What about fractals? These are infinite spaces that you can explore on a computer. Their instantiation in a computer realizes what was an abstract infinity. — hypericin
What is a potential infinitesimal — jgill
You mean without absolutely infinite sets, presumably. — sime
Who specifically do you have in mind as mentioning "absolute infinity" since Cantor's work was superseded by formal set theory? — TonesInDeepFreeze
approximating an ideal, abstract and causally inert mathematical derivative. — sime
[a potential infinitesimal] is the reciprocal of a potentially infinite number — sime
obscuring matters even further, e.g the hyperreals . — sime
You simply skipped the point that mathematical definitions are never circular or tautologies. — TonesInDeepFreeze
That article does not describe potentially infinite sets or sequences in a manner that is properly paraphrased as "finite entities of a priori indefinite size" or as "finite entities".
Moreover, most remarkably, in another thread:
https://thephilosophyforum.com/discussion/comment/653236
you say, "whose length is eventually finitely bounded". Again, there is nothing in the SEP article that states anything like that. — TonesInDeepFreeze
Moreover, you mention internal set theory, but internal set theory is an extension of ZFC, so every theorem of ZFC is a theorem of internal set theory. So, if one rejects any theorems of ZFC then one rejects internal set theory. Moreover (as I understand), non-standard analysis and internal set theory make use of infinite sets, and though (as I have read) there is an intuitive motivation of 'potentiality' in internal set theory, I do not find 'potentially infinite' defined in those papers or articles I have perused. Let alone that you are now combining non-standard analysis with potential infinity without reference to where that is in any mathematical treatment (possibly there is one, but you have not pointed to one). — TonesInDeepFreeze
ZFC is of course inadmissible for the purposes of this discussion — sime
the I,S and T axioms, divorced from the problematic axioms of ZFC — sime
appear to be a relevant fragment of some formalization of potentially infinite logic. — sime
Linear Logic — sime
Your whole line of argument has sputtered. As well as you still have not addressed that you terribly misrepresented Brouwer — TonesInDeepFreeze
For that matter, as I've already asked, what is your definition of "absolute infinity"? — TonesInDeepFreeze
For that matter, as I've already asked, what is your definition of "absolute infinity"? — TonesInDeepFreeze
Again, the article says nothing that is tantamount to describing potentially infinite sets or sequences as "finite entities of a priori indefinite size" or as "finite entities" or having "length [that] is eventually finitely bounded". — TonesInDeepFreeze
Analogously, in mathematics absolute infinity corresponds to interpreting the intensional description of a total function or algorithm as being synonymous with an exact limitless extension . . . — sime
in mathematics absolute infinity corresponds to interpreting the intensional description of a total function or algorithm as being synonymous with an exact limitless extension — sime
Absolute infinity refers to a semantic interpretation of a mathematical, logical, or linguistically described entity — sime
Absolute infinity arises when an analytic sentence is mistaken for an empirical proposition. — sime
total function — sime
Again, the article says nothing that is tantamount to describing potentially infinite sets or sequences as "finite entities of a priori indefinite size" or as "finite entities" or having "length [that] is eventually finitely bounded".
— TonesInDeepFreeze
It is practically equivalent to the common definition of potential infinity as being a "non-terminating" sequence that is never finished but occasionally observed after random time intervals. — sime
The important thing regarding the common definition of potential infinity is that in order to obtain a value or extension, the process constructing the sequence must be "paused" after a finite amount of time. — sime
when dx is declared to be infinitely small or x to be infinitely large. — sime
What is important to the definition of potential infinity is pausing a process to obtain a finite portion of a sequence, whereupon one might as well regard restarting the process as starting a new process. — sime
potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces a sequence with no last element, and where each individual result is finite and is achieved in a finite number of steps
There is no "pausing" and "restarting", only a reference to subsequences leading toward a limit definition. — jgill
Simply:
Where do you find "eventually finitely bounded" in Brouwer, or even any secondary source, on potential infinity? Please cite a specific passage. — TonesInDeepFreeze
.You said that your claims about the notion of potential infinity are supported by the article about intuitionism. Now you're jumping to non-standard analysis. You would do better to learn one thing at a time. You alrady have too many serious misunderstandings of classical mathematics and of intuitionism that you need to fix before flitting off fro them. — TonesInDeepFreeze
You say, "in mathematics". The ordinary mathematical literature does not use 'absolute infinity' in the sense you do. As I've pointed out to you several times, 'absolute infinity' was a notion that Cantor had but is virtually unused since axiomatic set theory. And Cantor does not mean what you mean. So "in mathematics" should be written by you instead as "In my own personal view of mathematics, and using my own terminology, not related to standard terminology". Otherwise you set up a confusion between the known sense of 'absolute infinity' and your own personal sense of it. — TonesInDeepFreeze
If a student asked you to explain "what is a non-terminating process?" what would your reply be, and how would you avoid running into circularity?
I cannot think of any way of explaining what is meant by a non-terminating process, other than to refer to it as a finite sequence whose length is unknown. Saying "Look at the syntax" doesn't answer the question. Watching how the syntax is used in demonstrative application reinforces the fact that "non-terminating" processes do in fact eventually terminate/pause/stop/don't continue/etc.
The creation of numbers is a tensed process involving a past, a present (i.e. a pause), and only a potential future. — sime
If a student asked you to explain "what is a non-terminating process?" what would your reply be, and how would you avoid running into circularity? — sime
I cannot think of any way of explaining what is meant by a non-terminating process, other than to refer to it as a finite sequence whose length is unknown. — sime
"non-terminating" processes do in fact eventually terminate/pause/stop/don't continue/etc. — sime
Where do you find "eventually finitely bounded" in Brouwer, or even any secondary source, on potential infinity? Please cite a specific passage.
— TonesInDeepFreeze
It is a logically equivalent interpretation of Brouwer's unfinishable choice sequences generated by a creating subject — sime
Martin[-]Lof's "The Mathematics of Infinity" — sime
Classical mathematics and Set theory conflate the notions of absolute with potential infinity, — sime
in computer science, where a rigorous concept of potential infinity becomes needed, and where ZFC is discarded as junk. — sime
Cantor does mean what i mean in so far that his position is embodied by the axioms of Zermelo set theory: — sime
The Law of Excluded Middle is not only invalid, but false with respect to intuitionism. — sime
The axiom of regularity (added in ZFC) prevents the formulation of unfinishable sets required for potential infinity. — sime
For example, in {1,2,3 ... } where "..." refers to lazy evaluation, there is nothing wrong with substituting {1,2,3 ...} indefinitely for "..." — sime
no honest mathematician knows what is being asserted [with the axiom of choice] beyond fiat syntax when confronted with an unbounded quantifier. — sime
The Axiom of Extensionality : According to absolute infinity, two functions with the same domain that agree on 'every' point in the domain must be the same function. — sime
Newton and many many other famous (and hence influential) mathematicians are lim brains. Lim as in limit. Lim as in they can't do math without starting with limited integrals. Hand them a set and ask them to do anything with it and they have a meltdown and rant about new ideas ruining everything. — SkyLeach
Mathematics is actually just a very precise language. It's possible to say almost anything but the less precise the definition and description the more statements it requires and error prone (anomaly prone in this context) it tends to be — SkyLeach
There is a working axiom in mathematics (not yet extremely popular but growing every year) that linear algebra should be the core axiom of mathematics. — SkyLeach
Traditionally the first axiom of mathematics is commutative principle. — SkyLeach
derived axiom — SkyLeach
the axiom that given any observable set we can assign whole numbers to that set as a form of measurement. — SkyLeach
Hand them ["famous and influential mathematicians"] a set and ask them to do anything with it and they have a meltdown and rant about new ideas ruining everything. — SkyLeach
1 infinity + 1 infinity = 1 infinity — Agent Smith
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