Talk:Indeterminate (variable)

Unclear definition

The definition "a quantity that is not known, and cannot be solved for" is not clear.

• Chaitin's constant is not known, and cannot be solved for. Does that mean it is an indeterminate?
• The text implies that an indeterminate is not a variable. This does not fit with the common use of the term "variable" as I know it.
• In the text "the polynomial $X-1$", X is not known and cannot be solved for. In the text "the polynomial equation $X-1=0$", X is not known but can be solved for. Does that mean X is an indeterminate in the polynomial by itself but not in the polynomial used in the polynomial equation?  --Lambiam 17:39, 24 July 2008 (UTC)

I agree with the criticism; this definition should be rewritten. In fact, I would say an indeterminate is a symbol that is not used to designate any (other) known or unknown quantity: it is neither an unknown (initially unspecified value, but which with some good luck can be solved for), nor a parameter (value supposed to be given and fixed in the problem at hand, but not explicitly known), nor a formal variable (as the bound variable in some syntactic construction like a summation, or the variable used for the argument in a function definition; in both cases such a variable stands for many different values at once). Note that parameters nor formal variables are known, and cannot be solved for either, showing again the defect of the current definition.

What distinguishes indeterminates is that by standing for nothing else, they become bona fide values in their own right. So for instance by constructing the polynomial ring Z[X], the symbol X acquires the status of indeterminate (but many other formal constructions can introduce indeterminates as well). This is a rather fundamental conceptual step that ought to be described properly in this article. But to answer your last question: indeed, if a polynomial is used in a polynomial equation, its indeterminates lose their status, and become unknowns; regarding $X-1=0$ as an equation of formal polynomials it just states an unconditional falsehood (both members are distinct polynomials, period). Marc van Leeuwen (talk) 10:51, 2 August 2008 (UTC)

• I find this entry confusing. In case 2, doesn't

  2 + 3x = a + bx

  actually mean

  '2 + 3x' = ˹Φ + ψx˺

  where the funny brackets are quasi-quotes and Φ and ψ are meta-variables?Adam.a.a.golding (talk) 01:57, 21 December 2009 (UTC)

• The final example given in the section "Polynomials" (regarding 0-0^2=0, 1-1^2=0, with modulo 2) seems to lack an additional note for it in the main page which might cause some misunderstanding? We can find that another modulo 2 value of '-1' gives the form (-1)-(-1)^2=(-1)-1=-2, which results in the answer of -2 which in modulo 2 also equals 0 (yes, there is a small mistake in doing this which i'll get to soon, but the statement is not incorrect). There is a bit of trickiness in seeing that binding a range limit (ie. Must be an integer, and must have value equivalence after applying modulo 2) to some unknown x; is trying to give an example too complicated which a student might get confused (because it's perfectly fine for the intermediate X to be limited as well -- say 'must only be an integer').

Getting back to why (-1)-(-1)^2=(-1)-1=-2, isn't a concern; we need to remember that for our function, the modulo 2 property is a requirement of the Domain. The numerical output structure is the Range (which can be defined as something else). Our target output of 0 needs to make sense for what we define the Range to be. THIS is the reason why doing modulo 2 on such output is not correct. I believe what the author of the equation example tried to do was to say "hey.. there's some function that generates an intended result value with a source of unknown x (the structure of x is defined by some Domain. We are talking about particular Numbers here, not a fancy matrix) which generates more than 1 solution, but this unknown x is not an indeterminate X; because an indeterminate (with a structure defined by the Domain) requires that the function will ALWAYS generate the intended result value regardless of what value of X is." 60.240.106.190 (talk) 00:52, 11 February 2015 (UTC)

why is x the unknown

Perhaps it is worth to mention, why we use in general x as unknown. http://www.ted.com/talks/terry_moore_why_is_x_the_unknown.html --77.186.34.177 (talk) 21:12, 2 July 2012 (UTC)

definition is unclear

The following defines an indeterminate in terms of other undefined concepts: symbol, variable, placeholder, and in terms of what it is not. Can we do any better?

In mathematics, and particularly in formal algebra, an indeterminate is a symbol that is treated as a variable, but does not stand for anything else but itself and is used as a placeholder in objects such as polynomials and formal power series. In particular, it does not designate a constant or a parameter of the problem, it is not an unknown that could be solved for, and it is not a variable designating a function argument or being summed or integrated over; it is not any type of bound variable.--345Kai (talk) 22:07, 3 September 2018 (UTC)

Links contradict description

Many of the links provided say that indeterminates are synonymous with unknowns. Specifically the first three links. Farkle Griffen (talk) 20:28, 10 August 2024 (UTC)

Need a positive descriptive definition.

Currently this is only defined in terms of negatives

"It is not a variable" "It is not a function argument" "It is not a function parameter" "It is not an unknown in a equation"

The only descriptive term is "it is a symbol that doesn't represent anything except itself" which is extremely vague.

This article needs a positive definition, and explanations as to how it is different from all of those in the list. Farkle Griffen (talk) 16:15, 12 August 2024 (UTC)

Agreed. Recommend free variable and remove "It is not a variable". Placeholder is currently a disambiguation. Informal dummy variable is deprecated. Perhaps sources can be found to quote, motivating the obscurity implicit in this concept. — Rgdboer (talk) 22:30, 12 August 2024 (UTC)

Disparity between definition and usage

My previous edit trying to pin down a formal definition of an indeterminate was the consensus after asking the professors in the math department at my university (specifically, "A variable with no inherent domain") was reverted citing WP:Original research, which is fair as no sources defending this definition were cited. However, after finding sources that actually try to define the term, it seems to require that the coefficients need to be elements of some sub-ring, while X is an element of the higher ring, transcendental to the subring.

This is in contrast to how most algebra books seem to use the term, as they do not require X to be an element of any ring, and moreover, most do not define polynomials in terms of a “subring”. There does not seem to be a citable, formal definition that accurately describes how most mathematicians use the term, apart from the very informal description “A formal symbol”. A definition should be reducible to or constructed from the axioms of ZFC.

Additionally, much of the lead before my most recent edit was deleted due to being completely unsourced, or the source was too weak to be considered. Farkle Griffen (talk) 16:56, 23 August 2024 (UTC)

@D.Lazard The second book, An introduction to algebraic structures, specifically notes that "pi is an indeterminate over the rationals"; page 205. Farkle Griffen (talk) 18:10, 23 August 2024 (UTC)

Being unsourced does not mean that sources do not exist. The template {{citation needed}} must be used instead of deleting when no source is provided, and you do not have a source saying that the assertion is wrong.

Also personal communications of your professors cannot be used as a source for Wikipedia. Moreover, since the "domain" of a variable is nowhere defined, the sentence "A variable with an empty domain" is nonsensical.

Apparently, the word "indeterminate" is not used in Beckenbach's book. D.Lazard (talk) 18:24, 23 August 2024 (UTC)

I have provided three sources Farkle Griffen (talk) 18:29, 23 August 2024 (UTC)

Please provide here the exact quotation of your three sources, since I suspect that you misinterpret them. D.Lazard (talk) 18:34, 23 August 2024 (UTC)

You can see them for yourself:

[Introduction to Algebra] Page 160

[An Introduction to Algebraic Structures] Page 204

[Introduction to Modern Algebra] Page 139-140 (Page 142 uses the term "a transcendental")

The links have been modified to bring you to the exact page numbers, with the term "Indeterminate" highlighted. Though you may have to create a free account to view them. Farkle Griffen (talk) 18:54, 23 August 2024 (UTC)

This definition of an indeterminate seems specific to textbooks on (elementary) [[abstract algebra. However, this definition is not used outside this area of mathematics. So, I have added a paragraph in the lead to mention this alternative definition. D.Lazard (talk) 10:21, 24 August 2024 (UTC)

Can you provide any sources that have definitions to the contrary? As of now, none of the sources support the main definition provided in the article. I will start a discussion post about this Farkle Griffen (talk) 14:27, 24 August 2024 (UTC)

The definition of these textbooks implies trivially that every transcendental number (including ⁠$\pi ^{2}$⁠) and every nonconstant polynomial (including ⁠$X^{2}$⁠) should be an indeterminate. The fact that these silly assertions cannot be found an any mathematical text shows clearly that the definition given in the article is closer to the common practice of mathematicians.

Nevertheless, assertions of the lead about "what indeterminates are not" are not useful for defining an indeterminate, and the assertion that an indeterminate is not a constant is wrong, since indeterminates are constants of the theory of polynomials. So, I have removed all the items except the last on, which is sourced. D.Lazard (talk) 16:08, 24 August 2024 (UTC)

Sometimes silly assertions have some truth? I mean certainly as rings the polynomial ring $\mathbb {Q} [x]$ is isomorphic to the subring $\mathbb {Q} [\pi ]$ of $\mathbb {R}$, the field of fractions of the former can be given a natural order that makes it isomorphic as an ordered field to $\mathbb {Q} (\pi )$, etc. --JBL (talk) 17:58, 24 August 2024 (UTC)

@D.Lazard and @JayBeeEll, It seems even Dummit and Foote agree with this definition. (Though they take quite a while to get around to this)

In chapter 14, section 9, paragraph 2, they say:

"We generally reserve the expression “‘t is an ‘indeterminate’ over F’’, when we are thinking of evaluating t. Field theoretically, however, the terms transcendental and indeterminate are synonymous (so that the subfield $\mathbb {Q} (\pi )$ of $\mathbb {R}$ and the field $\mathbb {Q} (t)$ are isomorphic)." Farkle Griffen (talk) 18:27, 14 September 2024 (UTC)

Problems with the article

1) Problems with "Is used as a variable but does not represent anything but itself"

(1) This is not supported by any sources in the article

(2) It is unilluminating to anyone trying to understand the concept

(3) It cannot be easily formalized, as the most obvious meaning "A variable that represents nothing" does not satisfy the algebraic requirements of such an item, namely, equality of polynomials. Since any equation involving an indeterminate by this definition would vacuously form an identity, thus 3-2X = 4 is true. Since all resolutions of X (none) make this true; or more clearly, there does not exist a resolution of X which makes the equation false.

(4) This blatantly contradicts many of the sources provided in the article. Source 1 (Mathworld) says "Indeterminate" is a synonym for "variable". And contradicts the only three sources that seem to actually try to define the term. More on this:

2) Problems with sources

Source 1) Mathworld "The term "indeterminate" is sometimes used as a synonym for unknown or variable." (Article) "It is not an unknown that could be solved for."

By this source, this article should simply be a redirect to variable (mathematics)

Source 2) ProofWiki This does not even seem to even attempt to define the term; only saying that "The indeterminate of (S,ι,X) is the term X" (These two sources have been removed

Source 3) McCoy
This does not attempt to define the term beyond "just a symbol we will use in an entirely formal way", which is not a definition; it is clear that the exact definition is not of high importance to them.

Source 7) Herstein
They do not attempt to define the term either. And further, this term is used exactly once in their book.

As of now, the only sources that attempt to define the term are being used as a sidenote. If more sources are not found that define the term as anything other than what is described in those sources, I will begin moving the article to be more in line with those definitions. Farkle Griffen (talk) 15:14, 24 August 2024 (UTC)

There are thousands mathematical articles that use indeterminates in the sense given in the present lead. You can find many of them by searching Scholar Google with entries "indeterminates" (with quotes for avoiding adjectives), "indeterminates" algebra, commuting "indeterminates", noncommuting indeterminates, etc. Since this is mainstream mathematics that Wikipedia must describe, the definition of the lead must be kept as the main definition, even if slight improvements are possible. So, please, do not change the present definition into a definition that could astonish or confuse the readers of these articles. D.Lazard (talk) 20:12, 25 August 2024 (UTC)

What you have done is WP:Original research; Sure, your definition could support how other sources use it, but it is not supported by any sources. One could come up with several distinct definitions that could conscieveably be correct. Since you have not given any sources in the several weeks that I have asked, I will be removing this definition and replacing it with one that is actually sourced. Farkle Griffen (talk) 18:37, 14 September 2024 (UTC)

To editor Farkle Griffen: Please, do not change the present definition without a consensus here.

I have reverted your last edits for this reason, and also because

• It contains systematic misinterpretations of the sources you provide: most of them are coherent with the first paragraph, and I have added to it.
• If there is no formal definition of an indeterminate that is available in the literature, one must not try to give one, as this would be WP:Original research.

There are several other reasons of the revert, but these two are the main one. Nevertheless, one of the sources you provide define indeterminates in essentially the same way as the lead; so, I have restored it in the first paragraph.

Also, I removed the fact that an indeterminate may refers to itself since this is too technical and not useful for most readers: this is needed in logical frameworks where all variables that are not arguments of functions must be linked to some value. D.Lazard (talk) 10:48, 26 August 2024 (UTC)

I did not touch the definition in the first sentence. Moreover, this is not justification to remove 13 sources. If you believe I am misinterpreting, please simply edit the article. Please Revert only when necessary; this was not vandalism and included material that genuinely helped the article. My edits also contained edits outside of the lead, and an explanation of a technical term used in the lead.

My edits did not assert that there is a single formal defnition. Farkle Griffen (talk) 12:41, 26 August 2024 (UTC)

@D.Lazard, Simply saying there are errors is not justification for removing 13 sources. You did not mention what these errors are. Farkle Griffen (talk) 13:37, 26 August 2024 (UTC)

Farkle Griffen' edit

Recently, Farkle Griffen added a new section that I reverted as WP:Original synthesis. He added again the section with the edit summary "Everything added on this edit is taken nearly word for word from one or more articles".

Apparently, Farkle Griffen ignore that taking a quotation out of its context can change dramatically its meaning. In fact, none of the source that I have looked on support the related assertion.

The controversial section starts with "Some authors use the term to mean generaly variable". This may mean that, for these authors "indetrminate" and "variable" are synonyms. This may also be a confusing way to say that an indeterminate is a sort of variable; in this case, this would be a poor repetition of the first sentence of the article.

The section contains "[Lang] considers an indeterminate to be the generator of an infinite cyclic group", followed by a footnote saying that Lang do not use the trm "indeterminate".

"Other authors use it as a kind of syntactic entity, similar to punctuation, ...": "syntactic entity" is correct, but "similar to punctuation" is blatantly wrong.

I could continue, almost every sentence contains wrong assertions or confusing formulations.

Nevertheless, the main reason for removing the section is that it is WP:original synthesis since none of the cited authors claim to provide a formal definition of an indeterminate.

So, I'll revert this edit again. D.Lazard (talk) 18:07, 26 August 2024 (UTC)

Thank you for giving an explicit reason for your revision this time. I will address exactly the issues mentioned here. If you have other issues please mention those.

But first, let me reiterate, as you said, "Being unsourced does not mean that sources do not exist. The template {{citation needed}} must be used instead of deleting when no source is provided, and you do not have a source saying that the assertion is wrong." My edit was made in Good faith

• The controversial section starts with "[...]. This may mean that, for these authors "indetrminate" and "variable" are synonyms. This may also be a confusing way to say that an indeterminate is a sort of variable; in this case, this would be a poor repetition of the first sentence of the article.

If you believe this sentence sounds confusing, please feel free to clarify it. A confusing sentence is not usually justification for deleting a section. And repetition is of least importance to the article right now since much of it is still up for change given that this article still lacks consensus on the definition of the topic.

• "[Lang] considers an indeterminate to be the generator of an infinite cyclic group", followed by a footnote saying that Lang do not use the trm "indeterminate".

Yes, Lang uses the symbol in a way that matches the article's description, and matches how other authors use items they call an "indeterminate". If you believe this is too much of a stretch to be considered a source for that statement, feel free to add {{better source needed}}.

• "syntactic entity" is correct, but "similar to punctuation" is blatantly wrong.

I use the analogy to punctuation to get across the point that (by these authors interpretation) it is very much not a variable, and is used purely for syntax, hence the rest of that sentence "[...] similar to punctuation, in that it does not represent anything or have any algebraic properties."

If you dislike the analogy, feel free to replace it with a better one. A poor analogy is not usually justification for deleting a whole section.

If you have other issues you would like me to address please feel free to add them. Until then, I will assume we have reached a consensus. Farkle Griffen (talk) 19:04, 26 August 2024 (UTC)

You do not address my main objection to your edit: You try to extract a formal definition from sources that do not provide such a formal definition. This is WP:original synthesis, which is definitely forbidden in Wikipedia.

I disagree with almost everything you wrote in your answer. I do not list all my points of disagreement because this would be too time consuming. As, for the moment, there is no opinion of other users, there is absolutely no consensus despite the last sentence of your last post. Please, read WP:CONSENSUS. D.Lazard (talk) 14:23, 27 August 2024 (UTC)

Here, one can "formal" to mean a description which can be reduced to (or derived from) the axioms of ZFC (or any other foundation, really).

[Edit: this is a fairly common definition; see formal system, formal proof, formalism (mathematics), etc.]

Several sources given provide such a description. And it is not Original synthesis as these sources explicitly state these descriptions. (With the exception of one example that these authors describe polynomials in a ring as an infinite tuple, however, my example used a finite tuple for simplicity). If you are having trouble finding these descriptions, please tell me and I can provide you with a more exact location.

Moreover, I cannot resolve issues you don't mention. Unfortunately, I was not gifted with the ability to read minds. Farkle Griffen (talk) 15:54, 28 August 2024 (UTC)

@D.Lazard, do you still have issues with the edit? Farkle Griffen (talk) 19:39, 31 August 2024 (UTC)

If you mean "does my explanations convinced you?" the answer is definitively no. Moreover, explanations on the talk page must be explanation of the reasons of an edit, not explantion of the meaning of the new content. If a content needs explanation as you did by explaining your meaning of "formal", this imply that it must be reverted, since readers are not supposed to read the talk page.

In any case, I am tired to try to explain to you basic mathematics and basic rules of `wikipedia. I give up, and leave other editors fixing your disimprovements of the article. D.Lazard (talk) 20:38, 31 August 2024 (UTC)

For the analogy, a better example may be "similar to the 'd' symbol in a derivative". This seems more appropriate. However, I don't know if we can assume most readers of this article have taken Calculus. Farkle Griffen (talk) 15:44, 28 August 2024 (UTC)

The definition of indeterminate

As seen above, there's a bit of contention about how to define "indeterminate", however, given that the current definition is not supported by any sources, and in my effort to find a source defending it, I have found none, I will be editing the article [to remove what appears to be WP:Original Synthesis, and to*] be more inline with the sources which do define the term. The definition I will be using will be roughly that an indeterminate is a constant that is transcendental to a given ring. Here are the explicit quotations that I will be referencing:

[1] Dummit and Foote (2003) Abstract Algebra (3rd ed) p.645

"We generally reserve the expression “‘t is an ‘indeterminate’ over F’’, when we are thinking of evaluating t. Field theoretically, however, the terms transcendental and indeterminate are synonymous (so that the subfield $\mathbb {Q} (\pi )$ of $\mathbb {R}$ and the field $\mathbb {Q} (t)$ are isomorphic)."

[2]: Lewis, Donald J. (1965). Introduction to Algebra. p.160.

"Let x be an element of a ring $L$, and let $R$ be a subring of $L$ such that ax = xa for all a in $R$. We say x is algebraic over $R$ if there is a finite set a0, ax, ..., an in $R$, not all 0, such that a0 + a1x + ... + anx = 0. If x is not algebraic over $R$, we say x is a transcendental (or an indeterminate) over $R$. [...] The term “indeterminate” is customarily used when one is studying the existence of nonalgebraic elements over a ring $R$. However, as indicated above, the two terms “transcendental” and “indeterminate” may be used interchangeably."

[3] Landin, Joseph (1989). An Introduction to Algebraic Structures. p.204

"Definition 2. Let A[t] be a polynomial ring in t over A. The element t is an indeterminate over A if and only if ${\textstyle \sum _{i=0}^{n}a_{i}t^{i}=0{\text{ implies each }}a_{i}=0,0\leq i\leq n}$. Thus π is an indeterminate over the rationals, whereas ${\sqrt {2}}$ is not. "

[4] Marcus, Marvin (1978). Introduction to Modern Algebra. p.140–141

We say that x; is an indeterminate with respect to a ring R if there exists a ring with identity, S, containing x and a monomorphism i : R-> S satisfying the following conditions:

(1) xi(r) = i(r)x for each r in R. [Commutativity]

(2) Every element of S is of the form ${\textstyle \sum _{k=0}^{n}a_{k}x^{k}}$ where ak is in i(R), and n is in N U {0} .

(3) If ${\textstyle \sum _{k=0}^{n}a_{k}x^{k}=0}$, ak = i(rk), rk in R, then r0 = ... rn = 0 [Transcendence]"

Then, a few authors go further by trying to give an explicit construction which meets these criteria, by defining an indeterminate as a kind of infinite coordinate vector (or sequence), where the powers to the indeterminate are essentially the unit vectors in each direction.

[1] Lang, Serge (1987). Undergraduate Algebra . p.106-108

[This is over the scope of several pages, but the gist of it is:]

"Let R be a commutative ring. Let Pol_R be the set of infinite vectors

(a₀ , a₁, a₂, ..., ...)

with a_n in R and such that all but a finite number of a_n are equal to 0. Thus a vector looks like

(a₀, a₁, ..., a_d, 0, 0, ...)"

[...]

"Now pick a letter, for instance t , to denote

t = ( 0 , 1 , 0 , 0 , 0 , ...)."

[...]

"[...] in other words tⁿ is the vector having the n-th component equal to 1, and all other components equal to 0"

With two other supporting sources which define it similarly, and through multiple pages, so I won't bother trying to quote them, but I have provided links if you wish to read them yourself

[2] Sah, Chih-Han (1967). Abstract algebra. p.56-57

[3] Sawyer, W. W. (2018). A Concrete Approach to Abstract Algebra. p.64-67

(These two sources support the term "infinite sequence" rather than vector)

Given how common this definition appears to be, I will be mentioning it.

Does anyone have any notes? Concerns? Suggestions? If not, I will begin editing the article in this direction.

(* Edited) Farkle Griffen (talk) 02:17, 15 September 2024 (UTC)

With a notable mention from Herstein, possibly supporting the second definition, saying on page 153:

"Let F be a field. By the ring of polynomials in the indeterminate, x, written as F[x], we mean the set of all symbols "a₀ + a₁x+... + a_nxⁿ ", where n can be any nonnegative integer and where the coefficients a1, a2 , ..., an are all in F. [...] We could avoid the phrase "the set of all symbols" used above by introducing an appropriate apparatus of sequences but it seems more desirable to follow a path which is somewhat familiar to most readers."

While he decides not to use the "sequences" definition, it seems reasonable that he is endorsing something similar to the sequence/vector construction above. While I don't think this should be used as a source in the article, I think it simply deserves an honorable mention as supporting evidence for the latter definition. Farkle Griffen (talk) 03:33, 15 September 2024 (UTC)

About Farkle Griffen' version of the lead

Farkle Griffen is edit-warring for trying to impose his version of the lead.

The main change consists of expanding the sentence "There are generally two camps that attempt to formalize this term." This sentence is pure WP:original research. Indeed there is no source asserting the existence of two camps, and no source mention the need of formalizing the concept of an indeterminate. It follows that the description of the two "camps" is WP:Original synthesis and therefore does not respect a fundamental policy of Wikipedia.

Third-party opinions are really needed, since, apparently, Farkle Griffen consider that the lack of a third-party opinion is a consensus in his favoor: "Again, the discussion topic was posted nearly a week ago, currently you are the only one opposing this edit, and you have not not engaged in the discussion." D.Lazard (talk) 09:06, 21 September 2024 (UTC)

If your issue is with the phrase "there are generally two camps", that is something that can easily be changed without a revert of the whole thing [Edit: See WP:FIXTHEPROBLEM]. That sentence is just a transition into noting two common formalizations. Durbin (2008) [Modern Algebra pp. 160–161] notes both of these as possible definitions. Is your only issue with the phrase "there are generally two camps"?

And my issue is not with lack of 3rd party opinion; my issue was the lack of your opinion in the discussion post above. Farkle Griffen (talk) 05:15, 22 September 2024 (UTC)

My issue is also with "the attempt to formalize the term" and with the whole content that is introduced by this phrase.

Also the fact that I am tired to repeat my objections again and again does not mean that I agree with you. In fact, my answer to your wall of texts has already been given, and I must repeat it:

You do not address my main objection to your edit: You try to extract a formal definition from sources that do not provide such a formal definition. This is WP:original synthesis, which is definitely forbidden in Wikipedia.

I disagree with almost everything you wrote in your answer. I do not list all my points of disagreement because this would be too time consuming. As, for the moment, there is no opinion of other users, there is absolutely no consensus despite the last sentence of your last post. Please, read WP:CONSENSUS.

D.Lazard (talk) 10:36, 22 September 2024 (UTC)

I've tried to address this before but apparently I must have misunderstood. What do you mean by "You try to extract a formal definition from sources that do not provide such a formal definition"? Because, yes they do; the provide the definition exactly as I have written in my edit. Farkle Griffen (talk) 15:08, 22 September 2024 (UTC)

@D.Lazard, can you please explain what you mean by "You try to extract a formal definition from sources that do not provide such a formal definition"? Farkle Griffen (talk) 00:49, 25 September 2024 (UTC)

I am not here to teach English. Several of my posts contain essentially the same assertion with different words. D.Lazard (talk) 08:42, 25 September 2024 (UTC)

Yes, we are here to try to build an accessible encyclopedia, and snide remarks are not helpful in doing that.

If your objection is "No source provides such a definition", than this is wrong; all cited sources very explicitly support at least one of the definitions noted. If your objection is "No source asserts that these definitions are more formal", this is also wrong: Knapp, Goodman, and Grinberg explicitly note their definitions as more formal.

Nevertheless, to repeat from my previous post, the current definition is not supported by any sources; no source currently says "Indeterminates are variables", and in my effort to find a source, I have found none. I have asked several times for you to provide a source for your assertion, and you have provided none. Therefore it is reasonable to assume the current description is your WP:Original research. (Moreover, the source currently used as an attempt to justify the definition (McCoy) also notes the sequence definition to explain the meaning of "formal expression" in the following paragraph)

Further, the current description "just a symbol used in a formal way" is unenlightening; all symbols in math are "used in a formal way". This version is unhelpful to any reader who isn't already familliar with the topic.

I see no reason as to why this version should stay.

(Edit: @D.Lazard) - Farkle Griffen (talk) 16:12, 25 September 2024 (UTC)

@D.Lazard, can you please respond to this? Farkle Griffen (talk) 20:05, 27 September 2024 (UTC)

I do not like to repeat myself. D.Lazard (talk) 08:14, 28 September 2024 (UTC)

@Farkle Griffen – these links are course notes, but in general they don't seem to define indeterminate, and usage seems compatible with D.Lazard's preferred language. The first one has e.g. "a polynomial in one indeterminate with coefficients in F is an infinite sequence ...". I think it would be fine to mention in the sections about polynomials and formal power series that they are sometimes formally defined as sequences (or infinite sequences). –jacobolus (t) 20:03, 28 September 2024 (UTC)

"no source ..." – How about this footnote: "According to modern terminology, the unknown quantity or quantities within a polynomial (regarded as an expression) are called indeterminates, and they are called variables only when the polynomial is considered as a function. It is, however, widespread to retain the classical terminology and use the word “variable” in both expressions and functions." Toth (2001) doi:10.1007/978-3-030-75051-0_6 –jacobolus (t) 22:32, 29 September 2024 (UTC)

I came accross a few sources stating something similar to this, but its hard to say that this qualifies as a source asserting that "indeterminate" and "variable" mean exactly the same thing.

For instance, it could be very easily be interpreted instead as "variable" simply has multiple meanings. There is "variable" in the common or 'function' sense, and there is "variable" in the 'indeterminate' or 'expression' sense.

And there is reason to reject the former interpretation too: several authors wish to apply properties of indeterminates that don't apply to variables (most notably, two polynomials are equal if and only if their coefficients are equal) Farkle Griffen (talk) 23:38, 29 September 2024 (UTC)

Yes that is correct, the term "variable" is used broadly, including for this purpose. Sometimes authors specify "formal variable". –jacobolus (t) 23:55, 29 September 2024 (UTC)

That's fine, I don't have a problem with this, however, if this is mentioned in the article, it should be made very clear that is not the same as "variable" in the common sense as used in Variable (mathematics). And further, one would still have to define this second sense of "variable" in the article.

Though it seems like the article would become very confusing if this distinction is not very explicitly made. Farkle Griffen (talk) 00:23, 30 September 2024 (UTC)

I'm not sure where is most useful to interject in this and previous discussions, but here's a source intended for beginners: "A polynomial with coefficients in a commutative ring R is an expression of the form [...], x is a symbol called an indeterminate, [...]. The symbols ⁠$\textstyle x^{2},\ldots ,x^{n}$⁠ are powers of the indeterminate ⁠$x$⁠: that is, ⁠$\textstyle x^{2}=x\cdot x,x^{3}=x\cdot x\cdot x$⁠, etc." [...] "Functions defined by polynomials with real coefficients are familiar objects in calculus, for they are the easiest functions to differentiate and integrate. ¶ However, a polynomial with coefficients in a commutative ring R should not be thought of as a function described by its value at an indeterminate element of R, but rather as just a formal expression involving the symbol x and its powers. The reason for making this distinction between polynomials and functions has to do with when two polynomials are equal, compared with when two functions are equal." [...] "Detaching the Coefficients: We have defined polynomials in terms of an indeterminate, or formal symbol x, but it is possible, and sometimes more convenient, to define a polynomial strictly by its coefficients, without using x. The relevant information about the polynomial is the coefficients. Thus we can associate to [...] the 5-tuple [...]. Here we must agree on the order in which the coefficients appear, [...]." Childs (1995) A Concrete Introduction to Higher Algebra, Ch. 14 doi:10.1007/978-1-4419-8702-0.

Here's a source with a more extensive and considered discussion of this subject more specifically:

V. Indeterminates. The following formulae (without any qualifying or explanatory legends) belong to the theory of polynomial and rational forms.

(1) ${\frac {x^{2}-1}{x-1}}={\frac {x^{2}-x-2}{x-2}}={\frac {x+1}{1}}.$

(2) In the realm of integers mod. ⁠$2$⁠, the polynomial forms ⁠$(x+1)\cdot (x-1)$⁠ and ⁠$\textstyle x^{2}+1$⁠ are equal; the forms ⁠$x$⁠ and ⁠$\textstyle x^{2}$⁠ are considered as unequal.

(3) $x^{2}-9y^{2}=(x+3y)\cdot (x-3y).$

It will be noted that the statements V(1), (2) about rational forms are (for opposite reasons) quite unparallel to the statements II(1), (2) about the corresponding rational functions. In contrast to a function, a form is not a class of pairs of numbers and thus is not meant to be evaluated for any specific argument. In contrast to a numerical variable, the letter ⁠$x$⁠ in a form is not supposed to be replaced by specific numerals. Just as a complex number may be considered as an ordered pair of real numbers, a polynomial form containing one letter is completely characterized by the sequence of its coefficients; that is to say, the form may be regarded as a sequence of numbers belonging to a given field. A rational form is an ordered pair of such sequences. Forms thus are hypercomplex numbers of a certain kind that are equated, added, and multiplied according to well-known laws. For instance, if, in a self-explanatory way, the rational forms in (1) are denoted by

${\frac {(1,0,-1)}{(1,-1)}},\ {\frac {1,-1,-2}{1,-2}},\ {\frac {1,-1}{1}},$

then the first two are equal because $(1,0,-1)\cdot (1,-2)=(1,-1)\cdot (1,-1,-2).$ Operating with such sequences becomes more perspicuous if to each element of the sequence an indicator of its positions is appended. It is customary to affix such an indicator as a quasi-exponent and to use as its quasi-base the letter ⁠$x$⁠. The choice is motivated by the parallelism between rational forms and rational functions (the form ⁠$x$⁠ corresponds to the identity function ⁠$x$⁠), even though this parallelism is very incomplete, as shown by the contrast between V(1), (2) and II(1), (2). In a form, each letter ⁠$x$⁠ thus is, as it were, a holder of a place card (the quasi-exponent) describing the position of a coefficient. Such a letter in a form is called an indeterminate.

Menger, Karl (1956). "What Are x and y?". The Mathematical Gazette. 40 (334): 246–255. JSTOR 3609607.

–jacobolus (t) 21:56, 29 September 2024 (UTC)

Wilf (1994) generatingfunctionology doesn't use the name "indeterminate", but invokes the concept we are here calling an indeterminate (but just calls it a "variable"), and has some good reflections on the (ab)use of algebraic and analytic methods:

To discuss the formal theory of power series, as opposed to their analytic theory, is to discuss these series as purely algebraic objects, in their roles as clotheslines, without using any of the function-theoretic properties of the function that may be represented by the series or, indeed, without knowing whether such a function exists.

We study formal series because it often happens in the theory of generating functions that we are trying to solve a recurrence relation, so we introduce a generating function, and then we go through the various manipulations that follow, but with a guilty conscience because we aren’t sure whether the various series that we’re working with will converge. Also, we might find ourselves working with the derivatives of a generating function, still without having any idea if the series converges to a function at all.

The point of this section is that there’s no need for the guilt, because the various manipulations can be carried out in the ring of formal power series, where questions of convergence are nonexistent. We may execute the whole method and end up with the generating series, and only then discover whether it converges and thereby represents a real honest function or not. If not, we may still get lots of information from the formal series, but maybe we won’t be able to get analytic information, such as asymptotic formulas for the sizes of the coefficients. Exact formulas for the sequences in question, however, might very well still result, even though the method rests, in those cases, on a purely algebraic, formal foundation. [...]

But we cheated. Did you catch the illegal move? We took our generating function and evaluated it at x = 1, didn’t we? Such an operation doesn’t exist in the ring of formal series. There, series don’t have ‘values’ at particular values of x. The letter x is purely a formal symbol whose powers mark the clothespins on the line.

What can be evaluated at a particular numerical value of x is a power series that converges at that x, which is an analytic idea rather than a formal one. The way we make peace with our consciences in such situations, which occur frequently, is this: if, after writing out the recurrence relation and solving it by means of a formal power series generating function, we find that the series so obtained converges to an analytic function inside a certain disk in the complex plane, then the whole derivation that we did formally is actually valid analytically for all complex x in that disk. Therefore we can shift gears and regard the series as a convergent analytic creature if it pleases us to do so.

–jacobolus (t) 23:29, 29 September 2024 (UTC)

I'm a bit confused on what exacly you're advocating for by mentioning these quotes. Are you suggesting a certain definition? Farkle Griffen (talk) 23:42, 29 September 2024 (UTC)

I think our lead section might instead start something like:

In mathematics, an indeterminate or formal variable is a kind of variable (a symbol, usually a letter) used in mathematical expressions of some particular class, used as a purely formal placeholder but not intended to be substituted by any concrete value. The variable in an expression is considered to be an indeterminate when the members of a class of mathematical expressions, such as polynomials, rational functions, or formal power series, are treated as algebraic objects per se, elements of an algebraic structure, rather than taken to describe functions which might be evaluated. The distinction between variables, representing unknown values, versus indeterminates, placeholders with no value, is often elided or ignored, especially in mathematical analysis, but becomes important [...]

For example, a polynomial, an expression of the form $\textstyle a_{n}x^{n}+a_{n-1}x^{n-1}+{}$$\textstyle \cdots +a_{1}x+a_{0}$ is completely characterized by its coefficients ⁠$\{a_{i}\}$⁠, and can be exactly represented as a sequence of coefficients ⁠$\textstyle [a_{0},a_{1},\ldots ,a_{n}]$⁠, so that some algebra sources define a polynomial as the sequence. Polynomials can be added and multiplied by the appropriate combination of their coefficients without a need to evaluate the corresponding polynomial functions at any specific value of ⁠$x$⁠, following the structural identities of a polynomial ring. In some contexts, such as when working in a finite field, two distinct polynomials can represent the same polynomial function, so treating the polynomials as functions would lose information by conflating them. Thus the ⁠$x$⁠ in the expression can be taken to represent nothing more than a placeholder, making the expression into nothing more than a familiar and convenient alternate notation for the coefficient sequence.

(The above is probably not great, and may even be getting worse as I keep tweaking it; someone else can probably do better.)

I think your current version is less accessible and somewhat misleading compared to the previous version, but I agree that we could expand this article to better describe the history and uses of indeterminates. –jacobolus (t) 17:35, 1 October 2024 (UTC)

There are a few smaller issues I have with this that I can mention if asked, but, my biggest issue is that this uses a specific definition, and makes assertions that disagree with other definitions. (And also seems to be contradicting itself?)

For instance, the authors who support the transcendental definition or the sequence definition certainly would not agree that an indeterminate is a variable. And, according to Dummit and Foote, expressions in indeterminates are intended to be evaluated.

By WP:NPOV, emphasis should be made that there are several distinct (but isomorphic) definitions, and the lead shouldn't spend more than a sentence or two on any of them. Save the specific details for sections specifically dedicated to it, and make lead about properties common between them. Farkle Griffen (talk) 18:54, 1 October 2024 (UTC)

What you are calling the "transcendental definition" is not a different definition at all. I think you are misunderstanding what those authors are trying to say and the context in which they are saying it. The point, to my understanding, is that there's an isomorphism between transcendental field extensions, so that picking any arbitrary transcendental number doesn't change the algebraic structure you get when you add it to your field. In other words: a transcendental number is one which has no algebraic relationship to the numbers already defined, making it field-theoretically arbitrary. This is not the same as a definition of the term "indeterminate". What you are calling "the sequence definition" is a definition of polynomial, and the authors you linked who used it (among those I examined) did not define the term indeterminate at all but took its meaning for granted. –jacobolus (t) 19:04, 1 October 2024 (UTC)

Here's the Concise Oxford Dictionary of Mathematics (6th ed.):

indeterminate: A variable, in particular a formal variable, such as in the polynomial ring ⁠$F[x]$⁠ or the formal power series ring ⁠$F[[x]]$⁠ over a field ⁠$F$⁠. Such polynomials might be formally differentiated, but without reference to limits, and such power series multipled, but without reference to convergence.

Does that satisfy you that it's fair to call an indeterminate a type of "variable"? –jacobolus (t) 19:40, 1 October 2024 (UTC)

It certainly does add weight, and I'm willing to concede that a reasonable portion of authors consider indeterminates as a type of variable. For now, I won't argue with that definition; my goal is just to convince you that there are different definitions.

For instance, in Introduction to Algebra - Lewis, Donald, I don't see how this description (1) wouldn't qualify as a definition, and (2) could even be considered as describing a type of variable:

"Let x be an element of a ring L , and let R be a subring of L such that ax = xa for all a in R. We say x is algebraic over R if there is a finite set a₀, a₁, ..., a_n in R, not all 0, such that a₀ + a₁x + ••• + a_nxⁿ = 0. If x is not algebraic over R, we say x is a transcendental (or an indeterminate) over R. Thus x is a transcendental (or an indeterminate) over R provided there is a ring L such that (a) x in L, (b) R is a subring of L, (c) x commutes with each element of R, and (d) a polynomial in x over R is 0 only if all the coefficients are 0. If we are given a ring L with a subring R and we are classifying the elements of L relative to R, the term “transcendental” is customarily used. The term “indeterminate” is customarily used when one is studying the existence of nonalgebraic elements over a ring R. However, as indicated above, the two terms “transcendental” and “indeterminate” may be used interchangeably."

And as for the sequence definition, I would agree, that it doesn't define “indeterminate” itself, but it does define the more abstracted “polynomial in an indeterminate X over a field F”. They define it as an infinite sequence (a₀, a₁, … ), with a_k in F, and only finitely-many non-zero a_k’s, and one can write this instead as a₀ + a₁X + ... + a_nXⁿ. Basic Algebra - Knapp uses it like this. The point here is that, if a student were to ask, by this interpretation, “But what IS an indeterminate?”, the response “It’s a kind of variable” would not make sense.

Another example, A Concrete Approach to Abstract Algebra - W. Sawyer talks extensively about the distinction between “indeterminates” and “variables” in this sequence interpretation.

Given this, would you say it’s fair to say that not all authors would agree that an indeterminate is “a type of variable”? Farkle Griffen (talk) 17:45, 2 October 2024 (UTC)

The point here is that, if a student were to ask, by this interpretation, “But what IS an indeterminate?”, the response “It’s a kind of variable” would not make sense. – To the contrary, it's a perfectly reasonable and comprehensible response. The plain-language meaning of variable as used across written mathematics in this type of context is something like, "the symbol ⁠$x$⁠ as used in expressions such as ⁠$ax^{2}+bx+c$⁠." One way of interpreting that symbol might be as an unknown or changing element of some class of numbers such as ⁠$\mathbb {R}$⁠ or ⁠$\mathbb {C}$⁠ (for example, ⁠$x$⁠ might represent the time in seconds and the expression, taken as a function of time, might represent the vertical position of a flying projectile, measured in meters the ground); another way of interpreting that symbol is as a "formal variable" or "indeterminate", a place holder symbol without any particular value (this interpretation would make more sense when taking the expression to represent a formal polynomial per se, not implying any functional relationship). –jacobolus (t) 19:27, 2 October 2024 (UTC)

I would disagree with your definition of "variable", however, we can talk about the exact definition of "variable" later. This doesn't take into account the last author I linked in my previous comment, who spends multiple pages talking about how indeterminates in the sequence definition are distinct from variables. At the very least, you would have to agree that 'A Concrete Approach to Abstract Algebra - W. Sawyer' would not agree that indeterminates are variables.

And this also does not take into account the transcendental definition. For instance, no mathematician would call $\pi$ a 'variable' over the rationals, but multiple authors do explicitly call it an 'indeterminate' over the rationals.

So I ask again, given this, would you say it’s fair to say that not all authors would agree that an indeterminate is “a type of variable”? Farkle Griffen (talk) 05:25, 3 October 2024 (UTC)

This is very easy to deal with using one extra sentence along the lines of "Some authors reserve the term variable to only mean a symbol representing an unknown number, and strictly distinguish indeterminates from variables." This is a common type of situation, where a term with a broadly used meaning is given a more restricted specific definition in one author's work, for their own narrative convenience. –jacobolus (t) 06:52, 3 October 2024 (UTC)

Yes, but I believe you have it a bit backwards; I would argue the more common meaning is not the one you are suggesting, but is, in fact, the one you are implying is less common. Specifically, a variable is a symbol that represents a range of values.

In elementary mathematics,

College algebra - Beckenbach:

"Definition 1.6 A variable is a symbol representing an unspecified element of a given set"

Oxford English Dictionary, s.v. “variable (n.), sense 1.a,”

"Mathematics and Physics. A quantity or force which, throughout a mathematical calculation or investigation, is assumed to vary or be capable of varying in value"

Collins English Dictionary. Variable, (noun)

"mathematics a. an expression that can be assigned any of a set of values b. a symbol, esp x, y, or z, representing an unspecified member of a class of objects"

I don't have access to the 6th edition of "Concise Oxford Dictionary of Mathematics", however, the 4th edition defines "variable" as:

"variable An expression, usually denoted by a letter, that is defined for values within a given set. Can be used to represent elements of sets which are not numbers but frequently it relates to numerical quantities and functions defined in them together with the relationship between them."

Cambridge dictionary. variable noun [C]:

"MATHEMATICS a letter or symbol that represents any of a set of values"

---

In more formal contexts, specifically, First-order and Mathematical Logic, and for Formal languages, usually called "individual variables" (to distinguish it from Propositional variables) is defined very similarly:

"Individual variable" Encyclopedia of Mathematics. - Sobolev, S.K:

"A symbol of a formal language used to denote an arbitrary element (individual) in the structure described by this language."

And more specifically, variables should be able to be quantified over:

A Tour Through Mathematical Logic - Wolf

"The grammatical rules for the use of quantifiers are simple: if P is any statement, and x is any mathematical variable, then $\forall$xP and $\exists$xP are also statements." Farkle Griffen (talk) 16:52, 3 October 2024 (UTC)

@Jacobolus, for the point of an explicit consensus, given this, would you agree that not all authors support that an intdeterminate is a kind of variable? Farkle Griffen (talk) 04:58, 7 October 2024 (UTC)

No I think authors agree that an indeterminate is a type of variable, but some authors intentionally use a narrow/idiosyncratic definition of "variable" in their own work, for their own narrative convenience, which does not match what they understand "variable" to mean when they read it in general literature. My impression is that @D.Lazard's version is more or less correct and we should return to that especially in the lead section, but we can do a better job of adding sources and maybe explaining some subtleties in the body of the article. I still think you are somewhat misunderstanding/mischaracterizing your sources. –jacobolus (t) 05:22, 7 October 2024 (UTC)

Okay... this seems somewhat unreasonable to me to be honest. How exactly are you reconciling their version with the definitions of variables mentioned above? What set do they represent? If indeterminates are variables, are they free or bound? If they're free, what values can they be replaced with? If they're bound, how are they being quantified? Farkle Griffen (talk) 07:00, 7 October 2024 (UTC)

The basic problem is that you are not considering the context, and you are mashing together (more or less formal) definitions from different authors, different disciplines (logic, analysis, algebra, number theory, ..), and pretending they represent some kind of unified precise definition. But that's not really how language works.

In analysis, and roughly in all mathematics up until at least the 18th century, every "variable" was considered to be an "unknown", the idea of a "formal variable" or "indeterminate" was not yet current. Words like "unknown", "variable", "indeterminate", etc. were all more or less interchangeable. Indeed Euler explicitly defines "variable" as an "indeterminate".

At some point (perhaps Gauss, or perhaps some other authors in different terms, and with growing importance in the mid–late 19th century), in considering polynomials (etc.) as "forms" rather than as generic expressions, functions, changing quantities, scalar fields (in the physical sense), or the like, started to treat the "variable" of a function as a pure place-holder symbol. Since words like "variable" by that time had centuries of practical use, most authors, including in algebra and number theory, continued (and to this day continue) to use the word "variable" to indicate these symbols, only qualifying it or using an alternative term when they are trying to be very explicit and clear.

(Aside: we should have an article about the word form as used in this context, which can be linked from quadratic form etc.; currently Form (mathematics) redirects to Homogeneous polynomial, but that is both too narrow and also not very helpful.)

When you look at a generic definition for the word "variable", you are going to find one which is appropriate to introductory generic contexts, or to analysis. But algebra books, including those which adopt the term "indeterminate" (and with high likelihood including most if not all of the sources you have cited here), continue to use the word "variable" in practice all over the place. You might find a definition up front in a textbook somewhere which tries to carefully distinguish "variable" from "indeterminate", but if you skim through the rest of the textbook and the same author's research papers, etc., you are likely to find repeated uses of the word "variable" to mean both unknown quantities and indeterminate place-holder symbols. Moreover, one of the most useful features of mathematics is that somehow different interpretations of objects are generally consistent, and you will find "indeterminates" or "formal variables" profitably interpreted as unknown quantities and substituted for concrete values / treated analytically with analysis of convergence etc. all over the place in mathematics.

As such, making a pedantic "a variable and an indeterminate are fundamentally different" kind of approach, as you have done here, is fundamentally misleading to readers, and does not reflect mathematical practice. It's also much less accessible to novice readers who know what a "variable" is but don't know anything about abstract algebra or advanced number theory, etc.

Moreover, all of the variations in senses of "indeterminate" you have identified are essentially the same unified concept, of a generic/purely symbolic placeholder coming from outside / without a value meaningful within what Kronecker called a "domain of rationality". Making a big deal about the different contexts where such a concept appears being "different definitions" is substantially misleading to readers compared to a description of these as "essentially the same idea but adapted for slightly different contexts".

When I get the time, I'll revert to @D.Lazard's preferred version, but try to clean up the language a bit and add a sentence to clarify so we can hopefully avoid confusing anyone who comes across a source which defines and explicit separation between "variable" and "indeterminate". –jacobolus (t) 16:33, 7 October 2024 (UTC)

Let me try and explain myself a little better this time, and I will try to respect the current version as much as possible. I believe the issue with the current version is its focus.

For instance, take the section on Polynomials. In it it says:

"But the two polynomials $2+3X,\,5+2X$ are unequal, since 2 does not equal 5, and 3 does not equal 2. In fact, $2+3X=a+bX$ does not hold unless $a=2$ and $b=3$. This is because $X$ is not, and does not designate, a number."

This gives the impression that, "since X does not denote a number, then this clearly follows." However, this is purely philosophical justification, and would only confuse the reader more (or, at the very least, it confused me). And, moreover, no textbooks give such a handwavey justification: most have it as a definition that two "polynomials in indeterminates" are equal iff their coefficients are equal.

Given what I have slowly come to learn, I believe it would believe it would reasonable to approach it in one of two ways:

(1) If this is a property of the whole expression:

For instance, defining the whole phrase "a polynomial in an indeterminante" as string of symbols of the form $a_{0}+a_{1}X+...+a_{n}X^{n}$, where two strings are equal iff their coefficents are equal and (insert definitions of addition and multiplication)

In this case, it should have heavy emphasis that an individual "indeterminate" is not defined here, but rather the whole phrase "polynomial in an indeterminate" is being defined.

(2) If this is a property of the indeterminate

This is what I believe the authors who use the transcendental definition are trying to capture. If $a_{0}+a_{1}X+...+a_{n}X^{n}$ is a bona fide expression and "+" is bona fide addition and not just a symbol in a string, then it follows that X is transcendental over the specified ring.

For instance, the definition of transcendental is that $a_{0}+a_{1}X+...+a_{n}X^{n}=0$ iff $a_{0}=a_{1}=\cdots =a_{n}=0$,

So given that original definition of "a polynomial in an indeterminate", assume $a_{0}+a_{1}X+...+a_{n}X^{n}=0$ for some $a_{i}$; then, reasonably, $a_{0}+a_{1}X+...+a_{n}X^{n}=0+0x+...+0X^{n}$, so $a_{0}=a_{1}=\cdots =a_{n}=0$, thus X is transcendental.

So, given the reasonable assumption that $0+0x+...+0X^{n}=0$, it follows that X is transcendental.

This is what I mean by the focus. The article should be clear whether the expression should be viewed as an individual object, or whether the indeterminate should, and right now it seems to be mixing the two. Farkle Griffen (talk) 04:14, 9 October 2024 (UTC)

A few fundamental remarks must be taken into account for every modification of of the article.

• This article is about modern mathematics. All modern mathematics are founded over Zermelo–Fraenkel set theory (ZFC) and mathematical logic. So, apart from a history section, everything must be compatible with Zermelo–Fraenkel and mathematical logic. As far I know, within the metamathematics of ZFC, the definition of "variable" is borrowed from mathematical logic, and within internal ZFC, all objects are sets. If this does not use the matheamtical logic definition of variable, then the definition should be reducible to sets.
• There is a very common definition of variable that the "Variable (mathematics)" article uses which disagrees with the definition proposed here. If we wish to define "An indeterminate is a kind of variable", then this instance of the term should not link to that article.
• Because of the foundational crisis of mathematics and the development of mathematical logic in view of its resolution, every reference before 1900 must be taken with care outside a history section.

These remarks are of little help for improving the article without making it too WP:TECHNICAL for a general audience, but they are certainly useful for not making the article more confusing than presently.

Farkle Griffen (talk) 17:55, 7 October 2024 (UTC)

(1) No, all of mathematics is not ZFC, and claiming so is a substantially misleading assertion (and also completely off topic in the context of this conversation up to now). (2) The article variable (mathematics) can certainly be clarified and expanded; it discusses "indeterminates" and other types of variables, but that discussion could be more more explicit and extensive. (3) The foundational crisis you reference is not relevant to this topic. (4) I agree, your changes have made the article unhelpfully more technical and more confusing, which is why I think they should be substantially reverted. –jacobolus (t) 18:17, 7 October 2024 (UTC)

(1) ZFC is the most common foundation.

(2) Apart from indeterminates, all other types of variables listed are reducible to ZFC

(3) The foundational crisis is relevant to this discussion in particular. Its whole resolution was to provide a foundation, for which a definition here should be reducible to (or taken directly form) any common foundation

(4) What exactly is your issue with my version of the lead? The only difference between mine and D.Lazard's is I use "variable symbol" rather than "variable". Farkle Griffen (talk) 18:37, 7 October 2024 (UTC)

(1) I disagree strongly with your assertion that the only difference between your version and mine lies in the use "variable symbol" rather than "variable".

(2) "Everything must be compatible with Zermelo–Fraenkel and mathematical logic". No, no, no: by Wikipedia rules, everything must be compatible with the most common usage. This is completely different.

(3) Otherwise, I fully agree with Jacobolus who says things better than me.

(4) Please, stop deforming what I wrote in view of asserting an inexistent agreement (see (1)). D.Lazard (talk) 20:01, 7 October 2024 (UTC)

(1) In the lead specifically, apart from that, what exactly do see as different? I added an example, but I don't think that example is very controversial...

(2) Noted.

(4) That wasn't in attempt at "asserting an inexistent agreement"; the point was I was trying to respect your version as much as possible. Why are you assuming bad faith? I only edited the first paragraph, and this was to change "variable" to "variable symbol" and add an example. Was there something else in the lead I wrote that angered you? Farkle Griffen (talk) 20:57, 7 October 2024 (UTC)

As for field theory, I think it would be reasonable to include a section called something like "field theory", mentioning that transcendentals and indeterminates are field theoretically interchangeable (even if not in other contexts). It would be reasonable to mention that this usage was popularized by Leopold Kronecker, who credited Carl Gauss with first using "indeterminate" in this manner. –jacobolus (t) 01:17, 3 October 2024 (UTC)

I'd be willing to do this, however, I don't see a reason why it should be limited to field theory; unless I'm misreading, several authors use it in the context of Ring theory as well. As for your last sentence, do you have a source for this? A source for such a statement would be very helpful for a history section (or any history notes at all), which this article is severely lacking. Farkle Griffen (talk) 05:31, 3 October 2024 (UTC)

Kronecker, 1887: "[M]it der principiellen Einführung der 'Unbestimmten' (indeterminatae), welche von Gauss herrührt, hat sich die specielle Theroie der ganzen Zahlen zu der allgemeinen arithmetischen Theorie der ganzen ganz- zahligen Functionen von Unbestimmten erweitert. Diese allgemeine Theorie gestattet alle der eigentlichen Arithmetik fremden Begriffe, den der negativen, der gebrochenen, der reellen und der imaginären algebraischen Zahlen, auszuscheiden. Der Begriff der negativen Zahlen kann vermieden werden, indem in den Formeln der Factor ⁠$-1$⁠ durch eine Unbestimmte ⁠$x$⁠ und das Gleichheitszeichen durch das Gauss'sche Congruenzzeichen modulo ⁠$x+1$⁠ ersetzt wird. So wird die Gleichung ⁠$7-9=3-5$⁠ in die Congruenz ⁠$7+9x\equiv 3+5x\!\!{\pmod {x+1}}$⁠ transformirt"
[[W]ith the systematic introduction of 'indeterminates' which goes back to Gauss, the special theory of integers has expanded into the general arithmetic theory of polynomials in the indeterminates with integer coefficients. This general theory allows one to avoid all concepts foreign to arithmetic proper, that of negative, of fractional, of real and imaginary algebraic numbers. The concept of negative number can be avoided when one replaces in the formulas the factor ⁠$-1$⁠ by the indeterminate ⁠$x$⁠ and the sign of equality by the Gaussian sign of congruence. Thus the equation ⁠$7-9=3-5$⁠ will be transformed into the congruence ⁠$7+9x\equiv 3+5x\!\!{\pmod {x+1}}$⁠]

See Petri, Birgit; Schappacher, Norbert. "On Arithmetization". The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae. Ch. V.2, pp. 343–374. doi:10.1007/978-3-540-34720-0_12. and the rest of that book. Or do a search for kronecker indeterminate in the literature for more. (See also On the Concept of Number at the Wayback Machine (archived 2015-09-24).)

Kronecker was inspired by Gauss's Disquisitiones, e.g.: "Formam ⁠$axx+2bxy+cyy$⁠, quando de indeterminatis ⁠$x$⁠, ⁠$y$⁠ non agitur, ita designabimus ⁠$(a,b,c)$⁠." [The form ⁠$axx+2bxy+cyy$⁠, when the indeterminates ⁠$x$⁠, ⁠$y$⁠ are not at stake, we will write like this, ⁠$(a,b,c)$⁠.]

–jacobolus (t) 05:53, 6 October 2024 (UTC)

D.Lazard, Farkle Griffen: I tried making a second paragraph for the lead focused on a concrete example, thinking that might be more accessible to less technical readers. Does that help convey what the difference is between "unknown quantity" vs. indeterminate? I think some of Farkle Griffen's content / sources are probably worth restoring, but I've been camping for a few days and haven't thought too carefully about it yet. I agree with above discussion that we should try to include some bit of history, etc. –jacobolus (t) 02:49, 16 October 2024 (UTC)