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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.
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
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
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
The Greek word in the etymology in this article is illegible on this browser (Netscape) and looks like a sequence of question marks. Contrast this:
and this:
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)
"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)
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 (
talk •
contribs) 07:53, June 23, 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.
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)
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)
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)
OK, I've changed the article to discuss this point. Josh Cherry 23:50, 1 Dec 2004 (UTC)
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)
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The article now claims:
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)
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 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 ( talk • contribs) 05:54, June 3, 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)
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 ( talk/ contribs) 01:17, 22 October 2006 (UTC)
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)
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 ( talk • contribs) 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)
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))
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)
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 ( talk • contribs) 15:22, 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)
Both 'Axiom Schema' and 'Axiom Scheme' are used in this article. Are they both correct? Passingtramp ( talk) 09:45, 21 May 2009 (UTC)
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)
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)
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)
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)
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)
~~ BlueMist ( talk) 16:09, 21 December 2015 (UTC)" Dogma is a principle or set of principles laid down by an authority as incontrovertibly true."
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:
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:
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)
~~ BlueMist ( talk) 20:40, 21 December 2015 (UTC)
Of course, the issue being "closed" for Blue Mist does not mean that there are no improvements to be made.
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)
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?
Widefox;
talk
08:53, 26 September 2015 (UTC)
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)
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)
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)
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
Hello Community🙂❤️
Here below my edit aproach:
"In historical sciences/context you can see axiomata perhaps as more inclusive as it, for example the third reich here, is satisfying enough if it fully encovers the empathic aspekt abeling one with permission to go on with further assumptions. Inclusive in this way, that it is possible to work with prezise statements regarding empathic ideas without necessarly having great knowledge of formally required information normally required in this topic. So for example: "If ones an ex nationalsocialist it is appropriate for him as a way of caesura in his further life to go on with maybe dramatically social ideas maybe in a slightly risky kontext at least in generating this ideas mentally". So this axiomata is more inclusive as through fully satisfying every aspekt (at that point not yet formally) regarding empathy and through this at least this specific phrase in this specific context, if satisfying the empathic site enough automatically creates valid statements to work with. It is also inclusive in that way that in its origin, the messenger is able to collect information and knowledge for this specific case without explicit formal reputation/education so it enables the working with this discipline to a broader number of people."
So first my provided sources [1] The autor of this article says that the ability to work with statistical math requires the same way of thinking as with connecting to someones feelings. [2] The autor of this article works with our core sozial values as mathematical axioma and how society changes if changing this values. [3] This (also a bit more Professional) source says that with an AI, working with empathy and with this training it in social complex situations outperformes other AIs in methodological learning.
Also I tried writing down some context below, first in case of reposting my edit, but because I love wikipedia too much to get blocked, I switched to the Talk section so Here it is:
"First, I want to say, I love Wikipedia and sometimes I Love editing. Also english is not my native language and I know I make grammatical and orthographically mistakes. Second, I provided some Sources which stated that empathy have significantly Impact on mathematics. But thats just a Part of what I wanted to say. As said before I Love wikipedia and this article. Im not an expert, but first, I tried Just writing about things, that, as far as I guess am able to understand. So second I tried to say that when you, in the process of working with axiomatas, switch the things referred to in this process, to themes you are more familiar with, maybe for this parts of the process Its easier to work with making this Part of mathematics maybe more inclusive. So with this said maybe I can work on this edit to Point this Out more clearly. Second, I guess that, If just regarding the improved performance when work with themes, one is more familiar with, and, regarding my explicit example of working with the third reich in this context, which, is from a science aspect one of the few themes, that, are finished in some way completely and almost every statement is true working With an anti fascist View, maybe this theme gives somehow an extra boost. To be honest, surely, I don't know if the parts I describe in the process with axiomatas, even when performance is better, wether it is possible to get outcomes which are satisfying enough for this mathematical aspect even with, as it is with me also, little lackings of full knowledge of this aspect. To the critiques regarding non dictionary language I want to say that the people I wanted to reach are nonexperts explicitly so I wanted to decrease the limitations for gaining more safety in performing this theme a bit, mostly just as I do in thinking of this theme relatively correct, so it isn't even my aproach to reach people working, regarding this theme, in expressing their workings down to sheets of paper or professionals."
So if you read all this, I would love to hear your critiques, improving ideas, and, if you think all this is pure trash, let me hear your opinion, don't worry Im not THAT emotional unstable😁❤️ Materie34 ( talk) 03:00, 20 November 2023 (UTC)
References
Hello community,
while non-logical axioms (e.g., a + b = b + a) are substantive assertions about the elements of the domain of a specific mathematical theory, such as arithmetic.
I think additive commutative law are not considered as an axiom but an theory derived from the Peano Axiom? I did found some of the people call it an "axiom" in arithmetic. However, in early undergraduate analysis courses, it's often used as an example of basic reasoning to derive some laws in natural numbers from Peano Axiom. I doubt if it's a good example here. Alexliyihao ( talk) 02:03, 30 January 2024 (UTC)
The Peano postulates are not relevant in that context.
the point Alexliyihao madeis bogus, because it ignored the context and the wording. The text
For example, in some groups, the group operation is commutative,is clearly talking about Group theory, not about the Peano postulates. -- Shmuel (Seymour J.) Metz Username:Chatul ( talk) 15:21, 22 April 2024 (UTC)
The lead should [...] summarize the body of the article
not an axiom of group theoryIt is in the theory of commutative groups. So we add a word. Paradoctor ( talk) 20:50, 22 April 2024 (UTC)
Seriously, what are you reading?!?Obviously, the comments relating to Axiom#Non-logical axioms, NOT TO THE LEAD.
Alexlihiyao did not talk about group theory,. The text in contention,
For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom,Which part of
The only appearance of commutative is in #Non-logical axioms, where the context is group theory.did you not understand? And, yes, there were subsequent updates to the lead, but that has nothing to do with the validity of comments posted before them.
All he did is criticize the use of commutativity of addition as an example of a non-logical axiom in the lead,. Patently false: the lead, AT THAT TIME, did not contain any such text. -- Shmuel (Seymour J.) Metz Username:Chatul ( talk) 12:16, 25 April 2024 (UTC)
the comments relating to Axiom#Non-logical axioms, NOT TO THE LEADThen you're missing the point, because the discussion is about a passage from the lead. Paradoctor ( talk) 12:54, 25 April 2024 (UTC)
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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.
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
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
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
The Greek word in the etymology in this article is illegible on this browser (Netscape) and looks like a sequence of question marks. Contrast this:
and this:
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)
"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)
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 (
talk •
contribs) 07:53, June 23, 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.
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)
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)
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)
OK, I've changed the article to discuss this point. Josh Cherry 23:50, 1 Dec 2004 (UTC)
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)
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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. —
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The article now claims:
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)
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 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 ( talk • contribs) 05:54, June 3, 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)
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 ( talk/ contribs) 01:17, 22 October 2006 (UTC)
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)
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 ( talk • contribs) 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)
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))
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)
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 ( talk • contribs) 15:22, 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)
Both 'Axiom Schema' and 'Axiom Scheme' are used in this article. Are they both correct? Passingtramp ( talk) 09:45, 21 May 2009 (UTC)
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)
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)
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)
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)
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)
~~ BlueMist ( talk) 16:09, 21 December 2015 (UTC)" Dogma is a principle or set of principles laid down by an authority as incontrovertibly true."
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:
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:
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)
~~ BlueMist ( talk) 20:40, 21 December 2015 (UTC)
Of course, the issue being "closed" for Blue Mist does not mean that there are no improvements to be made.
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)
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?
Widefox;
talk
08:53, 26 September 2015 (UTC)
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)
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)
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)
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
Hello Community🙂❤️
Here below my edit aproach:
"In historical sciences/context you can see axiomata perhaps as more inclusive as it, for example the third reich here, is satisfying enough if it fully encovers the empathic aspekt abeling one with permission to go on with further assumptions. Inclusive in this way, that it is possible to work with prezise statements regarding empathic ideas without necessarly having great knowledge of formally required information normally required in this topic. So for example: "If ones an ex nationalsocialist it is appropriate for him as a way of caesura in his further life to go on with maybe dramatically social ideas maybe in a slightly risky kontext at least in generating this ideas mentally". So this axiomata is more inclusive as through fully satisfying every aspekt (at that point not yet formally) regarding empathy and through this at least this specific phrase in this specific context, if satisfying the empathic site enough automatically creates valid statements to work with. It is also inclusive in that way that in its origin, the messenger is able to collect information and knowledge for this specific case without explicit formal reputation/education so it enables the working with this discipline to a broader number of people."
So first my provided sources [1] The autor of this article says that the ability to work with statistical math requires the same way of thinking as with connecting to someones feelings. [2] The autor of this article works with our core sozial values as mathematical axioma and how society changes if changing this values. [3] This (also a bit more Professional) source says that with an AI, working with empathy and with this training it in social complex situations outperformes other AIs in methodological learning.
Also I tried writing down some context below, first in case of reposting my edit, but because I love wikipedia too much to get blocked, I switched to the Talk section so Here it is:
"First, I want to say, I love Wikipedia and sometimes I Love editing. Also english is not my native language and I know I make grammatical and orthographically mistakes. Second, I provided some Sources which stated that empathy have significantly Impact on mathematics. But thats just a Part of what I wanted to say. As said before I Love wikipedia and this article. Im not an expert, but first, I tried Just writing about things, that, as far as I guess am able to understand. So second I tried to say that when you, in the process of working with axiomatas, switch the things referred to in this process, to themes you are more familiar with, maybe for this parts of the process Its easier to work with making this Part of mathematics maybe more inclusive. So with this said maybe I can work on this edit to Point this Out more clearly. Second, I guess that, If just regarding the improved performance when work with themes, one is more familiar with, and, regarding my explicit example of working with the third reich in this context, which, is from a science aspect one of the few themes, that, are finished in some way completely and almost every statement is true working With an anti fascist View, maybe this theme gives somehow an extra boost. To be honest, surely, I don't know if the parts I describe in the process with axiomatas, even when performance is better, wether it is possible to get outcomes which are satisfying enough for this mathematical aspect even with, as it is with me also, little lackings of full knowledge of this aspect. To the critiques regarding non dictionary language I want to say that the people I wanted to reach are nonexperts explicitly so I wanted to decrease the limitations for gaining more safety in performing this theme a bit, mostly just as I do in thinking of this theme relatively correct, so it isn't even my aproach to reach people working, regarding this theme, in expressing their workings down to sheets of paper or professionals."
So if you read all this, I would love to hear your critiques, improving ideas, and, if you think all this is pure trash, let me hear your opinion, don't worry Im not THAT emotional unstable😁❤️ Materie34 ( talk) 03:00, 20 November 2023 (UTC)
References
Hello community,
while non-logical axioms (e.g., a + b = b + a) are substantive assertions about the elements of the domain of a specific mathematical theory, such as arithmetic.
I think additive commutative law are not considered as an axiom but an theory derived from the Peano Axiom? I did found some of the people call it an "axiom" in arithmetic. However, in early undergraduate analysis courses, it's often used as an example of basic reasoning to derive some laws in natural numbers from Peano Axiom. I doubt if it's a good example here. Alexliyihao ( talk) 02:03, 30 January 2024 (UTC)
The Peano postulates are not relevant in that context.
the point Alexliyihao madeis bogus, because it ignored the context and the wording. The text
For example, in some groups, the group operation is commutative,is clearly talking about Group theory, not about the Peano postulates. -- Shmuel (Seymour J.) Metz Username:Chatul ( talk) 15:21, 22 April 2024 (UTC)
The lead should [...] summarize the body of the article
not an axiom of group theoryIt is in the theory of commutative groups. So we add a word. Paradoctor ( talk) 20:50, 22 April 2024 (UTC)
Seriously, what are you reading?!?Obviously, the comments relating to Axiom#Non-logical axioms, NOT TO THE LEAD.
Alexlihiyao did not talk about group theory,. The text in contention,
For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom,Which part of
The only appearance of commutative is in #Non-logical axioms, where the context is group theory.did you not understand? And, yes, there were subsequent updates to the lead, but that has nothing to do with the validity of comments posted before them.
All he did is criticize the use of commutativity of addition as an example of a non-logical axiom in the lead,. Patently false: the lead, AT THAT TIME, did not contain any such text. -- Shmuel (Seymour J.) Metz Username:Chatul ( talk) 12:16, 25 April 2024 (UTC)
the comments relating to Axiom#Non-logical axioms, NOT TO THE LEADThen you're missing the point, because the discussion is about a passage from the lead. Paradoctor ( talk) 12:54, 25 April 2024 (UTC)