another sci.math post) is Smooth
Infinitesimal Analysis [see: http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf
]. It is much simpler than Non-Standard Analysis, and may be
valuable, if your desire is to become more Mathematically rigorous.
Due to its denial of L.E.M., I found it personally frustrating, but
you may have a different take.
LudovicoVan
> With respect to Constructivism, I certainly accept that beginning with
> their assumptions, you get Constructivist theorems. Personally, I
> find it less interesting, since so much is lost within such a
> framework. It is essentially a subset of what I already know.
On the contrary, constructivism is open to assumptions and closed in
the rules of derivation. What you get is *certifiably* correct, so an
*essential* subset of what can be conceived. (My take of course.)
-LV
Math1723
What "assumptions" do Constructivists have that differ from the usual
axioms? What do you mean by it being "certifiably correct"? Godel
proved that no such system can prove its own consistency, and also
that it is necessarily incomplete. So how is this different than any
other formal system? Also, what makes a subset "essential"? (Other
than the circular perspective that "essential" is merely those axioms
we already accept.)
LudovicoVan
> On Jan 6, 11:51 am, LudovicoVan <ju...@diegidio.name> wrote:
> > On Jan 6, 4:37 pm, Math1723 <anonym1...@aol.com> wrote:
>
> > > With respect to Constructivism, I certainly accept that beginning with
> > > their assumptions, you get Constructivist theorems. Personally, I
> > > find it less interesting, since so much is lost within such a
> > > framework. It is essentially a subset of what I already know.
>
> > On the contrary, constructivism is open to assumptions and closed in
> > the rules of derivation. What you get is *certifiably* correct, so an
> > *essential* subset of what can be conceived. (My take of course.)
>
> What "assumptions" do Constructivists have that differ from the usual
> axioms?
Nope. On the contrary...
> What do you mean by it being "certifiably correct"?
I mean that you are missing the core point: of enforcing some specific
restrictions *in order to* get some specific kind of results.
> Godel
> proved that no such system can prove its own consistency, and also
> that it is necessarily incomplete. So how is this different than any
> other formal system?
Reread what I said in my previous post, then restart from the very
first reply you got in this thread, by Aatu Koskensilta:
< https://groups.google.com/group/sci.math/msg/7185f8b8052e5352 >
> Also, what makes a subset "essential"? (Other
> than the circular perspective that "essential" is merely those axioms
> we already accept.)
*Essential* has got a precise definition, which is anything but so
vacuous as you assume.
E.g. in modal logic, again from A.K. in another context:
< https://groups.google.com/group/sci.logic/msg/4ad49aa4158fc893 >
(*Essential* does not prove God, but *essential* proves *essential*.
There are circles and circles!)
-LV
Virgil
<179eb515-42d2-4624...@30g2000yql.googlegroups.com>,
Han de Bruijn <umu...@gmail.com> wrote:
Mathematics is not now and never has been and never will be a "science",
since its truths, if any, are neither established nor destroyed by
physical experiment or physical evidence.
While the APPLICABILITY of certain mathematics to specific physical
problems may be tested experimentally, that in no way tests the "truth"
of the mathematics itself.
Virgil
>
> > As far as mathematics is a _science_, it should meet scientific
> > standards. There is simply NO scientific evidence for "infinity".
Math is not a science at all since its truths are not established or
falsified by physical evidence.
The legitimacy of applying a mathematical model to a physical situation
is quite a separate issue from the legitimacy of that mathematics in
isolation. And the criteria for testing those legitimacies are quite
different.
Virgil
<8b4c0639-b39b-4adc...@w17g2000yqh.googlegroups.com>,
Han de Bruijn <umu...@gmail.com> wrote:
I am unaware of any "scientific" evidence for any "number" at all.
What physical experiment establishes the existence of the number "one"?
Virgil
<9cd83fb0-5a70-42cc...@o4g2000yqd.googlegroups.com>,
Han de Bruijn <umu...@gmail.com> wrote:
Scientific standards require physical evidence of existence before
accepting such existence.
What physical evidence proves the existence of a number?
Math1723
>
> > > On the contrary, constructivism is open to assumptions and closed in
> > > the rules of derivation. What you get is *certifiably* correct, so an
> > > *essential* subset of what can be conceived. (My take of course.)
>
> > What "assumptions" do Constructivists have that differ from the usual
> > axioms?
>
> Nope. On the contrary...
??? Mine wasn't a Yes or No question. Perhaps I didn't phrase it
properly, so I'll try it again. In your post, you mention
"constructivism is open to assumptions". My question for you is: What
assumptions are those (aside from Axioms we already shared from
standard Mathematics)?
> > What do you mean by it being "certifiably correct"?
>
> I mean that you are missing the core point: of enforcing some specific
> restrictions *in order to* get some specific kind of results.
I appear to still be missing the point. How does one "enforce some
restrictions" mathematically, aside from formally stating them as such
(such as with an axiomatic system)? What "specific kind of results"
are you referring to? Are there particular theorems you are trying to
prove?
> > Godel
> > proved that no such system can prove its own consistency, and also
> > that it is necessarily incomplete. So how is this different than any
> > other formal system?
>
> Reread what I said in my previous post, then restart from the very
> first reply you got in this thread, by Aatu Koskensilta:
>
> <https://groups.google.com/group/sci.math/msg/7185f8b8052e5352>
I have done so. I am sorry to say, I still do not know what you mean,
or if this is somehow an answer to my question. Could you be more
specific in what you mean? (You apparently are seeing a relevant
point being made by this link which I am not.)
> > Also, what makes a subset "essential"? (Other
> > than the circular perspective that "essential" is merely those axioms
> > we already accept.)
>
> *Essential* has got a precise definition, which is anything but so
> vacuous as you assume.
>
> E.g. in modal logic, again from A.K. in another context:
>
> <https://groups.google.com/group/sci.logic/msg/4ad49aa4158fc893>
I am (apparently) unaware of the "precise definition" of your use of
"essential". I tried to Google "essential subset" and "mathematics",
as well as "modal logic" and "essential", but did not see anything
obviously related to our discussion. Could you explain, or perhaps
link me to, an explanation of what you mean here?
> (*Essential* does not prove God, but *essential* proves *essential*.
> There are circles and circles!)
Since we were not (to my knowledge) discussing theology, I will be
kind and view these two sentences as simply a non sequitur.
LudovicoVan
> > (*Essential* does not prove God, but *essential* proves *essential*.
> > There are circles and circles!)
>
> Since we were not (to my knowledge) discussing theology, I will be
> kind and view these two sentences as simply a non sequitur.
Feel free to get lost.
-LV
Math1723
I am sorry you have chosen to terminate this discussion, but I
certainly understand if you feel there is no value in continuing.
Good Luck to you.
Tim Golden BandTech.com
> On Jan 6, 10:35 am, "Tim Golden BandTech.com" <tttppp...@yahoo.com>
> wrote:
>
>
>
> > Yes, but by denying any freedom at the axiomatic level of construction
> > the chance at discovering new fundamentals withers.
>
> Formalists do not deny "any freedom at the axiomatic level of
> construction" since you are free to begin with whatever axioms you'd
> like. You may find that some axioms you take will result in an
> inconsistent formal system. Other times, you will end up with
> something which is equivalent to, or a subset of, a current system.
> Others still yield something different but not very interesting.
> Every now and again, you end up with something new and interesting.
>
> This is essentially how Non-Euclidean Geometry became investigated.
> It began with people trying to prove Euclid's Parallel Postulate, by
> starting with its negation, using that as an axiom, and hoping to
> yield and inconsistent system. To everyone's surprise, rather than an
> inconsistent system, new forms of geometries were discovered, provably
> shown to be no less consistent than Euclidean Geometry.
>
> > Here I see you've contradicted yourself. If a variation on old axioms
> > exist and that variation has extensions then the old axioms become
> > more dubious.
>
> I don't see how. Just because there are consistent Geometries with
> differing axioms with respect to the Parallel Postulate, how does that
> make the axiom (or its negation) dubious?
Well, I think in the case of geometries you are holding up, but if an
axiom becomes broken or unnecessary through some other changes, then
the new interpretation can challenge the old construction. This is to
say that we should remain open to practicing mistaken assumptions, and
should continue to scrutinize those old assumptions. Where you are
seeing a split into two acceptable systems then isn't your LEM
principle compromised? Perhaps there is a third or a fourth system
which will encompass them all under some more primitive
generalization. It seems that the main way to equate these systems is
simply to grant another dimension and constrain the behaviors in the
higher dimensional system. This raising of dimension behavior seems to
go on quite a lot as a means to advance things, but that doesn't make
it right. I seem to remember a Discover article in which Brian Greene
mentions 500 dimensional string theory or some such.
> By definition, an axiom is
> a statement that is taken as true within the corresponding model. As
> long as the set of axioms you start with are not inconsistent (for
> example, taking both PP and ~PP as axioms), the resulting theorems
> will represents truths of any model to which it applies.
> there is no value in continuing.
Sure. We are really on the same page I think. To me though there are
continuous levels of judgement which take place, and we typically
consider these in terms of 'strength' or some such human judgement. As
with you LEM concept this continuum level of judgement goes beyond the
discretely stated axiom and a boolean truth. Sometimes the conflicts
are not so straight forward a an exact inversion.
> > You see, whether you personally are willing to grant the
> > generalization of sign which has previously been restricted to a
> > modulo two property could be a fair instance by which a conservative
> > will deny, but a progressive will entertain.
>
> Again, as a formalist, I don't see this as a conservative vs.
> progressive issue. Although I am not familiar with these "polysign"
> numbers (I have never heard of them prior to recent posts on
> sci.math), but as long as you define them using a consistent set of
> axioms, feel free to see where they take you. If you end up with an
> inconsistency, then you merely need to reexamine your beginning set of
> axioms.
Yes, right, but as to who else will follow the freedom that one takes;
this becomes a boundary, and for instance in journals they work from
established concepts and many will not welcome a contradiction of
their primary assumptions; even if the topic is in their genre. The
crank label applies here on that continuum, where some are more
dismissive than others.
>
> > There are easier ways to attain crankhood here, such as merely
> > thinking freely.
>
> I disagree. Crankhood is marked by stubbornness and refusal to accept
> the results of beginning hypotheses (ironically, the very thing they
> think non-cranks are doing). Typically, it is found by an emotional
> embracing (or repulsion) of certain ideas, irrespective of their
> mathematical merit. There are cranks who refuse to accept, for
> example, the consistency of transfinite ordinals and cardinals, and
> will refer to Mathematicians as "Cantorians" (as if they are
> "followers" of Georg Cantor). But they are missing the point. You
> don't *have* to adopt the Axiom of Infinity (or any other ZFC axiom)
> if you don't want, for whatever private system you wish. But what you
> do have to do is accept the Mathematical proofs that say "If you start
> from these assumptions, the following will result." Such cranks have
> such an emotional attachment to the denial of the axioms (due to some
> very naive intuityions about infinity), they cannot bring themselves
> accept relative Mathematical consistency. They treat these as
> religious doctrine, rather than Mathematics.
I agree with the overemotional bit, but I would concede that anyone
with a sincere interest in the topic of mathematics deserves to be
admitted, especially here on usenet. Different people have different
levels of synthetic and analytic abilities. Some are trying their bad
ideas out here on usenet, but this does not exclude them from having a
good idea in the future. If anything the most active minds playing out
variations are the most likely to make a discovery, including the
'cranks' of which by many here I am considered to be one.
>
> As stated earlier, I believe in the Axiom of Choice. What could be
> simpler than taking one item from each of a collection of sets? Yet,
> I am happy to investigate systems using the Axiom of Determinacy (an
> Axiom which is inconsistent with A.C.). So what? I can leave my
> religion at home when I do Mathematics. I know many Mathematicians
> who object to the concept of Infinitesimals, and simply don't believe
> they "exist". Yet they all acknowledge the relative consistency of
> Non-Standard Analysis (whether they find it interesting or not).
> Cranks, on the other hand, do not dispassionately accept the results
> of whatever system you are in. They refuse to accept the logic of the
> matter, but would rather hold onto some emotional intuition which, in
> the end, cannot be consistently maintained.
>
> > > As for Constructivism, apparently I am not the right person to talk
> > > with. I sincerely desire to give an accurate accounting of this
> > > philosophy (even though I don't share it), but other presumed
> > > Constructivists have me I have "missed their point", so there you have
> > > it.
>
> > Well, you seem to have failed to reflect a notion of flexibility at
> > the axiomatic level which to me is very important to the notion of
> > construction.
>
> I am not sure what you mean by this. I would argue that Formalists
> are the most flexible with respect to the axiomatic level. Choose
> your axioms, be my guest!
I see now that you do accept the axiomatic freedom. But as to when the
training begins on this topic, well, it is very much lacking in the
academic system, where students will have a hard enough time simply
mimicing the accepted topics. The level of creative freedom versus the
quantity of assumptions we are trained to emulate is badly imbalanced.
>
> With respect to Constructivism, I certainly accept that beginning with
> their assumptions, you get Constructivist theorems. Personally, I
> find it less interesting, since so much is lost within such a
> framework. It is essentially a subset of what I already know. You
> get the same basic theorems of most things, but lose some of the most
> interestingly unusual objects (non-measurable sets, etc). Worse
> still, the removal of the Law of the Excluded Middle is particularly
> annoying, robbing me of a basic tool in Mathematics. But hey, you can
> change your logical axioms just as you can change you mathematical
> ones. If that's your thing, have at it.
>
> If you are truly interested in Constructivist Mathematics, one area of
> study you might enjoy (related to another sci.math post) is Smooth
> Infinitesimal Analysis [see:http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf
> ]. It is much simpler than Non-Standard Analysis, and may be
> valuable, if your desire is to become more Mathematically rigorous.
> Due to its denial of L.E.M., I found it personally frustrating, but
> you may have a different take.
I took a look at the paper by John Bell. I don't really think much of
the selection of the function f(0)=1, f(x)=0 otherwise. The idea of
microstraightness is interesting, but I think that under a graphical
interpretation the function f that challenges this may not actually be
a function if we connect f=1 to f=0. This graphical interpretation
provides a series of possible values at f(0) between 1 and 0. It is a
bad mix of discrete and continuous processes. If this 'function' was
on the integers I think it would be more acceptable.
I don't relate to the axiom of choice. I have poor appreciation of it.
To me the words
'axiom of choice'
should yield the constructive freedoms as a first axiom, as in we are
free to choose our axioms. But clearly this is my own interpretation,
and likewise the constructivist is pretty foreign. Though I see it all
connected in the wikepedia entries I guess I am ultimately a gut and
physical correspondence type of thinker. When we encounter a physical
correspondence in math form then your LEM thing starts to kick.
- Tim
Math1723
wrote:
>
> Well, I think in the case of geometries you are holding up, but if an
> axiom becomes broken or unnecessary through some other changes, then
> the new interpretation can challenge the old construction.
Axioms can become "broken" only if they are inconsistent with the
other axioms. If an axiom has been proven undecidable with respect to
the others, then you can add it or its negation freely, without ever
worrying about introducing an inconsistency. Again as I stated
before, the reason we have Non-Euclidean Geometry was due to some
people trying to "prove" that ~PP was inconsistent. Turned out they
were wrong. Neither PP nor ~PP was inconsistent.
And you needn't wait for an axiom to become "broken". You can, at
anytime, consider a formal system with a differing set of axioms.
> This is to say that we should remain open to practicing mistaken
> assumptions, and should continue to scrutinize those old assumptions.
My point is that there is no such thing as "mistaken" assumptions ...
just different assumptions.
> Where you are seeing a split into two acceptable systems then isn't
> your LEM principle compromised?
Not at all. I have no problem acknowledging the consistency of such a
system. It is equally valid from a Metalogical and Metamathematical
viewpoint. I just find it annoying.
> Sure. We are really on the same page I think. To me though there are
> continuous levels of judgement which take place, and we typically
> consider these in terms of 'strength' or some such human judgement. As
> with you LEM concept this continuum level of judgement goes beyond the
> discretely stated axiom and a boolean truth. Sometimes the conflicts
> are not so straight forward a an exact inversion.
Hmmm ... not sure we are on the same "level". I don't see "continuous
levels of judgment" here. I see formal systems each of which is
relatively consistent.
Now I grant you, there is the Platonist side of me that sees truth in
all of this. I would not find it interesting to, say, remove the
logical law of Modus Ponens (Given Propositions A->B and A, we can
conclude B). I just seems a waste of time. To me, Modus Ponens is as
self-evident as can be. But I think the same thing of Double Negation
(that from ~~A you can infer A), yet Constructivists disagree, so
there you have it.
> I took a look at the paper by John Bell. I don't really think much of
> the selection of the function f(0)=1, f(x)=0 otherwise. The idea of
> microstraightness is interesting, but I think that under a graphical
> interpretation the function f that challenges this may not actually be
> a function if we connect f=1 to f=0. This graphical interpretation
> provides a series of possible values at f(0) between 1 and 0. It is a
> bad mix of discrete and continuous processes. If this 'function' was
> on the integers I think it would be more acceptable.
I'm sure Constructivists won't care to hear this, but I'd love to see
Smooth Infinitesimal Analysis furthered with new axioms which do not
require the rejection of LEM. Nilpotent infinitesimals and
microstraightness are intriguing concepts, and I'd like to see them
made more accessible to the more mainstream mathematical world.
> I don't relate to the axiom of choice. I have poor appreciation of it.
> To me the words
> 'axiom of choice'
> should yield the constructive freedoms as a first axiom, as in we are
> free to choose our axioms. But clearly this is my own interpretation,
> and likewise the constructivist is pretty foreign. Though I see it all
> connected in the wikepedia entries I guess I am ultimately a gut and
> physical correspondence type of thinker. When we encounter a physical
> correspondence in math form then your LEM thing starts to kick.
More accessible reading (which you might find in a bookstore or
library) is "The Mathematical Experience" by Davis & Hersch [
http://www.amazon.com/Mathematical-Experience-Phillip-J-Davis/dp/0395929687
], a layman's look at Mathematics, with sections the Platonist/
Formalist/Constructivist debate, and interesting looks at different
topics in the field, including various Set Theories, infinities and
even Non-Standard Analysis.
Transfer Principle
> I'm sure Constructivists won't care to hear this, but I'd love to see
> Smooth Infinitesimal Analysis furthered with new axioms which do not
> require the rejection of LEM. Nilpotent infinitesimals and
> microstraightness are intriguing concepts, and I'd like to see them
> made more accessible to the more mainstream mathematical world.
http://en.wikipedia.org/wiki/Dual_number
"In linear algebra, the dual numbers extend the real
numbers by adjoining one new element epsilon with the
property epsilon^2 = 0 (epsilon is nilpotent). The
collection of dual numbers forms a particular
two-dimensional commutative unital associative algebra
over the real numbers. Every dual number has the form
z = a + bepsilon with a and b uniquely determined real
numbers. The plane of all dual numbers is an
'alternative complex plane' that complements the
ordinary complex number plane C and the plane of
split-complex numbers."
Nothing mentioned in the article suggests that the dual
numbers require the rejection of LEM.
Math1723
>
> http://en.wikipedia.org/wiki/Dual_number
>
> "In linear algebra, the dual numbers extend the real
> numbers by adjoining one new element epsilon with the
> property epsilon^2 = 0 (epsilon is nilpotent). The
> collection of dual numbers forms a particular
> two-dimensional commutative unital associative algebra
> over the real numbers. Every dual number has the form
> z = a + bepsilon with a and b uniquely determined real
> numbers. The plane of all dual numbers is an
> 'alternative complex plane' that complements the
> ordinary complex number plane C and the plane of
> split-complex numbers."
>
> Nothing mentioned in the article suggests that the dual
> numbers require the rejection of LEM.
This is true, but is epsilon an infinitesimal? It appears to be more
of an extra dimension. It also doesn't have the micro-straightness
characteristic of SIA.
Having said that though, Dual numbers look interesting, and I
appreciate the suggestion, as it may be a rewarding investigation.
Thanks.
Tim Golden BandTech.com
It's funny; this extra dimension stance is exactly the remark I made
to you on the development of curved spaces. If we use the surface of a
torus as a curved space, then when we add the third dimension, was the
third dimension actually necessary to construct the torus surface
initially? It seems that it was, so long as we constructed these
devices from the real number.
The fact is that when we speak of a 'one dimensional' number that we
actually speak of a bidirectional quality. This observation matches
the build of the polysign number where the ray is treated as
fundamental and so what is traditionally fundamental (the real number)
is composed of two balanced parts. These parts are more fundamental
than the whole, for they carry no sign. They are simply magnitudes,
and so we should consider the construction of spaces from rays as more
fundamental than constructions from the real line. What is the quality
that forms the geometrical line that we all accept of the real number?
This awareness is of an assumption that may never go questioned
because the assumption has been wired in from childhood. The
willingness to attempt such an investigation is a quality that a
conservative mathematician will not entertain, for this subject is
dead and done by many prior great thinkers, right?
There is an issue of physical correspondence, and no physicist will
ever give a perfect real value for anything measured, since there is
always a tolerance figure associated with such values, so that in some
ways there is already a physical correspondence with the dual number,
but the meaning is much simpler under this context, and is more
consistent with the concerns over infinitesimal slop.
- Tim
jbriggs444
> In article
> <9cd83fb0-5a70-42cc-bf62-ffe430fb9...@o4g2000yqd.googlegroups.com>,
> Han de Bruijn <umum...@gmail.com> wrote:
>
> > On Jan 6, 10:43 am, Tim Little <t...@little-possums.net> wrote:
> > > On 2011-01-06, Han de Bruijn <umum...@gmail.com> wrote:
>
> > > > As far as mathematics is a _science_, it should meet scientific
> > > > standards.
>
> > > It is not a science at all. It is an activity that may be *used* in
> > > the sciences.
>
> > Only if that activity meets scientific standards.
>
> Scientific standards require physical evidence of existence before
> accepting such existence.
>
> What physical evidence proves the existence of a number?
You don't usually get very far arguing about trivialities, but...
Suppose that we want to prove that apples exist?
We start with a definition of an apple (round, red, grows
from trees, has seeds but no pit, if you cut the fruit
crosswise there's a five pointed star in the middle,
it's not a Bartlett Pear, etc etc.).
Then we look for an exemplar. In this case, we need
not look much farther than the produce section or an
apple orchard.
Suppose that we want to prove that 3 exists?
Define 0 as a place on the carpet where a pile of
marbles could be.
Define S(x) as what you have after you add a marble
to x.
Define 3 as S(S(S(0)))
Now find a patch of carpet and a bag of marbles and
see if you can prove the existence of 3.
[Note that 0 and S() are traditionally undefined,
accordingly, there are no constraints on what
physical analogues one picks to model them]
In my opinion, that pile of marbles is an exemplar
of the number 3 just as surely as an apple in the
orchard is an exemplar of an apple.
Math1723
wrote:
>
> It's funny; this extra dimension stance is exactly the remark I made
> to you on the development of curved spaces. If we use the surface of a
> torus as a curved space, then when we add the third dimension, was the
> third dimension actually necessary to construct the torus surface
> initially? It seems that it was, so long as we constructed these
> devices from the real number.
I don't recall a discussion about curved spaces. Are you sure it was
with myself you mentioned this.
With respect to my use of the term "dimension", I was speaking
figuratively, due to the two variables, rather than a rigorous spacial
dimension. I meant that the Dual numbers of the form (a + b*e) seemed
more analogous (at least to me) of complex numbers (a + b*i) than it
did of reals with infinitesimals (a + b*e).
> The willingness to attempt such an investigation is a quality that
> a conservative mathematician will not entertain, for this subject
> is dead and done by many prior great thinkers, right?
Again, I tend not to think of this as a conservative vs progression
distinction. It's more a matter of how interesting the topic is to
any given research mathematician. I find Set Theory and Non-Standard
Analysis interesting, but Constructivist Mathematics not. There are
others who feel the reverse. And some both and some neither. The
same can be said here. If it's interesting to you, have at it.
> There is an issue of physical correspondence, and no physicist will
> ever give a perfect real value for anything measured, since there is
> always a tolerance figure associated with such values, so that in some
> ways there is already a physical correspondence with the dual number,
> but the meaning is much simpler under this context, and is more
> consistent with the concerns over infinitesimal slop.
The ideal Mathematical world is a different sphere altogether than the
Physical world. In Physics, you have to deal with the laws of Physics
preventing knowledge at the Quantum level, and what does it mean to
describe position with greater accuracy than Planck length, etc. etc.
Not being a particle physicist myself, I don't study this area.
People in sci.physics may be able to speak more eloquently on the
topic.
FredJeffries
> Smooth Infinitesimal Analysis furthered with new axioms which do not
> require the rejection of LEM.
That would be undermining the motivation for Smooth Infinitesimal
Analysis: There are two domains: the discreet and the continuous.
Neither can be reduced to the other.
Use the correct tool in the correct domain: Discreet Aristotlean logic
in the domain of the discreet, Continuous Intuitionistic logic in the
domain of the continuous.
It's not a question of platonism or formalism or constructivism. It a
question of reductionism.
Cauchy/Dedekind/Cantor/Weierstrass have defined a notion of continuity
in terms of the discreet. Smooth Infinitesimal Analysis is a-whole-
nuther notion of continuity altogether.
http://publish.uwo.ca/~jbell/Comparing%20the%20Smooth%20and%20Dedekind%20Reals.pdf
See page 3:
"The continuum in SIA differs in certain key respects from its
counterpart in constructive analysis CA"
Tim Golden BandTech.com
> On Jan 6, 7:13 pm, "Tim Golden BandTech.com" <tttppp...@yahoo.com>
> wrote:
> > Well, I think in the case of geometries you are holding up, but if an
> > axiom becomes broken or unnecessary through some other changes, then
> > the new interpretation can challenge the old construction.
>
> Axioms can become "broken" only if they are inconsistent with the
> other axioms. If an axiom has been proven undecidable with respect to
> the others, then you can add it or its negation freely, without ever
> worrying about introducing an inconsistency. Again as I stated
> before, the reason we have Non-Euclidean Geometry was due to some
> people trying to "prove" that ~PP was inconsistent. Turned out they
> were wrong. Neither PP nor ~PP was inconsistent.
>
> And you needn't wait for an axiom to become "broken". You can, at
> anytime, consider a formal system with a differing set of axioms.
>
> > This is to say that we should remain open to practicing mistaken
> > assumptions, and should continue to scrutinize those old assumptions.
>
> My point is that there is no such thing as "mistaken" assumptions ...
> just different assumptions.
If every curved space is constructed from flat space, then I believe
that this limitation of construction does deserve attention. I believe
that the correct alternative is to use continuous modulo principles
initially such that 1=0, or some such wrapping condition. We are
comfortable with this usage in terms of angular measure.
We seem to be comfortable with a finitely sized universe of 15 billion
light years. Should we see ourselves out there then the number of
times that we should see ourselves might be an open problem.
>
> > Where you are seeing a split into two acceptable systems then isn't
> > your LEM principle compromised?
>
> Not at all. I have no problem acknowledging the consistency of such a
> system. It is equally valid from a Metalogical and Metamathematical
> viewpoint. I just find it annoying.
Well, where you have constructed a split into two consistent systems I
claim that they are more consistent. You do not address usage of
additional dimensions as a means of bridging the two. The ability to
construct the non Euclidean from the Euclidean with additional
dimensions... I can accept parallel theories, and even parallel
theories with variations, but then those variations naturally cause
criticisms of the multiple options. I guess this goes toward
unification type thinking. This is for about the joinery of the
mathematics with the physical, which I suppose you are not so
concerned with.
>
> > Sure. We are really on the same page I think. To me though there are
> > continuous levels of judgement which take place, and we typically
> > consider these in terms of 'strength' or some such human judgement. As
> > with you LEM concept this continuum level of judgement goes beyond the
> > discretely stated axiom and a boolean truth. Sometimes the conflicts
> > are not so straight forward a an exact inversion.
>
> Hmmm ... not sure we are on the same "level". I don't see "continuous
> levels of judgment" here. I see formal systems each of which is
> relatively consistent.
Oh. Nice distinction. I think I understand.
>
> Now I grant you, there is the Platonist side of me that sees truth in
> all of this. I would not find it interesting to, say, remove the
> logical law of Modus Ponens (Given Propositions A->B and A, we can
> conclude B). I just seems a waste of time. To me, Modus Ponens is as
> self-evident as can be. But I think the same thing of Double Negation
> (that from ~~A you can infer A), yet Constructivists disagree, so
> there you have it.
Neat. I didn't realize that double negation stance. For Boolean logic
where A and B are strictly Boolean I see no way to alter the values of
A and B other than to oscillate them, which then introduces another
dynamic into the logic which goes beyond the simple form but does seem
consistent with the problems that you mention. Whether we can really
exclude the middle in these dynamic functions would challenge the
boolean qualities, which could then obfuscate the double negation.
I agree that Modus Ponens is strong, yet to assess its strength means
investigating its assumptions. As to whether A and B are symmetrically
independent, well, if they were prior to this restiction, then they
certainly are not upon its instantiation. To what degree the symmetry
of B to the more general A is valid depends somewhat on those
formalisms. I find the instantiation of variables to be a bit loose in
existing mathematics, and there are some mild contradictions there
which relate to dependencies like the one above. This can be brought
back around to the cartesian product, and the instantiation of two
Boolean(B) variables b1 and b2 in B then have we made usage of BxB
here? While
b1 -> b2
does not delete the full dimensional freedom it does seem to delete
half of it.
This cartesian product awareness is likewise dubious when we consider
that b1 and b2 are to be free and independent within the construction,
and so upon declaring any dependency upon them then we have challenged
the construction itself. This challenge arguably is initiated upon
insisting that b1 and b2 are orthogonal geometrically. Here I should
really switch over to real values, but I am fairly certain you will
follow this reasoning. If two entities are truly independent then no
relation can be made from one to the other. Thus to what degree the
space preexisted the construction must be considered. This then
weakens the cartesian product to merely being a representation of a
space of greater complexity in terms of copies of one of less
complexity, rather than an actual construction of that greater
complexity space. I apologize if my usage of the word 'space' is poor,
but I do mean to remain nearby to the physical interpretation of space
as much as possible.
>
> > I took a look at the paper by John Bell. I don't really think much of
> > the selection of the function f(0)=1, f(x)=0 otherwise. The idea of
> > microstraightness is interesting, but I think that under a graphical
> > interpretation the function f that challenges this may not actually be
> > a function if we connect f=1 to f=0. This graphical interpretation
> > provides a series of possible values at f(0) between 1 and 0. It is a
> > bad mix of discrete and continuous processes. If this 'function' was
> > on the integers I think it would be more acceptable.
>
> I'm sure Constructivists won't care to hear this, but I'd love to see
> Smooth Infinitesimal Analysis furthered with new axioms which do not
> require the rejection of LEM. Nilpotent infinitesimals and
> microstraightness are intriguing concepts, and I'd like to see them
> made more accessible to the more mainstream mathematical world.
Well, we are literally considering the exclusion of the middle here,
as in a graph which does not connect its parts together. It is a
premise of Bell's construction that is not consistent with any
infinitesimal. Still, I agree that it is interesting, but here the
challenge of constructing these things well will perhaps require
something more than the one dimensional unsplit function. So I offer
you a split function which essentially carries two envelopes and is
thick in its middle. There is the option to do analysis on the two
functions as traditional functions, but we should not exclude the
middle, where an area function exists more literally than in the
traditional integral. To what degree then the traditional integral
actually is a split interpretation, well, I think this whole
construction may be rhetoric, but then again, at the edge of
understanding we are forced to consider things which do not agree with
our assumptions. I do take interest in calculus works and am seeking
something fundamental here to unfold from polysign, but the thick
functions are fairly new to me, and it is interesting that you've
dovetailed to them indirectly.
After enough debate I believe that you and I may agree that what is
fundamental does require the ring of truth. I don't claim the above
argument to be such yet. I see that there is room for criticism of
existing mathematics, and that without this criticism the subject is
quite dead and done. We, as elements of spacetime do not necessarily
have access to its fundaments, and so we construct in hindsight. Its
preexistence or pregeometry is not an arbitrary concept, and I have
some very simple mathematics which is consistent with this claim. This
level of physical correspondence has not been done before; especially
not from a simple algebra. To ponder the physical basis versus physics
upon the physical basis may require this purity, as when we discuss a
topic we require a basis upon which to work e.g. the axiom to the
mathematician. This is somewhat the mathematical pursuit. Newton got
to calculus this way, but he also did number games with his bible, so
the story goes. These two disparate concepts share one common theme;
an ability to tread at the edge, which is an exciting portion of the
body of work; not the stale part. Here though we should admit that
there is a high probablility of failure, especially if we work upon
principles of variation.
>
> > I don't relate to the axiom of choice. I have poor appreciation of it.
> > To me the words
> > 'axiom of choice'
> > should yield the constructive freedoms as a first axiom, as in we are
> > free to choose our axioms. But clearly this is my own interpretation,
> > and likewise the constructivist is pretty foreign. Though I see it all
> > connected in the wikepedia entries I guess I am ultimately a gut and
> > physical correspondence type of thinker. When we encounter a physical
> > correspondence in math form then your LEM thing starts to kick.
>
> More accessible reading (which you might find in a bookstore or
> library) is "The Mathematical Experience" by Davis & Hersch [http://www.amazon.com/Mathematical-Experience-Phillip-J-Davis/dp/0395...
> ], a layman's look at Mathematics, with sections the Platonist/
> Formalist/Constructivist debate, and interesting looks at different
> topics in the field, including various Set Theories, infinities and
> even Non-Standard Analysis.
I'll try and get this book through ILL.
Did you know that Descartes never actually used negative values in his
graphics?
I have not witnessed this first hand, but have two third hand
confirmations of this.
Orthogonality is actually optional too.
- Tim
Math1723
wrote:
>
> Well, where you have constructed a split into two consistent systems I
> claim that they are more consistent. You do not address usage of
> additional dimensions as a means of bridging the two.
I don't know what you mean by "dimensions" here. I don't understand
the concept of "bridging the two" (actually three, as there are two
forms of Non-Euclidean Geometry). To me the axioms belong to a formal
system all its own, in complete isolation to anything else. If you
wish to conjoin mutually compatible systems, I suppose you can simply
union all the axioms of the systems together. But since PP and ~PP
are not mutually consistent, you cannot conjoin the Geometries this
way.
Now, there are submodels within a model (for example, Non-Euclidean
geometry can be modelled in terms of Euclidean Geometry). Is perhaps
this what you mean?
> I agree that Modus Ponens is strong, yet to assess its strength means
> investigating its assumptions. As to whether A and B are symmetrically
> independent, well, if they were prior to this restiction, then they
> certainly are not upon its instantiation.
A and B are independent if both (A and B) and (A and ~B) are each
consistent. There really isn't a strength issue here (at least as far
as I think you are using the term).
> I find the instantiation of variables to be a bit loose in
> existing mathematics, and there are some mild contradictions there
> which relate to dependencies like the one above.
There really isn't such a thing as a "mild" contradiction. If a
contradiction exists between the axioms of a formal system, then the
entire system is hosed. In fact (assuming the system uses the common
logical laws), an inconsistent system will have as theorems *every*
proposition (and its negation).
> > More accessible reading (which you might find in a bookstore or
> > library) is "The Mathematical Experience" by Davis & Hersch [
> > http://www.amazon.com/Mathematical-Experience-Phillip-J-Davis/dp/0395929687
> > ], a layman's look at Mathematics, with sections the Platonist/
> > Formalist/Constructivist debate, and interesting looks at different
> > topics in the field, including various Set Theories, infinities and
> > even Non-Standard Analysis.
>
> I'll try and get this book through ILL.
> Did you know that Descartes never actually used negative values in his
> graphics?
> I have not witnessed this first hand, but have two third hand
> confirmations of this.
> Orthogonality is actually optional too.
Interesting you mention that, as these same authors also have a book
called "Descartes' Dream [ http://www.amazon.com/Descartes-Dream-According-Mathematics-Science/dp/0486442527/
]. I haven't read this second one, so I can't speak to it at this
time.
Tim Golden BandTech.com
> On Jan 7, 11:40 am, "Tim Golden BandTech.com" <tttppp...@yahoo.com>
> wrote:
>
>
>
> > Well, where you have constructed a split into two consistent systems I
> > claim that they are more consistent. You do not address usage of
> > additional dimensions as a means of bridging the two.
>
> I don't know what you mean by "dimensions" here. I don't understand
> the concept of "bridging the two" (actually three, as there are two
> forms of Non-Euclidean Geometry). To me the axioms belong to a formal
> system all its own, in complete isolation to anything else. If you
> wish to conjoin mutually compatible systems, I suppose you can simply
> union all the axioms of the systems together. But since PP and ~PP
> are not mutually consistent, you cannot conjoin the Geometries this
> way.
>
> Now, there are submodels within a model (for example, Non-Euclidean
> geometry can be modelled in terms of Euclidean Geometry). Is perhaps
> this what you mean?
Well, perhaps I am just admitting that I don't fully understand the
development of curved spaces. Further how can they be called curved
without some flat reference? Have a look at
http://en.wikipedia.org/wiki/Non-Euclidean_geometry
and its first graphic with elliptic and hyperbolic curvatures. This is
wrong from the outset, for if we lay the same line down twice or
thrice it should not necessarily toggle as they would show it. So by
laying more than two parallel lines down we'll just have to admit that
there is another quality (a dimension) to those 'lines', which has
simply gone unspecified within the graphic.
>
> > I agree that Modus Ponens is strong, yet to assess its strength means
> > investigating its assumptions. As to whether A and B are symmetrically
> > independent, well, if they were prior to this restiction, then they
> > certainly are not upon its instantiation.
>
> A and B are independent if both (A and B) and (A and ~B) are each
> consistent. There really isn't a strength issue here (at least as far
> as I think you are using the term).
Yes but we're invoking Modus Ponens on A and B; an asymmetrical
operation.
I suppose the resolution that I would hate to spell out is to regard B
as of a different type than A, for B is now somewhat derived from A,
though not fully. A and ~B is not possible so long as we have admitted
A->B. I do see this as nitpicking, but this is more the point of the
conversation; we are free to query these things further, and
regardless of how many formalists have come before and been satisfied
with an argument it does not mean that all the successors must do the
same. The idea that there may be multiple variable types, even if all
of the same value set, is pretty clear in the way that B's symmetry
and freedoms have been constrained, whereas A's have not. We seem to
be aware of constant and variable as complete types. There may be some
more room, and if there is more room then the consequences could be
interesting, and in places like this what is stretching the truth
versus what is a valid new construction, well, this distinction is per
individual; an independent pursuit. This same realization should
probably occur across mathematics generally, so that the subject
remains open. We all are either cranky or cranks, so crank it up; we
just need to be open to errancy, and hopefully not share too much
crap, which this paragraph may be, but its purpose is not.
>
> > I find the instantiation of variables to be a bit loose in
> > existing mathematics, and there are some mild contradictions there
> > which relate to dependencies like the one above.
>
> There really isn't such a thing as a "mild" contradiction. If a
> contradiction exists between the axioms of a formal system, then the
> entire system is hosed. In fact (assuming the system uses the common
> logical laws), an inconsistent system will have as theorems *every*
> proposition (and its negation).
Growl... There are such things as mild contradictions, but it may be
true that they lay beyond the axioms and occur more as accepted habits
or assumptions, such as the usage of the real number as fundamental
unconditionally and such that when we speak of a magnitude we are
forced by tradition to call this thing the 'nonnegative real number',
which is poor construction, for the magnitude is simpler than the real
number and so deserves more fundamental treatment, which polysign does
give it. This happens to have physical correspondence as well. Sadly,
the arithmetic product of signed values has very poor physical
correspondence generally, but there is a sort of a half
correspondence, as in the classical force equations, which use a
product relationship. Signed products are rotational in nature, and
these rotational qualities are interesting. The real value happens to
have a discrete form, but beyond the reals the form can become
continuous.
>
> > > More accessible reading (which you might find in a bookstore or
> > > library) is "The Mathematical Experience" by Davis & Hersch [
> > >http://www.amazon.com/Mathematical-Experience-Phillip-J-Davis/dp/0395...
> > > ], a layman's look at Mathematics, with sections the Platonist/
> > > Formalist/Constructivist debate, and interesting looks at different
> > > topics in the field, including various Set Theories, infinities and
> > > even Non-Standard Analysis.
>
> > I'll try and get this book through ILL.
> > Did you know that Descartes never actually used negative values in his
> > graphics?
> > I have not witnessed this first hand, but have two third hand
> > confirmations of this.
> > Orthogonality is actually optional too.
>
> Interesting you mention that, as these same authors also have a book
> called "Descartes' Dream [http://www.amazon.com/Descartes-Dream-According-Mathematics-Science/d...
> ]. I haven't read this second one, so I can't speak to it at this
> time.
Well, their paperback version of the first book you mentioned has a
drawing of him on the front cover. I'd like to see his original work
that led to the naming of 'cartesian coordinates', but haven't found
it yet.
I posted a message on you NSA thread on modulo principles.
- Tim
Math1723
wrote:
>
> Growl... There are such things as mild contradictions, but it may be
> true that they lay beyond the axioms and occur more as accepted habits
<snip>
Formal systems have been studied pretty rigorously over the past
century, reaching perhaps some of its most surprising conclusions in
the 1930's by Kurt Godel. Suffice it to say that *ANY* inconsistency
amongst the axioms will doom the entire system, causing every and any
statement to be a theorem (since anything proceeds from a falsehood).
As for "habits or assumptions", there are no assumptions of the
systems other than the axioms (both logical and mathematical) and
rules of inference of the formal system. If no inconsistency lives
there, then there is no problem. Not sure what you mean by "habits".
Certainly there is notational and usage difference amongst various
mathematicians, but these could not be called "contradictions" in any
mathematical sense.
> ... such as the usage of the real number as fundamental
> unconditionally and such that when we speak of a magnitude we are
> forced by tradition to call this thing the 'nonnegative real number',
> which is poor construction, for the magnitude is simpler than the real
> number and so deserves more fundamental treatment, which polysign does
> give it.
<snip>
I don't see this "poor construction" you refer to. Although it is
true that some axiomatic systems (typically Algebraic) begin with the
reals already formed, whilst others (typically Set-Theoretic) like to
construct them out of simpler sets, none of this can be consider any
kind of contradiction. They just have different starting points.
(Compare for example Algebraic Topology with Set-Theoretic Topology.)
I can't speak to "polysign" as I know nothing about it.
> I posted a message on you NSA thread on modulo principles.
I will look forward to reading it.
David R Tribble
> Constructivism is the primary paradigm in Computer Science.
Computer programmers don't prove theorems with their programs.
(Unless, of course, they are writing actual theorem-proving inference
programs.)
A typical web application, for example, has nothing in it dealing
with derivations of proofs from axioms.
Just because applications contain well-derived statements
within a formal grammar does not equate them with mathematical
theorems (or existence proofs) within a formal axiomatic system.
David R Tribble
>> Yes, but by denying any freedom at the axiomatic level of construction
>> the chance at discovering new fundamentals withers.
>
Math1723 wrote:
> Formalists do not deny "any freedom at the axiomatic level of
> construction" since you are free to begin with whatever axioms you'd
> like.
We have to remember that axioms are, in essence, implicitly
prefixed with "If", so that for, to take an example, the Peano
Axioms, we get:
If zero is a natural, then <some theorem derived from PA>.
In other words, the axioms from which we choose to build our
particular system at hand are nothing more than assumptions,
and all statements (theorems) derived from them are contingent
on our assuming them to be true. "True" meaning only that
we accept them at face value within the framework of our
system.
Obviously, what we assume is true within one framework
does not have to be true (or even expressible or meaningful)
within a different framework. Each system is just a different
set of assumptions we start with.
David R Tribble
>> Scientific standards require physical evidence of existence before
>> accepting such existence.
>
Virgil wrote:
>> What physical evidence proves the existence of a number?
>
jbriggs444 wrote:
> You don't usually get very far arguing about trivialities, but...
>
> Suppose that we want to prove that apples exist?
>
> We start with a definition of an apple (round, red, grows
> from trees, has seeds but no pit, if you cut the fruit
> crosswise there's a five pointed star in the middle,
> it's not a Bartlett Pear, etc etc.).
>
> Then we look for an exemplar. In this case, we need
> not look much farther than the produce section or an
> apple orchard.
>
> Suppose that we want to prove that 3 exists?
>
> Define 0 as a place on the carpet where a pile of
> marbles could be.
>
> Define S(x) as what you have after you add a marble
> to x.
>
> Define 3 as S(S(S(0)))
>
> Now find a patch of carpet and a bag of marbles and
> see if you can prove the existence of 3.
>
> In my opinion, that pile of marbles is an exemplar
> of the number 3 just as surely as an apple in the
> orchard is an exemplar of an apple.
The difference being that a given apple is really an
actual apple, whereas a pile of three marbles is not
the actual number 3; it is merely an example of a pile
having the property of three-ness.
To quote John Derbyshire, you can't stub your toe
on a seven.
David R Tribble
>> Formalists do not deny "any freedom at the axiomatic level of
>> construction" since you are free to begin with whatever axioms you'd
>> like. You may find that some axioms you take will result in an
>
Tim Golden BandTech.com wrote:
>> If a variation on old axioms
>> exist and that variation has extensions then the old axioms become
>> more dubious.
>
Math1723 wrote:
>> I don't see how. Just because there are consistent Geometries with
>> differing axioms with respect to the Parallel Postulate, how does that
>> make the axiom (or its negation) dubious?
>
Tim Golden BandTech.com wrote:
> Well, I think in the case of geometries you are holding up, but if an
> axiom becomes broken or unnecessary through some other changes, then
> the new interpretation can challenge the old construction.
Not following you. How does an axiom become "broken"?
> This is to
> say that we should remain open to practicing mistaken assumptions, and
> should continue to scrutinize those old assumptions.
If by "assumption" you mean "axiom", then how is an axiom
"mistaken"?
FredJeffries
>
> Constructivism is the primary paradigm in Computer Science.
Science is to computer science as hydrodynamics is to plumbing.
-- Stan Kelly-Bootle, _Computer Language_, Oct 1990
Math1723
> But i feel that nonequatified mathematical operands should not be used
> in evaluation of mathematical expressions ...
<snip>
Can you explain what this means? To understand why you are objecting
to this, I have to know what 'this' is. Particularly, what is
"nonequatified mathematical operands"?
Tim Golden BandTech.com
No, you have not addressed the topic as I've described it.
It is in the usage of a complicated concept to describe a simpler
concept.
If we were to consider magnitude, or distance as one object or topic,
and we were to consider the real number as another topic or object,
then it is fairly clear that the simpler of the two is the magnitude.
That we are forced within the modern system to describe magnitude in
terms of a real number is a failing point.
Within polysign this is exposed more clearly, and this is how I come
to make this criticism, which does happen to be a matter of
construction whereby we have somehow gotten around the axiomatic
level. I have little doubt that these concerns can be brought back to
the axiomatic level, and that we just have not gotten there yet.
The real number can take the representation
s x
where s is a modulo two sign and x is a magnitude. Under this
construction sums of like sign yield
s1 x1 + s1 x2 = s1 (x1 + x2)
e.g. - 3 - 4 = - 7
The product behaves as
(s1 x1)(s2 x2) = (s1 + s2)(x1x2)
e.g. ( - 3 )( - 4 ) = + 12
where s1+s2 is a modulo two sum, and the x operations are merely
magnitudinal sum and product without any sign content. The balance of
the signs comes with the cancellation
- x + x = 0
e.g. - 5.2 + 5.2 = 0
and this balance is as much a geometrical concern as it is any ability
to perform mathematical computation. This is nearly like the dual
number that you were just studying with Transfer Principle, but the
cancellation allows the value to resolve to a singular expression; the
other component being nill. In effect these principles state that the
real number can be regarded as having a negative portion and a
positive portion
- a + b or (a,b)
though with the cancellation law we will always be able to do things
like
( - 1 + 2 ) = ( - 1 + 1 ) + 1 = 0 + 1 = + 1 .
so that the dual component is not actually inconsistent with the
standard representation. The reason to do all of this is because there
are three-signed numbers and four-signed numbers, and so on through
this same ruleset.
There are even one-signed numbers whose behavior are as controversial
as time is in their unidirectional and zero dimensional behavior.
Still, let's not forget that the term 'dimension' refers explicity to
the real line, and any solitary ray system must be less than one
dimension. This is a point at which I could actually use an opinion,
and it would be great to see if you have one.
Anyway, that magnitude is more fundamental than the real number
exposes that the real number can be built from magnitude, which is
more appropriate than defining magnitude per numerical system, where
for the real numbers we will typically see a definition of magnitude
as
sqrt( square( x ))
or some such sign manipulation.
The treatment of the ray as fundamental leads to very nice simplex
based geometry. The three-signed(P3) numbers are the complex numbers
in this new simplex based format. Four-signed(P4) have a tetrahedral
geometry, though it is nothing like the Fuller version, which seems
more and more like quackery to me, and I do try to expose them to this
version in which a general dimensional geometry is exposed all through
that very simple cancellation law which existed in the old traditional
real number:
Sum over s ( s x ) = 0
which expands to
- x = 0 (P1)
- x + x = 0 (P2)
- x + x * x = 0 (P3)
...
These statements above define the geometry and upon rendering an n-
signed value within its geometry this cancellation law is implied, or
inherently invoked. This balancing act need not be considered a
necessary operation within any sum or product algorithm. Numerical
values will continue to resolve without invocation of this law. Though
their values will grow, their balance will be maintained.
It is for this reason that I come to challenge the cartesian product:
it is unnecessary. This then also leads into abstract algebra and much
of existing mathematics like the real numbers as a subset of the
complex numbers, associative algebra, and so on. The polysign
construction is extremely primitive and so even if the claim of
parallel construction is valid within the existing math polysign is
the simpler way out. There is much much more to the scene though.
Polysign contains a behavioral breakpoint that is consistent with
emergent spacetime, with unidirectional time, which is where I seek
your opinion.
It goes something like this: traditionally a zero dimensional entity
would be regarded as a geometrical point. However polysign exposes the
unitary ray as a zero dimensional entity. In that we do freely
construct one dimensional lines in 2D and 3D space, why should we not
allow the construction of the unitary ray? To what degree was it
actually valid to construct the one dimensional line in the 2D space?
To what degree is the traditional 2D space merely constructed from two
1D lines? Hereabouts is the puzzle and where polysign constructs the
2D space from three rays, then it could be that the invocation of a P1
ray within that P3 space could compactify the P3 down to P2. It would
then follow that the standard equivalent requires the compactification
of the plane down to a 1D space when such an operation of imposing a
line in it occurs. There is trouble in here and I have no doubt that
you see I am confused, for I am confused. The details here reek in
other places such as abstract algebra. Dimensional collapse is also a
part of the higher sign systems under some products; denying field
status to P4+.
Sorry I wrote too much here. I should simply have pointed you to
http://bandtech.com/PolySigned
where this is more carefully laid out. I don't mean to drag you into
it if you don't want to apply yourself to it. I will accept a clear
rejection, though I will no doubt argue on the clarity, which I do see
is human based, and continuous. We are in a gray age. Boolean logic
can yield the oscillator, on which the Boolean logician will
necessarily vacillate. It is P2 discrete logic and P1 exists beneath
it. I am arguing for P3, which under the same constraints as P2 can
form a continuum, where parallel theories may exist. The constraint,
not unlike curved space is to limit the discussion to unit magnitude
values. Sorry to get all cryptic, but there is some sense there.
- Tim
Marshall
wrote:
> On Jan 7, 10:30 pm, Math1723 <anonym1...@aol.com> wrote:
>
> > I don't see this "poor construction" you refer to.
>
> No, you have not addressed the topic as I've described it.
> It is in the usage of a complicated concept to describe a simpler
> concept.
"Simpler" is usually a matter of opinion. Objective tests might
be designed, but coming up with a single metric for something
as complex as complexity is unlikely to be achievable without
building in a bias.
The existing number systems we have are very simple. The
consequences of these systems may not be to your liking,
but that is an altogether different matter.
If you want to try to show that polysign numbers are simpler
than real numbers, that might be interesting. If you want to
try to show that "magnitude" (whatever you mean by that)
is simpler than real numbers, that might be interesting.
To show one thing being simpler than another, you have
to be able to concisely define each; let us just say that
I am skeptical of your ability to do that, but you may take
that as a challenge if you like. Once we have a pair of
concise definitions, we can compare them by size, or
some other metric.
Marshall
Math1723
wrote:
>
> No, you have not addressed the topic as I've described it.
> It is in the usage of a complicated concept to describe a simpler
> concept.
> If we were to consider magnitude, or distance as one object or topic,
> and we were to consider the real number as another topic or object,
> then it is fairly clear that the simpler of the two is the magnitude.
It's not clear to me. In fact, I have an opposite perspective. What
you call "magnitude" (which I think is the same thing as what I call
measure), is a much more involved concept.
Perhaps you need to define what you mean by this "magnitude" concept,
as it's not really a term I use in Mathematics (at least in not the
way you are using it).
> That we are forced within the modern system to describe magnitude in
> terms of a real number is a failing point.
The more I read, the less I understand what you mean by this term.
> Within polysign this is exposed more clearly ...
<snip>
Again, I appreciate the polysign explanation, but not sure how that is
helpful here. You appear to be asking much more fundamental questions
about measure vs. distance, and your polysign example already assumes
more involved arithmetic, which builds upon these ideas.
, ... and this is how I come to make this criticism, which does happen
> to be a matter of construction whereby we have somehow gotten around
> the axiomatic level.
Not true at all. You certainly have your polysign axioms, you just
don't formally present them as such. Sadly though, they are not well
defined enough for me to comment. For example, announce rules such
as:
> s1 x1 + s1 x2 = s1 (x1 + x2)
> (s1 x1)(s2 x2) = (s1 + s2)(x1x2)
These look like things which do belong to an axiomatic system. But
you need to be more formal. For example, you start with:
> The real number can take the representation
> s x
> where s is a modulo two sign and x is a magnitude.
You have not defined what sign and magnitude are. Perhaps you want to
express it this way:
Every real number r can be expressed as (s,x) where r = |x|, s=0 for
r>0, s=1 for r<0. (Not sure exactly how you are defining s, as you
are not explicit enough, nor do you say wht s is if r=0.)
Then you can proceed in like manner. But again, you are laying down
axioms. You are just doing it in a very imprecise and non-well-
defined way.
> There are even one-signed numbers whose behavior are as controversial
> as time is in their unidirectional and zero dimensional behavior.
What are "one-signed numbers"? Are these standard reals? If so, I
disagree that there exists "controversy" about them.
In any case, there is no "controversy" once you explicitly define your
terms and lay down your axioms properly.
> It is for this reason that I come to challenge the cartesian
product:
> it is unnecessary. This then also leads into abstract algebra and much
> of existing mathematics like the real numbers as a subset of the
> complex numbers, associative algebra, and so on. The polysign
> construction is extremely primitive and so even if the claim of
> parallel construction is valid within the existing math polysign is
> the simpler way out. There is much much more to the scene though.
> Polysign contains a behavioral breakpoint that is consistent with
> emergent spacetime, with unidirectional time, which is where I seek
> your opinion.
Sadly, it's paragraphs like these that cause people to criticize you
and label you a crank. Cartesian products aren't subject to
"criticism". They fulfill their definitions consistently. Just
because you have created a separate axiomatic system does not obviate
other axiomatic systems. And your system already presumes the
existence of objects (real numbers, for example), which other systems
construct out of simpler ones. Set Theory is arguably the lowest
level, in which only certain very fundamental objects are axiomitized
(the empty set, set operations to build other sets, etc), and from
there natural numbers are constructed, and from them the ordinals, the
cardinals, integers, rationals, reals, complex, polynomials, and on
and on. This is far more fundamental than any polysign "construction"
you have outlined. Now, I am not saying that these ideas aren't
useful or consistent; they may be. It just this business that other
consistent systems are somehow "unnecessary" is just silly.
> Sorry I wrote too much here. I should simply have pointed you to
> http://bandtech.com/PolySigned
> where this is more carefully laid out. I don't mean to drag you into
> it if you don't want to apply yourself to it. I will accept a clear
> rejection, though I will no doubt argue on the clarity, which I do see
> is human based, and continuous. We are in a gray age.
Where you see "gray", I simply see ill-definition and the need for
greater mathematical education.
Aatu Koskensilta
> Set Theory is arguably the lowest level, in which only certain very
> fundamental objects are axiomitized (the empty set, set operations to
> build other sets, etc), and from there natural numbers are
> constructed, and from them the ordinals, the cardinals, integers,
> rationals, reals, complex, polynomials, and on and on.
In standard accounts neither ordinals nor cardinals are constructed
from the natural numbers. It's the other way around: naturals are
identified with finite ordinals (or cardinals).
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Math1723
> Math1723 <anonym1...@aol.com> writes:
> > Set Theory is arguably the lowest level, in which only certain very
> > fundamental objects are axiomitized (the empty set, set operations to
> > build other sets, etc), and from there natural numbers are
> > constructed, and from them the ordinals, the cardinals, integers,
> > rationals, reals, complex, polynomials, and on and on.
>
> In standard accounts neither ordinals nor cardinals are constructed
> from the natural numbers. It's the other way around: naturals are
> identified with finite ordinals (or cardinals).
Ah yes. In my haste to explain, I misremembered that detail. Thanks
for your eye to accuracy, Aatu!
quasi
wrote:
>> happen to be a matter of construction whereby we have somehow
>> gotten around the axiomatic level.
>
>Not true at all. You certainly have your polysign axioms, you just
>don't formally present them as such. Sadly though, they are not well
>defined enough for me to comment. For example, announce rules such
>as:
>
>> s1 x1 + s1 x2 = s1 (x1 + x2)
>> (s1 x1)(s2 x2) = (s1 + s2)(x1x2)
>
>These look like things which do belong to an axiomatic system. But
>you need to be more formal. For example, you start with:
>
>> The real number can take the representation
>> s x
>> where s is a modulo two sign and x is a magnitude.
>
>You have not defined what sign and magnitude are. Perhaps you
>want to express it this way:
>
>Every real number r can be expressed as (s,x) where r = |x|,
I think you meant "where x = |r|". Right?
>s=0 for r>0, s=1 for r<0.
>(Not sure exactly how you are defining s, as you are
>not explicit enough, nor do you say wht s is if r=0.)
quasi
Math1723
>
> I think you meant "where x = |r|". Right?
Oops, right you are, thank you! (It seems I can always count on
someone on sci.math to have my back in catching mistakes!) Thanks!
Tim Golden BandTech.com
Ha. You really don't like polysign I suppose. No, the point is that
what we start with are magnitude and sign, and yes, I can accept that
I have not formally defined magnitude. But you have completely failed
to recognize that this number without any sign content can exist. So I
should provide the gorilla conjecture here:
We can likely train a gorilla to sort sticks by length, but it will be
very challenging to get this same gorilla to perform signed
mathematics.
This statement exposes how much simpler magnitude is from the real
number, and I see that you refute this claim. It is a failure on your
part to admit that simpler things need not be composed from more
complicated things. It is likewise your maintenance of an attachment
to the real number, which polysign does break into.
Numerical representations such as
0.001, 1.001, 2.1
are adequate magnitude selections, though I have to admit that there
may be others that will fit physical reality better. Such values can
be ordered, as the gorilla can sort sticks, and this method of
distinguishing magnitudes as corresponding to stick lengths is more
appropriate since the lens of the modern mathematician does take the
real number as an assumed basis, but we will not see any real stick
even at length zero, let alone beneath there.
>
> Then you can proceed in like manner. But again, you are laying down
> axioms. You are just doing it in a very imprecise and non-well-
> defined way.
>
> > There are even one-signed numbers whose behavior are as controversial
> > as time is in their unidirectional and zero dimensional behavior.
>
> What are "one-signed numbers"? Are these standard reals?
No, P1 carry just one sign, so valid values in P1 could be
- 2, -0.01, -1.2
and these can be summed through the polysign rules
- 2 - 0.01 - 1.2 = - 3.21
or a product can be taken
(- 2)(- 3) = - 6
but these numbers likewise take on geometry through
- x = 0
which is the general cancellation law expressed on P1. That the ray
which defined P1 algebra is zero dimensional is consistent with
interpretations of time, and this is a new concept, for the point was
considered the zero dimensional concept in traditional geometry. It is
roughly in this area that I was hoping for an opinion, but I will
relax on that now.
> If so, I
> disagree that there exists "controversy" about them.
>
> In any case, there is no "controversy" once you explicitly define your
> terms and lay down your axioms properly.
Yes, you are of the persuasion that the real number is fundamental, so
I can move on.
Thanks for taking a look at polysign.
- Tim
Math1723
wrote:
>
> > Every real number r can be expressed as (s,x) where r = |x|, s=0 for
> > r>0, s=1 for r<0. (Not sure exactly how you are defining s, as you
> > are not explicit enough, nor do you say wht s is if r=0.)
>
> Ha. You really don't like polysign I suppose. No, the point is that
> what we start with are magnitude and sign, and yes, I can accept that
> I have not formally defined magnitude. But you have completely failed
> to recognize that this number without any sign content can exist.
"Not liking polysign" states more than is the case. I know little to
nothing about it, other than that which you have shared. I was merely
trying to create a foundation based upon preexisting notions. If you
object to that, you can certainly define them any way you wish. It
was merely my first stab at it knowing nothing else about it.
> So I should provide the gorilla conjecture here:
>
> We can likely train a gorilla to sort sticks by length, but it will be
> very challenging to get this same gorilla to perform signed
> mathematics.
Agreed, but your use of "magnitude" here is merely synonymous with
length (which is essentially a use of the more general concept of
distance). Since length is traditionally defined fairly
straightforwardly (subtraction of values of the end points), I am not
sure your gorilla would find polysign arithmetic any simpler.
> This statement exposes how much simpler magnitude is from the real
> number, and I see that you refute this claim. It is a failure on your
> part to admit that simpler things need not be composed from more
> complicated things. It is likewise your maintenance of an attachment
> to the real number, which polysign does break into.
I don't think I "admitted that simpler things need not be composed
from more complicated things", or had I, why it would necessarily be
"a failure on my part". I am also not sure what you attribute as "my
maintenance of an attachment to the real number", since I used it
merely to model your polysign numbers. If you wish to model it
differently, please feel free to do so.
> Numerical representations such as
> 0.001, 1.001, 2.1
> are adequate magnitude selections, though I have to admit that there
> may be others that will fit physical reality better. Such values can
> be ordered, as the gorilla can sort sticks, and this method of
> distinguishing magnitudes as corresponding to stick lengths is more
> appropriate since the lens of the modern mathematician does take the
> real number as an assumed basis, but we will not see any real stick
> even at length zero, let alone beneath there.
Perhaps my failure is seeing what is different between what you call
"magnitude" and what I see as the range of the non-negative real
numbers. Take a circle of radius 1, for example. It circumference is
pi. I assume you agree that pi is a "magnitude" since it is the
length of something. The Real numbers R are define to be a completely
ordered field, and it is proven that any other completely order field
would be isomorphic to R. Any measurable quantity would map to these
reals, so anything you are doing with "magnitude" would map to R, and
so I am not sure what you object to. Constructing this field so that
it includes all measurable values (including transcendental
irrationals like pi) is not trivial, so hence my use of R.
> Yes, you are of the persuasion that the real number is fundamental, so
> I can move on.
Again, that's saying more that I did. Real numbers can be constructed
from simpler objects (traditionally through Dedekind cuts of sequences
of rationals), but you may also just start by assuming they are there
without too much objection.
> Thanks for taking a look at polysign.
Again, "taking a look at" them may be overstating the case. I looked
at your use of them in our discussion, but really nothing more than
that. My recommendation would be to be more rigorous in your
definitions and explanations, and leave out "criticisms" to other
axiomatic systems which are provably equi-consistent (or at least no
less consistent) with yours.
Good Luck.
Brian Chandler
> On Jan 9, 8:46 am, Math1723 <anonym1...@aol.com> wrote:
> > On Jan 8, 10:53 am, "Tim Golden BandTech.com" <tttppp...@yahoo.com>
> > wrote:
> >
> >
> Ha. You really don't like polysign I suppose.
Well, Timmy old chum, the problem isn't that he or anyone else
"doesn't like" [the cutely article-challenged] polysign, the problem
is that you are a bumbling fool. You don't seem to have realised that
if you want people to be interested in polysign a good start would be
to make yourself look like one smart cookie [I hope that terminology's
correct]; mere silence might even be enough to rouse interest, but you
really need to display an ability to understand things to which you
wish to propose alternatives.
Instead of which you have produced nothing but pomo foam. For
instance...
> No, the point is that
> what we start with are magnitude and sign, and yes, I can accept that
> I have not formally defined magnitude. But you have completely failed
> to recognize that this number without any sign content can exist.
Have you heard of the natural numbers, I wonder? You don't suppose the
natural numbers have "sign content"? As always, from your position of
supreme ignorance you presume to tell those brighter than you what
they haven't understood.
> This statement exposes how much simpler magnitude is from the real
> number, and I see that you refute this claim. It is a failure on your
> part to admit ...
Using a gorilla as an example was probably a bad idea. Now the gorilla
is trying to lecture its trainer...
Enough.
Brian Chandler
Transfer Principle
Unlike concepts like adjacent infinitesimals and bigulosity,
which are difficult to define rigorously, it should be very
easy to define polysigned numbers rigorously. This is one
major difference between the polysigned numbers and the
other concepts proposed on sci.math.
I've presented the following attempt at rigorous definition
for polysigned numbers several times before, but it seems to
be rejected. I wouldn't mind if someone would please at
least tell me what's wrong with my definition.
You (Math1723) already mentioned the rigorous construction
of the real numbers, in the following sequence:
Sets
Ordinals
Naturals
Integers
Rationals
Reals
When discussing the polysigned numbers, I prefer the
following alternate order, proposed by David Ullrich on
sci.math and several others outside Usenet:
Sets
Ordinals
Naturals
Positive Rationals
Positive Reals
Reals
Ullrich used Dedekind cuts to advance from Positive
Rationals to Positive Reals, but one could have easily
used Cauchy sequences at this point as well.
Now the step from Positive Reals to Signed Reals is the
step on which I focus here, since it's a generalization
of this step which gives us the polysigned numbers. I
denote the set of Positive Reals (or magnitudes) by the
letter P. Now to construct R, we take the set P^2 of
ordered pairs in P, and define a pre-addition on these
ordered pairs simply as component wise addition in P.
Now we define a real number to be a certain equivalence
class of ordered pairs in P. There are two ways to
define the equivalence relation ~. One way is to say:
(a,b) ~ (c,d) iff a+d = b+c
The other is to say that (a,b) ~ (c,d) iff one of the
following occurs:
1) (a,b) = (c,d) (i.e., a=c and b=d)
2) (a,b) = (c+p,d+p) for some peP
3) (c,d) = (a+p,b+p) for some peP
I prefer the second since it's easier to generalize.
Now to define the polysigned numbers Pn, instead of
taking ordered pairs in P, we start with the set P^n of
ordered n-tuples in P. It's now necessary to define the
equivalence relation ~ on the set P^n of ordered
n-tuples in P, as follows: (a_1,a_2,...,a_n) and
(b_1,b_2,...,b_n) are equivalent iff one of the
following occurs:
1) (a_1,a_2,...,a_n) = (b_1,b_2,...,b_n)
2) (a_1,a_2,...,a_n) = (b_1+p,b_2+p,...,b_n+p) some peP
3) (b_1,b_2,...,b_n) = (a_1+p,a_2+p,...,a_n+p) some peP
The simpler rule for R, (a,b) ~ (c,d) iff a+d = b+c, is
harder to generalize for Pn. When n=3, one can try
writing it is:
(a,b,c) ~ (d,e,f) iff a+e+f = d+b+f = d+e+c
and for n=4:
(a,b,c,d) ~ (e,f,g,h)
iff a+f+g+h = e+b+g+h = e+f+c+h = e+f+g+d
Now you can see why I prefer "...some peP" instead.
One can show that addition on these equivalence classes
is well defined. They make Pn into a additive group,
with (1,1,...,1) the additive identity. Indeed, Pn is a
vector space over R, of dimension n-1.
A major part of the definition of Pn is how one can
_multiply_ two polysigned numbers. Rather than squeeze
the rigorous definition of multiplication into the same
post, I will leave it off, until there is agreement
over what I've written thus far about addition.
spudnik
it is only to note,
Newton didn't have a theory of light, although
he appears to have said that it is corpuscular,
which it clearly is not, viz the photoelectrical (electromag.) effect.
he also did not have a calculus in Principia,
which is relagated to the mere element, dxdy,
in Book2, Section2, Sholien2, etc.2
> <http://www.androcles01.pwp.blueyonder.co.uk/Algol/Algol.htm>
herr doktor-professor Albert was quite a racnonteur;
read all about it in _Einstein's Mistakes_.
anyway, both special & general rel., as natural
as they may be, are all bound-up in the hoary nothingness
of Minkowski's spacetime sloganeering,
nothin' but mere "3+1" phase-space.
--GMMXI, notmy IQ; yourn?
http://wlym.com
Han de Bruijn
> On Jan 6, 10:35 am, "Tim Golden BandTech.com" <tttppp...@yahoo.com>
> wrote:
>
> > Yes, but by denying any freedom at the axiomatic level of construction
> > the chance at discovering new fundamentals withers.
>
> construction" since you are free to begin with whatever axioms you'd
> like. You may find that some axioms you take will result in an
> something which is equivalent to, or a subset of, a current system.
> Others still yield something different but not very interesting.
> Every now and again, you end up with something new and interesting.
>
> This is essentially how Non-Euclidean Geometry became investigated.
> It began with people trying to prove Euclid's Parallel Postulate, by
> starting with its negation, using that as an axiom, and hoping to
> yield and inconsistent system. To everyone's surprise, rather than an
> inconsistent system, new forms of geometries were discovered, provably
> shown to be no less consistent than Euclidean Geometry.
Important note. Non Euclidian geometries can still be _embedded_ into
conventional Euclidian geometry. With other words: there are Euclidian
models of non Euclidian geometries. Thus they are not essentially new
in that respect, so to speak.
> > Here I see you've contradicted yourself. If a variation on old axioms
> > exist and that variation has extensions then the old axioms become
> > more dubious.
>
> I don't see how. Just because there are consistent Geometries with
> differing axioms with respect to the Parallel Postulate, how does that
> make the axiom (or its negation) dubious? By definition, an axiom is
> a statement that is taken as true within the corresponding model. As
> long as the set of axioms you start with are not inconsistent (for
> example, taking both PP and ~PP as axioms), the resulting theorems
> will represents truths of any model to which it applies.
>
> > You see, whether you personally are willing to grant the
> > generalization of sign which has previously been restricted to a
> > modulo two property could be a fair instance by which a conservative
> > will deny, but a progressive will entertain.
>
> Again, as a formalist, I don't see this as a conservative vs.
> progressive issue. Although I am not familiar with these "polysign"
> numbers (I have never heard of them prior to recent posts on
> sci.math), but as long as you define them using a consistent set of
> axioms, feel free to see where they take you. If you end up with an
> inconsistency, then you merely need to reexamine your beginning set of
> axioms.
>
> > There are easier ways to attain crankhood here, such as merely
> > thinking freely.
>
> I disagree. Crankhood is marked by stubbornness and refusal to accept
> the results of beginning hypotheses (ironically, the very thing they
> think non-cranks are doing). Typically, it is found by an emotional
> embracing (or repulsion) of certain ideas, irrespective of their
> mathematical merit. There are cranks who refuse to accept, for
> example, the consistency of transfinite ordinals and cardinals, and
> will refer to Mathematicians as "Cantorians" (as if they are
> "followers" of Georg Cantor). But they are missing the point. You
> don't *have* to adopt the Axiom of Infinity (or any other ZFC axiom)
> if you don't want, for whatever private system you wish. But what you
> do have to do is accept the Mathematical proofs that say "If you start
> from these assumptions, the following will result." Such cranks have
> such an emotional attachment to the denial of the axioms (due to some
> very naive intuityions about infinity), they cannot bring themselves
> accept relative Mathematical consistency. They treat these as
> religious doctrine, rather than Mathematics.
I don't believe the ZFC axioms are clear enough to derive _anything_
from these in a truly rigourous way. But my bias is "constructivism";
I'm basically a computer programmer, and I find that things which can
not be implemented in a computer program do not belong to my universe.
> As stated earlier, I believe in the Axiom of Choice. What could be
> simpler than taking one item from each of a collection of sets? Yet,
> I am happy to investigate systems using the Axiom of Determinacy (an
> Axiom which is inconsistent with A.C.). So what? I can leave my
> religion at home when I do Mathematics. I know many Mathematicians
> who object to the concept of Infinitesimals, and simply don't believe
> they "exist". Yet they all acknowledge the relative consistency of
> Non-Standard Analysis (whether they find it interesting or not).
> Cranks, on the other hand, do not dispassionately accept the results
> of whatever system you are in. They refuse to accept the logic of the
> matter, but would rather hold onto some emotional intuition which, in
> the end, cannot be consistently maintained.
Rather replace "intuition" by "experience", even "expertise" perhaps.
And that experience can originate in _another_ field than mathematics.
And that experience perhaps is contradictory to a mathematical result.
> > > As for Constructivism, apparently I am not the right person to talk
> > > with. I sincerely desire to give an accurate accounting of this
> > > philosophy (even though I don't share it), but other presumed
> > > Constructivists have me I have "missed their point", so there you have
> > > it.
>
> > Well, you seem to have failed to reflect a notion of flexibility at
> > the axiomatic level which to me is very important to the notion of
> > construction.
>
> I am not sure what you mean by this. I would argue that Formalists
> are the most flexible with respect to the axiomatic level. Choose
> your axioms, be my guest!
>
> With respect to Constructivism, I certainly accept that beginning with
> their assumptions, you get Constructivist theorems. Personally, I
> find it less interesting, since so much is lost within such a
> framework. It is essentially a subset of what I already know. You
> get the same basic theorems of most things, but lose some of the most
> interestingly unusual objects (non-measurable sets, etc). Worse
> still, the removal of the Law of the Excluded Middle is particularly
> annoying, robbing me of a basic tool in Mathematics. But hey, you can
> change your logical axioms just as you can change you mathematical
> ones. If that's your thing, have at it.
>
> If you are truly interested in Constructivist Mathematics, one area of
> study you might enjoy (related to another sci.math post) is Smooth
> Infinitesimal Analysis [see:http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf
> ]. It is much simpler than Non-Standard Analysis, and may be
> valuable, if your desire is to become more Mathematically rigorous.
> Due to its denial of L.E.M., I found it personally frustrating, but
> you may have a different take.
Han de Bruijn
Marshall
> On Jan 6, 5:37 pm, Math1723 <anonym1...@aol.com> wrote:
>
> > I disagree. Crankhood is marked by stubbornness and refusal to accept
> > the results of beginning hypotheses (ironically, the very thing they
> > think non-cranks are doing). Typically, it is found by an emotional
> > embracing (or repulsion) of certain ideas, irrespective of their
> > mathematical merit. There are cranks who refuse to accept, for
> > example, the consistency of transfinite ordinals and cardinals, and
> > will refer to Mathematicians as "Cantorians" (as if they are
> > "followers" of Georg Cantor). But they are missing the point. You
> > don't *have* to adopt the Axiom of Infinity (or any other ZFC axiom)
> > if you don't want, for whatever private system you wish. But what you
> > do have to do is accept the Mathematical proofs that say "If you start
> > from these assumptions, the following will result." Such cranks have
> > such an emotional attachment to the denial of the axioms (due to some
> > very naive intuityions about infinity), they cannot bring themselves
> > accept relative Mathematical consistency. They treat these as
> > religious doctrine, rather than Mathematics.
>
> I don't believe the ZFC axioms are clear enough to derive _anything_
> from these in a truly rigourous way. But my bias is "constructivism";
> I'm basically a computer programmer, and I find that things which can
> not be implemented in a computer program do not belong to my universe.
So your universe has no girls in it.
Perhaps more to the point, apparently your computer can't
handle FOL theories and proofs therein. How lame. You
might consider buying an ordinary computer at the computer
store and loading Prover9 onto it; this would expand your
capabilities immensely.
Marshall
Han de Bruijn
Oh well, I have Maple8 on it .. :-)
Han de Bruijn
FredJeffries
>
> One can show that addition on these equivalence classes
> is well defined. They make Pn into a additive group,
> with (1,1,...,1) the additive identity. Indeed, Pn is a
> vector space over R, of dimension n-1.
>
Do you mean that (0,0,...,0) is the additive identity?
I've only glanced at this and have no idea what polysigned numbers are
supposed to be, but I'm not convinced that your Pn are vector spaces
over R. Seems to me that they would be "half-vector-spaces" over the
positive reals. For instance, what is the additive inverse of (1,1,1)
in P3 ?
FredJeffries
Most humble apologies, Leonard. Mustn't try to read, comprehend AND
reply to a post in the five minutes before the bus comes. Realized
what an ass I had made of myself by the time I got to the bus stop.
Yes, (1,1,...,1) is the additive identity and to find the additive
inverse of something, get the max of the components plus one and
subtract the component.
Please proceed.
LudovicoVan
No no! Before he proceeds I need a layman terms explanation of what a
pair of real magnitudes has to do with a real number! (And how (1,1)
would be an additive identity for that, and an how to the inverses...?
Maybe a link to David Ullrich's original post would suffice?)
Please bear with the first year students...
Thank you,
-LV
LudovicoVan
OK, sorry, maybe I get it. E.g. -3.25 is (0.0, 3.25) or, equivalently,
(1.0, 4.25), etc. etc.
A link to David Ullrich's original post would still be welcome!
Thanks again,
-LV
Math1723
>
> You (Math1723) already mentioned the rigorous construction
> of the real numbers, in the following sequence:
>
> Sets
> Ordinals
> Naturals
> Integers
> Rationals
> Reals
>
> When discussing the polysigned numbers, I prefer the
> following alternate order, proposed by David Ullrich on
> sci.math and several others outside Usenet:
>
> Sets
> Ordinals
> Naturals
> Positive Rationals
> Positive Reals
> Reals
Hmmm ... the problem with that list is that you have no additive
inverse by the time you define "Positive Rationals", and thus
preventing them from being a field. In fact, with your list here, you
never construct Z or Q, only Z+ and Q+, and what we know of Z- and Q-
is as only subsets of R. This seems (at least to me) exceedingly
unhelpful, if not bizarre. Essentially, the smallest algebraic set
that -1 first belongs is R ??
You said David Ullrich proposed this?? I'd certainly like to hear his
rational for it, as a Mathematics Professor at Oklahoma State, he must
have some very good reasons (beyond what I can see).
Jesse F. Hughes
> I don't believe the ZFC axioms are clear enough to derive _anything_
> from these in a truly rigourous way.
We can formalize the ZFC axioms in first order logic and derive theorems
from them. In this respect, they are no different than the Peano
axioms. So, it's not at all clear to me what you mean by "rigorous".
> But my bias is "constructivism"; I'm basically a computer programmer,
> and I find that things which can not be implemented in a computer
> program do not belong to my universe.
As far as I know, cheese can not be implemented in a computer program,
and hence cheese does not belong to your universe. This must cause
considerable problems for a Dutchman.
--
"A recruitment consultant I know thinks the most important quality in
a winner is to be lucky. To avoid wasting his time with unlucky
applicants, he takes half the resumes piled on his desk and throws
them straight in the bin." -- John Ramsden
FredJeffries
>
> Hmmm ... the problem with that list is that you have no additive
> inverse by the time you define "Positive Rationals", and thus
> preventing them from being a field. In fact, with your list here, you
> never construct Z or Q, only Z+ and Q+, and what we know of Z- and Q-
> is as only subsets of R. This seems (at least to me) exceedingly
> unhelpful, if not bizarre. Essentially, the smallest algebraic set
> that -1 first belongs is R ??
For one thing, it makes working with Dedekind cuts much easier. Also,
it mirrors the historical development.
I believe the thread referred to is:
http://groups.google.com/group/sci.math/browse_frm/thread/cc4b18d39b4d4b62
Notice that Gerry Myerson
http://groups.google.com/group/sci.math/msg/58d61a07548443bc
points out that John Conway favors the same method in his "On Numbers
and Games"
http://books.google.com/books?id=tXiVo8qA5PQC&printsec=frontcover
page 25
Virgil
<a3b9ae08-4b34-44f7...@k30g2000vbn.googlegroups.com>,
Math1723 <anony...@aol.com> wrote:
For one thing, defining the arithmetic operations on Dedekind cuts is
ever so much simpler if you are only cutting the positive (or is it
non-negative) rationals than with all of them.
Bill Dubuque
> On Jan 10, 9:46 am, Math1723 <anonym1...@aol.com> wrote:
>>
>> Hmmm ... the problem with that list is that you have no additive
>> inverse by the time you define "Positive Rationals", and thus
>> preventing them from being a field. In fact, with your list here, you
>> never construct Z or Q, only Z+ and Q+, and what we know of Z- and Q-
>> is as only subsets of R. This seems (at least to me) exceedingly
>> unhelpful, if not bizarre. Essentially, the smallest algebraic set
>> that -1 first belongs is R ??
>
> For one thing, it makes working with Dedekind cuts much easier. Also,
> it mirrors the historical development.
>
> I believe the thread referred to is:
> http://groups.google.com/group/sci.math/browse_frm/thread/cc4b18d39b4d4b62
Here's an upate to my remarks in that thread regarding the isomorphism of
the two different construction paths, viz. N -> ring Z -> field Q vs.
semiring N -> semifield Q+ -> field Q. The discussion I was trying to
recall may be that in Remark 5.11 in [1]. Note in particular that the
general case is hairier and the two paths needn't yield isomorphic rings.
In any case these are standard results on semirings and semifields.
--Bill Dubuque
[1] Hebisch, Udo; Weinert, Hanns Joachim, Semirings and semifields.
Handbook of algebra, Vol. 1, 425--462, North-Holland, Amsterdam, 1996
http://dx.doi.org/10.1016/S1570-7954(96)80016-7
Tim Golden BandTech.com
There is a linguistic ambiguity here in that 'positive reals' imply
that the reals have already been constructed.
Magnitude is a superior term, and I do not believe that it will be
necessary to provide operators until sign is introduced. These are
merely ordered values, traditionally from zero to infinity; mimicing
the naturals but continuous.
I guess I see that you mathematicians are so accustomed to treating
the 'nonnegative reals' as fundamental, but still, I quibble over the
linguistic failure.
Otherwise, thank you so much TP for stepping in here to provide a more
traditional presentation. I don't personally find it attractive, but
you've clearly spent some energy on it and I do respect it. The
geometry is so important though. The simplex geometry needs to be
exposed for a good understanding of what these numbers mean. The real
numbers
-1, +1 (P2)
form two unit vectors on a two verticed simplex; zero being the center
of that simplex. The P3 values
-1, +1, *1 (P3)
form three unit vectors on a three verticed simplex; zero being the
center of that simplex. These are vector behaved spaces. These are
nonorthogonal coordinate systems. Dimension is sourced here, and
summing these unit vectors yields the origin:
- 1 + 1 * 1 = 0 (P3) .
Anyway, thanks again Transfer Principle.
- Tim
Tim Little
> Hmmm ... the problem with that list is that you have no additive
> inverse by the time you define "Positive Rationals", and thus
> preventing them from being a field.
You can branch off construction of the rationals, and identify them
with a subset of the signed reals later. In fact leaving sign
extension until later is closer to the historical development of
numbers.
> You said David Ullrich proposed this?? I'd certainly like to hear his
> rational for it, as a Mathematics Professor at Oklahoma State, he must
> have some very good reasons (beyond what I can see).
The introduction of signs early on complicates construction of the
reals. Leaving the signedness extension until the last step avoids
that messiness and allows the exposition to concentrate more on the
aspects that are actually relevant.
--
Tim
Transfer Principle
> > One can show that addition on these equivalence classes
> > is well defined. They make Pn into a additive group,
> > with (1,1,...,1) the additive identity. Indeed, Pn is a
> > vector space over R, of dimension n-1.
> Do you mean that (0,0,...,0) is the additive identity?
I've been thinking about this, and it would be a lot more
convenient (especially when it's time to define our
multiplication) if we could use 0's instead of 1's.
Of course, we used 1's because our construction is based
on the set P of positive reals. (Actually, I agree with
Golden that we should call these _magnitudes_ or unsigned
reals, not _positive_ reals.)
So I was wondering whether we should make it so that P
contains 0 as well. But P is formed from the unsigned
Dedekind cuts. You (Jeffries) and Dubuque already point
out the elegance of using unsigned D-cuts, including the
simple definitions of add and multiply for them. One
could try letting 0 be an empty set, but then the simple
definitions of add and multiply (i.e., setwise operations
on the element rationals) can't both work. Either we'd
have to add 0 and declare by fiat that 0+p=p and 0p=0 for
all peP, or switch to Cauchy sequences, where we already
have C-sequences of positive rationals that converge to 0
(like {1,1/2,1/3,1/4,...,1/n,...} of course). But then
again, how can we tell if a sequence is Cauchy unless we
have subtraction, which we don't have in Q+?
Also, I'm torn between whether the n-tuples in Pn should
be indexed starting with 0:
(a_0,a_1,a_2,...,a_(n-1))
or with 1:
(a_1,a_2,a_3,...,a_n)
It makes a difference when it's time for multiplication.
Golden usually starts with a_1, but starting with a_0 will
make the multiplicative identity be (1,0,0,...,0) rather
than the somewhat awkward (0,0,...,0,1).
In either case, we define for (n large enough):
-1 to be the n-tuple with a_1 = 1 and all others zero.
+1 to be the n-tuple with a_2 = 1 and all others zero.
*1 to be the n-tuple with a_3 = 1 and all others zero.
#1 to be the n-tuple with a_4 = 1 and all others zero.
Since +1 is already taken, the symbol for addition is no
longer +. Instead, Golden uses the symbol @. This symbol
can also serve as the "zeroth sign" if we begin the
indexing with 0 instead of 1, so we can define:
@1 to be the n-tuple with a_0 = 1 and all others zero.
Transfer Principle
> On Jan 10, 6:43 am, FredJeffries <fredjeffr...@gmail.com> wrote:
> > I've only glanced at this and have no idea what polysigned numbers are
> > supposed to be, but I'm not convinced that your Pn are vector spaces
> > over R. Seems to me that they would be "half-vector-spaces" over the
> > positive reals. For instance, what is the additive inverse of (1,1,1)
> > in P3 ?
> Most humble apologies
Apology accepted.
> Please proceed.
OK then, let's define multiplication now.
The key to Golden's multiplication is something called
"circular (or cyclic) convolution."
A Google search reveals the following link, which describes
cyclic convolution in detail:
http://www.lss.uni-stuttgart.de/matlab/cycconv/index.en.html
The link demonstrates the convolution:
[1,2,3,4][1,0,0,1] = [3,5,7,5]
This can be interpreted as multiplication in P4. Since the
link uses 0-based indexing, I do the same here.
The simplest examples of multiplication involve the elements
with a 1 in one position and 0's everywhere else:
(1,0,0,0) = 1 (can also be written as @1 or #1)
(0,1,0,0) = -1
(0,0,1,0) = +1
(0,0,0,1) = *1
To multiply these using cyclic convolution, we note that
the element with 1 in the ith position, multiplied by the
element with 1 in the jth position, gives the element with
1 in either the (i+j)th or (i+j-4)th position. (In other
words, the index is (i+j) mod 4, but some purists don't
like the use of "mod" in this fashion.) So we have:
(1,0,0,0)(0,1,0,0) = (0,1,0,0)
(1 in 0th pos. times 1 in 1st pos. gives 1 in 1st pos.)
(0,1,0,0)(0,0,1,0) = (0,0,0,1)
(1 in 1st pos. times 1 in 2nd pos. gives 1 in 3rd pos.)
(0,0,1,0)(0,0,0,1) = (0,1,0,0)
(1 in 2nd pos. times 1 in 3rd pos. gives 1 in 5th pos.,
but since there are only 4 positions, 5 - 4 = 1st pos.)
The last two can also be written as:
(-1)(+1) = *1
(+1)(*1) = -1
And (1,0,0,0) is indeed the multiplicative identity.
We can prove that this multiplication is welldefined on
the equivalence classes from above. The proof depends on
the fact that in the ring R^n, with vector addition and
cyclic composition as the operations, we have that the
subrng generated by
(1,1,1,...,1)
is an ideal.
Returning to the example given at the link, we have:
(1,2,3,4)(1,0,0,1) = (3,5,7,5)
which can be written as:
(1-2+3*4)(1*1) = 3-5+7*5
which reduces to -2+4*2 via equivalence classes.
I notice that at the bottom of the link, there appears
to be a Java applet which computes cyclic convolution
(and thus polysigned multiplication), but I haven't
figured out how to make it work.
Notice that in P2, cyclic convolution gives:
(1,0)(1,0) = (1,0)
(1,0)(0,1) = (0,1)
(0,1)(0,1) = (1,0)
corresponding to:
(+1)(+1) = +1
(+1)(-1) = -1
(-1)(-1) = +1
just as in standard R. Thus cyclic convolution really
is an extension of the sign rules in multiplication
of standard R.
Han de Bruijn
> Han de Bruijn <umum...@gmail.com> writes:
>
> > I don't believe the ZFC axioms are clear enough to derive _anything_
> > from these in a truly rigourous way.
>
> We can formalize the ZFC axioms in first order logic and derive theorems
> from them. In this respect, they are no different than the Peano
> axioms. So, it's not at all clear to me what you mean by "rigorous".
>
> > But my bias is "constructivism"; I'm basically a computer programmer,
> > and I find that things which can not be implemented in a computer
> > program do not belong to my universe.
>
> As far as I know, cheese can not be implemented in a computer program,
> and hence cheese does not belong to your universe. This must cause
> considerable problems for a Dutchman.
Sorry. (Didn't know that cheese is something mathematical ..)
Han de Bruijn
Jesse F. Hughes
Well, you didn't say that *mathematical* things which cannot be
constructed do not belong to your universe.
--
"Out of the rubbles of Trent Lott's house -- he's lost his entire
house -- there's going to be a fantastic house. And I'm looking
forward to sitting on the porch."
-- George W. Bush consoles Katrina victims, Sep 2, 2005.
Tim Golden BandTech.com
I rarely use the tuple form but would start with a_0; the identity
element; no different than the polynomial does.
Still, I suspect that we use the tuple a bit too loosely. Some would
like the tuple
(a,b)
where a and b are in some set to mean cartesian product, but this is
not so in polysign. This is why I stress that to someone who wants to
understand polysign the geometry has to be exposed. The 'ideal'
(1,1)
is merely admitting that
- 1 + 1 = 0
or
- x + x = 0
where x is in P2(the reals), or is a magnitude. The values in the
tuple are unsigned components and their position represents their
sign. The complaint on a P2 value having two components goes away when
we leave the tuple form and allow the balance to be reduced. Still,
this issue of additional components is present, and is not such a bad
thing for without it there will be no generalization of sign, and this
is the authentic meaning of polysign: to generalize the sign of the
real number such that numbers other than two-signed numbers may exist.
I mispoke here when I suggest that the magnitude need not have
operators. When we express a sum or product in the sx form as
s1 x1 + s1 x2 = s1 ( x1 + x2 )
then this implies that we do have summation of magnitudes within their
development. Likewise for products
( s1 x1 ) ( s2 x2 ) = ( s1 + s2 )( x1 x2 )
the product of magnitudes must exist.
I see a short move away from modulo principles within the tuple form,
but the familiarity of the real values signs need not be forgotten.
There is a short logical argument here that I see again on the tuple.
We did not need the tuple in the initial development of the real
number. Instead a sign structure was constructed, roughly be granting
an inverse to every value. This concept disappears when we study
magnitude, and it is unnecessary within the development of P3 and so
forth, for there is now a concept of balance rather than inverse. The
inverse does still exist, but it is not any longer a fundamental
concern. The inverse of
- 1 + 2 * 3 (P3)
is
+ 1 * 1 - 2 * 2 - 3 + 3
in algorithmic form, which reduces to
- 2 + 1 .
Beyond inverse within the geometry of polysign there can come a debate
on the simplex, and to some it will be built in n+1-space such that
the origin of the construction is off the graph. This is the cartesian
interpretation of the vertices
(1,0,0),(0,1,0),(0,0,1)
which admittedly forms the three verticed simplex in three-space, but
leaves the origin off of the graph. Because the real number has been
constructed convincingly without this level of complexity there is
this issue. The tuple interpretation needs to be carefully defined if
you are using that notation, and whether you imply a cartesian product
within its usage. I do not believe that this usage is necessary. P3
are inherently two dimensional, and to mistake them as a three
dimensional construct is consistent, but is not necessary. For some
this is just notation, but for others clean notation is an indicator
of a clean system. I admit that these parallel representations exist,
but do worry about the freedom with which the tuple usage is slung
about to mean many different things at different times. If we ever did
get to considering the behaviors of
P4 x P4
then we would see a conflict of representation if we were to use the
notation
((a0,a1,a2,a3),(b0,b1,b2,b3)) .
This is somewhat an argument for the raw signed form and remains
parallel with the real number. Yes, the tuple notation works, and will
be necessary in computers and for large sign on paper, but that tuple
takes too many meanings, and we've just added another one which rubs
against a prior one. This same criticism exists on AA's choice to use
the tuple as the polynomial representation.
I agree with TP's interpretation here as consistent other than this
potential nuanced conflict. As if I didn't drag this conflict out
enough here, I suppose it could be parried over to a criticism on
dimension, for the sole value
(a0)
within the representation as P1 can be discussed through this lens.
This is where I sought Math1723's opinion, and would be happy to hear
other's opinions here. The gist in the past has been that most are
happy declaring this singleton an exception, but I believe in time
this will turn around, just as it did for me. The geometry of the
situation does matter here, and we have a new object which is
consistent with time, in a form which can allow the unification of
spacetime without the weirdness of relativity theory. Because the very
word 'dimension' has relied upon the real number, and so much of
mathematics has relied upon the real number as fundamental, then this
consideration requires careful thought. Again I strongly proclaim that
the real number is not fundamental.
- Tim
Virgil
<6ec3e3a8-89f8-4cce...@l8g2000yqh.googlegroups.com>,
While you may have intended your "things which can not be implemented in
a computer program do not belong to my universe" to apply only to things
mathematical, that is not what you said.
And in mathematics, being excessively literal minded is more of a virtue
than a vice.
Math1723
>
> While you may have intended your "things which can not be implemented in
> a computer program do not belong to my universe" to apply only to things
> mathematical, that is not what you said.
The entire statement is silly as well, even when limited to
Mathematics. I am a software developer myself, and there is virtually
nothing I do that has any relationship to the foundations of the
integers, set theory or what have you, whether Constructivist,
Formalist or Platonist. It's absurd to suggest that a computer
scientist is somehow better served because there is a Constructivist
proof for, say, Euclid's proof for the infinitude of primes. The lack
of Axiom of Choice and non-measurable sets does not change whether I
can compile my code or not.
Also, I would argue that computer programming is not as in tune with
Constructivism as is being suggested. On my computer, boolean
variables have only two settings: true and false, and furthermore:
(not not a) == a. Constructivism, on the other hand, uses
Intuitionist Logic, which rejects the Law of Double Negation.
> And in mathematics, being excessively literal minded is more of a virtue
> than a vice.
Quite right.
Han de Bruijn
> On Jan 11, 4:15 pm, Virgil <vir...@ligriv.com> wrote:
>
> > While you may have intended your "things which can not be implemented in
> > a computer program do not belong to my universe" to apply only to things
> > mathematical, that is not what you said.
>
> The entire statement is silly as well, even when limited to
> Mathematics. I am a software developer myself, and there is virtually
> nothing I do that has any relationship to the foundations of the
> integers, set theory or what have you, whether Constructivist,
> Formalist or Platonist. It's absurd to suggest that a computer
> scientist is somehow better served because there is a Constructivist
> proof for, say, Euclid's proof for the infinitude of primes. The lack
> of Axiom of Choice and non-measurable sets does not change whether I
> can compile my code or not.
And THAT schizophrenia doesn't make you think about the foundations of
mathematics ? I think computers are the main reason why those "cranks"
are redundant in sci.math nowadays. They simply observe a discrepancy
between their daily practice and the pure mathematics they've learned.
> Also, I would argue that computer programming is not as in tune with
> Constructivism as is being suggested. On my computer, boolean
> variables have only two settings: true and false, and furthermore:
> (not not a) == a. Constructivism, on the other hand, uses
> Intuitionist Logic, which rejects the Law of Double Negation.
Intuitionism is not Constructivism in the first place. There are many
flavours of constructivism and intuitionism is only one of these (not
the most successful in Computer Science). It is _obvious_, by the way,
that much of the mathematics in CS shall be constructive. Programs ARE
constructive mathematics. Not?
> > And in mathematics, being excessively literal minded is more of a virtue
> > than a vice.
>
> Quite right.
Where the danger of overkill is overly present. As in our case, the
mere _context_ of 'sci.math' makes it already clear that the subject
is intended to be mathematical and not "cheese".
Han de Bruijn