I'm not sure what was funny about that except that it's perfectly computable and doesn't require choice at all. Did I understand that and/or get the math right? And on a practical level we could input the resolution of the printer or display device, and calculate exactly how many iterations of the curve would show up as solid black. And it would of course be a finite number, so definitely computable and not needing any mathematical foundations beyond counting to a large but finite number. That's way less than the Peano axioms. An ultrafinitist, someone who doesn't believe in the infinitude of sufficiently large sets, would be able to compute the space filling curve to the point that it appeared black on the display. I'd be willing to guess you don't need that many iterations. Your eye couldn't make out the lines, it would all black pretty soon. — fishfry

You know, no one I've told the story too ever thought about it like that. The thing that I found funny about it was that the other prof didn't doubt that the eccentric prof would go in MS paint and make a black square to represent a space filling curve to some approximation, he criticised the ability to represent it exactly constructibly. You know a-priori that a sufficiently computed space filling curve in the unit square is indistinguishable from filling in the unit square in MS paint.

You've forgotten about summation. It might be the case that "4" represents 4-ness, but "2+2" represents a particular instance of the general rule of summation, not 4-ness. — Metaphysician Undercover

You are looking behind the symbols to the mathematics they represent. They are certainly equal in this regard. But if you look superficially at the compound symbol "2+2" and the singular symbol "4" as ink squiggles on paper, they clearly are not the same. But, of course, that's not your perspective. I am simply giving an instance when two things are equal in one sense, but not the same in another sense. You and fishfry can argue ad infinitum it seems. :roll:

You know a-priori that a sufficiently computed space filling curve in the unit square is indistinguishable from filling in the unit square in MS paint — fdrake

Then there is the sequence of continuously differentiable functions that converge uniformly on [0,1] to that line segment, but whose arc lengths go to infinity:

${{z}_{n}}(t)=t+i\frac{1}{2\sqrt{n}}Sin\left( n\pi t \right),\text{ }t:0\to 1$

No matter what the magnification, once chosen, n sufficiently large produces a straight line one unit long on the computer screen.

You guys still chatting about whether 2 + 2 = 4 or 2 + 2 ≠ 4 ? :D
Maybe putting it to a vote could be interesting — in The Lounge, though.
Typing 2+2 and 4-2 into a pocket calculator could be an accompanying exercise.

You guys still chatting about whether 2 + 2 = 4 or 2 + 2 ≠ 4 ? — jorndoe

There is no question that 2+2=4. Not even a fool would deny that. The question is whether "2+2" represents the same object as "4".

You are looking behind the symbols to the mathematics they represent. They are certainly equal in this regard. But if you look superficially at the compound symbol "2+2" and the singular symbol "4" as ink squiggles on paper, they clearly are not the same. But, of course, that's not your perspective. I am simply giving an instance when two things are equal in one sense, but not the same in another sense. You and fishfry can argue ad infinitum it seems. — jgill

I think it's true that fishfry and I will never agree, but the disagreement between you and I appears to be as to what constitutes a "mathematical object", or "Platonic ideal". Mathematicians may have defined "object" in such a way that "2+2" represents the very same object as "4", but I disagree with this definition, as it does not properly represent what a Platonic ideal really is. So I believe that the definition was manufactured for the purpose of doing what the mathematicians wanted to do with it, rather than with the purpose of representing what a Platonic ideal really is. That is not an acceptable way of doing logic, to manufacture premises which will support the desired conclusion. It is a case of petitio principii

The problem is, as Aristotle demonstrated, that there is an inherent and fundamental incompatibility between a process (becoming), and an object (being). The object is intelligible, and the process is fundamentally unintelligible. So a symbol like "+", or any type of operator, or function, does not represent an object, and represents something which is fundamentally unintelligible. And if we allow this element of unintelligibility into the object, to say for instance that "2+2" represents an object, then the object itself becomes fundamentally unintelligible. This might be the Kantian perspective, to represent the noumena (intelligible object), as fundamentally unintelligible, but it is not the Platonic perspective, and you can see how it is inherently contradictory to say that the intelligible object is fundamentally unintelligible.

In other words, if the Platonic ideal is supposed to be an eternal Form, meaning a truth which escapes from the corruption of time, it cannot be represented as containing within itself a temporal process. So these representations of Platonic mathematical objects, which allow processes (functions) to inhere within the object, are not proper representations. They are not proper because any such process, which inheres within an eternal object, must have infinite temporal extension, being a never ending, never changing infinite process, such as Aristotle's perfect circular motion. But this metaphysical idea has been demonstrated as a false one. Ideal perfection cannot be granted to a process.

,

"mathematical object", or "Platonic ideal" — Metaphysician Undercover

Abstract quantities (is the phrase)

but the disagreement between you and I appears to be as to what constitutes a "mathematical object", or "Platonic ideal" — Metaphysician Undercover

You and I don't really disagree, MU. I enjoy reading your posts, learning of philosophical perspectives I never considered the years I was a practicing mathematician. And I enjoy reading posts by fdrake and fishfry when they pertain to elements of mathematical thought of which I have only superficial knowledge.

If I had known colleagues who were concerned about Platonic ideals, irrational numbers, or transfinite set theory I might have more to offer, but those issues were at best peripheral to our interests. As one of approximately 36,133 descendants of Weierstrass I have been happy using the notions he and Cauchy championed. :cool:

Abstract quantities (is the phrase) — jorndoe

A number such as what is indicated by the numeral "4", is an abstract quantity. In the case of "2+2" , there are two distinct abstract quantities signified with "2" and "2", along with an operation signified with "+". Therefore it is incorrect to say that "4" represents the very same abstract quantity as "2+2", because clearly there are two distinct abstract quantities signified by "2+2", related to each other by the signified operation "+", not one quantity.

If I had known colleagues who were concerned about Platonic ideals, irrational numbers, or transfinite set theory I might have more to offer, but those issues were at best peripheral to our interests. — jgill

We all have our own interests. Since one of mine is philosophy, and mathematics really is not, I approach this issue from the perspective of being concerned about Platonic ideals rather than the associated mathematics. Many others in the forum will say to me, forget about what Platonic ideals are, or whether "Platonic ideals" is a valid concept, take the axioms for granted, and discuss the mathematics. But if the substance of the axioms is Platonic ideals, I want to understand the validation.

You know, no one I've told the story too ever thought about it like that. — fdrake

I have that happen to me a lot.

The fact that a different process can be utilized to make an object indicates that the process is not the same thing as the object. — Metaphysician Undercover

Are you identifying an object such as the number 4 with the process that "creates" it? I think that's a pretty big stretch. I can't imagine Aristotle wouldn't have been that wrong, so I'm guessing your interpretation is.

In other words you seem to be associating 4 with the "process that created it," so that a 4 stamped whole on God's forge is different than one made by jamming 2 and 2 together.

This notion makes no sense to me. The number 4 is the number 4, and it's inherent in its nature that it can be represented many different ways. Are you claiming Aristotle said that a thing is actually "thing plus method of making it?" And what makes him right and everyone else wrong?

Are you identifying an object such as the number 4 with the process that "creates" it? I think that's a pretty big stretch — fishfry

No I'm not making such an identity. But I'm saying that if you assert that "2+2" is the same object as "4" you are claiming such an identity. The process of adding two with two will make four, but it is not four. I am saying that the process which makes four is not the same as the number four itself. A cause is not the same thing as its effect. You are insisting that the two are the same.

The number 4 is the number 4, and it's inherent in its nature that it can be represented many different ways. — fishfry

I really can't believe that you do not see the difference between what "2+2" represents and what "4" represents. The former indicates two quantities of two, with an operation of addition also indicated. The latter indicates one quantity of four. To interpret "2+2" as representing the number four is very clearly a misinterpretation.

The proof is that we can represent the relationship between two distinct 2s in many different ways, such as "I have 2 dogs, and 2 cats", "2,2", "2+2", "2X2", etc. In all such instances of representing two distinct quantities of two, they must be interpreted as two distinct quantities, to avoid misinterpretation. If you do not follow this simple rule of interpretation you completely disregard the application problem of adding apples and oranges. If two groups of two are automatically four there is no way to avoid the category mistake described as adding apples and oranges. In other words, you do not allow any provision for the reason why they were represented as distinct in the first place. Therefore, in insisting that "2+2" represents the same thing as "4", you are denying any valid reason for representing the two 2s as distinct in the first place, and nullifying that representation, of two distinct 2s, as an invalid representation.

But if you look superficially at the compound symbol "2+2" and the singular symbol "4" as ink squiggles on paper, they clearly are not the same. — jgill

It seems to me that 2+2 says more than just 4. It says how you can get 4 from starting with 2.

No I'm not making such an identity. But I'm saying that if you assert that "2+2" is the same object as "4" you are claiming such an identity. — Metaphysician Undercover

This is the same error @jgill pointed out that I made the other day. I do not assert that 2 + 2 and 4 are identical. They're obviously different symbol strings. They do refer to the same abstract object, as does 3.999..., 6 - 2, and the smallest positive integer $n$ such that there are two non-isomorphic groups of order $n$. All those descriptions refer to the same abstract object; same as in identical.

The "means" or "process" of getting to the abstract object is irrelevant. In fact given the abstract object represented by 4, all the other representations are already inherent in it. You deny that. That is the crux of our disagreement I believe.

The process of adding two with two will make four, but it is not four. I am saying that the process which makes four is not the same as the number four itself. A cause is not the same thing as its effect. You are insisting that the two are the same. — Metaphysician Undercover

The abstract number represented by '4', which I represent as 4 (but do not confuse the referent with the symbol), is not an effect. It stands alone, and in its very existence incorporates all of its representations and the processes that they bring to mind. But the processes are not the thing. There are multiple processes that may lead to the same abstract thing, as this example shows. But I don't think of this as a process leading to 4. I think of 4, as the primary object, that already incorporates all of the processes that could lead to it. 2 + 2, 3.9999..., etc.

I really can't believe that you do not see the difference between what "2+2" represents and what "4" represents. — Metaphysician Undercover

That's because I'm not hallucinating "causes" or "processes" where there are none. The causes and processes are secondary to the essential existence of the abstract number 4. You have your ontology backwards.

The former indicates two quantities of two, with an operation of addition also indicated. The latter indicates one quantity of four. To interpret "2+2" as representing the number four is very clearly a misinterpretation. — Metaphysician Undercover

You're stubborn, I'll give you that.

The proof is that we can represent the relationship between two distinct 2s in many different ways, such as "I have 2 dogs, and 2 cats", "2,2", "2+2", "2X2", etc. In all such instances of representing two distinct quantities of two, they must be interpreted as two distinct quantities, to avoid misinterpretation. If you do not follow this simple rule of interpretation you completely disregard the application problem of adding apples and oranges. If two groups of two are automatically four there is no way to avoid the category mistake described as adding apples and oranges. In other words, you do not allow any provision for the reason why they were represented as distinct in the first place. Therefore, in insisting that "2+2" represents the same thing as "4", you are denying any valid reason for representing the two 2s as distinct in the first place, and nullifying that representation, of two distinct 2s, as an invalid representation. — Metaphysician Undercover

Oh, proof. Why would I waste my time looking at, or even acknowledging that you've provided a proof? You don't offer me the same courtesy.

But again you're just repeating your confusion. The number 4 incorporates within it 2 + 2 cats or 3.999... or whatever. They all point to the same thing. They're not "ways of getting to" the thing. I can't imagine why you have such a strange idea.

Sheboygan, Wisconsin is the same identical city whether you get there from Milwaukee or Green Bay. 4 is the same number whether you "get there from" 2 + 2 or 3.999... Why you think these two cases are different I can't imagine. Unless you think that $\text{Sheboygan}_{\text{Milwaukee}} \neq \text{Sheboygan}_{\text{Green Bay}}$. Is that what you think?

You're stubborn, I'll give you that. — fishfry

Ah. Harkens back to the last words of the state as they executed the heliocentrist. Before they were disproved and later overthrown of course. How reminiscent.

Ah. Harkens back to the last words of the state as they executed the heliocentrist. Before they were disproved and later overthrown of course. How reminiscent. — Outlander

For sake of conversation, I don't think they actually executed any heliocentrists. If you're thinking of Giordano Bruno, Wikipedia points out that ...

"Starting in 1593, Bruno was tried for heresy by the Roman Inquisition on charges of denial of several core Catholic doctrines, including eternal damnation, the Trinity, the divinity of Christ, the virginity of Mary, and transubstantiation. Bruno's pantheism was not taken lightly by the church, as was his teaching of the transmigration of the soul/reincarnation. The Inquisition found him guilty, and he was burned at the stake in Rome's Campo de' Fiori in 1600."

Neil deGrasse Tyson somewhat misrepresented the Bruno case on his tv show. Bruno's fatal heresy was religious and not necessarily scientific.

Nor was Galileo executed. He was confined to house arrest. Even this story is more nuanced than commonly known. The Pope was not a science denier but was in fact an enlightened, scientific man; and Galileo was the Pope's good buddy. Galileo went out of his way to cause his old friend trouble and that's a large part of what happened to him.

And nor, as long as I'm here, does the earth revolve around the sun in any absolute way. If you stood outside the galaxy and plotted the paths of the earth and sun, they'd spiral around each other as the spirals moved through the universe. Only by drawing a coordinate system with the sun in a fixed position can we write down the "orbit" of the earth. And if we wanted to, we could assume a fixed earth and write down the orbit of the sun. A fable like "The earth revolves around the sun" involves simplifying assumptions that are not metaphysically true; unless by truth you mean historically contingent scientific convenience and consensus.

I'm thinking of taking up geocentrism. It might be the last sane stance left in a crazy world.

They do refer to the same abstract object, — fishfry

We're just going around in the same circle. By what principle do you say that this string of symbols "2+2" refers to one object? How would you distinguish which symbols refer to an object, on their own, and which objects require a string of symbols to represent them? Is it your rule that the left side of an equation refers to one object, and the right side of an equation refers to the same object?

The "means" or "process" of getting to the abstract object is irrelevant. — fishfry

It's not irrelevant because "2+2'" clearly refers to an operation, or ":process". There is an operation symbol "+", within the phrase. This is the fact you are ignoring, and are in complete denial of. How can you deny this fact? It's right there in black and white, "+". You look at "2+2" and read it as symbolizing the abstract object "4", refusing to interpret what the symbols actually represent. If this is not a very clear and obvious example of misinterpretation, then how else can you explain your ignorance of what is written? Do you actually recognize what is written but have some reason to deny it?

But I don't think of this as a process leading to 4. I think of 4, as the primary object, that already incorporates all of the processes that could lead to it. 2 + 2, 3.9999..., etc. — fishfry

Do you not see that this is a false way of looking at an effect? If there exists an object, and there is a multiplicity of processes which may have led to the existence of that object, only one of those processes is the correct process which caused the object. It is false to say that the object "incorporates all of the processes that could lead to it". This misconception leads to many ontological problems, because it allows that infinite possibilities, and infinite chains of processes inhere within an object, rendering an object as inherently unintelligible. To employ a system of mathematics which makes the existence of objects unintelligible, by incorporating infinity into the object's existence is self-defeating, if one's goal is to understand the existence of objects.

The causes and processes are secondary to the essential existence of the abstract number 4. — fishfry

I can agree with you on this point, the causes and processes are secondary. But are you ready to agree that since there is a "+" in "2+2",this phrase refers to a process?

But again you're just repeating your confusion. The number 4 incorporates within it 2 + 2 cats or 3.999... or whatever. They all point to the same thing. They're not "ways of getting to" the thing. I can't imagine why you have such a strange idea. — fishfry

Again, you are ignoring the significance of "+". Please read the phrase, and interpret it according to what the symbols actually say, not according to some preconceived notion of what you want the symbols to say.

Sheboygan, Wisconsin is the same identical city whether you get there from Milwaukee or Green Bay. — fishfry

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.

This is your problem, you are reading "2+2" as 4, instead of reading it as directions of how to get to 4. .

Is it your rule that the left side of an equation refers to one object, and the right side of an equation refers to the same object? — Metaphysician Undercover

Yes. My rule and everyone else's. That's what the equal sign means.

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."

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.

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? 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. 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?

The equal sign means that the two sides are equal — Metaphysician Undercover

Profound stuff.

After months of arguing this point, fishfry still seems to think that the equals sign means the same. Next, fishfry will offer me a proof of this. As you can see, we've been through it before.

How could he possibly offer a proof? It's a definition!

But yes, we have been through this before. You remain unconvinced, but that is about you, not about "=".

Fishfry claimed to have given me a proof of this, in the other thread, and continues to refer to this proof.

Do you accept this definition, that to be equal is to be the same? Are you and I the same, just because we're equal? Oh yeah, I remember now, you have no respect for the law of identity either, and you equivocate with "the same" in your interpretation of Wittgenstein's so-called private language argument. You think that if two distinct instances of sensation are similar, they can be said to be "the same" in the way that a chair remains the same chair if no one switches it out when you're not looking.

Do you understand the reason why Aristotle formulated and stated the law of identity in the way that he did? He did this to have a tool to be applied against such logical sophistry. When "the same" is used in a way which is not consistent with the law of identity, then we are not talking about objects. Objects are identifiable according to the law of identity. So this culture of mathematicians who use "same" in this way, and then proceed to talk about the mathematical "objects" represented, are engaged in this form of sophistry which we might call deception.

If we take jorndoe's suggestion, and say that numerals represent abstract quantities, we see very clearly that "4" represents one abstract quantity, and "2+2" represents two distinct abstract quantities with a mathematical operation of addition represented. If we replace "abstract quantity" with "object", there is no rule which dictates that "2+2" could represent one object. So this culture, which assumes that "2+2" represents an object, which is the same object that is represented by "4", just because two plus two is equal to four, is a culture of sophistry and deception.

Do you accept this definition, that to be equal is to be the same? Are you and I the same, just because we're equal? Oh yeah, I remember now, you have no respect for the law of identity either, and you equivocate with "the same" in your interpretation of Wittgenstein's so-called private language argument. You think that if two distinct instances of sensation are similar, they can be said to be "the same" in the way that a chair remains the same chair if no one switches it out when you're not looking. — Metaphysician Undercover

If we take jorndoe's suggestion, and say that numerals represent abstract quantities, we see very clearly that "4" represents one abstract quantity, and "2+2" represents two distinct abstract quantities with a mathematical operation of addition represented. If we replace "abstract quantity" with "object", there is no rule which dictates that "2+2" could represent one object. So this culture, which assumes that "2+2" represents an object, which is the same object that is represented by "4", just because two plus two is equal to four, is a culture of sophistry and deception. — Metaphysician Undercover

:D

There is no question that 2+2=4 — Metaphysician Undercover

"2+2=4" ⇔ 2+2=4

∎

Are you and I the same, just because we're equal? — Metaphysician Undercover

Good point, MU. :nerd:

Indeed; in the wrong hands, equivocation is a powerful tool.

The question then, is who is doing the equivocating? Meta apparently denies that 2+2 is the same as 4. It's not at all clear, as @fdrake, that he means something different to '"2+2" is not the same as "4"'.

SO, to be clear,

2+2 is the same as 4, and

"2+2" is not the same as "4"

It seems likely that @Metaphysician Undercover does not use quotes in the way the rest of us do.

But are you ready to agree that since there is a "+" in "2+2",this phrase refers to a process? — Metaphysician Undercover

This is your problem, you are reading "2+2" as 4, instead of reading it as directions of how to get to 4. . — Metaphysician Undercover

So if you follow the directions and complete the process, then 2+2 = 4?

2+2 is the same as 4, — Banno

it's quite obvious, that in no way is 2+2 the same as 4. That's the point. It's very clear that 2+2 is equal to 4, meaning that the two have the same value within some value system, but at the same time it is also very clear that these two equal things are not the same thing. That is the way that "equal" is normally used, to refer to two distinct things of the same value. We never say that because two things are equal, they are therefore the same thing.

If you are defining "equal" to mean "the same as", as fishfry does, this is an unacceptable definition for a logical system because it is not consistent with the law of identity

So if you follow the directions and complete the process, then 2+2 = 4? — Luke

Of course, 2+2=4, and this is an acceptable statement. As I said already, not even a fool would deny that. What is at question is whether "2+2" and "4" both refer to the same thing. In other words, is there a difference between being equal and being the same.

it's quite obvious, that in no way is 2+2 the same as 4. — Metaphysician Undercover

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

So again, you have a specialist use that is at odds with common use.

Just sayin'.

The law of identity, and logic generally, has moved on a bit since Aristotle. I've been unable to follow your argument that such is not consistent with the law of identity... SO, for example, it would generally be accepted that Hesperus = Phosphorus. Would you accept that?

In other words, is there a difference between being equal and being the same. — Metaphysician Undercover

There can be a difference; it depends how the terms "equal" and "the same" are being used. But the terms "equal" and "the same" can also be used synonymously. Is this the basis of your dispute?