# Talk:Axiom

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## "True" and "universal" hopeless dream?

In the twentieth century, the grand goal of finding a "true" and "universal" set of axioms was shown to be a hopeless dream by Gödel and others.

That's not what Gödel showed.  :-) --LMS

Well, that's a reasonable paraphrase of what he showed, which was that no set of axioms sufficiently large for ordinary mathematics could be both (1) complete, i.e., capable of proving every truth; and (2) consistent, i.e., never proving an untruth. Or to put it yet another way, there must exist some assertions that are true but unprovable. --LDC

Not even that... He showed that any formal system strong enough to have equality, addition, and multiplication, had those properties. (It didn't have to encompass all of ordinary mathematics.) [He also had a loophole that might have allowed non Omega consistant logics to bypass the problems, but Rosser closed that hole. Nahaj 01:57:56, 2005-09-08 (UTC)
That's first incompleteness theorem. There was also a second one, which is what LMS refers to — Kallikanzaridtalk 08:57, 6 March 2011 (UTC)

Hey... I was just following the style guidelines that said that I should leave something hanging! (I still stand be the statement that Incompleteness can be colloquially said to imply that there is no universal and true set of axioms. There are definitely complete and consistent systems such as real arithmetic, but they lack the power of, say, integer arithmetic and thus can be said not to be universal. Another way to read what I was saying is that Principia was a hopeless task and not just because of a few paradoxes that might someday be weaseled around. -- TedDunning

What do you mean by real arithmetic? Not arithmetic of real numbers, surely, because that includes integer arithmetic as a subset and so is just as powerful. -- Josh Grosse

Actually, real arithmetic does not include integer arithmetic as a subset. The reals include the integers, but logical systems built on the two fields are not equivalent. In particular, real arithmetic is generally taken as not including comparison while integer arithmetic has comparison. The exclusion of comparison is generally due to the complexity of the definitions of the reals. The completeness of the real system was proved (I think) by either Banach or Tarski in the middle of the twentieth century.

My own personal view is that Incompleteness is just a guise of the Halting problem. Since you can solve the Halting problem with real arithmetic where the reals are defined using bit-strings and you are allowed to look at and compare a finite prefix of any real. The trick is that the algorithm requires an initial condition that is not a computable real (TANSTAAFL!) -- TedDunning

## Illegible

αβ&gamma

and this:

${\displaystyle \alpha \beta \gamma }$

the first is also illegible on Netscape, but you can tell what was intended; the second is perfectly legible. Michael Hardy 18:45 Mar 10, 2003 (UTC)

Note also that the first isn't valid HTML. Character entities *MUST* have the closing ";" to be valid. (I.E. It should be "&gamma;" instead of "&gamma") Since browser behavior is (as far as the standards are concerned) undefined if the HTML is invalid, one ought not to expect that the first case do anything reasonable. That said, the use of the math markup is preferable anyway, in my opinion. Nahaj 01:54:53, 2005-09-08 (UTC)

## Self-evident?

"As the word axiom is understood in mathematics, an axiom is not a proposition that is self-evident."

The Liddell and Scott entry for (axioma) says the exact opposite --Dwight 15:36, 12 Apr 2004 (UTC)

That remark seems very silly. The only "Liddell and Scott" I've been able to find is a lexicon translating ancient Greek words into English. They would therefore be expected to write about what the word meant in Ancient Greek, not about what it means in the usage of modern mathematicians. Liddell and Scott are probably right, and the statement you quote above about use in mathematics is also right. They do not contradict each other; they are about two different things. Liddell and Scott do not appear to be mathematicians and cannot be supposed to have expertise in that area. I, on the other hand, am a mathematician, and I am quite familiar with both usages. I suggest you read the whole Wikipedia article, and you will see that there is no contradiction between these points. Michael Hardy 22:33, 12 Apr 2004 (UTC)

Defined by Websters as a "self evident truth." It is one of those things that you think up while sitting on the can, or when when you can't sleep at 3:30 in the morning and you have some huge presentation to give the next day. You know, it just sort of hits you, but you knew it all along. Not to be confused with an epiphany. —The preceding unsigned comment was added by 172.198.219.243 (talkcontribs) 07:53, June 23, 2004 (UTC)

Nor with the trivial and obvious, which are theorems =) 142.177.126.230 21:19, 4 Aug 2004 (UTC)
...except that there is are technical definitions of self-evident in epistemology. See self-evidence. Michael Hardy 01:41, 5 Aug 2004 (UTC)

Axiom and postulate are different things. Axioms are taken as self evident. Postulates are accepted because the theory that is derived from them is proven to be correct. Manuel, march 2008.

I'm a layman in logic, but from what I read in other aricles, treating axioms as self-evident is an outdated approach, they are rather considered starting points of theories. In fact, there are many different logics (classical logic, intuitionist logic etc.) so treating axioms as self-evident becomes philosophically moot. — Kallikanzaridtalk 08:54, 6 March 2011 (UTC)

The article has a logicistic attitude. It suggests non-logical axioms are not assumed to be true, but mathematical axioms are just as self-evidenct as logical axioms. — Preceding unsigned comment added by 72.238.115.40 (talk) 02:20, 11 December 2013 (UTC)

## Completeness?

Gödel's completeness theorem establishes that every deductive system with a consistent set of non-logical axioms is complete

Does it? Doesn't it apply to a certain set of logical axioms and rules of deduction? Take a typical deductive system and remove a logical axiom schema or modus ponens. You still have a deductive system with a consistent set of non-logical axioms--albeit one that would not ordinarily be used (except perhaps by an intuitionist)--but it's not complete. Josh Cherry 02:07, 24 Oct 2004 (UTC)

## Uniqueness of the reals?

We are fortunate enough to have that the standard model of "real analysis", described by the axioms of a complete ordered commutative field, is unique up to isomorphism.

This seems to say that there is a set of axioms that picks out the reals uniquely (up to isomorphism). What about the Löwenheim-Skolem theorem and such? I presume that although the reals are the unique complete ordered commutative field, completeness can not be expressed axiomatically, at least in systems to which the L-S theorems apply. Josh Cherry 02:58, 24 Oct 2004 (UTC)

L-S is about first-order theories, and this axiomization isn't one. You can come up with a first-order theory of all first-order sentences true of the real numbers using the ring functions of addition and multiplication, as well as the order relation, and then L-S would apply, and would tell us it has a countable model. We get the theory of formally real fields in this way, but not the real numbers uniquely. Gene Ward Smith 08:01, 1 Dec 2004 (UTC)

OK, I've changed the article to discuss this point. Josh Cherry 23:50, 1 Dec 2004 (UTC)

## Examples

This editorial text was removed from the end of the examples page and is reproduced here:

[OK. The later two are being presumed to actually be logical axioms, i.e. valid formulas. It would better be to say "valid formulas, as follows..." The proofs of these facts are definitely a technical issue, but interesting enough on their own.]

Hu 20:36, 2004 Nov 22 (UTC)

Tip: Some people find it helpful if these suggestions are shown on this talk page, rather than on another page. To do this, just add {{User:LinkBot/suggestions/Axiom}} to this page. — LinkBot 10:39, 17 Dec 2004 (UTC)

## When are axioms used?

The article now claims:

• Basic theories, such as arithmetic, real analysis (sometimes referred to as the theory of functions of one real variable), linear algebra, and complex analysis (a.k.a. complex variables), are often introduced non-axiomatically in mostly technical studies, but any rigorous course in these subjects always begins by presenting its axioms.

It seems to me this just isn't true. Most often, what is done instead is to present definitions inside of set theory, and the set theory used is normally naive set theory. Linear algebra uses axioms when it wants to talk about vector spaces over arbitary fields, but that is quite different than using axioms to define the integers, real numbers, or complex numbers. If this sentence isn't given a good defense I'll remove it. Gene Ward Smith 19:34, 29 April 2006 (UTC)

## Galois and geometry

While Galois theory was successfully applied to classic questions of geometry, the names here should be Gauss and Pierre Wantzel, not Galois. Gene Ward Smith 02:02, 2 May 2006 (UTC)

## I have a complaint.

I probably shouldn't be saying this. And this maybe should be deleted. But I'm a little annoyed how every equation in wiki makes absolutely no sense. I would think there should be easy and hard ones to demonstrate how it works. —The preceding unsigned comment was added by 208.186.255.18 (talkcontribs) 05:54, June 3, 2006 (UTC)

Firstly, could you identify yourself by user name so we don't need to do detective work with the edit history to find out who wrote this?
Secondly, your complaint is horribly vague. Please explain what in the world you mean and cite examples. Michael Hardy 00:31, 5 June 2006 (UTC)

I think what this persons complaint was trying to convey is the language and explanations provided assume the average person knows as much as you do, I ended up here in the process of reading about a prescription drug which in the study cited, refers to percentages from (n=(some number), so in my search to find what specifically they were referring to I wiki'd statistics, proceeded to standard deviation, than on to algebraic symbols, than epsilon, than summation, harmonic numbers (though that was out of curiosity). I appreciate that knowledge especially mathematical is built on a chain of previous knowledge and that people take the time to share this. I did find what I was looking for as well and maybe better off for the journey, however I have been discouraged by other articles which seem more technical and maybe partly driven by the types of debates I've read. I think accessibility and relevance should be priority over precision as these topic's tend to mirror the tangent's of the contributors who may be focused on something more technical than required for a basic understanding and maybe a little less accessible for most people. However if I had a sample I could show you a graph now. Thanks again. —Preceding unsigned comment added by 64.53.203.180 (talk) 17:23, 31 August 2010 (UTC)

## Lack of sources

Sorry, I forgot to post here after adding {{unreferenced|article}}. There's not a single reference in the entire article, so I think the tag is warranted until the problem can be addressed. Simões () 01:17, 22 October 2006 (UTC)

## abstracted 'truth' out of introductory paragraph

Assuming it is not controversial to make the point that an 'axiom' does not necessarily connote a notion of "truth" or actuality, (except perhaps in the realms of epistemology, deontology, etc.) and therefore axioms are subject to whatever motivation is deemed appropriate under the circumstances, intro paragraph should reflect this. drefty.mac 07:00, 28 October 2006 (UTC)

## Given

I came here looking for the goalkeeper Shay Given, and typed in "Given". Was redirected here. Obviously the disambiguation page for "axiom" was no use for me. Somebody might want to look into this. —Preceding unsigned comment added by 212.64.98.189 (talkcontribs) 22:41, March 7, 2007 (UTC)

The article states that

"...for nearly two millennia the fifth (parallel) postulate ("through a point outside a line there is exactly one parallel") was suspected of being derivable from the first four. Ultimately, the fifth postulate was found to be independent of the first four. Indeed, one can assume that no parallels through a point outside a line exist, that exactly one exists, or that infinitely many exist."

I think it might be a good idea to change part of this statement. It is true that the fifth postulate is independent of the first four, but the assumption that no parallels exist is not - it is a much stronger statement than the negation of Euclid's fifth and is inconsistent with the first four postulates. The difference between the two kinds of geometry with the first four postulates already given is that in non-euclidean geometry parallels are not unique, whereas in euclidean geometry they are. See "Euclidean and non-euclidean geometries" by Greenberg. Stephen Thompson 01:14, 29 April 2007 (UTC)

## φ, ψ, χ or φ, χ, ψ?

Under "Mathematical logic" the article says: "...φ, ψ, and χ can be any formulae of the language...". Are these letters in the correct order? The article "Greek alphabet" says the alphabetical order of these letters is φ, χ, ψ. (Complex Buttons 20:03, 4 July 2007 (UTC))

## acceptation?

Is this a word??? maybe Bush wrote it? —Preceding unsigned comment added by 128.232.238.250 (talk) 22:58, 24 November 2007 (UTC)

Yes, "acceptation" is a word. It means the generally recognized meaning or sense attributed to a word. It is a term frequently used in both philosophy (especially logic and epistemology) and linguistics. (I think the confusion here may be originating from a failure to maintain the distinction between words that do not exist and words that one simply does not know.) Mardiste (talk) 12:58, 28 January 2008 (UTC)

Snork Mardiste! :) 85.210.15.219 (talk) 23:29, 6 May 2011 (UTC)

I added a link to http://www.allmathwords.org/axiom.html to this article. It was removed. Wikipedia guidelines allow links to sites that provide something the Wikipedia article does not. The Wikipedia article 'Axiom' is written at a college level. The All Math Words Encyclopedia is written at a level for grades 7-10 (U.S.). It is much more accessable to middle school and high school students than the Wikipedia article. I feel this is sufficient reason to include the link to All Math Words Encyclopedia. —Preceding unsigned comment added by DEMcAdams (talkcontribs) 15:22, 8 October 2008 (UTC)

I don't think this external link should be included. According to WP:LINKSTOAVOID, links that duplicate content that the article would contain if it were a featured article should be avoided. The allmathwords page has very little content, and apart from different choice of examples its content is already included in the present article. — Carl (CBM · talk) 16:04, 8 October 2008 (UTC)
I'm not familiar with the EL guidelines for simple.wikipedia.com, but it may be more appropriate there. — Arthur Rubin (talk) 16:07, 8 October 2008 (UTC)

formally, axiom is just anything you want to call an axiom... you can define any system you want and call any random sequence of symbols the "axioms" of the system...

Just because every system we ACTUALLY USE are nice and useful and consistent and have nice little axioms as starting points doesn't mean we HAVE TO have axioms like that... we can just as well have an inconsistent axiom that totally screw up the entire system. Alternatively we can have an axiom that doesn't imply anything (for example in systems where there are no inference rules that can be used to derive theorems from the axiom)..

Basically it is quite possible to have "non-ideal" axioms... but the thing is, i'm not entirely sure if it'd be useful to point out that we can have these "non-ideal" axioms... most wikipedia readers are probably not going to find the comment useful... and it's probably confusing to non-specialists Philosophy.dude (talk) 01:19, 2 December 2008 (UTC)

btw, i think the comment that logical axioms are universally "true" is not entirely correct. .. there are plenty of formal axiomatic systems that does not assume truth at all... Philosophy.dude (talk) 01:26, 2 December 2008 (UTC)

The article already says, in the lede, "In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived." Does it say somewhere that axioms, in the contemporary sense, have to be nice in some way?
Regarding the use of the word "truth", I think the article seems reasonable. The contemporary definition of a "logic" involves both syntactic and semantical parts, and thus includes a notion of truth. Systems of mathematical interest with no semantics are extremely rare. — Carl (CBM · talk) 02:42, 2 December 2008 (UTC)

## Scheme / schema

Both 'Axiom Schema' and 'Axiom Scheme' are used in this article. Are they both correct? Passingtramp (talk) 09:45, 21 May 2009 (UTC)

## An error?

On the first line of the third paragraph there is an example given in the first set of parentheses 'e.g., A and B implies A'. This appears to be mis-written. Then again, what do I know. I will leave it up to editors of these types of pages. If I am wrong, please disregard. 220.255.1.30 (talk) 03:49, 16 January 2011 (UTC)

## Axiom / Postulate

The distinction between axoims and postulates is never explicitly stated, which is regrettable since Postulate redirectes here. Anybody up for rectifying this? --93.206.135.188 (talk) 08:02, 13 September 2011 (UTC)

technically, no position is taken on the truth of a set of axioms. they are merely premises that might be true. the formal process of deduction states that if the set of axioms is true than the set of deductions follows; a theorem is said to be true if the axioms that led to it's deduction are. but, the correctness of an axiom is neither discussed nor relevant. the relevant concept is consistency.

it's a subtle point, but glossing over it can have serious consequences. the rejection of absolute truth is the great insight of modern mathematics, an insight that has yet to work it's way to other fields. explicitly making this point whenever possible should be done to get the idea out and circulating, and aid in the abolition of superstition.

i may come back and do it myself, but i'm low on time and would prefer somebody else take the initiative, if they have the opportunity, please. — Preceding unsigned comment added by 70.53.24.43 (talk) 09:27, 27 January 2013 (UTC)

This seems to have been corrected in the current page. Anaxiomatic (talk) 10:58, 12 February 2013 (UTC)

Agreed. There are no absolute, universally true axioms. Sentence deleted. BlueMist (talk) 22:27, 19 December 2015 (UTC)

OK, look, the question is subtle and complicated. The IP contributor, 70.53.24.43, seems to be taking a formalist view, but this is by no means the only relevant modern view. This article needs to be balanced between formalism and mathematical realism. From a realist view, it is definitely of interest to inquire whether or not a given axiom is true.
I have no huge objection to User:Blue Mist 1's second change, this one; it does not make very much difference whether we say "accepted" or "accepted as true". However, as this is explicitly called out as the "classical" notion of axiom, we should probably say something about self-evidence; certainly classically self-evidence was an essential part of the notion of axiom.
As to the first change, this one, we have a different problem. It is true that it has been tagged with "citation needed" and I don't immediately have a citation for the exact claim. However I can definitely give sources for workers grappling with the issue of what it means for an axiom to be true. Here are two just from Penelope Maddy's bio: Maddy, Penelope (Jun 1988). "Believing the Axioms, I". Journal of Symbolic Logic. 53 (2): 481–511. doi:10.2307/2274520. Maddy, Penelope (Sep 1988). "Believing the Axioms, II". Journal of Symbolic Logic. 53 (3): 736–764. doi:10.2307/2274569. That's just one example.
This is somewhat of a recurring problem; it is often easy to source important thinkers who take a certain position, but more difficult to source the meta-claim that various important thinkers take the position. But it's a problem that we ought to solve for this article; this is an important article that should present all the significant views. --Trovatore (talk) 23:06, 19 December 2015 (UTC)

Trovatore, you're an experienced editor. You know not to revert an edit agreed upon by three other editors. Please undo your revert, and discuss the issue here, as required by Wikipedia, before enforcing your personal, unsupported preferences ! ~~ BlueMist (talk) 23:40, 19 December 2015 (UTC)

I am discussing it. I will not revert it. You made the bold edit, which I reverted. There is no consensus here, just two other editors from almost three years ago. --Trovatore (talk) 23:41, 19 December 2015 (UTC)
By the way, not at all "unsupported" — I supplied specific citations, for one point of view. I can get others for that point of view and for others. I don't know how easily I could find one specifically for "multitude of views"; that's the problem I mentioned above. However there are a multitude of views that I can source individually. --Trovatore (talk) 23:47, 19 December 2015 (UTC)

I am deeply unhappy with the claims made here that axioms are the same thing as premises and postulates.
Axioms are one leg of a system of reasoning; they are statements taken to be true for the purposes of that system. The other leg is the rules of inference.
Premises are statements that are not taken to be true; rather, reasoning that proceeds from premises leads to conclusions that depend on those premises. For example, if the reasoning leads to a contradiction, it follows that one or more of the premises must have been false. That is not possible for axioms; there is no kind of reasoning that can lead one to conclude that an axiom is false.
A postulate is a different thing again. It is a statement that someone believes to be a theorem, but for which they have so far failed to construct a proof. Some fields of mathematics rely on the truth of such an unproven postulate (i.e. they treat the postulate as if it were an axiom, while fully accepting that it is no such thing). A postulate can be proven false; it then ceases to be a postulate, and becomes simply a false statement.
I fear to attempt to edit this page, because it is full of this belief that these terms all mean the same thing, because so many editors seem to share this 'all the same' view, and because I can't imagine what kind of source would suffice to prove they are distinct (I could cite Beginning Logic, by Lemmon; but that would presumably not satisfy those who believe that logical axioms are just one kind of axiom - a claim I do not accept).
If they do all mean the same thing, it becomes dramatically harder to talk about reasoning. And why would one use any of these terms, if not to talk about reasoning?
On the truth of axioms: if one considers an axiom in the context of the system of reasoning in which it is embedded, then it cannot be considered to be true, as if it were a fact about the world. A purported fact can be shown to be false, but it is impossible to falsify an axiom, either by logic or by reference to reality. I am unable to check the OED reference for this: "As used in modern logic, an axiom is simply a premise or starting point for reasoning", but I seriously doubt the OED gives any such vague definition for this term. My Concise OED certainly doesn't say that. MrDemeanour (talk) 16:01, 22 March 2018 (UTC)

Perhaps these references will help:

— Carl (CBM · talk) 23:12, 20 December 2015 (UTC)

This question is extrinsic and philosophical, not intrinsic and mathematical. There are a number of competing supported positions that need to be in that paragraph, or none at all. It's a WP:NPOV requirement. Please also see my talk page. ~~ BlueMist (talk) 00:25, 21 December 2015 (UTC)

Blue Mist, they are all there. Not all in detail, of course (actually, not any of them in detail). But every single position on the question, including yours, is a position about "[w]hether it is meaningful (and, if so, what it means) for an axiom, or any mathematical statement, to be 'true'".
The previous wording that you challenged said instead "[w]hat it means for an axiom, or any mathematical statement, to be 'true'", and that also included your position, though I take it your answer would be "nothing". However I can see that it is possible to read that text as implying that it does mean something (the text didn't actually imply that, but it was possible to read it that way), so I changed it to make it clear that "nothing" was one of the answers being considered. I can't see how you can possibly object to this on NPOV grounds, since it includes all the possible POVs. --Trovatore (talk) 06:11, 21 December 2015 (UTC)

My position is that I agree with the above two editors, 70.53.24.43 and Anaxiomatic that the sentence needs to be deleted because it is biased in favor of dogmatism. I urged you to undo your revert of this deletion in compliance with Wikipedia:NPOV. ~~ BlueMist (talk) 07:07, 21 December 2015 (UTC)

How is it biased, and in particular in favor of dogmatism? You have as yet not explained that. --Trovatore (talk) 07:12, 21 December 2015 (UTC)

"Dogma is a principle or set of principles laid down by an authority as incontrovertibly true."

~~ BlueMist (talk) 16:09, 21 December 2015 (UTC)
Not responsive. I didn't ask for a definition of dogma. I asked how the text you challenged was biased, and in particular in favor of dogmatism. --Trovatore (talk) 19:41, 21 December 2015 (UTC)

On another note, I find in the history that this language was put into place in May 2015, and there were various attempts before we settled on this. This was one attempt of mine, and in some ways I still like it better than what we have now:

In modern mathematics, it is controversial whether and when to consider axioms "true" or "false" outside the context of a deductive system starting with those axioms. In broad strokes, mathematical realists, sometimes called Platonists, consider some axioms to be true or false in structures that they believe to be well-specified, whereas formalists may take formal derivability from the axioms as simply the meaning of the derived statements (making axioms "true" by definition).

Blue Mist, what do you think of that? I didn't spend really very long on it at the time, and no doubt it can be improved, especially the formalism half, but I think it is at least balanced between the two.
One possible quibble is that some formalists may consider axioms "true by definition" whereas others may simply decline to consider their truth. My take on this is that this is not so much a difference in substance as it is in preferred terminology. However certainly both versions could be mentioned.
Also, I'm not sure how readable the "formal derivability from the axioms as simply the meaning" language is. I know what it means, but then I wrote it. No doubt that section can be written more clearly. --Trovatore (talk) 07:30, 21 December 2015 (UTC)

I don't see how the following sentence from the article is biased in favor of any particular position:

Whether it is meaningful (and, if so, what it means) for an axiom, or any mathematical statement, to be "true" is a central question in the philosophy of mathematics, with modern mathematicians holding a multitude of different opinions.

The sentence does not even state a particular viewpoint on the issue, it only states that there are several of them. — Carl (CBM · talk) 11:39, 21 December 2015 (UTC)

Carl, I've stated my position clearly enough above. I deleted the sentence following the suggestion of two (2) other editors, in accordance with WP:Consensus, as it was judged to be in violation of Wikipedia:NPOV.
As an ex-admin, you know that I am not required at Wikipedia to defend my reasons. I am not on trial here.
That sentence is plain nonsense in philosophy of mathematics terms, as even the idea it suggests as "controversial" is untenable according to the very citation that was added there in its defense. Just click on the reference and read the freely available first page of that article to see. Therefore, the sentence should also be removed as also a WP:NOR violation.
The paragraph that follows explains axioms, reality, and truth fully enough that this disputed sentence adds nothing to the content of the article.

~~ BlueMist (talk) 20:40, 21 December 2015 (UTC)

I also fail to see how that sentence can be read as being "biased in favor of dogmatism". Blue Mist would you please address this specific point? How would you rewrite it so as to make it neutral? Paul August 20:45, 21 December 2015 (UTC)
It's the judgment of the first 3 editors. In fact. If you disagree, then the issue is closed. ~~ BlueMist (talk) 21:56, 21 December 2015 (UTC)
Your history is a little off here, Blue Mist. The disputed language dates to May 2015, and the discussion involved at least three editors at that time. The last contribution from the two other editors you cite (at least, from those accounts) was in February 2013, so they definitely did not pronounce themselves on that language per se. However, if the issue is closed for you, then that's OK with me. --Trovatore (talk) 22:05, 21 December 2015 (UTC)

Of course, the issue being "closed" for Blue Mist does not mean that there are no improvements to be made.

• The current text (which is mostly the work of User:Sammy1339, I think) should indeed be better referenced, if we keep it.
• I still think it might be good to call out specific realist/formalist/other positions, along the lines of my attempt reproduced above in my comment of 07:30, 21 December 2015. The sentence structure of that attempt definitely needs work, though.
• More should be said about the classical (e.g. Euclid) notion of axiom as self-evident truth. (I think that's the Euclidean notion but I am not a scholar of that era; I am open to correction on that point.)
• The overall flow of the lead could use work, especially the second paragraph, the one that starts "[i]n mathematics". The text that we are currently discussing is about mathematics, so in that sense perhaps it belongs in the second paragraph. On the other hand, if we (as I think we should) restore a discussion of the classical Euclidean notion to the first paragraph, then some discussion of modern views on truth might also make sense there. --Trovatore (talk) 22:14, 21 December 2015 (UTC)

## Unclear: "When an equal amount is taken from equals, an equal amount results."

This example axiom is unclear; what does taken mean?

I can see only two interpretations: 1) Subtracting equal amounts from equals produces equal amounts; then the quote should be revised to read "equal amounts result", as there are two quantities being compared 2) Subtracting equal amounts from two quantities and adding each to a third produces an amount equal to the first two; this is plainly false

So perhaps the quote should be revised.Anaxiomatic (talk) 10:57, 12 February 2013 (UTC)

## Reflist

I changed it to use {{reflist}}. This is pretty standard common stuff for > 10 refs (or more), maybe 40em would be better than 30em with these ref widths but what's the objection? 08:53, 26 September 2015 (UTC)

WP:CITEVAR The established reference formatted should not be changed randomly; there is no "standard" that requires multiple columns. [1] - User:CBM
My understanding of CITEVAR is that's more about "citation style" than use of reflist. I can only see in WP:CITEVAR WP:ASL that one may use either, so not compulsory either but improves by reducing whitespace.
70% of articles use reflist and don't flip randomly. 08:53, 26 September 2015 (UTC)
There is no "standard" formatting that applies across all articles. Moreover, there is no benefit to the multiple columns, because there is an unlimited amount of scroll space in the browser. Personally, I don't see any extra white space in the references - perhaps your window is too wide? There is no way to get consistent wrapping, because monitors, browsers, and font sizes vary too much.
So, from my perspective, there was no objective benefit for the change, and it was just based on someone's personal preference to have multiple columns. But for things that are personal preferences, there is a guideline, WP:CITEVAR, which is to leave the established style in place, rather than trying to standardize different articles. Otherwise, other people could go around removing the columns from other articles, or changing the number of columns to suit their taste, etc. — Carl (CBM · talk) 14:05, 26 September 2015 (UTC)
What, figure of speech "pretty standard" (from an edit summary), struck through and replaced by "common" here (for clarity), and it's still not clear the exact precise intended word was "common"?
There's an inevitability of increasing refs. Good luck, 12:32, 27 September 2015 (UTC)

## Mismatching parentheses in formula

The third formula in 3.2.1.1 Arithmetic seems to have mismatching parenteses, there is 1 unnecessary opening parentesis. Could someone who understands the formula well repair it?213.125.95.58 (talk) 14:36, 7 May 2017 (UTC)

Done --Bill Cherowitzo (talk) 14:52, 7 May 2017 (UTC)

## Axioms, premises and postulates

I am deeply unhappy with the claims made here that axioms are the same thing as premises and postulates.

Axioms are one leg of a system of reasoning; they are statements taken to be true for the purposes of that system. The other leg is the rules of inference.

Premises are statements that are not taken to be true; rather, reasoning that proceeds from premises leads to conclusions that depend on those premises. For example, if the reasoning leads to a contradiction, it follows that one or more of the premises must have been false. That is not possible for axioms; there is no kind of reasoning that can lead one to conclude that an axiom is false.

A postulate is a different thing again. It is a statement that someone believes to be a theorem, but for which they have so far failed to construct a proof. Some fields of mathematics rely on the truth of such an unproven postulate (i.e. they treat the postulate as if it were an axiom, while fully accepting that it is no such thing). A postulate can be proven false; it then ceases to be a postulate, and becomes simply a false statement.

I fear to attempt to edit this page, because it is full of this belief that these terms all mean the same thing, because so many editors seem to share this 'all the same' view, and because I can't imagine what kind of source would suffice to prove they are distinct (I could cite Beginning Logic, by Lemmon; but that would presumably not satisfy those who believe that logical axioms are just one kind of axiom - a claim I do not accept).

If they do all mean the same thing, it becomes dramatically harder to talk about reasoning; we would need a new word for "axiom". And why would one use any of these terms, if not to talk about reasoning?

On the truth of axioms: if one considers an axiom in the context of the system of reasoning in which it is embedded, then it cannot be considered to be "true", as if it were a fact about the world. A purported fact can be shown to be false, but it is impossible to falsify an axiom, either by logic or by reference to reality.

I am unable to check the OED reference for this: "As used in modern logic, an axiom is simply a premise or starting point for reasoning", but I seriously doubt the OED gives any such vague definition for this term. My Concise OED certainly doesn't say that. MrDemeanour (talk) 16:01, 22 March 2018 (UTC)

This is an eternal topic of discussion. To start the new thread, I have copied a couple sources below, and asked a question about the source you mentioned. — Carl (CBM · talk) 18:52, 22 March 2018 (UTC)
Here are the four OED definitions, verbatim, for the convenience of those who cannot check them.
1. A proposition that commends itself to general acceptance; a well-established or universally-conceded principle; a maxim, rule, law.
• b. Specially restricted by Bacon to: An empirical law, a generalization from experience.
2. Logic. A proposition (whether true or false).
3. Logic and Math. ‘A self-evident proposition, requiring no formal demonstration to prove its truth, but received and assented to as soon as mentioned’ (Hutton).
— Carl (CBM · talk) 18:39, 22 March 2018 (UTC)
The (Springer) Encyclopedia of Mathematics [2] includes:
"Axiomatic method: A way of arriving at a scientific theory in which certain primitive assumptions, the so-called axioms (cf. Axiom), are postulated as the basis of the theory, while the remaining propositions of the theory are obtained as logical consequences of these axioms."
This seems to conflate postulates with axioms. — Carl (CBM · talk) 18:42, 22 March 2018 (UTC)
Which part of Lemmon's "Beginning Logic" has an explanation of the difference he maintains between axioms, postulates, and premises? — Carl (CBM · talk) 18:50, 22 March 2018 (UTC)
Good question! I seem to have lost my old copy; I've had a pretty good hunt for it. Also, I'm sure I haven't read it for thirty years, so I guess my recollection of what he did and didn't say is pretty suspect.
Update: I found my copy of Lemon, and to my astonishment he doesn't discuss axioms at all.
Well, if I'm wrong, that's a big deal for me. There's a lot of stuff I'd have to review, if it turns out that an axiom is the same as a premise. MrDemeanour (talk) 19:28, 23 March 2018 (UTC)
No one is saying you're "wrong", exactly. You're just not "right" in quite the universal sense you might think. Different people use these terms different ways. You seem to be taking a certain POV, one of many possible. --Trovatore (talk) 21:33, 23 March 2018 (UTC)
I don't think anyone is wrong - I am simply curious to find out about another usage of terminology that I was not familiar with. As a logician I have always identified "axiom", "postulate", "assumption", etc. - in first-order logic there is no difference between them. There is at least one sensible way to distinguish "premise" as a hypothesis used by an inference rule in a proof, even if that premise is not itself one of the assumptions of the proof. But the terminology can become more interesting when we start looking at more "traditional" logic texts, especially those written before the dominance of first-order logic or those that are interested in reasoning that is not captured in systems like first-order logic. So I would genuinely like to know if there is some position that keeps a distinction between "axiom" and "postulate", even if that position is not common in modern texts. — Carl (CBM · talk) 13:07, 24 March 2018 (UTC)
MrDemeanour seems to be using "postulate" in the sense that mathematicians more commonly use "conjecture". I have the notion that this terminology is common among physicists, though I don't have anything to point you to to back that up. --Trovatore (talk) 20:11, 24 March 2018 (UTC)
Interesting - so that would be more like the "Riemann hypothesis" for mathematicians. As you know "postulate" is not used much in math anymore - it makes me think of Euclid. — Carl (CBM · talk) 20:25, 24 March 2018 (UTC)

## Flaws in The Elements

One of the discoveries since the ancient Greeks is that there are incorrect proofs in Euclid's Elements and that, in fact, some of the stated theorems do not follow from the stated axioms and postulates. Should this not be discussed when mentioning the Elements?. In that context, a reference to, e.g., Hilbert's Grundlagen der Geometrie, may be appropriate, Shmuel (Seymour J.) Metz Username:Chatul (talk) 02:07, 8 July 2020 (UTC)

It seems to me more about the Elements than it is about axioms, so I would tend to think it does not really belong in this article. However I wouldn't dig in my heels to prevent a passing mention, if you can word it so it doesn't sound out of place. --Trovatore (talk) 05:55, 8 July 2020 (UTC)
To clarify, the article already mentions the Elements in Axiom#Early Greeks; I'm concerned that the existing text promises more than the Elements can deliver. Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:26, 8 July 2020 (UTC)
Of course we should mention the Elements, as an early (maybe the earliest?) explicit example of the axiomatic method. That's about axioms. When you start talking about whether all the specific results in the Elements follow from the axioms in the Elements, then it seems to me that maybe you've left off talking about axioms, and started talking about the Elements. --Trovatore (talk) 15:02, 8 July 2020 (UTC)
Well, would you object to a short footnote, e.g., well-illustrated [a], and to a brief mention in #Modern development of adding axioms? Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:30, 8 July 2020 (UTC)

Notes

1. ^ Although not complete; some of the stated results did not actually follow from the stated postulates and common notions.