Here's something to consider. There is an age old line of thought which holds that the entire sensible universe must be recreated at each moment of passing time. I believe this idea is prominent in Hinduism, but it can be derived from the Platonic-Aristotelian tradition, and is consistent with the Thomistic notion of aeviternal, the type of temporality which angels have, being a sort of medium between God and the physical world. Angels account for the separate substance Forms, required as what maintains the continued existence of the sensible objects, the means of Providence. https://link.springer.com/content/pdf/bfm%3A978-94-010-2800-4%2F1.pdf

When we consider the reality of human choice, free will, we see the need to reject the necessity assigned to the continuity which comprises the substance of our empirical reality. This continuity is what is expressed in Newton's first law, the inertia of mass. It is theorized, based on empirical observation, that the inertia of mass will continue, unabated, without the action of an external force to interrupt. We can see though, that in reality this continuity is not necessary, and therefore requires a cause itself. So if Newton's law is inverted, we see that the temporal continuity of mass requires an internal cause.

Now, our common conceptions of space are woefully inadequate because we do not dimensionalize space in an internal/external format. All space is given equal status in mathematics (the same rules hold, big or small), and there is no way to properly relate an internal force to an external force. As a result, we can see that the "strong interaction" force of quarks for example, is independent of distance. So we might conclude that the force could act equally over an infinitely short, or an infinitely long, distance.

The problem which is very evident, if we account for the need to assume that spatial existence recreates itself at each passing moment in time, is that the established relationship between space and time, which forms the convention, is not at all indicative of reality. If spatial existence is recreated at each moment, then we need to assume something like a Big Bang occurring at each passing moment of time. But it's not one Big Bang, but a separate Big Bang at each point in space, and all these points must be related through some underlying Forms (the angels). And this is completely inconsistent with the continuity of massive existence which empirical observation gives us.

If a proposition by its very nature is a hinge, then it's not doubtable. — Sam26

The reason why the hinge is not doubted, is because it is unreasonable to doubt it. It is unreasonable because of what fooloso4 says, so much hinges on it, not because it has a certain relation to reality. I don't think Wittgenstein discussed "reality". What would you even mean by that, other than so much (what we apprehend as reality) hinges on it?

I'm thinking of working up an article on 'scientific idealism'. — Wayfarer

I perceive a deep divide between idealism and materialism which was propagated by Hegel. He laid the grounds for unabashed idealism to swallow up western science, while at the very same time the Marxist interpretation produced a strong materialism.

When you look at the world what do you see?

Is it concepts all the way down? — Harry Hindu

No, I'm dualist, I apprehend both, with a fundamental incompatibility between the objects which I see, and the concepts which I understand.

Do objects and their behaviors symbolize mathematical concepts or do mathematical concepts symbolize objects and their behaviors? — Harry Hindu

It goes both ways. Some scientists try model the behaviour of natural things using concepts, but artificial things are representations of concepts. Fundamentally, symbols always represent something mental.

We are basically pigs. — god must be atheist

Reminds me of Pink Floyd Animals
"And any fool knows a dog needs a home
A shelter from pigs on the wing."

So when you look at reality you see numbers and mathematical function symbols, not objects and their processes? F=ma refers to a state of affairs that isn't just more math. — Harry Hindu

When I look around, I do not see force, nor does "f=ma" refer to a state of affairs, it is a universal, which is a generalization. Force is a concept. And I do not think we can adequately differentiate between mathematical concepts and non-mathematical. Is "large" a mathematical concept? Are shapes mathematical?

You are close to what I think is a key unsettled issue for such exegesis: are language games incommensurable with each other? — Banno

If they are distinct and different language games then they are incommensurable because commensurability would produce one game. This is why equivocation is a fallacy. The logical relationship between a word's use in one game, and its use in another game, cannot be established.

However, the inclination is to assume that language, in general, is one game. But this assumption requires commensurability between the various games, to produce the one game of language. It's like the question of what does "3" refer to. Does it refer to three distinct and different objects, or does it refer to one object, the number 3? It depends on how you use it. But how could these two different ways be commensurable?

Not all language-games or all uses are correct. If I teach a child how to use the word pencil, and later the child points to a cat, and says, pencil, then their use of the word is incorrect, even if it's used in a particular language-game. — Sam26

I would dispute this "correctness" is determined relative to a language-game. There is nothing to indicate that one language-game would produce a more correct use of a word than another. So the game you play, when teaching your child the word "pencil" is just as correct as the other game which uses "cat" instead.

If that is the case, then the following is untrue as well. We can arbitrarily make up language-games, and derive meaning from those games.

However, this is not to say that all language-games have the same force, or that we can arbitrarily make up any language-game and derive meaning from it. The same is true of use, I can't arbitrarily use words the way I want without the loss of meaning. — Sam26

The radical skeptic (I'm referring to a specific kind of skepticism, not all skepticism) is not playing the game correctly. And, this must be viewed from outside our subjective view. It's viewed by looking at the community of language users, not one's personal interpretation. One's personal interpretation may or may not line up with the community, and this corresponds to the correct or incorrect interpretation. When I say correct and incorrect, I'm speaking generally, if it wasn't true generally, language would simply fall apart. — Sam26

This argument is untenable as well. There are no principles to determine what constitutes "playing the game correctly". It is a matter of your judgement, or my judgement, of whatever rules are apprehended as applicable. And this amounts to "one's personal interpretation". To step outside one's own personal interpretation, and get an objective view, or the view from "the community of language users" is impossible. So it really doesn't make any sense to assume such a thing as the correct or incorrect interpretation under these principles.

To make the judgement of correct interpretation, we would commonly refer to the intent of the speaker. But if dismiss this as a determining factor, and proceed toward a "game" system of modeling, there is no principle to determine the "correct" game, and its applicable rules.

So are you saying that the mathematical symbols don't refer to anything that isn't just more math? — Harry Hindu

That's what mathematicians claim, so I would think there is some truth to it. If there is anything more than math, being referred to, this is dependent on application.

What makes our universe more real than the others, or what makes us sure ours is the real one? — TiredThinker

That's the problem with describing reality in terms of possible worlds, we lose the premise whereby we distinguish what is actually the case, the real, from what is possible. There is no such thing as the real world in that description, because giving one world a special status would negate the premise which gives all possible worlds equal status as possible worlds.

This is similar to the is/ought gap. One might propose principles whereby we could designate what "is", but the two ways of mapping would remain fundamentally incompatible such that the two ways would not be on the same map.

Thanks. I don't agree with your rejection of platonic realism, however. As far as I know, Plato never placed dianoia - mathematical and discursive knowledge - at the top of the hierarchy of knowledge. It was higher than mere opinion, but didn't provide the same degree of certainty as noesis. — Wayfarer

What Plato placed at the top of the hierarchy is the study or knowledge of ideas and forms. But I do not think that "certainty" is the proper descriptive term for the higher levels of knowledge. Aristotle followed a similar division, and named the highest as intuition. You can see that there might be something amiss with describing this as certainty. The higher forms of knowledge lead us to higher levels of correctness, or good in Plato's world, but this is not really based in certainty.

Modern epistemology places too much emphasis on certainty, but certainty is just a form of certitude, which is an attitude. And this attitude is more properly associated with the lower levels of knowledge like opinion. We become certain of our opinions, but the true knowledge seeker always keeps an open mind.

Have you heard of Sabine Hossenfelder's book Lost in Math? She too agrees that mathematicism in physics, if we can call it that, is leading physics drastically astray, but that has nothing really to do with Platonism, as such. It is the consequence of speculative mathematics extended beyond the possibility of empirical validation. — Wayfarer

I haven't heard of that book, maybe I'll check it out when I get a chance. I believe the problem referred to is related to Platonism, because a misunderstood Platonic perspective is what validates the separation of logic (such as speculative mathematics) from empirical validation. When Forms are allowed completely separate existence, then a coherent logical structure need not be grounded in empirical fact. So we might construct an elaborate and very eloquent logical structure, which is even very useful for the purpose (good) that it serves, without having any real substance. That in itself is not bad, but amateur philosophers, and many common people will come to believe that it is saying something real and true about the universe, when in reality the whole structure is just designed to make predictions based on statistics, or something like that, and it's not saying anything real or true about the universe.

The aspect of platonism I focus on is the simple argument that 'number is real but incorporeal' and that recognising this shows the deficiencies of materialism, and also something fundamental about the nature of reason. How to think about the question is also important. I think there's huge confusion about the notion of platonic 'entities' and 'objects' and the nature of their existence. Most of that confusion comes from reification, which is treating numbers as actual objects when they're not 'objects' at all except for metaphorically. — Wayfarer

Recognizing the reality of the incorporeal is a very important step. The way I see it is that if anyone has any formal training in the discipline of philosophy, this recognition cannot be avoided. There are many self-professed philosophers who will not make the effort to properly train themselves, and will simply deny the reality of the incorporeal. For whatever reason, I don't know, they tend to deny the reality of what they have not educated themselves about. Perhaps it is simplicity sake, monism provides a nice simple approach to reality, and whatever aspects of reality it cannot explain, they can be ignored as unimportant to those materialistically minded people. But unless a person is ready to take on the task of informing oneself about the immaterial, having a personal reason to do so, some sort of interest, it seems unlikely that the deficiencies of materialism will ever become evident to such a person. It's like morality, unless a person has the attitude, the desire to be morally responsible, the person will never see the benefits of morality.

My response is to acknowledge that this timeline is empirically true, and that I concur with the evidence in respect of the timeline of human evolution. But I also point out - and this is the crucial point - that 'before' is itself a human construct. The mind furnishes the sequential order within which 'before' and 'since' exist. In itself the Universe has no sense of 'before' or 'since' or anything of the kind. — Wayfarer

This is well said, and I will extend this to point out that the whole concept of "the Universe" is just a human construct as well. So it makes no sense to argue from the premise that "the Universe existed before humanity evolved to see it". This is because, as "the Universe", is simply how we see things (as per Kant, phenomena). Therefore it assumes that our conception of "the Universe" correctly represents what existed before humanity, and this requires that temporal extrapolation which is doubtful.

This proposition, that "the Universe existed before humanity evolved to see it" is only justified if our conception of "the Universe" truly correlates with what existed before humanity, otherwise it's similar to the often quoted expression "have you stopped beating your wife", where you start by assuming something unjustified, likely a falsity, and say something about it. It's nonsensical.

Therefore, as philosophical skeptics, we ought to call into question, all the mathematical constructs, and the premises employed by the theories of physics, to determine their justification to assess our conception of "the Universe". And this is why platonic realism needs to be rejected. Platonic realism leads to the idea that mathematics provides us with eternal unchanging truths, and this supports the idea that "the Universe", as we conceive of it is a true conception, based in the eternal truths of mathematics. Then when the relationship between what is theorized about "the Universe" through mathematics, and what is actually observed empirically, becomes completely disjointed, (as in quantum mechanics wave/particle duality), platonic realism pushes us into this notion that mathematics (which is really a human construct), is the underlying fabric of the Universe which existed before humanity evolved to see it. This is because the underlying fabric or "substance" of the new conception of "the Universe" is no longer consistent with empirical observation. Therefore if "the Universe" is to represent something real, the mathematics must be real, because that's all that's there, mathematics with nothing empirically observable.

For this reason, it was refreshing to hear from CERN this week, that they may grudgingly have to admit that another previously unknown force may exist in nature. This may fit in with the long term concerns about our inability to detect something that should be everywhere - and in profusion - Dark Energy. — Gary Enfield

So, what's this new information?

The video which Tim suggested, does present such a distortion to preserve C by arguing, without evidence, that space is expanding - what more do I need to say? There is no proof that space is expanding. — Gary Enfield

The idea of spatial expansion is just an escape. When objects are observed to be moving faster than the speed of light, it is proposed that the substance which they exist in, is actually changing, so this doesn't qualify as "motion" in accepted usage. But what this does is introduce the concept of a changing or evolving substratum. And if the substratum, within which objects exist, is changing in this way, which is not accounted for in our normal modeling of motions, then this conventional modeling of motions is invalid. So what this assumption (spatial expansion) does is invalidate conventional models of motions.

If so, here's an exposition discrediting it - and if not, we can continue quarreling incessantly.

v=dsdtv=dsdt doesn't suffice herein - since it doesn't attain the velocity of a body on the fabric it's ensconced in, if the fabric migrates too. — Aryamoy Mitra

Yes, this is the problematic issue, the ideal that "the fabric migrates". Conventional modeling of motions does not account for the migration of the fabric. If it is true that the fabric is migrating then it is also true that conventional modeling of motions is incorrect, because the part of the motion which ought to be attributed to the migration of the fabric is unknown, and not accounted for.

I think you provide a very good argument. "Life" as we use the word, is defined by what we find here on earth. I've heard it said before that terrestrial life is carbon based, and there is speculation of the possibility of non-carbon life. But I don't think that this would qualify as "life" as we know life, and use the term.

The conclusion I think should be that the word "life" has a specific usage by us, to refer to certain forms of existence on this planet. And, if we hypothesize realistically about forms of existence in other parts of the universe, and desire to call them "life", then there must be something to indicate that such forms would be consistent in their physical constitution with the forms of life on earth, and this would indicate some sort of continuity in the form of a relation between here and there to account for that consistency. This is what we find here on earth, consistency and continuity between all life forms. When we find a form of existence, like a rock, which does not bear that continuity we do not call it "life". This principle ought to hold for discovery in other parts of the universe. If there is no continuity between the forms of existence on earth which we call "life", and the forms of existence discovered far away, there is no reason to call them "life", they need a different name.

So for example. when we speculate about physical existence in other parts of the universe, we establish a relationship between there and here through laws of physics, and we assume certain continuities to exist between there and here, such as electromagnetic activity, and fundamental atoms. Without this continuity of principles, forming a relationship between here and there, such speculation would be completely random and useless. Likewise, if we are to speculate about a specific type of existence which we find here on earth, as existing elsewhere in the universe, "life", it is completely useless and nonsensical to make such speculations without the assumption of some sort of relationship to establish a continuity between what is her and what is there, or else we are not really talking about "life" out there.

So you opted to suggest that I'm lying about the whole thing instead of just asking "Would you please provide some links?" — TonesInDeepFreeze

I still think you're lying. I don't believe there is any such thing as proof that "2+1" denotes the same object as "3" does. I think it's false, and I think you know it's false. But you're in denial, and you've come up with this proposition that the "method of models" provides a proof, as a ruse.

I seem to have this uncanny ability to sing in my head too. — Olivier5

If you had the ability to get a song out of your head I might be inclined to call that uncanny.

You tend to think irrationally or not at all. — TonesInDeepFreeze

I can't help it if your terminology is a little off the beaten path. You kept referring to a "method of models", and I couldn't even find that on google. Now I see you were really talking about model theory.

I took a look at your first reference. The book is directed at graduate students in mathematics, but it distinctly says in the preface that fundamental philosophical problems are not dealt with.

"Philosophical and foundational problems of mathematics are not systematically discussed within the constraints of this book, but are to some extent considered when appropriate."

I took at look at the second reference, and it does discuss "model theory", but I don't see how anything there can be used to prove that "2+1" denotes the same object as "3". The fact that mathematicians utilize that assumption does not prove that it is true.

I took a look at the third reference, and it tells me that in model theory the truth or falsity of a statement is understood to be dependent on the interpretation.

So, it appears to me, like you and I are both correct according to model theory. I interpret the statement "2+1" denotes the same thing as "3" as false, and you interpret it as true, and neither of us is wrong. We each interpret "2+1" differently and so, 'that "2+1" denotes the same thing as "3"' is false for me and true for you. Therefore we ought not even talk about whether it's true or false, because that's not something which could ever be determined. Is this conclusion correct? If so, then it clearly does not prove that "2+1" denotes the same object as "3".

I took a look at the fourth reference and it doesn't seem to be relevant.

I took a look at the fifth, and it just talks about structures as if they are objects, so it seems like this article simply assumes what you need to prove. By the way, most these articles you refer seem to have that problem. Your task is to prove that "2+1" refers to the same object which "3" refers to, not to show me instances where this is taken for granted. I already know, from your behaviour and the behaviour of others, that this is taken for granted. There is no need to prove that now.

And so I find the same problem with the sixth reference. It states right of the bat: "In this
course we develop mathematical logic using elementary set theory as given..."

What sort of proof is this, which takes what you are tasked with proving as a given? I think you are simply continuing with your fallacy of begging the question.

So reconsider what you said in the op now:

Silently hum a note in your mind. Now duplicate that same frequency aloud. I think some readings of the Private Language argument would say this activity is nonsense because there's no way to tell if the note you hum is the same as the note in your mind.. — frank

Let's start with the assumption that "there's no way to tell if the note you hum is the same as the note in your mind". Ask yourself is it necessary that the note you hum be the very same (numerical identity) as the note in your mind, in order for your humming to be significant, have meaning, or be a sensible activity. If the answer is no, as it clearly is, then the so-called private language argument has no bearing.

I find nothing about the "method of models" in my google search so I tend to think it is something you made up as a ruse, citing all these prerequisite subjects for understanding.

Under "Scientific Modelling" in Wikipedia I find this:

"Scientific modelling is a scientific activity, the aim of which is to make a particular part or feature of the world easier to understand, define, quantify, visualize, or simulate by referencing it to existing and usually commonly accepted knowledge."

Notice the explicit statement of "...to make a particular part or feature of the world easier to understand..". That's exactly what I said about the term "equal", it refers to a designated part, aspect, feature, or property of an object. Two distinct objects are said to be equal on the basis of modelling a part. The issue however, is how do you proceed from modelling a part, and concluding equality based on a model of that part, to making a conclusion about the whole?

I meant the same frequency. — frank

The problem is that it's not really "the same". Tones of the same frequency from two different instruments are not the same tone, for example, there are overtones and all sorts of other interference patterns. If you have a sound in your mind, from an instrument, and you try to match the pitch of that sound with your mouth, you are selecting a part of the sound, (the principal pitch) and trying to replicate it. If your goal is to produce that pitch you are not necessarily trying to replicate the sound exactly.

I don't see that. I think it attacks the British empiricist psychology of ideas and impressions: the narrative of a private construction of mind from sense-data. His argument seems to be that identity and similarity of the internals has no basis when asserted in private. I don't see any conflating of numerical identity with similarity. — bongo fury

The premise of the argument, is that one would take a sensation signify it as "S", and every time the same sensation occurs it is noted as "S". The problem is that reoccurring sensations are not the same sensation, each time it occurs, it is a new instance of a similar sensation, therefore it is distinct and in some sense different. To create the private language argument it is necessary to assume that the symbol "S" is supposed to denote the very same object each time it is employed. Then the issue is on what criteria is the use of the symbol validated. How does the user know that it is the very same object? But when we recognize that the symbol "S" is simply employed to signified a similar sensation, not the very same thing, and the criteria is completely subjective, and this is consistent with language use in general, then the so-called private language argument cannot be constructed.

So the issue pointed to by Wittgenstein is the judgement of similarity, it is not the issue of judgement of identity or sameness. He creates ambiguity by using the word "same" to refer to similar things, in the common way of usage known as qualitative identity, and allows the reader to create a private language argument through the assumption that "same" is being used in the sense of numerical identity. The latter would be a faulty interpretation. Then the question which follows ought to be, on what criteria do we judge similarity. If the judgement is base on private principles and there is no requirement for public input, then a private language is possible.

How do you know the early and later internals are the same? — frank

The resolution to this is issue is to realize that they are not ever "the same", but they are the same type. Unless one has absolute pitch it's just a tone (same type), and even with absolute pitch the two tones are not the same tone, but still the same type with a more precise definition or criteria for that type. The private language argument misleads us into thinking that we must recognize two things as being the same thing in order for such a recognition to be useful. But this is not the case, because we only need to recognize similarities, and hence types.

Thanks for the example --- not that I understand it.

Maybe the current crises in cosmology and physics vindicate Plato's original contention that matter itself is unintelligible. — Wayfarer

That's the way Aristotle designed his system of logic, from the premise that matter is unintelligible. Part of the physical reality is intelligible, form, and part is unintelligible, matter. It was evident that there are aspects of reality which cannot be understood because they appear to defy the three fundamental laws of logic. What Aristotle did was insist that we uphold the law of identity, and insist that we uphold the law of non-contradiction, but for that aspect of reality which appears unintelligible he allowed that the law of excluded middle to be violated under certain circumstance. So for example, in the case of future occurrences which have not yet been determined (the sea battle tomorrow for example), propositions concerning them are neither true nor false. And even after the event occurs, if it does, it is deemed incorrect to think that the proposition stating that it would occur was true prior to it occurring. In his Physics and Metaphysics, "matter" is assigned to this position of accounting for the real ontological existence of potential, that which may or may not be.

Any property? They're called bricks. Can you think of any reason why? And if your and my sevens are not the same, then I have some ones and fives I'll trade for your tens and twenties. — tim wood

I don't see your point. Your ones and fives, and my tens and twenties are physical objects like bricks. And each one of your ones is different from every other one of your ones, just like each brick is a different brick, despite the fact that they may all look the same to you. So why would you assume that there is some type of a one, which is the very same one as every other one, despite occurring in distinct situations?

Great, and where do those come from? Mind, now, nothing human here. — tim wood

They are proposed as Divine, therefore not from human minds.

And see if you can find one, any one, off by itself where no mind is to have it. — tim wood

If you read what I said, you'd understand it as saying that the separate Ideals, which are the property of a Divine Mind, are not found by human minds. Human minds are lacking in the perfection required for such Ideals.

And you're the guy who goes to the building supply store to purchase bricks. You're handed two bricks, one in each hand. You look at the one in your left hand and say, "That is one great brick!" And you look at the one in your right hand and say, "What the hell is that?!" There may be strange things in your philosophy - clearly there are - but nothing stranger than your philosophy. You can buy a brick, but not bricks. And I'm thinking that's a problem Plato would not have had. — tim wood

Clearly, despite the fact that I have two things both of which I call a brick, and I have a similar brick in my right hand to the one in my left hand, they are not both the same thing. The fact that the two things, called bricks, each have a different identity, is what the law of identity is meant to express. Do you agree with this?

You need to read a book or other systematic presentation of mathematical logic in which the method of models is explained step by step, including the notions: concatenation functions, formal languages, signatures for formal languages, unique readability of terms and formulas, recursive definitions, mathematical induction, et. al. And prerequisite would be understanding basic mathematical notions, including: sets, tuples, relations, functions, et. al." — TonesInDeepFreeze

Come on TIDF, it must be a simple proof, if it exists, just like in the Fonda example, we look at the person denoted by "father of Peter", and also the person denoted by "Henry Fonda", and see that they are the same person. Why does this proof require concatenation functions, formal languages, signatures for formal languages, unique readability of terms and formulas, recursive definitions, mathematical induction, sets, tuples, relations, functions, et. al?

We both have an idea of seven. I buy the notion that our several sevens are identical - and must be. — tim wood

I don't buy this, because you and I are different, just like two bricks are different. If two bricks are different why would you think that any one property that one brick has would be identical to the property of another brick. And if ideas are properties of human beings, why would you think that an idea which I have would be identical to your idea?

Do you know what identity is, according to the law of identity? It means the very same as, one and the same. Identity is proper to the thing itself, and it is shared with nothing else, because if something else had the same identity it would not be something else, but the same thing. So I don't buy the idea that the very same idea could be in your mind, and my mind. I think the evidence indicates that this is not true.

Idealism might propose independent "Ideas", as Ideals, which are independent from any human ideas. These Ideals are supposed to be the true immaterial objects. This is what Plato describes in The Republic. There is the divine Idea of a bed, the perfect bed. The carpenter attempts to replicate this Ideal with one's own idea of a bed, then builds a replication of that idea. Notice the two layers of representation between the artificial material object and the independent "Idea", as an object. The human idea does not obtain to the level of "object" with independent existence, because it is only a representation of that supposed independent, divine Ideal.

The whole point of what was to become form-matter dualism, is that the forms of things can be identical, or rather, particular things can ‘participate’ in a form. — Wayfarer

The theory of participation comes from Pythagorean Idealism. It can be argued that Plato actually refuted this theory. Through his analysis of this type of Idealism, he actually exposed its weaknesses. Although Aristotle is given credit for the actual refutation, he simply synthesized, in a more formal argument, the information provided by Plato's analysis.

A very good example of the theory of participation is fount in The Symposium. Beautiful things obtain their beauty by partaking in the Idea of beauty. What is evident here is that there is an independent Idea, also sensible things which partake in that Idea, and then human ideas which are produced from observation of the sensible things. Notice how the sensible world is a medium between the independent Ideas, and human ideas, because it is the sensible things which partake of the independent Idea, not the human ideas themselves. The human ideas are derived from the sensible things

Plato wanted to understand how the sensible particulars partake in the separate Ideas. The issue is that the Ideas must be prior to the sensible particulars in order to account for numerous particulars being part of the same Idea. This means the Ideas must be in some sense a cause. But from the human perspective, we get our ideas from the sensible things, so we see them as the cause of ideas. So from our perspective we see sensible things as active, and the ancient view was that sensible things actively participated in the separate Ideas. This makes the separate Ideas appear as passive (eternal, unchanging), and denies them causal capacity.

The key to turning this around is revealed in The Republic, as "the good". The good is the motivation for action, as the ground for intent. When we assign causal capacity to intent then we see the reality of human actions, and the fact that the sensible objects follow from the human ideas, necessitating that the idea of the artificial thing is prior to its sensible existence. Both Plato and Aristotle assign this order to natural things as well, making the independent or separate "Forms" the cause of natural sensible objects.

In The Republic, Plato removes the sensible object as necessarily the medium between the human idea and the divine Ideas. This is contrary to Kant, who makes all human ideas dependent on sensation. But for Plato, it appears like the good, in the sense of what is morally proper, cannot be derived from sensible existence, it is only apprehended by the mind. Therefore the human mind must have the capacity to be guided directly toward the divine Ideas, without the intervention of sensation.

Now wait just a minute. Isn’t the idea, in form-matter dualism, that ‘the mind perceives the Form, and the eye the Shape?’ Go back to the original metaphor of hylomorphism - a wax seal. The wax is the matter - it could be any wax, or another kind of matter, provided it can receive an impression. The seal itself is the form - when you look at the seal, you can tell whose seal it is (that being the purpose of a seal). That is the original metaphor for hylomorphism. — Wayfarer

I don't see the point in distinguishing shape from form. The shape is a part of the form, the part perceived by the eyes. But the eye cannot interpret the meaning in the shape.

Remember, in Aristotelean hylomorphism there is two distinct senses of "form". There is the form of the object which inheres within the object itself, combined with its matter, constituting its identity as the thing which it is, and there is the form which the human mind abstracts. These two are not the same, as the abstraction does not contain the accidents.

Ok, what do you mean by object? I assume you do not mean like screws or brick at the hardware store. — tim wood

Of course screws and bricks are objects. Why not? I take for the defining of "object", individuality, particularity, and this is described as a unique identity by the law of identity. So the conditions for being an object is to be a unique individual, and this means having an identity proper to itself.

This strange from you. Because what true means in this sense is not-true, and I'd have thought you'd be all over that. — tim wood

That's how my dictionary defines "true" and it seems to be how it is most commonly used. If you want to propose something different, I could look at that and we might hash it out, but I think you'd have a hard time changing my mind after I've spent so many years studying this.

The only other option I see as viable is to define "true" in relation to honest. Is that what you would prefer "truth" is a form of honesty or authenticity?

Not only is it not necessary, it is impossible, and it is irrelevant.

Admittedly very informally axioms are by default thought of as true, but we're looking more closely, or, I'm looking more closely because I think up above somewhere you got confused when you claimed that, — tim wood

How is it, that determining the truth or falsity of a mathematical axiom is not necessary, it is impossible, and irrelevant, yet axioms are "by default thought of as true". There is no honesty here. This is clearly self-deception, to think of a proposition as "true", when truth or falsity plays no role in its formulation.

From online, the axion of extensionality:
"To understand this axiom, note that the clause in parentheses in the symbolic statement above simply states that A and B have precisely the same members. Thus, what the axiom is really saying is that two sets are equal if and only if they have precisely the same members. The essence of this is: A set is determined uniquely by its members."

What about this requires the treatment of anything as an object ("object" awaiting you definition), and what does it have to do with Platonism and why is Platonism "required"? — tim wood

The word "members" signifies distinct and unique individuals, "objects" as per my definition. Since numbers are commonly said to be the members of sets, then numbers are objects with identity. Notice that the identity of a set is dependent on the assumption that a number, as a member, is an object with an identity.

The reason why Platonism is required is that this is the ontology which supports the assumption that numbers are objects, by designating this as true, i.e. in correspondence with reality. Here's a brief explanation. Let's assume we use the symbol "2" to refer to a group of two things, as the quantity of things there. Do you agree that this is a true description of how one would use the symbol?

In this case, "two" is what is said about the group of things, it is a predication, and the subject is the group. The group is a quantity of two. Here, it is impossible that "two" refers to an object, because it necessarily refers to a group of two objects. However, if we employ a Platonist premise, we can assume that this Idea, the quantity of two, is itself an object being referred to by the numeral "2", independently of any group of two things. Then we might use the symbol "2" to refer to this object, the quantity of two, independently of any existing groups of two. So when 2 is the member of a set, that is what the symbol "2" represents, an object, the number 2, which is independent of any group of two.

That's why Platonism is required for set theory because it provides the premise whereby the number 2, or any other number, exists independently of any quantity of things. By this ontology it is true that the symbol"2" refers to an object, the number 2. Without this premise, when "2" is used it would necessarily refer to two objects, not one object.

The point being that the world of ideas is different from the world of worldly objects. And that failing to keep the distinction in mind leads some minds astray. But let's see what he says. — tim wood

That's exactly the point I was arguing when you interjected. Altheist was offering a definition of "object" from semeiotics which would dissolve the distinction between ideas and physical things, making them both "objects" as what is denoted by a symbol, under that proposed definition.

Worth noting here - this is something I’m saying, I don’t know if the poster you asked will agree - that a number or geometric form is a noumenal object, that being an object of ‘nous’, mind or intellect.

So it’s not an object of sense, which is what is presumably implied by many of the question about what ‘object’ means in this context. It’s not a phenomenal or corporeal object, like a hammer, nail, star, or tree. You could even argue that the word ‘object’ is a bit misleading in this context, but if it’s understood in the above sense - as something like ‘the object of an enquiry’ or ‘the object of the debate’ - then it is quite intelligible nonetheless. — Wayfarer

I don't mind using the same word "object" to refer to a sensible object, and also an intelligible object, as an approach to these categories, so long as we maintain the separation between what it means to be an intelligible object and what it means to be sensible object. What I objected to was altheist's proposed definition of "object" which would dissolve this distinction, making sensible objects and intelligible objects all the same type of "object" under one definition of "object".

However, I find that when I employ adherence to the law of identity as the defining feature of an object, then it's difficult to maintain the status of intelligible objects as true "objects" under this principle. This presents the difference between the phenomenal and the noumenal. The human intellect apprehends the phenomenal, but we assume a perfection, or Ideal, which is beyond the grasp of the human intellect, like God is. This is where we derive the idea of the individual unity, and why it is impossible for the human intellect to grasp the completeness, or perfection, of the unique individual. And "object" is generally used to refer to a unique individual.

Yes, with the qualification that 'idea' in this context has determinate meaning, i.e. a real number or mathematical proof is an idea. Not simply an idea in the general sense of mental activity 'hey I've got an idea, let's go to the pub.' (Not that it's a bad idea.) — Wayfarer

Here's a problem to think about. At what point does an idea manifest as a full fledged "intelligible object"? What would be the criteria to distinguish a simple idea in the general sense, from an "Idea" or "Form" in the sense of a mathematical object?

Let's say there are two extremes, the bad idea and the good idea, with countless cases in between. The good ideas, like mathematical objects get designated as "Ideas", or "Forms", Platonic objects of eternal truth. The bad ideas are human mistakes. But what about all the things in between which are not so easy to judge? What about a human idea which gets accepted and becomes an object, like Euclid's postulates? Or on the other hand a proposed mathematical axiom which gets rejected as insufficient? Doesn't the distinction between a Platonic object, as eternal truth, and a human idea which may be mistaken, seem somewhat arbitrary?

To say that 2+1 and 3 are equal is saying that 2+1 is 3.

To say that John and Mary are equal (in the sense of equal rights) is not saying that John is Mary. Rather it is saying that the rights of John are the same as the rights of Mary.

These are very different uses of the word 'equal'. — TonesInDeepFreeze

As I explained, "equal" in both of these uses is based in a value system. If you truly believe that having an equal numerical or quantitative value justifies the assertion that the two things referred to, with the same value, are in fact the same thing, then you ought to be able to demonstrate to me your reasons for believing this. Suppose that I have two apples and you have two apples, are our apples the same, because they are equal quantity. Surely this is not the same as one of us having four apples despite the fact that there are four apples in that scenario. Are two objects which each weigh five kilograms the same object? I just don't understand where you get this idea that having an equal numerical value means being the same thing.

What do you intend the pronoun 'they' refer to there? — TonesInDeepFreeze

"They" refers to what "2+1", and "3", refer to it. I use "they" because it is plural, "2+1" refers to something, and "3" refers to something, hence there are two things referred to, and the plural "they".

(1) So in the empirical context, your objection was refuted. — TonesInDeepFreeze

This is incorrect, because there is no empirical object referred to by "2+1", or "3". So your act of introducing the empirical aspect of the Fonda example only makes the example irrelevant. To maintain relevance we must proceed, as I did, through logic only. Then to argue that the two phrases refer to the same object requires a question begging premise. This would make your argument invalid through that fallacy.

(2) In the mathematical context, numbers are not physical objects. And over the course of this discussion I said that we arrive at mathematical conclusions by mathematical proof or by performing mathematical procedures. You are not caught up in the discussion because you ignore and skip. — TonesInDeepFreeze

Right, there are proofs. Now I'm waiting for proof that "2+1" refers to the same object as "3". So far you've offered me only a false premise that if they refer to equal things, then they refer to the same thing. And you admit that you cannot back this up with any logical necessity, so it appears to me like you really recognize it as false. Of course you do, any rational human being of grade school education would recognize the falsity of that. Why argue so persistently that it's true?

You mean like screws at the hardware store, or bricks? What do you mean when you say, "treat numbers as objects"? — tim wood

I mean to assume that a number is an object.

Why? What does this even mean? — tim wood

Do you know what "true" in the sense of correspondence means? It means to correspond with reality. So take the law of identity for example, it states that a thing is identical with itself,. And this corresponds with the reality of things, as we know them. A thing cannot be different from itself. And from this we also derive the law of non-contradiction. If a thing were other than itself, then the required description of it would be contradictory, because it would correspond to a specific description, and not correspond with that description, at the same time. We see that these principles correspond with reality, i.e. that they are true. Do you agree with me on this?

However, we can state principles, laws, or axioms which are not true. And it is not required that they be true, i.e. correspond with reality, in order for them to be useful. So we can state useful axioms which are not true.

And it has to be said, from what you write, you apparently do not know what an axiom is. Nope. You apparently have no idea what an axiom is. Google "axiom." — tim wood

There are two common senses of "axiom", the philosophical sense, and the mathematical sense. In philosophy an axiom is taken to be a self-evident truth, like the law of identity and the law of non-contradiction. In mathematics, an axiom is a starting point for a logical system, like a premise, but it is not necessary that the truth or falsity of the axiom (whether it corresponds with reality) be evident. I've taken this from the Wikipedia entry on "Axiom".

And you claimed that you asked me a question I didn't answer. — TonesInDeepFreeze

If I did not ask it in the exact way that I repeated it, I apologize for the unclarity. But, here:

If you can show that equality is something other than a human judgement, then you might have a case. Otherwise the charge holds. — Metaphysician Undercover

Formal languages, including the language of identity theory, are more precise than natural languages. But the point I made was not so much about precision but that 'equality of human beings' in the sense of equal rights or whatever is a very different meaning of 'equality' in mathematics. — TonesInDeepFreeze

These two senses utilize the same principle. They establish a value system and judge equality according to that value system. What differs is the value systems employed. In law they have legal values, rights, and in mathematics they have numerical or quantitative values. Since each refers to the specific aspect which it is designed for, neither provides what is sufficient for a judgement of "the same". Quantitative value is a single predication, therefore it does not suffice for a judgement of "same" which requires taking account of all attributes.

True, but I don't mean it.
... Nope. I am not bringing the notion of logical necessity into play. — TonesInDeepFreeze

OK, so lets dismiss the notion of logical necessity. Let's assume that you say "if they are equal, then they are the same", and now you admit that you do not mean that they are "necessarily" the same, by any logical necessity. What good is such a principle? You apprehend things as equal, and you judge them as the same. But now you say that they are not necessarily the same.. So you are admitting now that your judgement of "the same" might in some cases be wrong.

Is this what you are arguing? You judge "2+1" as referring to the same thing as "3", because they are equal, but there is no logical necessity there, which proves that they are? If this is the case, then how do you know that they are the same? Don't you think that you might be mistaken just as often, or even more often then being correct in that judgement?

One can stipulate premises and then infer conclusions. That is not question begging. Also, we don't have to stipulate that Henry Fonda is the father of Peter Fonda, since we can arrive at that claim by empirical or historical evidence. — TonesInDeepFreeze

Sure, in that case we can refer to empirical judgement, but in the case of numbers we cannot, because we cannot sense numbers in any way. So your judgement that two equal things are the same thing is supported by no logical necessity, and no empirical evidence. Don't you think that this is a little flimsy?

We prove that 2+1 = 3. Then we prove that '2+1' and '3' have the same denotation by the method of models. I've told you that about a half dozen times now. — TonesInDeepFreeze

I don't recall such a demonstration. Can you show me through your "method of models", how you prove that "2+1" and "3" have the same denotation? Then I can judge the soundness of that proof.

On the contrary, within semeiotic the definitions of terms including "object" and "subject" are unambiguous and foster greater understanding. — aletheist

I made my decision through an assessment of the results of the semeiotic (Peircian) definition of "object". You have "objects" which violate the law of excluded middle, "objects" which violate the law of non-contradiction. And, vagueness, which ought to be represented as a feature of human deficiencies, inadequate principles, and inadequate application of principles, is seen as an objective part of the universe.

This is generally the most significant negative feature of Platonic realism. When mathematical principles, and other human creations like inductive generalizations, are apprehended as objective, unchanging aspects of the universe, we have no approach toward deficiencies, falsity or other defects within these principles. When mathematical principles are apprehended as the result of human activities then we view them as fallible.

Eh? What does that even mean? That the axioms "require" something to be axioms? Or that as axioms they mandate something? I'm not finding sense here. — tim wood

To be "true", an axiom must correspond with reality. We can make all sorts of useful axioms which do not correspond with reality. Usefulness does not entail truth because it is determined in relation to its purpose, as means to end. The reality of deception demonstrates very conclusively that usefulness does not entail truth. So usefulness, and pragmaticism in general, must be subservient to truthfulness, in a respectable metaphysics. This means that pragmatic principles cannot take top position in the hierarchy of decision making, because usefulness is determined relative to the end, so it does not necessarily provide us with truth. The further point which Plato himself indicated, is that the end is "the good", and Aristotle outlined the need to distinguish between "apparent good" and "real good".

What axioms, what objects? Just a simple example ought to suffice to demonstrate the necessity of Platonism. — tim wood

When an axiom, such as the axiom of extensionality, treats numbers as objects, then if this is true, the axiom will provide us with sound conclusions. If it is not true then the axiom will provide us with unsound conclusions. To be true, such axioms require that the ontology of Platonic realism is a true ontology.

Anyway, there are infinitely many previously unwritten mathematical equalities that humans just happened not to have made judgements on yet. — TonesInDeepFreeze

That itself is a judgement, that these unwritten equalities are equalities. Clearly equality remains a human judgement. See "equal" is a human concept. To say that there are equalities which humans haven't discovered, is to already judge them as equalities.

The case is that you can't read. I replied about the notion of human equality many many posts ago, and you skipped recognizing my reply, and I even mentioned a little while ago again that I had made that reply and you skipped that reminder too! — TonesInDeepFreeze

And I will continue to skip it because all you did was assert that equality in mathematics is more precise than equality in other subjects. The point being that in no subject does "equal to" mean "the same as", not even mathematics. As I explained the left side does not signify the same thing as the right. And your assertion of the precision of mathematics still doesn't get you to the point of being the same. Equal to, and the same as, are distinct conceptions.

I have not said that numbers are special regarding denotation. — TonesInDeepFreeze

This is exactly what you are saying. By insisting that "equal to" in the case of numbers means 'denotes the same object', you are saying that numbers have some special quality which can make two distinct but equal things into the same thing. You are claiming that numbers have a special status which makes equal things into the same thing.

.

I have not use the term 'necessarily' in this context since 'necessarily' has a special technical meaning that requires modal logic. — TonesInDeepFreeze

You don't have to use the word "necessarily", to mean it. When you say that being equal implies that they are the same, you refer to a logical necessity which dictates that if they are equal then they are necessarily the same. Otherwise it would be false to say 'if they are equal then they are the same'.
Which of course, is obviously false to say, because that necessity is based in a false premise.

'2+1' and '3' have different senses but not different denotations. No matter how many times I point out the disctinction between sense and denotation, and even after I linked you to an Internet article about it, you keep ignoring it. — TonesInDeepFreeze

I've already explained to you how you do not have the premise required to say that "2+1" denotes the same object as "3", when the two signify different things ("have different senses").

Remember your example? "The father of Peter Fonda" denotes a person in a particular relationship with Peter Fonda. That is the "sense". "Henry Fonda" also denotes a particular person. Again, that is the "sense". Now, you do not have the premise required to validly conclude that these two persons, indicated by those two senses, are the same person. The same thing is the case with "2+1" and "3". They signify different things (have different senses). Now, you do not have the required premise to conclude that they denote the same object. You can stipulate, as a premise, "Henry Fonda is the father of Peter Fonda", but that would be begging the question. Likewise, you can stipulate that "2+1" denotes the same object as "3", but that's simply begging the question. You are creating the premise required to support your desired conclusion, and that's a fallacy.

So no matter how many times you assert that despite the fact that "2+1" and "3" mean something different, they denote the same object, you have not produced a valid argument to prove this. All you've produced is the false premise that if they are equal then they are the same.

Again, you skipped my reply much earlier in this thread. It is in the method of models that we have that equality is sameness. — TonesInDeepFreeze

You mean that false premise?

Again, I explained to you many posts ago that '=' maps to the identity relation per the method of models. — TonesInDeepFreeze

Shouldn't we call this what it really is, the method of the false premise?

One example, please. — tim wood

Set theory, and the axiom of extensionality.

As far as I can tell, your only reason for rejecting them is that they are different from your preferred definitions, — aletheist

I conclude that you didn't read, or for some reason couldn't understand what I said then.

Perhaps you think that there cannot be causation between different kinds of object, and thus if our brain events cause our mental events this would be evidence that brain and mental events must be events involving the same kind of object. — Bartricks

Probably the best analysis of the nature of the soul, ever written, is found in Plato's Phaedo. The idea that the brain is the cause of the mind, is very similar to the harmony theory. The material parts exist in a way which creates a harmony, and the harmony is the soul. But this theory is demonstrated as deficient because it cannot account for the reason why the parts exist in such a way as to be in harmony rather than dissonant.

So the theory needs to be inverted such that each material part, in itself, as an organized existent, is a harmony, and the cause of that harmony is something immaterial. This is what Aristotle takes as his starting point in "On the Soul". A living being is an organized material body. The cause of the organization, which manifests as the material body, is the soul. We can conclude therefore, that the soul, being prior to the material body as cause of it, is immaterial.

The question though, do you see that I have very good reason to reject those definitions? They increase ambiguity, leading to equivocation and category mistake. In the interest of understanding, you ought to reject them as well. Don't you think?

You believe that equality holds based on human judgement. That doesn't entail that I said that equality or identity does. It's a strawman to represent me as saying something I did not say. — TonesInDeepFreeze

I asked you for an instance of equality which is not a human judgement. You didn't give me one. That's probably because you understand that such a thing is ridiculous.

Of course if 2+1 and are equal 3 then they are the same. — TonesInDeepFreeze

In case you're having a hard time to understand, I see this as very clearly false. You and I are equal, as human beings, but we are not the same. Therefore we cannot conclude that if two things are equal they are the same. You seem to think that numbers are somehow special, so that if they are equal they are necessarily the same. I'm waiting for you to attempt to justify this belief, which to my understanding is demonstrably false.

Again, I did not say that indiscernibility implies identity. I said the reverse direction of implication: identity implies indiscernibility. — TonesInDeepFreeze

Right, so the law of identity states that a thing is identical to itself. That's identity. Then we can proceed to say that a thing is indiscernible from itself, and this is consistent. Now, 2+1 is discernible from 3, so how do we conclude that they are the same?

Still yet again, I am not using indiscernibility to support that 'equal' means 'identical' or that 'equal' means 'the same'. Stop mixing up what I've said and then representing your own mixed up version as if my own. — TonesInDeepFreeze

Then how in hell are you supporting this obviously false assumption that "if 2+1 and are equal 3 then they are the same"?

In ordinary mathematics, 'equal' is not defined but rather is a primitive. It is the sole primitive of first order identity theory. In that context 'equal' and 'identical' are two words for the same undefined primitive. — TonesInDeepFreeze

The problem is that mathematicians do not use "=" in a way which is consistent with the law of identity. Therefore your "undefined" primitive is a violation of the law of identity. If the right and left side of the equation signified the exact same thing, as required by the law of identity and if equal signifies identical, then all equations would read like "X=X", or "Y=Y", or some other way of saying that the very same thing is represent on the right and the left. However, mathematicians use "=" to relate two distinct expressions with distinct meanings, which clearly do not signify the exact same thing.

And again the two different terms '2+1' and '3' refer to the same object. They have the same reference. However, of course, they do not have the same sense. — TonesInDeepFreeze

This is the contradiction which altheist was trying to impose on me. If "2+1" signifies something different from "3", then it is impossible that what they denote is the same object by way of contradiction. If they are supposed to be signifying different predications, so contradiction is avoided, then no object is denoted, just two distinct predications without a subject, predicated of nothing.

That is rich from someone who dismisses approaches in ordinary mathematics while insisting on remaining ignorant of understanding their fundamental principles or even reading a single page in a book or article about the subject. — TonesInDeepFreeze

That's nowhere near as bad as someone who routinely applies mathematics without recognizing the falsity of fundamental principles. I cannot understand the fundamentals because they are unsound. Contradiction or falsity make understanding impossible. But accepting contradiction, or falsity and proceeding to apply these principles is self-deception and misunderstanding.

I gave you a formal mathematical proof of this fact over two years ago, maybe three. You're telling a little fib here. — fishfry

Talk about begging the question. That's what your so-called proof did.

'S' stands for the successor operation.

def: 1 = S0

def: 2 = 1+1

def: 3 = 2+1

The proof in this case is utterly trivial, from the definition of '3'. — TonesInDeepFreeze

All I see is "=" here. Where's the proof that "=" means the same as?

No, these words are examples of signs whose objects--that which they denote--are general concepts. — aletheist

I know that's what you think, but I disagree. I think that you're way of looking at things creates ambiguity in the meaning of "object", which leads to equivocation between mental objects and physical objects, resulting in category mistake.

The interpretant of each individual word is the aggregate of all the different propositions that include it, which we attempt to summarize whenever we write a definition of it. — aletheist

Oh, come on. One cannot expect to consider all the different propositions that include a word, when interpreting that word. Many could be inconsistent or contradictory. In a logical proceeding there are stipulated propositions. But there are no objects, just subjects. The propositions make predications of subjects, not objects.

An object is defined according to the law of identity, as unique, primary substance, but a logical subject is not unique, as secondary substance. So the relations you refer to may be applied to multiple objects, as universals, because a subject is not limited to representing one object. But in calling these subjects "objects" you imply the uniqueness of an object, as required by the law of identity, Therefore in defining "object" in this way you loose the capacity to distinguish between whether the relation referred to is a unique relation, specific to a particular situation (object in my sense), or a universal relation, common to numerous objects (in my sense), represented by a single subject. In other words, by calling the subject an object, we loose the capacity to distinguish uniqueness, due to the ambiguity and the category mistake which will prevail.

An object is not necessarily something physical, and a subject is not necessarily something that we study. In semeiotic, an object is whatever a sign denotes, and a subject is a term within a proposition that denotes one of its objects. — aletheist

Do you believe in the law of identity and the uniqueness of an object? If so, then how can you allow that "whatever a sign denotes" is an object, when the same sign denotes different things in the minds of different people? Do you assume an independently existing Idea, or Form, as the ideal conception, or unique object denoted, separate from the less than perfect ideas of individual human beings, which are all slightly different?

So get these straight already:

(1) My explanation runs in this order:

Determine equality, then it is justified to assert that the terms denote the same.

(2) Equality implies indiscernibility. I did not opine one way or the other whether indiscernibility implies equality. — TonesInDeepFreeze

Equality is insufficient for a judgement of "same". That's very simple, clear, and obviously true, from all the instances where equal things are not the same thing. I'll expound on this below, but you ought to respect this principle instead of trying to deny it, and insist that equal things are necessarily the same thing.

I know about identity vastly more than you do. And your reply merely repeats your own thesis. And you did argue by strawman by trying to make me look as if I had said that identity holds based on human judgement. — TonesInDeepFreeze

If you can show that equality is something other than a human judgement, then you might have a case. Otherwise the charge holds.

Of course, people make judgements of equality. But at this particular juncture in the discussion, I am pointing out that the activity is not that of judging equality itself but rather judging whether the terms refer to the same thing. Those activities are related but different. — TonesInDeepFreeze

You said that from a judgement of equality you can infer that they are the same. I'll quote for the third time:
"Rather, we infer they share all properties from having first proved that they are equal."
You are clearly arguing that if they are equal then they are the same.

First we determine (by proof or whatever method) that 2+1 is 3. — TonesInDeepFreeze

Obviously this is false, 2+1 is equal to 3 but it is not the same as.

Sure it does. The indiscernibility of identicals is the general principle. Substitutivity is the formal application of the principle. — TonesInDeepFreeze

I explained why the indiscernibility of identicals does not support your assertion. Obviously 2+1 is not indiscernible from 3. Therefore you cannot use the indiscernibility of identicals to support your claim that they are identical.

What I said:

equality -> indiscernibility. — TonesInDeepFreeze

This is false because what "2+1" signifies is very clearly discernible from what "3" signifies. There are two numbers denoted, 2 and 1, while "3" only denotes one number. We've been through this countless times already and you are in denial of the truth. Admit the fact, 2+1 is not indiscernible from 3.

But you keep saying that I say:

indiscernibility -> equality

even after I've told you that is not what I say. — TonesInDeepFreeze

This is the only way that the principle of indiscernibility could be used to support your claim that equality means the same as. So I assumed that this is what you meant. The other way, the way you claim to be using it, would work if it were true, but it is clearly a false premise. Equal is not sufficient for indiscernibility. That's obvious from all the cases of equal things which are discernible.

Don't reverse the direction of my conditionals. — TonesInDeepFreeze

I assumed you were trying to make a sound argument. However, you've now corrected my to show that you are simply using a false premise. You admit now that your premise is that if things are equal they are the same. Therefore I'll take you back to what I asked days ago. Are you and I the same because we are equal? You have no special pleading now, for a special sense of "equal", which is supported by "indiscernible", because you've just admitted that you support "indiscernible" with "equal". By turning this around you have no special definition for "equal".

And to define that sense of "equality" with "value" doesn't help you because all senses of "equality" rely on a judgement of value. Quantitative value is no more special than moral value as an indicator as to whether or not two things are the same. The value which we assign to a thing is not a thing's identity.

In ordinary mathematics, we concern ourselves only with denotation, which is the extensional aspect of meaning. — TonesInDeepFreeze

See, you admit right here, that you only concern yourself with a part of what "2+1", and what "1" refer to. Therefore you ignore the other aspects, which are clearly different from each other, and you proceed to claim that what they represent is identical. What this really means is that they are the same in some aspect, but not in every aspect, so it is false to claim that they are the same.

That is because you evidently have an extremely narrow definition of "object" and refuse to accept how that word is defined as a technical term within the discipline of semeiotic. Anything whatsoever that is denoted by a sign is the object of that sign. — aletheist

Sorry aletheist, but I must inform you that technical definitions narrow down a word's definition. That is because a broader definition allows for ambiguity. So your attempt to broaden the definition of "object", is not at all an attempt at a technical definition. It's an attempt to create ambiguity, which might be useful for the creation of vagueness and equivocation. So I see your definition as completely misguided because it's not conducive for understanding.

Not at all, do some research into semeiotic (also called "semiotic" or "semiotics") and you will learn that what I have been discussing is a well-established field of study. — aletheist

I have, and I do not agree with the fundamental principles of that proposed field of study. It appears to be lost in ambiguity and category mistake. This opinion which I have, you might be able to detect. I am not one to dismiss things off hand, without some understanding of the fundamental principles.

The object of a term is whatever it stands for, while the object of a proposition is the collection of objects denoted by the terms that serve as its subjects. — aletheist

This appears to involve a fallacy of composition. And I think this is why your way of looking at "2+1" appears so incorrect to me. You say that "2+1" signifies something which is other than what "3" signifies, yet "2+1" denotes the same object as "3". You make this conclusion of denoting the same object through a fallacy of composition, concluding that the attributes of the parts within the statement "2+1" can be summed up into a collection, to make an object with the exact same attributes as 3.

That is not what I said. There is no such thing as pure signification (without denotation) in common usage. Likewise, there is no such thing as pure denotation (without signification) in common usage. Instead, in practice every sign both denotes its object (what it stands for) and signifies its interpretant (what it conveys about that object). This is most readily evident in a proposition, where the subjects (terms as names) denote the objects and the predicate (embodied as syntax) signifies the interpretant. The fundamental principle of semeiotic (following Peirce) as distinguished from semiology (following Saussure) is that a sign thus stands in an irreducibly triadic relation with its object and its interpretant, rather than there being only a dyadic relation between signifier and signified. — aletheist

Sure there is pure signification, in the case of any abstract use, a universal, like "temperature", "big", "good", "beauty" "green", "wet", and the list goes on and on. You just want to insist that these can only be used when describing an object, to support your special form of Platonism. But it's not true. We use all these terms as a subject when we say things like "temperature is a measurement", "big is a size", "good is desirable", beauty is what the artist seek", "green is a colour". These are phrases of pure signification, and to turn the subject into an object will most probably lead to category mistake. Because then we lose the capacity to distinguish between a physical object denoted, and a subject of study denoted. If these two are the same, as "object denoted", category mistake will prevail.