So two meals can't be identical? It is called qualitative identity rather than quantitative identity. — Bartricks

If we're talking about qualitative identity, then only one quality need be the same in order that we call it "the same", the same colour, the same weight, the same length, etc. Or even if two things appear similar we might say that they are the same. But you said that everything describable about the supposed "two" acts are the same, so you clearly weren't talking about qualitative identity.

Face the facts Bartricks, you're trying to put forward an argument which fails, as unsound, because it begins in an impossible premise.

Whether or not there can be anything real apart from any observer's perspective is not a question I am attempting to answer. My concern was with what is commonly meant by "real", and what is commonly meant just is something indepedent of any observer's perspective. So, who is not listening, eh? — Janus

This is what you said:

For me the real is what is, not (necessarily) what we experience. — Janus

You were clearly associating "what is" with something independent of any observer's perspective

What I pointed out is that it doesn't make sense to speak of the real as "what is", and try to separate "what is" from what we experience, because for something to be, (and this is what "what is" refers to), requires an experiential perspective. Furthermore, there is a large body of evidence, and philosophical arguments, which demonstrate that if anything did, or could, exist separate from any experiential perspective, it would be completely different from, and incompatible with, what we refer to with "what is".

All of that supports the conclusion that it is unacceptable metaphysically, to speak of "the real" as "something independent of any observer's perspective", whether or not this is common parlance. Common parlance is often inconsistent with what is acceptable within a field of study. Logic shows that there couldn't be anything without an observer's perspective, so it really doesn't make sense to define "the real" as that which is independent of any observer's perspective, because this would just be like saying that there is nothing real, no such thing as reality. That's why I said your claims are just an attempt to make the illusion (that there could be something independent of an observer's perspective) into reality.

Physicist Victor Toth answered the question, "What is a quantum field?" in this manner :
"But no, quantum fields do not interact with matter. Quantum fields are matter." ___ https://www.forbes.com/sites/quora/2017/12/20/what-is-a-quantum-field-and-how-does-it-interact-with-matter/#6c0495928c4a

That would also be my answer to "What is an Information Field?" : the information field does not interact with matter, it is matter. — Gnomon

The problem though, is that matter itself is just an idea, a concept. It was introduced by Aristotle as an attempt to substantiate logic, being faced with the scourge of sophism. In more modern times, physics has replaced matter with energy, which substantiates its logic. Perhaps it now turns to "fields". Aristotle however, laid down strict conditions, metaphysical principles, concerning the use of "matter" (as an idea). These were derived from a logical treatise on the nature of "change", his Physics. Those conditions have long ago been ignored, and have been superseded by metaphysical principles which do not adhere to such strict principles.

The issue being that "matter" was introduced into physics as necessary to account for the unintelligible aspects of change. These unintelligible aspects allowed sophists to argue the reality of absurdities But "matter" is defined metaphysically (because it is of the unintelligible). Under the direction of Aristotle, it is a concept formulated with specific logical rules, intended to keep physics "real", grounded. If we move away from these rules, without introducing new rules which are at least as rigorous as the old, there is nothing to keep physics real, or grounded.

Omg, this is just too painful. No, there can't be 'other factors'. The whole point is that the two acts are identical in every way apart from that one is right and the other wrong. — Bartricks

How can you not see that you're talking nonsense? If they are identical then they are not two acts but one and the same act. You are starting with an impossible premise.

Now, imagine two actions. These two actions have the same consequences (they both result in an innocent person's death, say). They are both performed with the same intentions. Now, do they both have to have the same morality? That is, if one is wrong, must the other be wrong too? — Bartricks

Good and bad are measured by degree, that's why there are differing punishment sentences for "the same" criminal act. It is the fact that each of these acts in your example, results in an innocent person's death which makes them both wrong. But the degree of wrongness may vary such that one is worse than the other.

Now, given that they are identical in terms of their intentions and consequences, must they have the same morality? — Bartricks

No, there could still be other factors specific to the circumstances which makes one worse than the other, therefore they would not have the same morality.

Virtually everyone - I mean, virtually everyone - gets the rational intuition that they do. — Bartricks

So this is false, by the argument above. I think virtually everyone would get the idea that one act would be worse than the other, depending on the circumstances.

But we can't do that in terms of moral properties. We do not seem able coherently to say that two acts can be identical in every way apart from that one was right and the other wrong. — Bartricks

Just as you did before, you are misrepresenting morality. Morality is based in judgements of degree, so it is not the case that every good act is morally equivalent to every other good act, nor is it the case that every morally bad act is morally equivalent to every other morally bad act. So the fact that two similar acts are both morally bad doesn't imply that they are morally identical.

In conclusion, just because two acts are bad doesn't mean that they are morally identical.

So, if act A is right and act B is wrong, then either act A was performed with a different intention or it had different consequences - and that explains why it is right whereas B is wrong. — Bartricks

But act A may differ morally from act B even if they are both bad. So you start from a false premise, that if two acts are bad, they are morally identical.

Now the point I was making to Janus is that if we divide events up, or put events together, which is what we have to do to say "this is a rock", or "that is a proton", this is just a human construct. So it makes no sense to talk about "what is", independent of human experience, because "what is" is a human construct, and there is no such thing as what is, without the human consciousness which constructs it.

That would be a fair point if I was speaking in terms of temporality, but I was speaking generally about what I think the common meaning of the term 'real' is. What is real is what is...what was real is what was...what will be real is what will be. The point is that I think we have a common understanding that what is real, as such, does not depend on us, or on our experience. — Janus

See this is evidence of what I said, you don't listen. Instead of addressing my point (that "is" necessarily implies temporality present, now) directly, you simply side step the issue.

If "what is real" , is what "is", and the existence of the present depends on us, as it appears in modern scientific theory that the present (now) is a subjective perspective (the observer's point of view), then how can there be a "what is real" independent of an observer's perspective? And an observer's perspective requires an "observer". All we have here is a glorified form of the relativism presented by the sophist Protagoras. You claim that "what is" gives you a perspective independent reality, but "is" requires a perspective.

I'll tell you, that this position of yours (concerning what "is") is denied by many modern metaphysicians who attempt to maintain consistency with relativity theory. In physics we describe reality in terms of motions, assuming that all is moving, and this is conducive to a philosophy of process, everything is changing, there is no rest, or what "is". in any absolute sense. This means that any description of "what is" is just that, a description, artificial, produced by the observer, and that which is supposed to be independent, being described, is incompatible, and therefore completely different and unlike any description, of "what is". Furthermore, this metaphysical position is strongly supported by Hegelian dialectics, which also supports dialectical materialism, and dialetheism. These positions accept violation of the contradiction law, such that reality must accept what is and what is not. That's because "becoming", change, motion and activity, which is what reality consists of for these metaphysicians, cannot be adequately described in terms of is and is not.

You may understand this incompatibility from the discrepancies between the ancient philosophies of Parmenides and Heraclitus. Parmenides described reality in terms of being, what is and is not, and this position supported the idealism of the Pythagoreans and others. Heraclitus described reality in terms of becoming, motion and change. Plato considered both of these perspectives, and found them to be incompatible.

From the Platonic/Aristotelian tradition, dualism is the solution to this problem. Reality must be described with reference to two distinct and incompatible aspects, the passive (what is), and the active (what is changing). The monist simplification which you propose, "the real is what is", only reintroduces this problem of incompatibility. So it is very easily demonstrated as unacceptable, inadequate, as a return to presocratic confusion.

I really really want to find fault with any or all of these insights ... Maybe Gnomon, Metaphysician Undercover, et al are up to that challenge. But damn, J, well done — 180 Proof

It's very easy to find fault with Janus' principles. But Janus doesn't listen, so I've given up on that. Here's some examples:

For me the real is what is, not (necessarily) what we experience. — Janus

There is no such thing as "what is". "Is" refers to what is "now", present tense, and time passes so fast, that by the time the future is present, it is past. One cannot say "what is", because by the time this is done, it is past. "The present" is an illusion, as is "what is", because all is future and past. So Janus' claim, that "the real is what is", is nothing other than a claim that what is an illusion "the present", is what is real.

It makes no sense to me to say that something could "exist" beyond the bounds of reality. — Janus

Reality, according to what it has been claim to be, above, is already an illusion, as nothing "is". Therefore everything must be beyond the bounds of reality, as defined. So it is not only the case that something "could" exist beyond the bounds of reality, it is necessary that if anything exists at all, it is beyond the bounds of reality, when reality is defined in that way.

But Janus only pretends to listen to reason, until it gets too difficult to maintain the principles which Janus holds in the face of reason which demonstrates the deficiencies of these, then Janus let's go and slips back into the unreasonable principles, and refuses to listen.

Two acts - A and B. They are the same in every non-moral respect. — Bartricks

Problem is, these are not two acts. You are referring to one and the same act, and calling it "two acts". That's why your premise is self-contradicting.

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

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.

Answering questions is not in the interest of spreading pro-Trump propaganda. — Echarmion

That is done by making statements of fact. That's all NOS4A2 does, states "the facts" over and over again.

If you put two cows on the commons, we should all move away from you when you go to the Pub for a beer after work — Banno

The real problem with the commons was not from the greed of the individual, who wants more than one's fair share of cows, it's a problem of too many individuals sharing the same resources. That's overpopulation. Overpopulation seems to be a natural tendency for any living species which is capable of dominating the others. Have you ever grown a culture on a petri dish? The thriving species will run rampant, rapidly overrunning and using up all the choice nutrients, then it dies because it hasn't the capacity to adapt: some might go into suspended animation (seeds or spores) waiting for another chance to dominate.

That's why the third option won't work. We haven't the capacity to adapt. I'd propose a fourth option, vegetarianism or something like that, and I think Plato suggested something like this in his Republic (which was supposed to be a communal society), saying that meat ought to be given up, as a relish. But again, I don't think we have the capacity to adapt. "Taste" might be the strongest of all instinctual motivators. We take oxygen for granted and don't need to taste it out, but food is not only fundamental to subsistence, it supports growth, loco-motion, and all the higher level activities like sensing and thinking. The variations between individual highly organized living beings, like the human, are probably closely related to taste.

Again: imagine two acts that are identical in every way apart from spatially and or temporally. Not hard. — Bartricks

Yes it is hard. I find it to be impossible. This is like saying imagine two people who are exactly alike except they are different. It's nonsense.

I mean, if I ask you to imagine a car identical to yours in every way apart from it is in another location, would you find that difficult? Would you say "er, but then it is not the same car" - yes, I know. Not the same car. But similar in every way - apart from it is over there. — Bartricks

Cars don't have intention. Anyway, I'd say that the two cars were not exactly alike, they have a different serial number to begin with, and they've both been used in different ways with different wear and tear.

Imagine two acts - two, not one, two - that are identical in every non-moral way apart from spatially or temporally. — Bartricks

Sorry, but I find that you are asking me to imagine the impossible, like a square circle. It's nonsense, it can be said, but not imagined.

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

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. — Zuhair

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) — Zuhair

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. — Zuhair

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.

No! Unless these differences are indescribable by formulation of the language. — Zuhair

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). — Zuhair

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"! — Zuhair

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. — Zuhair

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. — Zuhair

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?

A similar act in different circumstances cannot be called the same act. They may have different intent. Therefore it is possible that they could differ morally.

Doesn't your reason tell you that the two acts must be morally identical? — Bartricks

No, because the acts are carried out on different planets with differences between them so reason tells me that they are not identical. If they are not identical acts, then why say that they are morally identical?

It's not just humans though, each individual animal has its own way of being.

What do you mean by "numerically distinct"? If there are differences, what are the difference between them? These differences could amount to the act being morally unacceptable on one and morally acceptable on the other.

But it's the Republicans who really hold his fate in their hands. Several, including Romney, are saying that 'this is serious, it's an impeachable offense'. If that takes root amongst more Republicans and begins to snowball, then things could develop very quickly. — Wayfarer

The election is not far off. The Republicans need to seriously consider who their candidate for presidency will be.

Now imagine there is another planet exactly like this one - I mean, exactly like it in every physical respect. — Bartricks

If the planets were exactly the same, then by the principle of the identity of indiscernibles, they would be one and the same planet, and everything which is the case on one would also be the case on the supposed "other", because it wouldn't be "other", it would actually be the very same planet. So your question doesn't make any sense because you are talking about one and the same planet as if it were two distinct planets.

disagree because as I said different kinds of beings have different ways of being. — Janus

How would the difference between the different ways of being, which are proper to the different kinds of beings, be fundamentally different from the difference between the different ways of being which are proper to the different kinds of human beings? For example, beavers make dams while birds make nests, and engineers make dams while homemakers make nests. Each individual being has a different way of being.

That criticism is that if moral norms and values are the prescriptions and values of a subject (be they a god, gods or us) then they would not be immutable. They could vary over time.
Yet, in Plato's day as today, moral norms and values appear to be fixed. They are represented to be by our reason. Hence a problem. — Bartricks

I don't agree that moral values appear to be fixed, not today, nor in Plato's day. The criticism appears to be way off base.

Just engage with the actual criticism. — Bartricks

I did engage with the actual criticism. I pointed out that you were using "necessary" in a way which is inconsistent with the way that it is used in moral philosophy. You insisted that "necessary" means "cannot not be the case", which is some sort of logical principle that has nothing to do with morality, which deals with how people ought to behave. Then you went off on some tangent talking about different worlds with identical features.

If two worlds are identical in all non-moral respects, then necessarily they are identical in all moral respects? — Bartricks

You know that if they are two worlds, then they must differ in some way, or else they would be one and the same world. If they do not differ in non-moral respects, then they necessarily differ in moral respects.

"Consider yourself owned."

Otherwise the discussion would be really very poor. — Zuhair

Then why did you waste so much of your time discussing this with me?

However when logicians are speaking about equality in the sense of satisfying the substitution schema, then in reality they mean "identity", — Zuhair

I know enough but the substitution axiom to know that it deals with equality not identity. The difference is that two distinct things may be equal, but they cannot have the same identity. The substitution axiom allows that one thing may be substituted by another equal thing, so it clearly accepts that these are two distinct things, not one and the same thing, with one identity. 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. But clearly, that there are differences between them means that they are not one and the same thing, as required by the law of identity.

Judging by the op, I'd say you haven't read The Euthyphro. In it, Plato relates morality to the gods, not to Reason, or to worlds which are identical in some aspects (whatever that means).

Trump does not seem to be able to see beyond his own experiences. Hunter Biden made millions of dollars in the Ukraine therefore he must be corrupt.

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. — Zuhair

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

You need to read some foundation of mathematics book, then you can come a speak about it. — Zuhair

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?

If two worlds are identical in all non-moral respects, then necessarily they are identical in all moral respects? — Bartricks

How does that make any sense?

No, it is exactly how it is used in moral philosophy. You're the one using it incorrectly. — Bartricks

It seems you haven't read any moral philosophy. Do you recognize a difference between "is" and "ought". "Necessary" in the sense of "cannot not be the case" is based in what "is". "Necessary" in the sense of what is needed for some purpose is based in what ought to be done.

Or do you think that moral values are not invariable across time and space?

if so, what do you do with all those widely corroborated rational intuitions that represent it to be? Just reject them? — Bartricks

How could moral values be invariable across time and space when "ought" refers exclusively to future acts?

I am using 'necessary' to mean 'cannot not be the case'. — Bartricks

The point though, is that this is not how "necessary" is used in morality. It is used to indicate what is needed, what ought to be done for some purpose. If something is "necessary", there is a reason why it is necessary, it is deemed as needed for some purpose. So you are taking the wrong sense of "necessary", one not applicable to morality, and trying to make a moral argument out of it. That's nothing but equivocation.

You mention the Platonic form of the good - okay, so if this strange obelisk values things (a notion I can make no sense of whatsoever), why is it the case that it could not disvalue the things it values? — Bartricks

That's not what I said about 'the good", nor is it what Plato said about 'the good".

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. — Zuhair

All the information I've seen shows that the reflexive axiom and the substitution axiom are equality axioms. Why do you think they that are identity axioms?

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. — Zuhair

I saw no such rule, to dictate that "2" and "4" are "zero placed function symbols", on the page you referred. I think you're making this up. Any way "zero" would indicate an absence of objects.

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. — Zuhair

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?

1. If moral values are objective, then moral values with be contingent, not necessary
2. Moral values are necessary, not contingent
3. therefore, moral values are not objective — Bartricks

You are not reading "necessary" in the right way. Necessary here means "for the purpose of". And since there is something which makes them necessary, in this way, they are contingent on that thing.

It appears self-evident to the reason of most that moral truths are necessary, not contingent. How else do you explain why the Euthyphro is considered by virtually all contemporary moral philosophers to be such a damning criticism of subjectivist views??? — Bartricks

When you read "necessary" in the right way, it makes perfect sense to say that moral values are necessary. You've misinterpreted the word, then claimed the statement makes no sense.

From the Platonic perspective, the thing which necessitates moral values, "the good", is an object, a goal, (which because it does not exist as it is what is wanted, or lacking by the subject), is separate from the subject. This is what objectifies moral values.

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. — Zuhair

I looked through the Wikipedia page on the syntax of first order logic, which you referred me to above. This is what I think. In the expression "2+2=4", the "2", and "4" symbols are predicate symbols. A predicate symbol represents an "element" with a relational definition. Because of this, expressions like "2+2=4" cannot be considered as terms, and therefore there is no basis to the claim that any objects are represented here.

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. — Zuhair

That's clearly not the case. A function, or process is indicated by "+". There is nothing to indicate "the result" of the function. Consider for example cause and effect. The cause may be represented or described without representing the effect. When only the cause is represented, there is nothing to indicate what the effect is. Though it is true that "effect" is implied by "cause", unless one already knows that x cause has y effect, y would not be indicated by stating "the cause is x". "1+1" represents a process, it does not represent the effect or result of that process. This is evident from the fact that one might state "1+1" without knowing that the result is "2". And of course this is the natural process of summation, we write down the numbers to be summed before we know the result of the summation.

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. — Zuhair

No, absolutely not. It may be true that "+" denotes the process, but there must be something which is active in the process, or else you just have a type of process indicated, with no specifics. So the two "1" symbols denote the elements involved in the specified process, and there is nothing to indicate what is caused by the process, "the result" of the process.

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. — Zuhair

It is an interpretation, a faulty one. There is nothing in the rules to say that the expression represents "the result" of a process. You are making that up, back to your old habit of bullshitting again.

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? — Zuhair

According to the definitions on the referred page, these numerals are predicate symbols, therefore they denote elements. The expression is not meaningless though, it demonstrates a relation.

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. — Zuhair

Until you demonstrate how numerals like "2" and "3" denote objects rather than elements, as predicate symbols, which is clearly explained in the rules, I think we ought to stop saying that these denote objects.

hmmm...., let me think about that, I'm really not sure if "identity" really arise in mathematical system per se. — Zuhair

That's the point. Fishfry keeps trying to switch out "equality" for "identity", as if the two have the same meaning. But "identity" has a very specific, well defined meaning in philosophy, and no such definition in mathematics. So Fishfry's use is either an attempt at equivocation, to smuggle the philosophical meaning into mathematics as if "equality" means the same as philosophical "identity", Or else he just brings a non-defined word into a mathematical usage which would leave it meaningless.

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: — Zuhair

Perhaps you can do for me what I've asked of fishfry. Show me an axiomatic theory of identity which is proper to mathematics.

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. — Zuhair

I haven't yet been shown these identity axioms of ZFC. 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. It presupposes some sort of identity with the use of "same", but it doesn't stipulate what "same" means.

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). — Zuhair

Well you'll need to justify this claim. 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. It is just one of many possible interpretations, and interpretations are often illogical or incoherent.

Regardless, we seem to agree that the two 1s in "1+1" represent distinct objects, so what needs to be explained is how this function "+" makes these two objects into one. We cannot just stipulate that these two objects are one, because that would be contradiction, so the function must do something to avoid such contradiction. The function must represent a process which makes them into one.

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. — Zuhair

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. So 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"? Then wouldn't you say that "2", and "3", along with the process of addition results in "5"? But you really have no result of the process of addition until you state the sum. The process is something carried out by the thinking mind, not the symbol itself.

Good news. I'm working on a reply in case it takes a while. I do think you're failing to distinguish between:

* The philosophical question; and

* The mathematical question.

When you send me to SEP and make subtle (and interesting!) points about the nature of identity, that is part of the philosophical problem. About which I have already stipulated that I'm ignorant and open to learning. — fishfry

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.

Could you please repeat exactly what I said that you think is false? — fishfry

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. In mathematics there are theories of equality, and perhaps axioms of equality, but these are not laws of identity. So what I was asking for was if you know of a law of identity which states that things which are equal have the very same identity.

In set theory, 2 + 2 and 4 are strings of symbols that represent or point to the exact same abstract mathematical object. That's a fact. — fishfry

It's one thing to make these sorts of assertions, but another thing to justify such claims. This would require showing how this string of symbols "2+2" denotes the exact same abstract mathematical concept as this symbol "4".

I tend to do that, sorry. — A Gnostic Agnostic

Tend to do what, incite hatred? It's very obvious that everything in that post (truth or falsity being totally irrelevant) was clearly expressed with the intent to incite hatred.

Stop with the blind hatred... — A Gnostic Agnostic

Stop the hatred! Says the hypocrite who speaks with a clear design and purpose of creating hatred.

What the fuck?

Physical explanation replaces nouns with verbs. — Gnomon

The modern trend is for a metaphysics of becoming (process) to supplant or uproot the traditional metaphysics of being. There are numerous reasons for this shift, Hegel/Marx, Einstein/Whitehead, for example.

Which symbols the = links, the answer is that it links the expression 1 + 1 to the expression 2, so the = sign here represented an equality relation occurring between the objects denoted by these expressions. Since equality is a binary relation between objects, then the symbol for equality, which is "=", must be written as linking symbols that denote objects, since = links 1+1 to 2 then 1+1 must denote an object, and 2 must denote an object. That's why 1+1 must be an expression that denote an object. — Zuhair

OK, now we're back to the same problem. In "1+1" each "1" is a term denoting a distinct object. Therefore there are two objects denoted. How does it come about that "1+1" denotes a single object, as a term in itself?

It is always the case that relations are between objects, and so relation symbols must link terms, because terms are the symbols that denote objects, this is because the symbolization must copy what is symbolized. — Zuhair

I don't see that this is "always the case". Why can't a group of terms be related to another term through the same relation, like in your analogy, a group (Jesus and James) are related to Mary?

The first 1 denotes an object
The second 1 denotes an object
The string 1+1 denotes an object — Zuhair

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

The + sign is denoting a ternary relation that is occurring between the above three objects.
[Imagine that like the the expression "the mother of Jesus and James" here Jesus and James are denoting persons, the whole expression is denoting another person "Mary", "the mother of" is denoting a relation between objects denoted by Jesus, James and by the total expression above. — Zuhair

I explained why this analogy doesn't work. "Jesus and James" is analogous to "1+1". "The mother of" is a term, denoting an object, a person "the mother", who has an implied relation dictated by the definition of "mother". It does not denote a relation, it denotes a person (object) who has a specific relation.

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. — Zuhair

Hey thanks, I'm glad you liked it. Here's the problem though. In logic, we can learn the logical procedures, by playing the "empty symbol game". It is useful for that purpose of teaching procedure, but beyond that mode of practise, it's useless, like an activity not being applied, what Wittgenstein calls language which is idol, or on vacation. To be useful there needs to be substance, subject matter, the symbols must be applied (represent something), to have meaning, and allow the logical proceedings to actually do something.

Arithmetic is different though. It is based in symbols and relations, rather than procedures. So the foundations of arithmetic involve symbols (1,2,3, etc,) which represent objects and the relations between these objects, relations which are determined and inherent within what the symbol represents (its object). This is somewhat different from the foundations of formal logic, which involve what we can and cannot say about an object.

Because of this difference the notion of an "empty symbol game" in mathematics is illogical. Arithmetic and mathematics are structures of coherency, every symbol has its place in the structure according to what it represents, so that an empty symbol would not fit into the structure at all, having no place. Imagine if there really was a symbol in arithmetic, like "2" for example, which represented nothing other than itself. It could have no relationship with any other symbols like "1", "3", "4" etc. because if it did have such relations with other symbols, they would have to be inherent within what is represented by "2". Then '2" would not just represent itself, it would also represent these relations, as "2" actually does.

Think of the object represented by "the mother of...". Not only is a particular object represented, but inherent within that way of representation (how the term is defined) is also its relation to other objects. This is the way it is with the terms of arithmetic, inherent within the object represent by "2" is all the relationships with other arithmetical objects by the coherency of the definitions. This is very important in geometry because some fundamental definitions or axioms, (like the circle has 360 degrees for example) are very arbitrary. If all the definitions which follow, and build up the structure based on this axiom, are not consistent and coherent, then the conceptual structure is useless. The object represented by the symbol "circle" might be completely arbitrary, but it is necessary that there is an object which is consistent with the objects represented by the other terms, thus making the symbol useful, or else everything is incoherent and meaningless.

No problem with two 2's in 2 + 2 being denoting different objects, since they can be interpreted as denoting themselves and they are of course distinct. — Zuhair

I think, if the "2" is denoting itself then it is not denoting anything, it is simply a 2. If we say that it is a symbol, and therefore denoting something, then to say that it denotes itself is nothing other than to say that it denotes nothing. A symbol which denotes nothing is not a symbol, so the "2" which is supposed to denote itself is simply an object, denoting nothing.

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..

Now generally speaking when we are in a mathematical language we must specify which symbols are taken to refer to objects (even if to themselves) which we call as "terms" and which symbols are taken to refer to "relations between objects" we call them "predicate" or "relation" symbols. — Zuhair

A "term" which refers to itself, is not acceptable to me. It is not really a term as per the definition, but a subject employed for trickery, deception. "Term" as per the definition requires that the symbol represent something, and to represent itself is to represent nothing. So the use of a "term" which represents itself is a ploy to avoid the restrictions of the definition, which dictates that the term must represent something.

In nutshell relation symbols link terms. So for example = is a binary relation symbol, so it must occur between two term expressions, i.e. expressions taken to represent objects. — Zuhair

I see a problem here, the possibility of category error due to confusion between the object which is the symbol (term), and the object which the symbol denotes. You say that relation symbols link terms. Therefore, what is related are the terms, the subjects. And, we must remember that it is not necessary that the subjects represent objects, because the trickery employed which allows that a term represent itself, such that the represented object is really just the subject representing nothing.. This allows that the mathematician may be just playing with subjects, establishing relations between terms which do not represent anything, just like logicians do, but it is in defiance of the definition of "term". The definition of "term" disallows this, but the crafty mathematician has found a loophole.

Lets take (2 + 2 = 4)
Now for = to be a relation symbol it must occur between terms, so the totality of whats on the left of it must be a term and so is what's on the right of it, 4 is clearly a term, so 2 + 2 must be a term, otherwise if 2 + 2 doesn't signify a term (i.e. a symbol referring to object) then what = is relating to 4? either 2 + 2 is a relational expression (similar to 1<2) but those are not put next to relation symbols, image the string
1< 2 = 4, it doesn't have a meaning, it is not a proposition, or 2 + 2 might be neither a proposition nor a relation symbol, but this is like for exame 2+ = 4 here "2+" is an example of a string that is neither a term nor a proposition, it even cannot be completed with =4 to produce a proposition.

In order for "2+2" to be completed with "=4" to produce a proposition, then 2 + 2 must be a term of the langauge, and thus denoting an object, even if that object is the string of the three symbols itself!, otherwise we cannot complete it by adding to it a relation symbol and a term after it. — Zuhair

I agree with this, except that the term must denote an object, as I described above, the mathematician has found a way to employ terms which do not denote objects. Let's just say that the term is a subject, which is a special sort of object, one which is subject to the will of the logician, and it need not denote anything. I'm sure you've seen examples of formal logic, expressed in 'symbols' which do not denote anything. These examples are used in teaching, to demonstrate the logical process. The process is shown using terms which do not denote anything. 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.

Notice that not every string of symbols in a language are taken as well formed formulas of that language for example 2 + 2 = is a string of symbols, it is also incomplete, it doesn't represent a term nor a relation, even though it is composed of two terms (the "2") and another term (2+2) and a relation symbol =, but here it doesn't constitute a proposition and it is not itself denoting a term. When you add 4 to it of course it becomes a proposition. So not every part of a proposition is a term or a proposition, examples are 2+, +2, 2=, =4, etc.. all are neither proposition nor terms — Zuhair

So, lets start from the beginning, and enforce the rule of definition, in a rigorous manner. If "2" is a term, it must denote something. What it denotes must be something other than itself. This means that we must assign meaning to "2". That requires a proposition. We cannot just assume that "2" denotes itself, in order that it's properly a term, we must say what it denotes. I suggested "1+1=2" as a proposition which defines what "2" denotes.

2 is referring to an object (which is itself here), but to identify it in relation other symbols by using the particulars of a certain language (for example in arithmetic those mount to +,x,=,< etc.. symbols) then we'll need propositions, but those can only occur by relating it by a relation symbol to other term symbols so 2= 1+1 won't have any meaning if 1 + 1 was itself not a term of the language denoting some object (which can be taken here to be the string 1 + 1 itself), otherwise if 1 + 1 is not an expression denoting an object (i.e. a term) then how can we related 2 to it via the equality symbol = which is a binary relation symbol (sometimes called two place relation symbol), the whole string of symbols would be meaninging much like writing 2= 1<3 i.e. 2 is equal to (1 being smaller than 3), this is meaningless, it is not a proposition, same if we say 2 = 1 + 1 and envision 1 + 1 as a relational expression expressing a binary relation + occurring between 1 and 1, then we be saying ( 2 is equal to (1 having + relation to 1)) which is meaningless because an object is equal to an object and not to a relation. — Zuhair

Do you realize that this is all one sentence? Maybe you could express it in a more comprehensible way? As I explained in the last post, if "4" is defined by the proposition "4=2+2", and "2" is defined by "2=1+1", then we must turn to the definition of "1+1", as a term, to make "2" intelligible. And, it is necessary that each time the symbol "1" appears it denotes a different object, or else "1+1" is unintelligible.

Heck, we have done just fine with repairing or at least stopping the breakdown of the ozone layer. — ssu

The damage has been done. Less ozone means more UV radiation (high energy), reaching and warming the earth's surface, energy which would have been absorbed in the upper atmosphere and radiated to outer space, if the levels of ozone had been maintained. Warming at the surface, and cooling in the upper atmosphere has been observed. Further, UV is harmful to most if not all life forms.

Though some people claim that the ozone layer is "healing", differences between ozone levels in the south, and ozone levels in the north, and cycles of fluctuation, make it extremely difficult to say whether we've actually stopped the breakdown.

I thought we got over that point. I agreed with you that "=" is NOT necessarily the identity function, so why you are returning the discussion backwards. — Zuhair

I know you agreed to that, but I could trust you because of propensity toward lying. And, you've continued to argue that "2+2" is the same (in the sense described by the law of identity) as "4". So your agreement appeared to be worthless in that matter.

I agreed with you that if you interpret "=" just as an equivalence relation (as it is officially formalized in PA for example), then of course the object that the + operator send objects denoted by 3 and 5 to, is NOT necessarily identical with the object denoted by 8. We already passed this point. — Zuhair

OK, so I made my point, you now agree with me, and perhaps we don't have anything more to debate.

To me it is nothing but an assignment scheme, i.e. a sending rule, nothing more nothing less, it sends maximally two objects to a third object. — Zuhair

I really do not know what you mean by "sending" "two objects to a third object". You seem to think that sending is not a relationship, so what is it? How are Jesus and James "sent" to the Mother, when their true relationship to the Mother is that they are from the Mother?

Actually although I don't want to go there, one of the intended interpretation of arithmetic is as a closed syntactical system, i.e. non of its expression denotes anything external to it, so for example under that line of interpretation the symbol 2 means exactly that symbol itself, and so for example 2 + 2 has "distinct" symbols on the left and right of the + sign, and although they are "similar" in shape, yet they are two different objects since they occupy different locations on the page, each 2 is denoting itself only. — Zuhair

OK, I can agree with this.

Now also 4 denotes itself only, also to further agree with you 2+2 is denoting nothing but itself (the totality of the three symbols) and so it is NOT the same as 4, not only that every individually written 2 is not the same (identical) to the other, and the equality in 2+2=4 doesn't entail at all identity of what is on the left of it with what's on the right of it, its only an equivalence syntactical rule, and can be upgraded to a substitution syntactical rule without invoking any kind of identity argument at all, and the whole game of arithmetic can be understood as a closed symbolic game nothing more nothing less. — Zuhair

See why I accused you of lying when you were arguing something opposed to this? I knew you were intelligent enough not to actually believe what you were saying.

But still we need to maintain that expressions like 2 + 2 denotes an object while expressions like 2 > 1 denotes relations (linkages) between objects and such that expressions like 2 + 2 cannot be labeled as true or false since they are by the rules of the game not propositions, while expressions like 2 > 1 are propositions and they are to be spoken about of being true or false. — Zuhair

Why must "2+2" denote an object? Each symbol, "2" is itself an object. The two 2s are distinct symbols, distinct objects. Now the question is what is denoted by each of these symbols. In common language, a word has different meanings depending on the context, and this is how we know, without a doubt, that the different instances of what we call 'the same symbol", are actually different symbols, having different things denoted by each of them. Perhaps, the first "2" denotes something different from the second "2", and the "+" denotes a relation between these.

I agree that "2+2" on its own is not a proposition, but "2+2=4" is. So what is "2+2" on its own? It is just a part of a proposition, terms which need defining. If we add "=4" we complete the proposition and give some meaning to "2+2". Now, consider that as human beings, we require a proposition to identify an object. Yes, it is true that the object has an identity distinct from that given by us, and that is what is stated by the law of identity, (that the identity of the object is proper to itself), but we as human beings also want to give the object an identity, for our sake, and this cannot be done without a proposition. Therefore "2+2" cannot identify an object, because it is not a proposition, but "2+2=4" may be a proposition which identifies "4" as an object.

Now each of the 2s need to be defined, or identified, So we can define one 2 with "1+1=2", and the other 2 with "1+1=2". What defines "1" though? We have four different 1s here. In general, "1" signifies the fundamental unity, an entity, or an object, and each application of the symbol "1" is use to signify a different object. That's how we count, continually adding a new and different "1". Each instance of "1" is itself a different symbol, a different object, also denoting a different object, allowing us to count a multitude of different objects. And this is clear evidence that each instance of a symbol like "1", or "2", or "3", signifies a different thing, otherwise when we count, by continually adding "1", we would just be counting the same object over and over again and the count, the total, or the summation would be invalid because it requires that there are actually that many different things, for the total count to be valid.