Well, thing is... most folk will disagree. — Banno

Really, you think that two equal people, because they are equal, are the same person. I think you are very wrong to think that most people would disagree with me. I think that there is a very small, and distinct culture of people who claim that equal things are necessarily the same thing. You might happen to be a part of this culture, and you like to believe that most folk are like you, to make it feel like your culture is more important than it really is.

But the terms "equal" and "the same" can also be used synonymously. Is this the basis of your dispute? — Luke

My argument is that this use, to use "equal" and "the same" synonymously, is in violation of the law of identity, and therefore unsuitable for any system of logic.

Yeah, you are better off skipping the question and answering something else. Nice dodge.

Just so we are clear,

...it would generally be accepted that Hesperus = Phosphorus. Would you accept that? — Banno

Well, thing is... most folk will disagree. — Banno

Did you read my quote from Wikipedia?

"The equals sign or equality sign, =, is a mathematical symbol used to indicate equality. It was invented in 1557 by Robert Recorde. In an equation, the equals sign is placed between two expressions that have the same value, or for which one studies the conditions under which they have the same value." — Metaphysician Undercover

Since Wikipedia agrees with me. it seems highly unlikely that most folk would disagree with me. Notice how "same" is qualified with "value". The right and the left side of the equation have "the same value". Anyone who knows how to read English knows that this does not mean that they are the same.

SO, for example, it would generally be accepted that Hesperus = Phosphorus. Would you accept that? — Banno

I already went through this. If two symbols refer to the same thing, they are equal. But it is a fallacy of affirming the consequent to say that if two things are equal they are therefore the same.

Righto, then. All cleared up.

My argument is that this use, to use "equal" and "the same" synonymously, is in violation of the law of identity, and therefore unsuitable for any system of logic. — Metaphysician Undercover

What does the use of these terms have to do with the law of identity?

You seem to be saying that "equal" is a suitable term in relation to the law of identity, but "the same" is not. That is, you are attempting to prescribe the use of these terms to be non-synonymous. But why can't these terms be used synonymously in relation to the law of identity?

What does the use of these terms have to do with the law of identity? — Luke

The law of identity is a fundamental law of logic, established to prevent deceptive use of terms. such as equivocation. We are discussing the use of terms within a logical system, therefore the law of identity is applicable.

But why can't these terms be used synonymously in relation to the law of identity? — Luke

Yes, "equal" and "the same" could be used synonymously. But if we adhere to "the same" as defined by the law of identity, and consider "equal" as it is used in mathematics, they are not synonyms. That's the point, this definition provides a false representation. That definition is a false premise. The equals sign is not used to indicate that the right and left side of the equation refer to the same thing, it is used as described in my quote from Wikipedia, to indicate that the right and left side have the same value. Therefore, to define "equal" as indicating "the same", for the sake of a logical argument concerning the nature of mathematical systems, or "mathematical objects", is to start with a false premise. Such arguments which take this definition as a premise are unsound because this is not the way "equal" is used in mathematics.

It appears to me like some people participating in this thread have seen "equal" defined in this faulty way, so according to that definition they conclude that "equal" is actually used in this way in mathematics. Do you see the equivocal sophistry here? Because it is defined in this way, they are misled into the conclusion that it is used in this way. They clearly have not taken any time to investigate, observe, and notice the reality of how "equal" is actually used in mathematics. If they had, they would see that mathematicians use "=" to indicate that two things have the same mathematical (quantitative) value, not to indicate that the things designated by the two sides of the equation are the same thing. So when these people produce a logical argument about the nature of mathematics, or "mathematical objects" and start with this definition as a premise, their arguments are completely unsound, being based in that false premise, which is not representative of mathematics.

I explained to fishfry already, that if the two sides of an equation actually referred to the very same thing, equations would be totally useless in application. To resolve problems, equations are used to compare distinct situations, knowns are compared with unknowns, producing information about the unknown situation through comparison with the known. If both were exactly the same, such a comparison would be unnecessary because we would know that the supposed unknown is really exactly the same as the known, and therefore really known as the same. The equation would give us no new information.

mathematicians use "=" to indicate that two things have the same mathematical (quantitative) value, not to indicate that the things designated by the two sides of the equation are the same thing. — Metaphysician Undercover

You're arguing that both sides of an equation have equal (the same) value, but that they are different expressions of that value? Sounds reasonable.

You're arguing that both sides of an equation have equal (the same) value, but that they are different expressions of that value? Sounds reasonable. — Luke

What are you talking about? How can you be so daft in your interpretation, when I've explicitly stated over and over again what I am arguing. Your misinterpretation appears to be completely intentional, an intentional straw man.

I am arguing that what is represented by the right side of the equation has "the same value" as what is represented by the left side of the equation. But we cannot conclude that what is represented by the right side of the equation is "the same" as what is represented by the left side of the equation, if we adhere to the law of identity in defining what "the same" means. Do you understand the difference between "having the same value", and "being the same"?

I think you understand what I am saying, so don't bother replying with another straw man, fake interpretation. But if you have a relevant objection then I'd be glad to see it..

Leibniz put it beautifully: he had a book of Euclid's elements on his table. That book came from an earlier copy of that book, which in turn was copied from an earlier manuscript and so on...

So of course, there must be a first manuscript, by the hand of Euclid himself, else none of the other manuscripts could have any objective reality whatsoever.

Do you understand the difference between "having the same value", and "being the same"? — Metaphysician Undercover

Two diamonds have the same value, but are not the same. They are equal in value. They appear equal when viewed without high magnification. Just babbling, pay no mind. :roll:

Thanks jgill, the others just don't seem to get it. We could say the same thing about all forms of measurement. Two different things are two kilograms, or two meters long, so they would be equal in respect to the particular system of evaluation. Likewise, "2+2" and "4" are equal in respect to quantitative value, but this does not mean that the same thing is signified by each of these two, only that they are equal in that particular system of evaluation.

How can you be so daft

Sense of irony is not strong with this one, I see.

The equal sign means that the two sides are equal, just as "equal" indicates. Here's what Wikipedia says:
"The equals sign or equality sign, =, is a mathematical symbol used to indicate equality. It was invented in 1557 by Robert Recorde. In an equation, the equals sign is placed between two expressions that have the same value, or for which one studies the conditions under which they have the same value." — Metaphysician Undercover

Yes that's very cool. I do happen to know that Robert Recorde invented the equal sign. I keep hoping someone will ask me someday.

Clearly you are wrong to say that it's everyone's rule, that the equal sign means that the right and left sides refer to the same object. This rule is an expression of your idiosyncrasy. — Metaphysician Undercover

You're wrong mathematically and we are not getting anywhere. I tried to beg off the conv a while back but seem to be having difficulty executing on my intention. We're not making progress. I have nothing new to say.

I already went through your converse error, but I'll explain it to you again, as you don't seem to get it for some reason. I believe the formal fallacy is called affirming the consequent. If two symbols refer to the same thing, then there is necessarily equality between what the symbols refer to. But this does not mean that two equal things are the same thing. Do you understand this so far? — Metaphysician Undercover

I understand your belief that two things that are mathematically equal are not the same thing. You're wrong. If X = Y then X and Y necessarily refer to the same abstract object. There is no question about it.

Many things are equal, like two human beings, two dogs, or two cats, in the sense that the two distinct things can be given the same value. A human being might be equal to a dog if the evaluation criteria is being an animal. — Metaphysician Undercover

Yes, and this is called the fallacy of equivocation. Using the same word, equality, with two distinct meanings within the same argument. In human affairs, equality has a different meaning. When Thomas Jefferson wrote that "All men are created equal," he of course did NOT mean that they were mathematically equal, as 2 + 2 and 4 are equal; but rather equal under God and nature as human beings.

Mathematically, no two human beings are equal. They could be equivalent modulo various properties. If they both live in California or if they are of the same race or work in the same profession or so forth, we could call them equivalent. Or if we want to assert the desirability of the state of affairs in which they each have the same chances and possibilities in life, that would be another form of human equality.

But it's not mathematical equality. I'd like to say it's beneath you to stoop to such a low rhetorical trick. But I guess it's not beneath you after all. Frankly it's beneath ME to have to explain this in words, it should be obvious that mathematical equality and Jeffersonian equality are not the same thing.

And do you see that the equal sign means that the right and left side are equal, as the Wikipedia articles says? How can you conclude, without the fallacy of affirming the consequent, that two equal things are necessarily the same thing? — Metaphysician Undercover

Because that's what mathematical equality is. That's how mathematicians define equality. Ultimately you have the same set on both sides of an equation. Once again you erroneously take your ignorance of mathematics as profundity in philosophy.

If two expressions do NOT refer to the same abstract object, then they are NOT equal.

That is a mathematical fact. Though of course it is not a fact in the Jeffersonian sense of equality. Two distinct people could be equal under the law. Equivocation. Same word different meaning depending on context.

Right, now if you were in Green Bay, and followed the directions of how to get from Milwaukee to Sheboygan, you would not get there from Green Bay. Likewise, if you were at 6, and followed the directions of how to get to 4 from 2, i.e. "+2", you would not get to 4 from 6, following those directions. — Metaphysician Undercover

WHAT? That is completely nonresponsive to the point, which is that Sheboygan is still Sheboygan regardless of how you got there.

Nevermind. If you respond I'm going to try not to. Just for my own sanity. Interacting with you is fun in a tongue-in-my-sore-tooth kind of way, but I need a little break please.

You're arguing that both sides of an equation have equal (the same) value, but that they are different expressions of that value? Sounds reasonable.
— Luke

What are you talking about? How can you be so daft in your interpretation...

Do you understand the difference between "having the same value", and "being the same"? — Metaphysician Undercover

Yes, that’s why I said that both sides of the equation are different expressions of the same value. What part do you disagree with?

You're wrong mathematically and we are not getting anywhere. I tried to beg off the conv a while back but seem to be having difficulty executing on my intention. We're not making progress. I have nothing new to say. — fishfry

Why does Wikipedia agree with me if I am wrong? I can assure you that I didn't write the page.

If X = Y then X and Y necessarily refer to the same abstract object. There is no question about it. — fishfry

Why does Wikipedia disagree with you and say that under specific conditions the two things referred to have the same value, not that they are the same thing?

hen Thomas Jefferson wrote that "All men are created equal," he of course did NOT mean that they were mathematically equal, as 2 + 2 and 4 are equal; but rather equal under God and nature as human beings. — fishfry

Yes, two men are mathematically equal. One man is mathematically equal to another man. Each being "one" man, indicates that they are mathematically equal. Furthermore, each person, having the right to vote is mathematically equal to every other person, so they count the votes to see who wins the election. It is clearly the case that human beings are mathematically equal in a democracy, each having one vote. There is no other form of equality. "equal under God and nature as human beings" is just a fancy way of saying that under the condition of being human, we are all mathematically equal.

But it's not mathematical equality. I'd like to say it's beneath you to stoop to such a low rhetorical trick. But I guess it's not beneath you after all. Frankly it's beneath ME to have to explain this in words, it should be obvious. — fishfry

If you really think that two men are not mathematically equal to another two men, or five dogs are not mathematically equal to another five dogs, or an object of five kilograms is not mathematically equal to another object of five kilograms, or a five kilometer stretch of highway is not mathematically equal to another five kilometer stretch of highway, such that there is some other meaning to "mathematically equal", then tell me what that other meaning of "mathematically equal" is.

Because that's what mathematical equality is. That's how mathematicians define equality. Ultimately you have the same set on both sides of an equation. Once again you erroneously take your ignorance of mathematics as profundity in philosophy. — fishfry

Oh, now I see your problem. You think that because some set theory defines "mathematical equality" in this way, then this is what equality means in mathematics. As I explained to Banno, and to you already, this is a faulty definition, the one employed by the axiom of extensionality. It does not truthfully represent how "equal" is used in mathematics. It was devised for some other purpose, not for the purpose of representing how "equal" is used in mathematics. So, some set theorist defined "equal" in this completely faulty way, totally unrepresentative of how "equal" is actually used in mathematics, and you employ this definition as a premise in your argument against me. Your premise is a false premise, as evidenced by the Wikipedia quote. Your argument is unsound, being based in the false premise that mathematicians use "equal" in that way. They do not.

Yes, that’s why I said that both sides of the equation are different expressions of the same value. What part do you disagree with? — Luke

The different expressions represent different things with the same value. "2+2" says something different, it represents something different from what "4" represents, though we say that the two distinct things represented have the same value within the arithmetical system.' I don't really know what you would mean by "different expressions of the same value". That sounds like you are assigning value to the expressions themselves, rather than to the things represented by the expressions.

But it was not a representation of what I am arguing anyway, that's why I criticized it as a straw man. I am arguing that the different expressions are not different ways of representing the same thing, as in Banno's example of "Hesperus = Phosphorus". That is what fishfry is arguing, that you have the same thing represented on both sides of the equation. Even if we try to reduce "2+2", and "4" to being simple representations of "value", as you seem to want to do, we'd have to qualify what type of value. So we might say that they both represent "the same quantitative value", or "mathematical value". But we still cannot say that they have "the same value" in an absolute sense because there are all sorts of different value systems. That's why the Wikipedia quote refers to "the conditions under which they have the same value". "Value" is relative to a particular system of evaluation. So when it is said that "they have the same value", it is implied that they have the same value within a particular system of evaluation, the mathematical system.

Why does Wikipedia agree with me if I am wrong? I can assure you that I didn't write the page. — Metaphysician Undercover

I am hardly responsible for anything on Wikipedia, let alone your own interpretations of same. I didn't spend any time looking at that Wiki page. I know the math. Critiquing Wiki on this issue is not of interest to me at the moment. Some Wiki pages are better than others. Especially when it comes to the lead sentence or paragraph of a technical subject, many simplifications are made and many subtleties ignored. They're trying to give naive readers a sense of an idea. Summary and overview. Not to be taken literally and used as some kind of trump card in an online discussion. Jeez man you are reaching.

I think they operate quite rightly, provisionally treating the theory as if it is the thing is part of how it works I think. If the discussion we'd have is "what properties of a model can be treated as standing in for a property or behaviour of the thing", that'd be quite different from "are all models merely epistemic" - the first would actually be about the uncertainty principle, the second is a much broader realism vs anti-realism of scientific content debate. If you and I have to go through the latter to get to the former, that's fine with me, both are interesting. — fdrake

I'm really interested in diving into this but haven't had time to think about it much or read your other post to me on the subject. Instead of deferring my response to your posts indefinitely in the false hope of eventually saying something clever, for now I'll just retreat into what I think my position is.

The Planck scale is the scale at which our current physical theories can not be applied. We don't know what's going on at distances and times smaller than the Planck length and time, respectively. That's why I say it's epistemic.

I am not sure I follow the argument from Fourier series to saying that "therefore the Planck scale is ontic."

I don't think I'm conflating the problem of models in general with the idea of the Planck scale. Except that you can't apply a model outside of its domain of applicability, which is the same as with the Planck scale. Good question. With models in general, they don't so explicitly tell us what we can't know. If we have a model, people can disagree about whether it applies to a given situation. But with the Planck scale, everyone agrees on where the theory applies and where it doesn't. I'm afraid I don't have any better thoughts on this topic at the moment. But it is important to me because I have a strong belief that the Planck limits tell us what we can know, not what is. I wish I understood your argument better.

The different expressions represent different things with the same value... I don't really know what you would mean by "different expressions of the same value". — Metaphysician Undercover

Do you understand what you mean by it (in your first sentence of the quote above)? Why do you think I mean anything different? You’re arguing against yourself here.

So when it is said that "they have the same value", it is implied that they have the same value within a particular system of evaluation, the mathematical system. — Metaphysician Undercover

I thought we were all talking about “the mathematical system”?

Do you understand what you mean by it (in your first sentence of the quote above)? Why do you think I mean anything different? You’re arguing against yourself here. — Luke

Sorry Luke, I have no Idea what you're talking about here. I did not use "it", and you're not making clear what "it" refers to. I said something different from what you said, so I think it's highly unlikely that we both meant the same thing. Since we did not say the same thing, it makes no sense to use "it" to refer to what both of us said.

I thought we were all talking about “the mathematical system”? — Luke

Just making sure that you understood this. You're statements appeared somewhat misleading, as if you were talking about "value" in an absolute sense, implying that numbers express the value of something in an absolute sense, rather than a specific type of value.

Do you understand the difference between "having the same value", and "being the same"?
— Metaphysician Undercover

Yes, that’s why I said that both sides of the equation are different expressions of the same value. What part do you disagree with?
— Luke

The different expressions represent different things with the same value... I don't really know what you would mean by "different expressions of the same value".
— Metaphysician Undercover

Do you understand what you mean by it (in your first sentence of the quote above)? Why do you think I mean anything different? You’re arguing against yourself here.
— Luke

Sorry Luke, I have no Idea what you're talking about here. I did not use "it", and you're not making clear what "it" refers to. — Metaphysician Undercover

If you can't even follow the discussion, then never mind.

The different expressions represent different things with the same value. "2+2" says something different, it represents something different from what "4" represents, though we say that the two distinct things represented have the same value within the arithmetical system.' I don't really know what you would mean by "different expressions of the same value". That sounds like you are assigning value to the expressions themselves, rather than to the things represented by the expressions. — Metaphysician Undercover

:D

Since 4=2+2, 2+2 and 4 are interchangeable.
Doesn't matter if you write y=3x+2+2 or y=3x+4, though the latter is a bit shorter.
Which, by the way, google plots like so:



Hopefully it's not too much for you; from memory, it's something like late elementary school / early high-school material.

Why does Wikipedia agree with me if I am wrong? I can assure you that I didn't write the page. — Metaphysician Undercover

Prompted by our conversation I went and looked at what Wikipedia says. It 100% absolutely agrees with what I said. You're being quite disingenuous to claim otherwise.

From Equality (mathematics):

"In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object. [my emphasis] The equality between A and B is written A = B, and pronounced A equals B. The symbol "=" is called an "equals sign". Two objects that are not equal are said to be distinct. [my emphasis]"

That's exactly what I'm telling you. Baffled that you claim that Wiki says otherwise when plainly it says exactly what I'm saying

Since 4=2+2, 2+2 and 4 are interchangeable. — jorndoe

I don't see how that's relevant. Since you and I are both human beings, we're interchangeable when someone says bring me a human being. It really means very little.



Sorry, I don't normally use Wikipedia and I only looked at the page on "equals sign".

Nevertheless, what is at issue is whether a so-called "mathematical object" is an object identifiable according to the law of identity. It is not, because two equal, but different things, such as the addition operation of 2+2, and the number 4 are said to be the same object. Therefore, despite what the Wikipedia quote indicates, and many mathematicians might claim, these two different things, the operation represented by "2+2", and the number represented by "4", cannot be "the same" if we adhere to the law of identity, which denies that two distinct things are the same object... The mathematical axioms which state that these two distinct things are the same thing are nothing more than deception. I know you'll continue in your denial, but so be it.

I don't see how that's relevant. Since you and I are both human beings, we're interchangeable when someone says bring me a human being. It really means very little. — Metaphysician Undercover

Confusing quantity and predication (as well)? Try differentiating, see what happens.

Quantity is a predication. There is no such thing as quantity, without it being a quantity of something. I think that's half the problem here, some people seem to think that quantity is a thing in itself, rather than a predication, as all measurements are. That way, instead of looking at what "2+2" really represents, they just assume that it represents "a quantity".

Sorry, I don't normally use Wikipedia — Metaphysician Undercover

But you're not above trying to quote-mine an off-topic page to make a point.

and I only looked at the page on "equals sign". — Metaphysician Undercover

But the equal sign is not the topic of discussion. The concept of mathematical equality is. No wonder you name-checked Robert Recorde. You looked up the wrong thing.

Nevertheless, what is at issue is whether a so-called "mathematical object" is an object identifiable according to the law of identity. It is not, because two equal, but different things, such as the addition operation of 2+2, and the number 4 are said to be the same object. Therefore, despite what the Wikipedia quote indicates, and many mathematicians might claim, these two different things, the operation represented by "2+2", and the number represented by "4", cannot be "the same" if we adhere to the law of identity, which denies that two distinct things are the same object... The mathematical axioms which state that these two distinct things are the same thing are nothing more than deception. I know you'll continue in your denial, but so be it. — Metaphysician Undercover

Completely ignoring the point I just made. Anyway I just wanted to note that you looked at the wrong Wiki article and when pointed at the right one, which 100% supports my point of view, you change the subject.

2 + 2 and 4 point to or refer to or represent the exact same object. It's not possible to do math without that understanding.

But the equal sign is not the topic of discussion. — fishfry

It was when I first produced that quote. We were interpreting "2+2=4", a phrase which the equals sign is a part of.

2 + 2 and 4 point to or refer to or represent the exact same object. It's not possible to do math without that understanding. — fishfry

That is bull shit. Clearly I do not have that understanding, which you claim is necessary, in fact I insist it's a misunderstanding. Yet I can get four from two plus two. Therefore I can do math without that misunderstanding, which you say is required to do math. I don't deny that two plus two equals four.

I only assert the obvious, that "2+2", which represents an operation of addition, does not represent the same thing as "4", which does not represent an operation of addition. If you, and your mathematician friends, want to keep living your deluded lives of believing that they do represent the same thing, then so be it.

That is bull shit. — Metaphysician Undercover

Tell it to Wiki, which yesterday you claimed was the ultimate authority on these matters and which, once you realized it agrees with me, you no longer have any interest in.

so be it. — Metaphysician Undercover

So be it.

There is no such thing as quantity, without it being a quantity of something ... — Metaphysician Undercover

... or of anything/whatever, hence the utility of a calculator.
Say, a set of my left ear, that soccer match, the Moon, and the experience of vanilla taste I had the other day when eating icecream, comes to 4 in quantity; kind of trivial to count.

Quantity is a predication. There is no such thing as quantity, without it being a quantity of something. I think that's half the problem here, some people seem to think that quantity is a thing in itself, rather than a predication, as all measurements are. That way, instead of looking at what "2+2" really represents, they just assume that it represents "a quantity". — Metaphysician Undercover

You're now confusing quantity, predication, measurement, ...
Say, a set of you and I comes to a quantity of 2, $|\{you, I\}| = 2$; kind of trivial to count.
Say, where $\phi$ = is human (predicate), it so happens that $\phi(you) \wedge \phi(I)$, I assume.
You're making a wicked mess of things. :confused: