Re: is mathematics just a tautology?
donsto...@hotmail.com
Fred Head
"magic math tricks" <markde...@hotmail.com> wrote in message
news:1145844552.2...@g10g2000cwb.googlegroups.com...
| math is just a tautology. true or false?
|
Tutology or Taustology?
Jesse F. Hughes
> math is just a tautology. true or false?
>
That's easy. False. Mathematics is an academic study, a field of
research or some similar thing. Tautologies are particular kinds of
statements.
Since math isn't a statement, it clearly can't be a tautology.
Good question, Dr. Demers, multimillionaire and renaissance man.
--
"I am the barbarian at the gates, raw creative force, willpower, and
the will to fight for the truth no matter what, no matter who stands
against me, no matter how many of you band [...] together in your
weakness to fight against the math." -- James S. Harris
Jesse Hughes
> Every mathematical theorem is a tautology. True or false?
As stated, false.
For instance, take, say, the theorem that there are infinitely many
primes. This theorem is clearly not a tautology.
On the other hand, take a particular proof of this theorem. There are
a finite number of axioms that occur in this proof, say, A1,...,An.
Let's call the statement "There are infinitely many primes" P. Then
the proposition
(A1 & A2 & ... & An) -> P
is a tautology---at least one common sense of the word. That is, it
is a theorem of first order logic. (In other contexts, "tautology"
means "theorem of propositional logic". In still other contexts, it
means something else entirely.)
--
Jesse Hughes
Department of Technology Management
Section of Philosophy and Ethics of Technology
Technical University of Eindhoven
donsto...@hotmail.com
use/content, so it's not a pure tautology. An example of a
tautological dialog is:
"A tautology is a tautology."
"Is that a metatautology?"
"A metatautology is a metatautology."
"I never metatautology I didn't like."
- Free Will Rogers
matt271...@yahoo.co.uk
Jesse Hughes wrote:
> "magic math tricks" <markde...@hotmail.com> writes:
>
> > Every mathematical theorem is a tautology. True or false?
>
> As stated, false.
>
> For instance, take, say, the theorem that there are infinitely many
> primes. This theorem is clearly not a tautology.
Only if you assume that the primes, and the number of them that may
exist, have some "real" existence outside the abstract system that you
use to make the proof (which is IMO questionable). Otherwise, what else
is there but the tautological proof?
Shmuel (Seymour J.) Metz
04/24/2006
at 05:09 AM, matt271...@yahoo.co.uk said:
>Jesse Hughes wrote:
>> "magic math tricks" <markde...@hotmail.com> writes:
>>
>> > Every mathematical theorem is a tautology. True or false?
>>
>> As stated, false.
>>
>> For instance, take, say, the theorem that there are infinitely many
>> primes. This theorem is clearly not a tautology.
>Only if you assume that the primes, and the number of them that may
>exist, have some "real" existence outside the abstract system that
>you use to make the proof (which is IMO questionable). Otherwise,
>what else is there but the tautological proof?
You seem to be confused as to what a tautology is. A tautology is very
basic type of inference; a logical equivalence in propositional
calculus, independent of more complex rules of inference. Thus "if A
and B then B and A" is a tautology, but "all X are A and y is an X,
therefor Y is an A" is not a tautology.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
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matt271...@yahoo.co.uk
Shmuel (Seymour J.) Metz wrote:
> In <1145880544.2...@i39g2000cwa.googlegroups.com>, on
> 04/24/2006
> at 05:09 AM, matt271...@yahoo.co.uk said:
>
> >Jesse Hughes wrote:
> >> "magic math tricks" <markde...@hotmail.com> writes:
> >>
> >> > Every mathematical theorem is a tautology. True or false?
> >>
> >> As stated, false.
> >>
> >> For instance, take, say, the theorem that there are infinitely many
> >> primes. This theorem is clearly not a tautology.
>
> >Only if you assume that the primes, and the number of them that may
> >exist, have some "real" existence outside the abstract system that
> >you use to make the proof (which is IMO questionable). Otherwise,
> >what else is there but the tautological proof?
>
> You seem to be confused as to what a tautology is. A tautology is very
> basic type of inference; a logical equivalence in propositional
> calculus, independent of more complex rules of inference. Thus "if A
> and B then B and A" is a tautology, but "all X are A and y is an X,
> therefor Y is an A" is not a tautology.
>
Right... I meant "tautology" in the sense of saying something appearing
to be new but actually with no information content beyond what we
already had.
tadchem
A little of both, actually.
Like Bob Dole answered when asked about his preferred style of
undergarments; "Depends..."
<grin>
Tom Davidson
Richmond, VA
tadchem
Tom Davidson
Richmond, VA
Edward Green
all its tautologies are extremely non-obvious.
Jesse F. Hughes
>> For instance, take, say, the theorem that there are infinitely many
>> primes. This theorem is clearly not a tautology.
>
> Only if you assume that the primes, and the number of them that may
> exist, have some "real" existence outside the abstract system that
> you use to make the proof (which is IMO questionable). Otherwise,
> what else is there but the tautological proof?
What is it you tautology means?
Platonism/realism doesn't have anything to do with whether or not that
theorem is a tautology.
--
"I've been thinking about my problems with getting any kind of
admission that my math arguments showing the core error in mathematics
are correct, so I've gone to marketing books."
-- James S. Harris, on when mathematics isn't enough
matt271...@yahoo.co.uk
Jesse F. Hughes wrote:
> matt271...@yahoo.co.uk writes:
>
> >> For instance, take, say, the theorem that there are infinitely many
> >> primes. This theorem is clearly not a tautology.
> >
> > Only if you assume that the primes, and the number of them that may
> > exist, have some "real" existence outside the abstract system that
> > you use to make the proof (which is IMO questionable). Otherwise,
> > what else is there but the tautological proof?
>
> What is it you tautology means?
>
I think I'm using the word in a non-technical sense that probably has
little relevance to what others are saying in this thread.
Jesse F. Hughes
> Jesse F. Hughes wrote:
>> matt271...@yahoo.co.uk writes:
>>
>> >> For instance, take, say, the theorem that there are infinitely many
>> >> primes. This theorem is clearly not a tautology.
>> >
>> > Only if you assume that the primes, and the number of them that may
>> > exist, have some "real" existence outside the abstract system that
>> > you use to make the proof (which is IMO questionable). Otherwise,
>> > what else is there but the tautological proof?
>>
>> What is it you tautology means?
>>
>
> I think I'm using the word in a non-technical sense that probably has
> little relevance to what others are saying in this thread.
Actually, I saw your definition of tautology. I'd have to say that
realism still doesn't have much to do with whether the theorem is a
tautology.
Suppose there really are numbers, etc., (whatever that means) and that
among the things we know about them are the axioms needed to prove
Euclid's theorem. Then the axioms are among what we know and they
imply the theorem.
That's my plausibility argument for the claim that a realist may claim
that the theorem is tautologous in your sense. On the contrary, a
formalist may claim that the theorem is not tautologous, since the
axioms of PA are arbitrary choices (not given). Thus, a non-realist
can give an argument that the theorem is not tautologous.
Nonetheless, I see what you were getting at. But I don't think it's
right. Realism is orthogonal to this issue.
--
Jesse F. Hughes
"There are VERY FEW real mathematicians and I am one of them. Few of
you can handle the pressure of real mathematics, like being wrong,
while I demonstrably can." -- James S. Harris
Shmuel (Seymour J.) Metz
Well, tautology is the wrong word, but it is true that every
mathematical theorem is a consequence of the relevant axioms and rules
of inference. The validity of a proof has no implication as to the
theorem's utility in modelling physical systems. Mathematics is not
Physics.
That said, the consequences of a simple axiom system may be far from
obvious.
magic math tricks
but you know it when you see it.
magic math tricks
cells on?
Jesse F. Hughes
> tautolgy means "boring". It may not be possible to define a tautology,
> but you know it when you see it.
I see. You didn't give a shit about thoughtful responses.
What a surprise.
Evidently, "Mark Demers" is a tautology.
--
Jesse F. Hughes
"Well, you know as soon as you have a new number I will be happy to
add it to the list. Don't try those childish tit-for-tat games with
me." -- Ross Finlayson on Cantor's theorem.
Jesse F. Hughes
> In <1145905055.8...@e56g2000cwe.googlegroups.com>, on
> 04/24/2006
> at 11:57 AM, matt271...@yahoo.co.uk said:
>
>>Right... I meant "tautology" in the sense of saying something
>>appearing to be new but actually with no information content beyond
>>what we already had.
>
> Well, tautology is the wrong word[...]
It's pretty close to a fairly standard (if non-technical) definition
of the word.
From The Collaborative International Dictionary of English v.0.48 [gcide]:
Tautology \Tau*tol"o*gy\, n. [L. tautologia, Gr. ?: cf. F.
tautologie.] (Rhet.)
A repetition of the same meaning in different words; needless
repetition of an idea in different words or phrases; [...]
Not quite the same, since he's discussing analytic consequences rather
than analytic restatements, but close enough that I think his usage is
understandable.
--
"[I want to] stand at the pinnacle of human achievement with no one
else in all of history even close, no human being having faced what I
have--and survived. Because when all is said and done, make no
mistake, the simple truth is, I am better." --James S. Harris
magic math tricks
the kitchen.
magic math tricks
T.H. Ray
The only point I have ever seen to anything you've
written in this forum, is that you don't care about
knowledge and have some private grudge against those
who do.
At any rate, you don't know how to use the term
"tautology," so you shouldn't toss it about carelessly
as if to pretend that any word can have any meaning
that you happen to personally assign it.
If the issue is what mathematics "is,"
you should discover that mathematics is, at least, a
language. Like any language, it has a lexicon of shared
meanings by which its speakers communicate coherently.
Tom
Jesse F. Hughes
> tautology means no news.
I get it. Let me try my hand at a tautology.
"Mark Demers" is an utter jackass and a moron.
How's that?
--
"You lack the ability to reason, but instead get an idea in your head
and hold on to it against all evidence. I don't find you credible,
and reject your claims, as coming from a flawed source."
-- James S Harris shoots for Projection Post of the Year (2004)
magic math tricks
mack dimiers guy?