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'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 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
You guys still chatting about whether 2 + 2 = 4 or 2 + 2 ≠ 4 ? — jorndoe
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
"mathematical object", or "Platonic ideal" — Metaphysician Undercover
but the disagreement between you and I appears to be as to what constitutes a "mathematical object", or "Platonic ideal" — Metaphysician Undercover
Abstract quantities (is the phrase) — jorndoe
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
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 — fishfry
The number 4 is the number 4, and it's inherent in its nature that it can be represented many different ways. — fishfry
It seems to me that 2+2 says more than just 4. It says how you can get 4 from starting with 2.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
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
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
I really can't believe that you do not see the difference between what "2+2" represents and what "4" represents. — Metaphysician Undercover
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
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
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
They do refer to the same abstract object, — fishfry
The "means" or "process" of getting to the abstract object is irrelevant. — fishfry
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
The causes and processes are secondary to the essential existence of the abstract number 4. — fishfry
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
Sheboygan, Wisconsin is the same identical city whether you get there from Milwaukee or Green Bay. — fishfry
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
The equal sign means that the two sides are equal — Metaphysician Undercover
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
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
2+2 is the same as 4, — Banno
So if you follow the directions and complete the process, then 2+2 = 4? — Luke
it's quite obvious, that in no way is 2+2 the same as 4. — Metaphysician Undercover
In other words, is there a difference between being equal and being the same. — Metaphysician Undercover
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.