Talk:Equality (mathematics)

Identity

It is inadequate to characterize mathematical identity as equality of functions since mathematical logic recognizes equality not only of functions, which are N:1 relations, but of relations generally. See Identity_of_indiscernibles#Identity_and_indiscernibility. Jim Bowery (talk) 16:15, 27 September 2015 (UTC)

Antisymmetry

Given the definition of "antisymmetric" on the binary relation page uses "=", isn't it a triviality to list it as a property of "="? -- Tarquin

Yes, it's pretty trivial that equality is antisymmetric. From a certain POV (the one that holds that equality is a purely logical relation with the axioms shown here), the other properties are trivial as well (since they follow from pure logic) -- but this one is particularly trivial (since it follows from pure logic even if you don't take that POV). Still, it's worth knowing that equality is unique among equivalence relations as the only antisymmetric one, so the fact has some use. -- Toby 08:10 Dec 1, 2002 (UTC)

Equality in different branches of mathematics

As I understand it, each branch of mathematics defines equality independently. That is, the axioms of number theory (implicitly) define numeric equality as some particular equivalence relation over the set of integers, the axioms of set theory define set equality as some particular equivalence relation over the set of sets, etc.. But, aside from saying equality must always be an equivalence relation, mathematics seems to have nothing to say about equality "in general". Thus there seems to be no definition with which to make sense of the claim that, e.g. 3 = {1, 4, 9}. In addition, it seems I could define a class of mathematical objects without bothering to define any equality relation over them. Is any or all of this correct? If so, I have some changes in mind for the article. --Ryguasu 13:31, 9 Sep 2003 (EDT)

I think it fair to say (a) the = symbol is heavily overloaded in mathematical usage, and (b) typical manoeuvres such as passage to the quotient are carried out without regard to the existence of a computable equality relation.

Charles Matthews 09:05, 10 Sep 2003 (UTC)

not equal to

is there a not equal to page or should there be a section on this page? i was specifically looking for the different symbols, such as <> and != and ≠ - Omegatron 02:15, Jul 12, 2004 (UTC)

• ≠ redirects here now, so we should expand that section if there is anything to add. — brighterorange (talk) 15:00, 14 October 2005 (UTC)

circular definition of equality

Isn't it a circular definition to say that

1. equality is the only binary relation that is

  reflexive, transitive, symmetric and antisymmetric

and

2. a binary relation R is antisymmetric iff

  R(x,y) and R(y,x) implies equality of x and y.

When it is possible to define antisymmetry without refering to equality, fine.- 193.175.133.66

The first is not presented as a definition but as a property.--Patrick 23:47, 27 Aug 2004 (UTC)

Japanese usage

I have removed the following text from the article:

The equal sign that is currently used in Japan (・) also is used as a punctuation to separate
the first and last names when a western person's name is written in Katakana.

1. I cannot find any references for the use of ・ as an equality sign in Japan. The usual sign I see in Japanese texts is ＝.

2. Even if ・ has such a use, its relevance to a discussion of the symbol "=" is unclear, and the relevance of its regular use to the history of the aforementioned symbol is extremely unclear.

If someone can come up with a reference that describes or demonstrates this usage of ・ as an equal sign, it might be a good idea to add a section on "other equals signs around the world". (If there are any. Might Arabic or Sanskrit have their own?)

?

• "Instead of considering Leibniz's law as an axiom, it can also be taken as the definition of equality. The property of being an equivalence relation, as well as the properties given below, can then be proved: they become theorems." -- does that mean the axiom is equivalent to the definition?
• It should be noted somewhere in the article that equality as a logical notion is in a larger sense undefined. To go into philosophical discussions in an article on mathematics is always dangerous. Plus, many's the time that equivalence in one domain is used to define equality in another. For example, what's a fraction? --VKokielov 04:08, 25 October 2006 (UTC)

For the first question, it may depend on the further rules for equality. In the context of a concrete proof system, it may be the case that the system has a weak axiomatization for equality that does not allow to conclude from P(x) and x=y to P(y). Then the definition, which allows this inference, is stronger. But if you replace the first "if" in Leibniz's law by "if and only if", it is clearly equivalent; there is no logical difference between an axiom "P(x) iff Q(x)" and a definition "P(x) iff Q(x)". Also, the axiom of extensionality: f = g iff f(x) = g(x) for all x, as far as the "if" direction is concerned, is not an obvious consequence of Leibniz's law.

What exactly is the danger you refer to? The technical way to define the rational numbers is as equivalence classes as pairs of integers. Then equality of rational numbers is indeed plain equality, even if these classes are defined using equivalence in the base domain. All basic logical notions are "in a larger sense undefined", beacuse if we could define them, they would not be basic. Think of implication. Equality is in comparison very defined or at least definable.  --Lambiam^Talk 06:52, 25 October 2006 (UTC)

What bothers me is that this is all very technical, and people might be inclined to think that there's something magical about names of elements of a mathematical system, whereas it's not the names but the relationships between them that matter. --VKokielov 12:48, 25 October 2006 (UTC)

You speak like a category theorist! But I've removed the section on equality and isomorphism you wrote, because it was not really clear what it was trying to say, nor who the intended audience were. In my opinion it was too technical for the average reader, and not particularly informative for mathematicians. Personally I think the article, in particular the lead section, should be made more accessible to non-mathematicians, while the more technical stuff should be pushed to the back.  --Lambiam^Talk 14:55, 25 October 2006 (UTC)

I agree entirely with Lambiam on the point that this article is not accessible to the layman. I myself was looking for an explanation of equality and I do not have a large background in mathematics beyond high school. I was quite confused and put off by the article. I do agree however that it might be best to maintain the technical aspect of the article but push it to the back. Just as in articles about such topics like the Mint plant or Tea, the subject matter is summarized and put in non-technical terms for the layman or casual reader at the beginning of the article and then the technical aspects are gradually added with many heavily technical aspects saved for the end of the article. I myself do not have the knowledge to revise this article myself but I would be grateful if someone with that background would. --lady 2:13, 08 January 2009 (UTC) —Preceding unsigned comment added by 67.142.130.13 (talk)

Tagged as unreferenced

The article currently appears to have no references for verification. While the facts are presumably correct, it will need published citations for the various pieces of information, such as to verify that the exact phrasing of various mathematical definitions are properly phrased according to independently published texts. Dugwiki 22:05, 8 February 2007 (UTC)

Equality is Antisymmetric?

Is it true that equality is antisymmetric? Doesn't symmetry mean that if A = B, then B = A. Set inclusion is certainly antisymmetric: A C B, then B C A, in general, is false. However, a definition of set equality would be if A C B and B C A, then A = B. In this case, wouldn't we also say that B = A. If so, doesn't that mean that set equality is symmetric?

Bkv2k (talk) 14:47, 1 November 2009 (UTC)

The word antisymmetry doesn't mean "asymmetry." If you look at the definition, you'll see that antisymmetry holds trivially for equality.—PaulTanenbaum (talk) 16:40, 2 November 2009 (UTC)

Definition using Leibniz's law?

From the article:

A stronger sense of equality is obtained if some form of Leibniz's law is added as an axiom; the assertion of this axiom rules out "bare particulars"—things that have all and only the same properties but are not equal to each other—which are possible in some logical formalisms. The axiom states that two things are equal if they have all and only the same properties. Formally:

Given any x and y, x = y if, given any predicate P, P(x) if and only if P(y).

Having a look at this definition, it is true for any x and y to be not equal and still have all their predicates identical - therefore Leibniz's law doesn't seem to be used. --Abdull (talk) 22:16, 4 September 2010 (UTC)

Equality as a relation

This article spends too much time discussing the identity relation, which logically is not what equality is -- it's a reification of equality, but equality must be defined axiomatically. Before you can define a relation, you need to define the properties of sets, and to do so, you need the ZF axioms. Those in turn presuppose a notion of equality, in particular the axiom of extensionality; without that, there is no single identity relation (it expresses the notion that "two objects are the same if they have the same properties", specifically the same elements).

In Peano arithmetic, there's a similar situation. There, = is a symbol is no defined meaning. It is taken as primary and its meaning becomes clear through the axioms that define when x = y. Qwertyus (talk) 12:51, 7 October 2012 (UTC)

There is only one kind of equality

There is only one kind of equality. Mendelson 1964 page 5 says, We shall write "t = s" to mean that "t" and "s" denote the same object. Regarding the "algebraic" and "logical" kinds, an algebraic expression is just a name for some object, as is a symbol; so these are the same notion of equality. Regarding the "set-theoretic" kind, the Axiom of Extensionality tells us what a set is: a set is determined by its members; it has no other structure. The axiom is not introducing a new kind of equality. David Marcus (talk) 13:44, 22 September 2018 (UTC)

Things are not so easy. You say that a set is determined by its members. If you take this as a definition, you should conclude that all sets form a set.

You are right when saying that there is only one notion of equality, only if you are working in a fully formalized theory. Mathematics, in the whole is not fully formalized, and this is what makes its richness. In common mathematics, you have to distinguish between syntactic equality and semantic equality: for example $(x+1)^{2}$ and $x^{2}+2x+1$ are not syntactically equal (there are different expressions). However, they are semantically equal, as they define the same polynomial and the same function. This distinction is fundamental for logical programming. D.Lazard (talk) 15:08, 22 September 2018 (UTC)

I'm not saying that the set of all sets is a set, nor was I defining what a set is. I meant that the Axiom of Extensionality is telling us something about sets, not something about equality. The meaning of equality is the same for sets as for other things. Two things are equal if they are the same thing. Regarding "sytnax" and "semantics", there is a difference between a thing and its name (as the quote from Mendelson makes clear). In algebra, if we say that $(x+1)^{2}$ and $x^{2}+2x+1$ are equal, we mean that the numbers are equal. We don't mean that the strings are equal. If we wanted to refer to the strings, we would put quotes around them, as Mendelson does. It is no different than saying that 1 + 1 = 2. This doesn't mean that "1 + 1" and "2" are the same string. It means that the numbers named by these strings are the same number. The letter "$x$" in your expressions is just the name of a number (assuming we are doing algebra). Similarly, my name is "David" and it is also "David Marcus". So I could say that David = David Marcus. The people I am referring to by the two names "David" and "David Marcus" are the same person. The Axiom of Extensionality tells us that a set is determined by its members; a set has no other structure. So, when checking whether two sets are the same set, all we have to do is check that they have the same members. If you want to prove a certain set exists, you need the other axioms. I am not discussing logical programming; I am discussing what equality means in mathematics. David Marcus (talk) 14:25, 23 September 2018 (UTC)

There are many books that define equality, and Mendelson's is only one of them. As this book is cited only in section "Set equality based on first-order logic without equality", I suppose that your concern is only for this section. I agree that it deserves to be expanded for clarification and addition of comments on the context. Feel free for being bold and editing it, per WP:BRD. By the way, the natural audience of this article consist of all readers interested in mathematics. I have thus restructured it for moving toward the beginning everything that may be interesting for this general audience, keeping to the end the sections about mathematical logic (the layman is not supposed to know what is a predicate). D.Lazard (talk) 15:29, 23 September 2018 (UTC)

My concern is that the introduction is wrong because it says that there is more than one kind of equality in mathematics. As I explained, the three kinds of equality that the article claims are different are actually all the same. Most math books don't bother to define what "equality" means since it just means the things are the same thing. Mendelson's book is a logic book, so he says things that most books leave unsaid. David Marcus (talk) 15:47, 23 September 2018 (UTC)

I agree that, in formal logic, there is only one notion of equality. But in $\{1,2\}=\{n\in \mathbb {N} \mid 0\leq n\leq 1\},$ none of the members of the equality are represented by symbols. In $(x+1)^{2}=x^{2}+2x+1,$ there is no set, and none of the members are represented by symbols. So, unless in a fully formalized context, the three definitions are not equivalent. If you know a way to explain this in a way that remains understandable by non-logicians, and is closer to one of the formal definitions of equality (there are several), it would be welcome. D.Lazard (talk) 17:18, 23 September 2018 (UTC)

The way to explain this to non-logicians and non-mathematicians is to say that equality means that the two things referred to are the same thing. I don't know what you mean by the word "symbol"; do you mean a letter? "$\{1,2\}$" is a symbolic expression that denotes a particular set. It is not the set itself. I can tell you which set this is in many ways, e.g., using symbols or using words. If I write "Let $A=\{1,2\}$", then I have introduced another way of denoting this set. But, "$\{1,2\}$", "$\{n\in \mathbb {N} \mid 1\leq n\leq 2\}$", and "$A$" all refer to the same set. So, they are all equal (the sets, not the names). The words/symbols/letters that we write on paper are not the things themselves. They are names for the things. The equals sign means that the things named are the same thing. If $x$ is a number, then $(x+1)^{2}$ is a number and it is the same number as $x^{2}+2x+1$. That is what the equals sign means. In formal logic, we formalize this, but we shouldn't lose sight of what it is that we are formalizing. David Marcus (talk) 17:42, 23 September 2018 (UTC)

You are definitively wrong by asserting that there is only one kind of equality, and I'll show it. This is true that the two first definitions are special instances of the third one, but it is wrong that the third one can be deduced from the first ones. In fact, "evaluation" implies a computation, and thus a proof, which may depend of the axioms of the theory and of the logical framework in which they are written. More precisely, there are, at least, three kinds of equality: equalities that can be proven in the theory, equalities that are true in every model of the theory, (I remember that these definitions of equality are equivalent in ZFC, but ignore if this is true for every formal logic) and equalities that depend on the choice of the axioms of the theory. A well known example of the third category is the continuum hypothesis. A more elementary example is the equality 0.999... = 1, where the left member is the least upper bound, if exists, of the sequence 0.9, 0.99, 0.999. This equality is true in standard analysis, but may be wrong in some instances of non-standard analysis. See Talk:0.999... and Talk:0.999.../Arguments for being convinced that this is not only a problem of abstract logic. A third example is Fermat's Last Theorem, which may be expressed as a first order predicate

$\emptyset =\{(x,y,z,n)\mid x\in \mathbb {N} ,y\in \mathbb {N} ,z\in \mathbb {N} ,n\in \mathbb {N} ,n>2\}.$

Although this is an equality expressible in first order logic, no proof is known in this logic. The original proof involves an extension of ZFC with new axioms, and it is unclear for me if a proof in ZFC is known.

So, the definition of equality is not as simple as you pretend. The present lead is acceptable, as hiding the difficulties behind the word "evaluate". It could be useful to expand the article for giving indications of these difficulties, but this is a difficult task, and I am not ready for doing it myself. D.Lazard (talk) 05:20, 24 September 2018 (UTC)

I agree with D.Lazard (and presumably also with Davidjmarcus) that in the current lead, the two first definitions are special instances of the third one. For this reason, I'd consider it sufficient to give only the third (i.e. the most general) definition there. The second one might serve as an example definition of equality for a particular kind of "object", or "value"; another example would be the definition of equality of two rational numbers a/b=c/d if ad=bc (which is luckily not claimed to be a fourth kind of equality). — As for D.Lazard's elaborations on provability, I didn't understand their relevance in this discussion: there are many other relations (e.g. ≤) which are used in both provable and unprovable formulas, without considering them to be of different kinds. Maybe I missed some point? - Jochen Burghardt (talk) 11:19, 24 September 2018 (UTC)

Regarding the third definition, if by "expression" it means name, and if by "evaluate" it means figure out the thing that the name refers to, then it is what equality means, but it is certainly an odd way of describing something that is much simpler. The reason that 0.999... = 1 is that both "0.999..." and "1" are names for the same number in usual mathematics. The fact that if we are doing non-standard analysis, we may not be able to prove that the least upper bound of the sequence is 1 just means that in that context we don't know if the least upper bound is 1. If "0.999..." refers to the least upper bound, then in that context we don't know if 0.999... = 1. This doesn't change the meaning of equality. Equality is really very simple in mathematics. The odd thing about equality is that in mathematics we often have more than one name for the same thing. Note that what I am saying is exactly what Mendelson says: "x = y" means that "x" and "y" denote the same object. I.e., they are names for the same thing. I have yet to see an example where equality means something different. The equality of rational numbers is still the same equality. Just because you have a new kind of thing, doesn't mean you have a new kind of equality. Once you figure out what the thing is, you know what it means for those kind of things to be equal. With fractions, we can have two different fraction symbols that denote the same fraction. E.g., 1/2 = 2/4. The faction symbol, i.e., "1/2", is not the faction. It is a name for the fraction. Both "1/2" and "2/4" are names for the same fraction, which is why 1/2 = 2/4. David Marcus (talk) 16:02, 24 September 2018 (UTC)

That's my point. I'd be fine with Mendelson's definition. It reduces equality between mathematical expressions to equality ("the same") between mathematical objects. As for the latter, I guess every mathematical domain (like sets, standard real numbers, all kinds of nonstandard real numbers, rational numbers, ...) comes with its own definition of equality; all we can say in general about it is that it is the finest equivalence relation on the domain (even the finest congruence relation). Using the above example, while (0.9, 0.99, 0.999, 0.9999, ...) and (1, 1, 1, 1, ...) are different as sequences, they are (considered by definition) equal as real numbers. If you both agree, I'd like to add a few sentences along these lines after Mendelson's definition. - Jochen Burghardt (talk) 07:11, 29 September 2018 (UTC)

It is not true that every "mathematical domain" (whatever that is) comes with its own definition of equality. It is true that you must say what a set, real number, rational number, etc. is before you can say if two such are equal. You can't reason about things that are not well defined. First, you must define your things. Then equality means what it always means, "x = y" means x and y are the same thing. You are mixing up what equality means with what various things are. Once you separate them, you find that equality does not change. Regarding your example, you should not say that a sequence (of rationals) is a real number. Sequences and real numbers are different things. It is true that one of the ways to construct the real numbers is from Cauchy sequences. If you want to construct the real numbers from Cauchy sequences, then neither of your two sequences are real numbers. The real numbers would be the equivalence classes containing your sequences under a suitable equivalence relation. Equality of two such equivalence classes means that the two equivalence classes are the same equivalence class, i.e., the two things are the same thing. David Marcus (talk) 13:45, 29 September 2018 (UTC)

Perhaps you were referring to the limits of the sequences. The limit of a sequence of real numbers is a real number. You can have two sequences that are not equal (i.e., are not the same sequence) that have limits that are equal (i.e., the limits of the sequences are the same real number). As always, "equal" means the things are the same thing. David Marcus (talk) 01:08, 30 September 2018 (UTC)

You are right, equivalence classes (e.g. of Cauchy sequences) are used to construct new "domains" from given ones (e.g. real numbers from rational numbers). Equality isn't redefined for every new domain. It was my fault (inspired by computer science practice) - sorry for the confusion!

My problem with your definition x and y are equal if x and y are the same thing is that it seems to be circular to me, assuming "equals" and "same" to be synonymous. Alternatively, one could say that it reduces the mathematical notion of equality to the common-sense notion of sameness; however the latter can be defined only in a circular way, if I understood the SEP article https://plato.stanford.edu/entries/identity right. In a Wikipedia mathematics article, we should be able to do better than to appeal to the "basicness of the notion of identity in our conceptual scheme" (SEP, citing Quine.1964). I had thought about using Leibniz' definition (see Equality_(mathematics)#Logical definitions), but SEP says it leads to circularity, too.

Accepting the quotient set approach which you mentioned, we should explain to the reader when two equivalence classes are equal. Assuming set theory as the (most widespread used) foundation of mathematics, we should hence remind the reader to the set-theory definition of equivalence classes, and define equality of sets in a prominent place. Assuming everything is a set, and ∈ is a well-founded relation, a formally correct recursive standalone definition of set equality can be given (no.2 in the current lead). Not all of this needed to appear in the lead. Would that be OK? - Jochen Burghardt (talk) 13:38, 30 September 2018 (UTC)

Thank you for agreeing that equality is not redefined.

I did not say that "equal" and "same" are synonyms. I said that "x = y" means that x and y are the same thing. Here, "same thing" has its usual English meaning. I am defining what the equals sign means. Since the English meaning does not depend on the equals sign, the definition is not circular.

Consider this dialog: A says, "Look at the blue car." B says, "Do you mean the car by the tree?" A says, "Yes, that car." Then the blue car and the car by the tree are the same car. In other words, "blue car" and "car by the tree" denote the same object.

The article is about mathematics. Its title is "Equality (mathematics)". It is not about philosophy. Mathematicians do not take a course in philosophy before they can define things or determine if two things are equal. I don't think the SEP article will help anyone understand what equality means in mathematics. If I write, "Let x = 2", then x and 2 are the same number. I don't need to know philosophy for this, or that set theory may be taken as the foundation of mathematics.

I am more concerned with the elementary school and high school teachers who may read this article than I am with the philosophers.

We can say that "x = y" means that "x" and "y" denote the same object, as Mendelson says. Or, we can say that it means that x and y are the same thing. I think most non-mathematicians will understand it better if we say the latter, or say both.

You don't have to know mathematical logic to understand equality (many mathematicians do not know mathematical logic). However, it does help to understand the difference between a thing and its name. Explaining what quotes mean (i.e., the difference between x and "x") would help the reader understand this difference.

You don't have to know what an equivalence class is to understand what equality means. If you don't already know what equality means, you won't understand the point of equivalence classes.

It is true that ZFC may be taken as a foundation for mathematics. So, if we assume all mathematics is done in ZFC, then everything is a set, so we only need to know what equality means for sets. But mathematicians were doing mathematics for thousands of years before ZFC was developed. So, the concept of equality in mathematics does not need ZFC to be understood.

The article contains some other questionable statements. The statement that "When A and B are not fully specified or depend on some variables, equality is a proposition" is peculiar. I assume this is referring to logic formulas with free variables. A logician may understand what this is referring to, but other people will be confused. Equality is not a proposition. The whole "Equality as predicate" section could be deleted to improve the article.

The "Identities" section is questionable. It seems to be implying that you can view something as a number or a function as you wish. If it isn't clear whether something denotes a number or a function, then the article or book is poorly written. In general, algebraic expressions denote numbers, not functions. Confusing a number and a function is unfortunately quite common, especially in writings by non-mathematicians. For example, you often see people write "consider the function f(x)" when they clearly mean that f is the function. Mathematicians used to write like this before the concept of function was developed, but the non-mathematical world hasn't caught up.

The "Equations" section is questionable. Equality means the same thing in equations and identities. The difference comes from the words around the equation or identity, i.e., is it saying that the things are equal for all values or is it asking which values make the things equal?

The "Congruences" section is questionable. It is true that high school mathematics uses phrases like "equal angles" for what would be better worded "angles of equal degree" or uses "equal" when "congruent" is meant. While this is unfortunate, it should be interpreted as (poor) terminology, not a new kind of equality. A warning about this terminology would be appropriate in the article.

The article claims that rational numbers are equivalence classes of fractions and that 1/2 and 2/4 are distinct as fractions. This is wrong. The symbols "1/2" and "2/4" should be called "fraction symbols". It is true that these two fraction symbols are different. A reasonable definition of "fraction" is that the fractions are the nonnegative rational numbers. (This is what fractions are in Hung-Hsi Wu's books. Of course, if we are developing things in order, the fractions would come before the rational numbers.) If you want to construct the rational numbers (which is not required), then the rational numbers are equivalence classes of ordered pairs of integers (not of fraction symbols).

In the "Relation with equivalence and isomorphism" section, the article claims that two sets, each containing three elements, are isomorphic. Isomorphic as what? The section is confusing and doesn't help you understand equality. It could be deleted to improve the article. David Marcus (talk) 15:27, 1 October 2018 (UTC)

I took a closer look at the SEP article. It is not mathematics. David Marcus (talk) 21:16, 2 October 2018 (UTC)

I understand you want to explain mathematical equality (and the notation "=") in terms of the common-sense notion ("usual English meaning") of sameness. The latter is in fact not mathematics, and it is the subject of the SEP article. I referred to it only to show that the usual English meaning of sameness is not as clear as one usually thinks (or merely feels); see also Ship of Theseus#The thought experiment for a brief famous ancient example. Therefore, while referring to the English meaning can provide a first vague intuition to the reader, we can't rely on it for a precise definition of the mathematical notion. We should give the reader something s/he can work with, rather than just a vague feeling.

To make a concrete suggestion, what about deleting in the lead everything after "equals sign", and adding instead:

Equality is an equivalence relation such that, whenever A=B holds, replacing A by B in any assertion keeps its truth value. As a consequence, equality is the finest possible congruence relation.

After that, I'd like to add:

When set theory is used as foundation of mathematics, all mathematical objects are modelled by sets; equality of sets is recursively defined by: A=B if each member of A is equal to some member of B, and vice versa.

(All wording could probably be improved; I'm not a native English speaker.)

You are probably right that the equivalence class (and maybe ZFC) can't be understood before equality is known. I think this may be similar to geometry, where "point", "line", etc. can only be characterized by the relations among them (Hilbert's axioms, 1899, e.g. "For every two points there exists no more than one line that contains them both"), while their standalone-"definitions" (in Euclid's Elements, 300 B.C., "A point is that which has no part" etc.) are no longer considered part of geometry.

I suggest to discuss the flaws you found in the later sections in one or more new sections of this talk page. This will help to keep an overview. - Jochen Burghardt (talk) 11:11, 4 October 2018 (UTC)

The Ship of Theseus is irrelevant since mathematical objects do not rot. The fact that "x = y" means that x and y are the same thing is how mathematicians define equality and how they prove that things are equal. It is not a "vague intuition"; it is a precise definition, and the one that is always used.

I don't see how congruence relations are relevant until you say what algebraic structure you are interested in.

The fact that equality is the finest equivalence relation on a set is a triviality since the equivalence classes are singletons, and you can't get any smaller than that.

Regarding your sentence about set theory, the word "modeled" is incorrect: If we work in ZFC, then every object is a set. Why say "recursively" and "is equal to some member"? How will this help the average reader (who hasn't seen a construction of numbers from sets)? The sets A and B are equal if and only if they are the same set.

The main problem is that the article and your suggested revisions give the incorrect impression that equality is different for different things and needs to be defined for each. This misrepresents what mathematics is and how it is done.

The notion of "same thing" is common sense. And it is part of logic. We don't change our logic when dealing with different parts of mathematics, so we don't change what equality is.

We no longer feel that Euclid's definitions are adequate, but we can still teach geometry in high school. We don't use Hilbert's axioms because it is easier to define geometry in terms of numbers. Axioms and foundations are often chosen for convenience, not because they are the most natural way of thinking about something.

Why not write the article so that it helps people to understand mathematics rather than making mathematics seem more complicated than it is? David Marcus (talk) 19:09, 4 October 2018 (UTC)

This article is not intended for logicians, nor for experimented mathematicians (which of them will have the idea of reading this article for learning more on equality?). The audience of this article is non-mathematicians and beginners in mathematics. For them, the three definitions given in the lead are useful, although all are special cases of the last one. All must be kept, as all can be encountered in the literature (see references). I have edited the lead for making clear that although apparently different, they are equivalent when the two members of the equality are formally and fully specified. In practice, equalities that are written require generally using the context for being fully specified. This is a cause of several apparent paradoxes that may be confusing and should be clarified in this article. Here are some examples:

• Why two different common notations for $(x+y)^{2}=x^{2}+2xy+y^{2}$ and $(x+y)^{2}\equiv x^{2}+2xy+y^{2}$?
• The preceding identity is wrong if x and y represent non-commuting entities, such as square matrices.
• ⁠${\sqrt[{3}]{x}}^{3}=x$⁠ is true if x represents a real number, but wrong if it represents a negative real number viewed as a complex number.

Also, many people have access to computer algebra systems, and should know that "if a = b then" and "if is_equal(a,b)" (the syntax may depend of the computer algebra system) are not equivalent, the first one denoting the syntactic equality, while the second one denotes the semantic equality, or, in more mathematical words, the equality of expressions versus the equality of the objects that are represented.

It is with these points in mind that I have added a reference to expressions in the lead ([1]). Splitting the definition in three list items has been done by another editor ([2]). Although I watches this article, I did not remark that this formulation was formally incorrect. I hope that the clarification that I have just added will allow closing this (too long) discussion. D.Lazard (talk) 13:16, 5 October 2018 (UTC)

Your use of the word "represent" is incorrect in mathematics. In mathematics, you must define things first. If x and y are real numbers, then $(x+y)^{2}=x^{2}+2xy+y^{2}$. They are equal because they are the same thing. If x and y are other things, then $(x+y)^{2}$ and $x^{2}+2xy+y^{2}$ need not be the same thing.

I already explained the difference between a thing and its name, so I won't explain that again.

If we have equals, we can add equals to them and get equals. Or, multiply by equals. Or, apply any other operation. Or, apply any function. These statements are completely general; it doesn't matter what the things are. This is trivial to prove using the definition of equal as same thing, and should be understandable by a high school student or elementary school teacher. I doubt that someone would understand why these are true after reading the article with its "three kinds of equality" and discussions of logical technicalities and equivalence classes. But, maybe they will read the Talk page and find this explanation. David Marcus (talk) 16:02, 5 October 2018 (UTC)

Mendelson 1964 p. 5 is cited for "set equality". However, that is not what Mendelson says. He says, "A set is a collection of objects" and "We shall write 't = s' to mean that 't' and 's' denote the same object". He does not say that t and s are sets. MacLane & Birkhoff 1999 p. 2 is cited for "set equality". I have the 1988 edition. They say, "Since a set is completely determined by giving its elements, two sets S and T are equal if and only if they have the same elements". They are explaining what a set is, i.e., the Axiom of Extensionality. They do italicize "equal", so it can be taken as a definition of equality for sets, but that doesn't mean it is a new kind of equality. Neither Mendelson nor MacLane & Birkhoff say that there are three kinds of equality in mathematics. David Marcus (talk) 15:12, 6 October 2018 (UTC)

This discussion is useless if not aimed to improve the article. Now, everybody has given his arguments (not always contradictory), and it is time to conclude. Clearly you find the lead section not convenient. Other editors find it not perfect, but acceptable. I have edited the lead in an attempt for a compromise between your concerns, others opinions and Wikipedia rules (MOS:INTRO and MOS:MATH#Article introduction). Now, it is to you to propose a better lead, and I suggest that you propose, in a new thread, a new version of the lead. So, we could compare it with the present version, and discuss it, in view of a WP:consensus on what should be said in the lead. D.Lazard (talk) 16:51, 6 October 2018 (UTC)

I asked three mathematicians what "equals" means. They agreed with me. None said that there are three kinds of equality.

I did not say that the lead section is not convenient. I said it is wrong. I've never seen a math book or mathematician say that there are three kinds of equality.

The introduction should say the following: In mathematics, equality means that the things are the same thing. If we write "x = y" or "x equals y", then x and y are the same thing. To say it another way, "x" and "y" are both names for the same thing. David Marcus (talk) 16:44, 12 October 2018 (UTC)

So, you define equality as "two things are equal if they are only one thing". This tautology is certainly not the right definition, because it cannot explain why the concept of equality is so important in mathematics. Nevertheless, I agree that three different definitions are too much. In fact, reading the lead again, it appears that a general definition of equality has been given in the first sentence. The true issue is that the three items at the end of the lead are erroneously qualified of definitions, when, in reality, they are examples of the general definition given at the beginning. I'll edit the lead for clarifying this. D.Lazard (talk) 08:42, 13 October 2018 (UTC)

Two things are equal if they are the same thing. "Equals" just means is the same thing as. That is what mathematicians and math books mean by "equals". It is because "equals" and "=" mean "is" that we care about them. We want to know that 2 + 2 is 4, that ${\frac {2}{4}}$ is ${\frac {1}{2}}$, that if a number $x$ is such that $x+5$ is 8, then $x$ is 3. That is why equality is important in mathematics. David Marcus (talk) 13:08, 20 October 2018 (UTC)

Your edits have not made the lead correct. David Marcus (talk) 13:49, 20 October 2018 (UTC)

"2 + 2 is 4": This formulation is correct but not accurate. The accurate (but somehow pedantic) formulation is "the object (here a number) denoted by 2 + 2 is the object denoted by 4. This exactly what says the first sentence of the lead.

"What mathematicians mean": I am a mathematician. Apparently, you are also a mathematician, but we do not mean the same thing by equal. So "mathematicians mean" is an assertion that cannot be sourced, and cannot be used in Wikipedia, even in a Wikipedia discussion.

"Mean": Apparently, you confuse the meaning whicis the intuitive interpretation, and the definition, which should be expressible in a formal language. By the way your repeated assertion that there only one definition of the equality in mathematics is totally wrong, as the definition of equality is not the same in equational logics, where the equality is a primitive predicate, and other logics where equality is defined by a mathematical construction. In all these logics, although the definitions are completely different, the meaning of the equality is essentially the same. You should note that in all these logic, the equality is a relation between terms, that is between expressions. Proving an equality amounts to use the rewriting rules (axioms) of the logic for reducing the two members of the equality to the same term. This is exactly the formalization of the definition of the first sentence of the article. Thus, your assertion "Equals" just means "is the same thing as". That is what mathematicians and math books mean by "equals" is wrong, as the only books hat give a definition of equality are books of mathematical logic, and no book of mathematical logic defines equality in this way. D.Lazard (talk) 20:38, 20 October 2018 (UTC)

"Your edits have not made the lead correct": Please provide a source supporting your assertion. D.Lazard (talk) 20:38, 20 October 2018 (UTC)

Enough is enough. You are repeating ad libitum the same arguments. So, nothing useful can result from continuing this discussion. So I'll no more answer to your posts, unless you suggest a precise formulation for a sentence that should be changed, and you provide a WP:reliable source for this formulation. As I'll stop to read this this thread, this should be done in a new section with a heading that indicates clearly that this is not the continuation of this useless discussion. D.Lazard (talk) 20:38, 20 October 2018 (UTC)

Your statement "the object (here a number) denoted by 2 + 2 is the object denoted by 4" is missing the quotes needed to make it correct. You could say that the object denoted by "2 + 2" is the object denoted by "4", i.e., the objects are the same object, i.e., the objects are equal. Or, you could say that 2 + 2 is 4, equivalently 2 + 2 = 4. It would not be correct to say that "2 + 2" is "4", or "2 + 2" = "4". Mendelson 1964 page 5 states it correctly: We shall write "t = s" to mean that "t" and "s" denote the same object. Note that Mendelson does not say that t and s denote the same object. David Marcus (talk) 23:45, 20 October 2018 (UTC)

If "2 + 2" and "4" denote the same object, then 2 + 2 and 4 are the same object. Since 4 is a number, 2 + 2 and 4 are the same number. You say that 2 + 2 is not 4. Either you are confusing a thing and its name or you are not using the usual meanings of the words "is" and "are".

Equational logic is not relevant to whether 2 + 2 is 4, nor are esoteric details of mathematical logic appropriate for an article on equality in mathematics. However, I will comment. You are confusing the formal theory with its meaning. A formal theory is just symbols and rules for manipulating them. When we do mathematical logic, we study these formal theories using mathematics. If a first-order theory includes the usual axioms for the "=" symbol, then there will be a model where the meaning of the "=" symbol is the usual meaning of "is" (there will also be models where the "=" symbol does not correspond to equals in the model). The model is where the meaning is. In the formal theory, you just have symbols. David Marcus (talk) 12:57, 21 October 2018 (UTC)

The current article should have "Assuming 2 + 2 is not 4" added at the beginning. David Marcus (talk) 15:24, 3 November 2018 (UTC)

Leibniz's law's weakened form

From the Logical definitions section:

In this law, "P(x) if and only if P(y)" can be weakened to "P(x) if P(y)"; the modified law is equivalent to the original, since a statement that applies to "any x and y" applies just as well to "any y and x".

This argument doesn't seem valid to me: take x and y such that the set of predicates holding for y is a proper subset of the set of predicates holding for x (and therefore x ≠ y). Then the modified statement says that "x = y [which is false] if and only if, given any predicate P, P(x) if P(y) [which is true]", and therefore is false.

My question is then: do such x and y exist? In other words, do we consider "being equal to y" to be a predicate (which might give rise to a kind of chicken-and-egg issue)? If such is the case, then x and y as described above cannot exist: "x = y if y = y" would imply that x = y, a contradiction.

In any case, I feel like that remark can either be fixed or made clearer, but I'm not sure how.

Naim42 (talk) 14:48, 22 January 2020 (UTC)

I agree, and deleted the paragraph. The succeeding one was wrong, too (see Identity_of_indiscernibles#Identity_and_indiscernibility). - Jochen Burghardt (talk) 19:05, 22 January 2020 (UTC)

"Distinct (mathematics)" listed at Redirects for discussion

A discussion is taking place to address the redirect Distinct (mathematics). The discussion will occur at Wikipedia:Redirects for discussion/Log/2020 October 31#Distinct (mathematics) until a consensus is reached, and readers of this page are welcome to contribute to the discussion. –Deacon Vorbis (carbon • videos) 17:28, 31 October 2020 (UTC)

"≣" listed at Redirects for discussion

A discussion is taking place to address the redirect ≣. The discussion will occur at Wikipedia:Redirects for discussion/Log/2020 November 14#≣ until a consensus is reached, and readers of this page are welcome to contribute to the discussion. 𝟙𝟤𝟯𝟺𝐪𝑤𝒆𝓇𝟷𝟮𝟥𝟜𝓺𝔴𝕖𝖗𝟰 (𝗍𝗮𝘭𝙠) 12:09, 14 November 2020 (UTC)

Deriving symmetry and transitivity using substitution and reflection.

From the Equality_(mathematics)#Basic_properties section, the last sentence states symmetry and transitivity can be deduced from substitution and reflection. I think that's true if the substitution being discussed is "if a = b then (P(a) <=> P(b))". However, the substitution being discussed is "if a = b then F(a) = F(b)". I don't think symmetry and transitivity can be deduced from that version of substitution. — Preceding unsigned comment added by 2601:989:4401:3830::8763 (talk • contribs) 15:25, 17 July 2021 (UTC)

If $F(x)=(x=a),$ the substitution gives that if $a=b,$ then $(a=a)=(b=a).$ So, symmetry results from substitution and reflexivity. If $F(x)=(x=c),$ then the substitution gives that if $a=b,$ then $(a=c)=(b=c).$ So, transitivity results from substitution alone. In these formulas, the $=$ that appears inside parentheses must be viewed as a function that can return true or false. D.Lazard (talk) 14:05, 17 July 2021 (UTC)

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) 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. 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? Farkle Griffen (talk) 03:33, 17 July 2024 (UTC)

OK, 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. 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. 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? 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. 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 classes, you can edit the target article accordingly. D.Lazard (talk) 16:27, 22 July 2024 (UTC)

Section "In logic"

Section § 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. 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.

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 objects. A famous example is the set of real numbers: The constructions with Dedekind cuts and Cauchy sequences 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. 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 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. - 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. - 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: 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. 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 ($1.5=3/2$ is certainly not a syntactic equality).
• The substitution property is presented in section § 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. 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 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.

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. Farkle Griffen (talk) 03:29, 26 July 2024 (UTC)

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? 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. Farkle Griffen (talk) 21:45, 8 August 2024 (UTC)

@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. 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 "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:

• Substitutivity: for every a and b, and every operation $f(x)$, if a = b, then f(a) = f(b). Informally, this just means that if a = b, then a can replace b in any mathematical expression or formula without changing its meaning.
  For example:
  • Given real numbers a, and b, if a = b, then $2a-5=2b-5$.
  • Given unary real-valued differentiable functions ⁠$g$⁠ and ⁠$h$⁠, if $g=h$, then ${\textstyle {\frac {d}{dx}}g={\frac {d}{dx}}h}$. (Equality of functions is retained by the derivative operation.)

If restricted to the elements of a given set $S$, those first three properties make equality an equivalence relation on $S$. In fact, equality is the unique equivalence relation on $S$ whose equivalence classes are all singletons. On every algebraic structure, all four properties together make equality a congruence relation, if it is defined at all.

References

Comments are welcome. - 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. 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". 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). 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). - Jochen Burghardt (talk) 20:51, 18 October 2024 (UTC)