## There is only one mathematical object

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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.

You're confused. If the left and the right refer to the same thing, then the formula is true (or satisfied). And when the left and the right refer to different things, then formula is false (or not satisfied. The fact that we can write a false identity formula doesn't vitiate that.

0 = 0 is a true equation.

0 = 1 is a false equation.

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.

You don't know anything about it. You've never read a single page in a textbook on the subject. What I wrote is correct. You may have your own philosophy about things, but when you make claims about what happens in mathematics, you are prone to be flat out wrong and posting disinformation.

The left and the right may refer to the same thing or to different things, but the equation is TRUE if and only if the left and right refer to the same thing; and the equation is FALSE if and only if the left and right refer to different things.
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Here's a simple example GrandMinnow: '2+2=4'. On the left side there is a specific operation represented. On the right side there is no operation represented. Therefore it is very obvious that what is represent on the right is not the same thing as what is represented on the left, and '=' does not signify identity.
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No, the left side does not represent an operation. The left side represents the value of an operation with an operand pair.

The value of the operation + applied to the operand pair <2 2> is 4. Thus the equation 2+2=4 is true.

That's the way it works in mathematics. Your philosophy about things does not refute mathematics. Meanwhile, if you wish to continue to ignore how mathematics actually works and instead insist on your philosophy, then you would do better to present a systematic development of the subject with your alternative premises, definitions, and notations listed, and not continue to post disinformation about mathematics you know nothing about.
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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.

Gosh, you are so confident!

Wikipedia on Equation: "There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variable. A conditional equation is only true for particular values of the variables."

This is how most practicing mathematicians (possibly not those engaged in foundations) see things. Show us your background in mathematics, please.
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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.

By this argument, no continuity of (the Aristotelian notion of) any substance can occur, for any physical object will have accidental differences between itself at any time t and t'. Yet (the Aristotelian notion of) substance - as I best understand it - is precisely that with is identical relative to itself over time; more precisely, that which survives accidental changes (implicitly, over time). In much the same way, the concept of tree remains identical relative to itself over time; i.e., it survives accidental changes, or differences, over time.

The issue becomes even more problematic when considering personal identity over time.

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.

When we say “tree” and a Spaniard says “arbol” are not the concepts denoted by each different term identical - this despite possible accidental differences in the two term’s connotations? As in: the concept of tree, T, is the same as the concept of arbol, A. Hence T = A.

Given that the definitions of each will utilize different words, the English definition of “tree” and the Spanish definition of “arbol” might very well not be identical; but both definitions will define an identical concept. Again, one that survives accidental changes, including those of possible differences in connotations.
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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.
— GrandMinnow

(1) There is nothing incorrect in what you quoted.

(2) Since not just Wikipedia (which itself is not a reliable source on mathematics and certain other subjects) mentions a usage that distinguishes between an 'identity' and a 'conditional equation', fair enough, I should not have allowed an impression that I claim that such usage does not exist, and I was incorrect to dispute that some people use that basis of distinction. But from a brief perusal on the Internet, I see that that usage is found mainly in high school level algebra texts (and "college" level that is seen in the examples to be really review of high school level). It is often wise to be wary of high school level explanations and terminology that need to be made rigorous and even corrected by rigorous mathematical treatments (for a salient example, the definition of 'function'). Meanwhile, I have never seen that rubric mentioned by Wikipedia used in rigorous mathematics at upper division and early graduate level, including the basic ordinary subjects: mathematical logic, set theory, abstract algebra, analysis or topology. In such subjects, the notion of an equation is as I have mentioned it, and it is at least implicit in texts that include introductions recognizing the logical and set theoretical foundations. I can't claim to a certainty that the rubric is not found anywhere in serious mathematics, but I am skeptical that it is.

(3) The distinction that Wikipedia mentioned is different from the example you gave. You gave an example of a true statement with no variables versus an equation with variables that is not true on all values for the variables. The Wikipedia article refers to a distinction between formulas that are true for all values for the variables versus formulas that are true only for certain values for the variables.

The distinction you mentioned is correctly given a precise explanation in my previous posts. For the Wikipedia sense, we can somewhat expand to make explicit the distinction between:

Valid formulas. Formulas are valid if and only if they are true (or satisfied) in all structures/variable_assignments.

and

Formulas that are not validities. Those include those that are true in some structures/variable_assignments and those that are logically false as they are not true in any structure/variable_assignment.
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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.

Sorry, I was referring rather poetically to the Aristotelian 'essence'.

In philosophy, essence is that which makes a thing be what it fundamentally is. It is often called the “nature” of a thing such that it possesses the necessary characteristics or properties in contrast with merely accidental or contingent ones. It is often considered a specific power, function, or internal relation (or set of relations) which again makes the thing be the kind of thing that it is. The notion of essence has acquired many slightly but importantly different shades of meaning throughout the history of philosophy, though most of them derive in some manner from its initial use by Aristotle.

I think that you are equating, or conflating, ‘essence’ and ‘identity’.
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The value of the operation + applied to the operand pair <2 2> is 4. Thus the equation 2+2=4 is true.

OK, on the left side is an operation with a value, and on the right side is something which is not an operation, which is assumed to have the same value. Therefore it is very obvious that the right and left side represent different things which are assumed to have the same value through some principles, or mathematical axioms. Clearly, '=' does not represent identity, it represents equal value according to those principles, in a way very similar to the way that you and I are equal, as human beings, according to some principles of value, but we are clearly not the same..

That's the way it works in mathematics. Your philosophy about things does not refute mathematics. Meanwhile, if you wish to continue to ignore how mathematics actually works and instead insist on your philosophy, then you would do better to present a systematic development of the subject with your alternative premises, definitions, and notations listed, and not continue to post disinformation about mathematics you know nothing about.

You do not seem to know much about philosophy. I do not need to present a better system to expose problems in the existing system. Finding deficiencies, and resolving them, are two distinct activities. A single person might not be adept at both finding the privation and fulfilling the need. That's the way it works in philosophy we apprehend the value of the "division of labour".

By this argument, no continuity of (the Aristotelian notion of) any substance can occur, for any physical object will have accidental differences between itself at any time t and t'. Yet (the Aristotelian notion of) substance - as I best understand it - is precisely that with is identical relative to itself over time; more precisely, that which survives accidental changes (implicitly, over time). In much the same way, the concept of tree remains identical relative to itself over time; i.e., it survives accidental changes, or differences, over time.

In Aristotelian physics temporal continuity is provided for by matter. Matter is what persists, unchanged as the form of a thing changes, and substance contains matter. Today, this is represented by conservation laws, energy and mass. Accidentals are formal, as part of a thing's essence. The problem with representing "the concept" in the same way, as having temporal continuity, is that it seems to be immaterial. So it seems like we need a principle other than the physical "matter" to account for any temporal continuity of a concept. We might try 'information' to account for the identity of a concept, but that doesn't remain constant over time, so identity of the concept would be completely different from identity of an object, if we were to develop such a principle.

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When we say “tree” and a Spaniard says “arbol” are not the concepts denoted by each different term identical - this despite possible accidental differences in the two term’s connotations? As in: the concept of tree, T, is the same as the concept of arbol, A. Hence T = A.

No, the concept denoted must be different, because the Spaniard and the Anglophone are two distinct people, with two distinct backgrounds, so the meaning will be different to each, just like the concept of 'tree' is different for you and me.. No two people would have the exact same idea of what "tree" is. We assume that there is such a thing as "the concept", for simplicity sake, because we do not understand the complexities of the mind. This allows us to carry on in our linguistic endeavours as if we know what we're talking about, when we use "concept", when we really don't. In philosophy we approach these issues with the intent of understanding, so we cannot just gloss over the complexities of these mental activities assuming that the mental activities exist as 'concepts'.

Given that the definitions of each will utilize different words, the English definition of “tree” and the Spanish definition of “arbol” might very well not be identical; but both definitions will define an identical concept. Again, one that survives accidental changes, including those of possible differences in connotations.

That there is "an identical concept", is just an assumption made to facilitate communication. So it is justified only on a pragmatic basis. It allows us to group together a whole lot of distinct (mental) activities, without any understanding of them, and talk about them as "the concept". So this idea, that there is such a thing as "the concept" is supported only because it facilitates, in that respect. Relative to the goal of understanding the true nature of reality, it is a hinderance.

It is often wise to be wary of high school level explanations and terminology that need to be made rigorous and even corrected by rigorous mathematical treatments (for a salient example, the definition of 'function').

Actually, what we need to be wary about, is when we learn the fundamentals, the basics, within a field, in high school, and then we proceed to the higher levels in that field, and find that what is taught in the fundamentals is contradicted in the higher levels. This happens in physics for example, when we learn about wave motion, as activity within a medium. Then we get to the higher levels and they want you to believe that there's wave motion without a medium. Such discrepancies are good cause for healthy skepticism.

I think that you are equating, or conflating, ‘essence’ and ‘identity’.

If there is a conflation of 'essence' and 'identity', it is Aristotle who makes this conflation. And, since Aristotle is often consider the author of the law of identity, then the so-called conflation is what is intended by the law of identity. Therefore the mistake is on your part, in rejecting it.

Maybe you could reread that post I made concerning Aristotle's Metaphysics Bk.7. Or even better, read the primary source, perhaps a couple of times because it's quite difficult. Also, it might be necessary to read "On the Soul" to have adequate background information.

"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.
...
"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.
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Aristotle's Metaphysics, 1032a -- Each thing itself, then, and its essence are one and the same in no merely accidental way
And this is supported by reference to Plato's Socratic discussion of 'snub nose' and form of 'Snubness' at 1037a?

Unfortunately, Aristotle was a logician and not a foundational mathematician like Plato, and distinctions implicit in Plato's discussions directed at Pythagorean mathematicians were lost in the translation.

My take is that Aristotle's metaphysics requires flat single level 'nominalist' logic as further developed in the first half of the 20th century. More recently this has been implemented as relational database systems. Plato used two-level hierarchical logic where higher level forms inform many lower-level particulars. In the Dialogues, Plato attempts to define Forms by induction from bottom up, and also conducts pathetic witch hunts for sophists from top down in a hierarchical database schema.

I can try to formalize this, as what is A=A for an Aristotelian is just A for Plato's forms and A>{a1,a2,a3,...} for his particulars.
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There are no mathematical truths. 1 plus 1 equals 1 plus 1, not 2 (which doesnt exist) so math is a bunch of semantic tataulogies masquerading as ontology. Someday it will be surpassed by those who don't equate feeling with being, and math will end as medieval philosophy fell to science
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In Aristotelian physics temporal continuity is provided for by matter. Matter is what persists, unchanged as the form of a thing changes, and substance contains matter. Today, this is represented by conservation laws, energy and mass. Accidentals are formal, as part of a thing's essence. The problem with representing "the concept" in the same way, as having temporal continuity, is that it seems to be immaterial. So it seems like we need a principle other than the physical "matter" to account for any temporal continuity of a concept. We might try 'information' to account for the identity of a concept, but that doesn't remain constant over time, so identity of the concept would be completely different from identity of an object, if we were to develop such a principle.

I'm going to push this issue a little.

Aristotle defines X's matter as "that out of which" X is made.[1] For example, letters are the matter of syllables.[2]

From such quotes I interpret Aristotelian matter to be fairly synonymous with composition. A material cause is a compositional cause, for instance, one whose effects are bottom-up and concurrent with the composition as cause.

So Aristotelian matter need not be physical (as we moderns interpret it to be). For a somewhat easier example by comparison to a concept, a paradigm's Aristotelian matter is, or at least can be, the sum of ideas from which it is composed. This in the same way that a syllable's matter is the sum of letters from which it is composed.

I know this breaks with common and traditional interpretations, but how do you find that Aristotle himself would have disagreed with what I've just outlined in relation to matter?

No, the concept denoted must be different, because the Spaniard and the Anglophone are two distinct people, with two distinct backgrounds, so the meaning will be different to each {...]

As someone who speaks two languages fluently, I wholeheartedly disagree with this. Yes, some concepts do not translate in a single word, if at all. But basic concepts (again, generalized ideas), such as that of "tree", are the same across multiple cultures regardless of the language via which they are addressed (given that the populace is exposed to concrete instantiations of trees in its environment).

[...] just like the concept of 'tree' is different for you and me.

As to the concept of "tree" being different for you and me: these visual scribbles we term letters, syllables, and words are meaningless in the absence of the concepts they convey. The complexities of language aside, if no such scribble could convey the same (essential) concept between two different people, how would communication of anything be possible?
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But from a brief perusal on the Internet, I see that that usage is found mainly in high school level algebra texts (and "college" level that is seen in the examples to be really review of high school level).

Of course. In my years as a prof the only times I recall discussing the equal symbol is in elementary set theory. If we had had an actual course in foundations or even set theory It's not likely I would have taught the course. I looked in an advanced text, Introduction to Topology and Modern Analysis by Simmons, which I consider an exemplary work of clarity, and he only gives the definition of "=" at the beginning for sets.

It is often wise to be wary of high school level explanations and terminology that need to be made rigorous and even corrected by rigorous mathematical treatments (for a salient example, the definition of 'function').

Thanks for the advice. A bit late, however. :cool:
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"Clearly, then, each primary and self-subsistent thing is one and the same as its essence." ~ Aristotle

What is included in the category of 'primary and self-subsistent things'? Do tools and artefacts belong in that category? Do they have 'an identity' according to this criteria?
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Unfortunately, Aristotle was a logician and not a foundational mathematician like Plato, and distinctions implicit in Plato's discussions directed at Pythagorean mathematicians were lost in the translation.

Plato a mathematician? Come on, have you read Plato's dialogues? What are you going by, the doctrine of recollection? I guess we're all mathematicians then.

From such quotes I interpret Aristotelian matter to be fairly synonymous with composition. A material cause is a compositional cause, for instance, one whose effects are bottom-up and concurrent with the composition as cause.

I don't think that you can call the parts of a thing as the cause of its composition. Notice Aristotle's full description of material cause, "that out of which a thing comes to be, and which persists...e.g. the bronze of the statue". The cause of the bronze being composed as a statue would be something other than the bronze itself.

So Aristotelian matter need not be physical (as we moderns interpret it to be). For a somewhat easier example by comparison to a concept, a paradigm's Aristotelian matter is, or at least can be, the sum of ideas from which it is composed. This in the same way that a syllable's matter is the sum of letters from which it is composed.

I agree, that matter need not be physical, but what is at issue is temporal continuity. Notice that the same matter which a thing is composed of, exists prior to the thing coming to be, and after it has come to be. This is why matter is described as potential, it provides the potential for the object, and even after the object exists the matter has the potential to be something else. So the problem I see with positing ideas as 'the matter' of a concept, is that ideas come and go; if they are forgotten, or replaced by something better, they disappear forever. Furthermore, within the Aristotelian conceptual structure, ideas are formal, so if we could find some temporal continuity within the ideas which make up a concept, as its parts, there is still a big inconsistency here.

Therefore, I believe we need to go much deeper than conscious ideas, right through the emotions, into the subconscious level of human existence, to find the true 'content', the 'subject matter' of ideas and concepts, the underlying substance. Since we haven't been able to determine this element which could account for temporal continuity in concepts, thereby providing that a concept has identity, we haven't been able to demonstrate that a concept is an object. In the case of a material object, we say that the object continues to be the same object because it is made of the same matter, despite the fact that its form undergoes changes as time passes. However, we need to respect the fact, that the belief that there is an underlying matter is just an assumption. Aristotle assumed that there was matter so that he could say that an object has an identity, and to insist that it continues to be the same object despite changes to it. This was an argument against philosophers like Heraclitus who would say that all is flux, becoming, disputing the idea that there even is any real objects.

As someone who speaks two languages fluently, I wholeheartedly disagree with this. Yes, some concepts do not translate in a single word, if at all. But basic concepts (again, generalized ideas), such as that of "tree", are the same across multiple cultures regardless of the language via which they are addressed (given that the populace is exposed to concrete instantiations of trees in its environment).

The point is, that if I were to define "tree", and you were to define "tree", I'm quite sure that we would not define it in the exact same way. In fact, I'm quite sure that I would define it differently myself, depending on the circumstances. Remember, that to be the same object, by the law of identity requires that it is the very same. Now, we know that with temporal continuity, an object changes as time passes, and the law of non-contradiction states that it cannot have contradictory properties at the same time, but what about the difference between your concept of "tree", and mine, which exist at the same time? You might say that these are not contradictory, but surely the concept of "identity" which I have contradicts the concept of "identity" which some others in this thread have. And even in numbers we see contradictions between natural numbers, rational numbers, real numbers, imaginary numbers. Therefore even the concepts of numbers cannot be objects, because defining any particular number in a way which would encompass all usage of that number, would involve contradiction.

The complexities of language aside, if no such scribble could convey the same (essential) concept between two different people, how would communication of anything be possible?

I get this question asked of me over and over again on this forum, and the answer is very simple. Communication clearly does not require that different people convey the same concept. All sorts of animals communicate without the use of concepts. We develop concepts to facilitate a higher understanding, but these concepts are developed through the use of communication, not vise versa. I believe that this is an important part of the issue, communication came first, then concepts were developed. And this is why it is so hard to establish the temporal continuity of ideas and concepts, they are forms, and forms come into existence and go out of existence.

What is included in the category of 'primary and self-subsistent things'? Do tools and artefacts belong in that category? Do they have 'an identity' according to this criteria?

I believe a self-subsistent thing is a thing which is primary, having nothing which accounts for its existence, as prior to it, like Forms. In Bk.7 Ch.6, it is said that a substance is the same as its essence, but he then proceeds to question "self-subsistent" Forms to see if this would be true for them. Things even Forms, if they are self-subsistent as some Platonists argued, would have to be the same as there essence. Otherwise the essence of good would be different than the good-itself, and the good-itself would not have the essence of good and without the essence of good, it could not be good. So the conclusion is that all things, whether a material thing as substance, or a self-subsistent Form, must be the same as their essence.

The question of the difference between the coming-to-be of artificial things, and the coming-to-be of natural things is addressed in in Ch.7-8. This is where it gets quite complicated. He is at this point carrying on with the problem of trying to account for the existence of accidentals, which is what that discussion of self-subsistent things was related to. So he now comes to matter, and says that in an artificial thing, the matter to be used might be stipulated as part of the formula, a bronze sphere for example. Therefore the form of the artificial thing comes from somewhere other than within the matter, and this is the soul of the artist. He then proceeds in Ch.8 to compare natural things to artificial things, and concludes that the process must be similar, the form which the material thing will have, must come from somewhere other than within the matter. But he claims there is no need for a self-subsistent Form, only that the thing which begets is of a similar type to the thing begotten.
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So does a hammer meet ‘the condition of identity?’
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I don't think that you can call the parts of a thing as the cause of its composition.

Here, you’ve misconstrued what I was saying. I wasn’t saying that a given’s summation of parts *causes* the given’s composition/matter. What I was suggesting is that its summation of parts, or constituents, *is* its composition/matter. This such that “matter” and “composition” can be used interchangeably. Hence the reason why the bronze statue can dent—for one example—rather than shatter or burn, is its composition/matter of bronze (rather than the same statue-form being composed of stone (which can shatter) or wood (which can burn)). Reworded, the bronze statue is dent-able (rather than shatter-able or burnable) due to its composition as the cause of its dent-ability. Again, such that composition and matter are in the addressed Aristotelean context interchangeable.

Aristotle assumed that there was matter so that he could say that an object has an identity, and to insist that it continues to be the same object despite changes to it. This was an argument against philosophers like Heraclitus who would say that all is flux, becoming, disputing the idea that there even is any real objects.

As regards continuity, forms, and matter, I think it’s a complex minefield. However, this is to me an interesting Aristotelian tidbit that might (?) clash with the gist of your affirmations regarding the impermanence of forms vs matter: the teleological unmoved mover is taken to be pure form sans any and all matter. And, as the unmoved mover, it neither comes into being nor goes from being, remaining as permanent as permanence can get, this while being deemed the mover of everything hylomorphic, with the latter taken to be in states of change, i.e. flux.

So, in my quirks of interpreting Aristotle, if we’re looking to affix identity strictly to that which is permanent, unchanging, then this cannot be matter but instead can only be form: specifically, that matter-less/composition-less form which specifies the identity of the unmoved mover as telos.

But again, to me this is a complex field to enquire into. And, for the record, no, I’m not denying that forms (other than that form which specifies Aristotle's teleological prime mover) change over time, including by appearing and disappearing. I simply don’t associate identity to that which is necessarily unchanging.

As regards concepts, it looks like we disagree on a lot of small points that, in short, add up to a large disagreement overall. I won’t nitpick, preferring the let this issue of concepts be for the time being.
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So does a hammer meet ‘the condition of identity?’

Sure, all things have an identity, by the law of identity. The point of the law of identity is that the identity of a thing is within the thing itself, it is not the word "hammer", nor the humanly applied definition of the word, because the thing's true identity needs to include all the accidentals proper to each individual hammer. To us, the accidentals appear to inhere in the matter of the thing which is the principle that we use to account for a thing's indefiniteness.

I'm sorry I made a mess of my reply when you said that I equate essence and identity, with those quotes I produced. I believe there are two distinct senses of "form" in my interpretation of Aristotle. One refers to the universals, which are the ideas by which we understand things, and the other refers to the form of the individual. "Essence" and "form" are very similar in meaning, so "essence" is ambiguous being used commonly in both ways.

The reason why we need to assume a form or essence which is proper to the individual, comes out earlier in the "Metaphysics". It is explained that when a thing comes into being it must necessarily be the thing which it is, or else it would be something other than it is; which is impossible, that a thing is something other than it is. This is the basis for the law of identity. It is impossible that the thing is something else, something other than the thing that it actually is. And, the important point is that a thing is not random matter, it is matter which is structured, 'organized' in a particular way. This means that the form of the particular, individual thing, must be prior in time to the thing's material existence, to ensure that it is the thing that it is, and not something else.

We have a good example of this in the case of living beings. The soul of the individual must be temporally prior to one's material existence, to account for the organization of the material body. This is the sense in which the matter must be included in the formula. This is the passage: "... 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;..."

If we proceed further in his "Metaphysics", understanding his so-called cosmological argument is the real key to making sense of all this. The cosmological argument demonstrates how "form" through its defined nature as actual, must be prior in time to "matter" in its defined nature of potential, in an absolute sense. So, according to what is described previously in the book (above), this Form (I capitalize it to signify its independence from matter) is the form of the individual.

You'll see that this is consistent with the Neo-Platonists who assign individuality to the independent Forms, "One", "the soul". Further, in Aquinas there is described a complete separation between the forms as universals which are dependent on the human mind, and therefore not separate from matter, and the independent or separate immaterial Forms such as God and the angels. Understanding this separation is important to understanding Aquinas. When I first started reading Aquinas he was talking about how the forms, as universals, intelligible objects are not immaterial, being dependent on matter, and I could not understand what he was talking about. I had to go back and reread a lot of Aristotle, specifically Metaphysics Bk.9 which contains the cosmological argument. Then it all made sense.

What I was suggesting is that its summation of parts, or constituents, *is* its composition/matter. This such that “matter” and “composition” can be used interchangeably.

What I did, was argue against what you said here. We cannot equate "matter" (or parts), with "composition". This is because the particular arrangement of the parts is just as important to the composition as is the parts themselves. So I am disagreeing with your use of "composition". I think it is misleading, implying that we can remove the particular arrangement of the parts as inessential to the composition.

Reworded, the bronze statue is dent-able (rather than shatter-able or burnable) due to its composition as the cause of its dent-ability. Again, such that composition and matter are in the addressed Aristotelean context interchangeable.

"Matter" in the Aristotelian context, can always be broken down, given a formula. So if we want to compare wood, and bronze, we proceed to the form of "wood", and the form of "bronze". Then we see that each of these consists of parts, atoms which are arranged as molecules with a molecular structure which is the form of wood, or bronze. Therefore the "shatter-able", "burnable", or "dent-ability", of the substance is still accounted for, by giving what was supposed to be "the matter", a form. In modern physics, they bring the form to deeper and deeper levels, in an attempt to understand the lower levels.

What is inevitable with this process of reduction of the matter, is the appearance of infinite regress. This is because "matter" is inherently indefinite, by its definition, so we must determine its form to understand it. But the conceptual structure which we're locked into is that if there is a form, there must be an underlying matter. So whenever we determine a lower level form, it is necessary that there is matter underlying this form. But the matter is necessarily only intelligible by its form, so we need to determine another level of form, and this appears like it might go on ad infinitum.

This is why we need to turn things around, as Aristotle did with the cosmological argument, and recognize the necessity of assuming immaterial forms which are prior to material existence. This is a way to break the infinite regress. However, it requires a distinctly different definition of "form", hence a dualism of form. The bottom-up form, which is properly an immaterial form, as responsible for the cause of material objects, is the form of an individual, rather than a universal form.

So, in my quirks of interpreting Aristotle, if we’re looking to affix identity strictly to that which is permanent, unchanging, then this cannot be matter but instead can only be form: specifically, that matter-less/composition-less form which specifies the identity of the unmoved mover as telos.

Yes, I believe that this is the right direction to take in understanding Aristotle. The teleological form, associated with intention and final cause is the bottom-up cause. We find this explained in "On the Soul". The living being, as an organized material body, must come into being as organized matter. So the matter of this body is organized to the lowest levels of its existence. This implies that even when this matter comes into existence as "matter", it must already be organized by a form which has prior existence, the soul.

What this principle does, is that it takes the indefiniteness away from matter. Matter is necessarily organized, there is no such thing as "prime matter" according to what the conceptual structure dictates. The cosmological argument demonstrates that the reality of prime matter is impossible. Now, the reason why "matter" is designated as indefinite is because that is the way that it appears to us, human beings who have deprived, or imperfect intellects. In our attempts to understand the parts of objects, there is always something at the bottom which appears unintelligible to us, as indefinite. Aristotle assigns "matter" to this. However, as "indefinite" is just the way that it appears to us, in reality there are immaterial Forms which underlie the matter, making it really organized and structured. This is what validates the law of identity. But with our deficient intellects we do not have the capacity to grasp these immaterial forms, and the true identity of material things.
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So I am disagreeing with your use of "composition". I think it is misleading, implying that we can remove the particular arrangement of the parts as inessential to the composition.

No such implication was intended:

1. The act of putting together; assembly.
2. A mixture or compound; the result of composing. [from 16th c.]
3. The proportion of different parts to make a whole. [from 14th c.]
4. The general makeup of a thing or person. [from 14th c.]

One one hand, these definitions are in accord with Aristotle's definition of matter as ""that out of which" X is made". One the other hand, my current more formal definition of composition is "a given's synergy of parts"

My reason for using "composition" rather than "matter" as a determinant (as in: material cause) - this in what I'm currently working on - is that "matter" nowadays commonly denotes that which constitutes the physical, whereas composition does not. The latter being more in-tune with what Aristotle meant. So, the synergy of ideas which constitutes a paradigm (say evolutionism rather than creationism, or vice versa) is the paradigms composition (its matter in Aristotelian terms). This synergy of parts, in this case of ideas, then sets the limits or bounds of what form the paradigm can take and of what changes it may or may not undergo so as to remain the same paradigm. This synergy of parts is then the paradigm's compositional determinant (the paradigm's material cause). This even though, in today's terminology, neither the paradigm nor the ideas from which it is composed are material - rather, both, to most, are deemed immaterial.

What is inevitable with this process of reduction of the matter, is the appearance of infinite regress.

Here, and in related passages, we seem to agree in full.

The bottom-up form, which is properly an immaterial form, as responsible for the cause of material objects, is the form of an individual, rather than a universal form. [...] The teleological form, associated with intention and final cause is the bottom-up cause.

This part to me is a bit confusing. Are you saying that formal causation is a bottom-up causation? Or that a hylomorphic given's form is the result of material causation, with the latter being bottom-up? Or something other?

As a little bit of background: To me a form, as the term was traditionally intended, can be reexpressed as a whole, as a given's entirety (of being). So, in a maybe oversimplified manner, one can contrast a given whole with the same given whole's synergy of parts. (Yes, each individual part is its own whole ... but this leads into different avenues of investigation, ones you've already touched upon). The given's whole, or form, results in the given's formal cause upon its synergy of parts. Whereas the given's synergy of parts results in the given's material cause upon its form (upon that which makes the given a whole given).

At least when viewed this way, formal causation is to me always top-down, rather than bottom-up, even if it is deemed to be of primary importance relative to any identity: for it is the whole's determination of the synergy of parts from which it as a whole is constituted. (And I acknowledge this is a very complex subject to embark upon; but, notwithstanding, it still strikes me as a synchronic top-down determination). Whereas it is material causation - the synergy of parts' determination of what the whole is - that strikes me as bottom-up causation.

So, here, everything both material and immaterial (modern usage) is hylomorphic, save for the unmoved mover. But the latter is a telos which moves everything teleologically. So its being as form need not have a synergy of parts. Not being itself a hylomorphic being, it holds neither top-down (downward) nor bottom-up (upward) determinations, but instead is solely composed of teleological (what I currently term "pull-ward") determinations ... which affects everything (including all efficient causes) either directly or indirectly.

I likely expressed more than a mouthful. Of course, feel free to critique my interpretations, but I am curious why you express formal causes to be bottom-up rather than top-down IF this is indeed what you here intended.
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I have mentioned this essay before, but you might find it worthwhile - Meaning and the Problem of Universals, Kelly Ross. (The author is a retired academic.)

if we’re looking to affix identity strictly to that which is permanent, unchanging, then this cannot be matter but instead can only be form: specifically, that matter-less/composition-less form which specifies the identity of the unmoved mover as telos.

Minefield, indeed. Buddhism says that there is nothing that constitutes an 'ultimate identity' in this sense whatever. Perhaps we could venture that Kant's antinomies are a corrective to the tendency in Aristotelian metaphysics to try and absolutise 'essence' etc.

I also question the tendency to 'absolutize' the forms. I think they're real on a specific level, viz, that of the 'formal realm' which 'underlies' the phenomenal realm but they can't be pinned down or ultimately defined.

Further, in Aquinas there is described a complete separation between the forms as universals which are dependent on the human mind, and therefore not separate from matter, and the independent or separate immaterial Forms such as God and the angels

I query this analysis. That would make Aquinas a conceptualist - 'Conceptualism is a doctrine in philosophy intermediate between nominalism and realism that says universals exist only within the mind and have no external or substantial reality.' Aquinas was not a conceptualist, but a scholastic realist, whom by definition accepts the reality of forms.

if the proper knowledge of the senses is of accidents, through forms that are individualized, the proper knowledge of intellect is of essences, through forms that are universalized. Intellectual knowledge is analogous to sense knowledge inasmuch as it demands the reception of the form of the thing which is known. But it differs from sense knowledge so far forth as it consists in the apprehension of things, not in their individuality, but in their universality.

...“Abstraction, which is the proper task of active intellect, is essentially a liberating function in which the essence of the sensible object, potentially understandable as it lies beneath its accidents, is liberated from the elements that individualize it and is thus made actually understandable. The product of abstraction is a species of an intelligible order. Now possible intellect is supplied with an adequate stimulus to which it responds by producing a concept.”

From Thomistic Psychology: A Philosophical Analysis of the Nature of Man, by Robert E. Brennan, O.P.; Macmillan Co., 1941.
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One one hand, these definitions are in accord with Aristotle's definition of matter as ""that out of which" X is made

I don't think so. #1 refers to an efficient cause. #2 refers to the complete substance, matter and form. #3 is a form, or formula. And #4, referring to a "general" make up, must also be formal. I really don't see how "composition" can refer to matter. I can see "what a thing is composed of" as being consistent with "matter", but neither the noun or verb form of "composition" seems to be the same.

The latter being more in-tune with what Aristotle meant.

I don't agree, I think "that which constitutes" is closer to what Aristotle meant than "composition".

This part to me is a bit confusing. Are you saying that formal causation is a bottom-up causation? Or that a hylomorphic given's form is the result of material causation, with the latter being bottom-up? Or something other?

What I'm saying is that I believe that final causation, intention, will, is bottom-up. Formal cause, which we apprehend as acting top-down, is distinct from final cause. What I tried to explain in the last post, is that material cause appears unintelligible to us ultimately, as leading to infinite regress. We can assign intelligibility to it, claim that matter is necessarily intelligible, by positing an immaterial Form as the cause of matter. The example Aristotle gives is the soul, which is the first cause of existence of living matter. This type of Form can be apprehended as acting from within the matter, teleologically, as final cause, and is distinct from formal cause. It must be a bottom-up cause.

I query this analysis. That would make Aquinas a conceptualist - 'Conceptualism is a doctrine in philosophy intermediate between nominalism and realism that says universals exist only within the mind and have no external or substantial reality.' Aquinas was not a conceptualist, but a scholastic realist, whom by definition accepts the reality of forms.

I was of that mind as well, until I actually read Aquinas. In reality, he argues that the forms which we know, as universals, are the product of abstraction, whereby the intellect uses the body, through the means of sensation, to produce the "intelligible species". Notice the use of "species" here, as he is very thorough to follow Aristotelian terminology, instead of the Platonic "intelligible objects". Because the human intellect is dependent on the body for its knowledge, it cannot grasp separate, or immaterial forms.

If you go to the Summa Theologica, the section on "Man", you'll find a part on the Knowledge of Bodies, At Q.84. Art.4, you'll see "Whether the Intelligible Species are Derived by the Soul from Certain Separate Forms". Following "I answer that..." you'll see a significant writing about Platonic Forms, followed by a discussion of Aristotle, Avicenna, and the active intellect. At the end: "We must therefore conclude that the intelligible species, by which our soul understands, are not derived from separate forms". This conclusion is brought about by the fact that the intellect is united to the body as a power of the soul. He explains this further in the following articles of Q.84. In Q.85, we have "Of the Mode and Order of Understanding". Here he explains abstraction, referring to the role of sensation, "phantasms", and the reality that the soul is united to the body. In Q.85 Art. 1 he explains that Plato did not properly respect the consequences or conclusions derived from the intellect being united to the body. Further along, you'll find Q.88, "How the Human Soul knows What is Above Itself". Here he looks into the possibility that we could have knowledge of separate substances, immaterial forms. In Art. 1, you'll see a number of detailed arguments and the conclusion: "Hence in the present state of life we cannot understand separate immaterial substances in themselves, either by the passive or the active intellect."

This is all a product of the separation which Aristotle established between the forms which the human intellect understands, (abstractions, universals, concepts), and the separate, immaterial Forms, which are proper to the divine realm. For the very same reason that we cannot properly know God, we cannot know the other immaterial forms, the angelic forms.

Intellectual knowledge is analogous to sense knowledge inasmuch as it demands the reception of the form of the thing which is known.
...
“Abstraction, which is the proper task of active intellect, is essentially a liberating function in which the essence of the sensible object, potentially understandable as it lies beneath its accidents, is liberated from the elements that individualize it and is thus made actually understandable. .”

I quoted the two sentences from the passage above, which are misleading. The active intellect does not receive the form from the sensible object, reception is a passivity. The active intellect actually creates a representation of the sensible object. This is the importance of phantasms and imagination in the intellect: Q.84 Art.7."And, therefore, for the intellect to understand actually its proper object, it must of necessity turn to the phantasms in order to perceive the universal nature existing in the individual."

So it is incorrect to say that the intellect receives a form from the sensible object. The form of the sensible object is a particular, and is united to that object, just like the soul is united to the man. But the intellect produces a universal form, through the use of phantasms, by means of which it understands the individual. That's why there is a separation between these two types of "form".
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I have mentioned this essay before, but you might find it worthwhile - Meaning and the Problem of Universals, Kelly Ross. (The author is a retired academic.)

Thanks. Briefly skimmed some of it for now. Will look further in it in a few days. Looks to be up my alley.

Buddhism says that there is nothing that constitutes an 'ultimate identity' in this sense whatever.

I have a great deal of respect for Buddhism in many regards, this being one of them.

That said, the Buddhist notion of Nirvana, though different in many ways to that of Aristotle's unmoved mover, to me does share a number of similarities. The utterly, literally, selfless state of awareness (hence, a state of awareness devoid of all duality) which is Nirvana - more correctly in this context, "nirvana without residue" - seems to be interpretable, to me at least, as the ultimate identity of all sapient beings (and at least some schools of Buddhism seem to hold of all sentient beings; this being an inference gathered from those Buddhists that take an oath to enlighten all sentient beings). And - again imo - it is from this vantage of what our ultimate identity is that the no-self principle of Buddhism can be derived.

Don't know if its just me, but I take it that anything which can be identified holds an identity, a discernible form. Though we're accustomed to thinking of all identities as being finite forms that are constituted of parts, the state of being which is Nirvana certainly is utterly devoid of parts, is stated to be infinite in the sense of being devoid of limits or boundaries, and is identifiable. Hence, Nirvana is a discernible form of being. So while this will probably be a bit irksome, I can interpret Nirvana to of itself be the ultimate identity. This in parallel to what was previously discussed about the Aristotelian unmoved mover as the ultimate identity ... or of what can be said in relation to the Neo-platonic notion of "the One".

(The perennial philosophy parts of me like to believe that all three are different interpretations of the same metaphysical given.)

Curious to hear your thoughts on the just given musings.

I also question the tendency to 'absolutize' the forms. I think they're real on a specific level, viz, that of the 'formal realm' which 'underlies' the phenomenal realm but they can't be pinned down or ultimately defined.

Right. I tend to agree in this for all forms save for "the One".

I don't agree, I think "that which constitutes" is closer to what Aristotle meant than "composition".

In truth I can't find much of any meaningful different between a constituent and a component of a given, so I'm perfectly fine with rephrasing material causes as a "constitutional determination" rather than a "compositional determination". I was initially hesitant in so doing due to "constitution" being so readily interpretable in the senses of government and law. But I suppose the term's contextual use would suffice to clarify the intended meaning. Thanks for that.

What I'm saying is that I believe that final causation, intention, will, is bottom-up. Formal cause, which we apprehend as acting top-down, is distinct from final cause.

While I find reasons to disagree, thanks for the explanation.

To illustrate what I mean, the goal, or objective, one has in mind while engaged in making a decision will be the decision's final cause: it determines what will be chosen in so far as what will be chosen will be so chosen for the reason of best obtaining the objective. Bottom-up addresses synchronic occurrences, yet the goal pulls the momentary act of choice making toward a potential future that has yet to be objectified. So, to me, there is a type of temporality involved with a telos. This to me stands in contrast to both bottom-up and top-down determinants, both or which strictly occur in the specified moment of time without any temporal extensions into the future.
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So, to me, there is a type of temporality involved with a telos. This to me stands in contrast to both bottom-up and top-down determinants, both or which strictly occur in the specified moment of time without any temporal extensions into the future.

I don't think we can correctly say that anything occurs in a moment of time without any temporal extension. All occurrences require duration. Therefore I do not think we can exclude "bottom-up" and "top-down" from a temporal analysis.

The problem I see is in the way that we commonly represent space. In simple conception we see space as representable with a 3-d coordinate system. This makes the thing represented, (i.e. the reified space, and there necessarily is a reified space to allow for the reality of motion), the same everywhere. So all of space is fundamentally exactly the same everywhere, under this conception, no matter how big or how small, and it might be infinitely big or infinitely small. This is a problem because it provides us with no principles to distinguish the real spatial difference between inside and outside, which is implied by the concept of spatial expansion. And, such a principle is required if we want to validate the real existence of objects. An object is defined by its boundaries, and this allows us to distinguish properties of the object itself from what is external to the object, in order to maintain consistency with the three basic laws of logic.

Because we have no such principles, we cannot properly differentiate between a force which acts from the inside, and a force which acts from the outside of an object. So I see the top-down/bottom-up distinction, when it's applied to causation, as based in the global/local distinction which is applied in physics. The problem though is that there are no real principles to distinguish inside from outside, so these distinctions are somewhat arbitrary, and gauge theory for example is just a mess. It's as if the whole of gauge theory is an attempt to deal with anomalies brought about by failing spatial conceptions.
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I don't think we can correctly say that anything occurs in a moment of time without any temporal extension. All occurrences require duration. Therefore I do not think we can exclude "bottom-up" and "top-down" from a temporal analysis.

In one sense I agree, but in this sense all four of Aristotle's causes co-occur (an Aristotelian variant of codependent arising). Which is not the case when each cause-type is addressed individually.

To address this via example, if a wooden table’s burnability holds as its material cause the wood out of which the table is constituted, what duration occurs between a) the material cause of wood and b) the table’s intrinsic potential to burn?

So far, to me (a) and (b) seem to be necessarily simultaneous, with no duration in-between, while standing in a bottom-up relation.

Nice concepts on the issue of space, btw.
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To address this via example, if a wooden table’s burnability holds as its material cause the wood out of which the table is constituted, what duration occurs between a) the material cause of wood and b) the table’s intrinsic potential to burn?

So far, to me (a) and (b) seem to be necessarily simultaneous, with no duration in-between, while standing in a bottom-up relation.

Matter being potential in Aristotle's conceptual structure, I think that (a) and (b) are just different ways of saying the same thing.

There is an issue with matter, which I sort of explained earlier. When we start saying anything about matter, we are always referring to a specific form of matter. That is because matter is defined as an indefinite aspect of things. We can try to get away from this by making the most general statements possible about matter, but since matter is indefinite such attempts would still end up being in a sense statements about forms. For example, "an object is composed of matter", and "matter has inertia", each in its own way states something formal, despite being attempt to say something strictly about matter. Even when I say "matter is the indefinite aspect of things", I say something formal, by referring to a privation of form.
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To do a Galileo like thing: but still the table can burn due to being constituted of wood rather than marble.

You are of course correct in respect to the notion of primary matter. Yet in practice we, for example, have to build houses whose bricks are constituted of solid matter rather than, say, some sponge-like material. I was addressing this more practical view of a hylomorphic given's constituency when addressing bottom-up determinacy.
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The perennial philosophy parts of me like to believe that all three are different interpretations of the same metaphysical given.

I'm with you on that. Not that they're all saying the same, or heading to the same destination, but that they're agreeing, and disagreeing, about the ultimate state.

The utterly, literally, selfless state of awareness (hence, a state of awareness devoid of all duality) which is Nirvana - more correctly in this context, "nirvana without residue" - seems to be interpretable, to me at least, as the ultimate identity of all sapient beings

Obviously a major digression from the OP (for which I'm responsible) - but, yes, this is the East Asian teaching of the 'Buddha Nature', tathagathagarbha. A very beautiful philosophy.
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I'm glad to hear that there isn't any significant disagreement (if any) in relation to Nirvana as "ultimate identity".

Finally got around to reading Kelly Ross's manuscript which you linked to. I’m envious of the clarity and simplicity with which complex concepts are expressed. I don’t fully agree with some of the concluding inferences. But, for the sake of this thread, I’ll skip all of this. (And for what its worth, despite his many shortcomings as a philosopher (what philosophy can ever be “perfect”?), I continue to greatly admire Hume for many of his insights. :razz: But anyways …)

For anyone interested in furthering the issues of identity already discussed in this thread, taken from about a third of the way in in Ross's manuscript:

Concepts, or predicates, are always universals, which means that no individual can be defined, as an individual, by concepts. "Socrates," as the name of an individual, although bringing to mind many properties, is not a property; and no matter how many properties we specify, "snub-nosed," "ugly," "clever," "condemned," etc., they conceivably could apply to some other individual. From that we have a principle, still echoed by Kant, that "[primary] substance is that which is always subject, never predicate." On the other hand, a theory that eliminates the equivalent of Aristotelian "matter," like that of Leibniz, must require that individuals as such imply a unique, perhaps infinite, number of properties. Leibniz's principle of the "identity of indiscernibles" thus postulates that individuals which cannot be distinguished from each other, i.e. have all the same discernible properties, must be the same individual.

If my interpretation of it is valid, to me Leibniz's principle of "identity of indiscernibles" can equally apply to substance theory and to bundle theory - the latter standing in contrast to the former, with the former being typified by the first portion of the quoted passage.

If so, curious to hear what would be wrong with the following: an individual object's identity of itself consists of a gestalt form that results from the synergy between all relevant properties as parts. In this manner, hybridizing substance theory with bundle theory in relation to identity. (I'm toying around with this notion at present). Hence, there here would be no inherent, independent substance (primary or secondary): all substances being emergent byproducts of properties. On the other hand, the gestalt is that to which all its properties are predicates of.

An individual apple's identity would then be the gestalt that results from all of the individual apple's properties, including those of its spatiotemporal placement (which is a predicate of the apple).

An individual number's identity (say, the number 2) would then likewise be the gestalt that results from all of its properties: this gets far more tricky due to the degree of abstraction, but maybe including those of duality, its placement within the appropriate context of other numbers (e.g., greater than 1 but lesser then 3), and so forth.

Edit: I'm aware that the Wikipedia article on bundle theory makes a skimpy mention of "bundle theory of substance". More musings on this issue can be found here https://plato.stanford.edu/entries/substance/#BundTheoTheiProb. All the same, if anyone is interested in debating the notion of identity as a gestalt form emerging from a bundle of properties as parts, I'm curious to see in which ways this would be critiqued.
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An individual number's identity (say, the number 2) would then likewise be the gestalt that results from all of its properties: this gets far more tricky due to the degree of abstraction, but maybe including those of duality, its placement within the appropriate context of other numbers (e.g., greater than 1 but lesser then 3), and so forth.

Edit: I'm aware that the Wikipedia article on bundle theory makes a skimpy mention of "bundle theory of substance". More musings on this issue can be found here https://plato.stanford.edu/entries/substance/#BundTheoTheiProb . All the same, if anyone is interested in debating the notion of identity as a gestalt form emerging from a bundle of properties as parts, I'm curious to see in which ways this would be critiqued.

The problem here is that the number 2 is a property itself. We take a group of two and look at it as a single thing, and say that this thing has the property of consisting of two. There's no fundamental problem in saying that the group is not a true object, it's arbitrary, and arguing therefore that the only true object is the property which is assigned. However, if we have no way to distinguish a true object, then the number 1 is invalidated as a false property because it cannot be truthfully assigned, and so the falsity of 2, as a property follows, being dependent on the truth of 1.

A very similar problem will appear with bundle theory. If an object is a bundle of properties, then there is no real principle whereby we might judge if this property is part of a specified bundle, or another bundle. Then we could not claim any objects as real objects, except perhaps a property itself. But this will prove to be completely incoherent because contradictory properties, as objects themselves, will be all over the place, and if the contradictory properties are not properties of the same thing, then we can't reject them by way of the law of non-contradiction. So we'd have all sorts of contradictory properties with no way to reject contradictions.
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We take a group of two and look at it as a single thing, and say that this thing has the property of consisting of two. There's no fundamental problem in saying that the group is not a true object, it's arbitrary, and arguing therefore that the only true object is the property which is assigned.

Given what we've been through in terms of prime matter being pure potential and all givens being identified by their forms, why would the abstract form of "2" be deemed arbitrary rather than a "true (abstract) object"? Seems to me that basic numbers are not arbitrary, despite their very abstract nature; else, for example, 2 + 2 could equal 5 in certain cases.
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Given what we've been through in terms of prime matter being pure potential and all givens being identified by their forms, why would the abstract form of "2" be deemed arbitrary rather than a "true (abstract) object"? Seems to me that basic numbers are not arbitrary, despite their very abstract nature; else, for example, 2 + 2 could equal 5 in certain cases.

Any concept which cannot be substantiated (grounded in substance) is an arbitrary concept. Unless we have a principle as to what constitutes a whole, an entity, or an object, all concepts with numbers would be arbitrary. If there is nothing to distinguish one object from two objects, then 2+2 might just as well equal 5 as 4, or any other random number, because it really doesn't make any difference, as number itself would be fundamentally meaningless.
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