Very amusing, MU. — Jamal

Reality is actually very amusing, that's why metaphysicians are generally of a good humour.

It is the logical structure underlying language and not mind that is a check against illogical thought. I take this to mean that any illogical thought or propositions would evidently involve a contradiction.and would not be accepted. — Fooloso4

The problem with this perspective is that illogical thought is actually quite common, and even illogical speaking cannot be ruled out. So the reality that there is not necessarily a logical structure underlying language must be respected. The lack of an underlying logical structure is the position Wittgenstein moved on toward in the Philosophical Investigations, with "family resemblance", and the idea that boundaries (the prerequisite for logic) are created as required, for the purpose at hand.

It is not my representation. It is what Wittgenstein says. I cited it. Unless you are claiming that he means something else by the term 'transcendental. — Fooloso4

I apologize for the misunderstanding, but you really did say "I think...", and you did not mention Wittgenstein in that opening paragraph at all, so I assumed you were stating what you believe. As much as you went on to cite the Tractatus, this is what you said in your opening paragraph:

I think this misses the mark. It is logic rather than language which is transcendental. Logic is the transcendental condition that makes language possible. Language and the world share a logical structure. Logic underlies not only language but the world. It is the transcendental condition that makes the world possible. — Fooloso4

One more interesting thing to note is that Kant and Wittgenstein are similar not only in their transcendental perspective on human beings, but also in their use of this perspective to show that most philosophy hitherto has gone astray by asking questions that cannot be asked. — Jamal

This is an interesting statement. With "asking questions that cannot be asked", you seem to imply that philosophy is doing the impossible. And when I wonder, "what do you actually mean by this?", because "doing the impossible" doesn't make any sense, I come up with two completely different possibilities. One would be "asking questions that cannot be answered", and the other would be "trying to ask questions which cannot actually be asked". Since the former is rather boring, implying a sort of unintelligible type of question, I assume you would mean the latter.

But even "trying to ask questions which cannot be asked" is difficult to understand because of the different senses of "possible" which we use, and the variety in types of limitations which are evident relative to the different types of possibilities. So for instance, we have a relationship between language and logic, which with a formal understanding of "logic", would make logic dependent on language unlike @Fooloso4's representation of "Logic is the transcendental condition that makes language possible." But if we restrict the definition of "logic" in the way that I just proposed, we still need to come up with terms to describe the type of thinking which transcends logic.

This is not at all difficult, because we have the means to talk about irrational, and unreasonable thought, and the thinking of different animals which is not logical. This is the way language works, it is not limited by the activities which it empowers, and this is why it is extremely powerful. Language transcends logic, it can also transcend knowledge to talk about the unknown, and it further transcends all forms of thinking and thought, to talk about things which cannot even be thought about (we can speak nonsense). That's what the "private language" demonstrates, the transcendent capacity of language. This implies that there is no such thing as "questions which cannot be asked", leaving only "questions which cannot be answered".

This puts language in a very special, unique place, which seems counterintuitive, and many would argue against it. This is the place of infinite possibility, absolutely limitless, implying that there is nowhere tht language cannot go, nothing which cannot be said. To understand this phenomenal position of language, all one needs to do is take a look at the language of mathematics. The natural numbers are limitless, infinite, and this provides the capacity to count any quantity. This is indicative of the way that language is, in general, it is "designed" so as to give the user the capacity to go beyond any limitations, therefore to speak about anything whatsoever. Now we must rule out the second option "trying to ask questions which cannot actually be asked", because anything can be asked, that is simply the limitless capacity which language is.

So we're back to the first possibility, "asking what cannot be answered", in our interpretation of "most philosophy hitherto has gone astray by asking questions that cannot be asked". Since language can go anywhere, and it is designed to speak about anything, and therefore ask any question, why would we say that philosophy has gone astray by asking questions which cannot be answered? Isn't this exactly the job of philosophy, to venture into the unknown, and ask what cannot be answered? Didn't Socrates say that philosophy begins in wonder? Putting the transcend nature of language to work, utilizing its limitless capacity, to ask what cannot be answered, is exactly the role of philosophy.

The logic is not merely supposed to be rigorous. It is rigorous in these senses: (1) The axioms and rules of inference are recursive, thus, for a purported proof given in full formality, it is mechanical to check whether it is indeed a proof, i.e., merely an application of the inference rules to the axioms. (2) It is proven that the logic is sound, i.e. that a formula is is provable from a given set of formulas only if the formulas is entailed from the set of formulas. — TonesInDeepFreeze

Since soundness requires true premises, and the logic you are talking about proceeds from axioms, which are not truth-apt, instead of from true propositions, how do you propose that it could be "proven that the logic is sound"?

mathematics, in ordinary context, 'x=y' is true if and only if x and y are the same object, which is to say 'x=y' is true if and only if what 'x' stands for is the same as what 'y' stands for. The claim that there are no such objects is not properly given as an objection to the fact that '=' stands for identity, since we would still have '=' standing for identity if the objects were physical, concrete, fictional, hypothetical, 'as if', abstract, platonic, etc. — TonesInDeepFreeze

The point I made, is that the sense of "identity" you use here, is not consistent with the sense of "identity" used in the law of identity. So it doesn't really matter that you insist that "=" stands for "identity". Anyone can make up one's own personal sense of "identity" and have a symbol for it, state the axiom, and persuade others to use the axiom, and even create a whole "identity theory", but that doesn't make that sense of "identity" consistent with the law of identity.

* Sets are not determined by an order in which the members happen to be mentioned. If I say, "What are the members of the set of books on your desk", then if you say, the set of books on my desk is all and only the books 'The Maltese Falcon', 'Light In August' and 'The Stranger', then no one could say "No, that's wrong, the set of books on your desk is actually all and only the books 'Light In August', 'The Stranger' and 'The Maltese Falcon'!" — TonesInDeepFreeze

This is an excellent example of why your sense of "identity" is not consistent with the law of identity. By the law of identity, if the identified thing is "the books on your desk", then everything about that thing, including the order of the parts, must be precisely as the books on your desk, to satisfy the criteria of "identity". Stating an order other than what the books on your desk actually have, would not qualify as an identity statement, because that specific aspect, the order of the parts, would not be consistent with the thing's true identity.

No law of identity is violated there. — TonesInDeepFreeze

I have become fully aware that you are not at all familiar with the law of identity. Therefore your statements about the law of identity, and whether it is violated under specific conditions, I simply take as off-the-cuff remarks of a crackpot.

Nobody says that the set of items on a desk is different depending on the order you list them. — TonesInDeepFreeze

Anyone with any degree of common sense recognizes that the identity of the specified thing, "the items on a desk" includes the ordering of the mentioned items. If describing those items as a "set" means that the ordering of the mentioned items is no longer relevant, then you are obviously not talking about the "identity" of the specified thing, which is "the items on the desk". You are talking about something other than "identity" as defined by the law of identity.

Not quite. Whether the object of fear is known is irrelevant to Heidegger's distinction between fear and anxiety. Instead, the source of the phenomenon (within the world or not within the world) determines whether the phenomenon is fear or anxiety.

That in the face of which one has fear is always an entity within the world while that in face of which one has anxiety is not an entity within the world. See Being and Time at 230-231, (Macquarrie & Robinson).

Simply put, "the forest and the trees" is not a good analogy for understanding Heidegger's distinction between fear and anxiety. — Arne

I do not think I would agree with this Heideggerian distinction between fear and anxiety. It is not the "entity within the world" which is the source of fear, but what is known about the entity. When the entity is known to be a threat, or even a risk, there may be fear.

So I agree that I placed "fear" in the wrong category, saying that it concerned the unknown, when really there must be a large degree of "known" involved, to provoke fear. "Anxiety", on the other hand requires "unknown", and it is associated with risk. So I think anxiety is best represented as the other side of the same relation to the same object. You can see "fear" (cowardice), and "confidence" (courage), as the two opposing aspects of the known part of the relation, while "anxiety" and "complacence" are the two opposing aspects of the unknown part of the relation. "Relation" here would indicate the person's attitude toward the specific "entity within the world".

I would not agree with Heidegger as you describe him, because I do not think we can make a clean break between fear and anxiety, as proposed, such that one would require an object, and the other not. Sometimes fear is directed at an object (rational fear) and sometimes it is not (irrational fear), and likewise with anxiety. The undirected fear, as irrational fear, directed at an imaginary object, is still fear rather than anxiety, but it is directed at an imaginary object. So fear and anxiety are the two sides of the same emotion, the known side, and the unknown side. The irrational aspect is due to faulty knowledge where the imaginary takes precedence, but it must still be classified as the "known" because the subject thinks oneself to know. If there is an object involved, then fear and anxiety are the two relations to that same object, and it makes no sense to say the fear has an object but the anxiety does not.

Notice also, the other possibilities. If, one has fear, and also complacence, there is no anxiety. And, if instead of fear, one has confidence, there could still be anxiety, or complacence along with the confidence. If we were to make the separation proposed by Heidegger it would leave anxiety as completely unintelligible, and that would render it as completely bad.

I agree. But what makes me wonder about how Fosse wrote the book is whether the silence is a reference to death (his parents and sister passed away and he feels alone) or the inability to say to them that he wants to go back to Norway. In this novel, the silence is a key factor and, most of the time, is confusing because even the protagonist feels scared of why his family remain in silence at the pier. — javi2541997

I wouldn't see "silence" as "death" from what I've read. Silence is the language of the imagination, and imagination is very much alive. In the case of schizophrenia the boundary between what is imaginary and what is real is blurred. That means that the silent language (imagination) becomes a very real language, with actual communicative power. We all share this to some extent, and it's an essential aspect of artistry, our imaginations can communicate something very real to us, and we transform this into artistic expression.

I believe the fundamental issue here is the way that you relate to "silence". There's two sides to this silence, the silence of the described scene (Fosse's silence), and the silence of the reader. Remember, silence is used by Fosse as a tool, to provoke the imagination of the reader. I believe Fosse would have intentionally left a blank (silence) concerning key aspects of Lars' family relations. From what I understand family relations can play a key role in the development of schizophrenia, and Fosse was probably not in a position to adequately understand those relations, so he would leave that to the imagination of the reader.

I would say that the family remaining in silence at the pier would be an implication of Lars' inability to communicate directly with them, expressing that much of his communication with them was through his imagination. This would indicate that them being dead is sort of irrelevant because he always communicated with them through his imagination anyway. Perhaps its a demonstration that they never provided for him the words that he needed from them. And this is what allows him to continue to communicate with them at the pier, regardless of them being dead.

Martin Heidegger: Analogy with a tree and a forest - anxiety is like the entire forest of trees. You don't see individual trees, what they are and how they are. Fear is a specific tree, one or two. But you don't see the whole forest, the connections, interdependence, sensuousness, and what is behind the next tree. Being and time. Do you see those other perspectives? — MorningStar

To continue what I pointed out yesterday, that I think you have things backward, I'll address the problem with this passage in the op. What you "see", meaning sense through the perceptive power of sight, is individual trees. The "forest" is a conception of thought, an abstraction which is not "seen" except in the more metaphoric sense in which the mind sees things.

The object of fear is the unknown, in a sense there is no object, and that produces the fear. So we cannot say that "fear is a specific tree", because that would imply that the fear is a thing seen and known as "a tree". Even if you see a particular object and there is fear relative to that object (a menacing animal or person for example), it is the unknown, what that object might do, which produces the fear.

The conception, "the forest", is always to some degree incomplete, it does not directly include every single tree. Therefore within the conception there are elements of the unknown, and this is the seed for fear and anxiety. So within your conversion of the sense image, (a whole lot of individual trees), to an object of thought, (the abstraction "forest"), there is a whole lot of the unseen, consequently unknown, which is allowed to inhere within the conception, and this propagates anxiety. You apprehend "a forest", but within this forest there is a whole lot of unseen trees.

The act, by which the abstraction is produced, is commonly and often habitual. You look ahead of you and see "a forest". You do not consciously think, 'I'm seeing some trees and I'm concluding there's a forest', so the unknown, and therefore the seed for fear and anxiety, is allowed right into your thought without you even knowing it. Once that seed for anxiety is there, you cannot recognize it because it is an unknown which has already been incorporated into the known. Your thoughts are the known, yet the unknown inheres within. That is why it is important to keep track of your thoughts as they arise, and assess them methodically. There are various ways of doing this, such as the skeptical method, to prevent the fear and anxiety from growing within.

What do you folks think? — javi2541997

I'll express an opinion on this, even though it's difficult to say anything confidently without knowing the greater context. Also, I'm really not familiar with any of the author's material, so I really do not know his style at all. Furthermore, i do not know your translation technique, nor the accuracy of your translation.

What exactly is the context of the father walking toward the pier? Is it possible that this is imaginary, a sort of daydream of the narrator? Notice that the father asks "if I'm not doing well", and this is completely different from asking "how are you doing" or something like that, because it implies that the speaker is already aware that the person is not doing well. So the father is already 'into the mind' of the son, as if a product of the son's imagination. That is a very strong indication that the son is doing very poorly, wants to go home, and is imagining, and hoping, that his father is coming to take him home. Instead, the imaginary father walks right into the water, dashing all hopes of taking him home, showing him a completely different direction, suicide.

Remember the use of silence, and the power of the imagination which we discussed earlier in the thread. Silence is the cue to use your own imagination, fill in all the blanks, where the author led you toward something, but did not explicitly say it. The silence says it better. So if the narrator is on the pier daydreaming, using his imagination, then it's better that you the reader, use your imagination to better understand the situation described.

Of course, I do not know the context of the expressed passage, so further explanation would better reveal whether my interpretation is what is intended by the author, or not.

As you also say:

In set theory, there is no constant nicknamed 'infinity' (not talking about points of infinity on the extended real line and such here). Rather, there is the predicate nicknamed 'is infinite'.
— TonesInDeepFreeze — RussellA

I wonder what "nicknamed" would imply in supposed rigorous logic.

To the Lounge with this rubbish. — Jamal

That's better, it's more relaxed in the lounge, and may serve to lower the tension by a few foot-pounds or something like that. I hope there's no drinks available here though, or things might go the opposite way.

The law of identity is:

Ax x=x

That is one of the axioms of identity theory. — TonesInDeepFreeze

As I indicated earlier, the issue is with the way that x=x is interpreted. Unless the interpretation employed by "identity theory" is consistent with the way that the law of identity is stated in its original formulation, "a thing is the same as itself", then the meaning of "x=x" which is employed by "identity theory", is not consistent with the law of identity.

What is required now, is that you state the interpretation of "x=x" which is employed by "identity theory", and more specifically "set theory", such that we can judge it for consistency with the law of identity, "a thing is the same as itself".

Goodness has two historical meanings: hypothetical and actual perfection. The former is perfection for (i.e., utility towards) some purpose (e.g., a good clock is a clock that can tell the time, a good car can transport things, a good calculator can perform mathematical calculations, etc.); and the latter is perfection in-itself (i.e., a good organism, clock, phone, plant, etc. is one which is in harmony and unity with itself). The former is pragmatic goodness; and the latter moral goodness. — Bob Ross

I don't think we can make the distinction you require, in this way. The problem is that activities which are conducive to "harmony" are judged as "good" because they are for that purpose, harmony. Now you've introduced "harmony" as the ultimate goal, the final end which all things strive for in perfection. So it's really just reducible to a pragmatic "good", but the ultimate pragmatic good, like Aristotle's proposal of "happiness".

But "harmony" refers to the relationship between things, and you characterize it as a perfection of the thing-in-itself, a relation which a thing has with itself. So you've mischaracterized "harmony" to say that it is a perfect relationship between a thing and itself, when really it is a relationship between distinct things. This indicates that to find out what constitutes the true perfection which a thing might have, as "in-itself", we need to look for something other than harmony.

it can persist even during action. Not only during passivity but also during activity, a person can feel the weight of anxiety. I’m not sure of the exact term for this phenomenon. Perhaps you are referring to intrusive thoughts—those persistent, unwelcome ideas that can cause distress. — MorningStar

I agree that anxiety will persist even during activity. This could be anxiety in relation to the end, or goal, or it could be in relation to something completely different, perhaps a goal not pursued by the current activity. I think those would be instances of intrusive thoughts. I find that the more negative form of anxiety is completely undirected by the conscious mind, so that it appears to have no source, and it is therefore more difficult to quell. We cannot characterize this anxiety as intrusive thoughts because it is prior to, and independent from the intrusive thoughts. The anxiety produces the intrusive thoughts, which are thoughts that cannot be directed toward the current activity.

During an anxious episode, emotions often override rationality, leading to a struggle between reason and intense feelings. It’s like a battle of emotions and logic. Slowing down, conscious breathing, and reminding yourself that you are in control can indeed be helpful strategies. Remember, those intrusive thoughts are not truly you; they are just passing mental events. — MorningStar

Consider thinking to be a form of activity. It is borderline between the conscious activity of physically moving oneself, and the subconscious inclination toward moving oneself (how I described anxiety). The indecisive person will think rather than move, but the thoughts may still be directed by the conscious mind in a way similar to the way that movement is directed by the mind. However, since anxiety, or the inclination to act, arises from the subconscious, it can cause undirected thoughts, or thoughts which the conscious mind has difficulty directing, because the source of the anxiety is unknown, and different types of anxiety would produce different types of thoughts. This would be what you call "intrusive thoughts".

So I wouldn't class the intrusive thoughts as "not truly you", I would class them in the opposite way, as "the true you". This is because the conscious mind is just the very top level of "you", and the vast majority of your activity, all your bodily systems for example, are in the subconscious. Therefore anxiety, and the intrusive thoughts which spring from it are actually "the true you", and you need to learn how to deal with them as such. This means that you cannot use your conscious mind to suppress and make your anxiety go away. Attempting to do this would be to pursue an impossible goal, and that would produce more anxiety in relation to that goal.

So i think the better strategy is to recognize the limits of your control. To quell the anxiety you must allow the activity which it is inclined to produce, thinking. And you cannot control the thoughts until after they're being produced. What I find has been a good strategy for me is to have various "objects" (which may be various goals or constructive things to think about), each of a different category or type, and depending on the type of thoughts which are inclined to be arising, I can send them into the appropriate category for direction. I think it is important to have the required goals or categories which are suited to the types of thoughts which you get, or else you would get the confusion and anxiety of attempting to direct your thoughts in a way which is unsuited to them.

You claimed that axiom of extensionality is inconsistent with identity theory. I proved it is not. You evade that, because you know virtually nothing about identity theory, the axiom of extensionality or consistency. — TonesInDeepFreeze

That's a good example of a crackpot reply. You are avoiding the issue, by switching to "identity theory" rather than the law of identity. The problem I brought up is that "identity" in set theory is not consistent with "identity" in the law of identity. Whether or not "identity" in set theory is consistent with "identity" in "identity theory" has no bearing on the problem I've exposed.

So an hourglass changes its identity as each sand grain drops. — Banno

No, the law states "a thing is the same as itself". Nothing here says that the thing cannot change as time passes. But all those changes are necessarily a part of the thing's identity. That's one of the important features of the law of identity, it allows for a true understanding of the temporal continuity of things, and the reality of change itself, by allowing that a thing maintains its identity despite changing.

Have you no familiarity with the law of identity? It seems to me that you've only been exposed to misrepresentations, proposed by logicians who want to reformulate it to support their own proposals. I do not argue that it is without problems, like the one presented by The Ship of Theseus example. And as I said earlier, some philosophers propose that we reject the law of identity altogether. But that would give us no principles for understanding the reality of temporal continuity. You see, there is an incompatibility between eternal unchanging Platonic "Ideas", and the temporal continuity of objects which are constantly changing. The law of identity refers to the latter, and the identity which a set is said to have refers to the former.

You're making claims about the axiom vis-a-vis identity. So it is very relevant what the axiom proves regarding identity. — TonesInDeepFreeze

I don't think so. A proposition (or axiom) needs to be judged by the principle it states, not based on what can be proven through the use of it. If you accept an axiom because it can prove what you want it to prove, that is just begging the question.

the main crank — TonesInDeepFreeze

I'm starting to like that handle, it makes me feel powerful like the driving part of a magnificent machine. Do you think it would be suitable for me to change my name?

The issue I'm discussing is identity. It's only indirectly related to the op, so if you do not want to discuss this, that's fine. What you can "prove from the axioms" is irrelevant, when it is the acceptability of the axioms which is being questioned.

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.