## There is only one mathematical object

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Any concept which cannot be substantiated (grounded in substance) is an arbitrary concept.

You're using substance to denote something different than what I'm denoting by it: for you, it seems, substance is only that which is empirically cognized via the physiological senses. For me it is any whole that can be cognized - perceptually or otherwise, such as via the understanding - which is constituted of parts, any hylomorphic given. In this latter sense, then, every concept is itself a substance. This as per Aristotle's philosophy, wherein concepts are secondary substances. Even so:

Unless we have a principle as to what constitutes a whole, an entity, or an object, all concepts with numbers would be arbitrary.

I don't yet understand why you presume that basic numbers are not substantiated via that which is empirically cognized? We perceive quantities. And we express these perceptions of quantity via numbers. Thereby making basic numbers (e.g., 2), as well as their basic relations (e.g., 2 + 2 = 4), non-arbitrary.
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Thereby making basic numbers (e.g., 2), as well as their basic relations (e.g., 2 + 2 = 4), non-arbitrary.

I haven't followed the posts in this thread for a while but just so you know, @Meta doesn't think 2 + 2 and 4 refer to the same mathematical object. You may be assuming too much if you think he agrees with the rest of the world about this.
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Yea, I get that. For right now I think the argument centers on whether or not there are any mathematical objects to begin with.
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Well, that's certainly a serious undertaking. Much hinges upon your findings.
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Dude! I'm trying to figure out MU's position a little better. Thanks for the ignoble backing, all the same.
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they do not have independent existence, i.e. they are not separate entities in any sense

Mathematics is not the entire reality but only an minor aspect of it. Trying to define the concrete reality as made up of numbers is as dumb as defining it as colors. Thanks Kant for that.

Any "mathematical model" of anything - and especially regarding human society - is just a metonym, not a substantive description. Those who do not realize this, even if they have a Nobel Prize, are illiterate.
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I'm trying to figure out MU's position a little better.

Good luck.
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:lol:
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You're using substance to denote something different than what I'm denoting by it: for you, it seems, substance is only that which is empirically cognized via the physiological senses. For me it is any whole that can be cognized - perceptually or otherwise, such as via the understanding - which is constituted of parts, any hylomorphic given. In this latter sense, then, every concept is itself a substance. This as per Aristotle's philosophy, wherein concepts are secondary substances. Even so:

I think "substance" has its meaning relative to logic, and it refers to whatever grounds any particular system of logic, as what underlies it to support it. So it is quite clear to me, that "any whole that can be cognized" is not an acceptable definition of substance, because it allows that fictitious objects may be substance, or have substantial existence. And this is clearly inconsistent with any logically rigorous definition of "substance", as that which provides truth to the logic.

In Aristotle's logic, secondary substance is what grounds the logic as the most specific within the system. So if we claim a concept, the genus, "animal", that genus might be grounded in the more specific, concepts "man", and "horse" for example. That is secondary substance. The primary substance is the individual, the particular horse or man, while the secondary substance is the type that the individual is. If, for example, there was proposed a species, like "man" and no particular example of that species could be found, the proposal would be unsubstantiated (in the sense of primary substance). Likewise, if someone proposed a genus, "animal", and no species could be shown to be a member of that genus, the genus would be unsubstantiated (in the sense of secondary substance).

Aristotle's conception of "secondary substance" does not allow that "every concept is itself a substance". Only a more specific concept, which grounds a more general concept is a secondary substance. But the secondary substance itself still needs to be grounded in particular individuals, which is primary substance.

I don't yet understand why you presume that basic numbers are not substantiated via that which is empirically cognized? We perceive quantities. And we express these perceptions of quantity via numbers. Thereby making basic numbers (e.g., 2), as well as their basic relations (e.g., 2 + 2 = 4), non-arbitrary.

It is only when you disavow the object to which "2" is attributed as a property, that numbers are necessarily arbitrary. That's what you've been talking about isn't it, claiming that there is not need for the physical object which substantiates the number? Didn't you mention bundle theory? Physical groups of two things, is what substantiates the non-arbitrariness of 2, just like physical instances of animals substantiates the genus "animal". The physical group, which consists of two, is the physical object, the particular, the substance in this instance, and "2" is a property of that physical object.

If we remove that object, that physical group of two, as the primary substance, we might substantiate the concept "2", in secondary substance, with the concept of "1", and the concept of "+", or something like that, like we might substantiate the concept "animal" in the species of "horse" (secondary substance). However, we still need to substantiate "1" in primary substance, like we need to substantiate "horse" in individual horses, or else any designations of one, or unity, or whole, are arbitrary. And, since "2" as a concept has been grounded in the secondary substance of "1" as a concept, and "1" is arbitrary if it's not grounded in primary substance, then so is "2"
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Mathematics is not the entire reality but only an minor aspect of it. Trying to define the concrete reality as made up of numbers is as dumb as defining it as colors. Thanks Kant for that.

Any "mathematical model" of anything - and especially regarding human society - is just a metonym, not a substantive description. Those who do not realize this, even if they have a Nobel Prize, are illiterate.

That's the problem with Platonic Realism, in general, which is more appropriately called Pythagorean Idealism. Unless we provide the required separation between the model, and that which is modeled, then we are misguided into the rather silly and naive notion that the universe is constituted of mathematical objects.

We cannot provide for the separation simply by referring to the symbols, as if the symbols themselves are the model. In reality, we have the two levels of separation. We have the reality, we have the mathematical model which represents reality, and we have the symbols which represent the model. So, we have two levels of interpretation to work through, 1) to understand the model based on an interpretation of the symbols, and 2) to understand how well the model represents reality. We cannot deny the importance of either one, nor can one be reduced to the other.
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I think "substance" has its meaning relative to logic, and it refers to whatever grounds any particular system of logic, as what underlies it to support it. So it is quite clear to me, that "any whole that can be cognized" is not an acceptable definition of substance, because it allows that fictitious objects may be substance, or have substantial existence. And this is clearly inconsistent with any logically rigorous definition of "substance", as that which provides truth to the logic.

I see your point here. A unicorn as concept is not substantiated by primary substances (which I still maintain can only be empirically known). The claim that “unicorns are objectively real” thereby being unsubstantiated. Yet the claim that “the concept of unicorns occurs within western thought” would be substantiated. Granted. Yet for me the concept of unicorns is itself hylomorphic and as such has an identity as a whole given, as a form, which is itself composed of parts, i.e. has a constituency.

It is only when you disavow the object to which "2" is attributed as a property, that numbers are necessarily arbitrary. That's what you've been talking about isn't it, claiming that there is not need for the physical object which substantiates the number? Didn't you mention bundle theory? Physical groups of two things, is what substantiates the non-arbitrariness of 2, just like physical instances of animals substantiates the genus "animal". The physical group, which consists of two, is the physical object, the particular, the substance in this instance, and "2" is a property of that physical object.

In relation to both quotes:

For what it’s worth, I personally don’t take the laws of thought, the law of identity included, to be grounded in anything physical. I instead interpret these to be grounded in metaphysical aspects of reality that then, via awareness, govern how we interpret that which is physical. This in a Kantian-like manner. It’s a can of worms - the details of which I’d rather skip - but, for instance, the absolute unity which can be conveyed by the numeral “1” cannot be found in physical givens: for any one physical given is itself less than perfectly integral—being, instead, in constant flux, change, regarding its constituency, with smaller components always coming in and out, with these leading all the way down into zero point energy. So I take it that the integrity, wholeness, of physical givens is only relative to their context, rather than absolute, and that a perfect wholeness, or unit, is what we experientially project onto the world perceptually. In short, to me, the law of identity isn’t substantiated by physical reality; instead, it of itself governs, and in this sense substantiates, that which we deem to be integral wholes within physical reality.

I mention this because, at the end of the day, our different takes on the law of identity - and maybe on laws of thought in general - seems to play a crucial role in why we disagree about the nature of mathematical objects.

I think I get where you’re coming from, however. More or less, the position that all concepts need to be substantiated in empirically known to be real particulars in order for the concepts to be non-arbitrary, and thereby true to reality. If I’m indeed interpreting you correctly, when applied to most contexts, I would be in agreement with this. The only main, but subtle, disagreement would be that the empirical itself is, to me, governed by metaphysical properties (these including what is formalized as the law of identity, in addition to other Kantian categories such as those of space and causation): thereby making the empirically known reality of the physical itself, in one sense, substantiated by that which is purely metaphysical.
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I instead interpret these to be grounded in metaphysical aspects of reality that then, via awareness, govern how we interpret that which is physical.

Noticed this yesterday. I think this is the book I have been looking for in regard to philosophy of arithmetic.

In his first book, Philosophy of Arithmetic, Edmund Husserl provides a carefully worked out account of number as a categorial or formal feature of the objective world, and of arithmetic as a symbolic technique for mastering the infinite field of numbers for knowledge. It is a realist account of numbers and number relations that interweaves them into the basic structure of the universe and into our knowledge of reality. It provides an answer to the question of how arithmetic applies to reality, and gives an account of how, in general, formalized systems of symbols work in providing access to the world.

all concepts need to be substantiated in empirically known to be real particulars in order for the concepts to be non-arbitrary, and thereby true to reality.

That is basically modern realism. It calibrates all philosophy against the a priori presumption of the reality of the empirical domain; whereas what you're advocating is much nearer Platonic or scholastic realism or Kantian transcendental idealism.
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For what it’s worth, I personally don’t take the laws of thought, the law of identity included, to be grounded in anything physical. I instead interpret these to be grounded in metaphysical aspects of reality that then, via awareness, govern how we interpret that which is physical.

OK, this is a good starting point. What has the capacity to govern how we interpret "that which is physical"? Suppose we could interpret the physical in absolutely any possible way. Then our interpretations would be arbitrary, or random. But we want to say that there are real constraints on the way that we may interpret the physical, so that our interpretations are truly consistent with the physical reality. So we proceed to make some metaphysical or ontological conclusions about physical reality. These are what we use to govern how we interpret the physical. Don't the three fundamental laws of logic qualify here, as fundamental conclusions concerning "that which is physical"?

for instance, the absolute unity which can be conveyed by the numeral “1” cannot be found in physical givens

But "1" does not signify any absolute unity. It is divisible, and infinitely so, by the accounts of many. So how could it signify an absolute unity?

In short, to me, the law of identity isn’t substantiated by physical reality; instead, it of itself governs, and in this sense substantiates, that which we deem to be integral wholes within physical reality.

I can't understand what you're trying to say here. If I said that the law of identity is substantiated by physical reality, I would mean that it is made true by the conditions present in the physical reality. So, if you say that the law of identity governs us as to what we can deem "a whole", aren't we really both saying a very similar thing, in slightly different ways? You are saying that the law of identity governs what we can say about physical reality, and I am saying that the reason why it governs what we can say about physical reality is that it already says something true about physical reality. The only difference is that you are not moving along to see the reason why the law of identity has the capacity to govern what we say about physical reality. It gains that capacity to govern, by saying something true about physical reality.

The only main, but subtle, disagreement would be that the empirical itself is, to me, governed by metaphysical properties (these including what is formalized as the law of identity, in addition to other Kantian categories such as those of space and causation): thereby making the empirically known reality of the physical itself, in one sense, substantiated by that which is purely metaphysical.

I don't see how you can say this. The "empirical" is fundamentally sense experience. Therefore it is a very base level of knowledge. How could it be "governed by metaphysical properties" which is a principled, and therefore higher level of knowledge? The most basic must always govern the higher, as the most basic has a higher degree of certainty. The lower substantiates the higher, and the empirical is the lowest. So the metaphysical cannot substantiate the empirical, it must be vise versa.
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That is basically modern realism. [...]

Agreed.

Don't the three fundamental laws of logic qualify here, as fundamental conclusions concerning "that which is physical"?

For me they apply to all forms, including fictional ones, and not only to that which is physical. That Harry Potter is not a unicorn is true - addresses a reality that stands in its own context of fictional concepts - this via the laws of thought, including the law of identity.

But "1" does not signify any absolute unity. It is divisible, and infinitely so, by the accounts of many. So how could it signify an absolute unity?

"Oneness" can be readily defined as the state of being undivided, of being a whole. As to 1's infinite divisibility, remember that I take the concept of one to be a hylomorphic whole, a form endowed with constituents. But one constituent does not of itself equate to the given whole. A whole given is taken to be undivided as form, hence - for me at least - can be represented by the number 1. As one example, one horse can only be represented by the number "1", and not by any division. Yes, a horse can be divided into parts ad nauseam, all the way down into zero point energy. But its multiple parts are not the horse as a whole, which is in a state of being undivided. As a more abstract example, one grouping of two or more givens is, as a grouping, itself one whole. As one example, "animal" can be conceived of as a grouping of givens, yet the concept of "animal" is itself one whole - distinct, for instance, from the concept of "plant".

As to the adjective "absolute" we neither innately perceive nor contemplate "a horse", for example, to be a relative whole - a whole that is only so due to its relativity to some other given(s). We innately identify it as a complete, unmitigated, whole.

The only difference is that you are not moving along to see the reason why the law of identity has the capacity to govern what we say about physical reality. It gains that capacity to govern, by saying something true about physical reality.

Again, the law of identity pertains to all conceivable givens, and not just those of physical reality. One three-headed dragon - say one that a person saw in an REM dream - cannot at the same time and in the same respect be both green and not-green. This, to my mind, is so because it would then break with the law of identity. At any rate, a three-headed dragon holds an identity despite it not being a physical given.

I don't see how you can say this. The "empirical" is fundamentally sense experience. Therefore it is a very base level of knowledge. How could it be "governed by metaphysical properties" which is a principled, and therefore higher level of knowledge? The most basic must always govern the higher, as the most basic has a higher degree of certainty. The lower substantiates the higher, and the empirical is the lowest.

Two points:

The empirical is just one aspect of awareness, not the only, nor, imv, the most important. Take the sense of understanding. Without an understanding of that which is perceived via the physiological senses, that which is perceived would be meaningless. We also experientially know of things such as being ourselves happy or sad, and some such states of personal being of which we are aware are in no way obtained via the physiological senses. Hence, I maintain that awareness, and not that which is empirical, is fundamental to knowledge.

Secondly, knowledge of metaphysical realities has nothing to do with whether or not these metaphysical properties occur. Same as with physical reality. Take a preadolescent child or a lesser animal as example. Their awareness operates via the law of identity without them having any knowledge of the law of identity. Or else take adult humans prior to Aristotle's formulation of the principle. They too where governed by the law of identity thought they had no propositional knowledge of it.

That said, to me these metaphysical realities are intrinsic aspects of awareness - again, irrespective of whether the awareness addressed has propositional knowledge of them. We do not, and cannot, create them. We can only discover them. As such, we do not govern metaphysical realities, this just as we don't govern physical realities. We, as aware beings, are predetermined by the former. And, though in different ways, we are likewise determined - bounded/limited - by the latter.
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How could it be "governed by metaphysical properties" which is a principled, and therefore higher level of knowledge? The most basic must always govern the higher, as the most basic has a higher degree of certainty. The lower substantiates the higher, and the empirical is the lowest. So the metaphysical cannot substantiate the empirical, it must be vise versa.

Giving the game away, MU. No Platonist - or Aristotelian - worth his/her salt ink would say such a thing.

It is the universal view of ancient philosophy that the 'empirical realm' which is taken by moderns as the sine qua non of the real, is in fact a treacherous illusion, which the hoi polloi do not see as their minds are contaminated by worldly passions, which blind them to the higher truths. The real can only be grasped by reason, and its truths are invariant and never subject to decay. Whereas everything in the sensory domain is subject to constant change and degradation through the ravages of time.
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For me they apply to all forms, including fictional ones, and not only to that which is physical. That Harry Potter is not a unicorn is true - addresses a reality that stands in its own context of fictional concepts - this via the laws of thought, including the law of identity.

We're right back to the initial problem. If we apply the law of identity to all forms, we see that universal forms cannot have an identity. And by that law, "identity" is something that "things" have, so we can use the law to exclude universal forms from the category of "things". This is the intent of that law, to distinguish two distinct types of form, and this is why Aristotle is properly called dualist. There is a categorical separation between primary substance and secondary substance. In your example, the "reality" you refer to is that of secondary substance. Not being "a thing", secondary substance has no identity.

"Oneness" can be readily defined as the state of being undivided, of being a whole. As to 1's infinite divisibility, remember that I take the concept of one to be a hylomorphic whole, a form endowed with constituents. But one constituent does not of itself equate to the given whole. A whole given is taken to be undivided as form, hence - for me at least - can be represented by the number 1. As one example, one horse can only be represented by the number "1", and not by any division. Yes, a horse can be divided into parts ad nauseam, all the way down into zero point energy. But its multiple parts are not the horse as a whole, which is in a state of being undivided. As a more abstract example, one grouping of two or more givens is, as a grouping, itself one whole. As one example, "animal" can be conceived of as a grouping of givens, yet the concept of "animal" is itself one whole - distinct, for instance, from the concept of "plant".

The issue here, is whether the numeral 1 can represent a unified whole, which is not a physical object. In your example, I assume one horse is meant to be a physical object. As I explained when we refer to one horse, we are stating something about it, that it is one, so "one" is used as a property of that thing. Now you need to justify your assertion that the symbol "1" refers to a unified whole, independently of the object which is said to be one. My argument is that since there are numerous number systems, natural numbers, rational numbers, real numbers, imaginary numbers, and so forth, there are numerous different conceptions of "one", and no single mathematical system unifies these into one concept. Therefore one as a concept, is not a unified whole.

Again, the law of identity pertains to all conceivable givens, and not just those of physical reality.

You don't seem to be grasping the fact, that when we apply the law of identity to forms which are other than material particulars, we are forced by the precept of that law, to exclude certain forms from the category of having an identity. This is because we can use language to refer things without any identity. You don't seem to be grasping the intent of the law, which is to prevent the situation where we assume that just because we can talk about it, it is a thing with an identity. You completely misinterpret the law if you claim that a fictitious thing has an identity, because the law of identity puts the identity of a thing into the thing itself, rather than what we say about the thing. The fictitious thing has no existence independent from what we say about it, therefore it cannot have an identity.

Sure, you can say that "the law of identity pertains to all conceivable givens", but unless you abide by that law, and acknowledge that some conceivable givens do not have an identity, then you step outside that law and you enter into hypocrisy.

One three-headed dragon - say one that a person saw in an REM dream - cannot at the same time and in the same respect be both green and not-green. This, to my mind, is so because it would then break with the law of identity. At any rate, a three-headed dragon holds an identity despite it not being a physical given.

This is not a breaking of the law of identity, it is an issue with the law of non-contradiction. And, if we do not limit what we can logically predicate of a subject, to what is actually possible in the physical world, there is no need for a law of non-contradiction at all. If relevance to the physical world is completely unnecessary, why not allow all sorts of contradictions?

Hence, I maintain that awareness, and not that which is empirical, is fundamental to knowledge.

"Awareness" implies being aware of something. I believe that being aware of the external is prior to being aware of the internal, and being aware of the external requires some form of sense. Of course this is a difficult issue to discuss logically because the terms are naturally slanted in my direction. To "be aware of" implies being informed, and this seems to refer to information from the external. But if we allow that information may come to us from an internal source, then ultimately the empirical is not fundamental. This is why Aristotle defined "the soul", as the first principle of actuality, or first form, of the living body. But if the soul is the first principle of actuality, and it is also the inner most thing, as well as the thing which is "aware", then there is nothing more inner that it could be aware of, and primary awareness is necessarily of the external.

Secondly, knowledge of metaphysical realities has nothing to do with whether or not these metaphysical properties occur. Same as with physical reality. Take a preadolescent child or a lesser animal as example. Their awareness operates via the law of identity without them having any knowledge of the law of identity. Or else take adult humans prior to Aristotle's formulation of the principle. They too where governed by the law of identity thought they had no propositional knowledge of it.

I don't agree with this at all. I think it is incoherent, so perhaps I misunderstand. First, how could one have knowledge of something which is independent of whether that something occurs? If the something does not occur, yet someone is claimed to have knowledge of it, this is not knowledge at all. It is misunderstanding masquerading as understanding. Second, the law of identity is extremely difficult even for human beings to understand (as evidenced by this thread), it is set up as a defence against sophism. So I don't see how children or lesser animals could be applying the law of identity as a defence against sophism. I believe you continue to misrepresent "the law of identity".

That said, to me these metaphysical realities are intrinsic aspects of awareness - again, irrespective of whether the awareness addressed has propositional knowledge of them. We do not, and cannot, create them. We can only discover them. As such, we do not govern metaphysical realities, this just as we don't govern physical realities. We, as aware beings, are predetermined by the former. And, though in different ways, we are likewise determined - bounded/limited - by the latter.

This is the pivotal point of how Aristotle applies the cosmological argument against Pythagorean Idealism (and some forms of Platonism). He analyzes what is involved in "discovering" such principles. Would you agree with Aristotle, that when the geometer produces geometrical constructs, and discovers geometrical principles, this is an act which is properly described as the mind actualizing the principles. The principles exist in potential, prior to being actualized by the mind.

Giving the game away, MU. No Platonist - or Aristotelian - worth his/her salt ink would say such a thing.

You've already demonstrated how you misinterpret Aristotle. This is quite understandable because he has a massive volume of material and some is quite difficult. I was in the same position until I read Aquinas, and found that his interpretation of Aristotle was inconsistent with mine. Then I had to go back and read much of Aristotle (Metaphysics, On the Soul) all over again, some of which I had already read two or three times, to see what I was missing.

Have you looked at any of that material which I referred you to in Aquinas, how the human intellect is united with the body, and how this union with the body effects the way that the intellect understands? These principles are taken directly for Aristotle's "On the Soul". The intellect is a power of the soul, just like the other powers of the soul such as self-subsistence, self-movement, and sensation. As such, it operates through the means of the bodily organs. Therefore there is necessarily a privation in the knowledge which the human intellect holds. We can say that the bodily organs taint out knowledge because the intellect is dependent on them for the acquisition of knowledge.

If we ignore, or circumvent this reality, we can assert that the human intellect has direct access to the independent, immaterial Forms. But then we have no principles by which we might demonstrate that any claimed a priori knowledge is actually, truly deficient. We can understand this in Kantian terms. The a priori intuitions of space and time are responsible for the tainting of our knowledge. These are manifestations of the physical constitution of the human body, which serve as the conditions for any ideas, concepts, or knowledge in general. Unless we acknowledge that what Kant calls a priori intuitions are properties of the human body, and are therefore fallible principles rather than eternal, immutable, truths, we have no approach toward arguing the deficiencies of them.

Plato himself demonstrated the deficiencies of the theory of participation, which provides the ontological support for the independent Ideas of Pythagorean Idealism. This is why he turned to "the good" to support the existence of Ideas. Notice that it is "the good" which supports intelligible objects for Plato, not "the idea of good", as often represented. It is sometimes argued that Plato himself was not a Platonist, and this is due to the common misrepresentation of Plato's philosophy in modern discussion which produces the common notion of "Platonism". It is very important to approach this ancient philosophy, as much as possible, without bias, if one is intending to develop a true understanding.

It is the universal view of ancient philosophy that the 'empirical realm' which is taken by moderns as the sine qua non of the real, is in fact a treacherous illusion, which the hoi polloi do not see as their minds are contaminated by worldly passions, which blind them to the higher truths. The real can only be grasped by reason, and its truths are invariant and never subject to decay. Whereas everything in the sensory domain is subject to constant change and degradation through the ravages of time.

In this passage you demonstrate significant ambiguity. The "empirical realm", if we use Kantian terminology, is phenomena. And this is how the mind apprehends the sensible objects. However, this refers to all ideas, and concepts which may be employed by reason toward apprehending truths. So all understanding is necessarily "a treacherous illusion" according to what you have stated. Therefore if a philosopher desires a true understanding, one must find a way out of this trap. The way out has been discovered by Plato, through the means of "the good", (implying final cause), and it has been developed by Aristotle, and carried forward by others like Augustine and Aquinas.

You ought to accept that "reason" refers to an activity, not a thing. It is carried out by the human intellect through the use of tools, ideas, and concepts. The human intellect is fundamentally deficient, necessarily so, because it is dependent on the human body. Therefore the ideas and concepts which the human intellect grasps are not the real, higher, invariant truths, which we might assume are out there somewhere. The things which the human intellect grasps receive their intelligibility relative to "the good", which is not necessarily the truth.
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Given that, as you say, we are going around in circles, it might just be that we might need to agree on disagreeing.

All the same I'll give a reply.

If we apply the law of identity to all forms, we see that universal forms cannot have an identity.

The conceptual form of "griffin" is not the same as the conceptual form of "unicorn". I take it we agree in this. How could this be so if neither has an identity? (an issue further addressed below)

My argument is that since there are numerous number systems, natural numbers, rational numbers, real numbers, imaginary numbers, and so forth, there are numerous different conceptions of "one", and no single mathematical system unifies these into one concept.

You would need to establish how the concept of "one" holds a different meaning in each of these systems to make this affirmation. What I find is that - even though they use the foundational concept of "one" in different ways - the concept of "one" remains the same. It's a given whole, a concept requisite for any such system of mathematics to manifest.

This is because we can use language to refer things without any identity. You don't seem to be grasping the intent of the law, which is to prevent the situation where we assume that just because we can talk about it, it is a thing with an identity. You completely misinterpret the law if you claim that a fictitious thing has an identity, because the law of identity puts the identity of a thing into the thing itself, rather than what we say about the thing. The fictitious thing has no existence independent from what we say about it, therefore it cannot have an identity.

Sure, you can say that "the law of identity pertains to all conceivable givens", but unless you abide by that law, and acknowledge that some conceivable givens do not have an identity, then you step outside that law and you enter into hypocrisy.

You are conflating identity with primary substances (with empirically known to be physically existent givens).

If I were to ask you for an example of a thing language can refer to that is devoid of any identity, you would likely identify givens that are not "empirically known to be physically existent" ... but you would be identifying them all the same, i.e. disclosing their identity. This in the same breath with which you'd affirm that they lack any identity.

If you believe you can sidestep this contradiction, please provide an example.

This is not a breaking of the law of identity, it is an issue with the law of non-contradiction.

It is commonly accepted that the law of noncontradiction is a derivative of the law of identity, and that the former is meaningless without the latter.

and primary awareness is necessarily of the external.

We again disagree. A different issue, though. But by this I take it that to you the laws of thought can only be external to awareness. A view which stands in utter contradiction to my own.

I don't agree with this at all. I think it is incoherent, so perhaps I misunderstand. First, how could one have knowledge of something which is independent of whether that something occurs?

Yes, my statement was misunderstood: What is real is regardless of whether or not it is known. One does not need to know what is real in order for what is real to be. This applies to what is metaphysically real just as it applies to what is physically real.

Second, the law of identity is extremely difficult even for human beings to understand (as evidenced by this thread), it is set up as a defence against sophism. So I don't see how children or lesser animals could be applying the law of identity as a defence against sophism. I believe you continue to misrepresent "the law of identity".

My dog can identify me (e.g., as not being another member of the household or some stranger). Nor does my dog behave as though me is not the same as me. My dog doesn't need to have a cognitive understanding of the law of identity in order to do so. He just does. This is what I meant by saying that the law of identity is intrinsic to awareness, i.e. that it governs all awareness - irrespective of whether there is propositional knowledge of it.

Would you agree with Aristotle, that when the geometer produces geometrical constructs, and discovers geometrical principles, this is an act which is properly described as the mind actualizing the principles. The principles exist in potential, prior to being actualized by the mind.

For the example just provided, I would not agree - though there might be other examples for which I would agree. The geometric principles, say those pertaining to a triangle, exist as actuality prior to their discovery. Awareness of them is a potentiality that becomes actualized. But the geometric principles can only be so, in actuality, prior to their discovery.

MU, since we disagree on so many issues, I'm OK with leaving things as they are. Of course, feel free to critique my reply, but I might not reply in turn. Benefited from the discussion all the same. Thanks.
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Javra, I understand if you no longer wish to discuss this subject. But please read through my post, specifically where I discuss the two senses of "identity". I believe we haven't been properly separating these two distinct senses, and this is at the root of our disagreement.

The conceptual form of "griffin" is not the same as the conceptual form of "unicorn". I take it we agree in this. How could this be so if neither has an identity? (an issue further addressed below)

There's a very straight forward answer to this. A conceptual form is not an "identity", by the law of identity. That's what I've been arguing is the intent of the law of identity, to elucidate this difference.

My belief is that the law of identity puts "same" into a different category from "different". Identity, which is indicated by "same", allows for difference, so that "same" in the context of this law, does not mean lack of difference. This is evident, because the same thing undergoes changes, becoming different from what it was, while still maintaining its identity as the same thing. This is the temporal extension of a thing, and the law of identity allows it to have the property X at one time, and not have the property X at another time, yet still be the same thing. So difference is allowed within the meaning of "same", not excluded by it.

Relating this to your example, what is dictated by the law of identity is what 'same" means, not what "different" means, and different is not opposed to "same" in this definition. So in your example, the fact that the two forms are different, doesn't say anything about identity. Difference is a concept which is related to something other than "same", when "same" is defined by the law of identity.

You would need to establish how the concept of "one" holds a different meaning in each of these systems to make this affirmation. What I find is that - even though they use the foundational concept of "one" in different ways - the concept of "one" remains the same. It's a given whole, a concept requisite for any such system of mathematics to manifest.

That's very simple as well. For instance, in the natural numbers "one" is not divisible, in the real numbers it is. It's easy to say "one" represents a "whole", but the divisibility of that whole is part of the concept. And then there is the possibility of having a negative whole, which is part of the concept as well. It's very clear that the meaning of "one" is quite different in these different systems, and you do not accurately represent the concept of "one" when you say it represents a whole. Using "one" in different ways implies necessarily different meanings, because the meaning is dependent on the use.

You are conflating identity with primary substances (with empirically known to be physically existent givens).

If I were to ask you for an example of a thing language can refer to that is devoid of any identity, you would likely identify givens that are not "empirically known to be physically existent" ... but you would be identifying them all the same, i.e. disclosing their identity. This in the same breath with which you'd affirm that they lack any identity.

If you believe you can sidestep this contradiction, please provide an example.

It appears like you are not quite grasping the law of identity clearly, and you are equivocating between two senses of identity, sometimes known as "numerical identity" and "qualitative identity" (check Stanford for an explanation). A type is identified by qualitative identity, but that is not what the law of identity refers to, which indicates one and the same thing. When language refers to 'a thing' which is devoid of identity (by the law of identity), it is referring to a type of thing, so it has qualitative identity, but this is distinct from "identity" as signified by the law of identity. So there is no contradiction, only a misrepresentation as to what "identity" means according to the law of identity, and possibly replacing this with "identity" in the sense of qualitative identity.

What is real is regardless of whether or not it is known.

Ok, maybe we have grounds for agreement here, and if we try we might be able to proceed. Do you believe, that just like "what is real is regardless of whether or not it is known", a real thing also has an identity regardless of whether or not the thing is known. This is what I am arguing that the law of identity states, the identity of the thing is within the thing itself, regardless of what is known about the thing. Can you agree with that? Further now, when we use "identity" in reference to what we know about a thing, this is "identity" in the other sense, qualitative identity. We describe the thing in terms of qualities, which are types.

This forms two distinct uses of identity which are consistent with the two distinct senses of "same". We might say that two different people have the same car, meaning the same make, model, year, colour, etc.. This is qualitative identity, reducible to type, really meaning the same type of car. On the other hand, we have to maintain that each car is distinct, because we know that the two cars are not the same car. This is numerical identity, what the law of identity refers to. If we are allowed to say, in strict logic, that two things are the same thing, because we have described them both in the same way, this creates many false conclusions.
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It appears like you are not quite grasping the law of identity clearly, and you are equivocating between two senses of identity, sometimes known as "numerical identity" and "qualitative identity" (check Stanford for an explanation).

Since you’ve pointed me to SEP, you’ll notice that the entry on identity is in no way unequivocal about what identity is. But taken from the introduction:

A distinction is customarily drawn between qualitative and numerical identity or sameness. Things with qualitative identity share properties, so things can be more or less qualitatively identical. Poodles and Great Danes are qualitatively identical because they share the property of being a dog, and such properties as go along with that, but two poodles will (very likely) have greater qualitative identity. Numerical identity requires absolute, or total, qualitative identity, and can only hold between a thing and itself. Its name implies the controversial view that it is the only identity relation in accordance with which we can properly count (or number) things: x and y are to be properly counted as one just in case they are numerically identical (Geach 1973).

Given this distinction - and the plasticity of the term "thing", which can reference a concept – how is your own concept of “griffin” not numerically identical?

(That it might change over time equally applies to any physical thing. Moreover, it can only change so much as concept while remaining the same concept of “griffin” – this, again, in parallel to the numerical identity of any physical object: e.g. the numerical identity of a flower between the time it is a bud, or earlier, and its full wilting, or later.)

Notice that qualitative identity requires the sameness of qualities that pertain to two or more things. By comparison, the concept of “griffin” is one thing - a given whole that as form is undivided - and not two or more. It is a hybridization of different animals – an eagle and a lion – true; but the hybridized given is nevertheless singular.

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As to the ontology of identity, my own views are fringe. In all fairness, I’d rather not get into them right now.
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Given this distinction - and the plasticity of the term "thing", which can reference a concept – how is your own concept of “griffin” not numerically identical?

I cannot locate, or in any way, point to this supposed concept, within me, to demonstrate to myself, it's existence as a thing. I might conjure up an image, but that's not really a concept. The closest I can come to finding a concept, is definition. I might assume the definition is "a thing", but the definition is just a bunch of words, which in that form is a multitude of things, each needing to be interpreted as signifying a concept itself. So this is really going in the wrong direction. It's heading toward an infinite regress of words associated with an assumed "concept", but no actual concepts. To find "the concept", if it exists as a thing, I need to take a different approach.

I could look for the boundaries of that thing, and this is how we distinguish things with our senses, or I could look for a unity of meaning, signified by the definition, or just the unity of meaning signified by the word "griiffin". Each of these, boundaries and unity, can be an indication of "a thing", and ultimately the most conclusive evidence would be to find both. But based on my understanding of the difference between sensible objects and concepts, I think the latter would be the most productive course. This is regardless of the modern trend in philosophy to look at definitions as boundaries created by rules of language, which I think is a dead end route. It is a dead end for the reasons stated above, the rules need to be interpreted, which would require rules for interpretation, resulting in an infinite regress of words, without finding any real boundaries. So the proper course appears to be to seek the unity of meaning, if I want find the object which is "the concept".

This is where the real difficulty begins, requiring an understanding of the use of words to signify meaning. Notice that we have to differentiate between using words to name a thing, as proper nouns, and using words to signify meaning. So to make a long story short (I could give you a vast multitude of examples of words referring to properties, which indicate meaning rather than an object), it is my understanding that when we use words to indicate meaning, what is signified is a type, rather than a thing. We might signify a type of property, or a type of thing.

So, we have a unity principle indicated here, which designates a type, as a category, a classification principle, like a "set" in mathematical terms. To understand this unity, and determine whether it is "a thing", we need to grasp the meaning of the principle which is supposed to render the unity. I suggest that the intent of this unity principle is to group numerous things together, which have some property in common, to aid in understanding. We could say that we attempt to create "a thing" in this way, and the thing which would be created exists as the category itself, which allows us to group numerous other "things" within that category.

Notice that I have determined two very distinct usages of "thing" now. One refers to the category itself, the principle of classification, the other refers to the objects to be classified. In effect, I have developed a principle of classification to distinguish two very different types of "thing". Therefore I need principles which distinguish them as separate. The one category I assign "identity" to. The things which will be classed, and categorized have an identity. The other category, the principles by which things are categorized, or classified, I cannot assign "thing" to, nor can I assign "identity" to, because I've already used those terms in the other category, and this is a philosophical endeavour, and my goal is to maintain logical rigour. It is true that there are many conventions in common vernacular which would call these categories "things", and say that they have "identity", but the goal here (being philosophy) is to maintain the validity of logic, and therefore avoid the equivocation and category mistake, which would inevitably result from such a duality of meaning.

The conclusion now, is that those so-called 'things", which I've now excluded from the category of "things", those concepts, which are more properly referred to as principles by which a unity is to be created, are not themselves properly called unities. This is because each one fits into a higher unity, and is not itself a proper whole, receiving meaning from the higher whole. So Socrates refers to man, man refers to animal, refers to living being, etc.. Therefore no individual concept is a complete unity, it always refers to something outside as a source for meaning. It is a part which is not itself a whole, because it is wholly dependent on something external to it for its meaning. I believe this is the point of the op, the meaning of the axiom is always derived from something else, so the object is not complete as an object until we determine the whole.

This is why we need the highest unity, "the Ideal", what the Neo-Platonists called "the One", to allow that a concept is a thing. This highest unity, this Ideal, is supposed to be the complete conceptual unity, being deferred to no higher whole to complete its meaning and unification. It is by that assumption, "whole", and therefore a thing with identity. This Ideal, "the One", is assumed as a concept which is also a thing with identity.

By comparison, the concept of “griffin” is one thing - a given whole that as form is undivided - and not two or more. It is a hybridization of different animals – an eagle and a lion – true; but the hybridized given is nevertheless singular.

I don't think this is quite true though. You are simply assuming that the griffin concept within your mind is one thing. But if you analyze this concept within your mind, you'll see that "griffin" refers to a type of thing. And if you assert that this "type of thing" is a thing itself, that assertion needs to justified. That is what I tried to do above. It leads to an infinite regress of meaning, requiring the assumption of a fundamental unity such as "the One", which unites all conceptual structure as one united whole. This is why we ought not allow any contradictions within knowledge, no matter how far apart the fields of study are, it negates the possibility of a united whole,
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Therefore no individual concept is a complete unity, it always refers to something outside as a source for meaning. It is a part which is not itself a whole, because it is wholly dependent on something external to it for its meaning.

A very informative post. Thanks for it. To let you know a little more of where I’m coming from:

There is the philosophical notion of holons: givens that are simultaneously both wholes and parts. Although my views are not identical to those addressed in the article, I do have great empathies toward the views therein expressed.

Any animal - as a whole token - is itself in part determined by its environment: from that of its ecological environment to that of the world’s natural laws as environmental givens. As one example, a mammal would not be in the absence of air it inhabits just as a fish would not be in the absence of water it inhabits; in both cases the occurrence of the former is *in part* determined by the occurrence of the latter. An individual animal can thereby be construed to be a part-holon of its environmental-holon.

It’s a complex ontological approach, but then an animal's parts, say its lungs, has an identity, just as the animal itself has an identity, just as the animal’s environment, say a particular forest, has an identity.

Using the notion of holons, then, to me each concept is itself a holon - constituted of parts that are themselves holons, and is itself a part of greater concepts that are themselves holons.

While this synopsis will not address all conceivable issues related to this approach, I get that we will likely disagree in our basic approaches. No harm in that though. Save for a few disagreements here and there. :grin:
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Using the notion of holons, then, to me each concept is itself a holon - constituted of parts that are themselves holons, and is itself a part of greater concepts that are themselves holons.

I'll see if I can interpret this in a way which is consistent with what I said in the last post. Let's say that there are two aspects of the holon, one which makes it in some sense an individual, free and independent, and another aspect which makes it necessarily dependent on a larger whole, so that it is united to other concepts by this aspect. You can see that these two aspects are fundamental incompatible, the one making it a distinct individual with its own independent identity, the other wanting to negate this existence as a distinct individual, giving it an identity, as a part of a larger whole. So they must be distinct aspects of the concept. This would mean that the holon partakes in both the categories I described, as a material individual, and also as part of a type, receiving meaning from a higher order. This is comparable to Aristotle's matter and form.

If a concept is an object, as a holon, then it cannot be purely immaterial, it must also have a material aspect, which is responsible for it being, in some sense, an independent individual. If a concept is strictly formal, then its complete existence is dependent on the larger whole. If it has any sort of individual, independent existence, then it must have a material aspect to account for this separate identity.

What Aristotle does, is seek the source of this material aspect, as that which accounts for the existence of individuals. In many ways, the material aspect is indefinite and therefore unintelligible. So the concept, being essentially something intelligible, must also have an aspect of it which is unintelligible, just like any material thing. This is the deficiency which Aquinas spoke of. Due to the fact that the intellect is united with the material body, and is not properly a separate substance, the intelligible forms which it deals with are also deficient.. I don't mind looking at concepts in this way. It makes more sense than saying that a concept is a completely separate, and immaterial object.
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Seems like we’re approaching a common ground in respect to the hylo-morphology of concepts. Cool.

BTW, to me there’s a parallel between Aristotle’s prime matter and today’s notion of zero-point energy. Both seeming to hold the properties of pure potentiality and unintelligibility while underlying all that is intelligible matter. As we were previously discussing, the intelligibility of actualized identity is always brought about by forms - including the forms of intelligible matter. And, in Aristotelian terms, the ultimate form is that of the teleological unmoved mover, which is singular as form in being devoid of constituents and, therefore, devoid of matter. Please remind me if there were any disagreements between us in the aforementioned.

What criticism would you give to the proposition that every intelligible form is, and can only be, cognized as a whole (for context, where every whole - save for the unmoved mover - is itself a hylomorphic holon). Thereby making the concept of a whole, i.e. of an entirety, and the concept of a form fully synonymous.

As background, I find this issue to be pertinent to the context of the Aristotelian category of formal causation. Which is distinct from, though entwined with, teleological causation (as might be evidenced in Aristotle’s coinage of entelechy as term for addressing actualized things).
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Seems like we’re approaching a common ground in respect to the hylo-morphology of concepts. Cool.

I used to hold a very similar understanding, but I've progressed toward what I consider to be a deeper understanding. If we maintain that concepts have a material aspect, then we need to account for the manifestation of that material aspect. In Aristotle's hylomorphic structure, matter accounts for the temporal continuity of the object, its capacity to persist, and therefore its identity as a continuation of being the same object. This is described as mass or inertia. This is also the aspect of the object which is unintelligible to us, in physical objects we simply take it for granted.

The temporal continuity of a concept can be seen to be provided for by the physical existence of the symbols. This is how we communicate, allowing the same concept to exist for generation after generation, and amongst different individuals. Also, we can see a similar situation within the mind of an individual, where memory relies on symbols, or some other sort of representation such as an image. The representation is essential to the temporal continuity of memory. However, this implies that the symbol is an aspect of the concept itself, as its material element. It is an accidental, as we see that the symbol may be arbitrary, but still the medium employed (as the matter) has an effect on the essence of the concept through the capacity for temporal continuity. So for example, there is a big difference in the potential for temporal continuity, between written symbols (as we find in mathematics), and spoken words. The idea that mathematical symbols represent eternal objects is derived from this capacity for extended temporal continuity. But when we look at spoken words we find that the concepts represented by symbols are actually quite fluid. So the forms (concepts) are changing as time passes, and even the mathematical ones change as knowledge evolves, but through a slower process.

Regardless of those revelations, as philosophers we may have a commitment toward further analysis of "the concept". If a concept is composed of matter and form, then to fully understand the nature of a concept we need to produce a proper separation between the matter and the form, so as to distinguish the intelligible from the unintelligible Matter has been designated as fundamentally indefinite and unintelligible.. Allowing the unintelligible (matter) to enter into the intelligible in our apprehension of "the concept", assuming that it is an inherent part of the intelligible (as a concept is fundamentally intelligible), is to provide for misunderstanding. So we still need a technique to identify and expel misunderstanding. Therefore we still need to proceed toward separating the material aspect (the representation), out, and attempting to understand strictly the immaterial, intelligible aspect of the concept.

BTW, to me there’s a parallel between Aristotle’s prime matter and today’s notion of zero-point energy. Both seeming to hold the properties of pure potentiality and unintelligibility while underlying all that is intelligible matter. As we were previously discussing, the intelligibility of actualized identity is always brought about by forms - including the forms of intelligible matter. And, in Aristotelian terms, the ultimate form is that of the teleological unmoved mover, which is singular as form in being devoid of constituents and, therefore, devoid of matter. Please remind me if there were any disagreements between us in the aforementioned.

The cosmological argument, and the teleological unmoved mover, are the means by which Aristotle brings matter itself into the realm of the intelligible. In Bk.7 Metaphysics, the part I referred to already, you can see how he begins this process by showing how the matter is a part of the formula in creative art and production. Ultimately, in the cosmological argument, Bk.9, it is shown that actuality must be temporally prior to potentiality, therefore form is prior to matter. This is the form of final causation which determines the matter in creation. Accordingly, there is no such thing as "prime matter", because matter is created as required by the purpose, so all matter is inherently formed by the purposeful act which creates it, therefore it is fundamentally intelligible. The fact that matter is by definition indefinite, and unintelligible, is an epistemological fact. The deficient human intellect does not have the capacity to understand temporal continuity, so this part of reality is designated as indefinite and unintelligible (matter). But this is the fault of the human intellect, temporal continuity, matter, is actually quite intelligible to a higher intellect, according to Aristotelian principles.

What criticism would you give to the proposition that every intelligible form is, and can only be, cognized as a whole (for context, where every whole - save for the unmoved mover - is itself a hylomorphic holon). Thereby making the concept of a whole, i.e. of an entirety, and the concept of a form fully synonymous.

This I think, is the problem evident in the hylomorphic approach to concepts. In the case of conception, such a whole is never quite complete, therefore an invalid "whole". This is the example I provided with the regress into unclarity: the concept of "Socrates" refers to "man", which refers to "mammal" which refers to "animal" which refers to "living being", and so on. The more specific is defined by the more general, and the more general becomes increasingly vague and ill-defined, such that we can never claim completion of a whole in conception. And, we see the same thing in describing the existence of physical objects. An object on the earth requires reference to the earth, and the earth requires reference to the sun, the milky way, etc.. We do not get the closure of a whole, even in our descriptions of physical objects.

So we might produce an Ideal, "the One", as a proposal of a valid whole. The One is what both closes the conceptual expansion into vagueness, and also validates a particular physical whole as everything, the universe. Aristotle proposed circular motion, and the divine thinking, which is thinking on thinking, as the closure of wholeness, but this proved to be insufficient. Aquinas stipulates "God", but it is asserted by him, that the human intellect, being dependent on the material body, cannot obtain a proper understanding of God.

So I would say that wholeness is what is required by the intelligible form in order to be completely and absolutely intelligible, but human conceptions lack this. This is quite evident in the most fundamental mathematical principles. The natural numbers are infinite. The spatial point is infinitely small. A line is infinitely long, etc. This is evidence that human conceptual forms, as intelligible objects, are fundamental lacking in wholeness. This is why I prefer not to call them "objects". However, as I said above, in our attempts to understand physical objects we are met with the same deficiency of wholeness. But this might just be due to the deficiency in our capacity to understand. We clearly sense boundaries of closure, we see objects as closed wholes. So it may be the case that we simply misunderstand what we see, and there may actually be some truth to the wholeness of an object which we perceive, but we just have not developed a proper understanding of it.
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This I think, is the problem evident in the hylomorphic approach to concepts. In the case of conception, such a whole is never quite complete, therefore an invalid "whole". This is the example I provided with the regress into unclarity: the concept of "Socrates" refers to "man", which refers to "mammal" which refers to "animal" which refers to "living being", and so on.

"Refers" is an inadequate term here. "Socrates" refers to Socrates, and not just any man. Likewise "animal" refers to animals, and not just any living being (plants, for example).

So I would say that wholeness is what is required by the intelligible form in order to be completely and absolutely intelligible, but human conceptions lack this. This is quite evident in the most fundamental mathematical principles. The natural numbers are infinite. The spatial point is infinitely small. A line is infinitely long, etc. This is evidence that human conceptual forms, as intelligible objects, are fundamental lacking in wholeness. This is why I prefer not to call them "objects". However, as I said above, in our attempts to understand physical objects we are met with the same deficiency of wholeness.

I'm glad that this is evident. In short, when in search of absolutes - such as in a complete and absolute intelligibility, to paraphrase from this quote - absolute wholeness does not occur for givens, be they conceptual or physical. Nevertheless we cognize givens as bounded entireties. For example, a rock is cognized as a bounded entirety, as a whole given. Not as two or more givens; and not as an amorphous process. Even "a process" is cognized as a bounded entirety, and can thereby be discerned to be one of two or more processes.

Maybe you're looking for the absolute, fundamental nature of individual things that dwells behind our awareness of them, so to speak. Whereas I'm addressing the very nature of how we cognize givens: by cognizing each individual given to hold the attribute of oneness.

But I find that this following statement might be pivotal to our disagreements in large:

In Aristotle's hylomorphic structure, matter accounts for the temporal continuity of the object, its capacity to persist, and therefore its identity as a continuation of being the same object.

What then do you make of formal causation?

I also note that while a flower is neither an unopened bud nor the stem off of which all petals have fallen, it yet remains the same (numerically identical) flower throughout the time period in-between, despite considerable changes in its matter over this span of time. Its identity nevertheless remains static in its form - again, despite the changes in its matter - such that form accounts for the temporal continuity of the object, and therefore its identity.
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Refers" is an inadequate term here. "Socrates" refers to Socrates, and not just any man. Likewise "animal" refers to animals, and not just any living being (plants, for example).

You misunderstand. The word refers to the object in some circumstances, but we're talking about the concept here, the concept of "Socrates". To say what Socrates is, the concept of Socrates, is to refer to "man". To say what a man is, is to refer to "animal". The issue was whether or not a concept is a whole. And since each concept is defined by something more general, which makes the definition get more and more vague, without closure, as we proceed in defining the terms, a concept cannot properly be called a whole.

I'm glad that this is evident. In short, when in search of absolutes - such as in a complete and absolute intelligibility, to paraphrase from this quote - absolute wholeness does not occur for givens, be they conceptual or physical. Nevertheless we cognize givens as bounded entireties. For example, a rock is cognized as a bounded entirety, as a whole given. Not as two or more givens; and not as an amorphous process. Even "a process" is cognized as a bounded entirety, and can thereby be discerned to be one of two or more processes.

The problem with this is that we see a physical object as a bounded whole. The spatial boundaries are very evident to our vision, so we have some reason to believe that there is some reality to these boundaries. Therefore we assume that physical objects are wholes. In the case of concepts we cannot find those boundaries. One concept is defined in relation to another, which is defined in relation to another, and so on, and we cannot find any true boundaries. So we might insert arbitrary ones. This is the same with "a process". If there is a temporal duration, we cannot really discern when one process stops and the next starts, so we insert arbitrary boundaries. Temporal processes are like that. When we look for the beginning or ending of a particular, identifiable object, the precise moment that it starts to exist or ends existing is arbitrary.

So I think that our conceptions of boundaries are based in spatial concepts, being derived from our sensations of spatial boundaries, and we don't really have any real, applicable principles toward understanding temporal boundaries. We can readily understand that the present makes a boundary between the past and future, but we have very little if any understanding of what this means.

Maybe you're looking for the absolute, fundamental nature of individual things that dwells behind our awareness of them, so to speak. Whereas I'm addressing the very nature of how we cognize givens: by cognizing each individual given to hold the attribute of oneness.

I must admit that I am not familiar with your use of "givens", and I don't think I understand what you mean with it. Maybe you can explain.

What then do you make of formal causation?

I would describe formal causation as the restriction imposed on the possibility of change, by the actual physical conditions present at the time. So at any given time, any situation is describable in formal terms. The describable physical conditions which are present act as a constraint on the possibility of future situations, therefore this present form, is in that sense, a cause of future situations.

I also note that while a flower is neither an unopened bud nor the stem off of which all petals have fallen, it yet remains the same (numerically identical) flower throughout the time period in-between, despite considerable changes in its matter over this span of time. Its identity nevertheless remains static in its form - again, despite the changes in its matter - such that form accounts for the temporal continuity of the object, and therefore its identity.

I think this is incorrect. The flower changes its form. The form is what is describable, and the changed form is describable. The matter only changes to the extent that such changes, material changes, are describable, but if they are describable, then they are formal changes. So by definition, the matter does not change. As Aristotle says in his Physics, it is what a thing comes from, and persists afterward. So by definition it is what does not change. Only the form changes. If there was a prime matter, it would be the fundamental elements or particles out of which all existing things are made. These fundamental particles would never themselves change, they would just keep existing in different configurations, and this would account for all possible change. However, this is the conception which Aristotle demonstrates as fundamentally incoherent in his Metaphysics. He shows how matter itself must come to be from some type of teleological form, therefore we need to seek the Divine Will, as the cause of matter and temporal continuity.

Its identity nevertheless remains static in its form - again, despite the changes in its matter - such that form accounts for the temporal continuity of the object, and therefore its identity.

The form of a thing does not remain static, it is always changing, and is by definition what is "actual". The matter, as "potential" is static, because despite changes the potential remains the same (conservation laws in modern physics). And by definition, the matter is what existed prior to a change, and persists after the change.
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What then do you make of formal causation? — javra

I would describe formal causation as the restriction imposed on the possibility of change, by the actual physical conditions present at the time. So at any given time, any situation is describable in formal terms. The describable physical conditions which are present act as a constraint on the possibility of future situations, therefore this present form, is in that sense, a cause of future situations.

We are oceans apart. A culture's form (imperfectly) determines the nature of the individual, constituent, human psyches it, as a culture, is composed of - language and its semantics as one example. But nowhere does a culture have "describable physical conditions".

He shows how matter itself must come to be from some type of teleological form, therefore we need to seek the Divine Will, as the cause of matter and temporal continuity.

This puts a big damper on things for me. I cannot logically appraise Aristotle's teleological unmoved mover to be "Divine Will" - in part because will itself is always teleological motivated by an outcome it seeks to accomplish, and it is thus always in motion. Maybe this is a/the primary source of our disagreements - with most other issues regarding identity being derivatives.

In any case, I'm respectfully bowing out of the conversation.
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We are oceans apart. A culture's form (imperfectly) determines the nature of the individual, constituent, human psyches it, as a culture, is composed of - language and its semantics as one example. But nowhere does a culture have "describable physical conditions".

I thought we were sticking to "form" in the sense of Aristotle's hylomorphism.
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