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== Proposition to edit Basic Properties Substitution property ==

This section seems unclear, and takes much more research to unravel. 

So, in spirit of clarity, I would like to change this section to something along the lines of:

Substitution property: For any quantities a and b and any well-formed expression F(x) or formula φ(x), if a = b, then F(a) = F(b) and φ(a) ⇔ φ(b)

Some specific examples of this are:
For any a, b, c ∈ ℤ,  
(a = b) →  (a + c = b + c)  
(Here, F(x) = x + c);

For any a, b, c ∈ ℚ,  
(a = b) → (a^2 + 2c + 1 = b^2 + 2c + 1)  
(Here, F(x) = x^2 + 2c + 1);

For any a, b, c ∈ ℝ,  
(a = b) → (0 ≤ a ⇔ 0 ≤ b)  
(Here, φ(x) is x ≥ 0)

Given a set S with a strict partial ordering (<), and any a, b, c ∈ S,  
If a = b and b < c, then a < c
(This is given by φ(x): x < c) [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]]) 04:52, 28 June 2024 (UTC)
: That looks to me very much less likely to be understood by an average user of the encyclopaedia than the current version. For example, most people won't know what ⇔ means, or what a, b, c ∈ ℚ means, etc etc. [[User:JBW|JBW]] ([[User talk:JBW|talk]]) 21:07, 3 July 2024 (UTC)
::You're right, but the examples given are kinda confusing since F is only shown as a function. So, for instance, if I want to show that: "Given a partial order on a set P, if a = b, and b < c, then a < c". This is something that should be obvious, but the examples given wouldn't give someone confidence that they can describe F as "F(x) is (x < c)", since it's not a numerical operation as the others are. And moreover, the link to "expression" rather than the more formalized "formula" article (especially when the former link specifically distinguishes between the two) sort-of convolutes the message being sent to anyone who actually wants to look further (like I did).
::I've talked to quite a few people about this and there is a lot of differing opinions on the specifics, but they all agree that something just isn't quite right with the way it is now.
::Can we change the examples to include the example I gave here (or some other non-numeric example). And maybe link to "formula" from the word "expression" so those looking for more rigor have it, but those just reading over aren't more confused? [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]]) 03:33, 17 July 2024 (UTC)
::: OK, {{u|Farkle Griffen}} I agree with you. Previously my attention got caught by the notation, rather than the content of what you said. I have made a small change in the direction you suggest. There is more to be done, but I don't have time now. [[User:JBW|JBW]] ([[User talk:JBW|talk]]) 16:31, 18 July 2024 (UTC)
::::Looking at that section again, I think the notation of "F(x) = F(y)" is far too removed from the original meaning of '=' and anyone reading through would only be left more confused. 
::::For instance, as shown above in the Talk section, If we want to prove the transitive property from the substitution property, we would use F(x) = (x = c), and we have given a = b and b = c then (a=c) = (b=c). Does that last jumble of symbols then prove that a = c? How would an average reader know that? If your answer is in terms of truth values, why not just use standard logical notation and/or jargon? 
::::This article is also constantly conflating 'expressions' and 'formulae'. The article links to 'well-formed formula' multiple times, but that article only really talks in detail about logical formulas and I can't find anywhere on here that defines a 'well formed ''mathematical'' formula'.
::::I agree, the word 'expression' is probably more accessable to the average reader, but we shouldn't be mixing the refferences. If "formula" is more correct, all links should be in terms of that. If 'expression' is more accessible, then let's just use the word 'expression' but link to formula articles. [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]]) 23:35, 21 July 2024 (UTC)

== Suggestion: Proof that Set equality satisfies the substitution property. ==

The article starts off talking about the basic properties of equality, but talks a lot about set equality. It would be nice if there was a proof that set equality satisfies the axioms first stated since it's not really trivial, and the proof doesn't seem to be on any other article. 

I'm just worried it it might take up too much space and distract from the topic since the proof has to go into the definition of a "well formed formula". 

I can set up a basic proof, but it likely won't be pretty.

Thoughts? [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]]) 14:15, 22 July 2024 (UTC)

== Basic Properties section should not be defined as sets. ==

The links to reflexive and such, link to specifically set relations. Equality should be defined more broadly than elements of a set, since we want to be able to declare equality between objects that can't form a set. The classes of Cardinals and Ordinals, for instance. [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]]) 15:43, 22 July 2024 (UTC)

:There is nothing in the present formulation that restricts the definition of equality to sets. Specifically, the basic properties are introduced with "for every ''a'' and ''b''", and not "for every ''a'' and ''b'' in a given set". The names of the properties are linked to the Wikipedia article where the properties are defined. If you are not satisfied that the definitions do not include [[proper class]]es, you can edit the target article accordingly. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 16:27, 22 July 2024 (UTC)

== Section "In logic" ==

Section {{alink|In logic}} is confusing by not distinguishing two different concepts of equality, the equality in philosophy and classical logic, and the equality in mathematics and in mathematical logic. It is true that the mathematical concept has been modeled after the philosophical concept, but it is false that the mathematical concept can be reduced to the philosophical one. For example the priciple of [[identity of indiscernibles]] is not true in modern mathematics, where two objects that share all their properties are [[isomorphic]] rather than equal. In other words, two equal objects share all their properties, but the convers is false.

So, the section must be completely rewritten to clarify the relation betwee philosophical equality and mathematicaal equality. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 15:06, 24 July 2024 (UTC)

:I agree, as it was, it wasn't very clear. I've updated the section to be closer in the direction you suggest. It still needs further improvement.
:However, it should be noted that isomorphism as you suggest does not satisfy the axioms of equality as stated.
:For instance, given structures A and B that are not equal, with an isomorphism F from A to B, we can take the statement φ(X): F is an isomorphism from X to B. If A and B were equal, we should have that φ(A) → φ(B) (F is an isomorphism from B to B). But this is false. Thus A and B are not equal by the axioms stated. 
: [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]]) 16:54, 24 July 2024 (UTC)
::Your example is wrong, since a statement cannot contain a free variable (''F'' in your example). Nevertheless, I agree to restrict what I wrote above to the case of objects that are equal [[up to]] a unique isomorphism, such as [[initial object]]s. A famous example is the set of [[real number]]s: The constructions with [[Dedekind cut]]s and [[Cauchy sequence]]s produce two different sets, but, since the results satisfy the priciple of identity of indiscernibles, one '''identifes''' both with '''the''' set of real numbers, although this is formally wrong. Note that Barry Mazur's article cited in the article page discusses these questions in details. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 17:29, 24 July 2024 (UTC)
:::I'm inclined to generalize your real-numbers example by claiming that, in modern mathematics, each theory comes with its own notion of equality. The theory's author actively prescribes (what you called "identifies") which of the objects one shall be able to distinguish, and which shall be indistinguishable. At least, this view applies to the well-known constructions of more sophisticated number sets (integer, rational, real, complex) from simpler ones. Cauchy identifies any two sequences that differ only by a sequence of limit 0 - i.e. he doesn't allow the theory users to distinguish them; this is his deliberate decision, and an important part of his theory.
:::If this general view can be accepted, I'd like to state something like {{tq|In a given mathematical theory, equality is the finest equivalence relation that is available; it is prescribed/constructed/defined by the theory's author}} (wording is likely to need improvement as I'm not a native English speaker). This also would avoid circular definitions like defining "equal" by "same", which might be ok in the lead but is unsatisfactory in more advanced sections. - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 19:07, 24 July 2024 (UTC)
:::When mathematics is based on set theory, equality prescription is usually achieved by factoring w.r.t. some equivalence relation - a well-known construction. However, e.g. with a category theoretic foundation, I doubt that there is any other way than explicitly stating "We'll identify any two objects that are isomorphic". My point is that both ways serve the same purpose: ''defining'' the equality associated with the mathematical theory under consideration. - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 19:24, 24 July 2024 (UTC)
:::I was a bit short in my last reply, and I apologise for that. Let me try again and explain where I'm coming from.
:::First, as a defense of the axioms as stated, they ''are'' the axioms of equality in mathematics and mathematical logic. See: [https://encyclopediaofmath.org/index.php?title=Equality_axioms&oldid=46837 Springer Encyclopedia of Mathematics]. In fact, this is the ''only'' definition of equality that I have found. The hyperlinks to the axioms are those specifically talked about by mathematitians like Bertrand Russell and Leibniz. 
:::Second, after reading the article you mention, I agree it is useful for describing how mathematitians use equality categorically, but I would disagree that his interpretation should be the center focus in this article specifially. Equality as a concept, especially in mathematics, is a foundational concept. Defining equality in terms of Category theory would be a chronology issue, as if Category theory supports equality, what supports Category theory? Following the links of definitions from the category theory article, you always get back to set theory and logic. You then have to deal with accusations of circularity. 
:::Lastly, to mention your example, let me offer an analogy. A pair of tomatoes is an '''instance''' of the number '2'. That does not mean that the number 2 is ''equal'' a pair of tomatoes, for instance, the tomatoes are fruit, and the number 2 is not a fruit. The number 2 is an abstraction, it is represented by any kind of pair of objects. It is not equal to any specific instance of pairs of objects, but it can be shown to exist using specific constructions of it, in this case, a pair of tomatoes. 
:::Similarly, the real numbers are an abstraction; the real numbers are defined as any ordered field satisfying some given axioms. Both of the constructions you mentioned are '''instances''' of real numbers, but are not '''the''' real numbers. For instance, given one of those constructions, I can take the union of the sets defining 1 and 2. But for the real numbers 1 and 2, I can't. They are instances proving that such a construction exists, but they are not themselves the abstraction.
:::This is an article where pedantry over the specific details of "equality" is not only tolorated but required. 
:::If you want this article to be about the categorical use of equality, that's fine, but given the number of articles in foundations that link directly to this one, it comes at the cost of either rewriting all of those articles to not mention equality, or creating a new article that defines equality foundationally. 
:::What I've written for the article so far provides both an axiomatization of equality, which allows any mathematitian to choose their grounding (whether it be classical logic, mathematical logic, set theory, or otherwise), along with the specific construction supported by mathematical logic and philosophy of mathematics.  [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]]) 03:39, 25 July 2024 (UTC)
::::Some fundamental remarks that must be taken into account for every modification of of the article.
::::*This article is about equality in mathematics. All modern mathematics are founded over [[Zermelo–Fraenkel set theory]] (ZFC). So, except for the section on logic, everything  must be compatible with Zermelo–Fraenkel definition of equality. As far as I know, in ZFC, every object is a set, and the only axiom that defines equality is the [[axiom of extensionality]].
::::*There are two different notions of equality that must not be confused the semantic equality and the syntactic equality. This is for clarifying this fact that I have recently edited the lead (<math>1.5=3/2</math> is certainly not a syntactic equality). 
::::* The substitution property is presented in section {{alink|In logic}} as an axiom of logic. For being accurate, it must be said that this is an axiom for syntactic equality only in [[higher-order logic]], since is contains a quantification over a predicate. In [[first-order logic]] this is only a [[axiom schema]] that defines infinitely many axioms. Moreover there are logical theories with equality that do not include this axiom schema.
::::* 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 an history section.
::::These remarks are of few help for improving the article without making is too [[WP:TECHNICAL]] for a general audience, but they are certainly useful for not making the article more confusing than presently. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 13:47, 25 July 2024 (UTC)
:::::You're right. I will try to keep this article up to the standards you've set here.
:::::However, there are a few things I somewhat disagree with. First, not everything should be compatible with ZF. For instance, the introduction of this article defines equality as a "relationship between two quantities", however ZF defines equality as an assertion declaring that they have the same elements, and moreover, ZF does not define an object type "quantity" at all. Meaning that the introductory definition is just wrong by ZF standards. However, as I'm sure you'd agree, the introduction should not use the ZF definition. Why? Because it is less accessible to readers, maybe, but more than that, it is not how most mathematicains think of equality.
:::::Second, as it is written, the Substitution property is not an axiom of syntactic equality, it is specifically for semantic equality. That is what the "well-formed" part of formula means. To show an example you use, the identity (x+1)^2 = x^2 + 2x + 1 is not a syntactic equality, but it is a semantic equality. x is defined as an arbitrary element from a given domain. Functions are defined as sets, specifically, a set of ordered pairs of all elements from their domain to their codomain. So the functions on both sides of this equality reduce to the same set, and thus they are equal. An identity just means an equality of these function sets.
:::::You might say that "These have diffrent properties, for instance, they have a diffrent number of terms", but ZF has no fundamental notion of "strings" or "terms". If you try to define some property that distinguishes the two, you will notice that this property is not [[Zermelo–Fraenkel set theory#Formal language|well-formed in ZF]], since both sides reduce to the same set, any well-formed formula in ZF cannot distinguish between them.
:::::This brings me to my last note, I do believe that what I have written in the In logic section does meet all of the criteria you have mentioned.  I've taken out most of the mentions to first-order logic and grounded the statements in ZF. It should be noted that the "asioms" I mention in that can be proved within ZF.
:::::My goal of what I have written is to describe the framework in which mathematicians view equality. Mathematicians don't usually see equality as set equality but rather a notion of same-ness between two mathematical objects, which they may not necessarily think of as sets. Most mathematicans take the statements "x=x" and "x=y implies you can replace x with y" as axioms anyway. For instance, if I were to ask you why equality is transitive, you might say "Well if a=b then a and b are the same thing, so if b=c then we can just rewrite it as a=c" But this is exactly the substitution property. In reality, the transitive property of equality is not very trivial to prove from the basic axioms in ZF. But given the substitution property, it becomes much easier.
:::::What I have now is an alternate framework; a place for mathematitians to point to justify their intuitions. It is still grounded in ZF, but now one can use and teach the substitution property as an axiom without worrying about rigor. Let me be clear on what this section ''doesn't'' say. It does ''not'' say that those two axioms form a complete axiomatization of equality, and it does ''not'' assume the identity of indiscernibles. It assumes the ''indiscernibility of identicals''.
:::::Identity of indiscernibles: "If Fx implies Fy for any formula F, then x = y". This is ''not'' assumed.
:::::Indiscernibility of identicals: "If x=y, then Fx implies Fy, for any fomula F". This ''is'' assumed. It states that if two things are the same, then you can't distinguish between them, which should be an obvious truth.
:::::As an incomplete set of axioms, it is not the case that if an object satisfies those axioms, then it is equality. It just proves statements about equality, like transitivity, and can prove things aren't equal. The only diffrence between standard set equality and this reinterpretation is that set equality is capable of declaring indiscernibles as unequal, which this isn't. It doen't say they ''are'' equal, it just can't say they aren't since they don't violate either axiom.
:::::<STRIKE>This is why I don't like the title of the section "In logic". As it is now, it is not founded in standard logic, but rather ZF. I will work on editing that section to make this more clear, but until then, I hope you will consider changing the section title. I would change it myself, but that feels dishonest since you were the one to put it in place. [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]]) 03:29, 26 July 2024 (UTC)</STRIKE>
:I've rewritten that section for clarity in the ways we have talked about. Are there any parts of that section you still consider confusing or pressing issues? [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]]) 15:51, 28 July 2024 (UTC)
:The section has been completely rewritten as we talked about. Are there any specific issues you believe are still too confusing for most readers? I am planning on removing the "confusing" tag on by August 11th, if no response is given. [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]]) 21:45, 8 August 2024 (UTC)
:@[[User:D.Lazard|D.Lazard]], I would like to apologise; I somehow misread this comment multiple times, reading it as the opposite of what is meant. I read this as saying, roughly, "The indentity of indiscernibles is true in mathematics, and two objects that are isomorphic should be equal". Rereading this time, this is clearly not what is meant.
:Though I believe I can see where the confusion came from. First, let me clear one thing up: that section has never said the ''identity of indiscernibles'', it said the ''indiscernibility of identicals'' is true. To see the diffrence:
:'''''Identity'' of indiscernibles''': "If two things share all properties, then they are equal"
:'''''Indiscernibility'' of identicals''': "If two things are equal, then they share all properties"
:The former is certainly controversial, but the latter is almost certainly true. However, because their spellings are so similar, it is very easy to mistake one for the other when reading, and I believe that is what happened here. Because of this, I replaced all instances of "indisceribility of identicals" with the more distinct "substitution property". 
:I apologise for any frustration this caused. [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]]) 15:05, 17 September 2024 (UTC)

== Congruence relation? ==

While checking the 2 most recent edits, I found that in section "Basic properties", item "Operation application" is a special case of item "Substitution" (should better be "[[wiktionary:substitutivity|Substitutivity]]", to have an adjective there, too). "Operation application" handles expressions of height 1 (or 2, if variables are counted, too), while "Substitution" handles expressions of arbitrary height. Therefore, by a simple induction on expression height, "Substitution" follows from "Operation application", so both properties are equivalent. Moreover, due to the single property "Substitution"/"Operation application", equality is a congruence relation on every algebraic structure where it is defined at all. 

Btw: The 2nd subitem of "Operation application" is wrong for ''b''=0.  In the 3rd subitem, both ''g'' and ''h'' need to be differentiable, and it should be clarified that the premise means ''g''=''h'' (equality of functions), not ''g(a)''=''h(a)'', which can be misread as equality of particular function values at input ''a''. 

To sum up, I suggest to change the current text as follows:
<BLOCKQUOTE>
*''[[Substitution (logic)#Mathematics|Substitutivity]]'': for every {{mvar|a}} and {{mvar|b}}, and every [[Operation (mathematics)|operation]] <math>f(x)</math>, if {{math|1=''a'' = ''b''}}, then {{math|1=''f''(''a'') = ''f''(''b'')}}.{{Efn|This generalizes to functions ''f'' of higher ([[countable]]) [[arity]]: for each ''n''-ary function ''f'', whenever {{math|1=''a'' = ''b''}}, then {{math|1=''f''(''x''<sub>1</sub>,...,''x''<sub>''k''-1</sub>,''a'',''x''<sub>''k''+1</sub>,...,''x''<sub>''n''</sub>) = ''f''(''x''<sub>1</sub>,...,''x''<sub>''k''-1</sub>,''b'',''x''<sub>''k''+1</sub>,...,''x''<sub>''n''</sub>)}} for all {{math|''x''<sub>''i''</sub>}} and each ''k''.}}<ref>Equality axioms. Springer Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equality_axioms&oldid=46837</ref> Informally, this just means that if {{math|1=''a'' = ''b''}}, then {{mvar|a}} can replace {{mvar|b}} in any mathematical [[Expression (mathematics)|expression]] or [[formula]] without changing its meaning.<br>For example:
**Given [[real numbers]] {{mvar|a}}, and {{mvar|b}},  if {{math|1=''a'' = ''b''}}, then <math>2a-5 = 2b-5</math>.{{Efn|In detail: {{math|1=''a'' = ''b''}} implies {{math|1=2''a'' = 2''b''}}, which implies in turn {{math|1=2''a''-5 = 2''b''-5}}.}}
**Given unary real-valued [[differentiable function]]s {{tmath|g}} and {{tmath|h}}, if <math>g = h</math>, then <math display=inline>\frac{d}{dx}g = \frac{d}{dx}h</math>. (Equality of functions is retained by the [[derivative]] operation.)

If restricted to the elements of a given [[set (mathematics)|set]] <math>S</math>, those first three properties make equality an [[equivalence relation]] on <math>S</math>. In fact, equality is the unique equivalence relation on <math>S</math> whose [[equivalence class]]es are all [[singleton (mathematics)|singleton]]s. On every algebraic structure, all four properties together make equality a [[congruence relation]], if it is defined at all.
{{talkref}}
{{notelist}}
</BLOCKQUOTE>
Comments are welcome. - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 11:19, 18 October 2024 (UTC)

: Is the word "Substitutivity" actually used in English by mathematicians? I've never encountered it, and even the Wikipedia article that you have linked it to doesn't mention the word. The wiktionary article that you link to appears to confirm that the word exists and is used in philosophy, but no mention of mathematics. [[User:JBW|JBW]] ([[User talk:JBW|talk]]) 12:51, 18 October 2024 (UTC)

: Also, what do you mean by "to have an adjective there, too"? Both "substitution" and "substitutivity" are nouns; like almost all nouns in English "substitution" can be used as an adjective, and presumably so can "substitutivity". [[User:JBW|JBW]] ([[User talk:JBW|talk]]) 13:36, 18 October 2024 (UTC)
::Oops, you are right, "substitutivity" is indeed a noun. What I meant was, that it is derived from an adjective "substitutive" in the same way as e.g the noun "reflexivity" is derived from the adjective "reflexive". However, while "substitutive" has an entry in Wiktionary ([[:wikt:substitutive]]), its meaning given there ("serving as a substitute") doesn't match the meaning used in my suggestion. "Substitutive" in the latter meaning is used e.g. by Milner (1990, Sect.3.4, p.1220-1222).<ref>{{cite book | author=Robin Milner | contribution=Operational and Algebraic Semantics of Concurrent Processes | pages=1203&ndash;1242 | isbn=0-444-88074-7 | editor=Jan van Leeuwen | title=Formal Models and Semantics | publisher=Elsevier | series=Handbook of Theoretical Computer Science | volume=B | year=1990}}</ref> After he has shown in Prop.3.2.8 on p.1217, that ~ is an equivalence relation, he continues in the lead of Sect.3.4 on p.1220: ''"We must demonstrate that ~ is indeed a congruence relation, i.e. that it is substitutive everywhere."''  - I guess, a justification for that meaning would be "a relation R is called substitutive if it preserves substitution, i.e. if ''b'' can be used as a substitute [replacement] for ''a'', then ''f''(''b'') can be used as a substitute for ''f''(''a'')". In [[wikt:Substitutivity]], the explanation seems hardly comprehensible (it might even be flawed?), but the Quine citation matches "my" meaning exactly (Quine was an analytical philosopher, who, like Russell, used mathematical logic to a great extent in his philosophy). - [[User:Jochen Burghardt|Jochen Burghardt]] ([[User talk:Jochen Burghardt|talk]]) 20:51, 18 October 2024 (UTC)
{{talkref}}

== Suggestions for further improvement ==

@[[User:Farkle Griffen|Farkle Griffen]], I saw you asked for review of this article elsewhere and for suggestions for further improvement. 

First, good work so far on the article: I think it's definitely much improved.

As one next improvement I'd suggest adding a "background" to the "In logic" section, mirroring the "background" for the "In set theory" subsection, that discusses how equalities were the defining predicates for the category of "quantity" in Aristotle's ''[[Categories (Aristotle)|Categories]]'' and then (very briefly) the development of mathematical logical equality to Leibniz from the Scholastic study of the ''Categories'' as part of the ''Organon''. I don't immediately have a source for that in mind, but one or two shouldn't be hard to find. [[User:RowanElder|RowanElder]] ([[User talk:RowanElder|talk]]) 20:35, 22 January 2025 (UTC)

:@[[User:RowanElder|RowanElder]], The bit about Aristotle is intresting, and seems relevant enough to the section; apart from that, I'm not sure how much more there is to say. From what I can find, Scholastics generally didn't really care much about quantitative equality, at least not in a way that seems relevant enough to mathematics to justify mentioning in the article. Does what I've added so far work? <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 21:21, 23 January 2025 (UTC)
::Yes, what you've added should roughly work, though it may need a minor rewording to avoid creating the impression that Aristotle conflated identity and equality.
::The stuff about the Scholastics is harder to find and it's not as high a priority as Aristotle's ''Categories''. Mathematician-philosophers like [[Richard Swineshead]] were crucial for the development of the ideas of equalities of intensities, i.e., the development of intensive quantities (forces, pressures, temperatures) in contrast to the extensive quantities (lengths, areas, volumes) ubiquitous in Euclid and Aristotle. This was studied by historians of math and science like [[Marshall Clagett]]. The Scholastic developments involved some subtle reasoning about quality and quantity that proved crucial to the development of, e.g., quantifiable temperature (one of Clagett's thesis topics), and thereby also to the way in which Leibniz conceived of the properties relevant to the substitution principle. 
::That material can be a bear to wade through, though, so no rush: your existing addition will hold the eventual place for it well. If you're on the verge of nominating this as a good article and it's still a pain to find this even with the Clagett and Swineshead pointers, I'll be happy to go find some up-to-date summary sources for this article myself, just say when (and give me about a week to get around to it). [[User:RowanElder|RowanElder]] ([[User talk:RowanElder|talk]]) 22:06, 23 January 2025 (UTC)
:::@[[User:RowanElder|RowanElder]], sorry for the delay, it's been busy.
:::Your pointers were very helpful, and I've done my best to try to summarize the sources I could find, but I may over/underemphasize details. Is what I've added satisfying? 
:::I wouldn't say I'm on the verge. There's still a few details I need to iron out, better sources for some parts, and I'm not sure what I'm going to do about [[Homotopy type theory]] yet, so if there's any issues, there's time to get it worked out. <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 20:25, 10 February 2025 (UTC)
::::No rush at all on my end. I appreciate what you've added and I'll try an edit to sharpen it up in a little while. I've been wanting to make time to finish work I started at [[I. Bernard Cohen]] first, though, so it might be a week or two unless you'd like this done sooner rather than later. I hope you enjoy figuring out what to do with HoTT! [[User:RowanElder|RowanElder]] ([[User talk:RowanElder|talk]]) 20:40, 10 February 2025 (UTC)
:::::@[[User:RowanElder|RowanElder]], I've done more or less what I wanted to do. I think I'm going to hold off on an HoTT section for now unless theres a real push for it during GA review (I don't think its necessary for GA, but I'll add it if I go for FA).
:::::I'd like to talk about the paragraph on the Scholastics. Looking at it again, I'm a bit lost on how exactly it relates to equality. "Equalities of intensities", sure, but that can just mean a number with units. Was there something fundamentally about equality in its definition that I'm missing here? <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 05:02, 1 March 2025 (UTC)
::::::Sounds good re HoTT. Re the Scholastics, the fundamental thing there is that they worked out what it meant for intensities of qualities to be equal to each other and thereby what it meant for those intensities to be quantifiable (since equality was the predicate proper to quantities in the Categories). We now take it for granted that each is "just a number with units," but historically it was a difficult advance. [[User:RowanElder|RowanElder]] ([[User talk:RowanElder|talk]]) 17:46, 1 March 2025 (UTC)

== Section on Type theory ==

I'd like some opinions on this. I've been back and forth with myself for a month or so about what to do.
Should this article include a section on Type Theory? On the one hand, [[Homotopy type theory]] has quite a bit (and quite interesting things) to say about equality, enough to fill a small section. On the other hand, it's somewhat obscure in that a mathematician could go their whole career never hearing about it. It makes it hard to determine how much is [[WP:DUE]]. 
I'd appreciate any opinions on this. <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 20:36, 10 February 2025 (UTC)

:I think a small section is warranted, personally. It has a decent footprint in computing, especially in functional programming circles, in addition to its pure mathematical audience. It has come up in conversations at every recent AI conference I've attended. [[User:RowanElder|RowanElder]] ([[User talk:RowanElder|talk]]) 20:46, 10 February 2025 (UTC)

== Peer review ==
{{Wikipedia:Peer review/Equality (mathematics)/archive1}}
<!-- Please comment in the transcluded page, thanks! [[User:AnomieBOT]] 15:59, 1 March 2025 (UTC) -->


==Article update==
Hi {{ping|Farkle Griffen}} How goes it. I would start by finding references for each note entry. Its an auto-fail of GA before it even starts  if they are missing.   '''<span style="text-shadow:7px 7px 8px black; font-family:Papyrus">[[User:scope_creep|<span style="color:#3399ff">scope_creep</span>]]<sup>[[User talk:scope_creep#top|Talk]]</sup></span>''' 09:23, 3 March 2025 (UTC)

:Gotcha. I can work on that <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 02:27, 4 March 2025 (UTC)
:I took a slightly different route. How's it looking? <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 07:36, 4 March 2025 (UTC)
::{{ping|Farkle Griffen}} The two remaining still need references. Next thing which is low hanging-fruit is to go around each image and add an alt tag.  '''<span style="text-shadow:7px 7px 8px black; font-family:Papyrus">[[User:scope_creep|<span style="color:#3399ff">scope_creep</span>]]<sup>[[User talk:scope_creep#top|Talk]]</sup></span>''' 10:06, 6 March 2025 (UTC)
::{{ping|Farkle Griffen}} On the references on many of them, you are missing location, publisher (e.g. ref 72) edition fields, isbn,  page numbers (e.g. ref 19), most egregiously, authors, . There is no oclc links. You get that field from worldcat which links the the ref to the worldcat entry. These are quick fail at GA.  '''<span style="text-shadow:7px 7px 8px black; font-family:Papyrus">[[User:scope_creep|<span style="color:#3399ff">scope_creep</span>]]<sup>[[User talk:scope_creep#top|Talk]]</sup></span>''' 12:06, 6 March 2025 (UTC)
:::I see you have listed it at [[WP:PR]]. It will take 3 to 6 weeks before somebody comes acround. It might be longer due to the specialism of the subject. '''<span style="text-shadow:7px 7px 8px black; font-family:Papyrus">[[User:scope_creep|<span style="color:#3399ff">scope_creep</span>]]<sup>[[User talk:scope_creep#top|Talk]]</sup></span>''' 12:10, 6 March 2025 (UTC)
::: @[[User:Scope creep|Scope creep]] What are oclc links?  <span class="nowrap">– [[User:Farkle Griffen|Farkle Griffen]] ([[User talk:Farkle Griffen|talk]])</span> 00:20, 25 March 2025 (UTC)
{{Talk:Equality (mathematics)/GA1}}

==Did you know nomination==
{{Template:Did you know nominations/Equality (mathematics)}}