## What is the difference between actual infinity and potential infinity?

• 126
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.

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.
• 6.3k
No! Unless these differences are indescribable by formulation of the language.

That's the key point, the limitations of the language. The law of identity puts identity of the thing within the thing itself, such that even if the human being cannot discern the differences (due to deficiencies of sense, language, whatever), but can still recognize two things as distinct, we can say that the two are distinct. Therefore the law of identity represents a recognition of the limitations of the language system, the inability of the human being to adequately identify certain objects.

When we approach mathematical axioms with the recognition that equality is not identity we uphold this principle which represents the limitations of the language. If we ignore this principle, and insist that equal things are identical we become ignorant of the limitations of the language, and we will start to believe that mathematics is capable of doing what it is not capable of doing. Belief that a tool is capable of doing what it is not capable of doing is a dangerous belief.

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).

Sure, within the theory there is no difference between the two objects. But in application, and theories are useless unless applied, there is a difference between the objects which the theory is applied to. Because of this, within the theory the two objects are said to be "equal". Therefore the rules of the theory recognize that the two objects are not actually the same, and express this recognition by using the word "equal" and not "identical". But within the theory, the objects are treated as if they are identical and this is a deficiency of the theory. If we ignore, or even deny this deficiency, we are in a world of self-deception.

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"!

Whichever things that are said by the premises to be true about x are also true about y. But that does not mean that the two are identical, it just means that they are treated equally by the theory.

. 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 add a strong principle, but a strong principle which is false (as yours clearly is) is a deceptive and dangerous principle.

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.

You see why I claim there is contradiction in the very first principles? What point is there in making exceptions to the first principle, because you know it is wrong? 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?
• 126
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?

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.
• 126

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.
• 6.3k
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.

OK, let's say equality is a "qualified" identity. This means that it is a relative identity. In relation to the specified theory, the two objects are identical when they are said to be equal. But we all know that they are not really identical, that's an artificial simplification which is theoretical only.

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.

What would be the point of this though, really? Let's say that this "quasi-identity" relation is "the nearest possible relation to identity that the theory in question can describe". How near this equivalence relation actually is to true "identity", would be completely dependent on the theory's capacity to describe. And unless we had some way of determining true identity, and comparing the identity produced by the theory, we would never know the theory's capacity, or how close the quasi-identity is to true identity. And if we had a way of determining true identity why would we be using the theory which employs quasi-identity.

The real issue I think, is what I explained to fishfry earlier. The purpose of equations in mathematics is to compare similar things in an attempt to determine the differences between them. So we find all the ways in which they are the same, "equal", and we are left with the differences. If the right and left side of the equation both represented the very same thing, then there would be no difference between the things represented, and the equation would be useless.

So I don't think that equality even aims for identity. If the two equal things were really identical, then we wouldn't be employing equality to determine this. We employ equality when we know that the things are different and we want to understand the differences between them. That's why the principle of identity is actually completely different from the principle of equality. But if we had some way to quantify the difference between "identical" and "equal", then we'd have the basis for accurately determining the difference between equal things.

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)

Let me refresh your memory. I didn't say that apprehending those symbols as predicate symbols is conventional, I said it's what I think, meaning it's the way that I see them. That's my interpretation, not the conventional interpretation. As you may have noticed, I don't see things in the conventional way.

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.

Thanks for the encouragement Zuhair, but following common conventions is really not what I enjoy, I find that rather boring. So I like to look for those bits of meaning which are omitted by the conventions. Generally, they are omitted because they are what's taken for granted. But what's taken for granted, is left as an unknown, like when the religious take God for granted. So for instance, Newton's laws of motion take inertia for granted, so what inertia is, its nature, is left without an approach, and it remains in the realm of the unknown. 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.
• 126
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.

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!
• 126
equality is a "qualified" identity

Yes!!!
• 6.3k
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.

OK, I've reconsidered. I recognize that making stupid comments about conventions which one is totally ignorant of is not good for a person's integrity, so I think I will take some time to educate myself on some of those basic conventions you've referred to. Thanks Zuhair.
• 848
I feel badly misunderstood, but hey, this is the internet...

I misunderstand many things. My apologies if that is the case in this instance. LOL at the Annie Hall reference, one of my faves.
• 848
This is the false premise you stated:
1.1 We have the law of identity that says that for each natural number, it is equal to itself.
— fishfry

That is not the law of identity. The law of identity is the philosophical principle which states that a thing is the same as itself.

You deliberately re-quoted exactly the line that I apologized for, explained as a typo, and corrected in my previous post. Why? You do know you're strenuously arguing against a typo for which a correction has already been issued, don't you?

You know this is a philosophy forum don't you? So it's likely that you should expect that we are discussing a philosophical issue. If you want to discuss a mathematical issue, maybe a different forum would be better.

Weren't you the one who originally made a mathematical claim, which I am refuting?

You claimed that in ZFC they misuse the law of identity in some way. I challenged you on that and you have not produced evidence.

YOU are the one who made a strictly mathematical claim about ZFC. And who can't defend it with facts.

Beyond your factual incorrectness, I found this a very patronizing and hostile remark. Did I misread it?

You made a specific claim about ZFC, an abstract mathematical system. I challenged you on your mathematical claim. You then say I have no right to talk mathematics? What kind of low-end game is that?
• 6.3k
You deliberately re-quoted exactly the line that I apologized for, explained as a typo, and corrected in my previous post. Why? You do know you're strenuously arguing against a typo for which a correction has already been issued, don't you?

It couldn't have been a typo because you continued afterward, to make the same mistake. Look:

I know of no instance in which mathematical equality is anything other than set identity and logical identity.

You claimed that in ZFC they misuse the law of identity in some way

No, I said that equality in ZFC is not based in the identity of the law of identity. I explained why this is the case.. You insisted that the equality of ZFC is based in identity, so I asked you for a citation of a law of identity which ZFC is based in. I'm still waiting.
• 3.8k
This one is so good.

• 848
It couldn't have been a typo

I see no good faith on your side (claiming my typo isn't a typo??) but I'll stipulate that you disagree. I'm done with this thread. You should, in the fullness of time, go back to the detailed proof from the Peano axioms that 1 + 1 and 2 are identical. You would learn something. The fact that you refuse to engage with that proof makes this thread irrelevant to my life. You asked for a proof, I gave you a proof and now you want to quibble that you have no obligation to read it? For weeks on end? Is that your idea of honest discussion?

Regarding the proof: That's how you show that 1 + 1 = 2 and that moreover, IF you believe that = means something other than identity, that 1 + 1 is identical to 2. I deny that mathematical equality differs from identity in set theory, except in a handful of casual conventions that can easily be rigorized on demand. You CLAIM they have different meanings but have not even attempted to defend or explain your claim but only seem to be avoiding the question. I use set theory because YOU are the one who invoked ZFC, claiming, and after all this time without evidence, that in ZFC two things that are not identical are asserted to be equal. I categorically deny that (except for as usual a handful of conventions such as embedding the integers in the rationals in the reals etc.) You made a claim about ZFC. I tell you that you are factually incorrect. You have failed to produce an example of your claim. You choose not to engage on the Peano proof, which contradicts your belief about 1 +1 and 2. I see no logical continuation of this dialog.

Again I do understand that you don't see things this way. That's what makes horse races.

Have a nice evening.
• 6.3k
You CLAIM they have different meanings but have not even attempted to defend or explain your claim but only seem to be avoiding the question.

This is ridiculous. I've explained numerous times in this thread how equality differs from identity. "Identity" applies to one thing, the same thing, its identity. So "identity" relates to what makes one specified thing other than everything else. "Equality" applies to two distinct things which are judged to be "the same" in a specific way. You might consider that "identity means "the same" in an absolute way, whereas "equal" means "the same" in a qualified, relative way. The example I gave is that two human beings are equal because they have the same rights, but they are not the same, because they each have a distinct identity.

Your task, as it has been since we first engaged in this thread, is to demonstrate that in mathematics "equality" is "identity".

I deny that mathematical equality differs from identity in set theory, except in a handful of casual conventions that can easily be rigorized on demand.

That is the point you keep asserting, without justifying. All the information you referred me to speaks of "equality", and there is no axiom which indicates that equality is identity. Your so-called proof relied on the premise, that "=" means identical, but the axioms don't bare this out. The information you referred me to spoke of "equality" not identity, so that premise is taken as false unless you can justify it. So unless you can justify your claim, produce this information whereby it is dictated that equality is to be interpreted as identity, I will continue to conclude that you wrongly interpret these axioms.

And if you insist that this is "the conventional" interpretation, that is not a justification. All this means is that "the conventional" interpretation is wrong, as I've demonstrated.
• 848
Identity" applies to one thing, the same thing, its identity. So "identity" relates to what makes one specified thing other than everything else. "Equality" applies to two distinct things which are judged to be "the same" in a specific way. You might consider that "identity means "the same" in an absolute way, whereas "equal" means "the same" in a qualified, relative way.

I understand what you're saying. You're wrong about mathematical objects. The number 2 is identical to the number 2. The number 2 + 2 is identical to the number 4. Identical as in your definition. There is only one thing, the Platonic number 4, which we may denote as 4 or 2 + 2 or the positive square root of 16 or 3.999... and many other representations.

I understand what you are saying and I deny it.

I do of course recognize contexts in which what you say is right. For example in group theory, two distinct groups that are isomorphic are often taken to be the same with respect to isomorphism. And in univalent foundations, we take as an axiom that isomorphism is equality (I'm paraphrasing greatly here but that's the essence as I understand it).

https://en.wikipedia.org/wiki/Univalent_foundations

So ok for that aspect of things.

But in ZFC, the domain of discourse in which you originally claimed that identity differs from equality, I tell you that you are incorrect. But I have said nothing new, I've written the same things over and over.

I hear you saying that in math the axioms speak of equality but not identity. I take that as profoundly insignificant, you take it as profound. This I believe is where we differ.
• 6.3k
The number 2 is identical to the number 2.

I went through this with Zuhair already. It is impossible that the numeral, the symbol "2" represents the same object every time it occurs. If it did, then both the 2's in "2+2" would refer to the same thing, and 2+2 would not equal four because there would just be the same two.

This is fundamental to the nature of counting. Each thing referred to by "1" must be a different thing, or else there would not be a multiplicity. "1+1" must represent two distinct things, or else it would not equal to two things. And "1+1+1" must represent three distinct things or else there would not be three. And so on, and so forth, each occurrence of the same numeral "1", must represent a different thing when we count, or else there is no multiplicity, only the same thing over and over and over again; and the sum of the count would be invalid because there would only be one thing being counted,.over and over again.

But in ZFC, the domain of discourse in which you originally claimed that identity differs from equality, I tell you that you are incorrect. But I have said nothing new, I've written the same things over and over.

That's what I've been saying, you keep asserting the same thing over and over and over again, without justifying your claim. I've demonstrated how equality is different from identity. So unless you can demonstrate how it is that equality is identity, in set theory, all you are doing is demonstrating that you misinterpret.
• 848
It is impossible that the numeral, the symbol "2" represents the same object every time it occurs.

You've just swapped in the term numeral for number. That's a particularly low form of false argument. It's like saying that two isn't two because some people call it zwei or dos or deux.

If you deny that the number 4 is the same as the number 4 you are entitled to your opinion, but that kind of sophistry is of no interest to me.

I did want to add that earlier when you said that mathematical equality is not identity but rather only equality in a certain context, you are thinking of equivalence or isomorphism. Mathematical equality is identity, not mere equivalence or isomorphism. You're simply wrong about that.

The idea that equality means that two "different" things are "the same" is nonsense. Equality means that two distinct expressions or representations of a thing refer to the same thing. 2 + 2 = 4 is an identity. I can't help what your grade school teacher put in your young and uncomprehending head. It's tragic that by your own admission your mind is stuck in the third grade.

I've demonstrated how equality is different from identity.

You have never done so, If you had we could talk about it. You have indeed expressed belief in the false claim that mathematical equality does not express identity. Repeating a false claim does not constitute a demonstration. On the contrary. Mathematical equality DOES express identity.

If you proposed an argument rather than just a repeated false claim, we could talk about it.

But in the end you have now said, and not for the first time, that you don't believe the number 4 is the same as the number 4. There is no conversation to be had (at least on this topic) with someone who professes such an obvious falsehood.
• 848
ps -- Wiki agrees with me.

In its formal representation, the law of identity is written "a = a" or "For all x: x = x", where a or x refer to a term rather than a proposition, and thus the law of identity is not used in propositional logic. It is that which is expressed by the equals sign "=", the notion of identity or equality. It can also be written less formally as A is A.

https://en.wikipedia.org/wiki/Law_of_identity
• 6.3k
If you deny that the number 4 is the same as the number 4 you are entitled to your opinion, but that kind of sophistry is of no interest to me.

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. If the two 4's both signified the same group of four, there would not be eight, by putting together the two things represented by the two 4's, there would only be the same four. Therefore the two 4's in 4+4 must signify different things or else 4+4 could not equal 8.

If you have anything of relevance to say, address my post, show me how it is possible that when you count, and you add 1+1+1+1 etc., each instance of "1" signifies the same thing. If you cannot address this issue you are just blowing smoke, saying that "1", or "2", or "4", always represents the same thing.

But in the end you have now said, and not for the first time, that you don't believe the number 4 is the same as the number 4. There is no conversation to be had (at least on this topic) with someone who professes such an obvious falsehood.

To be clear, I do not believe there is any such thing as the number four. Aristotle decisively disproved this Pythagorean idealism (currently known as Platonic realism) many centuries ago. What is the case, is that we use this symbol, written as "4", and each time it is used it signifies something, usually something different from the last time. When it is used, it may or may not signify the very same thing as in another instance of use, but in the vast majority of instances it signifies something different each time. Therefore the symbol, or numeral "4", does not represent the number four. This is a false assumption.

ps -- Wiki agrees with me.

And if you insist that this is "the conventional" interpretation, that is not a justification. All this means is that "the conventional" interpretation is wrong, as I've demonstrated.
• 848
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.

I can not relate this sentence to anything that I know nor to anything that makes any rational sense. I hope you'll forgive me, I cannot continue this conversation. You are factually wrong about this and you are not making me understand your reasoning. In other words if I thought you were wrong but I could say, "Ok I see where he's coming from," that would be fine. But I can't even do that.

The '4' in s 4 + 4 = 8 signifies two different things else 4 + 4 would not equal 8?

I find that absurd beyond the point of my being able or willing to respond. Of course the '4' signifies the exact same thing every time it is used, namely the number 4. If you are not willing to stipulate that then we have no common basis for conversation.

I get that you are sincere in your beliefs. From my viewpoint you give me nothing rational to respond to.

But perhaps you could give me a reference that supports your view. Earlier I noted that Wikipedia supports my view that logical identity and mathematical equality are the same thing. I'm willing to grant that Wikipedia is often wrong. But at least I have one reference. Give me something to work with, else I can't respond.

ps -- I should add this so you understand why you are wrong. It's a basic principle of math that the same symbol means exactly the same thing each time it's used in an argument or equation. For example when we say that for all even natural numbers n, 2 divides n, then even though n ranges over all possible even numbers, in each particular instance n means the same thing each of the two times it's used.

Likewise when we say 4 + 4 = 8, it's basic to all rational enterprise that the symbol '4' refers to the exact same thing each time it's used. Without that, there could be no rational communication at all. Natural language is symbolic. If I say that today it's raining and today it's Thursday, and you claim I can't assume that "today" refers to the same day each time I use it, then we'd all still be in caves. You couldn't say "pass the salt" without someone saying, "What do you mean pass, what do you mean salt, what do you mean "the"? You are denying the foundation of all symbolic systems from natural language to computer programming to math.

What exactly do you mean that '4' refers to two different things in 4 + 4 = 8? The burden is on you to justify denying the entirety of scientific and indeed rational discourse.
• 6.3k
But perhaps you could give me a reference that supports your view.

It's not a matter of giving you a reference, it's just a matter of whether you understand the reason or not. Do you know how to count? Say you have "1", and that 1 signifies something. And, you have another "1", and that 1 signifies something. In order that these two 1s, when they are put together (1+1), can add up to two, they must each signify something distinct from the other. If each of the two 1s signified the very same thing, there would not be two things, only one. Do you understand the reasoning here?

When we count objects, each object is counted as one (1), and so each object is represented by the symbol "1". So we count them, 1 plus another 1 makes 2, plus another 1 makes 3, plus another 1 makes 4, etc.. Each "1", must necessarily represent a distinct and separate object from every other "1", or else we would not have the multiplicity implied by the count, "2" "3" "4", etc.. It's not the case that the fourth object counted, when we point to it and count it as "4", is represented as 4, each distinct object is represented as 1. And, that each 1 represents a distinct object is absolutely necessary, or else the count would be invalid.

ps -- I should add this so you understand why you are wrong. It's a basic principle of math that the same symbol means exactly the same thing each time it's used in an argument or equation.

You are clearly wrong, and have given this absolutely no thought, or else you would see how wrong you are. When I say I have 2 chairs at the table, and I need another 2 chairs at the table to have 4, so that I can accommodate my guests, it is very obvious that each instance of "2" must represent a distinct pair of chairs. If the two 2s represented the same pair, I could not get four chairs out of them. I would be stuck with only one pair of chairs.

Likewise when we say 4 + 4 = 8, it's basic to all rational enterprise that the symbol '4' refers to the exact same thing each time it's used.

Again, you are very obviously wrong here, and you have clearly given this no thought or else you would see immediately how wrong you are. In the equation "4+4=8", each "4" must represent a distinct group of four things, or else they could not produce the sum of eight things. If both the 4s represented the same group of four things, there is no way to get a group of eight things, which is what is signified by "8".
• 6.3k
ps -- I should add this so you understand why you are wrong. It's a basic principle of math that the same symbol means exactly the same thing each time it's used in an argument or equation. For example when we say that for all even natural numbers n, 2 divides n, then even though n ranges over all possible even numbers, in each particular instance n means the same thing each of the two times it's used.

Likewise when we say 4 + 4 = 8, it's basic to all rational enterprise that the symbol '4' refers to the exact same thing each time it's used. Without that, there could be no rational communication at all. Natural language is symbolic. If I say that today it's raining and today it's Thursday, and you claim I can't assume that "today" refers to the same day each time I use it, then we'd all still be in caves. You couldn't say "pass the salt" without someone saying, "What do you mean pass, what do you mean salt, what do you mean "the"? You are denying the foundation of all symbolic systems from natural language to computer programming to math.

Try looking at it this way fishfry. There is a difference between what a symbol "means" (as said in your fist paragraph above), and what a symbol "refers to", (as said in your second paragraph above). So we can say that a symbol must always have "the same meaning" in order that it be useful, but the symbol doesn't necessarily refer to the same thing each time it's used. I use the word "house", for instance, and we say that it has the same meaning each time I use it, but I use it to refer to many different things which are all houses, so it doesn't always refer to the same thing.

What is important to understand here is that the phrase "the same meaning" does not use "same" in a way which is consistent with the law of identity. "Meaning" is the type of thing which varies according to circumstances, matters of context and interpretation. So in reality, even though we think "that the same symbol means exactly the same thing each time it's used", and this is necessary for a symbolic system to work, the very opposite of this is what is actually true. There must be nuanced differences in the meaning of "house", each time that I use it to refer to a different house, or else people would always think that I am referring to the exact same house each time I use the word. So "same" here really means "similar", and this is a qualitative identity, which is not what is described by the law of identity.

Qualitative identity is used to say that two things are equal, or "the same" according to some principle, or inferred criteria of judgement, but it does not mean that they are "the same" in the sense dictated by the law of identity, which would require that they are not two distinct things, but one and the same thing.
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Thanks for the reference Zuhair, but I really can't read the symbols used. It's like learning a new language for me, and it's a type of language which is even more difficult than a normal language, which I can't learn a new one anyway because that itself is very difficult for me. I have enough trouble with English.

Having said that, I see that the paper takes the premise of Platonic realism, assuming that symbols like "2" represent a thing called a number. This is the premise which I've been explaining to fishfry is incorrect. I believe that to adequately understand the use of mathematics it is necessary to apprehend that each time these symbols are used, in different circumstances, they represent different things, dependent on the circumstances of the application. What the symbol represents is not "a number", but a specific and unique object determined by the application of the mathematics.

So for example, when we count something, there is necessarily something which is being counted. One might just count, and claim to be counting "the numbers", having no tangible objects being counted, but as I explained, this is not a valid count. If nothing is being counted except "the numbers", then the start and finish are arbitrarily chosen, and the conclusion of "how many", which is what is determined by a count, is also arbitrary. Therefore any such count (how many), cannot be properly justified, it is just a function determined by the rules of the count, which are arbitrarily chosen. This is just an exercise, a practise, to demonstrate an understanding of the rules, like practising logic (as we discussed), where the symbols do not represent anything. If one were really going to count "the numbers", the count would never be finished. Therefore a count of "the numbers" can never be a valid count.

In reality then, the symbols in such a practise do not represent anything. 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".
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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".

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.
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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.

OK, now the question here is why does "2" represent one object, and not two objects. Intuitively I would say that the first "1" in "1+1=2" represents an object, and the other "1" represents an object, and "2" represents two objects. By what principle do mathematicians assume that "2" represents a single object, which might be called the number two?

Platonism is the easiest way to go about mathematics.

Perhaps Platonism is the "easiest way", but it is really nothing more than a cheat. Instead of recognizing, and understanding that a phrase like "1+1=2" is completely useless, and therefore meaningless, unless it is applied towards some real objects, in a real situation, the mathematician wants to say that it is implied within the phrase itself, that real objects are referred to. But this is contrary to the nature of language itself. In no instance of language use, is it inherent within the particular instance of usage, that there is necessarily objects being referred to. That this is the case, that no word necessarily refers to an object, is what allows for the existence of deception. So, claiming, or asserting that there is necessarily objects referred to, with a phrase like "1+1=2", is itself an act of deception, because there is really no language which can necessitate that if the word is spoken there is necessarily a corresponding object.

Now, the key to understanding, I believe, is to recognize that using "2" is an act which makes two objects into one object. We refer to the pair as if they are one object, using the numeral "2", but we have to remember that what is really referred to is two distinct objects, which are only made into one object through this artificial process, this synthesis, which is accomplished by someone uniting them, putting them together as one object, simply by calling them "2". So if we assert that "2" stands for one object, the true essence of this object which it stands for, is that it is really two objects which is only one object because we say that it is, and we have made it thus (one object), simply by saying that it is.

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 what I dispute though. The platonic model does not really engage with what the mathematical statements are truly saying. It is simply a cheat, an easier way for the mathematician, a way to avoid analyzing and understanding what the statements are really saying. Look, "2" really says two objects, and the mathematician just says consider those two objects as one object. It doesn't matter to the mathematician that there are no real principles whereby the two are considered as one, we'll just take it for granted that the two are one, and this will allow me to make all sorts of neat axioms. So the mathematician might assert that "2" says one object, and this is "what those mathematical statements are saying" but in reality we all know that the meaning of "2" is two distinct objects. So what the mathematical statement is really saying is that there are two objects here. But what the mathematician is saying is just bear with me, and consider that these two are one, so that I can perform my magic.
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There is a difference between what a symbol "means" (as said in your fist paragraph above), and what a symbol "refers to", (as said in your second paragraph above).

That is very funny. As in ironic When I wrote those two paragraphs I used "means" and "refers to" as synonyms. You take me to be saying two different things. I take meaning and reference to be the same in this context. But even "same" is a loaded word for you so I hope you're not going to go down another rabbit hole here.

I read the rest of your post and some of the interaction between you and @Zuhair. I see that you're sincere and knowledgeable about ... something. I can't figure out what because you won't supply a reference. You said that in "4 + 4 = 8" the two occurrences of the symbol "4' do not refer to (or mean?) the same thing. From my point of view there is simply no further conversation to be had. You're clearly serious, you're not trolling me. But when I try to take you seriously, I can't understand what you're saying. There is only one referent (or meaning) of the symbol "4" in the context of elementary arithmetic. [There could of course be other contexts, such as modular arithmetic in which "4' means some equivalence class mod some other integer]. But in the context of elementary arithmetic, "4" means the number 4.

I am simply not prepared, either by philosophical erudition or even the slightest interest, in debating this proposition. If you didn't seem so learned I'd honestly think you're trolling me.

What would help would be a simply clear example of WTF you are talking about. If the two occurrences of "4" in "4 + 4 = 8" refer to (or mean) something different, TELL ME WHAT THEY MEAN. Don't just toss out more paragraphs of obfuscation. Show me what you are talking about.

And -- secondly -- why won't you engage on the specific disagreement we're having about mathematical equality? I claim it is logical identity. You claim it's what's normally called equivalence, congruence, or isomorphism. This is a point we could engage on but you won't engage.

You said a while back that a grade school teacher once told you that an equality is between two different things. I'm afraid that experience imprinted an incorrect idea in your mind. You must let it go. With a handful of well-understood exceptions, mathematical equality is logical identity. You have not presented any specific examples to the contrary.
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It's not a matter of giving you a reference, it's just a matter of whether you understand the reason or not. Do you know how to count? Say you have "1", and that 1 signifies something. And, you have another "1", and that 1 signifies something. In order that these two 1s, when they are put together (1+1), can add up to two, they must each signify something distinct from the other. If each of the two 1s signified the very same thing, there would not be two things, only one.

You should give me a reference. If this is from some branch of philosophy or some philosopher's idea, let me know what that is. As it stands I think you were just warped by your grade school teacher.

One thing I see on discussion forums is that sometimes someone is arguing a point of view but not being up front about it. Someone claims uncountable sets are incoherent and twenty posts later it turns out they're a diehard ultrafinitist of the crank variety. I don't care if they're a crank but if they'd just start by saying, "From an ultrafinitist point of view ..." then I could engage with them. But without that information, their point of view is not comprehensible.

If your ideas are original, say that. If they follow from someone's work, say that. Give me something to hang on to. Because as it stands you're just saying flat out incorrect things, and waving your hands instead of giving hard facts, evidence, and examples to support your point.

Do you understand the reasoning here?

To the extent I do, you're wrong. And to the extent I don't, I really wish you'd give me a reference so that I have some idea where you're coming from.

I want specifics.

You made the outrageous and on its face absurd claim that "4" means something different each time it occurs in "4 + 4 = 8". Tell me what different things they mean. If you can't articulate the difference, perhaps your ideas aren't as clear as you think they are. Surely that's fair.
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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.

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.
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I take meaning and reference to be the same in this context.

Let me explain what I mean by the difference between what "4" means and what it refers to. What "4" means to me is that there is four objects signified, which are classed together in a group. And "4" always has the same, or at least a very similar meaning to me every time I use it or see it used. It means four things grouped together. However, many different times when "4" is used, it is used to reference different groups of four. So for instance, someone says get me the four books off my desk, or the four bases in baseball, or get me four winter tires, etc., the "4" refers to different things in these different instances of use. Despite having a very similar (what we call the same) meaning every time it is used, it refers to different things.

You said that in "4 + 4 = 8" the two occurrences of the symbol "4' do not refer to (or mean?) the same thing. From my point of view there is simply no further conversation to be had. You're clearly serious, you're not trolling me. But when I try to take you seriously, I can't understand what you're saying.

So imagine there are four chairs, and we represent those four chairs with the symbol "4". Now we want to add four more chairs, so that we might have eight chairs, so we represent these four more with the symbol "4". We can express this as "4+4=8". You must understand that the first four chairs, represented by the first 4, are not the same chairs as the second four chairs represented by the second 4, or else there would not be eight chairs. Do you understand this? Whatever group of four objects which is referenced by the first 4 in "4+4=8" cannot be the same group of four objects which is referenced by the second 4, or else there would not be eight objects.

Let's apply this to the most fundamental level of arithmetic. Let's say that the symbol "1" represents an object. If we add another object, and represent this with "1", so that we can say "1+1", don't you see that each of these 1s must represent distinct objects in order that we could get two from this? If each of the 1s represented the same object, how could there be two?

There is only one referent (or meaning) of the symbol "4" in the context of elementary arithmetic.

When they taught you elementary arithmetic, back in primary school, didn't they show you a group of four objects, represented by the symbol "4", and then another, distinct group of four objects represented also by the symbol "4", and if you added these two groups together, there would be one group of eight, represented by "8". If there was only one group of four referred to by both instances of "4", how could you ever get eight? Clearly these two instances of "4" must refer to distinct groups of four, just like I was taught in primary school.

How did you ever get the idea that each instance of usage of the symbol "4" must refer to the same thing? Surely they did not teach you this in school, in elementary arithmetic. Where did you get that idea from?

What would help would be a simply clear example of WTF you are talking about. If the two occurrences of "4" in "4 + 4 = 8" refer to (or mean) something different, TELL ME WHAT THEY MEAN. Don't just toss out more paragraphs of obfuscation. Show me what you are talking about.

Sorry, but I am not an elementary arithmetic teacher. But weren't you already shown this in primary school? This 4 references this group of four objects, and that 4 references that group of four objects. Obviously they must have shown you that each 4 necessarily represents a different group of four objects, or else it would be impossible to add them together and get eight. When this is what we were taught in primary school, where does your notion, that each 4 must reference the same group of four objects come from? Surely you must recognize that arithmetic would not work if this were the case.

You should give me a reference. If this is from some branch of philosophy or some philosopher's idea, let me know what that is. As it stands I think you were just warped by your grade school teacher.

What reference do you need? It's so obvious, that if you cannot see it, I don't know what else to say. If the two 1s in "1+1" both represented the very same object, then there is only one object represented. Do you understand this? And if there is only one object represented, then it is impossible that there is two (or any other multitude) of objects here. Do you understand this?

And -- secondly -- why won't you engage on the specific disagreement we're having about mathematical equality? I claim it is logical identity. You claim it's what's normally called equivalence, congruence, or isomorphism. This is a point we could engage on but you won't engage.

I already explained to you the difference between identity and equality, more than once. You keep asserting that in mathematics the two are the same, providing absolutely no evidence to back this up. I've already demonstrated that you simply interpret the use of "equality" in mathematics as meaning "identity". And despite me asking for them over and over and over again, you have provided no principles to support this interpretation, just the same assertion, this is what "equality" means in mathematics. So I conclude that your interpretation is a misinterpretation.

If your ideas are original, say that. If they follow from someone's work, say that. Give me something to hang on to. Because as it stands you're just saying flat out incorrect things, and waving your hands instead of giving hard facts, evidence, and examples to support your point.

How many more examples do you need? Put an object on the table, represent it with "1". Represent the same object with another "1". Say this "1" added to that"1" gives me 2, and voila, see if you have two objects on the table? No you still have only one, the same object which was represented by both 1s. Try it with "2". Put two objects on the table, and represent them with the symbol "2". Represent the very same two objects with another 2. Put those two 2s together in "2+2", and say voila! 2+2=4, so I now have four . There seems to be a problem, you still only have two, the same two represented by both 2s.. Try it with "3", and the problem will just be getting bigger. I could go on with example after example, and watch the problem get bigger and bigger.

Now try it my way. Put one object on the table and represent it with "1". Put another, completely distinct object on the table, and represent it with another "1". Now put 1 and 1 together, say "1+1=2" and voila! you actually do have 2, for real this time. Problem solved! Each "1" must represent a distinct object, if 1+1 actually equals 2.

I'm sorry Zuhair, but I really can't follow your example. It's quite complex, and as I said I'm not good with symbols, so I just get lost trying to figure out what you're saying. Here's something to think about though, which might be relevant to the case of tribe S. There is nothing to prevent one from using "4" to signify the very same object (group of four) in multiple instances of use. The problem is that in many instances, like in "4+4=8", it cannot signify the same object. But in some cases, like "4=4", or 2+4=4+2, it can signify the very same thing. This is why we can say that a thing is equal to itself (identity is an equality), but we cannot say that two equal things are necessarily identical (equality is an identity). So we find that identity is a very special sort of equality. Perhaps it's an absolute, perfect, or ideal form of equality.
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