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. — Zuhair
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). — Zuhair
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"! — Zuhair
. 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. — Zuhair
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. — Zuhair
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. — Zuhair
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. — Zuhair
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) — Zuhair
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. — Zuhair
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
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. — Zuhair
I feel badly misunderstood, but hey, this is the internet... — bongo fury
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. — Metaphysician Undercover
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. — Metaphysician Undercover
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? — fishfry
I know of no instance in which mathematical equality is anything other than set identity and logical identity. — fishfry
You claimed that in ZFC they misuse the law of identity in some way — fishfry
It couldn't have been a typo — Metaphysician Undercover
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. — fishfry
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. — fishfry
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. — Metaphysician Undercover
The number 2 is identical to the number 2. — fishfry
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. — fishfry
It is impossible that the numeral, the symbol "2" represents the same object every time it occurs. — Metaphysician Undercover
I've demonstrated how equality is different from identity. — Metaphysician Undercover
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.
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. — fishfry
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. — fishfry
ps -- Wiki agrees with me. — fishfry
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. — Metaphysician Undercover
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
But perhaps you could give me a reference that supports your view. — fishfry
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. — fishfry
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. — fishfry
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. — fishfry
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
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. — Zuhair
Platonism is the easiest way to go about mathematics. — Zuhair
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. — Zuhair
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