the journey corresponding to RRL converges to the point corresponding to the golden ratio. — keystone
I'll use 'actualized' for what I mean. In my view, an object is actualized if it is present in the memory of a 'computer'. I am actualized because I am present in the 'computer of the universe'. I'm thinking of a purple cow so that purple cow is actualized because it is present in the 'computer of my mind'. This statement is actualized as I type because it is present in the memory of my laptop. When a memory of an object is flushed, it becomes 'potentialized'. — keystone
Do you consider this a valid function?
— keystone
I haven't the foggiest. — TonesInDeepFreeze
def endless_loop() while True: print("Looping indefinitely...") Return 1
(1) Print "Hello". Go to (2).
(2) Go to (1).
(3) Print "Goodbye". Halt.
That's an algorithm. The execution of it successively prints "Hello", and it never prints "Goodbye", and it never halts.
Algorithms may have an instruction that is never executed [...] — TonesInDeepFreeze
What I want is your view on whether it makes any sense to say what this function returns. — keystone
as described above the code is designed to return 1. — keystone
I would phrase this by saying that the output of the function is potentially 1, but it is never actually 1. — keystone
(1) Print "Hello". Go to (2).
(2) Go to (1).
(3) Print "Goodbye". Halt. — TonesInDeepFreeze
By contrast, in a language with lazy semantics such as Haskell, terms can be used and passed around in partially evaluated form. This means that real numbers can exist in the sense of partially-evaluated "infinite lists" consisting of an evaluated prefix and an unevaluated tail. These lazy languages allow runtime conditions to decide what rational value is used in place of a term of real-number type, which is allowed to vary during the course of computation and which corresponds more closely to the notion of "potential infinity". — sime
The solution to this problem is therefore not a number, but instead a vanishingly small pseudo-interval. — keystone
The solution to this problem is therefore not a number, but instead a vanishingly small pseudo-interval. — keystone
The claim that there are only countably many algorithms/programs does not imply that this view has gaps in the line. — keystone
The line is there in full from the start. — keystone
[bold ORIGINAL]There are no points on this line. — keystone
All that's really changed is the philosophy — keystone
I've moved on from talking about the output of an unending program — keystone
I really should not continue to reply when you so obnoxiously continue to apply the same fallacy, clothed differently, each time though I have explained it over and over and over and even asked you whether you understand, yet you don't reply even to that question itself. — TonesInDeepFreeze
You STILL don't get it. You just keep putting new clothes on an old pig. Every time, you reformulate but you retain the essential fallacy.
There is no single "vanishingly small pseudo-interval". There are only successively smaller pseudo-intervals on successive rows. — TonesInDeepFreeze
Except that I don't see that it improves the more simple approach — TonesInDeepFreeze
But it is the standard calculus depends on the completeness of the reals. — TonesInDeepFreeze
There are no points on this line.
— keystone
[bold ORIGINAL]
You seem to have no compunctions about insulting intelligence. — TonesInDeepFreeze
Changed from standard analysis? No, your proposal is radically different from standard analysis, from the start: Standard analysis uses infinite sets, you disclaim infinite sets. Standard analysis has a continuum; you don't. Standard analysis has uncountably many reals. With you, it's not clear how many comp-reals there are, since there are denumerably many real-ithms but you disclaim that there are denumerable sets (but maybe you could argue that there is not a set of all real-ithms but instead a program that itself generates real-ithms). — TonesInDeepFreeze
Then, define '<', '+' and '*' on real-ithms, and you'd be on our way to something.
In other words, for arbitrary real-ithms G and R:
G < R <-> [fill in definiens]
G+R = [fill in definiens]
G*R = [fill in definiens] — TonesInDeepFreeze
You want only finite objects, but you also want real numbers — TonesInDeepFreeze
I'm getting plenty of value out of this dialogue. — keystone
I don't think your infinite loop programming example of me not listening was a fair representation — keystone
The idea behind this proposal is that the fundamental object is the line, not the point. — keystone
This algorithm can be described as R RL[...] and I call it phi. — keystone
But doesn't that mean that standard calculus depends on there being no gaps on the line? — keystone
Infinite sets are so deeply embedded in your thinking that you're not even willing to imagine the possibility that points are not fundamental. — keystone
In my proposal — keystone
Are my proposed algorithms that different from Cauchy sequences? — keystone
Is my proposed line that different from the real number line? — keystone
Is my proposed line not continuous? — keystone
Standard analysis achieves length by having uncountably many points. — keystone
Is length not also achieved by having pseudo-intervals? — keystone
we must remember that calculus came before set theory — keystone
I'm only proposing a different foundational underpinning. If you don't think that's philosophy then sure. — keystone
Meanwhile, I've asked you three times now whether you understand this post:
https://thephilosophyforum.com/discussion/comment/806060
But you still say not a word about it. — TonesInDeepFreeze
Bringing in a concept of an initial object that is determined solely by -inf and +inf and then pseudo-intervals is extraneous. — TonesInDeepFreeze
Are my proposed algorithms that different from Cauchy sequences?
— keystone
Indeed they are! I EXPLAINED this. — TonesInDeepFreeze
Your use of 'line' is only a figure of speech. It's not a line. It has nothing on it — TonesInDeepFreeze
So what? It used infinitisitic methods. Set theory provided axioms to make those methods rigorously derived from axioms. — TonesInDeepFreeze
There is an algorithm, call it the 'k-S-B algorithm', that generates rows, starting with the base row, then to the next row that is row 0, ad infinitum. The k-S-B algorithm recursively exhausts all "turn decisions" of R and L. — TonesInDeepFreeze
Meanwhile, I've asked you three times now whether you understand this post:
https://thephilosophyforum.com/discussion/comment/806060
But you still say not a word about it.
— TonesInDeepFreeze
Sorry, I thought I was answering this question indirectly but let me be more clear. The successive outputs of a k-algorithm do not converge to any object. Ever. The S-B algorithm does not terminate (or to someone who believes in actual infinity - there is no bottom of the S-B tree). — keystone
Maybe one day you will see set theory as the mathematics of the bottom of the S-B tree…the bottom which you (rightfully) claim doesn't exist. Perhaps it is you who wants to eat your cake and have it too. — keystone
To continue to clean up some of the language: — keystone
Your use of 'line' is only a figure of speech. It's not a line. It has nothing on it
— TonesInDeepFreeze
k-lines are associated with k-functions that describe their infinite potential. — keystone
The k-line becomes important when exploring higher dimensions. — keystone
I think it's a matter of perspective by what one means by 'that different'. — keystone
I believe though is that if this approach ever gets formalized it's going to use a lot of similar language as Cauchy sequences. — keystone
I was asking whether you understand the post, which includes the various aspects of its explanations. Knowing your answer would let me know how much communication is taking place here. — TonesInDeepFreeze
I get to say, "Phi is the limit", because set theory proves there IS such a limit. You do not, because your framework PRECLUDES that infinitistic limit. — TonesInDeepFreeze
So what? Lots of things use similar language, but say RADICALLY different things. — TonesInDeepFreeze
A continuum requires that the ordering is complete, meaning that for every bounded set, there is a least upper bound. — TonesInDeepFreeze
You don't get to incorrectly say what I'm willing to imagine. I don't even have notion of "points are fundamental", let alone that I won't imagine that it's not the case….
Please don't make pronouncements about what I am willing to imagine. — TonesInDeepFreeze
Cauchy sequences have a limit. But if we somehow defined the limit of an algorithm, then that would be infinitistic — TonesInDeepFreeze
our use of 'line' is only a figure of speech. It's not a line. It has nothing on it; it's a placeholder only - as YOU said. It's not a line in the sense of geometry or analytic geometry. — TonesInDeepFreeze
And at no output does the cutting remotely resemble the continuum. First, at every output, there are only finitely many cuts and thus only finitely many rationals described. Second, there are no irrationals described. That is VERY different from the continuum that has both rational reals and irrational reals and altogether not just finitely many, but uncountably many, and proving a continuum. — TonesInDeepFreeze
Your "line", the k-line, has NOTHING on it, as YOU said. So 'continuous' is not even applicable. And there is no infinite set of cuts on the k-line that comes after all the rows. You just now admitted that. — TonesInDeepFreeze
Length is the absolute value of a difference. Even without irrationals, we have length with just rationals. Uncountability is not required to define length. Sheesh! — TonesInDeepFreeze
But even if it worked out, calculus needs more than just lengths. It seems you don't know what calculus is. Do you? — TonesInDeepFreeze
You keep wanting to have both only finite objects but also objects that exist only as provided as an end of an infinite process, while refusing in different forms to recognize that there is no such end hence no such objects. — TonesInDeepFreeze
We don't need the k-line. It is extraneous to capturing the information we want. We can just say a row is the set of cuts.
We don't need cuts. They are extraneous to capturing the information we want. We can just mention the fractions and their ordering.
I think the reason you want all that is to give the illusion that it amounts to a kind of pseudo-"continuum". But it doesn't. Essentially it's a big red herring. Toss out the red herring and simplify as I showed you, which is basically what you proposed yesterday. — TonesInDeepFreeze
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