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.

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

My points don't depend on whether equality is or is not independent of human judgement.

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. — Metaphysician Undercover

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'.

Your argument is ridiculous.

now you admit that you do not mean that they are "necessarily" the same — Metaphysician Undercover

I don't "admit" in the sense of conceding or retracting some earlier point. I just never stated regarding necessity to begin with, and I don't state now because it would require a discussion about modality that is not needed to present the basic mathematical framework I've mentioned.

What good is such a principle? — Metaphysician Undercover

It provides a clear and straightforward framework for doing mathematics.

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? — Metaphysician Undercover

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

I don't say that '2+1' and '3' are equal. I say that 2+1 and 3 are equal.

But in a context where '2', '+', and '3' were symbols not standing for the number 2, the addition operation, and the number 3, then it may not be the case that '2+1' and '3' denote the same number in that context. And it is not necessary that symbols always denote the same. Denotation of symbols is by stipulation or convention not by necessity.

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. — Metaphysician Undercover

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

(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.

Can you show me through your "method of models" — Metaphysician Undercover

I can't cram it all into a post or even several posts. 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

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?

(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". — Metaphysician Undercover

The question at that point was about Henry Fonda and names for Henry Fonda, not numbers and names for numbers. You objected to my Herny Fonda example onto itself.

You're lost in the conversation.

Now I'm waiting for proof that "2+1" refers to the same object as "3". — Metaphysician Undercover

You skipped again. I told you that proof of same denotation is finalized in the method of models. You don't have to wait around to find out about it - you can find it many books. As I said, as you also skipped:

"I can't cram it all into a post or even several posts. 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."

I wouldn't expect someone to teach me, in the confines of a posting forum, the subject of molecular biology. You would be foolish to think I should do it for you about mathematical logic. You could educate yourself!

I just received a spiritual message from Henry Fonda. He suggests you move on to Kirk Douglas, as Kirk feels neglected. :smile:

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?

Those are concepts instrumental to a firm understanding of the method of models.

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

Under "Scientific Modelling" in Wikipedia I find this:

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

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

I tend to think it is something you made up as a ruse — Metaphysician Undercover

You tend to think irrationally or not at all.

Below are some articles and a couple of online books. For a more systematic study, if I recall, earlier this thread I mentioned particular books I most recommend.

http://page.mi.fu-berlin.de/raut/logic3/announce.pdf

https://en.wikipedia.org/wiki/Structure_(mathematical_logic)

https://plato.stanford.edu/entries/model-theory/

https://plato.stanford.edu/entries/modeltheory-fo/

http://www.math.toronto.edu/weiss/model_theory.pdf

https://en.citizendium.org/wiki/Structure_(mathematical_logic)

https://faculty.math.illinois.edu/~vddries/main.pdf

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

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

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

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

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

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

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

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

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

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

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

I can't help it if your terminology is a little off the beaten path. You kept referring to a "method of models", and I couldn't even find that on google. — Metaphysician Undercover

So you opted to suggest that I'm lying about the whole thing instead of just asking "Would you please provide some links?" Another example of your jejune approach to mathematics and discussion about it.

And what I wrote:

You need to read a book or other systematic presentation of mathematical logic in which the method of models is explained step by step — TonesInDeepFreeze

[emphases added]

So you could have easily searched 'mathematical logic method of models' or just asked me. Instead you burden me with the suggestion that I'm lying about the subject because you can't be bothered to make a reasonable search. Then when you look at the links, you only skim a few parts of them, then pick a few items in them out of context and misconstrue them. And you're welcome, by the way, for the links I provided, even though your response to them is itself incorrect - including incorrectly summarizing their level, relevance, content and import.

The way to learn the subject is not by perusing articles and parts of books out of context, but rather by first starting with at an introductory level book on symbolic logic.

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

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