No; and that's why the order is irrelevant when determining if two sets are the same... — Banno

Yes, "the order is irrelevant when determining if two sets are the same". But the order of the elements is essential to determining the identity of a thing. And the law of identity is a statement about the identity of things. Therefore the identity of sets is not consistent with the law of identity. Understand?

There is no such thing as "THE" ordering for sets with at least two members. — TonesInDeepFreeze

Exactly, the ordering of the elements which make up "a thing" is essential to the identity of the thing. Therefore "identity" in set theory is not consistent with "identity" as stated by the law of identity, which is a statement about things.

We might go on and consider the supposed identity of an empty set as well. What type of "thing" has no elements in its composition? Well, that's not a thing at all, and it has no identity, because "identity" by the law of identity is a statement about things.

Further, we might consider whether a thing with infinite elements could really have an identity. That's a difficult philosophical question, which you might just take the answer for granted, because there's a serious lack of rigour in your concept of "identity".

The order of the elements is not part of what a set is. See ↪TonesInDeepFreeze — Banno

I know, and that's exactly the point, because order of a thing's elements is an essential aspect of identity. That's why if two sets are said to be "the same", they are not the same by the conditions of the law of identity, because the order of the elements is not included in that supposed (fake) identity..

How do you suppose that there is a thing which has an identity, yet that thing has no order to its elements? That's not a thing at all. And if it's not a thing it has no identity, by the law of identity, which is a statement about things.

Again, you have shown that there is no value in discourse with you. — Banno

Yes, as usual, I prove you to be wrong in your belief, and then you go off and ignore me for a period of time. The problem though, is that you never learn, and will come back later to argue what has already been demonstrated to you as wrong. Oh well, its no loss to me.

Or would it be more appropriate to say that advancing technology is good in virtue of something else? It's obviously much more common to argue the latter here, and the most common argument is that "technological progress is good because it allows for greater productivity and higher levels of consumption." — Count Timothy von Icarus

I think this is a good point. It is not technology itself which can be judged as good or bad, but the way that it is used, which is judged as good or bad. Technology can be used in bad ways as well as good ways, and certain technologies could be developed specifically toward evil ends. The point being that the purpose of human existence is not to produce technology, it is something other than this, so technology is only good in relation to this other purpose, regardless of whether we know what it is, or not.

I can also design trusses and figure pressure loss in pipelines. Doesn't that sound exciting. — Mark Nyquist

I'm interested to know exactly how pressure is lost in pipelines, if there is no leaks. I've heard that in the USA a huge amount of natural gas just goes missing. Where does it go?

Why would A=A imply that the order of the elements in B would need to be the same as A? — Banno

Jesus Banno, if A is the same as B, as implied by "A=B", (if "=" signifies identity, or "the same"), then the order of A's elements is the same as the order of B's elements, necessarily, as this is a part of "being the same"..

Order has nothing to do with this.

An ordering is a certain kind of relation on a set.

The axiom of extensionality pertain no matter what orderings are on a set. — TonesInDeepFreeze

That is the first, and most obvious piece of evidence which indicates that the axiom of extensionality does not state identity. Clearly "identity" by the law of identity includes the order of a thing's elements, as it includes all aspect of the thing, even the unknown aspects. So the ordering of the thing's elements is therefore included in the thing's identity, unlike the supposed (fake) "identity" stated by the axiom of extensionality.

Here is the axiom of extensionality:
If A and B are sets, then A = B iff every element of A is also an element of B, and vice versa.

Here is the law of identity
A=A

Set out for us exactly how these are not consistent. — Banno

To begin with, the obvious. "Every element of A is also an element of B" is insufficient for identity by the law of identity because "A=A" implies that not only the elements, but also the order to the elements of A and B would need to be the same. Furthermore, every aspect of what is named A, and what is named B, must be precisely the same, even the unknown aspects.

Quite simply, stating some feature such as "every element is the same", is insufficient to qualify as identity by the law of identity, because the law of identity, as "a thing is the same as itself", or "A=A", implies that every aspect of the thing must be the same to qualify as "identity.

I see no philosophy nor mathematics in your latest replies to me. It appears you've simply gone off the rails in your crackpot ways. Oh well, maybe next time you'll be able to stay on track and manage a reasonable discussion.

However, in more advanced mathematical contexts like set theory, "=" is sometimes used to signify identity, indicating that two objects or sets are the same in every aspect. — ChatGPT

Just be a mathematical antirealist and accept that “true” in the context of maths just means something like “follows from the axioms”, with the axioms themselves not being truth-apt. — Michael

Then you'd have to reject the axiom of extensionality, and all axioms which follow from it, and set theory in general. As I explained, allowing that there is an object (abstraction, conception, or whatever you want to call it), which is referred to by a description like "1+1" is "truth" by correspondence. So it would be hypocritical to accept axioms which are demonstrably based in "truth", correspondence, yet claim that they are not truth-apt.

You’re making a mountain out of nothing. — Michael

I'm not trying to make a mountain, just arguing a point, and points are "nothing". You are making points into a mountain by implicitly accepting Platonic realism.

It tells us how to use the "=" sign. It is an instruction, and so is not the sort of thing that can be false. You either follow the instruction or you do not. If you do not follow the instruction you are not participating in the logic of sets. — Banno

The problem occurs when that axiom is interpreted as indicated that when A=B, then A is "identical" to B, in the sense that "A" and "B" each signify the same thing, as @TonesInDeepFreezeargues. This would mean that there is a "thing", with an identity, which is represented by both "A" and "B", such as in the examples provided by @Michael and @TonesInDeepFreeze. However, since there is no necessity of order within a set, and also there is such a thing as an empty set, it is very evident that it would violate the law of identity to interpret the axiom of extensionality as indicating "identity".

Incidentally, I argued extensively with @fishfry, that to read the axiom of extensionality as indicating identity rather than as indicating equality is a misinterpretation. However, it seems like identity is the conventional interpretation, and there are further aspects of set theory which require that equal sets are the same set And that produces a problem.

[

It tells us how to use the "=" sign. It is an instruction, and so is not the sort of thing that can be false. You either follow the instruction or you do not. If you do not follow the instruction you are not participating in the logic of sets.

The law of identity has various forms, but in set theory it is that
A=B iff both A⊆B and B⊆A.
— Open Logic
This is a consequence of extensionality, not an axiom. — Banno

My argument is a very simple one, and I am not trying to build it into a mountain. The point is that the sense of "identity" employed in set theory is not consistent with, therefore violates, a proper formulation of "the law of identity" expressed as an ontological principle. That itself is not a big deal, many philosophers like Hegel for example, have argued that there is no good reason for logicians to have respect for that ontological law. Leibniz, on the other hand, for example, argued that this law, along with the related principle of sufficient reason, ought to be respected. You may portray this as "the law of identity has various forms", but if the forms are inconsistent with each other, that implies inconsistency in what we believe constitutes "identity".

The thing which irks me as a metaphysician, (and why I argue this point fervently), is when philosophers of mathematics insist that the sense of "identity" employed in set theory is consistent with the "law of identity", as stated in ontology. These philosophers will employ examples like Tones and Michael did, of the "identity" of a physical object, implying that the "identity" of an abstraction is analogous, through some misinterpretation of "extensionality".

The reason it bothers me is that the law of identity is the principal tool employed by Aristotle against the sophistry of Pythagorean/Platonic realism. If we allow corruption of that "law", and ignore the difference between "identity" as employed by set theorists, and "identity" as stated in the law, we give up the front line in that defence, effectively surrendering to Eleatic sophistry (ref. Plato, The Sophist)

What Meta is doing is refusing to use "=" in the way the rest of us do. — Banno

If any one of you would look at the evidence of what I've presented, the use of the equation in mathematics, they would see that "the rest of us" use "=" in the way that I describe. The common way, that of the applied arithmetic of the common people, and the applied mathematics of architects, engineers, and scientists, is the way I describe. It is only a select few, those immersed in the advanced mathematics of set theory, who desire, for the sake of this theory, that mathematical objects have an "identity", who choose to make "=" signify something different.

You ignore what ChatGPT told you in the other thread, common arithmetic and mathematics use "=" as equality not identity. And, in this thread GP said, that sometimes in "advanced mathematical contexts like set theory" ... "from the need to express relationships between objects", "=" will signify identity. Why do you refuse to accept what GP told you? That's because Tones told you 'don't to listen to that machine it doesn't support me', or something like that. But what is GP's account really based on? The "way the rest of us" use "=". Clearly, it's Tones who is "refusing to use "=" in the way the rest of us do", not Meta.

But that internal sensations cannot be treated in the way we treat other objects. — Banno

That's exactly the point. Objects each have an unique "identity", like Wittgenstein shows with the chair example. Even if two chairs might appear to be the same so that we couldn't readily decide which is which, we'd still know that through some temporal continuity each maintains its own unique identity. This ontological belief is expressed by the law of identity. Whether that law is actually true or not is not the point, it's just an ontological belief, and by believing it we assume that it's true. Internal sensations cannot be treated as if they have such an "identity". Therefore we make your conclusion, "internal sensations cannot be treated in the way we treat other objects". You seem to readily accept the conclusion which Wittgenstein comes to, without understanding the argument that he presents which produces it.

One may reject ideation and communication premised in abstract objects. But the notion of identity is not even limited to abstract objects. Whatever things one does countenance as existing, named by, say, T and S, we have T = S if and only if T is S. That is what '=' means when it is used in contexts of ordinary identity theory, logic, mathematics and other contexts to. If one wishes to use it with another meaning in another context, then, of course, fine. But that doesn't justify saying that in logic and mathematics it is not used just as logic and mathematics says it is used. — TonesInDeepFreeze

The sense of "identity" I am concerned with is that stated by the law of identity, "a thing is the same as itself". Do you agree with this formulation of the law of identity, and that if logic and mathematics uses "identity" in a way which is inconsistent with this, then logic and mathematics violate the law of identity?

Again, more exactly:

If 'T' and 'S' are terms, then

'T = S' is true if and only if T is S. — TonesInDeepFreeze

I would accept this as consistent with the law of identity, if we're careful to clarify that what we are talking about is the thing which "T" and "S" each signify. Clearly T itself, as a symbol, is not the same as S as a symbol.

And whether 'T' and 'S' stand for abstract things, abstract objects, values that are abstract things, values that are abstract objects, concrete things, physical things, or whatever things you are looking at right now on your desk. — TonesInDeepFreeze

The problem with this statement, is that a careful analysis and thorough understanding of what is here called "abstract things" will reveal that abstractions cannot be adequately understood as things with identity. So all these so-called "internal objects", conceptions, ideas, values, emotions, feelings, and everything else in this category, cannot be assumed to have an identity. This issue is extensively reviewed by Wittgenstein in The Philosophical Investigations. Particularly relevant is the part commonly known as the private language argument, where Wittgenstein provides the example of an attempt to assign the symbol "S" to a sensation. What is revealed is that "the sensation" cannot be known as having an identity. And this principle is extended by Wittgenstein to include all supposed "internal objects".

Due to what has been revealed by a large body of philosophical work in the past, I propose to you that if "T" and "S" are intended to stand for abstractions, conceptions, or anything else in this category commonly known as "internal", then "T" and "S" have no proper identity, as demonstrated by Wittgenstein. This was extensively covered by Aristotle under the concept of "substance", when he noticed the need to apply the law of identity against the sophistical arguments of Pythagorean idealists. Allowing that abstractions are identifiable things breaks down the categorical separation between ideas and things, allowing that the universe is composed of ideas.

Then, '1+1' refers the SUM of the number one with the number one. — TonesInDeepFreeze

This is incorrect, and this incorrectness I already explained to Michael. It is very clear that "1+1" refers to a specific operation which is indicated by "+". If we ignore this, and take a shortcut, assuming that the operation has already been carried out, and assume that "1+1" refers to the sum, then we ignore the role of "correctness" in the carrying out of the operation. Then one could stipulate any arbitrary expressions as referring to the same thing. I could say "1+1 = 8-2", and have my own private operations which produce this identity. In reality it is only through the means of carrying out the correct operation which is specifically signified by "+", that "1+1" can be said to be equal to "2". Therefore the meaning of "+" in that expression "1+1" is extremely significant to the meaning of the expression. It is intensional, and this intensionality cannot simply be taken for granted in the interpretation, to claim that the expression is extensional.

'1+1' does not stand for an operation. It stands for the result of an operation applied to an argument. — TonesInDeepFreeze

Obvious falsity. We read "1+1" as it is written, we don't read the implied result, "2". If what you said is true, then there would be no place for the learning of mathematics. We would not be able to account for the person who can read a mathematical expression, but cannot properly apply the principles required to produce the correct answers.

The truth of the matter is that the ability to correctly produce the answer, from the expressed operation, must be accounted for. It is simply not the case that a person goes from reading "1+1" as one plus one, to reading it as two, without a learning process, and that means acquiring the intensionality. The reality of this learning process, and how to properly account for it, is what Plato looked at in his theory of recollection, and what Wittgenstein looked at in The Philosophical Investigations.

And, I request that you please be honest with yourself. Do you really believe that you read the left side of an equation as the result of the expressed operation? That's simply not true, it's impossible because some operations are not carried out in the order that they appear. That is why I request honesty from you, and recognition that what is expressed by "1+1" is an operation to be carried out, not the SUM of that operation.

It is difficult to reason with someone about mathematics who doesn't understand that 1+1 is 2. — TonesInDeepFreeze

That goes two ways. When a person such as yourself, unwaveringly insists that the right side of an equation signifies the very same thing as the left side, despite a world full of applied mathematics as evidence to the contrary, then it becomes very difficult to reason with this person. The person simply refuses to look at all the evidence, and denies the evidential status of the evidence. The simple fact is that all the mathematical evidence supports what I say, so I am justified in my stance. But there is nothing but a stipulated "axiom" which supports your stance.

@Banno@MichaelThe issue we've encountered is that the axiom of extensionality is simply false. Of course, some will say that truth and falsity are not applicable judgements for mathematical axioms, and that is exactly why the axiom of extensionality is an ontological principle rather than a mathematical axiom.

What this so-called axiom attempts to do is to introduce truth and falsity into mathematics in the form of correspondence. It implies that there is an identified object which corresponds with the expressions of "1+1" and "3-1", replacing the true representation of 'correct answer' with this proposed corresponding "object". Now we'd have an "object" which corresponds with "1+1", as a form of truth, just like there is an object which correspond with "Mark Twain", as a form of truth.

This is why the axiom of extensionality is not a mathematical axiom, it is an ontological principle. Therefore it ought to be judged in a way which is appropriate to ontology.

The crank will mangle what I wrote, misrepresent it, presume to knock down strawmen of it. Likely, I won't have to time to compose a response, especially to the sheer volume of his confusions. — TonesInDeepFreeze

I don't see any strawmen, you just demonstrate a simple misunderstanding of how "=" is used in mathematics, and an equally simple refusal to seriously consider the evidence, resulting in a simple denial. Perhaps it would help you if we move on to more complex equations. Do you really believe that "2πr" signifies the very same concept as "the circumference of a circle"? Surely you recognize that "r" signifies a straight line, and "circumference" signifies a curved line, and by no stretch of the mathematical imagination do these two expressions represent the exact same thing. A curved line cannot be made to be compatible with a straight line, as indicated by the fact that pi is irrational.

What about our interest in crackpots like Tones?

he extensional reading of "1 + 1" is the number 2. — Michael

That's nonsense, you cannot read "1+1" as "2" because that's obviously a misreading. There is an operation signified by "1+1" and this implies that the reading of it must be intentional. It would absolutely be a misreading of "1+1" to read it as "2". And to get 2 out of 1+1 is intensional as well.

Also – and correct me if I'm wrong TonesInDeepFreeze – but "1 + 1" doesn't actually mean "add 1 to 1". Rather, it means "the number that comes after the number 1". And "3 - 1" means "the number that comes before the number 3". — Michael

See, this is proof that your reading of "1+1" is intensional. "The number that comes after the number 1" is clearly intensional, and that's how you read "1+1". You cannot read "1+1" as two because that would be a misreading. Only "2" gets read as two.

When you say "values" it seems you refer exactly to what is supposed to be the extensional reading of 1+1 or 3-1. So, if we are discussing values, saying that 1+1 is the same as 3-1 is correct, as both represent the same value, even if not the same operation. — Lionino

That's right, but Michael and I already went through this discussion. The values which are produced by "1+1"and "3-1" are only created by carrying out the operations referred to by "-", and "+". The expressions "1+1" and "3-1" refer to those procedures, not the values produced as a conclusion to the procedures. To conclude that "1+1" and "3-1" both produce the same value requires that the operations referred to be carried out correctly. Therefore, that "1+1", and "3-1" each produce the same value is dependent on correctly carrying out the operations which are represented by the expressions. What is represented by the expressions is the operations, not the values which result as a conclusion.

The problem is that both you and Corvus badly misrepresent Wittgenstein in an attempt to subjugate his name to your psycoceramics. — Banno

I like that description "psychoceramics". It makes me feel like I belong to a group, the psychoceramicists, rather than just a lone wolf.

But the result is that we are unable to have a significant discussion of constructivist views of maths. — Banno

Oh you poor little boys, can't keep yourselves from being distracted by the antics of a couple of psychocermacists.

I gave the Mark Twain / Samuel Clemens example as an illustration, not an argument, of the distinction between sense and denotation. — TonesInDeepFreeze

The problem being, that contrary to your claim, there are no things denoted in mathematics therefore mathematics is not "extensional" in the way of your analogy. @Michael agrees that mathematics deals with values rather than things. And since values are inherently intensional the mistake you made ought to be easily avoided by Michael.

If one rejects the view that abstract objects exist (and obviously, as abstractions, they don't exist physically), then, of course, the left term and the right term in an identity statement cannot refer to abstract objects. But that is a different objection than objecting to taking '=' as standing for the identity relation.

And if one objects to calling whatever mathematics refers to as 'objects', then we note that the word 'object' is a convenience but not necessary, as we could say 'thing' instead, or 'value of the term', or 'denotation of the term', or even none of that, and just say 'members of the domain of discourse' so that 'T = S' is interpreted as, for any model M for the language, M(T) is M(S). — TonesInDeepFreeze

It is not matter of whether abstractions exist as physical objects, it is a matter of whether abstractions exist as "objects", or "things" in any rational, coherent sense of the word. The law of identity states that a thing is the same as itself, and we can satisfactorily replace "thing" with "object", or vise versa, making them interchangeable for the sake of discussion. Now the issue is whether there is an identity relation (consistent with the law of identity) expressed by "=" in mathematics.

So, the demonstration and reason why, there is not a "thing" or "object" which is referred to by a numeral such as "1" or "2", and why that supposed "thing" would be incoherent and irrational if it was a thing which is referred to, is explained by my example of "1+1=2". If the two 1's both refer to the very same thing, then there is only one thing represent by those two 1's. Therefore no matter how many times we represent that same thing, we cannot have an equivalence with 2. So it ought to be very clear to you that "1" cannot refer to an object or thing because this would render mathematics as incoherent. Even the simple minded ChatGPT understood this example, and in the other thread where Banno presented this to it, it was very clear to say that in mathematics "=" commonly represents equality, "not identity".

Moreover, there is a difference between what is meant in mathematics by '=' and what one thinks mathematics should mean by '='. Whatever one thinks mathematics should mean by '=' doesn't change the fact that in mathematics '=' stands for identity. — TonesInDeepFreeze

This is exactly the problem which I've been repeating over and over. In common usage of mathematics, "=" signifies equality. GPT corroborated, even though you dispute its authority on common usage of mathematics. However, some mathematical theory, such as set theory defines "=" as signifying identity, regardless of how it is actually used in mathematics. This produces the problem you mention. Some people such as yourself, think that "=" should signify identity, because this would make it consistent with the theory they support, even though the fact remains that in mathematical usage "=" continues to represent equality rather than identity.

Do you agree, that when it is the mathematicians themselves, who are insisting on what "=" should mean, with complete disregard for how it is actually used in mathematics, there is a problem? This is a common epistemological problem demonstrated by Plato in the Theaetetus. Epistemologists have an idea of what "knowledge" should mean, 'JTB', and this supports their epistemological theory. However, as Plato demonstrated we cannot actually exclude the possibility of falsity pervading knowledge, so the T of JTB doesn't actually represent a true definition of "knowledge" according to what the word is actually used for. It simply represent what some epistemologists think "knowledge" should mean. Likewise, "=" does not mean identity in mathematics, it represents equality, despite the fact that some mathematicians think it should represent identity because that's what their theory states.


My criticism remains unaddressed. Let me put it more clearly. Since we are discussing values, not physical objects as in the case of your example, there is no such thing as an extensional reading of "1+1 = 3-1". That constitutes a misinterpretation.

I suggest you to read an elementary school book on set theory. There indeed are infinite sets and there can be a bijection between these sets. It's not just "mistake" like you think. — ssu

Bijection is a specific procedure. If you think that an infinite bijection can be carried out, such that you can produce a conclusion about the cardinality of a supposed infinite set, then you ought to be able to demonstrate this bijection. This would demonstrate that you have made a valid conclusion concerning the set's cardinality. And by "demonstrate" I mean to actually perform this bijection, not to simply represent it with a symbol or symbols, as if it has been performed. The latter does not qualify as a demonstration because one can make a symbol to represent any impossible conception, like a square circle, or whatever. Are you prepared to make that demonstration?

But an attempt at any such conversation in these fora would quickly be derailed by those who cannot grasp equality and those who misattribute and fabricate willy-nilly. — Banno

This I agree with. There is a serious problem with those who conflate equality and identity to "fabricate willy-nilly". We seem to be in much agreement in this thread, which is unusual. You have already pointed out the problem with people like Tones and Michael who claim to be doing mathematics when they are not. These two have displayed a need to refer to non-mathematical examples like Twain=Clemens, and the president of the United States, to demonstrate their supposedly "mathematical" principles.

Well, I can't explaining the mistake you're making in any simpler terms, so if you don't understand that then I can't help you further. — Michael

Like Tones' you refuse to stick to mathematics, committing the folly @Banno pointed to, a pretense of mathematics. Until you define and demonstrate how the distinction between extensional and intensional is relevant to a discussion of mathematical values, your reference to physical objects is completely irrelevant.

It's just language and just maths. — Michael

It's not maths, as both you and Tones have clearly demonstrated, by needing to refer to physical objects rather than mathematical values to support your claims of "identical".

You're conflating an extensional and intensional reading. To hopefully make the distinction clear, consider the below:

1. The President of the United States is identical to the husband of Jill Biden.

Under an intensional reading (1) is false because "X is the President of the United States if and only if X is the husband of Jill Biden" is false.

Under an extensional reading (1) is true because the person referred to by the term "the President of the United States" is the person referred to by the term "the husband of Jill Biden". — Michael

Sorry Michael, I cannot follow you. You've strayed from mathematics, just like Tones did with the example of Twain=Clemens. Your example, like Tones' appears to be completely irrelevant. To me, you've changed the subject and I cannot follow the terms of the change. If you want to continue this course, please demonstrate how it is relevant to mathematics. However, in the meantime I ask that you consider the following

es, and the values returned by both sides are identical. — Michael

Because of the issue with Platonism, It is not even proper to designate these values, the one produced by the right side, and the one produced by the left side, as "identical". Identity is what is assigned to an object, by the law of identity, "a thing is the same as itself". Notice it is a thing which is the same as itself, "identical".

When we recognize that the value produced by carrying out the procedure on the right side is "equal" to the value produced by carrying out the procedure on the left side, we implicitly acknowledge with the use of "value", that this is something within the mind, dependent on that mental activity of carrying out the procedure. If we use use "identical", instead of "equal" it is implied that what is really a value (something mind dependent) is an object with an identity. This is why Platonism is implied when we replace "equal value" with "identical value". It is implied that the value is an object with an identity.

The values returned are the same. What is represent by the right and left sides is not the value itself, but the operation. Therefore the "=" signifies an equality between two operations, it does not signify "the same".

We can go with that position if you want. It is irrelevant to the rest of the post, which demonstrates that "the value" of the right side, and of the left side is only produced by carrying out the procedure to its correct conclusion.

Given that 1 + 1 = 3 - 1, the value given by the procedure "add 1 to 1" is identical to the value given by the procedure "subtract 1 from 3" – that value being 2. — Michael

No that is clearly not the case, because these two procedures are completely different. They are said to result in the same value, 2, but the operations represented do not have the same value, nor are they identical.

Look at the two operations claimed to have an equal value. One is to take two distinct individuals and unite them producing a group of two. The other is to take a group of three and remove one individual, producing a group of two. Surely you cannot believe that these two procedures could have the same value. For example, if you had one dollar and someone gave you a dollar, that would be a far more valuable operation than if you had three dollars and someone took one dollar from you, even though they both result in you having two dollars.

And it is not the case that I equivocate with "value" here, because as I explained in the last post, the reality is that operators signify a different type of value from numerals. And, we must account for this if we are to assert that the value represented on the left side of the equation is identical to the value represented on the right side.

What we can see is that the conclusion of these two different operations results in the same value, 2. But it is clear that we do not have that "same value" unless we come to the correct conclusions in carrying out the procedures. So we have two very different operations each concluding with the same value as one another. The value, which is the same for both, is assigned to the conclusion, not the operation itself. But the operations are what is signified on the right and left sides.

If we assert that the two operations "1+1", and "3-1", each themselves have the same value, we neglect the very important fact that having the same value is really dependent on correctly carrying out the operations which are signified. Therefore "the same value" is attributed to the two conclusions, not to the two operations, themselves.

I propose that what you present here is a very sloppy analysis of what an equation actually is. The operation presented on the right side does not inherently have the same value as the operation presented on the left side, as you propose. What is really the case is that correctly carrying out the two operations, to their respective conclusions, produces the same value. I say it is very sloppy because it neglects the essential aspect of applied mathematics, which is to produce conclusions.

This sloppiness appears to be endemic to the philosophy of mathematics, and is very relevant to the issue of "infinite". The very meaning of "infinite" implies that there can be no conclusion to the operation. But the tendency in the philosophy of mathematics is to ignore the need for the human task of carrying out the operation (the consequence of Platonism which removes the requirement of human conception, I would argue), as you demonstrate with your example. So we find this mistake commonly with examples such as what @ssu suggested a bijection between the natural numbers. Obviously, by the conception of "the natural numbers", that they are infinite, it is impossible to conclude such an operation. Therefore it is impossible that there is such a bijection, or that it could produce a quantitative value.

We’re not saying that the symbol “A” is identical to the symbol “B”. This is where I think you are misunderstanding. — Michael

Of course, we are not talking about the symbols, we are talking about what the symbols represent. In your example, "A" represents something, and "B" represents something. The issue is, what "=" represents

In the context of maths, when we say that A = B we are saying that the value of A is equal to the value of B. The value of A is equal to the value of B if and only if A and B have the same value. — Michael

Right, A=B means that the value of A is equal to the value of B. This does not mean that A is identical to B, so the "=" signifies a relationship of equality, it does not signify a relationship of identity.

A non-identical but equal value makes no sense. — Michael

How could this be true? Two dollar bills are non-identical, but equal value. There is however, a very special relationship, which a thing has with itself, expressed by the law of identity (a thing is the same as itself), which is known as the identity relation.

Would you agree with me, that every identity relation (the relation a thing has with itself as expressed by the law of identity) is also an equality relation, such that a thing is equal to itself, but not every equality relation is an identity relation? In fact, in the vast majority of cases when things are said to have equal value (like two different dollar bills) they are two distinct things, and it is very rare, because it is rather useless, for a person to say that a thing is equal with itself.

We can skip right to the matter at hand, if you're prepared. Let's propose your example in slightly different terms, unambiguous terms which might better expose the issue. instead of saying "the value of A", and "the value of B", let's simply say that "A" represents "a value", and "B" represents "a value". Then when we say "A=B" we might claim that A and B both represent the same identical value.

But this creates a procedural problem in practice. Let's take the example "1+1=2". The value represented by "1+1" would be exactly the same, identical, to the value represented by "2". The problem is that "1+1"contains the representation of an operation, and "2" does not. And in order that an operation can fulfill what is intended by the operator, the operation must have a very special type of value. Because it is necessary to recognize this special type of value, that signified by the operator, it is impossible that "1+1" signifies the exact same value as "2", because there is no operation represented by "2". In other words the value represented by "1+1" consists of an operation, and the value represented by "2" does not, therefore they are not representations of the exact same value.

By a 'mathematical antirealist' I meant someone who thinks maths is invented, not discovered. Or someone who thinks that your "objects" in set theory only exist in our minds, or as pebbles or ink or pixels, etc. — GrahamJ

The issue is a complex one, but here's the simple explanation. If a numeral such as "2" signifies an object, then every time that symbol is used it must refer to the exact same object. However, if a "mathematical antirealist" believes that math is invented and these concepts exist only in human minds, then one must accept that the conception of "2" varies depending on the circumstance, or use. This is very evident from the multitude of different number systems. So for example, when a person uses, "2" it might refer to a group two things, or it might refer to the second in a series, or order. These are two very distinct conceptions referred to by "2". So, since "2" has at least two referents, it cannot refer to a single object. We could however propose a third referent, an object named "2", but what would be the point in that? The object would be something completely distinct from normal usage of the symbol.

That's why we decided to construct formal systems with prescribed definitions and axioms to ensure that our maths was consistent. — Michael

Big problem with consistency when the use of "=" is not consistent.

Yes, that's precisely right, and is why your talk of axioms being "false" is nonsense. Axioms aren't truth-apt; they're just either useful for their purpose or not. And given that the axioms of ZFC are the most prominently used, it stands to reason that they are considered to be the most useful. And that's all there is to say about them. — Michael

In the sense that axioms are a representation of what mathematicians are doing, they can be judged as true or false, just like any other description. However, as you rightly describe, a judgement of the truth or falsity of an axiom is not required to judge whether it appears to be useful or not.

So this is where self-deception enters the environment. If a mathematician accepts an axiom because it is useful, but it is not representative of what that individual is doing mathematically (and this I argue is the case with the axiom which makes the claim about the relation between identity and equality), then the usefulness of that axiom must be in relation to something other than mathematics. It has some other purpose than a mathematical purpose.

Regarding the "=" sign, it was invented in 1557 by Robert Recorde:

And to avoid the tedious repetition of these words: "is equal to" I will set as I do often in work use, a pair of parallels, or duplicate lines of one [the same] length, thus: =, because no 2 things can be more equal. — Michael

Notice "two things". Equality deals with two things, identity only involves one thing.

Isn't there a bijection between the set of natural numbers and the set of natural numbers? — ssu

That's a bijection which cannot be carried out, cannot be completed. It's a nonsensical proposition.

Was that early or also late Wittgenstein? Because I suspect late Wittgenstein wouldn't have read any metaphysics into mathematics or set theory. They're just a useful language game we play, not something that entails the realist existence of abstract mathematical objects. — Michael

Notice early Wittgenstein talking about representing the world in terms of "elements". Notice later Wittgenstein rejecting this as not representative of what is really the case in the world.

It should be evident to any well trained philosopher, that set theory is just terrible philosophy. I think that is what bothered Wittgenstein about mathematics, but he was a bit too timid to actually come out and state it.

So when the issue is set theory, isn't then more correct just to talk about a bijection? — ssu

I don't see any issue with bijection in principle. But when it is proposed that the quantity of a specific set is infinite, bijection would be impossible. The proposal of infinite sets presents numerous procedural problems. That is self-evident.

It's not that we use maths and then retroactively describe what the symbols mean and infer the axioms; — Michael

You have this wrong. A study of the history of mathematics will reveal to you that the axioms come about as a representation of usage. We could start with something like "the right angle", and see that the Egyptians were using that concept to create parallel lines and things like that, far before the axiom, the Pythagorean theorem, which represents this usage, was expressed.

As I recently explained in a related thread, since axioms are determined by choice, and used by choice, we must accept that axioms follow usage, they do not determine usage. People can produce whatever axioms they like, but if they are not useful they will not be used, nor become conventional. So, the axioms which become the convention are the ones best representative of what mathematicans are actually doing.

In the case of the axiom of extensionality, it is useful for a purpose other than mathematics. It's use is rhetorical, to persuade people of the usefulness of set theory. It is clearly not true though, because, for example, the order of the elements within a set is not accounted for. So, sets which are said to be identical may have the same elements in a different order. But in any true sense of "identity" order is an essential feature. Therefore the rhetorical use of this axiom is really a matter of deception.

The symbol "=" is defined in ZFC by saying that "A = B" is true if and only if A is B. — Michael

Yes, and as I've shown over and over again, that definition of "=" is not representative of how "=" is actually used in mathematics. Therefore it is a false definition, designed for some other purpose, foreign to mathematics.

I can only imagine a unicorn by picturing a unicorn. A picture requires a "concrete instantiation". A "concrete instantiation" can be on a screen or a piece of paper. Both a screen and a piece of paper are physical objects existing in the world. As physical objects in the world, I can sense them. — RussellA

This is clearly incorrect. We can imagine things without a concrete instantiation. That's how artists create original works, they transfer what has been created by the mind, to the canvas. It is also what happens in dreams, things never before seen are created by the mind.

For a mathematical antirealist, does any of this constitute hypocrisy?

(@Metaphysician Undercover mostly.) — GrahamJ

I can't see the relevance. Your game clearly involves real objects, pebbles, or in the case of your presentation, the letters. Would the antirealist insist that these are not real objects?

Apparently, people will also try to do mathematics without the mathematics. — Banno

Those are the people who say "=" signifies identity in mathematics. They claim to be doing mathematics when they say that "1=1" means that what left 1 signifies is the same as what the right 1 signifies. But that's obviously not mathematics. In mathematics, the left side of the equation always signifies something different from the right side, or else the equation would be useless.

It's one thing for non-mathematicians, who don't know any better, to think that what they are doing is mathematics, when it's not. But it's truly shameful when mathematicians claim to be doing mathematics when what they are doing is not mathematical. As I explained already, that's how they come up with false axioms.

I imagine a unicorn by picturing a unicorn. — RussellA

Similar, but I wouldn't call it "picturing". Anyway, the point is that this "picturing" does not require a "concrete instantiation", which I assume implies a physical object being sensed.

"1 = 1" is a mathematical expression. — RussellA

When you say "=" signifies identity, "1=1" is not a mathematical expression. Think about it. If "1=1" means that the quantitative value signified by the first "1" is equivalent to the quantitative value signified by the second, then this is a mathematical expression. And in the case of ordinals, if "1" signifies "first", and the expression means that the first is equivalent with the first, as first, then this is also a mathematical expression. But if "1=1" is meant to signify that the thing identified by "1" on the right side is the very same as the thing identified by the "1" on the left side, then it is not a mathematical expression. It is an expression of identity. And, the fact that it is analogous with Twain = Clemens, which is clearly not a mathematical expression is evidence that it is not a mathematical expression.



I could address your examples, but I do not see how they are relevant really. In set theory it is stated that the elements of a set are objects, and "mathematical realism" is concerned with whether or not the things said to be "objects" in set theory are, or are not, objects.

To play chess you must accept the reality of the pieces as objects in order to move them, therefore you must accept "chess reality" to play chess. Since it may not be stated in the rules that the pieces are "objects" the acceptance is only implicit, unlike set theory in which case the rule is explicit, therefore acceptance is explicit.

It seems to me that you do not understand "realism". Do you agree, that to be able to take "an object", manipulate it, move it, do whatever you please with it, or move it according to some set of rules, you need to accept that the object which you are doing this with is "real"? And this implies that believing the things which you are manipulating to be "objects", implies some sort of realism. Or, do you separate "realism" from "objects", so that realism has nothing to do with objects? In which case, what would you base "realism", and consequently "antirealism" in?

I already did above. The axioms of some given set theory are just rules that you must follow when using that set theory. Different set theories have different axioms and so different rules. Given that there's no connection between using some set theory and believing in the mind-independent existence of abstract mathematical objects, there's no hypocrisy in using some set theory and being a mathematical antirealist. — Michael

"Mathematical antirealist" is also a "rule". It states an ontological principle, or rule. It is a rejection of mathematical objects. The rules of set theory are inconsistent with this rejection of mathematical objects, because set theory assumes mathematical objects, as a foundational premise. Therefore you must assume mathematical objects, as a fundamental premise, to be able to follow the rules of set theory. This activity is contrary to the ontological belief stated as "mathematical antirealist", and is therefore hypocrisy for anyone claiming to be a mathematical antirealist.

Your position is like arguing that it's hypocritical to play chess if I do not believe that the rules of chess correspond to some mind-independent fact about the world. — Michael

I don't see the relevance. You do not need to accept the premise of "mathematical objects" to play chess. You do need to accept the premise of "mathematical objects" to follow the rules of set theory.

And how would you justify that claim?

You don't need to believe in Platonic realism to use set theory. — Michael

I agree. I didn't say you need to believe in the truth of the principles you employ. However, it's hypocrisy to say "I'm a mathematical antirealist" and then go ahead and use set theory. But that sort of hypocrisy is extremely commonplace in our world, it's actually become the norm now. Very few people make the effort to understand the metaphysics which they claim to believe in, and whether it is consistent with the metaphysics which supports the theories which they employ in practise.

Why? — Michael

Set theory begins with the assumption of mathematical objects, hence it is based in Platonic realism.

I wonder if mathematical realists and mathematical antirealists have different views about mathematical infinity. I'm a mathematical antirealist. I have no problem with mathematical infinity. The "existence" of infinite sets does not entail the existence of infinities in nature (whether material or Platonic). — Michael

The issue is with the premises (axioms) of set theory. If you are "mathematical antirealist" you ought to reject set theory on the basis of the axioms it employs. Your view on "infinity" would be irrelevant at this point. So how set theory treats "infinite" would not even enter into the reasons for your rejection of it.

Set theory is based in the assumption of "mathematical objects". And, the mathematical objects as elements of the sets, are allowed to have relations which physical objects, according to our knowledge of them, cannot have. So a "set" by set theory is a bunch of "objects", but since they are mathematical objects instead of physical objects, what can be 'truthfully' said (what is acceptable by the axioms) about that bunch of objects, doesn't have to be consistent with our knowledge of physical objects. So for example, there can be an empty set (a bunch of objects with no objects), and sets do not necessarily have an order (a bunch of objects without any order to them).

Exactly, you understand the concept using images. — RussellA

The key point here, is that imagination does not require sensation of whatever it is that is imagined. But you claim understanding a concept requires "concrete instantiation" and I assume that means something which is sensed. If I'm wrong here, and "concrete instantiation" means the production of an image in the mind, without the requirement of sensing it, then we might have something to discuss.

There are two different cases.

The first a case of identity where the two 1's refer to the same thing. The second a case of equality where the two 1's refer to different things.

The practical advantage of using identity rather than equality is to distinguish two very different cases. — RussellA

Yeah sure, you've indicated that in the first case "=" signifies identity. I agree, that's what you've stipulated. The point though, is that in the case where you used "=" to signify identity, it is not a mathematical usage. The "practical advantage" you refer to is rhetorical only, intended to persuade me. The usage is not mathematical, because in the application of mathematics "=" is not used to represent identity. That's the issue with Tones' example of Twain = Clemens, it just demonstrates that it's possible to use "=" in this way. However, it doesn't at all represent how people applying mathematics actually use "=" in the formulation of equations. So it's nothing but a rhetorical example, produced solely for the purpose of trying to persuade, in the mode of sophistry.

"A mythical animal typically represented as a horse with a single straight horn projecting from its forehead" describes an object, even through the object is fictional. — RussellA

I would definitely disagree with this. There is a big difference between seeing, hearing, touching, or otherwise sensing an "object", thereby describing what i sensed, and creating an imaginary "object". The latter does not involve an object, nor does it involve a "description" ( in the proper sense of the word) because it is an imaginary creation an invention rather than a description. A "fictional object" is not an object, that's actually what "fictional" means. OED #1 definition of object "a material thing that can be seen or touched". "Fictional", on the other hand, means exactly the opposite, invented by the imagination, therefore not able to be seen or touched.

In fact, from my position of Neutral Monism, all objects, whether house, London, mountain, government, the Eiffel Tower, unicorn or Sherlock Holmes are fictional, in that no object is able to exist outside the mind and independently of the mind. — RussellA

I suggest that your "position" is not consistent with common understanding. It might benefit you to give up on the monism.

My belief is that the mind cannot understand an abstract concept in isolation from concrete instantiations of it, in that, if I am learning a new word, such as "ngoe", it would be impossible to learn its meaning in isolation from concrete instantiations of it. — RussellA

There is no such thing as a "concrete instantiation" of a concept. Concepts are categorically different from concrete objects. To take your example, show me where I can find a concrete instantiation of beauty, 6, or the square root of 2. It is one thing to assert that there is a concrete instantiation of a six out there somewhere, but quite another thing to prove this. And if it is true, it ought to be easy to prove. Just point out this 6 to me, so i can go see it with my own eyes, or otherwise sense it.

If I wanted to teach you the meaning of the symbol "ngoe", which I know is a concept, how is it possible for you to learn its meaning without your first being shown particular concrete instantiations of it? — RussellA

This seems to be completely inconsistent with what you've already argued. You've already made the claim that you can make a fictious description, so why couldn't you also define a concept, thereby providing the means for someone else to understand it, without showing a concrete instance of that type of thing? I mean, you presented me with "a horse with a single straight horn projecting from its forehead", and i understand this image without seeing a concrete instantiation, so why take the opposite position now, and say that a person cannot understand the meaning of a concept without being shown a concrete instantiation of it.?

Given 1 and 1, if the second use of 1 refers to the same thing as the first use of 1, then the proper equation should be 1 = 1. The symbol "=" means identity

Given 1 and 1, if the second use of 1 refers to a different thing as the first use of 1, then the proper equation should be 1 + 1 = 2. The symbol "=" means equality. — RussellA

If this is the case, then what you have shown is logical inconsistency in the use of "1". In the first case, the two instances of 1 must refer to the very same thing, and in the second case, the two 1's must refer to two different things. If we simply say "=" means equality, then there is consistency between your two examples. Furthermore, there is no practical advantage to designating "=" as meaning identical in the case of "1=1", so you're just proposing logical inconsistency for no reason. That is simply illogical, therefore not a fair representation of the logic of mathematics.

Can there be a description without an object being described? — RussellA

Of course, that's known as fiction.

However we can think of the numbers 1. 6 and 10 as not only abstract mathematical objects but also as natural concrete objects. — RussellA

This evades me. How do you think of a number as a natural concrete object? Are you talking about the numeral, or the group of objects which the numeral is used to designate, or what?

That raises the question as to how we are able to think of something that is abstract, disassociated from any specific instance (Merriam Webster – Abstract). For example, independence, beauty, love, anger, Monday, ∞
∞
, 2–√
2
and the number 6.

George Lakoff and Mark Johnson in their book Metaphors We Live By propose that we can only understand abstract concepts metaphorically, in that we understand the concept of gravity by thinking about a heavy ball on a rubber sheet.

Thereby, we understand the concept of independence by remembering the feeling of leaving a job we didn't like. We understand the concept of beauty by looking at a Monet painting of water-lilies. We understand the concept of infinity by thinking about continually adding to an existing set of objects. We understand the concept of 2–√
2
by thinking about the number 1.414 etc etc. We understand the concept of 6 by picturing 6 apples.

IE, we can only understand an abstract concept metaphorically, whereby a word or phrase literally denoting one kind of object or idea is used in place of another to suggest a likeness or analogy between them (Merriam Webster – Metaphor). — RussellA

So why would we label an abstract concept an "object", as in "mathematical object", and speak of it as if it had an identity in the same way that a natural concrete object has an identity? If we only know abstract concepts through analogy, or suggestions of likeness, isn't it completely wrong to suggest that anything which only exists in this way, i.e. through metaphor, could have an "identity"?

This is the problem which @TonesInDeepFreeze is stuck on. Tones seems to think that just because one can show how "=" can be used to to show a relationship of identity between two distinct names for the same natural concrete object (]Mark Twain = Samuel Clemens), we can conclude that when "=" is used in mathematics, it's being used in that same way.

But of course in mathematics this is not true. There is no such natural concrete object which the symbols refer to, in theory. only abstract concepts. Natural concrete objects are only referred to through application. And in application the concrete situation referred to by the right side of the equation is never the same as the concrete situation referred to by the left side. So all that Tones has indicated is that there is two very different ways to use "=", the mathematical way, and the way which signifies a relation between two different names for the same natural concrete object. Therefore, we must be careful not to confuse the two different ways, or equivocate between them, because that would be misleading.

It is exactly the point that it is not a mathematical expression, so mathematics is not called on to account for its intensionality. More generally that ordinary mathematics is extensional, and we don't require that it also accommodate intensioncality. That is how it is relevant. — TonesInDeepFreeze

I interpret that as 'mathematics is extensional and that's how intensionality is relevant'. Whatever it is you are trying to say here, it appears to be just as irrelevant as your analogy was.

Later, hopefully, I'll have time and motivation to dispel a number of misconceptions in a catalog of them you've posted lately. — TonesInDeepFreeze

I'll be looking forward to that.

Doesn't this problem, soluble by set theory, assume "objects", such as the object "a person who can speak English"?

If the number "1" does not refer to an object, what does it refer to? — RussellA

The issue is a little more complex than how you represent here, but this is a good indication of why "set theory" is not applicable to mathematics. In your first question, "a person who can speak English" is a description, not an object. It represents a category by which we could sort objects. In the second sentence, the numeral "1" represents a specific concept, which can be described as a quantitative value. It is not a true representation of how we use numbers, to think of a number as itself an object. Set theory may represent a number as an object, but that's the false premise of set theory.

The issue is not whether or not some mathematicians define "=" as meaning 'is identical to', as a premise for a mathematical theory, or some other purpose, like debate or discussion. We've seen very much evidence here that some actually do this. So there is no question concerning that.

The question is how "=" is actually used in the application of mathematics. And anyone who takes a critical look at an equation in the application of mathematics will see that the right side never signifies the very same thing as the left side. In fact, it's quite obvious that if the right side did signify the same thing as the left, the equation would be completely useless. That is why many philosophers will argue that the law of identity is a useless tautology.

Since this is the case, we can clearly see that those mathematicians who define "=" as meaning 'is identical to' do not properly represent the meaning of "=" with that definition. Therefore we can say that they are wrong with that definition.

For example:

Mark Twain = Samuel Clemens — TonesInDeepFreeze

This is not a mathematical equation, so I do not see how it is relevant. You are trying to compare apples with oranges, as if they are the same thing, but the requirement that "Mark Twain = Samuel Clemens" is a representation of a mathematical equation renders your analogy as useless.

Please consider a real mathematical equation as an example, like how the circumference of a circle "is equal to" the diameter times pi, or the square of the hypotenuse of a right triangle "is equal to" the sum of the squares of the two perpendicular sides, for example. Be my guest, pick an equation, any equation, and we'll see if the right side signifies the very same thing as the left side. I think that an intelligent mathematician such as yourself, ought to know better than to argue the ridiculous claim that you have taken up.


The principal problem with set theory, as I indicated in my reply to @Banno above, which is evident from Chat GPT's statement, is that set theory is derived from a faulty Platonist premise, which assumes "mathematical objects". If we recognize as fact, that mathematics does not consist of objects, we must reject the whole enterprise of set theory, along with its fantastic representation of "infinite" and "transfinite", as completely unsound, i.e. based in a false premise.

Are you serious? — TonesInDeepFreeze

Lying requires intent, which GPT lacks.

In mathematics, equality and identity are the same. — TonesInDeepFreeze

Here's the example I gave Banno in the other thread. You and I are each one. Together we are two. We can symbolize this as 1+1=2. The two 1's here each represent something different, one represents you, the other I. Because the two each represent something different, the two together as 1+1 can make 2, meaning two distinct things. Also, we can say 1=1. But if the two 1's here both represent the same thing, then 1+1 could not make 2, because we'd still just have two different representations of the very same thing.

While "=" is commonly understood to denote equality in basic arithmetic and algebra, its use to signify identity in formal logic or set theory arises from the need to express relationships between objects or sets in a precise and rigorous manner. — Banno

That mathematics consists of "objects" with identity is Platonist metaphysics. In this metaphysical theory, mathematical ideas like numbers are objects, rather than quantitative values. Set theory is nothing but Platonist based mathematical theory. Notice that it is "theory", not mathematics in practise.

In the actual application of mathematics, values are assigned, and the left side of an equation must represent something different from the right side, or the equation would be useless, as I explained.

The conclusion we can make is that set theory does not represent mathematics, as mathematics is actually used. That's the problem, We can define terms, or in this case symbols, for theory, in a way which doesn't actually represent how they are used in practise. That's an idealist folly. I think Wittgenstein made a similar point.

However, in more advanced mathematical contexts like set theory, "=" is sometimes used to signify identity, — ChatGPT

As I argue, there is much inconsistency in mathematics. The use of "sometimes" here is very telling.

I mean that all makes sense, although my understanding was that the question of whether or not space-time is infinitely divisible was an open one. — Count Timothy von Icarus

It appears to me, like the quantum nature of energy demonstrates quite conclusively that the reality of space and time cannot be infinitely divisible. You see, the wave-function represents a continuity, but what it represents is not an observable aspect of reality. Observations indicate discrete occurrences of so-called particles (quanta), with not necessary continuity between the occurrences. If there is a true continuity, it is not represented by the wave-function, which represents possibilities. And, it is not the continuity of space-time, which fails at the quantum level. So it hasn't yet been determined.