This is how I represent "one dollar", like that or like this, $1. Something equal to a dollar is "ten dimes", or "four quarters". I really do not believe that you can't see the difference, I think you're in denial. — Metaphysician Undercover

I see. You are emphasising the difference in representation. This is no different to what I mean when I say that they are different expressions of the same value. The point is that both sides of a mathematical equation have the same value (again: the same quantity), despite being different "representations" of that value.

You seem to think that the law of identity has some bearing on mathematics, or that A=A is somehow relevant to mathematical equations. I fail to understand what the relevance is. There would be little point using mathematical equations to state, e.g., 4=4. It seems that you just enjoy the confusion you create by treating the law of identity like a mathematical equation, or vice versa.

This is no different to what I mean when I say that they are different expressions of the same value. — Luke

OK, good, you are clarifying what you mean by expressions of value. MY point is that "different expressions of the same value" refer to different things. "Ten dimes" refers to something different than "four quarters".

You seem to think that the law of identity has some bearing on mathematics, or that A=A is somehow relevant to mathematical equations. I fail to understand what the relevance is. There would be little point using mathematical equations to state, e.g., 4=4. It seems that you just enjoy the confusion you create by treating the law of identity like a mathematical equation, or vice versa. — Luke

Fishfry has been insisting for months now, that what you call "different expressions of the same value" refer to the very same thing. It is claimed that each side of the equation, "2+2", and "4", both refer to the same mathematical object. I believe this idea is derived from the axiom of extensionality.

MY point is that "different expressions of the same value" refer to different things. — Metaphysician Undercover

You repeat the confusion. "Different expressions of the same value" are different wrt their expressions (or "representations"), but the same wrt their value.

"Ten dimes" refers to something different than "four quarters". — Metaphysician Undercover

Yes, but "ten dimes" and "four quarters" have the same value; they both "refer" to a value of one dollar. You have already demonstrated how problematic and confusing this becomes with an example such as "2+2=4".

Edit: There is no philosophical significance in pointing out that a dime is different to a quarter, or that "2" is different to "4". It goes without saying. You clearly exploit those cases where the values are the same (but the expressions are different) merely to provoke a response.

A fifth term you've now confuzzled up well and good? :)

There's more than [...] — Metaphysician Undercover

I don't see how it's relevant — Metaphysician Undercover

There's the English language, and its use of the word "value". Everyone else understanding should tell you something. Yet, your response suggests that you do after all. Odd. By the way, you may also want to think of the word "variable" (as in, variables may take on sets of values, and be represented by symbols). But please don't make this a sixth term.

You repeat the confusion. "Different expressions of the same value" are different wrt their expressions (or "representations"), but the same wrt their value. — Luke

You're missing something, what the expression represents, it's meaning. So there are three layers, the expression "four quarters", what it represents four twenty five cent pieces, and a third thing, the value we give that collection of coins, one dollar. Likewise, the expression "2+2", what it represents, the number two added to the number two, and the third thing, the value, four.

In each of these expressions, "ten dimes" and "four quarters", what the expression represents is different, not the same, regardless of whatever value you assign. The "value" is something else assigned to it through some principle of equality, or equivalence. Notice that this principle is not stated in either of the expressions. So the expression "four quarters" does not say "one dollar". You only conclude that value by applying that principle.

es, but "ten dimes" and "four quarters" have the same value; they both "refer" to a value of one dollar. — Luke

No, they do not "refer" to a value of one dollar. That's a false assumption. That four quarters has the value of a dollar is a conclusion produced by a logical process. Neither "four quarters" nor "ten dimes" refers to a dollar, but you can take what is referred to, and infer one dollar, when the logical process is applied. Do you see the difference between what an expression refers to, and what can be inferred from what is referred to by the expression, through a logical process? The implied conclusion requires further premises (often taken for granted) which are not stated in the expression, nor referred to by the expression. So, that four quarters is equal to a dollar, is a logical conclusion which requires a further premise not stated within the expression "four quarters".

Here's another example. If you have a red car, and I say, "the roof of my house is red", you could infer that the colour of my roof is the same as the colour of your car. But in no way am I talking about the colour of your roof. Likewise, if I say "I have four quarters", you could infer that I have a dollar, but in no way am I talking about having a dollar. And, if I say "2+2", you could infer that I am taking about the quantity represented by "4", but in no way am I talking about the quantity represented by "4". You simply apply some logical premise and make that conclusion. But applying a further premise, not stated by the expression, and making a logical conclusion from this premise, to insist that this is what the expression is saying, is faulty interpretation. It is not what the expression is saying, it is a logic conclusion that you've derived from what the expression is saying, through the application of a further premise.

Edit: There is no philosophical significance in pointing out that a dime is different to a quarter, or that "2" is different to "4". It goes without saying. You clearly exploit those cases where the values are the same (but the expressions are different) merely to provoke a response. — Luke

What are you talking about? That a dime is different from a quarter is obvious. What I am stating ought to be just as obvious, that ten dimes is different from four quarters, regardless of whatever value you assign to these things. Do you accept this fact, or are you in denial of the obvious, like fishfry? Do you believe that four quarters is the very same thing as ten dimes, just because you can apply some logical principle which makes them equal?

If having the same value meant being the same as, we could assign any random thing the same value as any other random thing, and conclude therefore that they are the same thing. That's nonsense.

Everyone else understanding should tell you something. — jorndoe

Well jorndoe, it seems very obvious from this thread that many people misunderstand. Jgill seems to be about the only one who does understand. Maybe you need to change that statement, and your perspective.

And, if I say "2+2", you could infer that I am taking about the quantity represented by "4", but in no way am I talking about the quantity represented by "4". You simply apply some logical premise and make that conclusion. — Metaphysician Undercover

There is no "logical premise" involved; that's simply how we use mathematical equations: the equals sign means that the value on the left is equal to the value on the right. "2+2=4" is a mathematical equation. To make the case that each side of the equation is different in a way which is unrelated to their values, i.e. in their symbols, or in what those symbols refer to, is just being a troll. Obviously, they are different in that sense; just look at the bloody symbols. That difference does not need to be pointed out. You are trolling for a response, and I won't oblige you any further.

here is no "logical premise" involved; that's simply how we use mathematical equations: the equals sign means that the value on the left is equal to the value on the right. "2+2=4" is a mathematical equation. — Luke

Do you not understand that a logical principle is required to say that ten dimes is equal to a dollar? Suppose that you had not learned any arithmetic and some one showed you ten dimes and taught you how to count the dimes, such that you knew there was ten dimes. How would you know that the ten dimes is a dollar unless you learned this further principle? Likewise, if someone just taught you how to count, and then said now take one two and add it to another two, you would not know that this is four. You need to know the further principle of addition to know that two plus two is equal to four.

Therefore, you cannot say that it is the equals sign between "2+2" and "4" which makes these two equal, nor can you even say that the equals sign means that they are equal. It is by means of that logical principle that they are equal. Otherwise, I could write "3+2=4", and the equals sign would mean that the value on the left is equal to the value on the right. But this is not true, because the logical principle of addition is not followed and adhered to, in this representation.

To make the case that each side of the equation is different in a way which is unrelated to their values, i.e. in their symbols, or in what those symbols refer to, is just being a troll. Obviously, they are different in that sense; just look at the bloody symbols. That difference does not need to be pointed out. You are trolling for a response, and I won't oblige you any further. — Luke

Call it "being a troll" if you like. I look at it as a trivial matter. The problem is that the difference needed to be pointed out, because fishfry kept insisting that the very same thing is represented on the left side as the right side of the equation. And when fishfry ceased arguing this, you took up that position. Therefore, I was obliged to point this out to you as well. If you now realize how obvious this is, I am amazed that it took you so long, because I stated very explicitly what I was pointing out, over and over again.

It really seems more like you were trolling me, arguing against me just for the sake of arguing against me, until you realized that what I am arguing is an extremely obvious truth. Now you accuse me of trolling, for making you look like a fool for arguing against something so obvious. But you engaged me, so you really trolled yourself.

AFAIK the Schrodinger equation's time evolution is deterministic, but that doesn't make the states deterministic. — fdrake

You mean it doesn't make the states stable or uniform. Determinism is commutative, but results can be unstable or changeable.

Call it "being a troll" if you like. I look at it as a trivial matter. The problem is that the difference needed to be pointed out, because fishfry kept insisting that the very same thing is represented on the left side as the right side of the equation. — Metaphysician Undercover

It's not trivial. Refine your claims -- this is what I gather from Luke's pleas. Whenever one invokes a mathematical equation, he or she is bound by a mathematical conclusion. In a game of chess all moves are determined and defined and understood.
Instead of arguing pages after pages of the same thing without changing one variable in your claim (you are engaging in futility, which amounts to nonsense) --
use the theory of meaning if you want to make a point on the the distinction between "2+2" and the number 4. Please start there.

it seems very obvious from this thread that many people misunderstand — Metaphysician Undercover

Af far as I can see, you're the only one that don't understand typical use of "value" (e.g., as in variables that may take values, like some/any proposition p in non-contradiction ¬(p ∧ ¬p)). I suppose, if you don't even (want to) try, then so be it.

Af far as I can see, you're the only one that don't understand typical use of "value" (e.g., as in variables that may take values, like some/any proposition p in non-contradiction ¬(p ∧ ¬p)). I suppose, if you don't even (want to) try, then so be it. — jorndoe

This is a good example. To be fair, values can also be invalid.
But I think Metaphysician's difficulty here, and other points on this thread, is the context or application of theory. I think MU is not trying to argue about equivalence --> 2+2 = 4, a mathematical proof, rather it is referent that's being invoked here.

Right, I've said this numerous times already in this thread, there is no question of whether or not 2+2=4. It does, without a doubt. What is at issue is whether "2+2" refers to the same thing, or even has the same meaning, as "4".

Some here seem to believe that "having the same value" implies "being the same mathematical object", such that "2+2" represents the same mathematical object as "4" does. I have argued that since "2+2" represents a more specific configuration of the four things indicated, it does not refer to the same intelligible object as the more general "4".

I just can't see how the fact that a specific variable can be assigned different values, is at all relevant. That's simply the nature of a value, because value is relative there is a degree of arbitrariness. One dollar appears to be a constant value, but when considered within the context of the international market, it is variable. This arbitrariness of "a value" is just further evidence that having the same value does not imply being the same intelligible object. Otherwise I could arbitrarily say that a chair and a table have the same value to me, therefore they are the same intelligible object.

I just can't see how the fact that a specific variable can be assigned different values, is at all relevant. That's simply the nature of a value, because value is relative there is a degree of arbitrariness. One dollar appears to be a constant value, but when considered within the context of the international market, it is variable. — Metaphysician Undercover

You don't seem familiar with the mathematical use of the term "value", which can describe any number or any result of a calculation (such as "2+2"). If you are familiar with this term, then I don't understand why you would describe this type of value as being "relative" or "arbitrary". Nobody else is talking about "value" in terms of worth.

A mathematical value is a type of "worth", it is the value which something has (what it is worth) within that mathematical system. Therefore that value is "relative" to that system. Evidence of this is the fact that if a person has not learned the system they will not be able to assign the proper value to the thing. The principles that the system is based in, the arbitrariness of the system, is a further matter.

Therefore that value is "relative" to that system. — Metaphysician Undercover

This appears quite different to your previous comments, where the value was not relative to a mathematical system, but instead relative to you:

Otherwise I could arbitrarily say that a chair and a table have the same value to me, therefore they are the same intelligible object. — Metaphysician Undercover

You also demonstrated the same misunderstanding about "value" previously, where you asked and asserted:

Do you even know what "value" means? It refers to the desirability of a thing, or what a thing is worth. — Metaphysician Undercover

It is clear that you have had this meaning of "value" in mind the entire time, and have misunderstood the meaning of "value" as used in mathematics, and by most of us here.

The principles that the system is based in, the arbitrariness of the system, is a further matter. — Metaphysician Undercover

A "further matter" that you don't care to explain? You didn't claim that it was "the principles that the system is based in" which were arbitrary; you claimed that it was value itself. You said:

That's simply the nature of a value, because value is relative there is a degree of arbitrariness. — Metaphysician Undercover

In what sense is a mathematical value arbitrary? There's no need to answer, because you clearly weren't referring to mathematical value when you said this.

This thread is no longer about the OP. To me it sounds unreasonable to say a casual chain goes back forever and that this is it's explanation. If you can't explain it without going back to infinity, adding past eternity doesn't help. The series has no efficacy because it's based on nothing, every member of the series being intermediate. Some posit a supernatural God or Gods in another "order" (orders are above dimensions) to start the series, but I like combining Descartes with Heidegger. Keep Heidegger's "potentiality-for-being" but throw out anything that looks like absolute time. You have the casual change going back to the first movement, a potential. People have trouble with how it goes from potential to actual without something or Someone (?) acting on it. I don't see the problem. The potential is in everything and all there is is the casual changes of the universe. What causes what can be debated, but the series goes back to to first pull of force in the mechanistic sense. There is nothing prior to it because time only is definable within the series as it moves. And here is where Heidegger can come in with his ideas of how being becomes actual for us. I think I have a comprehensive position and one that is a fine alternative to the Thomistic position on God. So you can atomically look at every motion of the series and see how it goes back to the first motion. This is a philosophical position of course

This appears quite different to your previous comments, where the value was not relative to a mathematical system, but instead relative to you: — Luke

The point is that it is relative to something. Whether it is relative to my own personal decision, or agreed upon decision (convention), does not change the nature of what a value is, itself.

It is clear that you have had this meaning of "value" in mind the entire time, and have misunderstood the meaning of "value" as used in mathematics, and by most of us here. — Luke

No, there you go again with your uncharitable interpretation for the sake of straw manning. You, yourself, introduced ambiguity onto the meaning of "value", trying to distance your use of "value" from my use of value, for the sake of your straw man, when no such separation is warranted.

In what sense is a mathematical value arbitrary? — Luke

Exactly as I explained. The value exists only relative to the system of evaluation. The system of evaluation may be entirely arbitrary if it is not grounded by something substantial. Look, it is completely arbitrary that the symbol "2" represents the quantitative value which we call "two". To remove the arbitrariness we might assume an object, a number, which "2" and "two" refer to. The reality of this object, the number two, must be substantiated in order that the arbitrariness be truthfully removed. This is the meaning of "2", which is to signify two distinct things that are not the same thing. Without this meaning of "2", what "2" refers to, the value associated with that symbol is completely arbitrary.

The important thing to notice, which is relevant to my argument, is that the two distinct things referred by "2" are necessarily distinct and different things, or else there would not be two distinct things, and the meaning of "2" would be lost. So when we say "1+1=2", or proceed in the act of counting, by adding another "1", to say 1,2,3,4,etc., each "1" must represent a distinct thing. Therefore each time "1" is used we must allow that this symbol may represent a distinct thing, and not necessarily the same thing (such as the number one). To claim that the symbol 1 represents the number one, or that the symbol 2 represents the number two is a false claim, because if it were true, then the fundamental use of these symbols in the act of counting would be invalid, "2' would not represent two distinct things, it would represent two instances of the same thing, the number one, which is only one thing, not two.

Therefore this attempt to remove the arbitrariness, which assumes that "1" refers to a number called "one", and "2" refers to a number called "two", is unacceptable, and the arbitrariness of the mathematical system of evaluation cannot been overcome in this way. Finally, the arbitrariness of the mathematical value system can only be overcome by grounding, or substantiating it, in principles which establish separation, individuation, difference between things, rather than an assumed sameness or equality of things. That two distinct things are equal, and therefore have the same value, is inherently arbitrary, but that they are distinct individuals, allowing us to number, or count them individually, is grounded in real difference.

The point is that it is relative to something. Whether it is relative to my own personal decision, or agreed upon decision (convention), does not change the nature of what a value is, itself. — Metaphysician Undercover

How is, e.g. the set of natural numbers, relative to your own personal decision? Also, how can it be relative if your decision "does not change the nature of what a value is, itself"? You're talking out of both sides of your mouth.

there you go again with your uncharitable interpretation for the sake of straw manning. You, yourself, introduced ambiguity onto the meaning of "value", trying to distance your use of "value" from my use of value, for the sake of your straw man, when no such separation is warranted. — Metaphysician Undercover

The ambiguity exists in the language because the word "value" has more than one meaning. If you think that mathematical value, or the set of natural numbers, has anything to do with "the desirability of a thing", then you are plainly incorrect.

Look, it is completely arbitrary that the symbol "2" represents the quantitative value which we call "two". To remove the arbitrariness we might assume an object, a number, which "2" and "two" refer to. — Metaphysician Undercover

That's an arbitrariness of the symbols used to denote a value, not an arbitrariness of the value itself. For example, the different expressions "2+2" and "4" both have the same mathematical value: 4.

That two distinct things are equal, and therefore have the same value, is inherently arbitrary, but that they are distinct individuals, allowing us to number, or count them individually, is grounded in real difference. — Metaphysician Undercover

You're aware that we can count independently of counting things, right? The arbitrariness you seem to be referring to is in what things we consider to be the same, not in the numbering system.

How is, e.g. the set of natural numbers, relative to your own personal decision? — Luke

It's relative to human convention, or agreement. Whether it was your idea, or mine, or someone else's is not relevant

Also, how can it be relative if your decision "does not change the nature of what a value is, itself"? You're talking out of both sides of your mouth. — Luke

I don't see how you construe this. Value is relative to human decision, whether it's your decision, my decision, agreement between us, or an enforced law, doesn't change the fact that it is relative to human decision. Therefore these issues of who's decision it actually is, are irrelevant to the fact that "value" is relative to human decision. There's no "both sides of the mouth" here, it's human decision plain and simple. You just seem to think that whose decision it was is somehow relevant.

The ambiguity exists in the language because the word "value" has more than one meaning. If you think that mathematical value, or the set of natural numbers, has anything to do with "the desirability of a thing", then you are plainly incorrect. — Luke

No, that's clearly wrong, mathematics has to do with the desire to count and measure things. Counting and measuring are desirable things. Therefore contrary to your ignorant assertion, assigning a quantitative value to things is the result of the desirability of something, counting and measuring, because these are desirable things to do, there's a purpose to them.

You're aware that we can count independently of counting things, right? — Luke

Oh yeah!, Finally you've gotten to the point. What exactly is a count, independent of counting things? It's just an arbitrary ordering of symbols. If we learned how to count, without actually counting things, we'd just be learning an arbitrary ordering of symbols. And as I explained in the last post, if "1" refers to an object called "a number", then "2" cannot refer to two distinct occurrences of that same number, or else we would not have two, but only one still. So in the act of counting, if it were independent of counting things, the symbols would refer to nothing, because they cannot refer to "a number", or else the count would be invalidated. Therefore that act of counting independent of counting something, is just an exercise in remembering an arbitrary ordering of symbols. If, on the other hand, you claim that three is one more than two, then there must be something which is being counted to validate this claim. It cannot be numbers which are being counted because each "one" being added which makes three one more than two, and four one more than three, must represent a distinct object, or else the count is invalid. So "one" cannot represent a number, nor can any of the other numerals represent a number because this would invalidate simple arithmetic. Therefore the act of counting, independent of counting things, is nothing other than an arbitrary ordering of symbols.

This arbitrariness of "a value" is just further evidence that having the same value does not imply being the same intelligible object. Otherwise I could arbitrarily say that a chair and a table have the same value to me, therefore they are the same intelligible object. — Metaphysician Undercover

A mathematical value is a type of "worth" — Metaphysician Undercover

So you've really managed to confuzzle value up good and well.

No, that's clearly wrong, mathematics has to do with the desire to count and measure things. Counting and measuring are desirable things. Therefore contrary to your ignorant assertion, assigning a quantitative value to things is the result of the desirability of something, counting and measuring, because these are desirable things to do, there's a purpose to them. — Metaphysician Undercover

Did you even look at the Wikipedia page I linked to earlier on Value (mathematics)? It's quite short; here's most of it:

In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as π or an integer such as 42.

— The value of a variable or a constant is any number or other mathematical object assigned to it.
— The value of a mathematical expression is the result of the computation described by this expression when the variables and constants in it are assigned values.
— The value of a function, given the value(s) assigned to its argument(s), is the value assumed by the function for these argument values.

For example, if the function f is defined by f(x) = 2x^2 – 3x + 1, then assigning the value 3 to its argument x yields the function value 10, since f(3) = 2·3^2 – 3·3 + 1 = 10.

A mathematical value can be a number, like 8 or 163 or pi. A set of numbers, e.g. the set of natural numbers, is a set of values. The meaning of "value" in this sense is completely different to the meaning of "value" in the sense of "desirable things". There is a value in using numbers, no doubt, but there are also the numbers themselves, each of which can be called a "value". These are different meanings of the word "value". You were clearly not aware of this.

And as I explained in the last post, if "1" refers to an object called "a number", then "2" cannot refer to two distinct occurrences of that same number, or else we would not have two, but only one still. — Metaphysician Undercover

1+1=1?

Therefore that act of counting independent of counting something, is just an exercise in remembering an arbitrary ordering of symbols — Metaphysician Undercover

Right, sort of like remembering the alphabet. Are you claiming it's not possible? Just because we can count (and do simple arithmetic) independently of "things" does not imply that we cannot count things or that we never count things.

In general, a mathematical value may be any definite mathematical object.

This is what I object to. In no way can a value be an object. Otherwise any count would be invalidated, as I explained. That there's a cult of mathematicians who believe that values are objects, when in actual usage value is really something predicated, indicates that these people believe falsity.

1+1=1? — Luke

If "1'" refers to an object called a number, then "1+1" indicates two distinct instances of the same object, which is still just the same object. So "1+1" would signify only 1 object if this were the case. However, in common usage it is correct to say "1+1=2". Therefore, to remain consistent with common usage and adhere to true principles of numerology, we must accept the conclusion that "1" does not refer to a mathematical object called a number because this would allow the representation of two distinct instances of the same object "1" to be the same as "2". But according to common usage in counting, "2" cannot refer to a second instance of the same thing.

Right, sort of like remembering the alphabet. Are you claiming it's not possible? Just because we can count (and do simple arithmetic) independently of "things" does not imply that we cannot count things or that we never count things. — Luke

What is not possible, is the notion that counting is a completely arbitrary ordering of symbols. If it were then mathematics would not be as useful as it is. So we can conclude that there is some meaning to these symbols, and learning to count is not a simple matter of learning an arbitrary ordering of symbols. It is a matter of learning the meaning of the symbols. Likewise, learning the alphabet is not a simple case of learning an arbitrary ordering of symbols, it is a matter of learning the sounds represented by the symbols.

The point I am making, is that if in learning how to count (learning the meaning of the symbols), we learned that one represents an object called a number, and two represents an object called a number, then we could not proceed from this understanding toward learning simple addition for the reasons described. However, this is not what we learn when we learn to count, we learn that "2" represents two distinct objects, not an object with equal value to two instances of the object represented by "1". Therefore, if someone comes along at a later time, after we've learned how to count, and tries to convince us that "1" represents a mathematical object, and "2" represents a mathematical object with the value of two distinct instance of the number named "1", we ought to reject this as a false representation of how we've learned to use numbers, and therefore a false premise. .

If "1'" refers to an object called a number, then "1+1" indicates two distinct instances of the same object, which is still just the same object. So "1+1" would signify only 1 object if this were the case. — Metaphysician Undercover

How can "two distinct instances of the same object" amount to only one object?

Therefore, to remain consistent with common usage and adhere to true principles of numerology, we must accept the conclusion that "1" does not refer to a mathematical object called a number because this would allow the representation of two distinct instances of the same object "1" to be the same as "2". But according to common usage in counting, "2" cannot refer to a second instance of the same thing. — Metaphysician Undercover

This is like arguing over the rules of chess with someone who doesn't know the rules. I'm done.

How can "two distinct instances of the same object" amount to only one object? — Luke

Isn't this obvious to you? If I count the object as "1" at time x, then I count the very same object as 2 at time y, this is a faulty count, counting the same object twice. Two instances of seeing the very same object, therefore a faulty count if I say there's two objects.

I should be asking you the opposite question, how do you think that two distinct instances of the very same object qualifies as two objects?

This is like arguing over the rules of chess with someone who doesn't know the rules. I'm done. — Luke

That's a sad analogy. Any time someone is going to argue with you over the rules of chess, you're going to insist that they do not know the rules, or else they wouldn't be arguing with you about them. Now I argue the rules of counting with you, and of course, you think that I do not know the rules of counting. Obviously it's you who doesn't know the rules of counting, because you think that the same object can be counted twice, for a count of two. Rule number one, you must count distinct objects, you cannot count the same object twice. No matter how many times the same object appears in front of you, you still only have one object.

Rule number one, you must count distinct objects, you cannot count the same object twice. No matter how many times the same object appears in front of you, you still only have one object. — Metaphysician Undercover

But what is an "object"? And what is a "metaphysical object"? Is there an overlap? :chin:

I would say that since metaphysics is the discipline which addresses the issue of what is an "object", and metaphysics therefore determines the meaning of "object", there is no difference between "object" and "metaphysical object". Where I see a problem is that "mathematical object" as people on this thread have proposed, does not seem to be consistent with acceptable metaphysical principles. In other words, it appears to me like mathematicians have posited a type of "object" which is metaphysically unacceptable.

Isn't this obvious to you? If I count the object as "1" at time x, then I count the very same object as 2 at time y, this is a faulty count, counting the same object twice. Two instances of seeing the very same object, therefore a faulty count if I say there's two objects. — Metaphysician Undercover

So we can’t add 1+1 - is that your argument? Because “1” is identical to itself? All mathematicians are wrong? How is “1” an object anyway? I note this is your first introduction of time into the scenario. I thought you meant the same type of object, not the same object counted again some time later. Like how we count three apples; they’re three of the same type of object counted on one occasion, not one object counted on three different occasions. But how we got to this point was that you stated:

if "1" refers to an object called "a number", then "2" cannot refer to two distinct occurrences of that same number, or else we would not have two, but only one still. — Metaphysician Undercover

So your argument is about numbers, not objects. This implies we can’t even add 1+1 because there is only one “object” which is the number “1”. Okay pal, whatever.

So we can’t add 1+1 - is that your argument? — Luke

No, "1" is a symbol. So long as each 1 represents a different object there is no problem to add 1+1 and get 2. But if both 1s are supposed to represent the same object I don't see how you could get two out of that.

No, "1" is a symbol. So long as each 1 represents a different object there is no problem to add 1+1 and get 2. — Metaphysician Undercover

Does “1” refer to an object called “a number”?

In other words, it appears to me like mathematicians have posited a type of "object" which is metaphysically unacceptable. — Metaphysician Undercover

From Wikipedia: "The strong, classical view assumes that the 'objects' studied by metaphysics exist independently of any observer, so that the subject is the most fundamental of all sciences. The weak, modern view assumes that the 'objects' studied by metaphysics exist inside the mind of an observer, so the subject becomes a form of introspection and conceptual analysis."