Axioms and postulates

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Axioms and postulates
Do you agree with this article there is a difference, in the way the article describes?:
http://blogs.scientificamerican.com/roo ... postulate/
PhilX
http://blogs.scientificamerican.com/roo ... postulate/
PhilX
Re: Axioms and postulates
if there was a difference a couple of thousand years ago, there isn't today. And an axiom is no longer something that we see is true by observation. Rather, it's a statement accepted without proof in order to get a particular deductive system off the ground. The standard axiomatic system for all mathematics is ZermeloFraenkel set theory (ZF). ZF in its present form dates from as recently as 1922. Less than a century old. Nobody even knows for sure if ZF is consistent. A hundred years from now we might well have different axioms for mathematics, and perhaps even a difference concept of what an axiom is.
It's interesting to contemplate the ancient worldview implicit in idea that the axioms are "So true they don't even need proof, we can see their truth"; compared to the modern conception that axioms are arbitrary as long as they suit our purpose.
I see a connection in popular society, don't you? From Gods on High to moral relativism, reflected in the history of geometry.
It's interesting to contemplate the ancient worldview implicit in idea that the axioms are "So true they don't even need proof, we can see their truth"; compared to the modern conception that axioms are arbitrary as long as they suit our purpose.
I see a connection in popular society, don't you? From Gods on High to moral relativism, reflected in the history of geometry.

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Re: Axioms and postulates
A "postulate" is just an old term for meaning to assume ahead of time without concerning any prove of it. Math later used the term, "axiom", to differentiate between original input assumptions as opposed to those that were proven within the given system. I think many also interchange them when discussing things lightly but gets misinterpreted if one wants to be more specific. Basically, most interchangeably use "postulate", "axiom", or "assumption" to refer to the relative assumptions going into any particular proof. Only if you want to interpret the specific interpretation in context to a set of proofs within a whole system does it actually matter. And even more distinctions can be made. What actually matters is how the particular author of some proof or system of proof means anyways.
As to the article, what I think they seem to miss is that Euclid was intentionally designing his geometry without specifying a meaning to quantity itself. They were still using the Roman numerical system and in that day, questions regarding nothingness (or zero, as a meaningful number) is just as valid today with the confusion people even here on this site argue against understanding whether a nothing means anything or not to reality. As such, I think it would have even been worse in Euclid's day without a digression into such depths that only divert from the intention Euclid was trying to deal with. As such, I think he avoided terms regarding literal quantities in hopes of demonstrating through practice, just as modern science emphasizes the empirical process to try to avoid digression into the philosophical aspects that are too hard for most to follow.
So it isn't important for Euclid to have to postulate all particular angles. He opted for simply the right angle if only to be simple. The demonstrations also implicitly show how other angles are 'equal' throughout his works based only on this assumption. I find it odd how even in the Dover classics version of "The Elements" in three volumes spends a great deal of philosophical concern over things like whether parallel lines exist in nature. Who cares? I found this kind of concern ignores intent and contemporary atmosphere of the times. While interesting as an aside investigation, it is also often what turns many people off of the even trying to bother with reading the works.
If you want to be more particular for proofs and you are developing a set of proofs for which you want to base previous conclusions within that system to other proofs, then it is useful to differentiate a postulate from an axiom. A postulate would then be the initial assumptions from the first proofs and definitions and the axioms would refer to the input assumptions from past conclusions of the same system. And it would have to defer to the particular author using it. They may want further distinctions too which may aid in their intended proofs. This is all that matters.
As to the article, what I think they seem to miss is that Euclid was intentionally designing his geometry without specifying a meaning to quantity itself. They were still using the Roman numerical system and in that day, questions regarding nothingness (or zero, as a meaningful number) is just as valid today with the confusion people even here on this site argue against understanding whether a nothing means anything or not to reality. As such, I think it would have even been worse in Euclid's day without a digression into such depths that only divert from the intention Euclid was trying to deal with. As such, I think he avoided terms regarding literal quantities in hopes of demonstrating through practice, just as modern science emphasizes the empirical process to try to avoid digression into the philosophical aspects that are too hard for most to follow.
So it isn't important for Euclid to have to postulate all particular angles. He opted for simply the right angle if only to be simple. The demonstrations also implicitly show how other angles are 'equal' throughout his works based only on this assumption. I find it odd how even in the Dover classics version of "The Elements" in three volumes spends a great deal of philosophical concern over things like whether parallel lines exist in nature. Who cares? I found this kind of concern ignores intent and contemporary atmosphere of the times. While interesting as an aside investigation, it is also often what turns many people off of the even trying to bother with reading the works.
If you want to be more particular for proofs and you are developing a set of proofs for which you want to base previous conclusions within that system to other proofs, then it is useful to differentiate a postulate from an axiom. A postulate would then be the initial assumptions from the first proofs and definitions and the axioms would refer to the input assumptions from past conclusions of the same system. And it would have to defer to the particular author using it. They may want further distinctions too which may aid in their intended proofs. This is all that matters.

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Re: Axioms and postulates
In terms of usage, in math I've seen it common to use the term axiom, not postulate, to describe a set of assumptions for a system. It's also explained that the assumptions have their greatest power when you don't attach a particular meaning to them so you can apply them widely to many math systems.
PhilX
PhilX

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Re: Axioms and postulates
It's as I said though regarding assumptions and how authors uses them that count. Euclid's efforts describes geometry in a way that eliminates a need to be concerned specifically about numbers themselves. There is a practical use of this procedure in their day since they did not have the modern use of zero in their number system. What has come down to us, for instance, as a "quadratic equation" was based on Euclid's demonstration using rectangles of four sides to describe what has come to be of our present form using powers of some variable as Ax² + Bx + C = 0 for equations to solve for 'x'. Yet our modern language loses where this derived from unless you understand how Euclid actually demonstrated this using quadrilaterals. [See http://aleph0.clarku.edu/~djoyce/elemen ... opII6.html]Philosophy Explorer wrote:In terms of usage, in math I've seen it common to use the term axiom, not postulate, to describe a set of assumptions for a system. It's also explained that the assumptions have their greatest power when you don't attach a particular meaning to them so you can apply them widely to many math systems.
PhilX
I think that the words people preferred for assumptions is more related to the language of the particular authors and times of writing. Math proper often describes numerical assumptions, "axioms" while reserving "postulates" for the pure geometric uses. But I like using them both personally to distinguish different kinds of assumptions if need be. And since even many people confuse these, it might even be necessary to have to reword these to avoid people commanding some preference of interpretation upon some author intending it for their real use. People are not so flexible and still resist adapting to the particular language of the authors. I get this as it too gets abused if we are begged to have to defer to some use in common among communicating between each other.
Anselm's Ontological Argument was intended to prove the existence of "God". But while his argument was good to describe how an absolute universal may exist, he merely labeled the idea of this, "God", and thought that this should be transferable to the same "God" as understood in religious context. This is an illegitimate 'trick' but may hint how some may have actually used "God" as a label just for such a meaning originally.
I think the concern of that article is interesting but the author of it seems to be missing actual experience in reading it to learn of the context AND not recognizing the significant point that they used the awkward Roman Numerals for whatever reason that prevented advancing an argument that people could understand geometry without laying out a process that eliminates the frustrations of interpreting numbers. Also, the 360 degree division also illustrates a cultural favoritism based on what was an old assumption of the days of the year as a division. So Euclid's Elements was incredible to me as a means to argue for its day. We can glean value in his particular uses of postulates by at least recognizing that we miss the cultural evolution of the day and it can help us be able to see how we can adapt to using different terms to describe things arbitrary to the author's preference.
EDIT NOTE: I fixed the link above. I'm not sure what is happening as I post the link but notice that even though I added more words, something keeps including it as a part of the address link even when it is actually outside of the closing url tag. ?? Some bug?
Last edited by Scott Mayers on Sat Oct 03, 2015 11:59 am, edited 2 times in total.

 Posts: 1642
 Joined: Wed Jul 08, 2015 1:53 am
 Location: Saskatoon, SK, Canada
Re: Axioms and postulates
Remember too, that in the days of Plato and Euclid, the Pythagoreans were relatively interpreted as unusual as they appeared by outsiders as a 'cult' in its day. So approaching proofs the way Euclid did may have also been intended to reintroduce such intellectual values of such past efforts in a new language that was more acceptable and evaded associations with such groups. I notice this occurs here on this site too with regards to creating dissenting labels to past thinkers or scientists that one interprets upon them without fair context. How "Newtonian" or "Platonists", get used as terms of dissent and insult baffles me as it is hard to discern precisely what one has disagreement with. A "Newton" or a "Plato" no longer exist and while many of their ideas have been passed down to us using new and different terms, many can't seem to relate to their value in context of the times.
This was my point of concern with regards to how modern science has reintroduced the value of space as real thing to which it was in the past was called, the aether. People may appear to support the modern terms as if they've discovered something uniquely distinct when in actual meaning, they are the same as the older terms as understood by people back then. So in meaning, if one is favoring the new QM interpretation of space as having meaning AND yet also appear to be in agreement of the past disposal of the aether, this can be contradictory. And often it is only about preserving the historical virtue of authorities and theories based on cultural, economic, or political values in the present day.
This was my point of concern with regards to how modern science has reintroduced the value of space as real thing to which it was in the past was called, the aether. People may appear to support the modern terms as if they've discovered something uniquely distinct when in actual meaning, they are the same as the older terms as understood by people back then. So in meaning, if one is favoring the new QM interpretation of space as having meaning AND yet also appear to be in agreement of the past disposal of the aether, this can be contradictory. And often it is only about preserving the historical virtue of authorities and theories based on cultural, economic, or political values in the present day.
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