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.

Identity is a reflexive relation. And I never said that things are identical due to human judgement. You're resorting to strawman again. — TonesInDeepFreeze

Do you know the law of identity? It states that a thing is the same as itself. It says nothing about equality or equivalence. That two things are equal is a human judgement.

And you said:

Rather, we infer they share all properties from having first proved that they are equal. — TonesInDeepFreeze

See, no strawman. You prove that they are equal (human judgement), then you infer from this, that they are the same. Let me put it simply, proving that they share one property, "are equal" does not prove that they share all properties. You need another premise, which states that equal things are the same thing. But we know that premise is false because we see all sorts of equal things (equal volume, equal weight, etc.) which do not make two things the same. Equality is not sufficient for a judgement of same.

We don't judge two things are equal. — TonesInDeepFreeze

You very clearly stated "having proved that they are equal". Therefore you do judge that they are equal, that's what proving is, providing the justification for judgement.

No, the principle of the indiscernibility of identicals holds. It provides the method of "substitute equals for equals" that is fundamental in mathematics. — TonesInDeepFreeze

The indiscernibility of identicals does not provide the principle required for substituting equal things. Things are judged to be equal not on the basis that they are indiscernible. Clearly, as altheist agrees, and what ought to be obvious to you, what "2+1" signifies is not indiscernible from what "3" signifies. Since these two are judged to be equal, equal does not mean indiscernible. Therefore it is false to claim that the principle of the indiscernibility of identicals supports such a substitution. It does not.

You haven't shown that it is false that mathematics does not have the kind of vagaries of natural language in everyday discussion. — TonesInDeepFreeze

Clearly mathematics has an extremely vague notion of identity, one not consistent with the law of identity, allowing that similar things which are judged to be equal may be substituted as if they are the same thing.

Good, and they also both denote the same object with "did," which is the relation of doing. — aletheist

I really can't see how a relation is an object. I think you are making things up as you go.

We already went over this with "Henry Fonda" and "the father of Peter Fonda." These signs both denote the same object despite signifying different interpretants because it happens to be a fact that Henry Fonda is the father of Peter Fonda. — aletheist

This explains nothing. Words like "did" signify something. But you insist instead, that they denote an object. But you also allow that they signify things as well, and denote objects at the same time. On top of this you allow that two phrases might signify different things, yet denote the very same thing. This indicates very clearly that there are contradicting interpretations of the same phrases. One interpretation says that they are different, the other that they are the same. Yet you allow that the contradicting interpretations are both correct.

It seems clear that you are using a different definition of "object" than the one rigorously employed within the discipline of semeiotic. Again, anything that is denoted by a sign--real or fictional, existent or imaginary--is an object in that technical sense. — aletheist

Why then did you insist on a distinction between "signification" and "denotation" in the other thread, when here you want any signification to be a denotation?

The only signs that theoretically could signify something without denoting anything are pure icons, unembodied qualities that would only convey themselves as they are in themselves. Any sign that stands for something else denotes that other object. — aletheist

So, you insist on a distinction between signification and denotation, then it turns out that there is no such thing as signification in common usage. All instances of signification are assumed to be denotations of objects. What's the point?

If this were true, then the author could not create those "images of characters" in the first place, and we could not think or talk or write about them afterwards. Again, the sign "Hamlet" denotes the fictional character in Shakespeare's play as its object. — aletheist

Let's start with a clean slate then. There is no such thing as signification. Words do not have meaning, they denote objects. is that what you want?

Otherwise it's pointless for me to say that an expression has meaning (signification) and you just overrule and say no, that's not meaning, it's the denotation of an object. We will never get anywhere like that. What is your rigorous definition of "object" which allows you to claim that any instance of meaning is a denotation of an object? If you and I both read the same expression, and I interpret it as meaningful, without denoting any object, and you interpret it as denoting an object, what is your rigorous definition which makes you right, and me wrong, if you are not just assuming that anything with meaning denotes an object?

(1) I didn't make "vague references". Indeed, I posted an explanation of the notion of exentionsality vs. intensionality. And I gave references in the literature for you to read about it. Moreover, even if I had not done that, it is still the case that the notion of extensionality vs. intensionality is a well known basic notion in the philosophy of mathematics and philosophy of language. The fact that you're ignorant of such basics of the subject is not my fault and doesn't make my reference to them "vague", and especially not when I gave explanation and additional references in the literature anyway.

(2) I posted multiple times that proving that '2+1' and '3' denote the same object is the basis on which we justify claiming that they do. Or, for a better example (since the equation '3 = 2+1' has such a trivial proof), we say '6-3' and '2+1' denote the same object because we prove that they do. — TonesInDeepFreeze

I told you already, extensionality provides a false premise. False premises produce unsound conclusions, which do not prove anything. When a human being judges two distinct things as having the same properties, and says therefore that they are equals, this does not make them into the same thing. The law of identity stipulates that the identity of a thing is within the thing itself, not a human judgement of the thing.

Ordinary mathematics regards '2+1' and '3' as having the same denotation, because we prove

2+1 = 3

In general, for any terms T and S, we infer

T = S

when we prove it and then we may say that T and S have the same denotation. — TonesInDeepFreeze

Clearly you, (and extensionality in general) have this backward. If we start with the law of identity, "a thing is the same as itself", as a fundamental premise, and we compare this with equality, which is a property that we assign to things, you ought to see this. When we judge two things as equal, we cannot assume that they are the same thing, because we need to allow for the fact that human judgements are deficient in judging sameness. We may not be able to account for all the potential differences between them, and thereby over look some, making a faulty judgement of "same".

So, we can truthfully say that a thing will be judged to be equal to itself, but we cannot truthfully say that things which are judged to be equal are the same thing. Therefore, when you say that T=S, you say that T has the same value as S within that system of judgement, and this means that the symbols have the same meaning within that system, but it does not mean that they denote the same object.

Rather, we infer they share all properties from having first proved that they are equal. And whatever we prove, we do so from axioms. — TonesInDeepFreeze

This precisely, is the false premise. Being equal is a human judgement, and being equal does not imply being the same. We can proceed the other way, and say that being the same implies being equal, but we cannot proceed from being equal to an implication of being the same. This is because two distinct things can be judged as equal, when they are not the same. Therefore proving that two things are equal does not imply that they are the same (share all properties). It only implies that they share the properties by which they are judged to be equal. And there is your false premise.

This is a basic fact of the way that we use signs and symbols. We use the symbol "2" here, and we use it later in some other application. These are two distinct instances of that symbol, they are not the same thing. However, they have an equality in what they signify. Each distinct instance of using that similar symbol signifies "the same value". This means that the two instances have the same meaning. It does not mean that they denote the same object.

So, proving that two distinct yet similar instances of a symbol "2", have an equal value, only proves that they have the same meaning. It does not prove that they denote the same object.

Of course in natural language and everyday discourse there may be vagaries that make definitive determinations difficult or impossible. So if you want to demand a context in which there are no vagaries, then of course all bets are off with natural language usage. So to proceed with understanding certain principles, of course we must assume, for sake of discussion, some context in which we are not thwarted by such vagaries as you mention. — TonesInDeepFreeze

This is counterproductive. If in reality, language use is filled with vagaries, and we want to discuss the truth about language use, then we need to account for the reality of those vagaries. To assume a context without vagaries as your prerequisite premise for proceeding toward an understanding of certain principles of language use, is simply to assume a false premise. Therefore by adopting such a position we proceed toward a misunderstanding rather than an understanding.

Similarly, of course I take it for granted that we already understand that 2+1 = 3, either by proof or by common mathematical knowledge. It is from that understanding that we then observe that '2+1' and '3' denote the same number. — TonesInDeepFreeze

In the case of Henry Fonda, we have observed with our senses, the very object being referred to. In the case of numbers we have not observed any such objects. You are requesting that I simply assume such an object, a number, so that we can talk about it as if it is there. Obviously, there are no such objects, the numerals have meaning dependent on the context of usage, just like any other symbols. They do not denote any objects, and your so-called understanding is actually a misunderstanding.

But denotation in ordinary mathematics is fixed, so it remains a simple fact that '2+1' and '3' denote the same number. — TonesInDeepFreeze

Claiming a denotation when there is only meaning, is a false premise.

The claim that 'Henry Fonda' and 'the father of Peter Fonda' denote the same person is not an argument! It is a conclusion. It is a conclusion from the premise (however it has been established) that Henry Fonda is the father of Peter Fonda. No one every suggested otherwise! — TonesInDeepFreeze

Do you understand the fallacy of "begging the question", assuming the conclusion?

I am treating "2+1" and "3" as signs here, and I already acknowledged that their signification is different. At issue is whether their denotation is different. What "+" represents in isolation is irrelevant, all that matters here is that I can point to the same group of items and truthfully say both "that is 2+1 apples" and "that is 3 apples." — aletheist

To be clear, what is at issue is whether there is a denotation at all (when denote is defined as you do). Read the above.

The "+" is not irrelevant, it must be accounted for in your interpretation. You cannot simply leave words out of a phrase, in your interpretation, to make it say what you want it to say, or denote what you want it to. I really do not see any logic to your claim that two expressions can have distinct significations, yet denote the same object. I can see how "I did X", and "I did Y", both refer to the same object with "I", but each signify something different. Since each expression signifies something completely different, if we replace what is signified with "denoting an object" as you seem inclined to, then we do not come up with the same object. How do you come up with this idea that two phrases which signify something completely different actually denote the same object. I would call that contradiction.

As I stated clearly in the last post. A group of three apples does not truthfully represent "2+1". If that's not obvious to you, go back to grade school and find out how they represent "2+1".

Indeed, but as the Fonda example has brought to light, Metaphysician Undercover apparently confuses denotation and signification. The result is wrongly denying that two different expressions signifying different interpretants can nevertheless denote the same object. — aletheist

Actually it has become very clear now, that you and Tones are the ones confusing denotation and signification. Clearly, in our use of mathematics there is signification without denotation. You and Tones are seeing an object denoted by "2+1", when there is none. That is misinterpretation.

However, we can point at a collection of three apples and say both "that is 2+1 apples" and "that is 3 apples." — aletheist

That's not true, because the operation signified by "+" is not evident in the group of three apples, so it is not a true representation of "2+1". It is just a representation of "3". If you were teaching children you would not show them a group of three apples and tell them this is 2+1.

Moreover, we can substitute "2+1" for "3" in any proposition without changing its truth value or in any equation without changing its result. What should we conclude from this? — aletheist

We might say that expressions which signify equal value can be substituted, within that value system . We cannot conclude that because the expressions can be substituted, they signify the same thing. They are only "the same" in relation to that value assigned to them.

If I need assistance, and Tom, Dick, or Harry, will do, each having equal value for the task, I ask for Tom, Dick, or Harry, as they are interchangeable in relation to this value. They each make "the task will be done" true. This does not mean that each of them is the same thing as each other.

Simple--in semeiotic, anything that is denoted by a sign is, by definition, its object. Since all thought is in signs, anything that we can think about--real or fictional, existent or imaginary--is an object in this sense. — aletheist

To adhere to the distinction you made for me in the other thread, in much usage of signs, probably the majority actually, the signs have significance without denoting anything. For instance in "I'm going for a walk", the only object denoted is "I". And in your example of fictional writing, there are no objects denoted. The author simply builds up images of characters without denoting any objects.

Then the example is irrelevant to the issue we are discussing, that "2+1" denotes the same object as "3".

You joined the discussion a bit late, and seem to be missing the issue.

Again, in semeiotic a subject is a term within a proposition that denotes one of its objects. — aletheist

Sure looks like fancified Platonism to me, if a subject must denote an object.

No one is claiming that fictional, imaginary things are real. In fact, being fictional is precisely the opposite of being real. That which is fictional is such as it is only because someone thinks about it that way; Hamlet was the prince of Denmark only because Shakespeare created a story in which that was the case. By contrast, that which is real is such as it is regardless of what anyone thinks about it; Platonism is one form of mathematical realism in this sense, but not the only one. — aletheist

I really don't understand your position. You assume that fictional characters are objects, but you deny that they are real, and you deny that they are existent. How do you validate your claim that they are objects?

Again, what anyone knows or does not know is beside the point. Since it is fact that Henry Fonda is the father of Peter Fonda, by definition (in semeiotic) the two signs "Henry Fonda" and "the father of Peter Fonda" denote the same object. — aletheist

These are nonsensical assertions. You are asserting that it is a fact that these words refer to these objects regardless of how people use the words. The issue is whether or not "the father of Peter Fonda" and "Henry Fonda" necessarily represent the same object. That you can define them as representing the same object, and insisting that this is a fact, is irrelevant I don't see any point in discussing the soundness of a logical argument with someone like who, who simply insists that the conclusion is a fact, and that's all there is to it.

If you want to start with the premise, that Henry Fonda is the father of Peter, we can do that. But then the example is irrelevant to the question of whether "2+1" denotes the same thing as "3", because we are not starting with that premise. We are arguing whether or not this claim is true, or logically sound. So we cannot start with the premise that "2+1" denotes the same thing as "3" because that would be a fallacy of begging the question.

So, to make the example relevant, we must start with the two expressions, "father of Peter Fonda", and "Henry Fonda", and you need to demonstrate how they necessarily refer to the same object, without begging the question. Insisting that it is a fact is simply begging the question, and that is a logical fallacy. So your procedure up until now has been completely useless.

I never said anything about persons or equality. I merely made the point--which is utterly uncontroversial (in semeiotic)--that since Henry Fonda is the father of Peter Fonda, the two signs "Henry Fonda" and "the father of Peter Fonda" denote the same object, regardless of whether someone else knows it. — aletheist

Yes, yes, keep begging the question, it really doesn't bother me if you do. You're only fooling yourself.

That latter is a bit disingenuous. If I say Socrates is a Greek philosopher, someone might object because they think I might have meant Socrates the cat philosopher. That's not really a good objection, if you fully qualified everything there would be no end to it. — fishfry

Let me give you a more relevant example. Let's consider an experiment in quantum physics. Consider that a photon is emitted by an emitter, and a photon is absorbed by a detecting machine. Each instance involves an equivalent amount of energy, so the assumption is that the two are the same photon. Then comes the difficult task of determining the continuous existence of that photon between point A and point B, which is produced by the idea that they are the same photon. But there is no need to assume that the two are the same photon, likewise there is no need to assume a continuous existence of the photon between point A and point B. It is only this (what I call odd) way of looking at things, that if there is a quantifiable value here, then an equal value over there, these must represent the same thing, which promotes the idea of the continuous existence of a photon between these two point.

Henry Fonda IS the father of Peter and that's that. — fishfry

As explained above, to premise that "Henry Fonda IS the father of Peter" is no different from premising that "2+1 IS 3". But since what I am looking for is an indication that 2+1 really is the same thing as 3, some sort of logical argument, that's simply begging the question. So the issue is to demonstrate logically, how it is that the two distinct expressions "the father of Peter Fonda", and "Henry Fonda" both refer to the exact same thing, and how this is relevant to the case of "2+1" and "3".

No, it does not. Hamlet, the fictional character in Shakespeare's play, is the object of the sign that is the first word of this sentence. — aletheist

Where's your grammar? Fictional characters are known as subjects, not objects. Your claim that "Hamlet" refers to an object is unsupported by any conventional grammar.

Even if I grant you that fictional, imaginary things may be called objects, my point was that some form of Platonism, as an ontology is required to support the claimed reality of such objects. So this line of argument is not really getting you anywhere.

At the risk of belaboring the point, it is an all-too-common nominalist mistake to insist that if abstract objects are real, then they must also exist. These are two very different concepts--whatever is real is such as it is regardless of what anyone thinks about it, while whatever exists reacts with other like things in the environment. Again, there are varieties of mathematical realism other than Platonism. — aletheist

Trying to establish a separation between "real" and "existent" just muddies the water by creating ambiguity, and is counterproductive toward understanding. As well as being "real", ideas, concepts and abstractions are obviously "existent". They have a significant effect on the physical world as clearly demonstrated by engineering.

So defining "existent" as having causal interaction, then attempting to remove ideas from this category is a mistake because ideas obviously have causal interaction. Then this proposed separation between "real" and "existent", which would put ideas into some category of eternal inert objects which cannot have any influence in our world in any way, is just child's play. It's an imaginary scenario which in no way represents reality.

I agree that they signify different interpretants, but this does not preclude them from denoting the same object. — aletheist

As I said, they may denote the same object, but we do not have the premises required to conclude that they do. In logic we cannot assume other premises which are not stated. We have a person denoted as "the father of Peter Fonda" and we have a person denoted as "Henry Fonda". We have no other information. So the conclusion that they both denote the same thing is extremely unsound, because it is not derived from valid logic. It is invalid.

t is a fact that Henry Fonda is the father of Peter Fonda, so by definition, it is also a fact that the signs "Henry Fonda" and "the father of Peter Fonda" both denote the same object. — aletheist

In logic, assertions do nothing for you. They are proposals, propositions which must be judged for truth or falsity. TonesinDeepFreeze has been asserting that "2+1" denotes the same object as "3" does, in a similar way. They very clearly each signify something different. The only attempt by Tones, to support this conclusion with a premise, was a vague reference to extensionality. But a premise which states that two equal things are the same thing is clearly false, making that argument unsound, by having such a falsity as a premise. To say that the person denoted as father of Peter Fonda, and the person denoted as Henry Fonda, are equal, as human beings, does not justify the claim that they are the same person.

If you just keep asserting as a proposition "it is a fact that...", and you expect me to take that proposition as a premise for an argument, then you're wrong. I will not. You need to demonstrate the truth of it. Adding the emphasis "it is a fact" does nothing for your case. I am very certain that two things with the same value are not necessarily the same thing, as I can give you endless examples, so that is a false premise.

But here I find myself inclined to see his side of it. — fishfry

Fishfry! Never in a hundred years did I think I'd see this day. Let's go, I'll buy you a beer.

I might know who Henry Fonda is, but I might not know he's Peter Fonda's father. I can see Meta's point that the "father of Peter" description conveys more information than merely saying "That's Henry Fonda." — fishfry

The point is that "the father of Peter Fonda" gives different information from "Henry Fonda". The latter gives nothing, just the name of a person. The first expression also denotes a person, as well as the second expression denotes a person. But the information required to conclude that they are one and the same person is not provided. Even if we add the further premise, "Henry has a son Peter", the condition of reversibility, equality, is fulfilled, but we still cannot conclude that they denote the same person. There might be more than one Henry Fonda with a son Peter. Therefore there is still a possibility of error, which demonstrates why such conclusions are unsound.

." Again, "Henry Fonda" and "the father of Peter Fonda" denote the same object, even though what they signify about that object is different. — aletheist

Again, your argument that they denote the same object is fallacious. They may or may not denote the same object. They clearly signify something different, and we do not have the premises required to conclude that they denote the same object. Therefore your conclusion that they denote the same object is fallacious.

This is false, since it is not necessary for something to exist--in the metaphysical sense of reacting with other like things in the environment--in order to be the object of a sign. It does not even have to be real--it could instead be fictional, as some philosophers consider mathematical objects to be. — aletheist

This only supports my point. To justify calling an imaginary thing "an object" requires some form of Platonism.

The view that numbers are real, independently of any mental activity on a human's part, is what is generally known as mathematical platonism. The point is, this is unpopular in today's academy; there are many very influential mathematicians, who are far greater experts than I could ever hope to be, who are intent on showing that it's mistaken. But according to this article Benecareff's influential argument against platonism was made 'on the grounds that an adequate account of truth in mathematics implies the existence of abstract mathematical objects, but that such objects are epistemologically inaccessible because they are causally inert and beyond the reach of sense perception.' In other words, this argument denies that we can have the innate grasp of mathematical truths that Frege asserts in the paper mentioned above. That's the 'meta-argument' I'm trying to get my head around. — Wayfarer

That mathematical Platonism is unpopular in today's academy presents an odd dilemma for mathematicians. Platonist principles support a huge part of modern mathematical systems, underpinning extensionality and set theory, to begin with. These mathematical axioms require that a term signifies an object. Only Platonism can support this prerequisite. So, if there are proficient and influential mathematicians who openly deny Platonism, then these same mathematicians must be prepared to revisit, denounce and replace, all the fundamental mathematical axioms which are based in Platonism, or else they are simply being hypocritical.

Completely wrong, denotation and signification are two different aspects of a sign, corresponding respectively to its object and its interpretant. This is Semeiotic 101. — aletheist

That's bullshit 101. In logic, there is no object, we have subjects. To denote is simply to be a sign of.

I offered no argument at all, I simply stated a definition--if one sign can be substituted for another in any and every proposition without changing the truth value, then both signs denote the same object. This is also Semeiotic 101. — aletheist

Then you're not addressing the issue we've been discussing. We've been arguing the truth or falsity of of a very similar principle. The "Fonda" example was provided as an argument for the truth of it. As I've shown, it's a fallacious argument.

We were arguing the truth or falsity of the principle of substitution, which is the basis of extensionality. It is claimed that if two signifiers signify things of equal value, they are exchangeable, therefore they signify the very same object. It is very clear to me that this is a false principle because "equal" is assigned according to some system of judgement, so only the properties deemed significant within that system are accounted for, and this is insufficient for the conclusion of "the very same object". I find it utterly amazing, and rather distressing, the number of people in this forum who cannot apprehend this simple fact.

Now you are arguing a slightly different form of that principle. "Truth value" is something judged. Propositions are stated. Predications are of subjects. So within a logical system "truth value" concerns what we say about subjects, not objects. Unless absolutely every property of a given object is stated (a task humanly impossible), so that an infallible judgment can be made, your proposed principle: "if one sign can be substituted for another in any and every proposition without changing the truth value, then both signs denote the same object" Is clearly unacceptable as false. You have not provided the means for closing the subject/object gap.

Again, this confuses denotation with signification. — aletheist

Denotation is a form of signification.

In any and every proposition about "Henry Fonda," we could substitute "the father of Peter Fonda" without changing the truth value. — aletheist

As I said, this is only the case if there is a premise which states that Henry Fonda is the father of Peter Fonda. But that is begging the question, which is respected as a fallacy.

Therefore the argument that "the father of Peter Fonda" denotes the same thing as "Henry Fonda" is a fallacious argument, by means of begging the question. The argument relies on assuming the conclusion.

As I explained, they do not denote the same object. One denotes the father of a person called Peter Fonda. The other denotes a person named Henry Fonda. That they denote the same object requires a further premise, that the father of Peter Fonda is the person named Henry Fonda.

Without that premise, the conclusion that they denote the same thing is invalid. And adding that premise is to beg the question. So the argument that they denote the same object is fallacious.

Are you going to address the points I made or not, Tones?

Do you apprehend the flaw in your example, and the difference between what "the father of Peter Fonda" denotes , and what "Henry Fonda" denotes?

In itself, yes; but we can still "divide" it at will to suit our purposes. — aletheist

To identify a point on a line is not to divide that line. So it's not really a matter of dividing which you are talking about.

For example, we can conceive space itself as continuous and indivisible, but we can nevertheless mark it off using arbitrary and discrete units for the sake of locating and measuring things that exist within space. — aletheist

The proposed units would be arbitrary, but I do not think you could call them discrete. And, according to the issues brought forward by special relativity, the supposed "same unit" would be different depending on the frame of reference, or more precisely, we could not determine the "same unit" from different frames of reference. This makes the whole idea of measuring the continuum through the means of units rather difficult. I suggest that if the application of units works well for measuring space and time, they are probably not actually continuous.

The speculation which is the reverse of yours is that continuity is what is artificial. The continuum is something created by human minds, and physical existence contains no such continuity. Problems such as Zeno's paradoxes arise because we apply principles of continuity to a physical world which is discontinuous.

No, this is a confusion of "infinitely divisible" with "infinitely divided." The former means potentially having infinitely many parts, while the latter means actually having infinitely many parts. A true continuum is infinitely divisible, but this does not entail that it is infinitely divided. It is a whole such that in itself it has no actual parts, only potential parts. These are indefinite unless and until someone marks off distinct parts for a particular purpose, such as measurement, even if this is done using countably infinite rational numbers or uncountably infinite real numbers. A continuous line does not consist of such discrete points at all, but we could (theoretically) mark it with points exceeding all multitude. — aletheist

If this is true, what you describe here, then it is impossible that "a true continuum is infinitely divisible. If marking points on a continuous line does not constitute dividing it, then there is nothing to indicate that the continuous line is divisible at all. And if dividing it once would break it's continuity, then a continuum cannot be infinitely divisible because dividing it once would prove it to be discontinuous.

Therefore it is a contradiction to say "a true continuum is infinitely divisible". We ought to say instead, "if it were divisible it would not be a true continuum". A true continuum is indivisible.

It is infinitely divisible, but not actually divided. — aletheist

Again, this is contradiction. If it cannot actually be divided, then it is false to claim that it is divisible, in any sense.

These are designated the 'primary attributes' of objects, and distinguished, by both Galileo and Locke, from their 'secondary attributes', which are held to be in the mind of the observer.

...

And through the quantitative method of science, the ability to reduce an objective to its mathematical correlates, the certainty provided by logical prediction can be applied to phenomena of all kinds with mathematical certainty (which is, I think, the point of Kant's 'synthetic a priori). It's the universal applicability of these logical and mathematical procedures to practically any subject which opens access to domains of possibility which would be forever out of reach to a mind incapable of counting. — Wayfarer

If, the distinction between primary and secondary attributes is broken down in this way, so as to allow for the universality of mathematical applications, then why conclude that all is "of the object" rather than all is "of the mind". As javi2541997;511801 indicates, Berkeley demonstrated that all is "of the mind" is the more logical conclusion. The other conclusion, that all is "of the object", requires the unsubstantiated assumption made by Kant, of the thing-in-itself, noumena, an assumption rejected by skeptics.

I distinctly did NOT say that. And you put that misrepresentation in quotes to fabricate something I did not say. — TonesInDeepFreeze

I apologize then, I misunderstood. I thought you meant that "2+1" could be interpreted as eithe of the following, (1) or (2).

For a while, in order not to split hairs, I went along with your term 'process', even though you have not defined it. That was okay for a while, but I was concerned that it would cause confusion, since there are actually two different notions: (1) a function. (2) a procedure for determining the value of a function applied to an argument. (I did touch on this earlier.) — TonesInDeepFreeze

Now I realize you are insisting that it is neither.

'The father of Peter Fonda' denotes the value of the function (call it 'the father of function') applied to the argument Peter Fonda. That value is Henry Fonda. — TonesInDeepFreeze

This is clearly incorrect. "The father of Peter Fonda" denotes that there is a person who has the position, the special relationship of being the father of the mentioned person, and this person who is the father of the mentioned person is your subject. It does not say that this person is Henry Fonda, so you cannot jump to that conclusion. If you knew someone named Henry Fonda, it would be a logical fallacy to jump to the conclusion that this man is the referred subject. You have not made the required logical connection, to determine that your subject is the same person as the one you know as Henry Fonda.

For about the seventh time now: '2+1' denotes the value of the function.. — TonesInDeepFreeze

This is false as well, and you just don't seem to get it. Take a look at your example of "the father of Peter Fonda". The thing which you claim as "the value", is clearly not signified, because the premise required to produce the logical conclusion is not stated in the argument. We need a further "unstated" (that's the way I use quotations, to signify special significance) premise to make your assertion a valid conclusion. In your example, the required premise might be "the person you know as Henry Fonda is the father of Peter Fonda". Then you can validly conclude that when some one says "the father of Peter Fonda", this is the person you know as Henry Fonda.

The thing which you seem to have no respect for, is the fact that "the father of Peter Fonda" does not refer to "Henry Fonda". This is very clear from the fact that one stated premise in a logical argument cannot refer to a conclusion. "Socrates is a man" does not refer to the conclusion "Socrates is mortal". That is because the expression does not include everything required to make that reference. Nor does "2+1" refer to the value signified by "3", because it does not include everything required to make that reference.

When we jump to a logical conclusion without stating the required premises, error is possible. You know someone named "Henry Fonda"; you jump to the conclusion that this is the man referred to by "the father of Peter Fonda", and mistake is possible. Rigorous logic seeks to exclude the possibility of mistake, not to create the possibility of mistake. The principles you are arguing for create the possibility of mistake by removing the need for the statement of premises. If some premises can be taken for granted, and not stated, as you seem to believe, then those premises cannot be judged for truth of falsity, and error is possible.

But '2+1' is not a description of a procedure. — TonesInDeepFreeze

Well, clearly "2+1" does not refer to a value. That is an invalid conclusion as I explained above. So, if it does not refer to a procedure, as I think it does, is it possible that we can find a compromise?

The term itself doesn't denote that it has a result. — TonesInDeepFreeze

OK, so what in the expression "2+1" denotes that there is a result. Grand Minnow was insisting that the expression denotes a result. I don't see it in the signification. Now you're pretending to be someone else, so that your inconsistency is not so glaring. Grand Minnow can argue that a result is signified and Deep Freeze can argue that an operation is signified. How's that?

The usage "result of an operation" is an informal way of referring to the value of the function for the arguments. — TonesInDeepFreeze

A function is a process. Grand Minnow kept insisting that "2+1" does not signify a process. That's why I say there is inconsistency. But clearly an "operation" or "function" is a process, and that's what is signified with "+".

The "value" of the function is not signified, because it must be figured out by carrying out the operation which is signified. If I say add some sugar to water, and bring it to a boil, the value (result) of that operation is syrup. But I'm not telling you "syrup", I'm telling you the procedure to make it. To obtain that value, syrup, you must carry out the operation referred to first.

For a while, in order not to split hairs, I went along with your term 'process', even though you have not defined it. That was okay for a while, but I was concerned that it would cause confusion, since there are actually two different notions: (1) a function. (2) a procedure for determining the value of a function applied to an argument. (I did touch on this earlier.)

So I'm not going to go along with your undefined terminology 'process'. Instead I'll use 'operation' (meaning a function) and 'procedure' (meaning an algorithm). — TonesInDeepFreeze

OK, I'm fine with "operation", so long as you recognize that what is signified is a a procedure, or operation, and as you say, this is "a procedure for determining the value of a function applied to an argument". The value is not signified, the "procedure for determining the value" is what is signified. Do you agree?

Do you even know what the use-mention distinction is? I — TonesInDeepFreeze

Of course I do. In philosophy we use a different convention. I use " " to signify a concept rather than a physical thing. I'm trying to conform to your convention but I'm a bit sloppy and missed one. Call it a typo.

There are an infinite number of ways to refer to the number 3. That doesn't mean they don't refer! Your argument is so daft! — TonesInDeepFreeze

Here's your inconsistency. You distinctly said "2+1" refers to "a procedure for determining the value of a function applied to an argument."

Now it's my turn to ask you, do you understand the difference between a procedure (function, or operation), and an object? Aristotle demonstrated a fundamental incompatibility between these two. A procedure cannot be an object, and an object cannot be a procedure because of this fundamental incompatibility. If "3" refers to the number three, and this is an object, then the procedure for determining a value, referred to with "2+1", cannot be the same thing as what is referred to with "3".

No he doesn't. If he does, he's wasting precious billable seconds. Instead, he just goes ahead to add the numbers. — TonesInDeepFreeze

Have you ever seen a ledger? Every account must be stated and balanced. Call it redundancy if you want, but there must be no room for error.

Are you serious? Are you trolling? — TonesInDeepFreeze

Of course I'm serious. You just told me there is an infinite number of ways to say "2+1", and I assume an infinite number of ways to say "3+1", so I ask you what distinguishes one from the other? Why is "3+1" not just another one of the infinite ways of saying "2+1"? That you do not answer means that you do not know.

No, the term '500+ 894+202' already denotes 1596. — TonesInDeepFreeze

Here is your inconsistency. Above, you said things like "500+ 894+202" denote "a procedure for determining the value of a function applied to an argument", which I accept. Now you are claiming that it actually signifies the value. What you say now is clearly false, because the procedure must be carried out before that value is derived.

It's just that the accountant doesn't know that until he performs the addition. The term doesn't start denoting only upon the knowledge of the account. The term doesn't spring into denotation every time some human being or computer somewhere in the world does a calculation. — TonesInDeepFreeze

What the person knows, is that "500+ 894+202" signifies the operation required to determine the value. Your claim that "500+ 894+202" represents the value is nothing but a misrepresentation. And, if you proceed in a philosophical argument with that misrepresentation of what "500+ 894+202" is known to signify, you are guilty of equivocation.

I left it out because it is a nonrestrictive clause. Further, any necessary termination is for a reason external to the process itself, usually to make an approximation. — tim wood

I don't know what you're talking about tim. To say "X will necessarily be terminated" seems very restrictive to me. Obviously, the cause of termination of the process is external, that's Newton's first law. But how's that relevant?

Ok. "Is said" seems gratuitous. π, I'm told, in decimal expansion never ends. To use it as a number, it's usually truncated at some point. That is just a number, nothing infinite about it at all, potential or otherwise. But why confuse the two? One stands in for the other to get an approximation. What is the issue about "potential" anything? — tim wood

This was Ryan's term. You'd have to go back to see what Ryan was talking about. Essentially Ryan suggested replacing infinities with infinite processes. Since the supposed infinite processes could never be completed they are assumed to be potentially infinite. I argued that every supposed potentially infinite process will for some reason or another, at some point be terminated. If this is the case then it is incorrect to even call them potentially infinite.

"But a process which is said to be potentially infinite, cannot truthfully be said to be potentially infinite." Eh? Sure it can. Or do you mean that the never ending decimal expression of π actually ends? — tim wood

You left out the important phrase: "which will necessarily be terminated". The infinite process would continue forever, by definition. Since forever never arrives Ryan says we ought to call it potentially infinite. Ryan suggested that we could put an end to a potentially infinite process, by rounding of pi for example, yet still say that it is potentially infinite. Obviously though, if someone puts an end to a process, it is not potentially infinite. So Ryan proceeded to distinguish between the rule which produces the supposed potentially infinite process, and the process itself, trying to place the potential for infinity within the rule rather than the process.

I'm offering you help here, though I doubt you'll take it in. — GrandMinnow

If it makes sense, I'll take it. But so far all you've offered is inconsistency. Let me see if I can follow you.

'2+1' denotes the result of the operation — GrandMinnow

I see the number 2 denoted, and the operation + denoted, and the number 1 denoted. So there is clearly an operation denoted. What denotes that the operation has a result?

Would you agree that a finite operation is distinct from an infinite operation, the one having a result, the other not? If this is the case then there is a need to distinguish between an operation with a result, and one without a result.

You got it exactly backwards. Our method does not lead to '2+1' denoting infinitely many things. '2+1' denotes exactly one thing. On the other hand, 2+1 is denoted infinitely many ways:

2+1 is denoted by '2+1'

2+1 is denoted by '3'

2+1 is denoted 'sqrt(9)'

2+1 is denoted by '((100-40)/3)-17'

etc. — GrandMinnow

It appears to me, like 2+1 demotes exactly nothing then. You can say the same thing in an infinite number of different ways, but none of these ways refer to anything. Each expression simply say I am the same as the infinity of others. If one of them refers to anything real, then they all must refer to something real, and you have an infinity of equivocation, with an infinity of different things referred to by on signification. Even the numeral "3" must refer to the result of an operation, exactly as the others, so there is nothing to validate any object

So what makes 2+1 different from 3+1 then? Each can be said in an infinity of different ways, and there is nothing which is actually being referred to be either one. How can they differ?

If we want to know how much a company did in sales, the accountant starts by seeing that the company got 500 dollars from Acme Corp., and 894 dollars from Babco Corp, and 202 dollars from Champco Corp. Then the accountant reports:

500+894+202 = 1596 — GrandMinnow

This is incorrect. The accountant writes out 500+ 894+202=?, or x, or some other placeholder for the unknown, because the sum is unknown. But if it were like you say, that "500+ 894+202" already says 1596, then the accountant would not have to sum up the numbers, because the result of the operation would already be stated.

That's why your way of looking at things, if it were true, would render the equation completely unnecessary and redundant. By the time the left side was stated, (500+ 894+202) the right side would already be known, because you claim that the left side states the result of an operation. Clearly this is false, because equations contain unknowns, and this is how we solve problems, by carrying out the operations required to determine the unknowns. Obviously you are spinning a web of deceit.

One wouldn't honestly claim to know that the equation is true until one worked it out that it is true. Or to find a right side without '+' in it, then first one might have to perform the addition on the left side. This doesn't vitiate anything I've said. — GrandMinnow

How do you apprehend a need to work things out? If "2+1" says sqrt(9), how is there any need to work out any equivalencies?