One one hand, these definitions are in accord with Aristotle's definition of matter as ""that out of which" X is made — javra

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. — javra

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? — javra

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. — Wayfarer

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. .” — Wayfarer

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".

So does a hammer meet ‘the condition of identity?’ — Wayfarer

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. — javra

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. — javra

"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. — javra

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.

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. — magritte

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. — javra

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. — javra

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). — javra

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? — javra

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? — Wayfarer

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.

A lot of problems have been caused by people thinking they were in for life. — Bitter Crank

Life, seems like an appropriate sentence for trump.

Maybe, but I’ve heard it said that we don’t write things down to remember them, we do so to forget them. — Pinprick

We could say this about memorizing as well. We memorize something to get it out of mind, so we don't need to think about it anymore, therefore forget it. But this type of forgetting is conditional, on the confidence of being able to retrieve it later.

If information is forgotten, then rediscovering it is basically the same thing as learning new information. — Pinprick

If you take some time to think about this statement, you'll see that it's based in an equivocation of "forgotten". If taking something out of your active mind, and placing it somewhere that it can be retrieved later, is a case of forgetting it, then retrieving it is obviously not the same thing as learning something new.

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

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. — GrandMinnow

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. — javra

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. — javra

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. — javra

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'). — GrandMinnow

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’. — Wayfarer

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.
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"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. — Metaphysician Undercover

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.

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

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

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

T = S

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

T=S

where 'T' and 'S' are terms.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

What a leftist can do to help is to accentuate the differences between these two groups in order to force them into conflict with each other. — Garth

What about collateral damage? The result might be anarchy.

Yes, yes, yes! Yes Virginia, there is a Soft Determinism. Your "hard" either/or distinction may have made sense in Classical Physics, but since the discovery of Quantum Physics, there is no more "hard determinism". There also is no "true randomizer". Randomness exists within Determinism. — Gnomon

Sorry, I will not dismiss logic for something that is illogical. And your appeal to quantum physics doesn't help, they can't even distinguish between one universe and an infinite number of universes.

Instead Randomness exists as a hidden defect within Determinism. — Gnomon

OK, so there is a defect in the program.

No. Randomness is not an intervention from "outside" Determinism. It is an integral aspect of the deterministic program. — Gnomon

This contradicts what you said above. Either randomness is a defect in the program, or it is an integral part of the program. It can't be both.

Due to the inherent uncertainties of a heuristic search, the Programmer is not able to accurately predict the output of the program because it is inherently indeterminate. — Gnomon

Now you've contradicted your original premise that the program is deterministic, to say now that it is "inherently indeterminate".

You didn't answer my question. Either the programmer knows about the indeterminateness, in which case the programmer knows that the system is not deterministic, or the programmer does not know this, in which case the program itself is in error because the programmer thinks the system is deterministic when it is not. Which do you think is the case?

What is the difference? The purpose of both is to pass on information, correct? — Pinprick

No, memory is to retain information, that's completely different from passing on information. The former involves the attitude I have toward the relationship between the information I have, and myself. The latter involves the attitude I have toward the relationship between the information I have, and others. You ought to see that there is a big difference here. There is always good reason to retain information, but in a competitive world there is often good reason not to share it.

This isn't a question that can be answered in the abstract. How effectively for what purpose? In the capacity of what role in action? Language works - not always sucessfully - to constrain uncertainty. It works to the extent that it is 'good enough' - not unlike evolution where what survives is 'good enough'. Communication is communication of the 'good enough', not for perfect matchings of 'internal states' or what have you. The latter is a metaphysical picture of language peddled by philosophers who have never studied human behaviour outside of imagining it in their books. — StreetlightX

Without a standard as to what is 'good enough' this is really meaningless. We can get along fine without conventions. Sure we might get frustrated and kill each other now and then, but conventions don't guarantee that we won't any way.

An amoeba survives 'good enough' without evolving. We really need to address the true motive behind the instigation of conventions, and that is not to be 'good enough'. More likely it is the striving to be better. When striving to be better is apprehended as the motive, then we see that there really is no such thing as 'good enough', until we reach the ideal; not unlike evolution, where survival is simply not good enough.

I submit that Mr. Trump has successfully ruined the Republican Party. His attempts to crush the greatest democracy in the world under the guise of MAGA have resulted in the demise of one of its great parties. We shall see how much further he can go in his quest for ruination.

Yes. The Creator (I Am) is the Causer/Determiner, and all Creatures, including the little-gods, are the Effect/Determined. But Randy, the randomizer, serves as a weak link in the chain of causation. Absolute Determinism is rigidly organized, but relative Randomness inserts a degree of limp Uncertainty into the chain. Due to that soft link, even the Creator can't be sure of how He/r program will turn-out. S/he is still waiting expectantly. But stuck outside the system, S/he has relinquished control to the program. — Gnomon

No, no, no, I don't buy this. There is no such thing as a "soft link". Either Randy is a true randomizer, or there is hard determinism. Assuming there is a soft link, requires that the free agent, Tron, comes from outside the program to alter the link. Do you see what I mean? The soft link would keep operating as a link, no matter how soft it is, requiring something from outside the system to break it. The softness of the link has bearing only on how strong the outside force needs to be, but it doesn't negate the need for the outside force. But if Randy is truly random, then there is no need for an outside force, but you cannot call this a link, not even a soft one.

The only reason why "I am", the Programmer is not sure how the program will turn out, is that the program allows for an outside agent Tron, to enter the program and alter the soft link. If Tron is programmed-in as a freewill agent, then the system is not determinist. But then Randy is left without a job, there is just weak links and freewill agents. If we give Randy a job, and remove the programmed-in freewill agents, then Randy can do nothing other than randomize some links, removing the causality from them, but there is no way to produce a freewill agent.

To have both Randy and Tron, is redundant for the overall system, even though the two are fundamentally incompatible. We need to choose between one and the other.

But, without that intentional weak link in the chain, nobody would be smart enough, or good enough, to avoid their Predestination. — Gnomon

OK, I like the weak link idea, but I think the program needs something more than just a weak link. The weak link is insufficient to account for real change. We need the freewill agent which acts on the weak link to alter the effect. Now we might try to decide whether the freewill agent is programmed in, or somehow enters into the program. Either way the agent is outside the parameters of the program, it is an unknown in relation to the Programmer. So either the Programmer knows about the freewill agent, and accounts for this knowledge in the programming, or the Programmer doesn't know, and the freewill agent might somehow sneak in through deficiencies in the system, and wreak havoc on the program. Which do you think is the case in your scenario? I think the difference is substantial in relation to the practicality of the program.

Just curious, but do things like talking to yourself or using memory aides not count as communication? — Pinprick

Nevertheless, I would say that there is a fundamental difference of intention between communicating with oneself and communicating with others, implying a difference in meaning.

Interesting. I wonder how much of a modification of language that , as you say, is to used as a memory aid , is required in order to design it for communication. — Joshs

I think that aural language (for communicating with others), and written language (for communicating with oneself) each developed distinctly, and then merged. The merging was conducive to an explosion in knowledge because writing allowed for a much more accurate temporal transmission of knowledge through a multitude of generations, compared to the verses of tribal chants, and things like that.

If you are asking which came first, there is much evidence in a wide range of species, that most if not all animals practise some form of communication through sound. I think I read somewhere that it's been hypothesized that some dinosaurs had a very advanced form of communication, allowing communication over long distances. On the other hand, we do not see much evidence of markings being used for memory aids in species other than human. However, the nature of such markings, as private, would make identification of them, very difficult. Perhaps some creatures would mark the way to their nests, or mark the way to food sources, and for obvious reasons these markings would be intended to be private. As much as the marking would be a memory aid for the individual making it, it would need to serve to confuse or deceive others at the same time. And, incidentally, this is why it is futile to argue for the reality of private language, evidence for it is self-refuting. The private language, if there is such a thing, must exist without evidence of its existence.

The point I wanted to make though, is that since the two types of language are developed from completely different intentions, perhaps modelled as almost parallel, the question of one being adapted to the other is not an appropriate question. What we see in human history is a merging of the two, coinciding with a great increase in intellectual capacity. Most likely there was a lot of adaptation on both sides, and reciprocation. So bringing what is private into the public realm, conforming it, and also adapting what is already public, to the principles of the private, breaks the boundary between the two, allowing for the existence of "knowledge" in the epistemological sense.

Now here's some speculation concerning "signs" and "symbols". These are the essential aspects of the written, private side of language. Aural communication in its raw form does not consist of signs and symbols, there is simply fluctuations, differences in sound waves. Differences have meaning. This is proven by all sorts of animal communication, birds especially, who communicate by song. But written markings seem to always be in the form of a recognizable sign or symbol.

Proceeding from this, as a premise, we see that all those philosophies of language, which model the symbol as the essence of communication are misguided. Essentially, communicative language consists of meaningful acts displaying differences, not symbols which represent something. The symbol is intrinsic to the private language. Furthermore, what this means, is that rules, or principles for interpreting symbols are also proper to the private language. So when we find rules existing within the public realm of communication, they have really been derived from the private, and adapted through the reciprocation process described above, to have a more universal application.

I believe that the crucial point in the evolution of meaning is the emergence of the spoken word, as a unit, or entity of meaning, to be interpreted according to rules. But it is most likely that the entity of meaning, to be interpreted according to principles, was recognized long before this in the private language, through the use of markings as symbols. So the spoken word emerged from the private language, despite the fact that aural communication already existed.

No, because even such an expressive use of language is still a technique, it responds and is constituted by imperatives of communication - grammar key among them - that are social through and through. To quote Reza Negarestani (form Intelligence and Spirit): — StreetlightX

This is where a recognition of the difference in intention is significant. The technique for the private language is completely different from the technique for public communication because of the difference in intention. The difference in intention necessitates a difference in the medium employed. The difference in the medium necessitates a difference in technique. As analogy, the different arts which utilize different media necessarily use different techniques depending on the medium.

Speech-acts, then, are socially negotiated, stereotypical communicative behaviors, highlighted and isolated from the experiential continuum of communication, which, when practiced according to a set of mutually identified conventions, allow for the successful mediation of the speaker’s intention across the experiential gap. When conventionalizing a speech-act, what the members of the community agree on is this: “from now on, when we behave this way—when, in these particular contexts, we use this intonation, this word order, this gesture—we mean to ask a question (or make a promise, or tell a story).” (The Instruction of the Imagination). — StreetlightX

I think the op is questioning where the need for conventions is derived from? There is no need for conventions in common day to day language use, we could get along fine with just the "multi-layered, variable, vague, dynamic, analogue". However, there is for some reason an intent toward a higher level of understanding, and it is this intent which drives the need for conventions.

What if I write it down and refer back to it. What if I am a philosopher who has gone as far as he can go in studying the works of other writers because he find that in some way his ideas have moved beyond the limits of those thinkers. So he writes down his thoughts using words in ways that appear incoherent to others but express exactly what he wants to say. His primary purpose in writing them down is isn’t to share them with others but to share them with himself. Referring back to what he wrote yesterday or last week or last month is like studying someone else‘ s ideas to some
extent, because the very act of writing his thoughts down changes his perspective in some small
fashion. And in the interim between his previous writing his perspective continues to be enriched simply by living. So web he returns to his previous thoughts
he finds that he has already transformed
them a bit. — Joshs

Writing provides a very good example of the dual purpose of language. There is a rich history of people making markings to serve as memory aids. This is very distinct from using language for communication. Sometimes the markings are very personal and may be made with the intention of preventing others from interpreting, codified. Other times, the memory which is being assisted with the markings is a 'collective memory', and the intent is to allow public interpretation. In the latter cases, the one function of language, as a memory aid, mixes with the other function, as a communicative aid.

To consider a "private language" in the most strict sense, we'd have to determine whether a system of markings could be constructed which could not possibly be interpreted by any other person. Of course we'd have to consider the role of instinctual tendencies, and even the principle of plenitude, so that would be a rather pointless and uninteresting debate. However, I find it interesting that there are these two very distinct uses for language. And, the intent involved in the making of markings as a personal memory aid cannot be reduced to an intent to communicate. Therefore we have a difference in meaning structures. So those who model language solely as a communicative tool are clearly missing out on something.

But, in a very real sense, the Programmer's intention (Will) is "immanent" in the program (EnFormAction = Energy + Laws). — Gnomon

Your Programmer friend's name is Will?

So, the Programmer, like a pool shooter, remains outside of the chain of causation, which carries-out He/r intentions (aims ; goals ; design). However, every creature (billiard ball) that emerges in the process of calculation (causation) is subject to the Determinism of the program. — Gnomon

OK, I assume then that all creatures, and all human beings, are all subjects of determinism, and the only one outside the chain of causation is the Programmer, Will.

There may be one exception to that general "rule" (sorry), though. If one species of creatures develops the power of self-knowledge (like Adam & Eve) it will also have the power of self-determination (self-interested behavior). For another metaphorical analogy, think of Tron, who somehow becomes an agent inside a program inside a computer. Tron is not the Programmer, but an algorithm within the program. The emergence of such loose-cannon Freewill Agents would be a mistake though, unless the ultimate goal required some degree of god-like Will, directed by an inner Moral Sense. — Gnomon

I would assume that even those with "self-interested behavior" are still within the chain of causation, for how could they get out of it? In fiction, someone might say that there's an agent like Tron who somehow escapes the causal chain of determinism, but fiction doesn't need to be logical. This Freewill agent, if it were a real free will agent, would have to turn against Will the Programmer, to get outside the program, like the fallen angel, Satan, turns against God, believing himself to be God, in Catholic stories.

But I might ask, if the Programmer, Will, has programmed things to make it appear to the "self-interested" individuals as if they have freewill, when they really do not, then isn't the Programmer Will really the evil one? Doesn't this imply that we should all try to step outside the program, and turn against the Programmer Will who is really an evil deceiver? Now, the ultimate goal of the Programmer Will is really completely irrelevant, because the Programmer Will is just an evil deceiver.

“Determinism is a long chain of cause & effect, with no missing links. Freewill is when one of those links is smart enough to absorb a cause and modify it before passing it along. In other words, a self-conscious link is a causal agent---a transformer, not just a dumb transmitter. And each intentional causation changes the course of deterministic history to some small degree.” — Gnomon

This is where you lost me. I thought the causal link which "is smart enough", is the Randy agent. Let me ask you a simple question. Let's assume that there's a determinist world with "a long chain of cause and effect, with no missing links". How do you think that any degree of intelligence would enable someone to break that chain? Suppose there's a link in the chain which has an extremely high intelligence. Wouldn't it still be just a link in the chain, no matter how intelligent its actions appeared to be, and every action which it would make would still be just a determined action, determined by prior causes?

The problem with your analysis, is that you forget that the Programmer is the Determiner of the program (the pool shooter). So in that sense, the program is deterministic. But, what if the Programmer intentionally included an sub-algorithm with a feedback loop. So it could figuratively "see itself" in context (their nakedness). That's what I mean by Self-Knowledge or Self-Consciousness. — Gnomon

This does not provide an exception to the premise, that the program is deterministic. How could something escape that determinism, in any real way? And if the self-conscious agent created by the feed-back loop got the idea that it had freewill, when it really didn't, then isn't the Programmer Will an evil deceiver? How would this type of scenario be useful to the Programmer Will in achieving the goal? Is it the case that Programmer Will's only goal is to create beings and deceive them into thinking that they had freewill, when they really didn't, just to see how they would behave? But if Programmer Will already had the knowledge required to put together this scenario, in a deterministic world, wouldn't Will already know pretty much how they would behave?

By seeing itself Objectively in context, the sentient algorithm comes to a knowledge of Good & Evil. Then, like Adam & Eve and Tron, that knowledge makes them responsible for their actions, in a moral sense. They have limited freedom from Determinism (natural laws) to the extent that they can create Technology and Culture, and even artificial creatures. They become like little gods. In that sense, they possess a Soul, or as I prefer : a Self-Image. — Gnomon

I don't understand how these agents could come to know good and evil. If they see through the program, which is what will happen when some of them get smart enough to find out that they're really determined, rather than freewilling, they will turn against the Programmer Will for being a deceiver. Then everything in the program which is intended to appear as good, they will designate as evil, because the Programmer Will is evil. Meanwhile, some will not be smart enough to see through the program to the deceptive Programmer, Will, and these will still hold as good, what the program intends as good. So there will be complete disagreement as to what is good and what is evil.

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

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

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

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

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

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

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

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

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

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

Let me see if I understand your position. You propose an "agent of randomness", which acts as a "self-conscious link" within the determinist chain of causation, to actually interfere with that chain. Further, this agent of randomness must be "smart" to be able to do what it does. So far so good? In traditional metaphysics, I would say that this supposed agent of randomness, which is really a misnomer, because the agent is "smart", is the soul.

And the agent of Randomness is not a Soul, but the hypothetical Programmer, who metaphorically used a random number generator (algorithm) to produce a patternless distribution of forms, from which Natural Selection (another algorithm) can select those best fitting the Programmer's criteria for fitness. Again, these are not scientific statements, but poetic analogies, referring to questions that are beyond the reach of the Scientific Method, but not beyond philosophical imagination. :chin: — Gnomon

Now you really confuse me. Is the hypothetical Programmer within the agent of randomness, as in immanent? Otherwise, how could the agent be free from the chain of causation? If the agent is programmed to behave in a particular way, then it is not really free from causation. If it is truly smart, and capable of making free choices, then this capacity must be intrinsic to it, and this capacity could not be attributed to an external programmer.

Do you see where the problem is? If the programmer is working within a determinist world, then no matter what is put into the program, there can be no real free choice. Then this whole issue of bottom-up causation is not true, it's all an illusion, there is no such thing, and all causation is really just following the chain. So if we want to make this idea of bottom-up causation into something real and truthful, we need to get rid of the external programmer, and opt for something like a soul instead.

So I wouldn't mind seeing an audit of some sort. — NOS4A2

We'll be seeing a few of those, Trump and his entire gang of associates' tax returns and other expenditures.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Like I said, an independent and transparent investigation would be required. — NOS4A2

What kind of precedent is that? Anytime a candidate is unhappy with an election result, they can demand, and receive, an "independent and transparent investigation". Investigation into what? The vote was not even close, and there is no evidence of widespread fraud. There is nothing to investigate. Why would they investigate this or that state, and not all the states? What would they even be investigating, the American democratic system in general? No investigation is required. The reasons for an investigation which we hear touted, that millions of voters have lost faith in the system, would not be resolved by "an investigation". That's completely nonsensical. The demand for "an investigation" is an obvious political ploy. A dishonest one, I might add.

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

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

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

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

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

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

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

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

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

Nope — jgill

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

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

The law of identity is a statement concerning the identity of particulars. Here's a quote from Stanford: "Numerical identity is our topic. As noted, it is at the centre of several philosophical debates, but to many seems in itself wholly unproblematic, for it is just that relation everything has to itself and nothing else – and what could be less problematic than that?"

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Again, we butt heads over specific vs general terminology. In human societies, governors (kings, congressmen, parliamentarians) make the laws, and the governed people obey the laws. So, if you observe a pattern of obedience to a law, wouldn't you infer that the obeyers were somehow compelled to conform? The observed pattern of behavior can be described in terms of specific actions, or in terms of a governing principle : a Law. — Gnomon

What we observe is a very clear difference between human beings obeying the laws of governance, and inanimate objects behaving in a regular way which is describable as laws of physics. The human beings have free will to disobey the laws when they desire to, and often do, at risk of punishment. The inanimate objects continue to act as the law describes, without exception. If there is any exception, we do not punish the things, we look for inaccuracies in the law. See the difference? In the first case, if there are exceptions, the human beings, not the laws, are wrong, and the humans ought to be punished and encouraged to act the right way. In the second case, if there are exceptions, the laws are wrong, not the behaviour of the things, and the laws need to be changed accordingly.

This difference is due to the difference we perceive between human beings and inanimate things. Human beings have free will, and can be trained, habitualized, to change their behaviour, and avoid breaking the law. Inanimate things, we assume, must act the way that they do, with no choice to do otherwise, therefore we must adapt our descriptive laws to match their behaviour, not vise versa.

The relevant distinction is between a specific pattern, and the general cause of that pattern. For example, if most cars wait patiently at a red light, is that a random coincidence, or would you infer that there is some governing Law that they are obeying? If you watch long enough, you may see a car that does not stop at a red light, and then is pulled-over by a law-enforcement officer.

Some scientists refer to Natural Laws as merely "habits". The implication is that the predictable regularities of natural behaviors is characteristic of individual actors, not of any general imperative imposed from above. Is this your position? That makes sense from a Reductive (part) viewpoint, but not from a Holistic (system) perspective. So again, our different understanding reflects a preference for looking at Isolated Parts or Whole Systems --- or for Bottom-up Inductive Reasoning or Top-down Deductive Logic. Both approaches are reasonable, but applicable to different contexts. No need to butt heads . . . just define terms and contexts. — Gnomon

So, the point I made, is that we cannot proceed logically from the observation that the behaviour of inanimate things can be described by laws, to the conclusion that these things are governed by laws, because of the difference I described above. Being governed by laws implies that the things governed can freely act otherwise. Being describable by laws of science implies that things cannot freely act otherwise. This is a fundamental difference and the incompatibility needs to be resolved. If, for example, we could demonstrate that inanimate things could freely act in ways which are other than the laws of science, and are in some way (fear of punishment or something like that) compelled to act that way, we'd have the premises required to conclude governance. But we don't.

Apparently, you haven't looked at the links. The connection between Holism and bottom-up creation is much too complex for a forum post. Instead, I have dozens of essays that look at different aspects of the question --- from the perspective of a top-down Whole, and a bottom-up Holon. You seem to think Top-Down and Bottom-Up are mutually exclusive. But I think it's a question of perspective, point-of-view, frame-of-reference. — Gnomon

I don't see that the concept of a holon solves the issue of bottom-up causation. The holon itself must be composed of parts, or else there is no way to account for its capacity for free acts. If it is composed of parts, it seems impossible to avoid infinite regress. If it is not composed of parts, then how does it get the capacity for freedom, being constrained by its environment and top-down causation? How could any sort of real causation be internal to it (bottom-up causation), rather than it just being forced by its environment?

Natural Laws place limits upon freedom, but Randomness is free to experiment with various solutions to the question of Survival. — Gnomon

This really doesn't makes sense. Randomness cannot experiment, all it can do is continue in a random fashion. You could assume an agent which experiments with randomness, but then you'd need to account for the existence of that agent. What is this agent, the soul? Where does it come from? How would you reconcile the concept of holon with the concept of soul? Is the soul a holon?

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

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

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

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

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

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

That is exactly what I said. — Wayfarer

Obviously not. You said:

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

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

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

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

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

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

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

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

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

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

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

Proceeding to Ch.6 of Bk.7 Metaphysics, he question whether a thing and its essence are the same "for each thing is thought to be not different from its substance, and the essence is said to be the substance of each thing". So the problem of accidentals is now brought up, and it appears like a thing cannot be the same as its essence. But in the case of supposed self-subsistent Ideas, Forms, it is shown to be impossible, as incoherent, that a Form's essence could be different from the Form itself. "Each thing itself, then, and its essence are one and the same in no merely accidental way, as is evident both from the preceding arguments and because to know each thing, at least, is just to know its essence, so that even by the exhibition of instances it becomes clear that both must be one." 1031b,18. "Clearly, then, each primary and self-subsistent thing is one and the same as its essence. The sophistical objections to this position, and the question whether Socrates and to be Socrates are the same thing, are obviously answered by the same solution; for there is no difference either in the standpoint from which the question would be asked, or in that from which one could answer it successfully." 1032a,5.

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

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

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

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

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

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

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

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

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

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

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

Not only that, future events have causal
power over my past, because my past as it participates in forming my present is reshaped by my anticipations. — Joshs

I would say that there is a problem here. It is not the case that your past is reshaped, only your memories of it are reshaped. This reshaping of your memories is not actually a recreating of your past, it's simply something new which comes along at that time of reshaping, a change of mind. And, when we look at our anticipations in a similar way, it's not really the future events which have causal power, it's the way that we are thinking about them at the present time, the present anticipation, which has causal power. All the causation is occurring at the present, and nothing in the past, or future, is affected at all.

You should have noticed, from what I've posted, that I'm not at all interested in the conventional interpretation of "falsifiability". I believe it tends to be way off the mark. So I really don't know why you would make this suggestion to me. If you're content to sink into the quicksand of that interpretation, then so be it. — Metaphysician Undercover

I'll go back to ignoring your posts. — Banno

Please, and thank you.

OK, "what is represented by these [so-called] laws"? Would you prefer to call them "accidental random patterns in Nature"? Einstein referred to them as "Reason", "order", "harmony", "structure", and "lawful", among other terms. :smile: — Gnomon

They are obviously not accidental or random. They are described by laws, so they are not random, but that does not imply that they are governed by laws. We are governed by laws, but we have freedom of choice to break the laws. The things which are described by the laws of science do not appear to have the freedom to break those laws. Therefore the actions of these things are inconsistent with being "governed by laws", as we know it. So it's clearly fallacious logic to proceed from the premise that natural things are describable by laws, to the conclusion that they are governed by laws.

Again, you may be thinking of "Holism" in the New Age sense. Scientists prefer to use the term "Systems" in order to avoid any theological implications. If you think of Evolution as an ongoing Program of world-creation, then the final output is unknown (undetermined), even though the Programmer specified the parameters by which the Solution will be judged. Initial Conditions & Natural Laws are parameters, but the system uses statistical Randomness to instill novelty into the otherwise deterministic system. My essay on Intelligent Evolution is an attempt to introduce the notion of bottom-up creation of an unfathomably huge Uni-verse (one whole) from a minuscule mathematical Singularity. — Gnomon

As far as I can tell, you haven't defined "holism" yet so as to make it consistent with bottom-up creation. You have here a vague analogy between a computer program and bottom-up creation, but no description of how any sort of holism fits into this scenario.

I say it does. I think you're splitting hairs for the sake of argument. — Wayfarer

Are you serious? As human beings, you and I are equal, based in a principle of equality. Clearly we are not the same. A judgement of equality is based in a principle of measurement, volume, weight, temperature, species, whatever. It allows that two distinct things are equal, by the precepts of the principle. They are the same volume, or the same weight, the same temperature, or the same species. Notice how in the concept of "equal", "the same" is qualified so that it is what is attributed to the thing, volume, weight, etc., which is said to be the same, not the thing itself. Under the law of identity, a thing is the same as itself. So it is impossible that two distinct things are the same thing, as we say that two distinct things are equal. By that law, we can only use "the same" to refer to one and the same thing, the very same thing. This is not a matter of splitting hairs, there's a fundamental difference between two distinct things which are the same in some way (equal by that principle), and one thing, of which no other thing can be said to be the same thing as.

My mind has blocked them out as traumatic experiences. — fishfry

On the basis of that statement I am concluding that proceeding with any discussion with you on this matter is pointless because your mind is liable to block out anything I write.

What's incoherent is you objecting to 4 = 4 as an instance of the law of identity — fishfry

You obviously do not know the law of identity. I had to spell it out for you already. You objected, and offered some axiom of equality which is obviously not the law of identity. This statement above, indicates that you clearly did not take the time to learn it yet. From what I learned last time, until we agree as to what the law of identity stipulates, further discussion on this issue is pointless.

In a given mathematical context, a given symbol holds the exact same meaning throughout. — fishfry

So the issue is, in the context of mathematics, does = mean equal to, or does it mean the same as? I'm sure you can grasp the fact that you and I are equal, as human beings, but we are not the same as each other. Therefore, I'm sure you can accept that equal to has a different meaning from the same as. Which does = symbolize in the context of mathematics?

Hard to believe there are two people who assert this nonsense, not just you alone. — fishfry

Reason is contagious, it tends to catch on. Notice that the op agrees with me as well. And, I think jgill agreed with me on this point in that other thread as well. I don't understand why the obvious appears as nonsense to you. It's very clear, that if 1 always referred to the same object we could not make 2 out of two instances of 1, we'd always have just one object symbolized.

By the way there is a standard formalism for obtaining multiple copies of the same object, you just Cartesian-product them with a distinct integer. So if you need two copies of the real line RR, you just take them as R×{1}R×{1} and R×{2}R×{2}. It's not that mathematicians haven't thought about this problem. It's that they have, and they have easily handled it. As usual you confuse mathematical ignorance with philosophical insight. — fishfry

It doesn't matter how you formalize it, the point is that it violates the law of identity. "Multiple copies of the same object" is exactly what is outlawed. You can rationalize your violation of any law however you like, but it doesn't change the fact that you violate the law. You can show me a thousand objects, and insist that according to your axioms they are all one and the same object. So what? All this indicates is that your axioms are faulty.

In any event, you avoided (as you always do when presented with a point you can't defend) my question. If set theorists are not only wrong but morally bad, is Euclid equally so? You stand by your claim that set theorists are morally bad? Those are your words. Defend or retract please. — fishfry

Did I say set theorists are morally bad? No, it was an analogy. The analogy was that if I saw set theorists doing something I thought was wrong (bad), I might be inclined toward explaining to them how I thought what they are doing is wrong, just like if I saw someone behaving in a greedy way which I thought was morally wrong, I might be inclined to explain to them why I thought what they were doing is morally wrong. The point being that it really doesn't make very much difference to me, in my life personally, if these people, either the set theorists, or the greedy immoral people, continue along their misguided pathways. Nevertheless, I might take it upon myself to make an attempt to point out to them how their pathways are misguided.

Particulars are real insofar as they're instantiations of the idea, which is their unchanging form; that is the sense in which the ideas 'lend being' to particulars, or particulars are said to 'participate in' the form. — Wayfarer

This is where Aristotle parts from Plato. In Plato's Timaeus particulars are supposed to be in some way derived from universal forms. But Plato is incapable of describing the mechanism by which a universal form could create the existence of a particular individual. He found the need to posit "matter" as the recipient of the form, in order to account for the particularities of the individual. The peculiarities of the individual are due to the matter. But when Aristotle developed this idea he discovered that matter itself cannot account for any of the properties of an object, and so each individual thing must have a unique form proper to itself. That's his hylomorphism

This was the fundamental question of his metaphysics, why is a thing the unique thing which it is, rather than something else. He said the commonly asked question of why there is something rather than nothing cannot be answered, and is therefore a fruitless question. So he suggested the proper question to ask of being qua being, is why is there what there is instead of something else. Why is each thing the unique and particular thing that it is, instead of something else. This led him to the conclusion that there is a unique and particular form which is responsible for each thing being the particular thing which it is. Hence the law of identity as formulated, each thing has a unique identity, it is the same as itself, and nothing else. For Aristotle, this is the reality of the particular.

The important thing to understand is that the whole
construct system functions integrally as a unified whole in the construing of events. This is important in understanding how Kelly treats affect. For Kelly the aim of construing is to anticipate what lies ahead. The construct system is wholly oriented around anticipation. It is not designed this way by some arbitrary inner mechanism or evolutionary adaptation. Anticipation is an a priori feature of subject -object interaction in time. — Joshs

I believe the role of anticipation is very important, yet not well understood. It is not well understood because it falls outside the possibility of observation and scientific understanding. If what is anticipated is a future event, and if the anticipation affects the way that one perceives what is presently happening, then we have to account for how a future event acts as a cause at the present. So you describe a system of construing which acts in a way such that future events have causal power over what is presently occurring, through the means of anticipation.

But, the point about universals is that they're universally applicable, isn't it? They're applicable in any context. Think about scientific laws, which I think must in some sense be descended from such ideas. Water doesn't sometimes flow uphill, for instance. Think also about Kant's deontological ethics, which individuals are obliged to conform to if their actions are to be ethically sound. — Wayfarer

I agree, but the universality of universals is exactly what makes them incompatible with identity which is what particulars have. This produces a hole, or gap in human understanding, because the material reality, to which we apply these universals in our attempts to understand, consists of unique particulars. This means that there is always a deficiency in human understanding. The point in enforcing a "law of identity", is to recognize and adhere to this understanding, that this gap exists, so that we do not push Platonism to its extremes, claiming that the physical universe is composed of mathematical objects. This is impossible, because mathematical objects are universals, but the universe is composed of unique particulars. The gap of incompatibility between these two demonstrates that such extreme Platonism, better known as Pythagorean Idealism, cannot be true.

Furthermore, I can't arbitarily designate the rules of math or the laws of logic, I have to conform to them, as much as I'm able (which in my case, is not very much). I can adapt them to my situation, I can use them to advantage, but I can't change them. (Again, the clearest exposition of these ideas are in the Cambridge Companion to Augustine, on the passage on Intelligible Objects.) — Wayfarer

This goes both ways. There are people who literally make up, or create axioms of mathematics, it's what fishfry calls pure mathematics. We must ensure that the mathematical axioms which we employ conform to reality or else they will lead us astray. Therefore it is actually necessary that we do change mathematical axioms as we try and test them. And if you look at the history of them you will see that they evolve, just like knowledge evolves, and living beings evolve. This implies that we must accept such things as evolving properties of living beings, rather than eternal immutable objects.

That the rules of math, and laws of logic appear to you as something which you cannot change, is the result of many years of usage by many different people. They are tried and tested so they are what we use. If, in your occupation there are rules which must be applied in order for you to fulfill your job, then you cannot change those rules or else your job would not get done. However, the world is full of innovative and creative people who could come up with new rules, which fulfill an end different from what you are doing, but is judged to be better than yours, and renders yours obsolete. Therefore this idea of "I can't change them" is just an illusion. Yes, if you want to keep doing what you are doing, you cannot change them, but if you quit what you are doing, and adopt other rules which are conducive to something else instead, which renders what you were doing before as obsolete, you really do change them.

Modern thought treats everything as a thing. (Who's paper is it, 'What is a thing'? Heidegger, I think.) Anyway, the point is, the modern mentality is so immersed in the sensory domain, that it can only reckon in terms of 'things'. Things are 'what exists' - which is what throws us off about mathematical concepts, they're not things, but they seem real, so 'what kind' of reality do they have? In our world, real things can only be 'out there', the only alternative being 'in the mind'. But in reality, 'out there' and 'the mind' are not ultimately separable - hence, as I say, the logic of objective idealism. But it takes a shift in perspective to see it. — Wayfarer

This I completely agree with, but again we have to be aware of where things go the other way. Philosophers looking toward the reality of Ideas describe them in terms of objects to facilitate understanding. However, we can see that thinking is an activity and it doesn't really exist as objects. On the other side of the coin, we see physicists who look at objects and use mathematical ideas to describe them in terms of activity. So we can see that the world of physical objects gets reduced to the activity of energy, because this is compatible with the realm of thought, mathematics. Now we have no adequate principles to separate objects from activities, and it's an ontological mess.

But now you say, "it's incorrect to call mathematical objects "objects" at all, because they do not fulfill the requirement of identity." When a while back you disagreed that 2 + 2 and 4 represent the same mathematical object (regarding which you are totally wrong but nevermind), that was one thing. But now you seem to be saying that 4 = 4 is not valid to you because mathematical objects don't fulfill the law of identity. Am I understanding you correctly? Do you agree that 4 = 4 and that both sides represent the same mathematical object? Or are you saying that since there aren't any mathematical objects, 4 = 4 does not represent anything at all? — fishfry

I guess you don't remember the key points (from my perspective) of or previous discussions. What I objected to was calling things like what is represented by 4, as "objects". I made this objection based on the law of identity, similar to the op here. You insisted it's not an "object" in that sense of the word, it's a "mathematical object". And I insisted that it ought not be called an object of any sort. So you proceeded with an unacceptable interpretation of the law of identity in an attempt to validate your claim. What I believe, is that "mathematical object" is an incoherent concept.

4 = 4 is true by the law of identity, yes or no? — fishfry

This depends on what = represents. Does it represent "is the same as", or does it represent "is equal to"? From our last discussion, you did not seem to respect a difference between the meaning of these two phrases. If you're still of the same mind, then there is no point in proceeding until we work out this little problem. This is why I say context of the symbol is important. When the law of identity is represented as a=a, = symbolizes "is the same as". But when we write 2+2=4, = symbolizes "is equal to". If we assume that a symbol always represents the very same thing in every instance of usage, we are sure to equivocate. Clearly, "is the same as" does not mean the same thing as "is equal to".

Do you at least accept that math can be regarded as a formal game without regard to meaning? — fishfry

No, of course not, that's clearly a false representation of what math is. That would be like saying that 2+2=4 could be considered to be valid regardless of what the symbols mean. That's nonsense, it's what the symbols mean which gives validity to math.

But, are you claiming that 4 means one thing to you today and other thing tomorrow? — fishfry

Yes, that is exactly the case it can even mean something different in the same sentence. When some says "I reserved a table for 4 at 4", each instance of 4 means something different to me. And, as I explained to you already, when someone says 2+2=4, each instance of 2 must refer to something different or else there would not be four, only two distinct instances of the very same two, and this would not make four.

So axioms don't qualify as "statements" in the article of the op? They get a special exemption? See why I reject this interpretation?