The law of identity is an ontological issue concerning the nature of all things. Did you not read the Wikipedia, or Stanford quote I provided? Here's Wikipedia:

"In logic, the law of identity states that each thing is identical with itself."

See, the law of identity makes a statement about the nature of things. — Metaphysician Undercover

I think what Wayfarer keeps trying to point out is what I’ve highlighted in bold: the law of identity makes a statement about the nature of things within a closed system of thought. I don’t agree that the law of identity is meant to be ontological.

All three laws of logic aim to produce a closed system of thought - that’s what logic is. Quantum physics demonstrates the process of accurately aligning the significance of physical event structures within the same logical system, and the qualitative uncertainty that necessarily exists at this level. — Possibility

I think that many of the problems of interpretation of quantum mechanics are the results of the culture of non-conformity to the law of identity within the mathematical community, which is highly evident in this forum. If some energy is assigned a quantitative value, and the same quantity of energy is allowed to be interpreted as "the same object", regardless of the form in which it exists, then there are no features to distinguish it from any other energy of the same value. It is impossible to maintain the identity of any particular quantity of energy through a temporal extension, if one quantity of energy which has the same value as another quantity of energy, can be asserted to be "the same" energy. A photon is an object defined as a particular quantity of energy. If any energy of equal quantity can be said to be "the same" photon, because the law of identity is violated in the way that it is in mathematical axioms, then it's very obvious that temporal continuity of a photon, as an object cannot be maintained.

For this to be a logical statement, the symbols need to be expanded out to include a qualitative relation to their represented physical event structures: “I reserved a table for 4 people at 4pm AEST.” — Possibility

This is a mistake, and to make this assumption is a problem. Logical statements exist independently, and are valid independently, of the physical structure which they are applied to. That is why we have a distinction between being valid and being sound. The judgement as to the truth or falsity of the premises, which are the grounds by which the logic is actually related to physical structures, is a completely different type of judgement from the judgement as to whether the statement is "logical". That judgement of truth or falsity, is outside the so-called "closed system of thought" (logical system). Nevertheless, it is a crucial part of soundness, though not a part of logical validity.

So in relation to what we're discussing here, we can take the natural numbers as a "closed logical system", which provides the rules for counting objects. However, the system does not give rules for what constitutes "an object". Therefore, strictly speaking, the fact that the count is valid, cannot guarantee that the count is sound, or correct. The person counting might have had to make some judgements along the way, and there might have been some ambiguity within the criteria of what constitutes a countable object. Therefore the so-called "closed system" is not actually completely closed because ambiguity cannot be excluded from the defining principles.

The more effort and attention required to potentially align the senses and meanings of sender and receiver, the more accurately the significance of the relational structure must be described in the information to reduce uncertainty (eg. What date? What restaurant? What town?). Because the receiver of the message needs the most accurate information to align the potential of their own physical event structure to that of the sender, in order to produce a genuinely closed system of thought. — Possibility

So, when we're dealing with numbers, the fundamental "meaning" which must be aligned between sender and receiver, is the meaning of "1", a unit, or object which counts as a unit. Numbers inherently deal with individual units. We could say that they were designed that way, how they got designed is another question we can put aside, and just respect the fact that numbers deal with units. Because of this, we need a very clear, and rigorous definition of what constitutes "a unit", which is understood all around, and adhered to, or else work done with numbers becomes unsound due to ambiguity. Hence we have "the law of identity".

I think what Wayfarer keeps trying to point out is what I’ve highlighted in bold: the law of identity makes a statement about the nature of things within a closed system of thought. I don’t agree that the law of identity is meant to be ontological. — Possibility

To understand, and judge this statement we need to understand what comprises a "closed system of thought". The problem here is that no system of thought is truly closed, as demonstrated above. A system of thought is a feature of a living system, and living systems are fundamentally open, as evidenced by evolution. This is why Platonism (eternal unchanging, closed, rules) is often contrasted with, as being inconsistent with, evolution (changing rules). A closed system cannot evolve.

So we might understand a system of thought as consisting of different levels of rules, none of the rules, neither those at the bottom, the top, or middle, ought to attempt to close the system, as this would be unnaturally stifling to the evolutionary process. If we go to the rules at the base of epistemology, upon which logic and mathematics are constructed, we find the three fundamental rules of logic. The soundness, or veracity of these rules must be judged in relation to something outside the epistemological system which they support. These are the premises of the system, which need to be judged for truth or falsity to make sure that the system is sound. So the judgement of these rules which form the foundation of epistemological principles, must be an ontological judgement. Ontology supports epistemology. That's why I represent them as ontological. A premise is always in some sense a conclusion, being a judgement. So the three laws of logic are epistemological premises, but they are ontological conclusions.

The question is, how do you know which form you are contemplating?

In the Platonic dialogues, there's a connection between 'the forms' and 'vision'. So you're not simply thinking about something, but actually perceiving it .I think, from a modern perspective, Plato means by 'to see' what we would mean by 'to have a vision of'; he's not referring to sensory perception, but seeing 'with the eye of reason'. For this reason the classical tradition, imagination (thinking of images) and intellect (nous, which perceives meaning) are distinguished, whereas this distinction is often lost in modern thought. It's important to recall the Platonic epistemology, comprising different levels of understanding with their corresponding types of objects (see the analogy of the divided line.)

to grasp something with the intellect is not the same as to form a mental image of it. For any mental image of a triangle is necessarily going to be of an isosceles triangle specifically, or of a scalene one, or an equilateral one; but the concept of triangularity that your intellect grasps applies to all triangles alike. Any mental image of a triangle is going to have certain features, such as a particular color, that are no part of the concept of triangularity in general. A mental image is something private and subjective, while the concept of triangularity is objective and grasped by many minds at once. — Feser

It is largely the very peculiar kind of being that belongs to universals which has led many people to suppose that they are really mental. We can think of a universal, and our thinking then exists in a perfectly ordinary sense, like any other mental act. Suppose, for example, that we are thinking of whiteness. Then in one sense it may be said that whiteness is 'in our mind'.... In the strict sense, it is not 'whiteness' that is in our mind, but 'the act of thinking of whiteness'. The connected ambiguity in the word 'idea ...also causes confusion here. In one sense of this word, namely the sense in which it denotes the object of an act of thought, whiteness is an 'idea'. Hence, if the ambiguity is not guarded against, we may come to think that whiteness is an 'idea' in the other sense, i.e. an act of thought; and thus we come to think that whiteness is mental. But in so thinking, we rob it of its essential quality of universality. One man's act of thought is necessarily a different thing from another man's; one man's act of thought at one time is necessarily a different thing from the same man's act of thought at another time. Hence, if whiteness were the thought as opposed to its object, no two different men could think of it, and no one man could think of it twice. That which many different thoughts of whiteness have in common is their object, and this object is different from all of them. Thus universals are not thoughts, though when known they are the objects of thoughts. — Bertrand Russell, Problems of Philosophy, The World of Universals

Hope that helps. Not wanting to come across as pedantic, I'm in the middle of trying to understand all this myself.

It's unfortunate that the law of identity uses the equation symbol, = , since in math the equality symbol has two meanings: (1) 2=2, identical, and (2) 2x=x+3, conditional. I would think philosophers would use the identity symbol, ≡, which means both sides are always equal in a particular discussion.

This is a mistake, and to make this assumption is a problem. Logical statements exist independently, and are valid independently, of the physical structure which they are applied to. That is why we have a distinction between being valid and being sound. The judgement as to the truth or falsity of the premises, which are the grounds by which the logic is actually related to physical structures, is a completely different type of judgement from the judgement as to whether the statement is "logical". That judgement of truth or falsity, is outside the so-called "closed system of thought" (logical system). Nevertheless, it is a crucial part of soundness, though not a part of logical validity. — Metaphysician Undercover

In my book, they don’t exist independently. It’s a claim to independence - a closed system of thought is never really closed, but exists in potential relation to sentient beings, who are ignoring the potentiality of its relation to physical structures in order to examine only its structure of logical possibility. Nevertheless, my use of the term ‘logical statement’ was only to highlight the ambiguity of the original statement, which might make sense in conversation, but would not be useful in logic - it was not a comment on the validity or soundness of the statement itself. Sorry for the confusion - I can quickly get out of my depth in discussions on logic, but I think we are on the same page here.

If we go to the rules at the base of epistemology, upon which logic and mathematics are constructed, we find the three fundamental rules of logic. The soundness, or veracity of these rules must be judged in relation to something outside the epistemological system which they support. These are the premises of the system, which need to be judged for truth or falsity to make sure that the system is sound. So the judgement of these rules which form the foundation of epistemological principles, must be an ontological judgement. Ontology supports epistemology. That's why I represent them as ontological. A premise is always in some sense a conclusion, being a judgement. So the three laws of logic are epistemological premises, but they are ontological conclusions. — Metaphysician Undercover

I agree with this - my point was that those who consider themselves ‘logical beings’ do not typically ground their system in a larger ontological structure. This is a problem I encounter often. They don’t recognise or acknowledge a necessary relation to the broader structure of reality in which logic, for instance, does not reign supreme. So it seems that what you’re referring to is not so much logic’s Principle of Identity, but Leibniz’s Principle of the Identity of Indiscernibles, as a principle of analytic ontology?

Okay, so when I contemplate an object, I don't need identity conditions for it because my intellectual faculty perceives that object in its naked Being.

But what if somebody else comes up to me and says, "I, too, have contemplated mathematical objects, and I apprehend that there is no form of specific polygons, only a form of The Polygon, and all shapes, like squares and triangles, participate in it."

Then another person says, "I have also contemplated mathematical objects, and the opposite is true: there are separate forms for scalene and right triangles, and I have perceived them both."

Without positing some kind of identity conditions for abstracta, how do I even begin arguing with those two?

A photon is an object defined as a particular quantity of energy. If any energy of equal quantity can be said to be "the same" photon, because the law of identity is violated in the way that it is in mathematical axioms, then it's very obvious that temporal continuity of a photon, as an object cannot be maintained. — Metaphysician Undercover

Hmmm. :chin: Physicist around?

But in so thinking, we rob it of its essential quality of universality. One man's act of thought is necessarily a different thing from another man's; one man's act of thought at one time is necessarily a different thing from the same man's act of thought at another time. Hence, if whiteness were the thought as opposed to its object, no two different men could think of it, and no one man could think of it twice. That which many different thoughts of whiteness have in common is their object, and this object is different from all of them. Thus universals are not thoughts, though when known they are the objects of thoughts. — Bertrand Russell, Problems of Philosophy, The World of Universals

The point I would argue here, is that I would agree with Russell, that no two men think the exact same whiteness, exactly as described. And, many people think of whiteness at many different times. These I accept as true premises. Along with the law of identity as another premise, the proper conclusion, is that whiteness is necessarily not an object.

That which the many different thoughts of whiteness have in common, is similarity in the conditions of use. Russell's conclusion is wrong, he has been influenced by the Platonic realism inherent in the mathematics he has studied. There is nothing to indicate the existence of such an object. As Wittgenstein argues, it would have to be some sort of paradigm and none exist. However, evidence shows similar usage. Therefore we can conclude that what the thoughts of whiteness have in common is a similar application. This similarity is simplified in the notion of conventions.

So it seems that what you’re referring to is not so much logic’s Principle of Identity, but Leibniz’s Principle of the Identity of Indiscernibles, as a principle of analytic ontology? — Possibility

Logic's principle of identity is the one put forward by Aristotle as the law of identity, commonly expressed as "a thing is the same as itself". This is consistent with the Leibniz principle which says that if two named things have the exact same properties, they are in fact one and the same thing. If you study them both, you'll see that one is a sort of inversion of the other. Aristotle says that the only thing which is the same as a thing is the thing itself. Therefore the thing itself is the thing's own identity. Leibniz says that if you claim to have two things which are the very same in terms of properties, they are really one thing.

What I think is the important aspect of the law of identity, is that it places the true identity of a thing within itself, as the Stanford article I quoted says, identity is a relation which a thing has with itself, and nothing else. However, in our modern way of talking about identity, we think of identity as something we assign to the thing, we say that a person's identity, for example, is the name that we give them. As I explained earlier in the thread, Aristotle formalized the law of identity to get us away from this notion of identity, because it was being abused in sophistry. Here's the quote I produced from Aristotle's Metaphysics Bk.7:

"Clearly, then, each primary and self-subsistent thing is one and the same as its essence. The sophistical objections to this position, and the question whether Socrates and to be Socrates are the same thing, are obviously answered by the same solution; for there is no difference either in the standpoint from which the question would be asked, or in that from which one could answer it successfully." 1032a,5. — Metaphysician Undercover

In many modern schools of logic, the law of identity is simply expressed as A=A. Since it is often not explained exactly what the law of identity really is, it is sometimes simply assumed, that the meaning here is that the symbol A must always symbolize the same thing. But that is not an accurate representation of the law of identity. The law of identity stipulates that symbols cannot give the true identity of an object. The true identity is within the thing itself.

It's unfortunate that the law of identity uses the equation symbol, = — jgill

I would think that = is appropriate for equivalence in physics for symbols or quantities with mixed implicit or explicit units attached, as in E=mc^2. In some computer languages = might stand for arbitrary assignment of value to a variable, like x=3. Clearly, neither is an identity in either mathematical or philosophical meaning. ≡ might be too strict for philosophy?

In many modern schools of logic, the law of identity is simply expressed as A=A. Since it is often not explained exactly what the law of identity really is, it is sometimes simply assumed, that the meaning here is that the symbol A must always symbolize the same thing. But that is not an accurate representation of the law of identity. The law of identity stipulates that symbols cannot give the true identity of an object. The true identity is within the thing itself. — Metaphysician Undercover

:up:

The law of identity stipulates that symbols cannot give the true identity of an object. The true identity is within the thing itself. — Metaphysician Undercover

You’re missing the point of being able to abstract. Abstraction is at the basis of language, and you’re not getting it. Logic and language relies on representation, representing some [x] in symbolic form. You’re mistaking logic for soteriology

You’re missing the point of being able to abstract. Abstraction is at the basis of language, and you’re not getting it. Logic and language relies on representation, representing some [x] in symbolic form. You’re mistaking logic for soteriology. — Wayfarer

Logic and language relies not just on representation, but on a potential relation to the possible existence of some [x] as it is. Otherwise what IS the point of being able to abstract?

The point of abstraction is to be able to represent the properties of objects so that you can recognise similarities and dissimilarities, what is the same and what is different. When we say that ‘x’ is like or unlike ‘y’ or has qualities and attributes in common with ‘y’, then we’re abstracting, aren’t we?

The problem that I’m having with Metaphysician Undiscovered’s posts in this thread, is that he’s referring to ‘identity conditions’ in terms of ‘what really make some particular what it is’. He’s talking about the metaphysics of identity. Whereas I and others are saying that ‘a = a’ purely on the basis of abstraction, or in terms of the meaning of symbols. I’m leaving aside the metaphysical question of ‘what makes [some particular] what it really is.’ The question I asked was, doesn’t ‘the number seven’ have an identity? Which was a rhetorical question, in that I take the meaning of ‘7’ to be precisely ‘ the number that is not equal to everything that is not 7’, or, ‘7 = 7’. But somehow, this has given rise to pages and pages of metaphysical speculation.

If any energy of equal quantity can be said to be "the same" photon, because the law of identity is violated in the way that it is in mathematical axioms, then it's very obvious that temporal continuity of a photon, as an object cannot be maintained. — Metaphysician Undercover

Photons and other sub-atomic units of matter~energy are obviously ‘indiscernible’, in that they have no individual identity. All those with the same attributes - spin, polarity, etc - are indistinguishable from one another. They belong to the domain of the unmanifest, the unrealised. That is why ‘the observer’ plays a role - when you ‘see’ one, then it becomes particularised; hence the ‘observer problem’. ‘It from bit’ - Wheeler.

The problem that I’m having with Metaphysician Undiscovered’s posts in this thread, is that he’s referring to ‘identity conditions’ in terms of ‘what really make some particular what it is’. He’s talking about the metaphysics of identity. Whereas I and others are saying that ‘a = a’ purely on the basis of abstraction, or in terms of the meaning of symbols. I’m leaving aside the metaphysical question of ‘what makes [some particular] what it really is?’ The question I asked was, doesn’t ‘the number seven’ have an identity? Which was a rhetorical question, in that I take the meaning of ‘7’ to be precisely ‘not equal to everything that is not 7’, or, ‘7 = 7’. But somehow, this has given rise to pages and pages of metaphysical speculation. — Wayfarer

This relates to the point that he’s making, though: ‘the number seven’ is not identical to its value, so 7=7 risks equivocation. It reminds me of the children’s trick: ‘one plus one equals window’. It’s all very well to insist on a closed system of thought in which abstraction is all that matters, but it isn’t, and equivocating symbols with their value/potential leads to inaccuracy in terms of the meaning of symbols, and all sorts of interpretation issues when applying logic to both physics and philosophy. We need to be more conscious of methodologies employed in abstraction and interpretation that carelessly assume a closed system of thought.

The question I asked was, doesn’t ‘the number seven’ have an identity? Which was a rhetorical question, in that I take the meaning of ‘7’ to be precisely ‘ the number that is not equal to everything that is not 7’, or, ‘7 = 7’. — Wayfarer

Maybe your question is not well formed? To Plato, there ought to be only three forms of number, namely none, one, many.

7 is not a platonic form capable of formal identity but is derived from iterated copies of the One. An issue is that if 7 then why not 77 or 777 which lead to an explosion of copies of the One. But still, there is only one Form for One.

Logic and language relies not just on representation, but on a potential relation to the possible existence of some [x] as it is. Otherwise what IS the point of being able to abstract? — Possibility

Just to clarify the "potential relation to the possible existence of some [x] as it is", what is abstract and what exists in the following identities ?

A=A :: Cloud=Cloud :: Knowledge=Knowledge :: 9bananas=2apples :: Virtue=Wisdom

You’re missing the point of being able to abstract. Abstraction is at the basis of language, and you’re not getting it. Logic and language relies on representation, representing some [x] in symbolic form. You’re mistaking logic for soteriology — Wayfarer

We're talking fundamental laws of logic. This is not soteriology. How is that even relevant?

Despite the fact that the first law of logic is expressed in language, and is an abstraction, stating a general rule, a universal, it clearly makes a statement about particular things. Do you apprehend a difference between a universal rule, and a representation? Physics for example, is full of universal rules. Being universals, they are rules for the application of logical processes, just like mathematical axioms. Strictly speaking, they are not representations, they are rules of procedure. In the case of the law of identity, it is not the case that there is some [x] (thing) represented in symbolic form. If I stated it that way earlier, this was a mistake of sloppiness on my part. What there is, is a universal statement, a law, which makes a statement about any, and every [x] (thing) which might be represented in symbolic form.

When we move to the second law, there is another statement, another universal law, concerning what we can say about that [x] (thing) which is represented in symbolic form. This law is a statement concerning how we represent that thing, or object. We are forbidden from representing the object as both having and not having the same property.

The fundamental laws of logic are meant to ground logic in fundamental realities of the world, truth about substance, in Aristotle's terms. This is why they are ontological. The judgement of truth or falsity of the laws themselves is an ontological judgement.

One might argue, the ontological position that a universal is itself a type of object. From this perspective the law of identity is violated because we assume an object (the universal) which has no particular identity. This is the route which Peirce takes, and he proceeds to argue how these universals, as things, require exceptions to the laws of logic, resulting in his philosophy of vagueness. The boundaries which we assume to define an object, as an object, are extremely unclear when a universal is looked at as an object, (i.e. when a type is an object) and so there are various reasons to violate the second and third laws of logic. I see this as the approach of process philosophy in general, which is heavily influenced by Mathematical Platonism. An "object" is what mathematical axioms say an object is, and there is an incompatibility between this and the physical world of "becoming" such that the boundaries are necessarily vague, and there is no such thing as an object in the physical world. This renders the law of identity as completely ineffectual.

Hegel also argued that the law of identity ought to be rejected, as somewhat incoherent, and you can see my argument with Jersey Flight on this subject in the debates column of this forum. He approaches the law of identity from a slightly different ontology, which he called dialectics, arguing that the law is fundamentally incoherent. He also proposes that the distinct logical separation between being and not being are subsumed within "becoming", I think the term is "sublate". This renders the separation between opposing properties as a temporal separation. But there's a trend in the modern scientific community to look at time as an illusion, so Hegel's rejection of the law of identity leads to dialectical materialism, and dialetheism which openly propose violation of the law of non-contradiction.

He’s talking about the metaphysics of identity. Whereas I and others are saying that ‘a = a’ purely on the basis of abstraction, or in terms of the meaning of symbols. — Wayfarer

What I argue is that you misinterpret the law of identity, which does not say anything about the meaning of symbols. It says something about particular things. Look at the Stanford quote again:

Numerical identity is our topic. As noted, it is at the centre of several philosophical debates, but to many seems in itself wholly unproblematic, for it is just that relation everything has to itself and nothing else – and what could be less problematic than that? — SEP

You are making "identity" into something other than it is, as stated by the law of identity. In common parlance there might be such a thing as identity "in terms of the meaning of symbols", but this is not what "identity" refers to in logic, or philosophy in general.

The question I asked was, doesn’t ‘the number seven’ have an identity? Which was a rhetorical question, in that I take the meaning of ‘7’ to be precisely ‘ the number that is not equal to everything that is not 7’, or, ‘7 = 7’. But somehow, this has given rise to pages and pages of metaphysical speculation. — Wayfarer

The number seven does not have an identity, if we adhere to the law of identity. Only particulars have an identity and 7 refers to a universal. That is the point. You are appealing to something other than the law of identity, some colloquialism of "identity", to justify your claim that it does have an identity. And when I point out that you misunderstand the law of identity you get flustered, as if it would be a significant embarrassment, if true. It's not an embarrassment, because the vast majority of human beings, including high level mathematicians, and most physicists, do not understand it at all, having no respect for it. They do not understand it because it is a high level principle of ontology, or metaphysics, which requires that discipline to apprehend, and this is a very specialized field which is not taught in most university courses.

Photons and other sub-atomic units of matter~energy are obviously ‘indiscernible’, in that they have no individual identity. All those with the same attributes - spin, polarity, etc - are indistinguishable from one another. They belong to the domain of the unmanifest, the unrealised. That is why ‘the observer’ plays a role - when you ‘see’ one, then it becomes particularised; hence the ‘observer problem’. ‘It from bit’ - Wheeler. — Wayfarer

This is the point I made earlier. Failure to adhere to the law of identity in the mathematical, and scientific communities is what has resulted in the interpretation problem of quantum mechanics. It's a real problem, because without something real, a grounding in substance, the designation of "a unit", entity, or in this case "a quantum", as a photon, is based on a judgement of value rather than on a principle of identity. If equal value means the same entity, this is a failure in the rigours of logic. This is what I wrote:

I think that many of the problems of interpretation of quantum mechanics are the results of the culture of non-conformity to the law of identity within the mathematical community, which is highly evident in this forum. If some energy is assigned a quantitative value, and the same quantity of energy is allowed to be interpreted as "the same object", regardless of the form in which it exists, then there are no features to distinguish it from any other energy of the same value. It is impossible to maintain the identity of any particular quantity of energy through a temporal extension, if one quantity of energy which has the same value as another quantity of energy, can be asserted to be "the same" energy. A photon is an object defined as a particular quantity of energy. If any energy of equal quantity can be said to be "the same" photon, because the law of identity is violated in the way, such as it is in mathematical axioms, then it's very obvious that temporal continuity of a photon, as an object, cannot be maintained. — Metaphysician Undercover

This relates to the point that he’s making, though: ‘the number seven’ is not identical to its value, so 7=7 risks equivocation. It reminds me of the children’s trick: ‘one plus one equals window’. It’s all very well to insist on a closed system of thought in which abstraction is all that matters, but it isn’t, and equivocating symbols with their value/potential leads to inaccuracy in terms of the meaning of symbols, and all sorts of interpretation issues when applying logic to both physics and philosophy. We need to be more conscious of methodologies employed in abstraction and interpretation that carelessly assume a closed system of thought. — Possibility

The issue I see is that the law of identity has been openly challenged in modern metaphysics, starting with Hegel. Kant exposed the separation between phenomena, as what's in our minds, and the thing itself. When he designated the thing itself as absolutely inaccessible and unknowable (contrary to Plato), he rendered the law of identity as irrelevant, outside the domain of knowledge, as a statement about the thing itself. This allowed Hegel to unabashedly abuse and violate that law, because it appears like the law really doesn't make any difference. This is the modern attitude toward metaphysics and ontology in general, it really doesn't make any difference. But what has happened is that a huge gap has opened up between reality and what is represented in models, because the true nature of reality is seen as inconsequential, due to the Kantian belief that we have no access to it anyway (model-dependent realism for example). The Aristotelian concept of "substance", as that which substantiates logic, has been rejected due to this ontology which stipulates that logic cannot be substantiated.

We're talking fundamental laws of logic. This is not soteriology. How is that even relevant? — Metaphysician Undercover

Point taken, wrong choice of words on my part. I meant ‘metaphysic’.

Do you apprehend a difference between a universal rule, and a representation? — Metaphysician Undercover

I see the difference, but I also believe that representation would not be possible without abstraction, and abstraction in turn relies on generalisations that are grounded in universals. That is why I think nominalism is fallacious. Universals are basic to the mechanisms of meaning.

An "object" is what mathematical axioms say an object is, and there is an incompatibility between this and the physical world of "becoming" such that the boundaries are necessarily vague, and there is no such thing as an object in the physical world. — Metaphysician Undercover

Hence, the role of mathematics in quantification which is fundamental to scientific method. It singles out specifically those attributes of any object which are measurable and quantifiable as the ‘primary attributes’. That becomes the basis of physicalism.

When he designated the thing itself as absolutely inaccessible and unknowable (contrary to Plato), he rendered the law of identity as irrelevant, outside the domain of knowledge, as a statement about the thing itself. — Metaphysician Undercover

I had the idea Plato regards the sensory domain as inherently unknowable as lacking in real being, which only inheres in the formal domain.

when I point out that you misunderstand the law of identity you get flustered — Metaphysician Undercover

Not ‘flustered’ - at the beginning of this entire can of worms, the topic whether the two symbols in the expression a=a were the same and you saying ‘it depends on what they refer to’

Therefore Aristotle posited that the identity of any object is within itself. — Metaphysician Undercover

So, you’re saying that ‘identity’ is the same as ‘esse’?

To Plato, there ought to be only three forms of number, namely none, one, many. 7 is not a platonic form capable of formal identity but one of yours. — magritte

I’m still reading on the Forms, it’s a very deep and difficult topic. But generally speaking I’m arguing for ‘small-p platonism’ which is not necessarily the philosophy of Plato, but realism with respect to certain classes of ideas, which originated with Plato but was refined and qualified by subsequent generations.

Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets.

Platonism in the Philosophy of Mathematics, Stanford Encyclopedia of Philosophy.

The way I see it, rational judgements make constant reference to mathematical and logical concepts. Such judgements are ‘subjectivised’ in a lot of modern philosophy by being treated as being ‘in the mind’ or the products of human minds. But the key idea I take from Platonism is that there are ‘objects’ (using that term metaphorically) which can only be grasped by a mind, but are not the product of the individual mind. Whenever we say that something is ‘the same as’ something else, we’re abstracting the similar attributes of two or more different particulars and making a judgement. But that is internal to the nature of reason, so ‘transcendental’ in the Kantian sense.

I’m curious to know how you would address the following scenario via the law of identity:

--The concept of tree is the same as (is equal to; i.e., is identical to) the concept of tree … and is different from (is not equal to; i.e., is not identical to) the concept of rock.

Here, I’m addressing conceptual forms (which are naturally devoid of perceivable shapes: for, as a concept, i.e. as a generalized idea, it can take on multiple concrete, perceivable shapes …. None of which individually specifies what the concept, a generality, itself consists of in full). This, to me, is very much in tune to how a triangle, a geometric concept, can take on innumerable perceivable shapes without any such concrete shape being in and of itself the universal, abstract form of triangle per se. Likewise to how any number, itself an abstract concept, is identical to itself as number but not to any other number.

But my main interest here is in how you'd address the concept of tree as having, or as not having, an identity (albeit an inter-subjective one) as a concept - this as per the example mentioned. To be explicit, an identity via which it as concept can be identified.

Points to make clear:

(1) Ordinary mathematics, formally and informally, uses the law of identity. This is the use of first order logic with identity (sometimes called 'identity theory') that has the built-in semantics:

If 'T' and 'S' are terms then

T = S

is true if and only if 'T' and 'S' stand for the same object.

Said another way:

T = S

is true if and only if the denotation of 'T' is identical with the denotation of 'S'.

Said another way:

In any interpretation of a language, '=' maps to the identity relation on the domain.

Moreover, for proofs, identity theory is axiomatized by an axiom schema.

This is the very precise sense of the identity.

(2) Leibniz's law is taken as either the principle of the indiscernibility of indenticals (if x and y are identical then x and y share all properties) or as the conjunction of two principles - the principle of the identity of indiscernibles (if x and y share all properties, then x and y are identical) and the aforementioned principle of the indiscernibility of indenticals.

Identity theory, hence mathematics, adheres to both principles:

* If T = S, then T and S have all the same properties.

This is the indiscerbility of identicals and is expressible formally in a first order schema:

For any formula F,

x=y -> (Fx <-> Fy)

* If T and S have all the same properties, then T=S.

This is not expressible in a first order schema unless the number of non-logical constants in the language is finite. However, even if the number of logical constants is infinite, the principle is still upheld by the semantics of identity theory.

(3) Formulations of set theory may be based on first order logic with identity (so '=' is taken as primitive). And this holds even if you take out the axiom of extensionality. The axiom of extensionality is needed to prevent urelements, but it does not contradict identity theory, rather it is an addition to identity theory.

So there are three ways to handle identity and have extensionality in set theory:

1. '=' is primitive from identity theory, so we have:

theorem: x=y -> Az((zex <-> zey) & (xez <-> yez))

and we add the axiom of extensionality:

axiom: Az(zex <-> zey) -> x=y

2, dispense with identity theory and stipulate:

definition: x=y <-> (zex <-> zey)

axiom: x=y -> Az(zex -> zey)

3. dispense with identity theory and stipulate:

definition: x=y <-> Az(zex <-> zey)

axiom: Az(zex <-> zey) -> Az(xez -> yez)

In all three cases, we have the same set of theorems and the identity of indiscernibles and the indiscernibility of identicals. So we have Leibniz's principles to exact specification. Both syntactically and semantically.

(3) It was claimed that '=' has two different senses, for example:

2=2

vs,

2x=x+3

But those aren't different senses of '='. Rather they are examples of the difference between a formula with no free variables and a formula with at least one free variable. The first is true and the second is true or false depending on what value is assigned to the free variable 'x'. That doesn't entail that '=' has two different senses. It's a matter of understanding variables here, not any supposed difference (there is not one) in the meaning of '='.

:up: Thank you.

(3) It was claimed that '=' has two different senses, for example:

2=2

vs,

2x=x+3

But those aren't different senses of '=' — GrandMinnow

These represent two types of equations in mathematics. But you are correct in a fundamental sense of the symbol "=". (just another small reason I've stayed away from phil of math - angels dancing you know where)

In the vast ordinary sense in mathematics, an equation (an identity statement) is a formula of the form:

T=S

where 'T' and 'S' are terms.

The equations mentioned differ in that one has no free variables and the other has occurrences of free variables. One of them happens to be satisfied in all structures, while the other is satisfied in some structures and assignments for the variables but not in others.

It's as simple as that. There are no "angels on pins" involved.

(just another small reason I've stayed away from phil of math - angels dancing you know where) — jgill

It's as simple as that. There are no "angels on pins" involved. — GrandMinnow

Ah, but this is a philosophy forum. We like those kinds of problems. I read about the origin of that 'urban myth' about angels 'dancing on the head of a pin'. The original dispute was about whether two angelic (i.e. incorporeal) intelligences could occupy the same spatial location - which really is not such a daft thing to ponder, if you believe that there could be immaterial beings. (I began to wonder whether there was an analogy of sorts with the concept of 'super-position' which is the notion that a quantum entity can be in more than one location simultaneously - an inverse of the medieval's conundrum. One thing in two places, rather than two beings in one place. ;-)

I see the difference, but I also believe that representation would not be possible without abstraction, and abstraction in turn relies on generalisations that are grounded in universals. That is why I think nominalism is fallacious. Universals are basic to the mechanisms of meaning. — Wayfarer

I don't see how this relates to nominalism, but I don't agree that generalizations are grounded in universals. I think that they are both of the same category, essentially the same type of thing, and grounding requires reference to another category. So for instance, we can't ground the concept of red by reference to another colour. We might try to ground it by reference to colour, but understanding colour requires reference to something outside the concept of colour. This is the problem I had with the idea of a closed system of thought, mentioned above. A closed system would be ungrounded. By going outside we avoid the vicious circle, but then the possibility of an infinite regress appears. So Aristotle grounded his logic in substance.

I had the idea Plato regards the sensory domain as inherently unknowable as lacking in real being, which only inheres in the formal domain. — Wayfarer

I don't think Plato regarded the sensory realm as completely unknowable. Recall the divided line in The Republic. The visible realm is one half, so it does have some epistemological status. If I remember correctly, the higher knowledge of the visible realm is belief, and the lower is opinion, or something like that. More importantly though, for Plato, the visible objects partake in the Forms. A beautiful thing has beauty through partaking in the Idea of beauty. So we can come to know the visible objects through the means of the Forms, because we know the Forms, and the objects partake in the Forms.

So, you’re saying that ‘identity’ is the same as ‘esse’? — Wayfarer

I can't answer this because I'm not familiar with the word esse. I don't think it's English and it doesn't enter my translations. I am familiar with 'essence' and with 'essential' and they both have a range of usage. Even if you mean 'to be' by esse, it's not that straight forward. 'Being' sometimes is used as a verb, and sometimes as a noun.

With the law of identity, we are talking about a thing, not an activity, and that is what we assign the uniqueness of particularity to the thing. In naming it, the thing is represented as the grammatical subject. When we talk about activities, it's always types, universals, because activities are properties. An activity only becomes particular when we assign it to a specified thing, just like other properties. So we have to be careful when we use the word "being", to clarify whether we are talking about a thing, a being, or some activity which beings have in common.

The concept of tree is the same as (is equal to; i.e., is identical to) the concept of tree … and is different from (is not equal to; i.e., is not identical to) the concept of rock. — javra

The argument I've made, is that a concept is not an object, therefore the law of identity does not apply. The concept of tree is not the same as the concept of tree, because there are accidental differences in each instance that it occurs, therefore it violates the law of identity and cannot be an object.

But my main interest here is in how you'd address the concept of tree as having, or as not having, an identity (albeit an inter-subjective one) as a concept - this as per the example mentioned. To be explicit, an identity via which it as concept can be identified. — javra

Because the law of identity applies to objects only, and a concept is not an object, I don't think there is a valid way to say that a concept might be identified. Instead, we define concepts. If we proceed to state that a definition identifies the concept, then we are in violation of the law of identity. A definition exists as words, symbols, so now we'd be saying that the identity of the concept is in the words, but by the law, the identity must be in the thing itself. That's why a concept does not have an identity. However, if we assume an ideal, as the perfect, true definition of tree, an absolute which cannot change, then this ideal concept could exist as an object. Every time "tree" is used, it would be used in the exact same way, to refer to the very same conceptual object. But I don't think that this is realistic.

(1) Ordinary mathematics, formally and informally, uses the law of identity. This is the use of first order logic with identity (sometimes called 'identity theory') that has the built-in semantics: — GrandMinnow

This is only true, if numbers are objects. And we've seen already in this thread that they do not qualify as objects because in mathematical usage the law of identity is violated. Since the law of identity is violated in mathematical usage of numbers, numbers cannot be objects. So your formula just begs the question. You assume that a number is an object, therefore '=' means identity. But of course, as I've already demonstrated, '=' is not actually used that way. So your question begging premise is actually false.

In the vast ordinary sense in mathematics, an equation (an identity statement) is a formula of the form:

T=S

where 'T' and 'S' are terms.

It's as simple as that. There are no "angels on pins" involved. — GrandMinnow

OK, show me how T and S necessarily refer to the exact same object, as required by the law of identity. Please don't beg the question by asserting that the '=' means that they refer to the exact same object, because we already know that this is not true in the common usage of '=' in equations.

Ah, but this is a philosophy forum. We like those kinds of problems. I read about the origin of that 'urban myth' about angels 'dancing on the head of a pin'. The original dispute was about whether two angelic (i.e. incorporeal) intelligences could occupy the same spatial location - which really is not such a daft thing to ponder, if you believe that there could be immaterial beings. (I began to wonder whether there was an analogy of sorts with the concept of 'super-position' which is the notion that a quantum entity can be in more than one location simultaneously - an inverse of the medieval's conundrum. One thing in two places, rather than two things in one place.) — Wayfarer

If two distinct things occupied the exact same space at the exact same time, I think we'd have a true violation of the law of identity.

show me how T and S necessarily refer to the exact same object — Metaphysician Undercover

I didn't say that they necessarily refer to the same object. I said the formula is satisfied when they refer to the same object.

The fixed semantics for '=' is given in the method of structures for languages in mathematical logic. If you are not familiar with that method and subject matter, then you won't follow what I'm saying here.

asserting that the '=' means that they refer to the exact same object, because we already know that this is not true in the common usage of '=' in equations. — Metaphysician Undercover

In common, pervasive usage in mathematics, as I mentioned, a formula

T = S

is true (or satisfied) if and only if 'T' and 'S' refer to the same object.

The reason you are not familiar with that fact is that you are not familiar with rigorous mathematics and especially as mathematics is treated in mathematical logic.

the law of identity is violated in mathematical usage — Metaphysician Undercover

No, the laws of identity are not violated in mathematics, no matter what we take the ontological status of the referents to be. Or, please state precisely a mathematical text in which you find violation of the laws of identity. I mean a specific piece of mathematical writing; not just something that you imagine someone has meant somewhere or another.

your formula just begs the question. You assume that a number is an object, therefore '=' means identity. But of course, as I've already demonstrated, '=' is not actually used that way. So your question begging premise is actually false. — Metaphysician Undercover

What specific formula are you referring to? A formula is just a formula and it's not an argument, while question begging refers to arguments, so a formula itself is not question begging. And the rest of what you wrote there is double-talk ignorant of the subject of actual mathematics. The remedy for you is to get a book on beginning symbolic logic then on mathematical logic.

I didn't say that they necessarily refer to the same object. I said the formula is satisfied when they refer to the same object. — GrandMinnow

Yes, it's always the case, that if the very same thing is referred to on the right and the left, use of the '=' is valid. That is because a thing cannot be unequal to itself. But since there are many instance when the right and the left refer to something different, we cannot conclude that '=' signifies identity.

n common, pervasive usage in mathematics, as I mentioned, a formula

T = S

is true (or satisfied) if and only if 'T' and 'S' refer to the same object. — GrandMinnow

This is a false statement. It is very evident from the common use of mathematics, and even your example of "free variables", that the right and left side usually do not signify the very same thing.