You say there exists a number called pi with infinite digits and you use a truncated approximation of it when you calculate the approximate area of a circle.
I say that what exists is a (finitely defined) algorithm called pi that doesn't halt but you can prematurely terminate it to produce a rational number to calculate the approximate area of a circle. — keystone
The difference is that you are asserting the existence of an infinite object, something beyond our comprehension. My approach seems more in line with what us engineers actually do, so why bother asserting the existence of something impossible to imagine if you don't even need to? — keystone
Do you believe that 0+0+0+0+... can equal anything other than 0? If not, then how can you claim that 0-length points could be combined to form a line having length? — keystone
Sounds like double-think from 1984. There are no contradictions in wave-particle duality. — keystone
I'm talking about the philosophy of mathematics, not the application of it. — keystone
Yes, all reality is void of actual infinities. So why do we need to believe that reality is just an approximation of some ideal infinity-laden object that we can't comprehend or observe? Why can't we stop at reality?...
...They think it's possible only because modern math welcomes actual infinity. If mathematicians rejected actual infinity then I'm sure physicists would be less inclined to accept it. — keystone
Once again, calculus is about LIMITS, — jgill
True, but in many a calculus problem and theorem the limit IS infinity. — god must be atheist
Yep. So construction gets replaced by constraint. And then my point is you go the next step of seeing construction and constraint as the two halves of the one system. — apokrisis
We start with the highest dimensional continuum of interest.
— keystone
Which would be the "infinite dimensional" continuum — apokrisis
So I am saying I wouldn't deal with the metaphysics of the number line in isolation. It is illustrative of the far bigger conversation we need to have about how holism in mathematical conception plays out. The same principles have to cover mathematical structure in general — apokrisis
This is a rather basic level of discussion. Again, how could it even be a continua unless it could be cut? How could it even be a 0D point except as the positive absence of any dimensioned extension? — apokrisis
So if you want to apply the strength metaphysics to questions about mathematical structure, you have to count to three in terms of "fundamental things". — apokrisis
Or in other words, no matter how many times I cut up a piece of paper, never will it vanish to nothingness.
— keystone
But each piece also gets more pointlike. — apokrisis
The cut has to be sandwiched between the two ends of two lines. Each end of the line is a point. At what point does the point marking the cut – that is, the absence of a point at that point – get marked off from the other two points marking the starts of a pair of now separated continua? — apokrisis
So it is easy to picture just forever cutting a line. Or instead, just forever gluing points. — apokrisis
The thing is that we can't go the limit.
— keystone
But the fact that we can approach the limit – both limits – with arbitrary closeness is how we know they are there. The limit is precisely that which isn't reachable in the end. But it certainly defines the direction we need to keep going from the start. — apokrisis
With this parts-from-whole construction, objects are finite and processes are potentially infinite...and there are no paradoxes.
— keystone
Again, this suffers all the usual problems of an object-oriented ontology. Reality is better understood in terms of relations – processes and structures. — apokrisis
For me, pi is clearly a number. — T Clark
∞∞ isn't and object like for example an elephant or the number 10100 or the word "elephant", it's simply a shorthand for the procedure 1. n = 0; 2. print n; 3. n = n + 1; 4. go to 2]. :chin: — Agent Smith
I have a feeling that some ideas like ∞∞ and nothing cause brain damage - Cantor lost his mind (theia mania) and spent his later years in a lunatic asylum for instance. These concepts & paradoxes of which there are many seem to have a deletorious effect on the brain/mind - constantly mulling over them may lead to a nervous breakdown. Such ideas are more than our brains can handle at present. And yet ... there have been no reports of an epidemic of mental problems among mathematicians. Why I wonder. — Agent Smith
Never could a continuum be decomposed into points
— keystone
For physics, isn't that the driving force behind quanta, to put a stopper into space leaking out ? — magritte
With my view many paradoxes (Zeno, Dartboard, Liar's, etc) are easily resolved
— keystone
I take it you must mean dis-solved? — magritte
An object is at rest. It is not moving.
Now the object is moving at a velocity V.
How many different velocities did the object move at, to get from zero velocity to V velocity?
If your answer is not "infinite" then you don't deserve the name "mathematician". — god must be atheist
I'm basically warning against logicism — sime
the algorithm for approximating sqrt(2) to any desired degree of approximation can itself be used to denote sqrt(2) without being executed. — sime
You might be interested in Norman Wildberger on YouTube. He seems to hold positions on infinity similar to yours. — emancipate
I still don't get it. I don't see any advantage in your way of seeing things. For me, pi is clearly a number. — T Clark
Seems like you're asking for an abstract, human invention to match up with your understanding of reality. It doesn't work that way. As they say, the map is not the terrain. — T Clark
A number is not an object. It doesn't have a physical existence. Also, it's not beyond my comprehension. That way of seeing things has always made sense to me. — T Clark
Abstract entities, i.e. all human concepts, are always simplified reflections of the world. I can't think of any that aren't. That's why math is so wonderful. — T Clark
Particles and waves are different kinds of physical entities. One is extended, spread out, in space and the other is found in a specific location. That's contradictory, and, just like numbers, both are simplified, abstract ideas. The fact that they seem contradictory, at least to most people, is a failure of human imagination. — T Clark
I think, like many mathematicians, you are expecting math to have a precise correspondence with reality. That never works. — T Clark
That's kind of a circular argument:
You - Mathematics shouldn't include elements with infinite properties because that doesn't match reality. Nothing infinite actually exists.
Me - There are qualified people who believe that infinite phenomena exist.
You - They've been fooled by their reliance on mathematics which include infinite elements. — T Clark
You say that the idea of an absolute continuity as the alternative is offensive to the ontic intuition. — keystone
What is the dimension of purely empty abstract space? One might say that it is infinite dimensional, another might say that it is 0-dimensional. — keystone
No, no matter how many times you cut a continua it never becomes more point-like. In a similar way, no matter how many times you cut a string it never becomes 'nothing-like'...since it always remains 'something-like'. — keystone
I would draw it like this: ----o o---- (note that the o is like an open interval)
Of this diagram, the cut is this: o o (note there's nothing actually there, the point is not an actual object) — keystone
I can keep going down this route whereby each term gets smaller and smaller, but the overall sum of each line remains 5. Something evolves to something, no matter how many times you cut it up. Points are not the limiting case of continua. — keystone
What is the usual problem of an object-oriented ontology that I'm facing? — keystone
Pi is a ratio. Diameter~circumference. So it is actually an algorithm. And it can vary between 1 and infinity as it is measured in a background space that ranges from a sphere to a hyperbolic metric.
How all that actual physics translates to claims one might want to make about numberlines and irrational values is another issue. — apokrisis
Maths just defines it and gets on with it. And that is fine. It is what maths does. — apokrisis
But the fact that the real world undermines the simplicities of the metaphysics that maths finds useful is part of the epistemic game here. The more holes there are in the story, the more we can take it as all just a story about reality - that works with “unreasonable effectiveness.” — apokrisis
Somewhat related, my understanding is that the planck length applies to measurement, not space itself. — keystone
Some think that math somehow produces reality. That if math doesn't track common sense, everyday reality exactly, there needs to be an explanation. That something is wrong. — T Clark
Since we can't actually complete the computation of an infinite series, we never produce a number. So let's just say that pi is the algorithm. The beauty of the algorithm is that it's definition is entirely finite (I just wrote it in finite characters) and it's execution is potentially infinite (i.e. it would compute to no end). — keystone
If a computer can't do the math (in principle), maybe there's something wrong with the math. — keystone
A number is an object of computation. Computers do stuff with numbers. I don't think you can fill your head with all the digits of the decimal expansion of pi. The best you could do is fill your head with an algorithm for calculating pi. That's what I'm saying exists - the algorithm. — keystone
abstract objects are the ideals and reality is just an approximation of the ideal. — keystone
Imagine me flipping a coin. While it's in the air is it heads or tails? I'd say it's neither. Instead it has the potential to be heads or tails. Only when it lands does it hold an actual value. In between quantum measurements objects are waves of potential. When they are measured they hold an actual state. I see no reason why the potential should behave the same as the actual so I see no contradiction. In fact, I think if they behaved the same then change would be impossible. — keystone
The universe has a wonderful way of avoiding actual infinities. — keystone
runs its couplings — apokrisis
But physics tells us that this is not fundamental, just a passing stage. The Big Bang had quite a different kind of ontology. And physics has worked up a decent account of the maths required to track how each stage evolved into its next. — apokrisis
The fundamental forces run their couplings. They are all fractured and very different in the cold/large universe of today. But all their strengths and properties (probably) converge at the Planck scale in one simple Grand Unified Theory – a vanilla form of quantum action that is the contents of a general relativity spacetime container of smallest scale. — apokrisis
So as I argued, continuity is measurable as the absence of discreteness. The fact you can choose to truncate your decimal expansion in search of some specific numberline value only shows you didn’t exhaust its capacity for discreteness and thus also failed to demonstrate it is as securely continuous as you might want to assume. — apokrisis
What is the dimension of purely empty abstract space? One might say that it is infinite dimensional, another might say that it is 0-dimensional.
— keystone
I think the maths of manifolds and topology would want to give a more sophisticated answer than that. — apokrisis
But a string has a width. And so you can eventually chop it so much that the width exceeds the length. At which point, your analogy is in trouble. — apokrisis
So rather than finding a point on a line, we create two lines with a cut that leaves them with a point sealing their bleeding ends, and some kind of gap inbetween … that is not a point, just the absence of even points now? An anti-point perhaps? Or what? — apokrisis
The numberline instead always exhibits its twin reciprocal properties of being both limitlessly integrated and limitlessly differentiable. — apokrisis
The numberline could be other. It could be just a swamp of vagueness. It could be a fractal Cantor dust for instance, where you could never know whether you land on a cut or a line. — apokrisis
I’m just arguing that the numberline debate is another example revealing that any holist ontology has to be triadic. — apokrisis
What is the usual problem of an object-oriented ontology that I'm facing?
— keystone
It binds you to a monistic and reductionist conception of nature. — apokrisis
So zoom in on the Planck scale and you find the same metaphysics I have described. — apokrisis
In a quantum reality we can only talk about it's velocity when measurements were made — keystone
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