As human beings, you and I are equal, based in a principle of equality. — Metaphysician Undercover

Now who’s equivocating? Social equality means ‘treating everyone the same’. In that sense it means treating them as ‘equal’ but that is a specific use in a specific context. When we’re discussing the ‘=‘ sign we are by definition discussing a symbol which denotes strict identity. So in the expressions a=a and 2+2 = 4, it is meaningless to say that each occurrence of the symbol ‘a’ or ‘2’ refers to a specific object. It refers to a symbol, which is useful precisely because it can be applied universally.

Aristotle developed this idea he discovered that matter itself cannot account for any of the properties of an object, and so each individual thing must have a unique form proper to itself. — Metaphysician Undercover

I’m aware that Aristotle rejected the Platonic doctrine of forms, but he still maintained a role for universals. ‘In Aristotle's view, universals are incorporeal and universal, but only exist only where they are instantiated; they exist only in things.’ So they don’t exist in another domain or realm, which is what Platonism appears to propose. I don’t think there’s any ‘mechanism’ by which ‘forms’ are instantiated as individuals. Perhaps one way of conceiving it would be that for a creature to fly, then wings have to assume a certain form; you can’t fly using weight-bearing limbs. So wings evolve towards a certain form, because of the function that they need to realise, not because there’s an ‘ideal wing factory’ situated off in la-la land. The form is what something has to take in order to exist. (Actually there’s a really good Kelly Ross essay on this, Meaning and the Problem of Universals.)

He said the commonly asked question of why there is something rather than nothing cannot be answered, and is therefore a fruitless question. — Metaphysician Undercover

I’d agree with that, although I’d be surprised if it was really an FAQ in his day.

When we’re discussing the ‘=‘ sign we are by definition discussing a symbol which denotes strict identity. — Wayfarer

That's not really true. Wikipedia says the sign is "used to indicate equality in some well-defined sense... In an equation, the equal sign is placed between two expressions that have the same value, or for which one studies the conditions under which they have the same value." Clearly the '=' sign is not used in mathematics to denote strict identity. It might be defined in some axiom of set theory, as denoting strict identity, but that definition would not reflect how it is used, therefore that definition ought to be rejected.

Familiarize yourself with the law of identity. It states that "same" identifies one thing and only one thing. There cannot be two things which are the same. Then, take a look at how the = sign is employed in an equation. Clearly the right and left side of an equation cannot both represent the exact same thing, or else the equation would be completely useless. We'd have to ask, if there's something represented on the right side, and the exact same thing is represented on the left side, and we already know that we are just representing the exact same thing in two different ways, because use of '=' indicates that we know that the two are the exact same thing, then what are we doing with the equation? The equation would be doing absolutely nothing for us. But the fact is, that we represent something different on each side, we say that the two are equal, not that the exact same thing is represented twice. So '=' does not indicate strict identity. Therefore if someone proposes to you that "=" denotes strict identity, as a mathematical proposition, an axiom to be used as a premise, you ought to reject that premise as false because it will lead to unsound conclusions.

The form is what something has to take in order to exist. — Wayfarer

Do you recognize that in Aristotelian physics, each individual material object has a particular form which is unique to itself, and this is expressed in the law of identity? If so, then the point of the op is that universal forms do not have such a particular form, this would be incoherent. Therefore the law of identity is not applicable to universal forms, nor can we say that universal forms are particular objects.

Wikipedia says the sign is "used to indicate equality in some well-defined sense... In an equation, the equal sign is placed between two expressions that have the same value, or for which one studies the conditions under which they have the same value." — Metaphysician Undercover

That is exactly what I said.

Do you recognize that in Aristotelian physics, each individual material object has a particular form which is unique to itself, and this is expressed in the law of identity? — Metaphysician Undercover

It's not a matter of 'recognising it', this is something that I have only ever read in your posts. If you provide a reference I'd be obliged.

Familiarize yourself with the law of identity. It states that "same" identifies one thing and only one thing. There cannot be two things which are the same. Then, take a look at how the = sign is employed in an equation. Clearly the right and left side of an equation cannot both represent the exact same thing, or else the equation would be completely useless. — Metaphysician Undercover

There is something really absurd here. So, you're saying, that in the expression A=A, that this expression only refers to particular instances of 'A'? That in order for 'A' to be 'A' then we have to refer to a particular instance of 'A'? That when we say, 2 + 2 = 4, that you're saying 'hang on! Which individual instances of '2' are you referring to?'

The point about 'the law of identity' is that both sides of the equation absolutely represent the exact same thing, namely, 'A'. That doesn't make the equation 'useless', it is why it is meaningful.

I'm not going to pursue this, as I think life's too short to argue about the meaning of the very first element of logic.

And, I think jgill agreed with me on this point in that other thread as well. — Metaphysician Undercover

From the perspective of appearances of symbols you have a point. Clearly, 2+2=3+1 displays symbols on either side that are not the same as symbols on the other side. So the two sides are not "the same" in this sense. But this is a triviality among mathematicians - and the general public - who associate with each side a mathematical entity, the number 4. Likewise, Four=4 shows different symbols representing the same mathematical item. However, I believe your position exceeds these parameters and is somehow more "fundamental".

This seems like a silly game of distinction without a difference that could only appeal to intellectual descendants of medieval scholasticism. But I could be wrong.

FWIW, I am in agreement with you in this discussion, but I’ve been reluctant to muddy the waters with my own interpretation. It seems confusing enough for those arguing as it is.

I will say that logic, like mathematics, like Shannon information, is not about meaning - meaningfulness is assumed upon use. It’s about the relation between signs (not things) within a specific value system. The equation is ‘possibly meaningful’ only within that system, in which both sides represent the exact same value, regardless of any particular instance, and regardless of its possible meaning. So long as you assume a perfect alignment in instances of value structure and possible meaning, then both sides of the equation 4=4 are ‘the same’. In reality, it’s more like a six-dimensional ratio (0, 0, 0, 0, 4x, 0) = (0, 0, 0, 0, 4x, 0), with only some of the redundancy removed - this equation 4=4 is entirely redundant in logic, mathematics and Shannon information theory. It has meaning only when the sides are NOT identical.

So long as you assume a perfect alignment in instances of value structure and possible meaning, then both sides of the equation 4=4 are ‘the same’. — Possibility

That is exactly the meaning of ‘abstraction’.

Actually, what I think Metaphysician Undiscovered is talking about is personal identity. The ‘law of identity’ is a different thing altogether.

The two big objections to Platonism that arise from conversations like this are that Platonic objects lack clear identity conditions and that the ontology is profligate, a crowded slum, what Quine called Plato's Beard. Reducing every object to Math should answer both objections. — Pneumenon

Well, it is an answer, but why is it the answer? Why one object and not two or 42 or all of them? Why do you elect to be a lumper and not a splitter?

I think you need to back up a bit and tell us why the question matters. What difference would an answer make?

That is exactly what I said. — Wayfarer

Obviously not. You said:

When we’re discussing the ‘=‘ sign we are by definition discussing a symbol which denotes strict identity. — Wayfarer

"Strict identity" is what is defined by the law of identity. The "=" in mathematics signifies that two distinct things have the same value. It does not signify that what is on the right is the same thing as what is on the left, as "strict identity" indicates.

There is something really absurd here. So, you're saying, that in the expression A=A, that this expression only refers to particular instances of 'A'? That in order for 'A' to be 'A' then we have to refer to a particular instance of 'A'? That when we say, 2 + 2 = 4, that you're saying 'hang on! Which individual instances of '2' are you referring to?' — Wayfarer

I think you ought to take some time to study the law of identity. Check Wikipedia, Stanford, and Internet Encyclopedia of Philosophy to get a good consensus. Remember, "A=A" is just a formal representation, and the true representation of the law is stated as a proposition, one of the three fundamental laws of logic, including also noncontradiction, and excluded middle. What the law of identity says is that a thing is the same as itself. What this means, is that a thing is unique to itself, and is not identical to anything else. Check the Leibniz interpretation. He says that if we try to assert that two distinct things have the exact same properties, they are in fact one and the same thing.

So, what I am saying is that when '=' is used to symbolize the law of identity, it has a completely different meaning from when it is used in '2+2=4'. The representation, 'A=A' is just a convenience. The 'A' symbolizes an object, any object. The '=' symbolizes 'is the same as'. So the representation means that an object is the same as itself. I argue that it is a misinterpretation of what '=' symbolizes in mathematics, to assert that it means 'is the same as', as it does in this representation of the law of identity. Therefore we ought to respect the fact that what '=' symbolizes in the expression 'A=A', which is meant to represent the law of identity, is not the same as what '=' symbolizes in its mathematical context.

Anyway, let's leave this issue for now, and I'll answer your other question which is much more interesting to me.

It's not a matter of 'recognising it', this is something that I have only ever read in your posts. If you provide a reference I'd be obliged. — Wayfarer

OK, but this is a complicated issue, Aristotle is somewhat ambiguous, and there are numerous interpretations, so it will take some work on your part. I'll take the time to take you through a number of references which support my interpretation, I can only hope that you'll take the time to try and understand.

First, we look at Physics Bk.2 Ch.3, where "form" is defined in relation to the four causes. The form of a thing is said to be the thing's essence, or definition. At this point you need to adhere to Aristotle's description of "essence" and not be swayed by later interpretations which attempt to decisively remove accidentals from a thing's essence, giving us the term "essential".

Now let's proceed to Aristotle's extensive description of the particular individual, in Metaphysics, Bk.7. I suggest you read the section of Ch.4-11 numerous times, because it's not easy reading. However, it's very important, and the ambiguity will make you lean one way at one time, and another way at another time, possibly allowing previous biases to sway your overall interpretation. The goal I think ought to be to understand what is written, interpret it in a way which makes sense to you. There is really a need to refer to other writings, like Categories, and On the Soul, to fully understand his use of terms, but I'll try to guide you on these other references.

Starting at Ch.4. "The essence of each thing is what it is said to be 'propter se"'. The footnote to my translation (W.D. Ross) states that it is convenient to translate 'propter se' as "in virtue of itself". If we proceed, we find in Ch.5 how "essence" is related to "substance". The closing sentence of the chapter reads "Clearly, then, definition is the formula of the essence, and essence belongs to substances either alone or chiefly and primarily and in the unqualified sense." Referring to "Categories" Ch.5, Aristotle distinguishes primary and secondary substance. In the truest and primary sense substance is the individual. In the secondary sense it is the species within which the primary substances are included.

Proceeding to Ch.6 of Bk.7 Metaphysics, he question whether a thing and its essence are the same "for each thing is thought to be not different from its substance, and the essence is said to be the substance of each thing". So the problem of accidentals is now brought up, and it appears like a thing cannot be the same as its essence. But in the case of supposed self-subsistent Ideas, Forms, it is shown to be impossible, as incoherent, that a Form's essence could be different from the Form itself. "Each thing itself, then, and its essence are one and the same in no merely accidental way, as is evident both from the preceding arguments and because to know each thing, at least, is just to know its essence, so that even by the exhibition of instances it becomes clear that both must be one." 1031b,18. "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.

However, Aristotle leaves the door open to ambiguity here, by allowing that in an accidental way, a thing is not the same as its essence. So we need to proceed further, and understand the nature of accidentals, the existence of which appears to drive a wedge between a thing and its essence. So Ch.7 proceeds to question the nature of "comings to be" with a comparison made between natural things and artificial things. To fully apprehend Aristotle's position here it is necessary to understand how he defines "soul" in "On the Soul". The question here is the relationship between a thing's matter and its form. I had an extensive discussion with dfpolis a year or two ago, on this chapter. It is described by Aristotle, that in artificial things, the form of the thing which will come to be as a material thing, exists in the soul of the artist, and is then put into the matter. Aristotle compares this to natural things, and concludes that the process must be similar. The form of the individual thing must be prior to its material existence, and some how put into the matter. Df argued against this point, insisting that Aristotle's position is that the particular form which the individual thing will have, is already intrinsic to the matter. However, careful interpretation will reveal that this is rejected because of infinite regress.

In any case, the line we need to follow, is the idea that the "formula" precedes the existence of the material thing. Now the question is asked, to what extent is the matter a part of the formula (1032b-1033a). When this occurs, the suffix "en" is used to determine the matter, "brazen", "wooden", etc.. "The brazen circle, then, has its matter within its formula" 1033a,4. At this point, we can see how the accidentals of the individual, which are commonly attributed to the matter, may be transferred to the form, when the matter becomes part of the form. Notice however, that this is how artificial production is represented, and it is necessary that the form be supported by the soul of the artist. Without this soul, we lose the ability to separate the form completely from the matter, the separation which allows that the matter itself is part of the formula, and we are left with df's argument that the uniqueness supplied by the accidentals inheres within the matter.

So, I referred df to Ch.8, which ties together natural things, and artificial things, as being the same type of process. At 1033b, we can see that if the accidents of the thing which will come to be are accounted for as being within the matter, we have an infinite regress. Therefore we must assume a separation between the form and the matter, in both natural and artificial things, such that the form comes from someplace else, the soul in the case of artificial things. And, I'll interject here to remind you (as relevant to the subject we are discussing), that the matter may become part of the formula, so that the accidentals which are commonly attributed to the matter, are in reality, part of the form.

"It is obvious, then, from what has been said, that that which is spoken of as form or substance is not produced, but the concrete thing which gets its name from this is produced, and that in everything which is generated matter is present, and one part of the thing is matter and the other form."1033b 17.

The remainder of this section which I recommended, up to and including Ch.11, deals with the difficulties, which are abundant, involved in trying to understand this relation between form and matter. Read the following section carefully, and recall the distinction made in "Categories" between "primary substance" referring directly to the individual (Callias for example), and secondary substance, the species (man). Notice how he says that there is no formula which includes the matter because the matter is indefinite, but with reference to primary substance itself, (which is the individual itself, therefore the application of the law of identity), there is a formula which includes the matter.

"What the essence is and in what sense it is independent has been stated universally in a way which is true of every case, and also why the formula of some things contains the parts of the thing defined, while that of others does not. And we have stated that in the formula of the substance the material parts will not be present (for they are not even parts of the substance in that sense, but of the concrete substance; but of this, there is in a sense a formula, and in a sense there is not; for there is no formula of it with its matter, for this is indefinite, but there is a formula of it with reference to its primary substance---e.g. in the case of man the formula of the soul---for the substance is the indwelling form, from which and the matter the so-called concrete substance is derived; e.g. concavity is a form of this sort, for from this and the nose arise 'snub nose' and 'snubness'); but in the concrete substance, the matter will also be present, e.g. a snub nose or Callias, the matter will also be present." 1037a 21-32.

From the perspective of appearances of symbols you have a point. Clearly, 2+2=3+1 displays symbols on either side that are not the same as symbols on the other side. So the two sides are not "the same" in this sense. But this is a triviality among mathematicians - and the general public - who associate with each side a mathematical entity, the number 4. Likewise, Four=4 shows different symbols representing the same mathematical item. However, I believe your position exceeds these parameters and is somehow more "fundamental". — jgill

What seems to be neglected in mathematical Platonism, is that '2+2' signifies an operation, and '3+1' signifies an operation. The two operations are clearly not the same, though they are in some sense equal. The more complicated the equation is, the more complicated are the operations which are signified. The difference between equal operations can be quite significant. To reduce these complex mathematical operations to simple mathematical objects, and assert that substantially different operations are actually the same mathematical object is a dreadful misrepresentation of what mathematics really is.

This seems like a silly game of distinction without a difference that could only appeal to intellectual descendants of medieval scholasticism. But I could be wrong. — jgill

I think it's a matter of metaphysics, ontology. But what is really at stake here is the meaning behind mathematical symbols, and therefore an understanding of what mathematicians are actually doing. I think that fishfry for example, demonstrates a very naive understanding of what mathematicians are actually doing by insisting that mathematical operations could be carried out in the same way which they are, even if the symbols signified nothing. This may be true of some formal logic, but in mathematics, the possible operations are determined by the meaning of the symbols. So it is impossible to separate the operations from what the symbols signify, as is done in formal logic. Therefore the attempt to represent mathematics as a type of formal logic which provides a separation between the operations which are performed with the symbols, as distinct from what is represented by the symbols, is a common misunderstanding of the nature of what is actually symbolized by the symbols in mathematics. In reality, what is signified by the symbols is operations, not objects. Even the most simple mathematical symbols like 4, can be understood as denoting an operation of grouping four individuals, and the symbol 1 denotes an operation of individualization. So we cannot get beyond the fact that operations are intrinsic within, and essential to, the mathematical symbols. Therefore the attempt to separate symbols from operations would leave us no access to any operations, and no mathematics.

I will say that logic, like mathematics, like Shannon information, is not about meaning - meaningfulness is assumed upon use. It’s about the relation between signs (not things) within a specific value system. The equation is ‘possibly meaningful’ only within that system, in which both sides represent the exact same value, regardless of any particular instance, and regardless of its possible meaning. So long as you assume a perfect alignment in instances of value structure and possible meaning, then both sides of the equation 4=4 are ‘the same’. In reality, it’s more like a six-dimensional ratio (0, 0, 0, 0, 4x, 0) = (0, 0, 0, 0, 4x, 0), with only some of the redundancy removed - this equation 4=4 is entirely redundant in logic, mathematics and Shannon information theory. It has meaning only when the sides are NOT identical. — Possibility

What's your opinion on my reply to jgill above, Possibility? Do you agree that what the mathematical symbols represent are operations? So when we have an equation, we say that the operation on the right side has the same value as the operation on the left side. And when we say that 4=4, the symbol 4 refers to a grouping of individuals, and we say that one grouping of four has the same mathematical value as another grouping of four. Therefore relative to mathematical value, a grouping of four is "the same" as any other grouping of four, but relative to identity, the two groups are clearly not the same. Notice how I refer to the "grouping" of four, because this is an activity, an operation, carried out by the sentient being which apprehends the four individuals as a group of four. Likewise, to apprehend one thing as an object, an individual unity, is an operation (individuation) carried out by the sentient being which perceives it that way. This fundamental act of individuation is the basic premise for mathematics. Therefore the axioms of mathematics need to be well grounded in the law of identity which stipulates the criteria for being an individual.

The "=" in mathematics signifies that two distinct things have the same value. It does not signify that what is on the right is the same thing as what is on the left, — Metaphysician Undercover

So, again, you're saying that every occurence of 'A' is unique? I still think you're confusing the law of identity, with the meaning of individual identity, which are different subjects even if related.

I did take the time to read your argument on essence, substance and so on. As you note it is replete with difficulties, ambiguities and aporia. This is a deep problem with Aristotelian metaphysics, generally - the difficulty of arriving at any ultimate definition of the fundamental terminology, I think due to the inherent limitations in reason itself. But, it's still worth studying and I appreciate the time you've taken to spell it out. It's one of the subjects I'm trying to find time to understand better.

I haven't had the chance to read all of this in detail so apologies in advance if I repeat something already stated or otherwise step on anyone's toes.

The reason I mentioned the (non)distinction between sameness and identity is because it is pertinent to the question of mathematical objects. If we begin by thinking of identity as

To have an identity is to be identifiable as a unique and particular individual. — Metaphysician Undercover

We can only investigate identity by looking at the qualities, properties, parts, etc. of an object in order to identify it. The essence of the object, or its form, is never what differentiates it for this purpose. Instead, it is what is accidental to it that allows it to be identified.

Plato, I think, takes identity "all the way" and so sees this process of identification as moving these accidents (which allowed identification) into the essence of the categorically more specific object that is identified. For instance, I see a person, then by perceiving certain accidents of that person (beard, tall, male, etc.) I realize the person is my father. What was accidental to the person (and to fathers in general) is actually essential to the individual that is my father. But if we admit an essence that is my father, he loses his individuality since some other person with the same (identical?) properties would also be my father.

Certain proofs in mathematics hinge on the dissolution of separate identities. For instance, the proofs on this page about lines tangent to a circle presuppose the existence of points with certain accidents. It is through this method that the contradiction necessary for the proof is shown. This reflexively shows that the points themselves cannot have the accidents which were assigned to them and thus the essence of the points of tangency is grasped. The proof equivalently amounts to showing that these points are the same.

Plato's mistake, it seems, is not noticing that identity only arises insofar as objects are not the same. It is an instrument of abstraction or speculation. Its persistence indicates an indefinite understanding. This implies it is never really present in complete understanding, actuality, truth, etc. Perhaps he was disturbed by the thought that his own philosophy suggested that we do not really have individuality or self-ness. It may have also threatened some of his assumptions about Ethics.

Kant also comments on this in Critique of Pure reason, Transcendental Doctrine of Method, Chapter I, Section I:

Philosophical cognition is the cognition of reason by means of conceptions; mathematical cognition is cognition by means of the construction of conceptions. The construction of a conception is the presentation à priori of the intuition which corresponds to the conception. For this purpose a non-empirical intuition is requisite, which, as an intuition, is an individual object

(my emphasis)

Kantian intuition therefore must involve this process of construction and dissolution of identity, not as sameness but as arbitrary differences which ultimately prove insubstantial for the concept.

Later, in section 4, he writes

Analytical judgements (affirmative) are therefore those in which the connection of the predicate with the subject is cogitated through identity; those in which this connection is cogitated without identity, are called synthetical judgements.

Kant seems to use Identity to mean sameness, or more specifically that to deduce two things as the same is to show that they share the same identity. This is further supported by Division I, Endnote 1. So even Kant doesn't really distinguish sameness from identity.

I have more to say but I've run out of brain juice...

Plato, I think, takes identity "all the way" — Garth

Plato was concerned with the identity of the transcendent soul, the identity of Forms in relation to particulars, and the identity of abstract parts with the whole. Accidents, essence, object are not in Plato.

We must ensure that the mathematical axioms which we employ conform to reality or else they will lead us astray. Therefore it is actually necessary that we do change mathematical axioms as we try and test them — Metaphysician Undercover

You appear to suggest that mathematical axioms are similar to theory in physics. String theory, however, seems un-testable at present. Does it then lead us astray? If you were to say it does, how could you possibly know? How might you test the Axiom of Choice?

It's interesting to read perspectives of mathematics that I suppose could be called pre-foundational to distinguish them from formal foundation theory that fishfry is good at explaining. These are notions I never entertained while active as a mathematician. Of course, I didn't spend time looking into formal foundations either.

fishfry refers to math as a game, and it certainly is that. But a practicing mathematician may lose that perspective and math may assume a kind of non-physical solidity and seem "real", even when it's not obvious that it may be related to physical phenomena. Similarly chess probably seems "real" to serious devotees. Incidentally, MU, "pure mathematics" simply means not immediately applicable to the physical world. I've dabbled in this sort of math for decades.

So, again, you're saying that every occurence of 'A' is unique? I still think you're confusing the law of identity, with the meaning of individual identity, which are different subjects even if related. — Wayfarer

It seems like you're not familiar with the law of identity.

Here's Wikipedia: "In logic, the law of identity states that each thing is identical with itself. It is the first of the three laws of thought, along with the law of noncontradiction, and the law of excluded middle."

Now here's Wikipedia on Leibniz' identity of indiscernibles: " The identity of indiscernibles is an ontological principle that states that there cannot be separate objects or entities that have all their properties in common. That is, entities x and y are identical if every predicate possessed by x is also possessed by y and vice versa; to suppose two things indiscernible is to suppose the same thing under two names."

The law of identity refers to all things, and I'm not sure what you mean by "individual identity". The law of identity is a fundamental ontological principle which represents the uniqueness of a thing. So, if 'A=A' is meant to represent the law of identity, then A represents an object, and both instance of A represent the exact same object, and '=' signifies "is the same as". A thing is the same as itself.

I did take the time to read your argument on essence, substance and so on. As you note it is replete with difficulties, ambiguities and aporia. This is a deep problem with Aristotelian metaphysics, generally - the difficulty of arriving at any ultimate definition of the fundamental terminology, I think due to the inherent limitations in reason itself. But, it's still worth studying and I appreciate the time you've taken to spell it out. It's one of the subjects I'm trying to find time to understand better. — Wayfarer

I strongly recommend reading Aristotle's "On the Soul". At first glance, it appears outdated, but it is very well written, not difficult, but quite informative, giving good examples of Aristotle's usage of fundamental ontological terms like matter, form, actual, and potential. Then read his "Metaphysics", because much in his metaphysics will seem incomprehensible without the background provided for by "On the Soul".

But if we admit an essence that is my father, he loses his individuality since some other person with the same (identical?) properties would also be my father. — Garth

The point though, which makes identity a real and true principle, is that there will not be some other person with the same, identical properties. This is what Leibniz' principle says, such is impossible. And, for simplicity sake, we can see that if spatial-temporal location are considered as properties of a thing, it is truly impossible that two distinct things have the exact same properties.

Certain proofs in mathematics hinge on the dissolution of separate identities. For instance, the proofs on this page about lines tangent to a circle presuppose the existence of points with certain accidents. It is through this method that the contradiction necessary for the proof is shown. This reflexively shows that the points themselves cannot have the accidents which were assigned to them and thus the essence of the points of tangency is grasped. The proof equivalently amounts to showing that these points are the same. — Garth

Clearly, each point on the circumference of a circle is distinct, not the same, having a different identity from every other point. If this were not the case, then there would be no definable angles between distinct radii.

Plato's mistake, it seems, is not noticing that identity only arises insofar as objects are not the same. It is an instrument of abstraction or speculation. Its persistence indicates an indefinite understanding. This implies it is never really present in complete understanding, actuality, truth, etc. Perhaps he was disturbed by the thought that his own philosophy suggested that we do not really have individuality or self-ness. It may have also threatened some of his assumptions about Ethics. — Garth

There are two ways of looking at this. You suggest that identity is not ever an aspect of "complete understanding, actuality, truth, etc.". You might believe that human beings possess "complete understanding...", and therefore identity is not anything real. I believe that identity is real, and human beings cannot ever possess "complete understanding...".

Kantian intuition therefore must involve this process of construction and dissolution of identity, not as sameness but as arbitrary differences which ultimately prove insubstantial for the concept. — Garth

This is right, the accidental differences are insubstantial for the concept, but for Kant, we cannot ever know the thing in itself. So Kant is consistent with me, identity is real, but human knowledge will always be incomplete, because those particular aspects of the thing in itself cannot enter into the concept.

Kant seems to use Identity to mean sameness, or more specifically that to deduce two things as the same is to show that they share the same identity. This is further supported by Division I, Endnote 1. So even Kant doesn't really distinguish sameness from identity. — Garth

By the law of identity, identity is sameness, but two distinct things cannot share the same identity because it is incoherent to say that two distinct things are the same thing.

You appear to suggest that mathematical axioms are similar to theory in physics. String theory, however, seems un-testable at present. Does it then lead us astray? If you were to say it does, how could you possibly know? How might you test the Axiom of Choice? — jgill

I think that some theories and axioms take a long time to test. The problem is that our testing capacity is very limited compared to the wide range of possible situations for application. So we test theories within a very limited range, and when they work within that range, we proceed to apply them to a much wider range where we do not have the capacity to adequately test the results. So for example, we take theories like Einstein's relativity, and we tested them around earth in a very limited range of spatial temporal relations, which we might call the midrange. Then we apply them to the furthest spatial distances in the universe, and the tiniest temporal durations in quantum physics, where we haven't tested them nor can we test them. We have no reason to believe that the theories are giving us accurate results in these conditions because we are operating on the assumption that what is true at the midrange is also true at the extremes. This is explained well in physicist Lee Smolin's book, "Time Reborn", in the chapter "Doing Physics in a Box".

So, in my analogy which suggests mathematical axioms are similar to theories in physics, we could consider the same principle. The axioms might prove themselves very well in all sorts of common applications, but when we get to the extremes, like when infinities enter the equations, they might really be failing us. Boundary conditions for example are very curious things. We could stipulate them arbitrarily, and when we apply them they confirm themselves, through the act of application, as long as we see no reason to doubt them. So it appears like there is a real boundary anywhere that the boundary conditions are imposed, because the mathematics is designed to treat the observations that way through the application of the boundary conditions. .

fishfry refers to math as a game, and it certainly is that. But a practicing mathematician may lose that perspective and math may assume a kind of non-physical solidity and seem "real", even when it's not obvious that it may be related to physical phenomena. Similarly chess probably seems "real" to serious devotees. Incidentally, MU, "pure mathematics" simply means not immediately applicable to the physical world. I've dabbled in this sort of math for decades. — jgill

I don't consider any such human activity as a game. Games are played for entertainment, and in general, the goal is to win. You must use a different definition of "game". I'd say you're playing a different game from me, and that would serve to demonstrate that we are not playing a game, because how could we be engaged in the same game, yet playing different games?

Anyway, fishfry goes beyond your definition of "pure mathematics" to claim that "You can, if you like, view the entire enterprise as an exercise in formal symbol manipulation that could be carried out by computer, entirely devoid of meaning. It would not make any difference to the math." This is what I disputed above, because the possible manipulations are determined by the meaning of the symbols. The most fundamental being the meaning of the unit, 1. Furthermore, the meaning of the symbols has been developed over years of application. So there is really no such thing as "pure mathematics" by your definition, unless one is starting with all new symbols never before used, because the symbols employed have already derived there meaning through application.

What's your opinion on my reply to jgill above, Possibility? Do you agree that what the mathematical symbols represent are operations? So when we have an equation, we say that the operation on the right side has the same value as the operation on the left side. And when we say that 4=4, the symbol 4 refers to a grouping of individuals, and we say that one grouping of four has the same mathematical value as another grouping of four. Therefore relative to mathematical value, a grouping of four is "the same" as any other grouping of four, but relative to identity, the two groups are clearly not the same. Notice how I refer to the "grouping" of four, because this is an activity, an operation, carried out by the sentient being which apprehends the four individuals as a group of four. Likewise, to apprehend one thing as an object, an individual unity, is an operation (individuation) carried out by the sentient being which perceives it that way. This fundamental act of individuation is the basic premise for mathematics. Therefore the axioms of mathematics need to be well grounded in the law of identity which stipulates the criteria for being an individual. — Metaphysician Undercover

I’ve found that the term ‘object’ - denoting a consolidated focus of thought or feeling - is often freely applied to physical objects, events or concepts. I find this ambiguity leads to much confusion, and I’ve had numerous discussions with other contributors to this forum regarding the dimensional distinctions between the relation of self-consciousness to, say, an actual object, an operation/event (eg. grouping), a symbol for the concept that represents the value/significance of an event, and meaning prescribed to that symbol.

But mathematics and logic, like computer information systems, are often treated as closed conceptual systems, with any qualitative relations (necessary for the system to be understood) assumed and consolidated: ignored, isolated and excluded. So a ‘mathematical object’ refers to the ‘individual’ symbol for a concept that represents consolidated value/significance of an event - any instance of which is a subjective, temporally-located relation between an observer/measuring device and qualitative relational structures of measurement/observation. But within the isolated conceptual system of mathematics (which effectively assumes and then ignores an alignment of underlying relational structure by abstraction), a ‘mathematical object’ would abide by the law of identity. This from the Wikipedia entry on Law of Identity, referring to violation:

“we cannot use the same term in the same discourse while having it signify different senses or meanings without introducing ambiguity into the discourse – even though the different meanings are conventionally prescribed to that term.”

The Law of Identity applies only in a logical, abstract (closed) system of thought or language. Any ‘mathematical object’ is interpretable in reality only by a self-conscious observer in a qualitative potential relation to both the symbol (to prescribe qualities of meaning) and the event (to attribute qualities of sense or affect). The moment you relate the Law of Identity to anything outside of logic - ie. once you cannot assume an alignment of sense or meaning in discussion - you risk violation.

I don't consider any such human activity as a game. Games are played for entertainment, and in general, the goal is to win. — Metaphysician Undercover

Sometimes games are played for money or prestige. The professional mathematician finds his activities entertaining, frequently fascinating, and he definitely likes to arrive at a result before others. He likes to win.

You must use a different definition of "game" — Metaphysician Undercover

Nope

Anyway, fishfry goes beyond your definition of "pure mathematics" to claim that "You can, if you like, view the entire enterprise as an exercise in formal symbol manipulation that could be carried out by computer, entirely devoid of meaning. It would not make any difference to the math." — Metaphysician Undercover

That's one way of looking at it. It's not the way I perceive the discipline.

It seems like you're not familiar with the law of identity. — Metaphysician Undercover

I wish you would stop saying that. I think it's your interpretation that is idiosyncratic.

So, if 'A=A' is meant to represent the law of identity, then A represents an object, and both instance of A represent the exact same object, and '=' signifies "is the same as". A thing is the same as itself. — Metaphysician Undercover

Precisely. 'A' represents an object. So, it's an abstraction. As are all symbols and numbers.

Where this all started was your nonsensical claim that:

Clearly, "is the same as" does not mean the same thing as "is equal to". — Metaphysician Undercover

When it obviously does, despite your obfuscation.

The law of identity refers to all things, and I'm not sure what you mean by "individual identity". The law of identity is a fundamental ontological principle which represents the uniqueness of a thing. — Metaphysician Undercover

What I mean by ‘individual identity’ is ‘the identity of individual particulars’. It’s exactly what you mean by it. So take two individuals, Peter and Paul. They’re equal in one sense, i.e. equal before the law, but that doesn’t mean they’re the same. You can’t say that Peter and Paul 'are equal', but you can say 'they’re equals'. (Unless they’re being graded for a sport, and Paul is graded higher than Peter, in which case Paul is ‘greater than’ Peter in respect of that ranking i.e. a better tennis player.)

But that has no bearing on the symbolic representation 'A=A' because in that case, we're not referring to particular beings, but to symbols. Same with mathematics. Symbols are abstractions, but due to our rational ability, they have bearing on the world.

What I mean by ‘individual identity’ is ‘the identity of individual particulars’. — Wayfarer

The law of identity is a statement concerning the identity of particulars. Here's a quote from Stanford: "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?"

This is what we are discussing, the identity of individual particulars. The point was that an abstraction, a Platonic Form, does not fulfill the conditions of the law of identity ("the identity of individual particulars), therefore it is not an individual particular, and ought not be called an "object". A Platonic Form does not have an identity as an individual particular. An object has an identity as an individual particular. Therefore a Platonic Form is not an object.

But that has no bearing on the symbolic representation 'A=A' because in that case, we're not referring to particular beings, but to symbols. Same with mathematics. Symbols are abstractions, but due to our rational ability, they have bearing on the world. — Wayfarer

We are referring to particular objects, you agreed above, 'A' represents an object. An object is an individual, a particular. The phrase 'A=A' is commonly used to represent the law of identity ("the identity of particulars"), which states that a thing is the same as itself. This is what Stanford refers to as "numerical identity", it means 'the very same', 'absolute identity', the "relation everything has to itself and nothing else".

You'll see that Stanford also refers to "qualitative identity", which means that things share properties. The law of identity is not concerned with qualitative identity. When we say two things are "equal" we are using qualitative identity.

An abstraction is a concept, an idea, which is within a mind, as a product of a mind, unless we provide for transcendent existence such as Forms which allows the abstraction to exist outside the mind. So symbols are not themselves abstractions, they are physical things which are meant to signify something. Symbols cannot provide for that transcendent existence of the Forms unless we show that the meaning, the abstraction, somehow inheres within the symbol itself.

But a symbol might signify an object, like a name or a proper noun does, or a symbol might represent an abstraction, as '2' does. Now, the case of 'A=A' is tricky. 'A' represents an abstraction meaning an object, any object. It does not directly represent a particular object. So in that sense, it represents an abstraction. However, that abstraction is as a universal law, a general statement such as an inductive conclusion. The statement is a proposition about particular individuals. Now, 'A=A' is a symbolic representation of a proposition concerning particular individuals.

Therefore, strictly speaking 'A' represents a specific part of an abstraction, it represents all objects. And, we are employing qualitative identity to make a statement about what all objects have in common. However, that statement concerns what distinguishes all objects as different from each other. Therefore the quality which all objects share (they are the same in this respect) is that they are different from each other. This is why I described in the other thread, that identity is a special type of equality. It is an equality which a thing has with itself, and all others are excluded from. It may take on the appearance of contradiction to some (Hegel), but what it really does is separate "same" as categorically different from, rather than opposed to "different". In a sense then it appears as an irrational equality, by that appearance of contradiction, but this is precisely why the accidental properties of objects are unintelligible to us. And it isn't really contradictory, as I said.

Now, the problem arises if we try to exclude those accidental properties as differences which don't make a difference, or something like that. When we say that '2', as a symbol, represents a "number", and that number is itself an object, we have assumed that the abstraction is an intelligible object. But the accidentals which distinguish one instance of '2' from another have been excluded because we assume that each instance of '2' refers to the very same object (as per identity principle). Therefore we do not have a true representation of an object here, because the difference between one instance and another has been excluded for the purpose of claiming "the same object". So, because the fact, or inductive truth about objects, which is pointed to by the law of identity is circumvented, to claim that '2' represents an object, we avoid a true representation of what '2' means. And in reality, "the number" does not fulfill the identity conditions of the law of identity, which are required of every object, as individual particulars.

This allows for the possibility of equivocation. Anyone who insists that "a number" is an object would likely proceed to equivocation. Since '2' does not refer to an object with a particular identity, it will have a distinct meaning depending on the context of usage. But if someone insists that it refers to an object, then it is asserted that it necessarily has the same identity in each instance of usage, as referring to the same object. To insist that the symbol '2' refers to the same object in each application, when it really has a distinct meaning, is to equivocate.

I’ve found that the term ‘object’ - denoting a consolidated focus of thought or feeling - is often freely applied to physical objects, events or concepts. I find this ambiguity leads to much confusion, and I’ve had numerous discussions with other contributors to this forum regarding the dimensional distinctions between the relation of self-consciousness to, say, an actual object, an operation/event (eg. grouping), a symbol for the concept that represents the value/significance of an event, and meaning prescribed to that symbol. — Possibility

As I described at the end of the last post, misuse of "object" allows freedom for equivocation.

But mathematics and logic, like computer information systems, are often treated as closed conceptual systems, with any qualitative relations (necessary for the system to be understood) assumed and consolidated: ignored, isolated and excluded. So a ‘mathematical object’ refers to the ‘individual’ symbol for a concept that represents consolidated value/significance of an event - any instance of which is a subjective, temporally-located relation between an observer/measuring device and qualitative relational structures of measurement/observation. But within the isolated conceptual system of mathematics (which effectively assumes and then ignores an alignment of underlying relational structure by abstraction), a ‘mathematical object’ would abide by the law of identity. This from the Wikipedia entry on Law of Identity, referring to violation: — Possibility

It appears like modern information theory and systems theory have converted our concepts of "information", such that the word now refers to the symbols directly, rather than what the symbols mean. This was discussed in the other thread on information.

The Law of Identity applies only in a logical, abstract (closed) system of thought or language. Any ‘mathematical object’ is interpretable in reality only by a self-conscious observer in a qualitative potential relation to both the symbol (to prescribe qualities of meaning) and the event (to attribute qualities of sense or affect). The moment you relate the Law of Identity to anything outside of logic - ie. once you cannot assume an alignment of sense or meaning in discussion - you risk violation. — Possibility

I think that the law of identity is actually an attempt to produce a closed system of thought. It is a prescriptive rule as to how we ought to use terms. Of course, as soon as a rule is imposed, there will be violations, that's the point of producing the rule, to distinguish violation from non-violation, and attempt to clear things up. But without the law of identity being enforced, there is freedom of ambiguity, and equivocation, as you describe.

Sometimes games are played for money or prestige. The professional mathematician finds his activities entertaining, frequently fascinating, and he definitely likes to arrive at a result before others. He likes to win. — jgill

What would you say is the "object" of that game, the goal? To get a "result" does not qualify as the object of the game, because anything could be construed as a result. If people are playing the same game, then they hold the same goal as the object of that game. If all mathematicians do not have the same goal, then they are not playing the same game, and we cannot describe mathematics as "a game"

Pure mathematics is more like an art. And art cannot be described as a game, because it breaks the rules which attempt to constrain one's goals. It's actually quite contrary to game play.

Nope — jgill

Sure looks like it to me. I think that a game has a clearly defined goal, and play without a clearly defined goal is not properly called a game. Mathematics does not have a clearly defined goal and is therefore not a game. I guess we'll have to just disagree.

I don't know what that means. Your example was triangles. Triangles are identified up to similarity by their angles; and up to congruence by the lengths of their sides; and identified uniquely by their congruence class and position and orientation in space. — fishfry

Is there a distinct Platonic form of the isosceles triangle, or just of triangles in general? Take the exact shape that my shoe has at some instant (since the particles in it move around). Is there a Platonic form of that shape?

I understand what you mean by mathematical structuralism. But relations between objects are only identifiable if you have identity conditions for the objects between which the relations obtain.

If people are playing the same game, then they hold the same goal as the object of that game. If all mathematicians do not have the same goal, then they are not playing the same game, and we cannot describe mathematics as "a game" — Metaphysician Undercover

How pleasantly wrong you are, MU. There are cliques within the broad structure of math in which participants work towards common goals. I was in such a clique.

Pure mathematics is more like an art. — Metaphysician Undercover

Since leaving my clique years ago, this is how I perceive math. I was never a good game player since I enjoyed going off in imaginative directions and doing my own thing.

Here's a quote from Stanford: — Metaphysician Undercover

The quote you provided does not support the point you made.

The point was that an abstraction, a Platonic Form, does not fulfill the conditions of the law of identity — Metaphysician Undercover

We've been discussing the nature of symbolic expressions, such as a=a, with some tangential discussion of the platonic forms.

However, I do agree that numbers and the like are not actually 'objects', but that the use of the term 'object' is metaphorical in this context.

Platonic form of the isosceles triangle... — Pneumenon

Platonic forms are not shapes per se. Triangles and circles are used as examples because they're simple.

Consider that when you think about triangularity, as you might when proving a geometrical theorem, it is necessarily perfect triangularity that you are contemplating, not some mere approximation of it. Triangularity as your intellect grasps it is entirely determinate or exact; for example, what you grasp is the notion of a closed plane figure with three perfectly straight sides, rather than that of something which may or may not have straight sides or which may or may not be closed. Of course, your mental image of a triangle might not be exact, but rather indeterminate and fuzzy. But 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, Some Brief Arguments for Dualism

There are cliques within the broad structure of math in which participants work towards common goals. I was in such a clique. — jgill

Those in such a clique might be said to be playing a game. Therefore the game is limited to that clique. Bit that's insufficient for the the claim that mathematics in general is a game.

Since leaving my clique years ago, this is how I perceive math. I was never a good game player since I enjoyed going off in imaginative directions and doing my own thing. — jgill

So you recognize that mathematics in general cannot be said to be a game then? Maybe we do have a similar definition of "game".

We've been discussing the nature of symbolic expressions, such as a=a, with some tangential discussion of the platonic forms. — Wayfarer

We have clearly been on different pages here. The discussion as far as I am concerned, has been the law of identity, and how it relates to so-called "mathematical objects", it has not been "the nature of symbolic expressions". The law of identity is not concerned with symbolic expressions, because it stipulates that the identity of a thing is within the thing itself. Symbolic expression is excluded from identity, as not an aspect of identity. Hence the quote from Stanford: "that relation everything has to itself and nothing else". The relation between object and symbol has been exclude from identity by the law of identity, and the symbolic expression rendered irrelevant to identity. This is how Aristotle dealt with the false identity asserted by the sophists, by removing the assumption that the identity of an object is something we create with a symbol.

The law of identity is not concerned with symbolic expressions, because it stipulates that the identity of a thing is within the thing itself — Metaphysician Undercover

Yes, but that is a much deeper problem, in some ways. You're talking about ontology, the nature of being. But the debate started with the argument over whether, in the expression a=a, that the 'a' on both sides of the '=' is the same. I'm saying, of course it is, and that the identity of 'a' is fully explained by its definition. I'm not talking about the being or essential nature of a, because 'a' is a symbol.

I've gone back to the previous page. You said:

You know that this ('a=a') is just a symbolic representation of the law. So the symbols need to be interpreted. What 'a=a' represents is that for any object, represented as 'a', that object is the same as itself.

Therefore it's not saying that 'a' doesn't have an identity, it's saying that the identity of the object represented by 'a', is the object represented by 'a'. — Metaphysician Undercover

The question I was asking at the time was whether numbers (etc) meet 'identity conditions'. And actually your answer was 'yes, but this is not relevant to the 'law of identity'. And that's because you're treating the 'law of identity' as an ontological issue concerning the 'essential nature of beings.' The point remains, however, that in the domain of symbolic logic, maths, and everyday speech, the identify of the symbols used - letters, numbers and so on - is fixed in relation to a domain of discourse. Therefore, letters, numbers, and so on, have an identity, which is fixed by their meaning. That was the only point at issue in my view.

We have clearly been on different pages here. — Metaphysician Undercover

Well, glad that is cleared up. :-)

I'm going to start at the heart of our misunderstanding:

And that's because you're treating the 'law of identity' as an ontological issue concerning the 'essential nature of beings.' — Wayfarer

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.

Yes, but that is a much deeper problem, in some ways. You're talking about ontology, the nature of being. But the debate started with the argument over whether, in the expression a=a, that the 'a' on both sides of the '=' is the same. I'm saying, of course it is, and that the identity of 'a' is fully explained by its definition. I'm not talking about the being or essential nature of a, because 'a' is a symbol. — Wayfarer

I don't think anyone was ever talking about the status of the symbol, 'A', I sure wasn't. You must have misunderstood. Sorry if I didn't express myself well.

However this is what is inconsistent with the law of identity: "the identity of 'a' is fully explained by its definition". The whole point of the law of identity is to affirm that the identity of a thing is within the thing itself, not some description or definition which someone gives. That's what gave Aristotle argumentative power over the sophists. If the identity of a thing is guaranteed by the definition employed, then we have no defense against unsound logic carried out on faulty definitions. Therefore we need a law of identity which takes identity away from the definition, to protect us against, and expose the inevitable false identities which will arise if the identity of a thing is whatever someone claims that it is with a definition.

The question I was asking at the time was whether numbers (etc) meet 'identity conditions'. And actually your answer was 'yes, but this is not relevant to the 'law of identity'. — Wayfarer

No, numbers do not meet identity conditions, that's the whole point. Identity conditions are stipulated by the law of identity. Since whatever it is which is referred to by the numeral 2 varies from one application to another, as 2 is meant to have universal application, then whatever it is which this symbol signifies, does not fulfill the identity conditions of the law of identity. That's the point, numbers do not fulfill identity conditions. Objects fulfill identity conditions, numbers do not. The conclusion which we need to make is that we have to look toward some other principles to understand what a number is. This is the highest division in Plato's divisions of knowledge, addressing directly, and attempting to understand the nature of, the so-called 'intelligible objects', philosophy. The second category is using these 'intelligible objects', such as mathematics.

The point remains, however, that in the domain of symbolic logic, maths, and everyday speech, the identify of the symbols used - letters, numbers and so on - is fixed in relation to a domain of discourse. — Wayfarer

This is very obviously not true, and I've argued it extensively elsewhere, enough to know that if a person is not inclined to see the reality of this, they are not likely to change. However, I'll provide a brief explanation. I gave an example of "I reserved a table for 4 at 4", already in this thread, but here's something a little more technical for you. Consider that there are natural numbers, rational numbers, real numbers, and we could even throw in imaginary numbers. In each different case, what the numeral means is slightly different because the operations which can be carried out are different. If mathematics is "a domain", then clearly what the symbols mean is not fixed within the domain. You might start breaking down mathematics into multiple domains, and say that the meaning is fixed within a specific domain, but this is not true, because people cross the boundaries, and this is why there is such a thing as equivocation.

The real existence of equivocation ought to be enough evidence for you to see that what the symbol signifies is in no way fixed by the domain of discourse. And if one is under the illusion that it is fixed, and even accepts the premise that there are fixed objects of meaning like Platonic Forms, that person will no doubt be deceived by the equivocation when it occurs. The first step in the defence against the malicious form of equivocation is to understand how it is enabled. This allows one to be wary of the conditions.

I don't think anyone was ever talking about the status of the symbol, 'A'. — Metaphysician Undercover

I was, although the original question was about numbers, not letters.

I think that the law of identity is actually an attempt to produce a closed system of thought. It is a prescriptive rule as to how we ought to use terms. Of course, as soon as a rule is imposed, there will be violations, that's the point of producing the rule, to distinguish violation from non-violation, and attempt to clear things up. But without the law of identity being enforced, there is freedom of ambiguity, and equivocation, as you describe. — Metaphysician Undercover

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.

If we go back to your simple example of “I reserved a table for 4 at 4”, the symbol 4 is the same, but the physical event structures they represent as a value are not. 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.” What results is akin to a wavefunction: describing the significance of a four-dimensional relational structure between the significance of two measurement events within the same logical system. 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.

I think ‘closed system of thought’ is what I was trying to get at with ‘domain of discourse’. Good explanations, by the way.