axiom vs. postulate!!!
Ian Beardsley
they both seem to be something self-evident and they went as far as to
say that an axiom was a postulate. I remember, I think, that in geometry
there are axioms and there are postulates, but what is the difference?
Ian
Russell Smith
"Ian Beardsley" <ianbea...@webtv.net> wrote in message
news:15314-3EE...@storefull-2134.public.lawson.webtv.net...
I think what you are thinking of is a "Postulate" and a "Theorem" Theorems
need to be proven because basically they are thoeries. Postulates are
statements that are accepted to be true and have been proven. An "Axium" is
a postulate.
I hope this helps
Russell
Jon and Mary Miller
Logically, they are the same.
By and large, I'd say that postulates are "self-evidently true" and
axioms are things we assume in order to do our work. This clearly has
psychological meaning rather than logical meaning. To physicists, it's
a huge difference (an axiom is really a hypothesis, that we accept or
reject based on physical evidence, while a postulate is a mathematical
truth), but when you get down to brass tacks, it's really the same.
And a postulate keeps its name for a long time after it's ceased to be a
self-evident truth and becomes merely another axiom to be accepted or
discarded at will. In fact, Playfair's Axiom is the same as the
Parallel Postulate.
Jon Miller
Ken Oliver
"Ian Beardsley" <ianbea...@webtv.net> wrote in message
news:15314-3EE...@storefull-2134.public.lawson.webtv.net...
In Euclid's time, say, and for some time after that, a postulate was a
self-evident truth.
In the early 19th century, the notion of what a postulate was changed a bit
and became more in the sense of the assumptions made in the creation of a
system. For most of us Euclid's 5th postulate of one parallel line through
a point not on a line is "self-evident." But in non-Euclidean geometries,
the Euclidean parallel postulate is replaced by a postulate of either no
parallel lines or an infinite number of parallel lines through a point not
on a line. While to most of us these are not self-evident, they are indeed
valid assumptions and lead to consistent and useful geometries.
George Cox
Euclid certainly has both axioms and postulates but I think that they
have almost equal logical weight, something like "a thing to be assumed
from which all subsequent claims will be deduced". I write "almost"
because I think that the postulates were geometrical and the axioms are
pre-geometrical logical assumptions. If I'm right, does that mean that
Euclid thought that geometry went beyond "mere" logic? If so, went
beyond it how? In having an empirical rather than a purely logical
underpinning?
GC
>
> Ian
Ian Beardsley
Re: axiom vs. postulate!!!
Group: alt.math Date: Tue, Jun 17, 2003, 5:50pm (PDT+7) From:
georg...@btinternet.com (George Cox)
the elements and I think what you are saying is it. Euclid doesn't write
on it himself, but quotes Proclus as having said Aristotle explained
that
"whenever that which is assumed and ranked as principle is both known to
the learner and convincing in itself, such a thing is an axiom e.g. that
which are equal to the same thing are equal to one another....When again
what is asserted is both unknown and assumed even without the assent of
the learner, then, he [Aristotle] says, we call this a postulate,e.g.
that all right equals are equal."
Looking at the examples, there seems to be a very subtle difference. I
can't fathom why anyone would say that all right angles are equal since,
if it is a right angle, then it is a right angle. Anyway, axioms and
postulates are both, as Euclid quotes from Aristotle,
"first principles...those the truth of which it is not possible to
prove."
And to add an interesting tid bit, Euclid says that Aristotle says
"Every demonstrative science must start from indemonstrable principles,
otherwise the steps of demonstration would be endless."
It is at this point that I wish I understood Godel's Theorem and could
learn a proof that doesn't go beyond Calculus. I am hoping to get that
in the book Godel, Escher, Bach that I just ordered.
Ian
George Cox
>
>
> Re: axiom vs. postulate!!!
>
> Group: alt.math Date: Tue, Jun 17, 2003, 5:50pm (PDT+7) From:
> georg...@btinternet.com (George Cox)
> Ian Beardsley wrote:
> I checked the deifinitions of axiom and postulate on dictionary.com and
> they both seem to be something self-evident and they went as far as to
> say that an axiom was a postulate. I remember, I think, that in geometry
> there are axioms and there are postulates, but what is the difference?
This is what I wrote :-
> Euclid certainly has both axioms and postulates but I think that they
> have almost equal logical weight, something like "a thing to be assumed
> from which all subsequent claims will be deduced". I write "almost"
> because I think that the postulates were geometrical and the axioms are
> pre-geometrical logical assumptions. If I'm right, does that mean that
> Euclid thought that geometry went beyond "mere" logic? If so, went
> beyond it how? In having an empirical rather than a purely logical
> underpinning?
> GC
-:
> Ian
> As it would turn out, I have book I and book 2 of the thirteen books of
> the elements and I think what you are saying is it. Euclid doesn't write
> on it himself, but quotes Proclus as having said Aristotle explained
> that
>
> "whenever that which is assumed and ranked as principle is both known to
> the learner and convincing in itself, such a thing is an axiom e.g. that
> which are equal to the same thing are equal to one another....When again
> what is asserted is both unknown and assumed even without the assent of
> the learner, then, he [Aristotle] says, we call this a postulate,e.g.
> that all right equals are equal."
Ah, thank you for the quote. That seems a rather good way to use the
words (apart from the formal use of "axiom" in modern mathematics).
GC