However, let's do something different. We take the same sets N and E. We know that N has the even numbers. So we pair the members of E with the even numbers in N. We can do that perfectly and with each member of E in bijection with the even number members of N. What now of the odd numbers in N? They have no matching counterpart in E.

Doesn't this mean N > E? — TheMadFool

Not it doesn't. Because it doesn't meet the definition of ">", here is the definition:

X > Y if and only if there is an injection from Y to X, but there is no injection from X to Y.

So when you show me two apples and say that they are one object, you might as well show me three or four apples, a thousand, or a million apples, and each time you are showing me the exact same object — Metaphysician Undercover

No that is not correct. If you show me three or more apples, the totality object would be some OTHER object. I just showed you two particular apples (those that I've bought today), and I asked you simply if by today when I bought them, do I have an object that is the whole of both of them, and I explained this object in terms of Part-whole relationship, an object such that each apple (that I bought today) is a part of it, and such that it doesn't have a part of it that is disjoint (doesn't share a common part) of both these apples. This object is the smallest object that has both of these apples as parts of. It is simply this object that I've asked you to tell me whether it exists or not. I didn't speak about a Naming category (like the one you've spoken about) nor did I pose the problem of changing material of a set. I said at the moment when I bought both of these apples is there the object that I've defined or not? To me this 'whole' or 'totality' object of these two particular apples, to me I say, is as concrete as the existence of each apple, and it is an object as each apple is an object. Of course this object can be ruined with time, as each apple can be, and actually only when one of the apples constituting it would start to ruin. But this is another question. My simple question is whether such an object exists in the first place.

You seem to think that you can randomly point to a bunch of objects, say "abracadabra", and suddenly there is another object, which exists as the unity of those objects you have pointed to. Sorry, but that's not reality. — Metaphysician Undercover

So in your sense if I bought two applies today, then I only have two objects, that is the apples themselves, there is no other object that is the totality of these two applies, i.e. the sum material of these two apples, i.e. an object such that each of these two apples is a part of it, and that doesn't have a part of it that is disjoint of these two apples. To my naive understanding, I see it obvious that there is that object.

As explained, this is absolutely false — Metaphysician Undercover

Well you seem to refuse tribe as an object, well this is a deep point. Anyhow to me a tribe, a herd, a bunch, etc.. all of those are objects, and they are well specified objects as long as each individual member of them is a well specified entity. Anyhow I don't think I can discuss refusal of such clear kinds of objects.

That definition of "||" is not so strict as you seem to think. You describe it as "the relationship || between tribes", as if it is necessarily a relationship between a plurality of "tribes". — Metaphysician Undercover

Hmmm.. I see the confusion here, OK, when I said a relationship between tribe(s), I only meant that each of its arguments is a tribe. That's all. It doesn't indicate plurality. It doesn't indicate that those arguments must be distinct from each other. For example identity is a relationship between object(s), it doesn't mean that the arguments of identity are distinct objects, of course not.

As regards false grouping argument of yours, and that tribes are not objects etc.... I object to this argument. A tribe is a well specified entity, it refers to the totality of specified individuals. And in the example I've outlined that each group of 50 men and 50 women that go and register themselves in the registry of the country as a tribe, then those would be called a tribe, so a tribe in this case is the totality of all those individual objects so registered. This is a well specified entity. Now whether || is not sensitive to "internal" relationships within a tribe versus "external" relationships, and that this would blur up the boundaries of tribes, etc.. argument of yours, this is not correct. Yes definitely || is not sensitive to internal x external relations between tribes, however this doesn't entail that the tribes are not well defined entity, it only means that the relation || is not sensitive to boundaries of tribes, that doesn't mean that the tribes don't have clear cut boundaries. Clearly each tribe is a well defined entity and what is external to it is very well demarcated, it is what is not a registered member in it, and what is a member of it and what is not, is well defined in the registry of that country. So each tribe is SHARPLY demarcated, and in that sense it is indeed an object, although a plural kind of object rather than a singular kind. It is this insensitivity of || to boundaries of tribes that cause it to be able to occur inside a tribe and for other tribes to be in-between them (outside each of them to the other), yes that's what cause it to be a relationship that can be between something and itself for some objects and also at the same time can occur between something and other things for some objects.

So we do have S||S, each of S represents the SAME tribe (which is indeed an object), and yet || is not a relationship of identity! So the occurrence of a same symbol on either side of a relation symbol doesn't entail that each occurrence must stand for distinct object.

The point is that the symbol "||" refers to a different situation in S||S than it does in A||B. Therefore the rule produces ambiguity in the use of that symbol, and the possibility of equivocation. If we assume that "||" has the same meaning in each case, we are deceived by equivocation. Therefore the rule is a faulty rule, and ought not be accepted. — Metaphysician Undercover

S||S is a particular case of A||B; also C||D when C, D are disjoint tribes is also a particular case of A||B.
To complicate the situation we may even allow for asymmetric gender partial Overlaps between tribes, like in saying there are tribes K, L where 20 men are shared between tribes K,L and say 12 woman are shared between tribes L,K, still we can get the same rule applicable to them! So there is a spectrum of possible overlaps, all those would be cases of A||B. Of course in each specific case there will be additional features that discriminate this case from others, for example the particularities of the tribes themselves, also the particularity of the number of actual marriages between the tribes, etc.. all of these doesn't matter, since they all meet the definition of ||. This is like variation in particularities of objects fulfilling a predicate, for example the predicate "is a circle", now not all circles are really a like, they might vary in their size for example, in their colors, etc.., that doesn't affect them all being circles. No equivocation at all. Similarly the relationship || between tribes has strict definition, and whenever that definition is met, then the relationship holds between the respective tribes, variations in particularities of individual actualization of that relationship are immaterial as immaterial is the size of the circle in meeting the definition of a circle. A circle is a circle whether its big or small in size, similarly A||B holds whenever tribes A,B fulfill the definition of ||, whether the actual marriages between the two tribes is 100, or 50, (or any other number in case of partially overlapping tribes). No equivocation at all. Equivocation might arose only when || is APPLIED in a manner that doesn't depend on the mere definition of it, that confuses different applications of || and attributes the same consequence to these as if they were the same, but that's something that has to do with APPLICATION of ||, and actually with a kind of non-careful application, i.e. an erroneous application of the relationship ||, it has nothing to do with the mere definition of || at all.

The whole matter began when I wanted to coin a relation that can exist between something and itself other than the identity relation! So the relation || as I defined in the example can occur between a tribe and itself, and also can occur between distinct tribes, so its not the identity relation. As far as the "application" of relation ||, there is no equivocation at all.

So identity is not the ONLY relation that can occur between something and itself.

But

Identity is the ONLY relation that can ONLY occurs between something and itself.

he phrase "is married to" will mean something different in "tribe A is married to tribe B", from what it means in "tribe S is married to tribe S". — Metaphysician Undercover

No! you are confusing matters. Notice my original statement:

RULE: For every tribe A for every tribe B (A || B if and only if for every male a of A there is one female b of B such that: a m b, and for every woman a of A there is one male b of B such that: a m b). — Zuhair

Notice the "if and only if", the above statement is a DEFINITION of "||". Notice that it was symbolized by another symbol from "m" which was given to marriage between individual.

Marriage between tribes (symbolized by ||) has NO meaning by itself, it is just a string of letters, the country gave it a meaning by the statement after the "if and only if" above. So you cannot say it leads to equivocation of meaning or anything like that, because its meaning is understood to be fully traceable to the specifications building it posed by the rule, in other ways that rule is a DEFINITIONAL RULE. Without it you have no meaning of tribal marriage at all.

In those rigid kinds of definitions, there is no room for equivocation or the alike. These are strict rule following machinery. Equivocation is out of question here.

You are refusing to acknowledge the equivocation in your use of "AND" in the rule. — Metaphysician Undercover

Yes, I'm refusing this. "AND" here is "logical conjunction", it specifically means a function from the truth value of each statement linked by "AND" to the truth value of the whole statement in such a manner that the truth value of the whole statement (i.e. the two statements linked by "AND", and "AND" itself) is positive (i.e. is true) if and only if both statements linked by "and" have positive truth values. So "AND" here has a specific role assigned to it, and that role serves as its meaning. It has nothing to do with the imagined mixed roles you are speaking about. So back again, all of what our rule is saying is that the following:

IF the statement (50 men of tribe A are married to 50 women of tribe B) is TRUE
AND the statement (50 women of tribe A are married to 50 men of tribe B) is TRUE
THEN the statement "tribe A is married to tribe B" is TRUE.

AND is specifically the logical conjunctive article, nothing more nothing less. Now let apply this to tribe S were 50 men of tribe S are married to 50 women of tribe S, we have the antecedent:

The statement (50 men of tribe S are married to 50 women of tribe S) is TRUE
AND
The statement (50 women of tribe S are married to 50 men of tribe S) is TRUE.

Then by the above rule it follows that

tribe S is married to tribe S.

There is no equivocation whatsoever!

What I'm speaking about is simple blind following of the rules, nothing more nothing less. "AND"
is given a constant meaning throughout applications of it whether A is B or whether A is not B.

No equivocation!

Your use of "AND" as a conjunction between the two expressions above provides the necessary ambiguity for your equivocation. "S is married to S" can refer to one situation only. Yet you use two distinct expressions. Since you allow that "S is married to S" represents the two distinct situations expressed above, the charge of equivocation is justified. — Metaphysician Undercover

Not it is NOT justified! Because we are using the "AND" in the GENERAL case of definition of marriage between any tribes A,B (whether A, and B are the same tribe or not), the general rule is:

IF
[50 men of tribe A are married to 50 women of tribe B
AND
50 women of tribe A are married to 50 men of tribe B]
THEN
A || B

I only applied that rule to the case of tribe S, where 50 men of them are married to 50 women of them and 50 women of them are married to 50 men of them as well. Just substitute S instead of A and S instead of B, and you get the conclusion S || S. No equivocation at all.

n other words, there is equivocation in the meaning of "S". Do you see this? "S is married to S" doesn't mean the whole tribe is married to itself, as consistency with "A is married to B" would imply, it actually means that half the tribe is married to the other half. Therefore each S in this case signifies half the tribe, whereas "S" was originally used to represent the whole tribe. — Metaphysician Undercover

Honestly I failed to see the "equivocation" you are referring to. "S" represents the WHOLE tribe, it represent all 50 woman and 50 men, i.e. it represents the collection of 100 persons, 50 of which are women and 50 of which are men, and this meaning remained consistently throughout the application, it NEVER changed at all. So I don't see any equivocation at all.

"50 men of tribe S are married to 50 women of tribe S,
AND
50 women of tribe S are married to 50 men of tribe S."

this completes all the required conditions for fulling "marriage" between tribes per the rules of that country. So accordingly the proposition S || S (i.e. S is married to S) is true.

ou have divided S into the subgroups MS and FS, and you ought to say that MS is married to FS. And now the mathemagician's shell trick of equivocation has been exposed. You claim that the same thing lies under each S, but in reality half of tribe S is under one S, and the other half of tribe S is under the other S. This is the only way to speak of S being married to S. This is verified from the fact that Q, the progeny of this union, is only half of C, the progeny of the union of A and B. — Metaphysician Undercover

No, the laws of the country doesn't specify a tribe of one gender, tribes can only be named if they have 50 women and 50 men. Notice the definition of marriage between tribes doesn't say what's the total number of marriages, so although you have 50 marriages between tribe S and itself, and 100 marriages between tribes A and B when they are different, still both cases are concealed by the laws, and both receive the same description of being "married tribes". The other point is that for the case of S and S each couple would given birth to 2 children one male and one female, and that would make the progeny tribe made of 100 people 50 men and 50 women and so would constitute another tribe according to the rules of the country, which is tribe Q. While each married couple of tribes A and B only give birth to ONE child, but totally they'll have equal amount of girls and boys. That's how the country breeds! Those are fixed game rules. So yes there are differences even in how the resulting tribe came into existence upon marriage of the tribes, but still the rules of the country are insensitive to those difference and thus collects them under the same parcel. So the nutshell is that we'll have the same treatment of S married to S as of when A is married to B, despite the inner differences.

Of course there would be some hidden details no doubt, but the point is that there are indeed hidden difference, but since the definitions involved are blind to those differences they would pass the same. Like when we say for example "MAN" this denotes a lot of grown up males, but there are still many differences but all fall under the same SHELL.

So definitely sets, tribes, etc.. do conceal differences, that's the point of them really.

as I said I'm not good with symbols, so I just get lost trying to figure out what you're saying. — Metaphysician Undercover

"Lost in symbols, hey!", me too really, I wonder if one can can get rid of that symbolic approach to mathematics and use instead of them understandable words within some rigorous language.

The problem is that in many instances, like in "4+4=8", it cannot signify the same object — Metaphysician Undercover

Yes, it can! If you followed by tribe example, you'll see that you can do that! consider 4 to be the name of the set of all four member sets, now you join any member of that set with a member of it that is disjoint form it, and the result of all such unions would be 8 member sets that would be collected in a different set named as 8. It is very similar to the tribe condition. If you have the patience (which I agree is difficult) to follow the whole example I gave, you'll see the analogy. It would be solved in a very nice manner through conceiving numbers as denoting "sets", well actually sets of sets.

Of course the concept of having sets of sets is not a nice concept and not easy at all, but it can be interpreted in hierarchical labeling of collections, but that's another story.

I just explained this. When the symbol "4" is used twice in "4+4=8", it must signify a different thing in each of the two instances, or else 4+4 would not equal 8. — Metaphysician Undercover

I understand the general difficulty in having both 4 symbols in "4+4" representing the same object, its indeed not that easy to fathom. I'll try to give here a situation were this can be understood. Its a hypothetical scenario to clarify that this can be the case.

Let's say we live in a country were people live in tribes, now each tribe exactly has 50 men and 50 women, and the progeny of each tribe are separated from their fathers to constitute another tribe, the law dictates that marriages must be fixedly arranged between "tribes" that is if a man of say tribe A marries a woman from tribe B, then all men from tribe A must marry woman from tribe B only, and the same applies for woman, i.e. if a man from tribe A marry a woman from tribe B, then every woman from tribe A must marry a man from tribe B. Now lets fix that when a tribe is married to another tribe, then the result is also 50 girls and 50 boys, and that those would be separated from the parent tribes and so constitute another tribe.

Now the country sets two kinds of descriptions, one is Predicative description, and the other is Functional description.

The predicative description given by the country is the predicate "||" to signify "is married to" and this occurs between TRIBES. While the predicate "m" is used to signify marriage between persons. So the general statement in a laws of that country is:

RULE: For every tribe A for every tribe B (A || B if and only if for every male a of A there is one female b of B such that: a m b, and for every woman a of A there is one male b of B such that: a m b).

Now we have the situation: A || B to mean tribe A is married to tribe B (according to rules above).
Now this is a predicative formulation, why, because A||B is a "proposition", it something that can be true or false, and the symbol || is denoting a "binary relation", so it is a "predicate" symbol.

Notice that we can have the situation were tribe S can marry itself!!! so we can have S || S
Notice that S occurred twice in the proposition "S || S" but still it denotes ONE object, although this object is a totality of many individuals, however that whole of many individuals is considered here as one object. So repeated occurrence of the symbol symbol in an expression doesn't denote different denotation, no here S repeatedly occurred in "S || S" but it still carries the same denotation, namely tribe S.

This also shows that we can have a binary relation between something and itself OTHER than identity, for the expression S || S could have been false? while S is S is always true! Of course this is understood, for example we can have " \not [Sarah hate Sarah]" this is an expression having two occurrence of a symbol that is "Sarah" and yet it refers to the same object, and the binary relation between them that is "hate" can be negated (i.e. its negation is the true statement). And clearly the relation "hate" is not the same as the relation "identity", although it can occur between objects and themselves.

Now the country further uses the following notations to express "is the progeny of tribes", that is:

"P (A || B)", this is read as "The progeny tribe resulting from marriage of A to B"

Notice that expression "P(A||B)" is a "denoting" expression, it denotes a TRIBE. So the expression "P(A||B)" is NOT a predicative expression, since it clearly does NOT constitute a proposition, it is not something that can we can say of being true or false. "P(A||B)" is denotative and not declarative.

But we need a declarative statement "i.e. a proposition, or a predicative expression" about what that denotative expression "P(A||B)" is about? Here were "=" will trip in, to complete the picture and turn it into a proposition. Here the country stipulates:

P(A||B) = C

Now this is a proposition, it is say that the progeny tribe of tribe A married to tribe B , is , tribe C.

Notice here that in that country tribe S is married to itself, and it resulted in tribe Q, so we'll write that as:

P(S||S) = Q

Now we have two distinct occurrence of the symbol S on the left, but still it has the SAME denotational coverage! Both symbols of S denote the same object that is " TRIBE "S" ".

So we can have the same object undergoing some process with ITSELF to resent in other thing, like what happened with S.

The problem with expression 4 + 4 = 8 , is that it in some sense "hides" information, it should have been written as: R(4+4)=8, to mean "the result of adding 4 to 4, is, 8", that would have been more informative. Anyhow mathematicians and logicians shorten that to just 4+4, but what is actually meant is R(4+4). The expression 4 + 4 is deceptive, it gives the impression that "+" is a binary relation occurring between what's denoted by symbol "4" on either side of it, as if it is declaring that "4 is added to 4", which is not what's intended, the foundational mathematicians stipulate "+" as a two place function symbol, and they mention it in the rules of the language, which are often not written explicitly in many contexts, and so it would be considered understood that when they write 4+4 then they mean a denotative expression and not a declarative one, and that 4 + 4 actually means "the result of adding 4 to 4". Anyhow.

Of course you can object to the notion that A,B,S,etc.. here when used to denote "tribes", then they are not actually denoting "individuals", and of course that is correct, they are denoting "multiplicities", but still when B is used it always denote the SAME multiplicity. Whenever we hear B the specific 50 men and 50 woman in that country that were recorded under name "B" would come up into our minds. So all occurrences of B have the same denotational value! or lets say "coverage". IF we accept a totality of multiple individuals as ONE object that is the sum object of all of those, then B would be said to denote ONE object along that understanding.

Logicians recognize this when they practise the laws of logic using symbols which do not stand for anything. They know that using such symbols is just an exercise to help them learn the laws of the system. But for some reason, mathematicians like to say that such symbols actually stand for objects (Platonic), things that they call numbers, and such. But we all know that such objects are just imaginary, and have no real existence whatsoever. So we ought to recognize that these mathematicians are just fooling themselves, claiming the real existence of non-existent imaginary objects, immersing themselves into this fantasy world which the paper calls "model theory". — Metaphysician Undercover

Well, you definitely have some point of view here. But model theory is not altogether useless. It's easier to understand mathematical theories as speaking about abstract models, since those models do not contain properties that the mathematical rules do not entail by themselves, so its prudent to say that 1 + 1 = 2 is speaking about some process working on some abstract objects, since there is nothing in 1+1=2 to confer additional properties to what 1 and 2 represent. Yes one can certainly use the above rule in applications like in adding an apple to another to get two apples, but the properties of Apple like it having a seed for example, a DNA, etc.. all those are particulars that are not inferred from 1+1=2, so we need to abstract away those properties. Moreover if we speak in the strict formal sense then 1+1=2 can stand by itself as a syntactical game prior to any application, and so the abstract model of it would indeed provide nearer semantics to the formal essence of 1+1=2. Platonism is the easiest way to go about mathematics. But of course that does't necessarily entail that its true. Indeed as you suggest mathematical statements has their value and probably "justification" in being applied to something external to them, something that is not arbitrarily chosen. So their semantics might break down to their multiple applications, each at a time. From the philosophical point of view this applicative reduction might look more prudent, but from the pure mathematical point of view, definitely platonic models would be preferable, since they are more direct engagements of what those mathematical statements are saying.

This is a good reference: Fundamentals of Model Theory.

Why would you add when you can delete and get the same result? — TheMadFool

Plain deletion is a partial answer. You won't get the same result! dichotomous optimistic realism adds hope and value, something that just deletion won't confer.

I don't know if you can call it a fact. It makes sense, much like evolution does. — god must be atheist

Evolution is a fact. But what you've mentioned is a conjecture. Is there strong evidence to support that energy levels will even out?

The sad truth is that we do. Entropy will nullify all movements in the entire world. — god must be atheist

Is this a FACT of physics? or just a hypothesis?

Mathematics in general does not require a "foundation" at all, and certainly need not be treated as Platonic, as if its objects "exist" in some immaterial realm. — aletheist

Yes, I agree to that. Anyhow, my point was that if one wants to understand applicability of formal systems, then definitely you'll need people outside of math to consult. The reductionist approach (usually called foundation) is nice technically, it can guild development of pure math. But I think guiding development of applicable mathematics is by far much more precious. Just an opinion.

That is not the business of pure mathematics, but of applied mathematics within all the other sciences--including philosophy. — aletheist

Yes! I'm speaking about those of course. I'm not speaking about tracing the bulk of mathematics technically into one MATHEMATICAL system, which is often called as foundation, that's not enough, I'd better call that as Reductionism. For a true foundation of mathematics the philosophers and various other disciplines must have their say in thinking whats the best foundation for the kind of mathematics that would have applications. In other words what's desired is a foundation for applicable mathematics, and not just pure platonic mathematics.

Evolution indeed can be seen as a function of this movement, but eventually entropy will take over, and everything will end up in a quiet, luke-warm, uneventful and smooth movementless world. — god must be atheist

We'll, I think the "real answer" to that is that WE DON"T KNOW. So in the face of our ignorance of what would happen, we are free to be optimistic and even hope for a dichotomous natural processes behind good and evil that produces a natural evolutionary process that ends with the triumph of the GOOD! The dichotomy about natural good and evil I think is important otherwise we'll end up in mythical accounts about ethics and worst confused ones where good produce evil and the irony it said to be 'wisdom'? Quite ridiculous!

I generally agree with that. But if we want the kind of mathematics that would be applicable, we'd better link it to the world.

equality is a "qualified" identity — Metaphysician Undercover

Yes!!!

Here, in set theory, identity is taken for granted, so what it means to be "the same" is left in the realm of the unknown. — Metaphysician Undercover

Yes, I agree. I generally agree with ALL of that posting really. And sorry for confusion about your stance from the conventions, I see know what are you trying to do, but I honestly see that for one to decipher those hard subjects, then one must read at least some of the conventional work done by foundational mathematicians on that. But again that's fine. Also I agree that concepts like 'identity', 'set membership' and even 'natural number' , and 'part-whole', are all very hard concepts when one try to dig down into their basis. Not easy at all. Nice correspondence!

I should add, that I'm really amazed by the last posting. Really strange. In this posting you appear to know exactly the official stuff about identity, and in-depth really! So your account was excellent. And the correspondence was indeed fruitful (at least for me). Unfortunately your account on other aspects of the syntax of first order logic and of arithmetic like not knowing that 0,1,2,.. are CONSTANTs, and that those are terms of the language, i.e. symbols denoting objects, and that the expression 1 + 1 is the value of the function + on arguments 1 and 1, and thus 1 + 1 being a TERM of the language also is denoting an object (besides the objects denoted by the two 1's in it), etc.. Also you not discriminating between a predicate (relation) symbol and a constant symbol, so you thought that 0,1,2,.. are held conventionally as PREDICATE symbols (although one can indeed make a formalization that can interpret them as such, but this is not desirable, and definitely not the convention), those aspects of your response were really very poor, and reflects great shortage of knowledge regarding the common conventions held by foundational mathematics regarding the main logical language which is first order logic and one of the most formal languages that are directly connected to mathematics, that is the first order language of arithmetic. Anyhow your account on equality was very good, I hope your knowledge increase one day about the syntax of first order logic, and of Peano arithmetic and set theory, etc.. so that we can have correspondence would be by far more fruitful and productive.

Why not just admit that the principle is not a principle of identity, but a principle of equality, it doesn't have the strength which you desire it to have, and get on with the use of the system, understanding that it has its weaknesses, instead of trying to hide its weaknesses and disguise them to create the illusion of strength? — Metaphysician Undercover

OK, what you are saying in this last posting is understandable, I in some sense agree with most of it.
There is something nice in your conception about 'equality', you view the substitution schema to mean 'equal' treatment given by the theory to the related objects, and not as indiscriminability which is the synonym of identity. So equality is not just an equivalence relation, but also a substitutive equivalence relation. That's nice. But I'd say that this is very near to identity, since WITHIN the theory they are seen "identical" [and not just being equivalent], but outside the theory that can be discriminated, so you want to give a term that describes what's going inside but at the same time alerts us that this is not necessarily what's going outside! That's fine. No problem. But again I would consider such a kind of "equality" relation far stronger than just being an "equivalence" relation, i'd consider it as some kind of quasi-identity relation, i.e. some equivalence relation that is the nearest possible relation to identity that the theory in question can describe. I agree that to be on the safe side, it's better to term it as "equality", although I still maintain that the primary intention was to capture "identity" that failed for first order theories "externally" [but not internally]. I'll officially use "equality" but I'll metaphorically use "identity", because sometimes "equality" can be understood as merely an equivalence relation, which is way weaker than what you are describing here.

The substitution axiom allows that two distinct things, with differences between them, which don't make a difference to the purpose of the logician, may be substituted as equals. — Metaphysician Undercover

No! Unless these differences are indescribable by formulation of the language. Once you are in a logical theory then what decides identity of something in it should be in relation to what the theory can describe. Indescribable difference are immaterial inside the theory, and the two objects would be considered identical by the theory because it cannot discriminate between them by its language, so it considers them "IDENTICAL", it sees them as identical (not just equal). The substitution scheme says that if we have x = y then whatever is true about x is true about y and whatever is true about y is true about x, which mean that "all equals are identical"! More precisely speaking all equals are indiscriminable. "Equality" in the sense of being just an equivalence relation, has nothing to do with this principle, why should it demand something like substitutivity? for example bijection (i.e. equality of sizes of sets) is an equivalence relation and a logical theory do discriminate between bijective objects. Clearly there is no need to demand something like the full substituition principle if we are just thinking of the equality relation as some equivalence relation. But if we are thinking of equality as indiscernibility and thus "identity" from the inner perspective of the theory, then we'd add such a strong principle. Of course you can raise the point that this is just indiscernitibility of identicals, while the other direction which is identity of indiscernibles is not granted by the substitution principle, which is true of course, but that is only because of the weakness of first order logic. So the intention from the first order logic point of view is that first order identity theory (reflexsive + substitution axioms) that its trying its best to capture the *identity* principle.

I just wanted to add, that first order identity theory does not allow adding to it objects that can obey the substitution principle and yet be non-identical. So it in some sense does imply identity of indiscernibles. For suppose for a proof by contradiction that we can add to identity theory two primitive constants x,y, and suppose we add the schema

x =/= y and [phi(x) \iff phi(x|y)]

in other words x is distinct from y and yet they are fully substitutive.

This cannot be because: let phi(x) be the formula x=x, now let phi(x|y) be the formula x=y, i.e. substitute only ONE of the occurrence of x in x=x by y, then we'll have

x =/= y and [x=x \iff x=y]

by modus Ponens we have

x=x
x=x implies x=y
-------------------
x=y

then we'll be having: x=/=y and x=y, A contradiction!

This assures that first order identity theory does speak of = as identity and not just some equivalence relation.

Are you still unwilling to accept a difference between equality and identity? I thought we agreed to that difference a long time ago. — Metaphysician Undercover

Equality is in the least sense an equivalence relation and that's it, it doesn't necessarily satisfy the substitution schema. However when logicians are speaking about equality in the sense of satisfying the substitution schema, then in reality they mean "identity", so for example if you formalize peano arithmetic on top of axioms for = that are the reflexive axiom and the substitution scheme, then in here what you mean by = is exactly "identity" and 2 + 2= 4 exactly means the OBJECT denoted by expression '2 + 2' is the object denoted by expression 4. If you don't formalize Peano arithmeetic on top of substitution scheme but you keep the reflexive axiom and add to it symmetry and transitivity axioms, then here you are using a weak notion of equality which is just some non-specific equivalence relation, here the substitution scheme doesn't work because its not there, and you can have the object denoted by 1+1 not being identical to the object denoted by 4.

Most mathematical logicians would like equality to be interpreted as identity because the substitution scheme makes matters easier.

And again and again symbols like 0,1,2,3,... all these symbols are CONSTANTs of the language, they are defined constants (except 0 which is a primitive constant symbol) they are TERMS, they are in reality zero place function symbols (i.e., function symbols without arguments), each of those denote ONE object in the universe of discourse. = is a relation symbol, and + is a two place function symbol, which is in turn a process that sends object(s) referred to by two occurrences of symbols to an object referred to by a third expression. And it is always the case that functions and relations are associating objects denoted by terms of the language.

That's the convention, it is simple and crystal clear, it is not involved in any contradiction at all.

Arithmetic is in the LEAST just an game played with empty symbols, and so symbols are just subjects to the rules of the game that's it; in the MOST they are about something other than those symbols, like about some platonic world of mathematical objects.

Mathematics as a whole range between strict formalism to outright Platonism.

I personality hold that the formalist side is the purest form of mathematics, when mathematical systems are trying to capture some particular subject matter, here is it becomes what I call as "applied mathematics" in the most general sense, and most mathematics is coined as applied mathematics to a platonic realm, however usually those are considered as pure forms of mathematics and what is considered as applied mathematics is reserved to their application in empirical sciences.

Good Luck!

this is also a good book. Look how does he use "identity" interchangeably with equality.

https://www.math.uni-bielefeld.de/~frettloe/papers/wikibuch.pdf

Why would I want to waste my time doing that, when I find inconsistencies and contradictions in the conventional interpretations of the very first principles? — Metaphysician Undercover

What? that's really strange. You need to first read it and then know about it then you should decide whether its worthy or not. You need to get a good book on mathematical logic like Mendelson's, or Shoenfield's, or Suppes's logic, then you can read Peano arithemtic, and then Set theory. You need to read them carefully, solve the exercises , etc.. it is not something that you'll manage to know on one glance or so.

After you manage to learn about mathematical logic, Peano arithmetic and Set theory, then you can start discussing matters about them, or matters that they are used to be a foundation of, which is most of known mathematics. Otherwise the discussion would be really very poor.

We agree then, that there are no objects denoted by "2+2=4"? On what basis would you claim that "2+2" is identical to "4" then? — Metaphysician Undercover

No of course we don't agree. The usual formulation is for 2 to be a constant (zero place function symbol), that's the usual convention. Now there are some second order logic theories that can interpret arithmetic in a manner that 2, 1, etc.. are predicate symbols, but those do have equality of predicates axioms in them. I'm not willing here to discuss these versions because they are un-important. The usual ones especially for peano arithmetic is for 1,2,.. etc to be constants and so they are terms of the language denoting an object in the universe of discourse. You need to read PA and first order logic very well. from my discussion with you, it is clear that you are so ignorant of even very well known conventions. You are simply discussing matters that you don't know much about, for example you don't know that constants are zero place function symbol, which tell a lot about how much you know of commonly held syntax. You need to read some foundation of mathematics book, then you can come a speak about it.

The one which fishfry steered me to, the axiom of extensionality is clearly stated as an equality axiom. So if it is taken to represent identity, I think that's a faulty interpretation. — Metaphysician Undercover

I wrote to you the identity relation in ZFC. I already wrote that explicitly it is the reflexive and substitution axiom schema, those are the identity theory of first order logic, and ZFC is *usually* formulated as extension of the rules of first order logic and those axioms of identity theory. However *sometimes* ZFC defines "identity" in the following manner "it also uses the symbol = to mean "identity" and not equality"

Define (=): x = y \iff \forall z (x \in z \iff y \in z)

However, this approach is not favored a lot, the majority would define axioms of ZFC as an extension of identity theory (i.e. the reflexive and substitution scheme).

This is what I think. In the expression "2+2=4", the "2", and "4" symbols are predicate symbols — Metaphysician Undercover

No this is wrong! It seems you didn't read it well, the expression 2, 4 are called zero placed function symbol, or simply constants, and those are TERMS of the language and they denote objects. That's the usual presentation in Peano arithmetic and most mathematical system. However, we CAN formalized 2 and 4 as predicates that's not a problem at all, this can be done. But it is not the usual thing.

The law of identity is the philosophical principle which states that a thing is the same as itself. In mathematics there are theories of equality, and perhaps axioms of equality, but these are not laws of identity. — Metaphysician Undercover

hmmm...., let me think about that, I'm really not sure if "identity" really arise in mathematical system per se. But if you consider first order logic as a kind of mathematical system, since it is a part of "mathematical logic", then of course there is a theory about identity. I'll speak about formal difference between an axiomatic theory of identity and an axiomatic theory of equality if that helps.

An axiomatic theory of equality basically presents equality as an equivalence binary relation, it basically contain the following three axioms:

1. Reflexive: For all x (x = x)
2. Symmetric: Forall x,y (x=y implies y=x)
3. Transitive: For all x,y,z (x = y and y=z implies x=z)

That's all.

Now an axiomatic theory of "identity" stipulate identity as a substitutive binary relation, most of the times it uses the symbol "=" to signify "identity" and not just equality, it basically contain the following axioms:

1. Reflexive: For all x (x = x) [in English: everything is identical to itself]

2. Substitution axiom schema: if phi(x) is a formula in which the symbol x is free and never occur as bound and in which the symbol y never occurs, and if phi(y|x) is the formula obtained by merely replacing some or all occurrences of the symbol x in phi(x) by the symbol y, then all closures of:

for all x,y (x=y implies [phi(x) \iff phi(y|x)]), are axioms.

That's all.

So the axiomatic systems of these two notions are clearly different!

Now identity theory proves all axioms of equality theory, but the converse is not true, i.e. every two identical objects are also equal to each other, since identity relation is an equivalence relation, but not every equality relation is an identity relation. For example the relation "bijection" which means that two sets can have their members linked to each other in a one to one manner, this relation is clearly an example of an equivalence relation (i.e. equality) because it fulfills the three axioms of equality theory, yet bijection is not the identity relation, since we can have two distinct sets that are bijective to each other, and so it violates the substitution scheme.

Identity relation simply states sameness of objects, and it does that by fulfilling the above axioms of identity theory especially the substitution axiom, which mean that any description of the object x also holds if we replace the symbol x and the symbol y, i.e. it is also a description of the object y, and vice verse, so the identical objects are said to be indiscernible from each other using our language, i.e. relative to our language they appear identical.

However in mathematical systems and even in logico-mathematical systems like PA, I see = axiomatized after equality theory! However in more deep formal systems like set theory and Mereology the = symbol is usually taken to represent "identity" and not just equality, and usually ZFC and Mereology are formalized as extensions of first order logic with "identity" rather than with just equality, although most of the time these terms are used interchangeably in set theory and Mereology but vastly to mean "identity" and not just equality, since the axioms about them are those of identity theory and not just of equality theory.

. This would require showing how this string of symbols "2+2" denotes the exact same abstract mathematical concept as this symbol "4". — Metaphysician Undercover

In set theory, yes that is the case. But of course you can reject set theoretic interpretation of arithmetic. But fishfry is saying the truth about set based implementations of arithmetical concepts. In set theory the symbol 2 denotes the von Neumann ordinal { {}, {{}} }, also the expression "2 + 2" is a functional expression that denotes the von Neumann ordinal { {}, {{}}, { {}, {{}}}, {{{}}} }, and also the symbol 4 is taken to represent exactly the set { {}, {{}}, { {}, {{}}}, {{{}}} }. This follows from the definitions given to those symbols and ZFC, and of course the inference rules and axioms of ZFC. Those are the usual interpretations held in ZFC.

o in your example of "2+3" we have an object denoted by "2" and an object denoted by "3", and the process, "addition" denoted by "+". There is no result of this process denoted, and therefore no third object denoted, just the process. Perhaps the third object you had in mind is "5"? — Metaphysician Undercover

No! There is! Please see my message that just precedes this, where I've pin pointed where is that object and I showed you all denotations involved. But to just reply to this here. I'll say:

In "2 + 3" we have an object denoted by "2" and an object denoted by "3", and the process, "addition" denoted by "+", and also we have an object denoted by the total string "2 + 3" itself. I didn't mean 5 at all, since 5 is not shown in the expression "2 + 3". The reason is because "+" is stipulated by the rules of arithmetic and underlying logic to be a FUNCTION, and by rules of the game any function symbol if written with its argument 'terms', then the whole expression of that function symbol and its argument would be denoting of an object. We don't have any mentioning of 5, yet, it is the rules of arithmetic that later would prove to you that the object denoted by 5 is equal to the object denoted by "2 + 3". Remeber equality is a relation between OBJECTs.

I've never seen it stipulated that the "+" is a binary function symbol. Nor have I seen it stated that when a binary function symbol is used with two terms, that the whole expression must be taken to represent one object. That such and such convention interprets things in this way does not mean that this is a fixed rule of mathematics — Metaphysician Undercover

Those are present in Peano arithmetic in a very clear manner. You can review a full treatment of them. That they are not fixed rule of mathematics, might be, but they are fixed rules of first order logic that function symbols represent an object and these can take complex form and not just the constant or the unary form.

I agree that "+" cannot denote a relation. It must denote a process, or function, as you call it. But I disagree that it signifies "the result of addition", it signifies the process of addition itself, not the result of the process. — Metaphysician Undercover

Of course + denote a process, more precisely a "function", of course it doesn't denote the result of a process, I never said that. What I'm saying is the whole expression of "1 + 1" is what is denoting the result of a process, and for that particular string it denotes the result of adding 1 to 1. I'm not sure if you are understanding what I'm saying. I'll try to break it down for you, take the above expression, i.e. "1+1, lets take its parts and see their denotation abilities:

The first symbol 1 : This is a denoting constant (i.e. a zero place function symbol), it denotes an object.
The "+" sign: this doesn't denote an object, but it denotes a process, more specifically a "function"
The second symbol 1: This is a denoting constant, it denotes an object
The string 1 + : This is an "incomplete expression" it doesn't denote an object
The string +1: This is an incomplete expression also, not denoting an object.
The string 1 + 1: This is a functional expression: IT DENOTES AN OBJECT.

So '+' denotes the process of addition itself, but "1 + 1" denotes the object that results from applying the process of addition on two "1" symbols.

It is definitely a rule of the game in logic that the total expression of 1 + 1 (i.e. the three symbols in that sequence) is denoting an object, that's definite, because it represents the result of a functional process. You cannot change this. This is NOT an interpretation of the symbols, to say that they are illogical, equivocal, erroneous, NO! It IS a rule of the game of arithmetic and the underlying logic.

And it makes full sense, because equality is a "binary relation" symbol between OBJECTS, so the symbol for equality, i.e. "=", must LINK two term symbols, i.e. two symbols that denote objects, otherwise the syntactical expression won't copy the semantic content. so you have the structure

"term - binary relation symbol - term"

Here we have 1 + 1 = 2

so = is linking the expression '1+1' to the expression '2', so '1+1' must denote an object. Otherwise the whole expression would be meaningless, it would be equality between what and what?

I tend to think (I'm not sure) that you think that 1 + 1 is an "instance of the process of addition" i.e. the process of addition is itself a big process it doesn't only work on 1's but also on any two naturals, now "1 +1" is just one instance of this addition process. It appears to me that what you have in mind is the following interpretation:

"instance of a process - binary relation symbol - term"

so here possibly you are thinking that 1 + 1 = 2 means that

(the process of adding 1 to 1) = 2

And that this would be a proposition that defines 2 by it being equal to an specific instance of the process of addition.

If I got you right, I think this interpretation can possibly work, I'm not sure, but definitely it is against known rules of the game, because relation symbols link terms and not expressions denoting process instances to terms, unless the "process instances" are understood as some kind of objects? and so expressions denoting them would be "terms"! Well that would be another issue.

In nutshell, the conventional way of understanding 1 + 1, well lets say in first order logic extended with the symbols of arithmetic, is for it to be the object that results from the function + operating on the two 1 symbols.

That + is a function, see this

So the third is the one which needs to be justified. How does "1+1" denote a single object? — Metaphysician Undercover

I read the whole of your message about why symbols must denote other objects etc.. Its nice and very DEEEEEEEEEEP, indeed; and I won't differ with you about those points (nor necessarily fully agree with them). I'm concentrating on the conventional terminology specially the syntactical ones.

To many mathematicians (well at the least those who stick to formalism) they are willing to understand mathematics as just a game of empty symbols because that would figure out the deductive streaming in those syntactical games we call as mathematical systems. So we have games with specified rules, we cannot go against them because these rules are fixed, and all of what's in the game is in reality "subject" to these rules.

Now one of the rules is to regard an expression like 1 + 1 to be denoting a single object. There is no justification for that at all, it is a rule of the game, you might object to it being unsubstantial, etc.., yes you may, but that won't change anything, the rules of arithmetic stipulate that 1 + 1 IS a functional expression, and a functional expression always denote an object, whether it does it in a substantial manner (i.e. denote an object other than itself) or whether it does it in vacant manner (like it being denoting itself), it doesn't matter, in both scenarios its a fixed rule of the game that 1 + 1 refers to some object.

The reason is because the "+" operator is stipulated before-hand to be a primitive "binary FUNCTION symbol" And by fixed rules of the game of logic and arithmetic when an n-ary function symbol is coupled with its n many arguments in a formula (which must be terms of course) then the *whole* expression is taken to denote some object (that is besides the objects denoted by its arguments which are shown in the formula). So binary functional expressions for example have 3 object denotations and not two as it appears, in general any n-ary functional expression has n+1 object denotations, although you will only see two objects (arguments) written in the expression. For example lets take the successor function S of peano arithmetic, here S(2) means "The successor of 2" which is a functional expression, you have 2 denoting an object and you have the whole expression i.e. "S(2)" also denoting an object, which is equal to 3. But you don't see it in the expression. Actually this is the main difference between "functional" and "relational" expressions. In a relational expression you see some n-ary relation symbol with n-many terms and this would be a proposition, and there is only n many object denotations, it would be a proposition because its speaking about a relation between those n many objects. While with functional expressions you'll also see an n-ary function symbol with n-many terms, BUT this won't be a proposition, because its not speaking about a relation between n many objects, no, actually its referring to an (n+1)_th object linked to the n many objects symbolized by the n many term symbols you see in the expression, so the total number of denotations is n+1 many object denotations and not n many object denotations as it is the case with relational symbols.

If 2 + 3 was denoting a relation between 2 and 3 and that's it, then it would be a proposition, because either 2 has the relation + to 3 or it doesn't have it, one of these two situations must be true, so it would denote a proposition, but clearly this is NOT the case, we don't deal with 2 + 3 as a proposition at all, we don't say it's true or false, so 2 + 3 must not be something that denotes a relation occurring between two objects, so what it is then? by rules of the game 2 + 3 is short for "the result of addition of 2 to 3" that's what it means exactly, and so 2 + 3 is referring to an object resulting from some "process" applied on 2 and 3 and that process is addition, that's why we call it as a functional expression, because its there to denote something based on a process acting on its arguments, and not to depict a relation between the two objects denoted by its arguments.

I think that's the best I can do in explaining common usage of these expressions.

I'm only explaining the rules of the syntax and its relation to denotations, which is something controlled by the rules of the game. See: rules of syntax of first order logic

You need to review these rule for yourself and see whether I'm telling the truth or not, since I know that you don't trust me!

Mathematics is supposed to be more rigorous, requiring that an object be represented. It is intrinsic to mathematics that objects be represented because if no objects are represented the distinction between numbers is meaningless — Metaphysician Undercover

This is indeed a plausible stance! But I think formalists won't agree. I still think that we can have distinctions between numbers even if they are meaningless. There are still meaningful matters that the formalist would hold to, that is the deductions carried in the system, those are non-trivial pieces of knowledge. But again it would be difficult for the formalist to account for the success of some mathematical disciplines in science and various applications, that's where it hurts when it comes to formalism.

If we bring this object into a logical operation, it is now a subject. It is a subject because we can move it around at will, use it as we please, it is subject to the will of the logician who uses it. What the subject "denotes", is dependent on how the logician uses it. and this is determined by definitions. As denoting something, the subject is a symbol, and it may denote anything, object, relation, etc., but in logical proceedings it need not represent anything.. — Metaphysician Undercover

That's a fantastic explanation of Formalism. I know that you don't like it, well, but by the way its really a nice account explaining my intentions. Yes the whole of arithmetic can be interpreted as just an empty symbol game, and saying that a symbol represent itself is next to saying that it is not representing anything, I agree. You may say an empty symbol is not a symbol, well its a character and that's all what we want, we may call it as "empty symbol", its a concrete object in space and time (even if imaginary) and it serves its purpose of being an "obedient subject" to the wimps of logicians and mathematicians. I really like it.