Thanks and sorry for posting a topic which is not a typical discussion — keystone
Not a mention of God or Jesus or climate change. — jgill
To Cantor, his mathematical views were intrinsically linked to their philosophical and theological implications – he identified the Absolute Infinite with God, and he considered his work on transfinite numbers to have been directly communicated to him by God, who had chosen Cantor to reveal them to the world. — Wikipedia
God = ∞. — Agent Smith
The tone of the OP does not suggest Cantor's theological nonsense. — jgill
I have never used infinity as anything more than unboundedness. — jgill
Cantor's theological nonsense — jgill
I accept ideas like the set of real numbers and associated cardinalities — jgill
I have never used infinity as anything more than unboundedness. To all intents and purposes my mathematics has been infinity free. — jgill
The forum has had a number of discussions about this topic, but that's no reason for you to avoid bringing it up in a new thread or resurrecting an old thread. — jgill
The tone of the OP does not suggest Cantor's theological nonsense. — jgill
Calculus is all about infinity. — T Clark
I would argue that calculus done right (with limits) is all about potential infinities. — keystone
I find it hard to imagine — keystone
All mathematics is about "potential" entities. So what we gonna do? Round pi off to 3.14? 3.14159? How many decimal places do I need to get to the real pi? — T Clark
History shows that is a bad standard by which to judge a concept. — T Clark
Perhaps I should have written that I believe it is impossible to imagine assembling points to form a continuum. A bit of magic is needed to make the leap from a finite collection of points forming nothing to an infinite collection of points forming a continuum. — keystone
That makes the real numbers a challenging and intriguing subject. — jgill
I was speaking of currently accepted set theory, not challenges of it. — jgill
Why can't we just say that pi is not a number? Instead, it is an algorithm (e.g. pick your favorite infinite series for pi) used to generate a number. This algorithm is potentially infinite in that we can never complete it, but we can certainly interrupt it to generate a rational number. If you interrupt it, maybe you'll get 3.14. Actual infinity only comes into play if you claim that the algorithm can be completed, in which case it would generate a real number - a number with actually infinite digits. This is what I would like to challenge. — keystone
Perhaps I should have written that I believe it is impossible to imagine assembling points to form a continuum. — keystone
A limitation of that conceptualisation, is that it asserts what might be considered an unnecessarily rigid ontological distinction between functions (intension) and data (extension), which is surely a matter of perspective, i.e the language one uses. — sime
Also, recall incommensurability; the length of diagonal lines in relation to square grid have a length proportional to sqrt(2). The decimal points of sqrt(2) are only "infinite" relative to the grid coordinates. — sime
That said, it could be argued that the concept of exact and correct computation, whereby a computer program or function specification is translated by man or machine to a precise and correct result of execution, is an ideal platonistic notion that is incompatible with the austere epistemic and metaphysical conservatism of finitism. — sime
from 0D points to 1D lines – doesn't fix the deeper issues. You just set yourself up for the same puzzle at the next geometric level — apokrisis
I mean it doesn't even make sense to talk about 0D points except in the context, or in contrast, with the presence of the 1D line, right? — apokrisis
This would see the discrete and the continuous as being each others limiting case. — apokrisis
Why can't we just say that pi is not a number? Instead, it is an algorithm (e.g. pick your favorite infinite series for pi) used to generate a number. This algorithm is potentially infinite in that we can never complete it, but we can certainly interrupt it to generate a rational number. If you interrupt it, maybe you'll get 3.14. Actual infinity only comes into play if you claim that the algorithm can be completed, in which case it would generate a real number - a number with actually infinite digits. This is what I would like to challenge. — keystone
Perhaps I should have written that I believe it is impossible to imagine assembling points to form a continuum. — keystone
We need a more subtle metaphysics. We need an intuition that itself sees parts and wholes, the discrete and the continuous, as the two emergent parts of the one common rational operation. — apokrisis
Like many who are philosophically inclined, I am happy to accept actual infinities as a useful mathematical simplification – an epistemic trick – but not something that makes proper ontological sense. — apokrisis
So the idea of 0D points – some kind of absolute notion of discreteness – is offensive to the ontic intuition. But the same should apply to its dichotomous "other", the idea of an absolute continuity as the alternative.
We need a more subtle metaphysics. We need an intuition that itself sees parts and wholes, the discrete and the continuous, as the two emergent parts of the one common rational operation. — apokrisis
What does this mean for number lines? It says that while we must think of the 1D whole being constructed of 0D points, that claim must be logically yoked to its "other" of each 0D point existing to the degree the 1D continuity of the line has in fact been constrained. — apokrisis
I mean it doesn't even make sense to talk about 0D points except in the context, or in contrast, with the presence of the 1D line, right? — apokrisis
I am proposing that instead of constructing the whole from the parts, that we construct the parts from the whole. — keystone
We start with the highest dimensional continuum of interest. — keystone
Might the same apply to objects in the abstract world? Might continua be fundamental instead of points? — keystone
Or in other words, no matter how many times I cut up a piece of paper, never will it vanish to nothingness. — keystone
The thing is that we can't go the limit. — keystone
With this parts-from-whole construction, objects are finite and processes are potentially infinite...and there are no paradoxes. — keystone
objects are finite and processes are potentially infinite — keystone
objects are finite and processes are potentially infinite — keystone
Most interesting — Ms. Marple
That can't be true. Calculus is all about infinity — T Clark
That can't be true. Calculus is all about infinity — T Clark
I was thinking the same thing. — god must be atheist
There are physicists who believe the universe is infinite — T Clark
Never could a continuum be decomposed into points — keystone
Once again, calculus is about LIMITS, as my mathematical genealogical ancestor, Karl Weierstrass would have explained. — jgill
Once again, calculus is about LIMITS, — jgill
I don't understand why you want to challenge this. I use approximations to pi all the time. When I want a quick and dirty approximation of the area of a circle inscribed in a square with sides x, I use 3/4 * x^2. I can round pi off anywhere I like depending on the precision I need. To say that irrational numbers are not really numbers doesn't make any sense to me. Of course they are. — T Clark
I really don't get this. I have no problem imagining continuity arising from discreteness. I learned, saw it, got it, in 6th grade algebra. — T Clark
Holding two apparently contradictory ideas in your mind at the same time is a required skill, e.g. waves and particles. It's no big deal. I learned that, saw that, got that in 12th grade physics. — T Clark
What advantage is there in seeing things your way. Expecting abstract concepts such as mathematical entities to have some sort of ontological reality doesn't make sense. Mathematicians love math for math's sake. Engineers such as me just want something that works - no ontological interpretation necessary. I assume the same is true for most scientists. How does your way work better than the way it is handled normally? — T Clark
It makes ontological sense to me. I do agree that is a useful, abstract simplification. Really, all math is. All reality is. — T Clark
This may be true, but I don't think everybody qualified to have an opinion agrees with you. There are physicists who believe the universe is infinite. That doesn't really make sense to me, but a lot of things that don't make sense to me turn out to be true, so I'll remain agnostic. — T Clark
Can you explain this to me from a computer programming perspective? In your comparison, is the data the output of the function? A function can return a function, but it can also return another object type, like a string. In the latter case, there is a type distinction between between the function and its output, but I don't see how this is unnecessarily rigid. I suspect I'm missing your point. — keystone
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