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.

By the way if called on to do so, I could drill that symbology down to an identity of sets. The thing on the left and the thing on the right are the same thing. — fishfry

I think we're making some progress, and I think this is the heart of the disagreement between us. So maybe we could start with this statement about "symbology". Do you agree with a distinction between the symbol and what is signified by the symbol? The symbols themselves are objects which we read, and these objects, the symbols, have differences between the right and the left, such that they are not the same symbols.

Furthermore, according to the law of identity, two instances of what we call 'the same symbol", are not actually the same. So this "S" is not the same as this "S" (by the law of identity), considering them both as objects. So when we use "same" in this way, to say that they are the same symbol, we are using "same" in a way which is not consistent with the law of identity. This sense of "same" is based in some principle of similarity, not in the principle of identity. As a consequence of this, we have instances where the same word (as "the same symbol") has a different meaning, depending on context. And equivocation can result. This is clear evidence that the phrase "the same symbol", uses "same" in a way which violates the law of identity. Do you agree that this sense of "same", by which we say that this "S" is the same symbol as this "S", is inconsistent with the law of identity?

Please just substitute "identity" or "logical identity" in my argument. My apologies. — fishfry

I would have to find out what you mean by "logical identity". In general, the sense of "identity" employed by logicians is not consistent with the sense of "identity" described by the law of identity. If you read Stanford on "identity", you'll see two distinct senses. One they call qualitative identity, and the other they call numerical identity. Numerical identity is what is described by the law of identity, identity is specific to each and every thing, a thing is the same as itself. "The same" means one and the same thing, and this is identity. Qualitative identity identifies through qualities, a description. If two things have the same description, they are treated as the same. This sense, qualitative, is the sense generally employed in logic, because logic proceeds with descriptive terms, and predication. But a careful logician will recognize the difference between subject and object, and that predication deals with subjects rather than objects. The law of identity deals with objects rather than subjects, while the logician deals with subject. So when the logician employs "logical identity" it is an identity based in qualitative identity, (the subject is identified through some sort of description), rather than numerical identity which is what the law of identity deals with.

Only that I'm disappointed at a personal level that I took the trouble to work out an immaculate technical proof; and you are just totally disinterested in actually following and engaging with the argument. — fishfry

I told you the problem with your proof. The very first premise is false. So it doesn't matter how immaculate your proof is, it is unsound by that false premise. There is absolutely no sense in me wasting my time following all the points of your argument when you start with a premise which is clearly false. Until you fix that premise, or show me an argument which does not require it, there is no point.

Not that Metaphysician Undercover will be happy with any cavalier embrace of equivocation. — bongo fury

That pretty much sums it up.

The only inconsistency (or incoherence), MU, is you refering to the Past "two years Ago" which is synomous with two years Before "the present" yet dissociating Before and After from Past and Future. You use the former in terms of the latter, MU. Res ipsa locquitur. Saying "We cannot convert this before and after into past and future" is ... nonsense. — 180 Proof

The problem with this claim of yours, that two years ago is synonymous with two years before the present, is that it requires a further premise which states that the past is before the present. When you derive your concept of before and after from the second law of thermodynamics your are not provided with such a premise. Before and after are defined in terms of entropy. There is no reference to "the present" here.

Now we have to refer to concepts of "past, present and future" to determine whether "past" necessarily means before the present. So let's say past means gone by in time, having already occurred, and future means expected to occur, or going to occur. We need a principle by which we can say that things which have already occurred are "before" things which are going to occur, and vise versa, things which are going to occur are "after" things which have already occurred.

We can say that before means earlier in "time", and after means later in "time", But our concepts of "past, present, and future" have not produced a concept of "time", so this doesn't provide the needed principle. To derive a concept of "time" form these concepts of past, present, and future, requires that we take notice of an exchange between these, things expected to occur, do occur, and become things which have occurred, for example. This relates to the concept of "change", and we say that things are changing. Notice, that what we observe is a direction to change, future things become past things, and never the opposite. This is the principle which supports conversion of past and future into before and after. So, your claim that "past" is synonymous with "before the present", is supported by this empirical claim, an inductive conclusion, that there is a direction to change. "Earlier and later", "before and after" presuppose change, and a direction to change.

The issue I raised, is that there is a gap between the concepts of "past, present, and future", and "the directionality of change and time". These are distinct sets of concepts (conceptual structures) which have not been made compatible with each other. Take "past, present, and future" for example. We have the distinction between already occurred and expected to occur, and this is an intelligible difference. There is nothing here to indicated that what is expected to occur might not have already occurred in the past, and nothing to indicate that what has already occurred might occur in the future. These assumptions are supported by referring to the other conceptual structure, that of the directionality of time. That inductive principle places those restrictions. Likewise, within the conceptual structure of the directionality of time, there is nothing which provides a basis for distinguishing between events which have already occurred (past), and other events which are just expected to occur (future), because it provides no principle for dividing one section of time from another.

So what I was asking for, was the principle by which we relate these two distinct conceptual structures. It's obvious that the relation is made through 'the present", so that "past" becomes before the present, while "future" becomes after the present. But if this is the case, there are some ramifications. It appears like change could only occur at the present, and as I explained in my reply to Janus above, this is problematic.

Because the rules of arithmetic and and the arbitrary definitions dictates that! I showed you how formally this can run in a prior comment on a system that is by far much easier than PA. I'll re-present it here: — Zuhair

No you did not show me that. You showed me "ternary relation". You did not show me how one object and the "+" function makes a completely different object.

2 is Defined as the object that the + operator would send the single object 1 to. (this is: 1 + 1 = 2)
,,,
This is the principle of "Set", a set is what turns multiple objects into one entity, it turns Jesus and James — Zuhair

See, you have inconsistency here. First you are talking about sending one object to a completely different object, then you are talking about turning two objects into one object.

So for example we can view "Mother" in the above sentence as a "sending rule" it sends the pair {Jesus, James} to another object which here happens to be Mary. EXACTLY a similar thing is happening here the "+" operator is sending the pair {3,5} to another object which by rules of arithmetic and arbitrary definitions this other object is enforced to be 8. — Zuhair

OK, but this does not satisfy what is required in order to say that "3+5" denotes the same object as "8", which is what you are arguing, that these two are identical objects. All it shows is that the two objects "3" and "5" are related to a third object "8", like Jesus and James are related to Mary. It does not make Jesus and James identical to Mary. Your example introduces a fourth object, " the mother of". Mother is not a sending rule at all, it is a named object.

I didn't say that + would make 3 and 5 into one object, I said it will send them to one object, if I did say that it makes them into one object, then I only meant that it would send them to one object. + is not a merging process, it is an assignment scheme. — Zuhair

The debate is whether "2+2" denotes the same object as "4". If the sending rule only "sends" "3+5" to the third object "8", and does not make those two objects into the object denote by "8", it does not fulfil the requirement of saying "3+5" denotes the same object as "8".

There is (implicitly) "sum of". (Not that the analogy follows through completely, as Zuhair points out.) — bongo fury

The analogy is ill, and unacceptable. What acts as the sending operator in "Mother of Jesus and James", is the :"and". It sends both "Jesus" and "James" to the same object, "the mother of". Zuhair wants to make "mother" the sending operator, sending "Jesus" and "James" to "Mary". In reality though, "Mary" is just another name for the object called "the mother of" and the analogy is way off base.

But some theories I think would fare far better if they do that, for example Set Theory, here to say that the set X defined for example as: for all y ( y in X if and only if y=empty set ), this is usually symbolized as {{}} or as {0}. — Zuhair

What are you saying, that an empty set is the same as a set having only one member, "0"? You know that 0 is a mathematical object, just like any other integer don't you? therefore the set is not empty.

Zuhair thinks that mathematics would be so much better, stronger, if instances of "equal to" really meant "identical to", so Zuhair will forever argue that this is actually the case, in order to create the illusion that mathematics is stronger than it really is.

But 8 also refers to a completely different object from objects referred to by "3" and "5", and "8" is also not at all equivalent or the same as "3 and 5". I see the analogy is perfect! Why you say it fails? — Zuhair

It fails because you are arguing that the "+" makes "3" and "5" into one object, an object which is the very same as "8", but there is no way to make "Jesus" and "James" into one object which is the very same object as "Mary". Do you see what I mean?

(notice that 3 and 5 is not the same as 3 + 5, 3 and 5 is the totality of the objects referred to by 3 and 5, it is not what the + operator sends 3 and 5 to. The totality of the object referred to by 3 and the object referred to by 5 is NOT equivalent to the object referred to by 8, those are different objects, the latter is refers to an individual object, the former refers to a totality of two separate objects, so they are not the same nor are they equal). — Zuhair

I know that, but it's not the issue. The issue is how does the "+" turn two distinct objects into one? And further, how is that created object the very same as "8"? Your analogy does not turn Jesus and James into one object. And even if we could somehow understand Jesus and James unified as one object, this object would in no way be the same as Mary.

The mother of Jesus and James = Mary — Zuhair

There is nothing in "3+5" to take the place of "mother", there is just Jesus and James. If you remove the "and" from "Jesus and James", and replace it with "mother", you have "Jesus mother James". You have nothing to show whether the mother is of Jesus, James, both, or neither. So it is really the "and" in "Jesus and James" which acts as the function, to show that one object "mother" is related to both Jesus "and" James. So let's replace the "+" with "and", then we have "three and five", just like "Jesus and James", and the analogy is good. What unites Jesus and James into one object, like you claim three and five are united as one object? Clearly it is not "mother". It is some logical principle which allows us to speak of multiple things as one object.

By applying this logical principle we can create objects through some sort of synthetic unity. Consider "2" for example. It denotes an object, a number, but within the meaning of "2", there is two distinct things signified, united within one object, 2. This is a synthetic unity, the object 2, is created by a principle of union which unites two other objects..

Yes + sends objects denoted by the symbols it occurs between, to some object. The objects denoted by the symbols the symbol of + is written in between (in infix notations) would be sent by the + operator to an object as specified by the rules of arithmetic. Just because + occur between two symbols doesn't mean that the objects those two symbols are referring to are distinct objects No. For example "2 + 2" here the first and second "2" which are linked by + symbol, both of those do refer to exactly the same object, why? because 2 is a "constant" term of the language, so it can only refer to a single object in the universe of discourse. — Zuhair

You don't see the problem here? Let's say each "2" in "2+2" denotes the same object. Also, "+" sends objects denoted to some object. We have only one object denoted, "2". There are no other objects denoted, just "2" and the operator "+". If "+" sends "2" to any object other than "2", it could be any object, randomly selected. There are no rules of arithmetic which would allow that the object "2", and the operator "+", could give us an object other than "2".

Now the + operator would refer that single object (symbolized by "2") into the object referred to by the symbol "4", that's it. — Zuhair

This is what I mean, that selection of "4" is a random choice. Why not "5", or "8", or any other of an infinity of possible objects? Why does that + operator send the single object "2" into the object "4", and not some other object?

If I use the word same instead of equal does that satisfy you? — fishfry

No, that's the point you cannot validly substitute "same" for "equal". It will produce equivocation. You don't seem to understand this.

The fact that you say you read my post and this is your complaint means we're done. You have no substantive reply? I showed a proof from first principles that 2 + 2 and 4 are identical. You ignore it? — fishfry

You showed no such proof. You showed that they are equal, according to equality theory, and you claimed that equality is based in the law of identity which it is not. Therefore you have no proof. Do you understand? Your proof requires that the law of identity says something about equality, which it does not. Therefore you have no proof.

There is no case in math of an equality meaning anything other than identity; whether of abstract objects (logical identity) or sets (set identity or equality). Set identity is the same as set equality. — fishfry

We're just rehashing the same thing. It may be the case, that in math equality means identity as you claim, but as I explained to you, this is not "identity" as defined by the law of identity. So you (and all other mathematicians who say this) are using "identity" in a way which is inconsistent with the law of identity. Do you understand that the law of identity states that a thing is identical to itself? So to use "identity" as equality and then claim that equality is supported by the law of identity in that way, is equivocation, plain and simple.

If you don't agree that's your right, even if you haven't and can't show a single example to support your claim. But in all this time you have not presented an argument. And you have never engaged substantively. And from me to you, you're factually wrong. All the best. — fishfry

My example is the law of identity. It is very simple, and very explicit. A thing is the same as itself.

So for example, let's define equal to, as identical to, as you are wont to do. Then we'll adhere to the law of identity for our definition of identical. Clearly "2+2" is not identical "4" according to the law of identity, therefore "2+2" is not equal to "4" by your own definition of equality. That is a substantive argument if I've ever seen it. Now, Zuhair has wasted pages trying, to no avail, to demonstrate that "2+2" actually is identical to "4". But Zuhair is just digging a deeper hole, recognizing the truth that "2+2" is not identical to "4", neither is what "2+2" signifies identical to what "4" signifies, so Zuhair tends to lie in an attempt to get out of the hole.

Equal is the same as same. — fishfry

How can you say this? All human beings are said to be equal, but no two are the same. Many things are said to be equal which are not the same. Do you recognize this? If so, how can you say that "equal is the same as same"?

That's really strange. Just see the example of 'The mother of Jesus and James", this sentence is denoting a single object that is Mary, also Jesus in it is denoting an object and James too and those objects are different from Mary. Just because the whole sentence is denoting a different object from what some of its parts are denoting, it doesn't mean that it annihilates the existence of the objects denoted by its part. This is like saying if the above sentence denotes Mary the it annihilates the existence of an object denoted by "Jesus", and an object denoted by "James". — Zuhair

I can't see how the analogy is relevant. Jesus and James denote distinct objects. The two 2's denote one object. Even if we take "3+5" which clearly denotes two distinct objects, and say that "+" makes this into one object, this is completely different from "Jesus and James", because "Mary" does not make Jesus and James into one object. "Mary" refers to a completely different object with a relation to both Jesus and James. So if "+" makes "3+5" refer to a single object with a relationship to the objects "3" and "5", like the relationship which "Mary" has to "Jesus and James" how could this object be the same as "8"? Mary is a completely different object from Jesus and James, and not at all equivalent or the same as "Jesus and James". So the analogy really fails.

Do you see what I mean? It's very clear that if "+" makes "3+5" into a single object with the same relation to the two mentioned objects "3",and "5", that "Mary" has with "Jesus and James", this newly created object cannot be said to be the same as "8", because "8" has a completely different relationship to "3+5" from the relationship which "Mary" has with "Jesus and James"..

But the situation here is much more difficult, with a deeper, more fundamental layer of complexity. We were talking about "2+2", which you claim indicates two instances of the same object. This contradicts the law of identity already. So, before there is any point to discussing how "+" makes two objects into one, you need to demonstrate how it is consistent with your principles to treat two occurrences of "2" as denoting two different objects. According to the law of identity, there must be two distinct objects denoted here, but you claim that mathematical principles deny this. So, before you can talk about the "+" making two objects into one, we need to deal with whether "2+2" denotes two objects or one.

Those are not my claims. Please read about the syntax of first order logic which is the background logic used in foundational systems of mathematics. Please read what it means to be "terms" of the language, and also read about "functional terms" in particular and how to differentiate it from relational expressions. — Zuhair

Sorry but you've lost my interest, too much lying and too much nonsense..

That said the symbol 2 is taken to denote a single object in the universe of discourse because 2 is a constant symbol, while the expression "x" is a term that ranges over many objects of the universe of discourse, this means that x can be substituted by many objects of the universe of discourse. — Zuhair

I explained the problems with this position. If the numeral 2 always denotes the same object then the only relation between "2" and "2" is identity. So when I write "2+2" both 2s denote the same object and the expression is meaningless unless "+" symbolizes identity. Then "2+2" would say 2 is identical to 2. But "+" does not denote identity, so either the 2's denote distinct objects or the "+" is nonsense.

Can you explain what the "+" denotes in a way which is not nonsensical, adhering to your stated principle that the symbol 2 always denotes the very same object? You now have claimed that "+" is a function which denotes an object. So with "2+2" you have one object, another object, then the first object again. That's meaningless nonsense. Or, "+(2,2), which denotes one object and two instances of a different object, still meaningless nonsense. how could there be two instances of the same object?

Any binary function is a ternary relation, please read the syntax and rules of first order logic. — Zuhair

I hate to have to bring this to your attention, but you are getting further and further from showing that "2+2" is the same as "4", because you are making "2+2" more and more complex, while "4" is simple. Clearly they are not the same object, and your demonstrations are simply proving this.

I feel that your problem is that you were thinking of the "+" sign as a binary relation symbol linking two terms of the language and so the expression 2 + 3 would NOT denote an object. Which is wrong!

By convention the "+" sign is a binary function symbol linking two terms of the language, and so the expression 2 + 3 would BE denoting an object. — Zuhair

The problem is that you have repeated stated that "2" denotes an object. Unless the "+" annihilates the existence of the object denoted by "2", to create a new object, then "2+2" cannot denote an object as well as "2" denoting an object at the same time, without contradiction. So if "2+2" denotes an object, by what means is the object denoted by "2" annihilated in favour of this new object denoted by "2+2"? And, if "2" no longer denotes an object its meaning is lost, such that "2+2' can no longer be equal to "4".

The scientific method has been widely applied and has produced vast and seriously impressive results. That's what supports it. — S

It's sad that I have to tell you this, but this is false. Success only supports the trials of trial and error. And the scientific method is a bit more advanced than trial and error. Don't you think? The scientific method, as a way of acting, is supported by principles, not by successes. I think what you are really trying to say, is that scientism as a metaphysics, is supported by the successes of the scientific method.

So are you suggesting that there aren't any non-metaphysical - methodological - grounds for attempting to explain unexplained states-of-affairs? — 180 Proof

Right, the desire to explain what is unexplained is a subject of metaphysics.

It's supported by the vast results it has produced, which Platonic metaphysics hasn't come anywhere near to producing. — S

A method is a way of proceeding in activities. It may be supported by a system of guidelines, rules or something like that. That a specified activity has produced favourable results may be cited as justification for the method, only after the fact. Since this cited success is necessarily posterior to proceeding into the activities employing the method, it is impossible that this is what supports the method. To account for what supports the method is to account for the foundation of its existence. What supports the method is what inspires one to proceed into the activity employing the method, and this is necessarily prior to the success of the method, as a cause of its success.

I'm talking about a critical method of examining the world, irrespective of whether or not you would class it as metaphysics, and I'm contrasting it with Platonic metaphysics, and I was questioning the worth that Wayfarer spoke of in regard to Platonic metaphysics in light of this. That's when you decided to chip in. In political terminology, I would say that Wayfarer is a reactionary: decrying modernity and showing favouritism towards an ancient metaphysics. — S

OK, but metaphysics is a necessary support for any epistemology, and even the claim that it is not necessary is itself metaphysics. So any "critical method of examining the world" must be supported by metaphysics. If Platonic metaphysics provides a better support than modern metaphysics then Wayfarer is correct to value Platonic metaphysics in that way.

Change is temporal movement, which we conceptualize as past, present and future. — Janus

We do not conceptualize change as past, present and future, we conceptualize it as before and after. And, this before and after may be conceptualized as completely in the past, completely in the future, or part past and part future. There is no place for "the present" in the conceptualization of change.

...the present is where change happens, no not static... — Janus

Change requires time, it takes time for an event of change to occur. If a change were happening at the present, then when the change starts, part is in the future. Likewise, by the time the change is finished part would be in the past. If you assume a midway point, then part of the change is in the future and part is in the past. So a change cannot occur at the present because it has temporal extension, and part would be in the future, and part in the past, with none of it at the present.

This ought to become evident to you when you recognize what I said above, that we conceptualize change in terms of before and after. There is a before the change, or prior to the change, and there is an after the change, posterior to the change. That's how we understand change, before and after. And if you say that there is a duration of time during which the change occurs, then there is a prior part, and a posterior part of that duration with a point dividing them. if the prior part is in the past, and the posterior part is in the future, there is never any part which is present. The present is always a dividing point between future and past.

And also left unaddressed was my point about the impressive results which have been brought about through modern methods which ancient Platonic philosophy would have no hope of coming anywhere close to matching. — S

I have seen no impressive modern metaphysics, when compared to the ancients. Can you provide an example? We are talking about metaphysics, are we not?

Is it possible you missed this? — fishfry

No, I didn't miss it, we went through it already, your first premise is false:

1.1 We have the law of identity that says that for each natural number, it is equal to itself. — fishfry

The law of identity states that a thing is the same as itself, not that a natural number is equal to itself. This is the problem, you keep asserting that the law of identity says something about equality, when it does not. It says something about identity. So you wrongly proceed to claim that mathematical principles which say something about equality are base in the law of identity. That is why I kept asking you to back up this claim, that there is a law of identity which states something about equality.

The operator + here is a relation between three objects, one expressed by 3 and the other by 5 and the third by the expression "3 + 5". — Zuhair

So it's now becoming clear to you that "3" and "5" in "3+5" denote separate objects, just like "2" and "2" denote separate objects in "2+2".

So the + sign here is understood to be a ternary relation that links objects denoted by the terms of the language which are "3","5","3+5". — Zuhair

Right, we have a multiplicity of objects denoted.

and '3+5' is also taken to denote ONE object (because 3 + 5 is a binary function symbol and so it is a term of the language, so it denotes one object (despite having parts of it that denote other objects)). — Zuhair

You are claiming that a "binary function" is an object. I see no justification in this. A function is an activity, or a relation. Each of these may be a property of an object, or a relation between objects, but is not an object itself.

So to be more precise the operator + in "x + y" means a ternary relation between the object denoted by x in the first role, and the object denoted by y in the second role and the object denoted by "x + y"
so + sign in "2 + 2" means a ternary relation that links the object denoted by 2 in the first role and the same object in the second role and the object denoted by "2 + 2". — Zuhair

You haven't justified your claim the "2+2" is an object, nor your claim "+" represents a ternary relation. I think you have fallen back into your habit of lying.

I don't agree that eternalism denies the phenomenological temporal movement which is understood in terms of past, present and future. — Janus

The issue, as I explained to 180 is that the understanding of past present and future does not produce any understanding of temporal movement. There is a part of time called the past. there is a part of time called the future. There is a division between them called the present. There is no temporal movement implied here. But what we observe is change and to understand change we assume a temporal movement. However, you can conceive of the present as static, eternal, with the entirety of the physical world changing relative to the static present. The static present has no movement, so there is no temporal movement, only a changing world relative to the 'God's eye view'. But there is a fundamental incompatibility between the two perspectives, the one has "temporal movement", the other denies it. the entropy perspective, expressed by the second law of thermodynamics, necessitates a temporal movement, as the arrow of time. so that law produces an understanding of time which is inconsistent with the understanding of past, present, and future.

But evidently it's not treated quite the same now as it was back in its heyday, which was kind of the point. I wasn't implying that no one reads the books or that we have all of the answers. It has very largely been superseded, because it has lost prominence and a different methodology which has more to do with his pupil, Aristotle, and some who came before him such as Democritus, has largely taken over. — S

It wasn't prominent though. Socrates, and Plato, were two people who expressed dissatisfaction with the sophistry which was prominent at the time. Aristotle attempted to resolve some of the problems raised by Plato, so he has been often quoted. Now Aristotle has dropped from the forefront of metaphysics. And similar sophistry has made a resurgence and is abundant today, so there is a real need for Platonic dialectics.

I've written plenty to you already that you haven't engaged with, including a proof directly from first principles that 2 + 2 and 4 are identical. — fishfry

We went through this. You provided no such proof, it was just an assertion.

all of this is wrong. + is a binary function symbol which means it is a ternary relation symbol, it is a relation between three occurrences of symbols, it relates the first two symbols to a third occurrence of a symbol, here it relates the two occurrences of 2 to a third occurrence of a symbol which is 4, this is explicit when you write it in relational terms as +(2,2,4), but when it is written in functional terms here the confusion would raise since you don't see the third occurring symbol (which is 4) you only see two occurrences of 2 linked by + sign in between, here it means that + is relating the two occurrences of symbol 2 to the symbol '2 + 2', you see here the expression '2 + 2' is acting as a symbol denoting an object of the language. — Zuhair

Why the change? We were talking about objects, now you switch to symbols. Let's maintain consistency. We were talking about what the symbols denote. Tell me what "+" denotes, not how it relates one symbol to another. If it denotes a relation, then what is it a relation between, distinct objects, or one and the same object?

in your views '2 + 2' represent two distinct objects operated upon by the + operator. While the common view is that '2 + 2' denotes the natural number that results from running the + operator on two occurrences of 2. — Zuhair

If we have two occurrences of 2, then 2 here is a symbol. That symbol represents one object, the number two. If we have two occurrence of the same natural number, 2, then the only natural number represented is 2. What does your operator do, show that 2 is the same as 2?

Similarly '2 + 2' is an expression that mentions denotations of objects by two occurrences of the symbol 2 and an operator running on them, yet the total expression (i.e. all three symbols in 2 + 2 in that sequence) is denoting non of those, what is denoted by the total expression '2 + 2' is a single object that can be what is denoted by '4' if you interpret '=' as identity, or it can be another object that is related by some equivalence relation to the object denoted by 4, anyway the whole expression of "2 + 2" is not denoting multiple objects, no , it is denoting a single object, because + is a FUNCTION. — Zuhair

This all depends on what your operator "+" represents. You cannot jump to conclusions without explaining what the symbol represents. We have two occurrences of the same object, 2. We have an operator which expresses a relationship between them. As I told you already, the only reasonable relationship between the same object is that of identity. So is the operator "+" expressing a relationship of identity between the two occurrences of the same object? If not, then I think that the two occurrences of "2" are not symbolizing the same object. If there is an expressed relationship between two objects, which is not an expression of identity, then the two objects must be distinct objects.

Is there a critical basis behind your seeming favouritism towards old Platonic metaphysics? Maybe it has been forgotten for a reason. Maybe it has long since been superseded. — S

It hasn't been forgotten, because the books are plentiful and many read them. Some people though, do not, and therefore do not learn platonic metaphysics. It hasn't been superseded because the issues raised have not been resolved

In the context of your full reply, its almost a trick question for me: yes when addressed epistemologically, but no when addressed ontologically - ontologically they're two different facets of the same overall process. — javra

What I was trying to show is that the difference is ontological. But we have to approach a demonstration of ontology with epistemic principles in order to make sense, so the ontological difference I was describing appears epistemological. The point was that the determinacy of the future is known by a completely different epistemological process from the way that the determinacy of the past is known, and therefore we can conclude that these distinct types of knowledge, knowledge of the past, and of the future, have distinct objects.

If we are to address the present epistemologically, the present is that portion of time in which we (in part) hold direct awareness of everything that is not past and future. I’ve bracketed “in part” because, on one hand, the present is also where we intend things (with intentions always extending toward the future) as well as – hopefully not making this overly complex – being a time-span during which we are also aware of the past (memories) and the future (expectations). Still, when I’m aware of a bird chirp in the present, for example, this awareness pertains to neither the past nor the future. — javra

I don't completely agree with this, because I don't see how you jump to the position of drawing any conclusions about the present. The point of the thread was to approach the present from the position of recognizing a difference between future and past. When it becomes necessary to conclude that there is a difference between these, then the conclusion of a present, as necessary to complete the separation between them becomes justified.

You have suggested that we have direct awareness of something which is neither past nor future, and this amounts to the present. But until we established accurate principles for things past, and things future, we are just wasting time thinking about what kinds of things might be neither. Take your example of a bird chirp. You know that sensation takes time, don't you? So by the time that you are aware that you've heard a bird chirp, that bird chip is in the past. Therefore, I don't think it is possible that you could actually be aware of a bird chirp in the present.

Looking at it from this perspective, "the present" becomes quite difficult to grasp. We might be able to isolate and distinguish the way that we relate to the past, from the way that we relate to the future, in theory, but in practise these two are completely intertwined. So the theory tells us that the future is different from the past, and therefore we ought to conclude a present which separates these two. But in practise, our relations with the past are all mixed together with our relations with the future, and suddenly it becomes evident that there is no sharp division between the two. Then we ought to consider the possibility that what we call "the present" is really just a mixing of the future and the past.

You see an oasis in the dessert; at this moment, your drinking of water in a little while (the future) is plausible because the present experience currently isn't contradicotry to the past. But once you arrive there and there is only sand, you now know that the experience of the oasis was only a mirage - because this conclusion is now the only one that is not contradicotory to the entirety of your solidified past. — javra

I'll take this example of drinking water, and explain why I see an ontological difference between past and future. We have a difference between having drank water in the past, and, will drink water in the future. The truth, or reality of having drank water in the past relies solely on the accuracy of the memory. I remember having drank water, and we say that there is a truth or falsity to this memory, which is dependent on my capacity to remember. With respect to the future, I anticipate drinking water. But the truth or falsity to whether or not I will drink water is beyond my control, as your mirage example demonstrates. The way Aristotle explains this (the sea battle tomorrow is a famous example) is that there is no truth or falsity in relation to future events. So he explains with a number of examples, very articulately, how it doesn't make any sense to assume that there is any truth or falsity to whether or not specified future events will occur.

Because we are talking truth and falsity, it appears like this is an epistemological issue. But it is really an ontological issue which has epistemological ramifications. What it says is that the world is such, or the reality of being, existence, is such that we can make true and false statements concerning events of the past, but we cannot make true or false statements concerning events of the future.

And entropy determines the direction of the arrow of time, the notion of which is intelligible only in terms of past and future. — Janus

I think that this is demonstrably false. Entropy and the arrow of time are compatible with eternalism. "Past and future" only have meaning in relation to the present, and the present is not compatible with eternalism. Therefore entropy and the arrow of time do not rely on past and future for intelligibility. As I explained, they are based in before and after, which is distinct from future and past.

I conceptualize it differently. Something more akin to stratifications along a determinancy-indeterminacy spectrum. But I greatly doubt I'd be able to properly explain myself in the soundbite form that forum discussions require. — javra

But don't you agree that the determinacy of the future is distinct from the determinacy of the past, being grounded or justified in a different way? Determinacy of the past is grounded in consistency of memories which produces a certainty in the idea that something specific actually occurred. So determinacy of the past is complex due to the necessity of coherency in "what happened". Determinacy of the future is based in an assumption of continuity, the idea that things will continue to exist as they have, if not caused to change. And since things are caused to change, determinacy of the future is made complex by the need to understand causation.

Both forms of determinacy are complicated, but they are made complicated by different elements. So we cannot make one determinacy-indeterminacy spectrum, we would need two, one relating to the past and one to the future.

Why not? Please explain.

In the OP you asked "what type of knowledge allows us to say that there is a difference between future and past" not for "a principle". Entropy isn't merely "a principle" but a physical theory (re: statistical mechanics, thermodynamics, information theory ...) — 180 Proof

Entropy refers to the ordering of the descriptive quantities which describe a specified system. The second law of thermodynamics states that within a closed system entropy cannot decrease as time passes. So there is a number of problems here. First, we cannot find, or create a completely closed system so the theory cannot be properly tested. But more relevant, is the fact that if we take an increase in entropy as an indication of a later time, this produces the basis for a before and after. And we cannot derive past and future from before and after because past/future requires a principle which is not available within the concept of before/after. Before and after implies a changing time, movement in a particular direction, the supposed "arrow" of time. But future and past implies a specific point in time, dividing one section of time from the other.

So one (entropy) implies the notion of a changing time, while the other implies a static division between two distinct sections of time. The difficulty in understanding time is to establish consistency between these two ideas, as they both appear to be well grounded principles which are fundamentally inconsistent with each other. The approach of "entropy" makes the point in time (the present for example, but not necessarily the present) a moving target. The approach from future/past makes the point in time a static divisor. If the point is static, the idea of determinate periods of time is well supported. But if the point is moving, then the idea of determinate periods of time cannot be supported.

Btw, MU, stating that entropy (which describes the disordering of closed systems) can "distinguish between before and after" - that is, relations among discrete system-states [micro] - seems to entail differentiated magnitudes, or degrees, of disorder of closed systems in their entirety [macro], wherein Minimum Disorder corresponds to "past" and Maximum Disorder to "future" (i.e. Arrow of Time); and so, for consistency's sake, either entropy is "insufficient" for both - this I hope you'll explain - or sufficient for both (in different ways) which is epistemologically warranted (e.g. beginning with what I've sketched here). — 180 Proof

Why entropy is insufficient is that "past and future" cannot properly refer to the division made at any random point in time, which can be properly referred to with "before and after". For example, if we take a point in time two years ago, there is a before and after relative to that point. We cannot convert this before and after into past and future though, because "past and future" implies "the present" as the dividing point. And, it is quite obvious that this is inconsistent with the premise which clearly states a point in time two years ago. It's very clear that to call this point in time two years ago "the present" is a falsity.

.

Divine Choice or Will is an actuality in the sense of a "live option". — Gnomon

Care to explain what you mean by "live option"?

OK, that's fine. OF course just to make it more precise. I said almost all of mathematics before the era of set theory can be formalized as an extension of first order logic with identity where the symbol "=" is taken to mean "identity" i.e. "being the same as". Actually this is a well known result, actually most of that kind of mathematics can be formalized in second order arithmetic, you can read about it in reverse mathematics which also can be re-formalized as an extension of first order logic with identity. Actually ZFC itself can be formalized as an extension of first order logic with identity, and ZFC is way stronger than almost all of mathematics before the era of set theory. This is a very well known result. — Zuhair

And, of course, this re-formalizing is a logical fallacy, equivocation, as I've demonstrated.

what is that fundamental aspect that enforce us to interpret = sign as some equality relation other than identity. — Zuhair

What is on the right side of the equation is not the same as what is on the left. For example, in "2+2", there is as you've described, an "operator" signified, which is not signified by "4". Therefore it is very clear that what is on the right side is not the same as what is on the left side and so to interpret the "=" as signifying "the same" is wrong.

If there is something fundamental to mathematics against the use of = symbol in it to represent identity, then how PA is formalized as such??? How ZFC is formalized as such and it is generally regarded by many as the official foundation of mathematics? Both are indeed formalized with = in them understood as identity. — Zuhair

I don't believe that PA is actually formulated as such. None of the websites which fishfry referred me to, to support this claim supported that notion. Those websites described PA as based in equality theory, not identity. I think that those people such as yourself who insist on this notion are practising sophistry. I said that way back.

What you are saying is that the current foundational systems of mathematics are committing a fundamental error? (notice that most of those are coined as extensions of first order logic with identity) According to your account they must instead represent the = as an equivalence relation that can hold between distinct objects, and that the object denoted by 2 + 2 must be considered as a distinct object from that denoted by 4. This is strange? why? — Zuhair

No, I think those who interpret, or "coin" these systems as based in identity are committing a fundamental error. We discussed the axiom of extensionality, it defines equality, not identity. It is incorrect to say that this axiom of equality is an axiom of identity.

just want you to answer this question

does the expression "2 + 2" denotes two objects or one object?

I know that it contains in it the symbol 2 twice, that is clear, but do you think just because of this containment, then it ought to "denote two objects" — Zuhair

It is clear that "2+2" must be interpreted as denoting two distinct objects or else the "+" symbol is left meaningless. This is obviously a problem for the ontology of mathematical objects. We commonly think that every instance of the symbol "2" would denote the very same object. This would be a Platonic object, a number, symbolized by the numeral "2". But if this were the case, then we'd have a problem interpreting the "+" symbol. You'd have two instances of the same sign, representing the same thing, "2" with the "+" symbol representing a relationship between an object and itself. You cannot add an object to itself, so what could this "+" sign possibly represent as a relationship between an object and itself, other than identity, "2" is the same as "2"? But this is not what "+" represents. It represents a relation between what is represent by the first "2", and what is represented by the second "2", and if there is a relationship between these two which is not a relationship of identity, they must be distinct objects. Therefore what is represented by the first "2" in "2+2" is necessarily a distinct object from what is represented by the second "2".

2 + 2 is equivalent to the expression "The result of summation of 2 and 2" — Zuhair

But "the result of summation of 2 and 2", refers to two objects. There cannot be a summation without a multitude. You just want to say that "2+2" means "4", but it doesn't, it means something different than "4" and you don't seem to be capable of respecting this.

2 + 2 means "the natural number that results from adding 2 to 2" — Zuhair

You're so ridiculous that it's becoming funny. "The natural number that results from adding 2 to 2" is "4". If someone wanted to say "the natural number that results from adding 2 to 2" they would just say "4". Instead, they say "2+2", because this means something different from "4".

But the formalization would be more cumbersome, because you are holding to a weaker concept than identity, you'll loose all the merits of identity, which shortens formalization to a great extent. — Zuhair

I think I now see why you lie about this matter. Identity is seen by you as a stronger position. You want your position to appear stronger. So you will continue to lie in the claim that you are in the stronger position, attempting to deceive people into believing that your mathematics is stronger than it actually is.

By "semi-(in)determinate" I basically wanted to emphasize that not all future events are fully indeterminate. — javra

That's right there is a fundamental continuity, expressed in a very simply form as Newton's first law, inertia, which makes future events somewhat determinate. According to this law though, inertia may be interrupted by a force. This makes the determinateness of the future rather complicated because the continuity expressed as inertia is always being interfered with by forces. So the determinateness of the future is really reliant on the determinateness of forces, and "force" is a complicated concept.

Here's the basic problem with "force". By Newton's fist law, a force is what interrupts the continuity of predictability. In Newton's second law, the force itself is described as being predictable according to the principles of the first law. However, the predictability of the force itself may be interrupted by another force. This produces a potential infinite regress, exposing a fundamental indeterminateness. This indeterminateness indicates that we do not really understand the nature of force.

Of course they can be formalized as an extension of first order logic with identity, — Zuhair

Obviously this cannot be done validly, because "=" does not always mean "same", something you refuse to acknowledge, for some strange reason.

Anyhow almost all of traditional mathematics before the era of set theory and modern mathematical logic, nearly all of it can be re-formalized as extensions of first order logic with identity systems, and of course the "=" in them would be understood to represent identity. — Zuhair

Sure, you can interpret "=" as identical, but that would only produce a false presentation based in the fallacy of equivocation, as the website which you referred me to clearly states: "Note that an axiom like "(x)(x=x)" or "(x)Ixx" is not logically valid because there are interpretations of "=" or "I" that do not take the meaning of identity."

I don't know why you keep assuming that I'm lying? — Zuhair

The evidence shows that you know what you are asserting to be false. This is lying.

(2,2) ---+---> k

Now "2 + 2" is that object k, in other words "2 + 2" is not denoting the ordered pair (2,2), No! '2 + 2' is denoting the object that the operator + send the pair (2,2) to, and that object, i.e., k is exactly the natural number denoted by the symbol 4. In other words "2 + 2" is denoting exactly the same object that 4 is denoting. That's the easiest way to understand it. — Zuhair

There is no "k" though. What is symbolized is "2+2", two objects and an operator, not one object "k". So this object represented by "k" is not represented by "2+2", it has been wrongly created by you mind, false imagination, nothing here represents it.

That this is the case is evident from the fact that you proceed to say "that object, i.e., k is exactly the natural number denoted by the symbol 4". There is no symbol "4" in the expression "2+2". The object "k" is only on the right side, where the operator sends the ordered pair. So I can ask you, what does "k" really represent? Does it represent what "4" represents, as you say here, or does it represent what "2+2" represents, as you say above? You are only contradicting yourself.

Or perhaps that little arrow represents the same thing as "=", and all you are doing is stating "2+2=4".

However you look at it, there is no operator signified by "4", so it is very clear that "2+2" does not represent the same thing as "4". You've just stated my case for me. It is your repeated demonstration that you clearly understand that "2+2" signifies something different from "4", though you assert the opposite, which makes me say you are lying.

Perhaps you can clarify this point for me then. The law of identity is that a thing is equal to itself. — fishfry

The law of identity doesn't say that a thing is equal to itself, it says that a thing is the same as itself. In formal logic, "the same as" is represented by "=". So when the law of identity is expressed in formal logic as "a=a" or some such thing, the "=" represents "the same as". Zuhair is arguing that all mathematical axioms can be interpreted as "=" representing "the same as", but this is equivocation plain and simple. I am arguing that no mathematical axioms can be interpreted in this way because it is fundamental to mathematics that the two sides of the equation represent distinct things, while the law of identity indicates that "the same" refers to one and only one thing.

Well PA is a mathematical system. Most formal mathematical systems nowadays are stipulated as extensions of logical systems, in particular first order logic with identity. And it is about those mathematical systems that I was speaking. — Zuhair

Only specific mathematical systems are based in first order logic, perhaps ZFC is one of them. Now, the question is where does ZFC derive its meaning of "=", from traditional mathematics, or from the law of identity. As we seem to agree, though you are intent on making an impossible reduction, the two meanings of "=" are distinct. Since the axioms of ZFC do not mention the law of identity, but mention a theory of equality, I think it is quite obvious that ZFC derives its meaning of "=" from traditional mathematics, and not the law of identity.

I've shown you the axioms of first order logic with equality and you replied that the equality sign in them is not about identity, when I showed you that this is just a terminology preference, and that it is also named as first order logic with identity and I showed you the rationale behind those axioms and its relationship to the informal notion of identity, you replied that this is not mathematics. — Zuhair

It is not a terminology preference. In traditional mathematics "equal" means having the same value (and I'm sure you are fully aware of this), implying that two distinct things may have the same value. In the law of identity "same" means one and only one thing.

It is very clear that ZFC derives its meaning of "equal" from the traditional meaning of "equal", and not from the law of identity, because ZFC does not cite the law of identity, and as we've seen, it allows that two distinct things are "equal". Therefore "equal" in ZFC cannot mean "same" as determined by the law of identity.

In reality all older mathematical systems that you know of can be formalized as extensions of first order logic with identity, and in those systems the symbol = is taken to represent identity. — Zuhair

There you go, continuing with your lies. You are fully aware that this is not true, being the well-educated individual that you are. Yet you assert it anyway! Why lie? What's the purpose?

Now the question is what about older systems that are not formalized as extensions of first order logic with identity... — Zuhair

Didn't you just say "all" older mathematical systems can be formalized as systems where "=" represents identity? And now you ask about those which cannot. Oh what a tangled web we weave when first we practise to deceive.

But anyway your argument that the expression '2 + 2' is taken to represent two objects is outright false, even in ordinary math the expression '2 + 2' is taken to denote a single natural number that is sent to by the + operator from the pair {2,2} [more precisely one must write it as (2,2) since it is an ordered pair], it doesn't denote two natural numbers as you think, because + is a FUNCTION. — Zuhair

Exactly, an "ordered pair". And an ordered pair is two objects. Why say that this is false? Your propensity for lying never stops amazing me.

No! Equality rules are spoken as Identity rules by mathematicians, it just happens that equality is used more: see this site on terminology: — Zuhair

Your site provides the terminology of first order logic, not mathematics. The use of "=" is not the same in first order logic as it is in math. To equate these two is to equivocate and that is a fallacy of logic. That the symbol "=" means identical in first order logic does not demonstrate that the symbol "=" means identical in mathematics. Here's a quote from the section of the referred site on "identity" : " Note that an axiom like "(x)(x=x)" or "(x)Ixx" is not logically valid because there are interpretations of "=" or "I" that do not take the meaning of identity."

In all the mathematical sites which fishfry referred me to, none of them spoke of equality as identity. So you are just continuing with your hollow assertions.

So the theory that fishfry and I are mentioning is about "identity", yes its known as equality theory, other sources name it as identity theory, but basically it is about 'identity" as indiscernibility under substitutivity, and it is certainly not about equality as common reference (which is what you think it is about), it doesn't make sense to think of it as being about common reference, why should we have a law about indiscernibility of objects that has common value under certain functions?? — Zuhair

So you and fishfry just decided to change the name of the theory from "equality" to "identity", for some arbitrary reason. Or was that done for the purpose of deception?

In mathematics when = is used it is meant to symbolize "identity", i.e. sameness of objects, and not assignment to a common value as you think. — Zuhair

Now I'm convinced, you've changed the name from "equality" to "identity" for the purpose of deception, and you are flat out lying here. Why do you seek to deceive?

So you seem to be arguing that since '4' is not denoting that the object it denotes is an object that is divided in half, then it follows according to your reasoning that 4 is denoting an object that is not divided in half. This is an error. — Zuhair

Please reread this, and I hope you can see your mistake. Is '4' denoting an object? Yes. Is it denoting that the object is divided? No. Therefore '4' denotes an object not divided. Look, '4' does not denote that the object denoted is divided in half. That is clear. Therefore what is denoted is an object not divided in half. That the object, like any object, may potentially be divided in half, divided some other way, manipulated in any other way, or be converted into an infinite number of other objects, is irrelevant to what is denoted. What is denoted is an object not divided, and you are lying when you say that to interpret in this way is an error, because I can tell from the contorted way that you've written the passage that you are trying to disguise the truth. Why lie?

Not claiming something doesn't mean that you are claiming its negation. I'm not claiming that my son would pass the exam, it doesn't follow from this that I'm claiming that my son will not pass the exam. — Zuhair

We are not talking about claims, we are talking about denotations. If '4' denotes an object, it is impossible that the object denoted by '4' is divided in half or else it would not be an object denoted, but two other objects, the halves.

So 4 not denoting that what it denotes is dividable in half, doesn't mean that 4 is denoting an object that is not divisible in half. — Zuhair

What is denoted is what "is" denoted. If what is denoted is an object then that object is not divided in half, or else it would be two objects, regardless of whether the object is divisible. I hope you understand that it is contradictory to talk about one object which is two objects.

Absence of denotation doesn't mean denotation of absence.

Absence of denotation just signal incompleteness of information. — Zuhair

This is utter nonsense. What is denoted is what is denoted. It's completely nonsensical and illogical to say that the denotation could possibly include an infinity of other things which are not actually included in the denotation. Can't you see that you're just blabbering nonsense in an attempt to cover up your lies? The denotation is of something specific and what is not included in the denotation is not denoted. It's nonsense to argue that something else could have been denoted, therefore we should allow that what could have been denoted is part of what was actually was denoted

2 + 2 only shows some extra-information about what it denotes more than the constant symbol 4 shows about what it denotes. That doesn't mean that what they are denoting is not the same object. — Zuhair

But '2+2' denotes two objects, each with a value of two. What do you think the '+' sign is there for, decoration?

I can say that Barack Obama is one of the presidents of the united states. Another time I can say that Barack Obama is one of the presidents of the united states that has a Nobel price. The first expression did NOT denote that Barack Obama had a Nobel price, yet I didn't deny it! It is only the case that the second sentence had more information, but both are speaking exactly of the same person. In a similar manner 2+2 and 4 are denoting exactly the SAME object, but 2+2 is denoting more information about that object than 4 does, but again 4 is not denying what 2+2 is denoting. — Zuhair

You're not even talking about identity here. Do you know what predication is? What you call "more information" is predication. "Barack Obama" is one subject. What you predicate of that subject is something different from what the name denotes. Let's say our subject is "Barrack Obama". We can say many things about this subject, many different predications, one of which might be "is one of the presidents of the United States that has a Nobel Prize." How could this lead to the conclusion that "one of the presidents of the United States that has a Nobel Prize" is the very same identity as "Barack Obama"? Likewise, let's assume '4' as our subject. We can say many things about 4, many different predications, one of which might be "is equal to '2+2'". How does this lead to the conclusion that 2+2 is the very same identity as 4? That's nonsense. "More information", or predication, is something completely different from identity.

It's a fine philosophical distinction. Of course, in the real world Potential & Actual occur in pairs : Voltage & Amperage. But, the voltage in a battery can exist unrealized for years, until a circuit is completed by the user (plug it into a device and close the on-off switch). So, in Eternity & Infinity, transcendent Potential could theoretically exist independently, until triggered by a choice, an intention, which completes a circuit from Ideal to Real and back to Ideal again. In this analogy, G*D is both battery and user, both potential and actualizer. The device is our universe. — Gnomon

Do you see the "choice", as an actuality which is distinct from both the voltage and amperage? If the voltage is potential, it could sit there forever without an actuality (choice in this case) to actualize it. This is the problem with infinite potential. If the potential is infinite, it cannot be limited by any actuality, and this includes the actuality of the being who would make the choice, or any other actuality which might actualize the potential. So if there ever was infinite potential, there would always be infinite potential, which is not what we observe.

My understanding may be erroneous or naturalists (e.g. scientists) may misunderstand what they doing. — 180 Proof

Why do you class scientists as naturalists? Scientists might seek to understand the aspects of the natural world which are proper to their field of study, without reference to metaphysics, but they do not, in general, seek to understand the entirety of the natural wold. So they do not attempt, through their various fields of study, to explain the natural world, only to explain specific aspects of it. Many aspects of the natural world might be understood without employing metaphysics, but understanding an aspect of the natural world does not render the natural world as explainable. And this is where metaphysics is called for.

Do you consider, for instance, that merely assuming 'the natural world is explainable' is a "recourse to metaphysics"? — 180 Proof

Yes I do, because the statement of "the natural world" implies that the entirety of nature is one entity, a unity, and this is a metaphysical assumption. Furthermore, I think that to assume that something which has not yet been explained, is explainable, is a metaphysical assumption.

Also, more precisely, the empirical / computational concept of Entropy ... :death: — 180 Proof

I don't think Entropy provides us with a principle to distinguish between future and past. It may distinguish between before and after, but this is insufficient to distinguish future from past.

A very simple technique would be memory. We don't have memories of the future but we can remember what has happened. The part of reality that is now in the past imprints itself onto our memory and we can recall certain events with varying degrees of clarity. The future, being unexperienced, hasn't had a chance to imprint itself on our memory and so can't be remembered. This would be a simple method of distinguishing the past from the future. — TheMadFool

We touched on this briefly already. It's true that we remember past things, yet we might imagine future things. How do you think we distinguish, within our minds, remembered past things from imagined future things?

The past as memory is grounded in coherency between all memories. This is applicable both intra-self and between selves. When memories result in logical contradictions, something is amiss and we infer that something about our specified set of memories is wrong. Its only when all recalled memories flow effortlessly into themselves that we hold confidence in them. This applies just as well when we interact with each other. Our history is, experientially, composed of intersubjective memory. To the same extent that our memories, both personal and interpersonal, are found to be fluidly coherent and, thus, devoid of logical contradictions, our past is then determinate for us – unchangable. — javra

OK, this is good, consistency, lack of contradiction, corroboration, personally and publicly is an indication that what is in the mind is a memory, and not imaginary.

Intentions are all goal driven. In Aristotelian terms, telos guided. Add the premise of limited freedom of will to a) choose between different alternatives toward that goal(s) aimed for and b) to choose between different goals and the intention facet of the future becomes to the same extent (semi-)indeterminate. Add the fact that the future is partly created by the intentions of multiple selves, and this same indeterminate aspect of the future becomes even more so. — javra

This is the future part, goals and expectations, and I think it is much more difficult than the past, because of the doubt and uncertainty which you mention. But maybe this uncertainty is key to recognition of the difference between past and future.

Let's say that with respect to the past, it is easy to establish consistency and certainty in relation to what has happened, but in relation to the future it is more difficult due to uncertainty. This produces the distinction between determinate and semi-determinate which you referred to. But why do you think that the future is semi-determinate, not completely indeterminate? Doesn't this confuse the distinction, making it unclear? What produces the idea that the future is in some way determinate?

Returning to the consistency and lack of contradiction which we find in past memories, we find this also in our predictions for the future. However, predictions are very different from memories, and to be true they rely on the fulfilment of certain conditions. These are conditions of continuity. It is this continuity which gives determinateness to the future. If things continue to be in the future, the way they have been in the past, the predictions will be true. So the determinateness of the future is distinct from the determinateness of the past, because it relies on the condition of continuity, whereas the determinateness of the past is based in a corroboration of memories.

1.1 We have the law of identity that says that for each natural number, it is equal to itself. — fishfry

This is our point of disagreement. The law of identity does not say this, you are claiming this. If there is a law in mathematics which states that each natural number is equal to itself, it is not the law of identity. it is a law of equality. So what I am asking is, on what grounds do you say that this law of equality is a law of identity, and supplant the real law of identity with this one?

This puts the matter to rest. The expressions 2+2 2+22 + 2 and 4 44 refer to the same number. — fishfry

It shows me that they are equal, it doesn't show me that they are the same. That's the disputed point, that being equal means being the same.

These are strings of symbols manipulated by formal rules. — fishfry

Your mistake is in your citation of the 'rules'. The law of identity does not say that a number is equal to itself. I suggest you revisit the difference between identity and equality. "Identical" means the same whereas "equal" means having the same value. Do you see a difference between these two? When two things have the same value they are not necessarily the same thing. Being the same thing is what is necessary to fulfill the conditions of the law of identity.

On the math there is no question. 2+2=4 2+2=42 + 2 = 4 is an identity derived directly from the law of identity, the Peano axioms, and the definitions of the numbers and of + ++. As I say it's practically a definition. — fishfry

You keep saying this, that it is "derived directly from the law of identity", and you refer me to websites which discuss equality. Nowhere have I found the law of identity mentioned in this discussion of equality. So I really think that it is just you (and perhaps many others) who mistakenly believe that equality is derived from identity, and I am trying to point this out to you. Perhaps I am the one who is wrong, and equality is really derived from identity, but if so, where is the evidence of this?

So just go to PA to fill in the missing part, you'll see that for yourself. — Zuhair

Well fishfry seems to have done this already, but it doesn''t show identity, it shows equality. That is the point being discussed, you and fishfry seem to think that in PA equal things are the same thing.

the + is a two place function symbol, it is an assignment that sends pairs of objects to single objects per each pair — Zuhair

Right, this is what I said,. By showing parts, '2+2' indicates a particular division of the object, unlike '4' which indicates no such difference. So '2+2' denotes an object divided in a particular way, in half, whereas '4' denotes no such division. Therefore '2+2' denotes a different object from '4'.

When we way 2+2 = 1+3 we (in mathematics) mean that the single object that 2+2 denotes is "identical" to the single object that 1+3 denotes, that's what is meant. It means identity of denotation, that's all. — Zuhair

This is not true, '=' means equal, it does not mean identical. You are arbitrarily replacing what '=' really denotes, with "identical" and this produces a false statement. When you arbitrarily change the meaning of symbols in your interpretation, you create false statements.

I can exactly mirror you argument to say that "The Sun" and "The nearest star to Earth and Jupiter" do not denote the same object? since the first is just involving one object, while the later is involving a process of two things being near to a third object, and it involves the meaning of star, earth, and Jupiter, so it is speaking of TWO entities with a relation from them (near) towards a third entity that at the end points to that third object, so the denotation of those two expressions is distinct, which is WRONG. — Zuhair

This is totally irrelevant. What is WRONG, is to arbitrarily claim that two equal (having the same value) things are identical (the same).

And by rules of arithmetic (say PA) it PROVES that the single object denoted by 2+2 is exactly identical to (i.e. the same as) the single object denoted by 4. — Zuhair

This is what fishfry claimed to show above. But the demonstration does not show that the two are exactly identical, it shows that they are equal. Then fishfry states a misrepresentation of the law of identity, claiming that the law of identity states "that for each natural number, it is equal to itself". Where is your understanding of the law of identity?

We need first to agree on what constitutes a "denotation" of an expression, and then we can argue its identity. — Zuhair

What we need to agree on is definitions of "equal", and "same".

Equality axioms:
1. for all x (x=x)
2. if phi(x) is a formula in which x occur free, and never occur as bound, and y doesn't occur, and phi(y|x) is the formula obtained from phi(x) by merely replacing each occurrence of the symbol x in phi(x) by the symbol y, then all closures of — Zuhair

This is proof of your's and fishfry's mistake. You cite "equality axioms". Equality axioms are not identity axioms. You and fishfry both arbitrarily replace "equality with identity. Sophistry rules!

You are missing the power of potential. If a potential is not capable of causing anything, it's not potential, it's impotent. — Gnomon

I think you are twisting the distinction between actual and potential. Do you accept this division? If so, do you see that it is necessary for a cause to be actual? How can you say that a potential can cause something if you uphold the distinction between potential and actual and see that an act is required as a cause?

By definition, the cause of our world possessed the creative power to cause a world to exist. — Gnomon

Sure, but do you see that possessing the power to cause a world to exist is different from actually causing the world to exist? I possess the power to do all sorts of different things, but I don't necessarily do them. That's the point of contingent existence, 'potential' always refers to a multitude of possibilities, but the fact that one thing is actualized rather than some other possibility is only explained by causation. And, the cause must be something actual. So, we cannot account for the existence of our world, simply by saying that there was something which had that creative capacity, we need to also account for how that particular creative capacity was actualized.

Voltage is not a property, it's a prediction. — Gnomon

No, voltage is a description, not a prediction. According to Wikipedia it is the difference in electric potential between two points.

But that doesn't by itself entail that what they are denoting is not identical! — Zuhair

Right, and I've explained how what is denoted by '2+2' is different from what is denoted by '4'. Principally, '2+2' denotes two units of two whereas '4' denotes one unit of four. So, there are things which we can say that are true about 2+2 which are not true about 4, and vise versa.

The expression "The sun" and the expression "Nearest star to earth" are also not identical, the first contains two words, the last contains four words, but they do denote exactly the same object. — Zuhair

Sure, but '2+2' denotes two objects whereas '4' denotes one object. And, even if you construe '2+2' as one object, that object is divided in a vey specific way, in half. No such division is specified by '4'.

Now in PA the symbol 2 is meant to denote the object denoted by the expression S(S(0)), for simplicity let us use the notation || phi || where phi is a functional expression, to denote the OBJECT denoted by phi, so we have:

phi denotes || phi ||.

so according to that 2 is denoting the object || S(S(0)) ||.

Also 4 is denoting the object || S(S(S(S(0)))) ||

Now PA proves that the expression 2 + 2 is denoting the object || S(S(S(S(0)))) ||, which is the same object that expression 4 denotes! So by the meaning given to phi=pi in PA, PA proves that:

2+2=4

The proof of that is present in PA. — Zuhair

You seem to have left something out. You've taken the '+' for granted. You've shown me what '2' represents, and you've shown me what '4' represents. Then you claim that '2+2' magically represents the same thing as '4'. But all I see is a claim that S(S(0)) +S(S(0)) represents the same thing as S(S(S(S(0)))). Sorry to have to inform you of this, but you haven't provided the premise required to draw your conclusion. Consequently, you have no proof.

However to veer to YOUR side, one can in some sense use a terminology that separates identity from equality, you can stress that identity is full matching, i.e. even with expressions, those would be identical only if every property associated with one of them is also to be associated with the other whether at the language level or the meta-language level, and so you'll demand that everything must match between them even the way how those expressions are written. OK, by this we can say that equality is identity of denotation, and that identity is full matching. If we adopt such terminology then of course 2+2 won't be identical to 4, but 2+2 would be equal to 4, since there is identity of denotation of those expressions. This might be plausible, but it is not often used, well as far as I know of, but it might have its virtues. not sure though. — Zuhair

You really don't seem to understand the difference between equal and identical. Here's some principles which might help.
1. Two distinct things may be equal. For example, distinct human beings are said to be equal.
2. Two distinct things cannot be identical, "the same". "Same" refers to one and only one thing, (Leibniz principle for example, if x is the same as y then x is y, there is only one thing).
3. We do say sometimes, that a thing is equal to itself, as well as being the same as itself.