Philosophy Explorer wrote:Points are dimensionless. ...
If point are dimensionless then you've already answered your own post so what paradox?
But they don't need gaps - points can lie next to one another on a line with no gaps, but with distinct identities (this involves the notion of limits that calculus handles well).
Surely if something is 'dimensionless' it can't lie anywhere?
If math is beyond you, what are you doing in this category? ...
I think the title of this category needs an inclusive or and not a conjunction but I could say the same to you as most of what you post is not Philosophy of Mathematics but Mathematics and should not be here but on a Maths forum. which also applies to most of your posts in the Phil of Science category.
Are you saying every branch of math is beyond your understanding? ...
Depends what you mean by understanding? But pretty much all Maths beyond basic Algebra is outside of my grasp or interest.
What are you looking for?
Logic and to read some Philosophy of Maths.
Even though points are dimensionless, they can exist as parts of groups, i.e. two-dimensional or three-dimensional space which are composed of infinite numbers of points. The paradox is regardless of the size of space in question, they all have the same cardinal number of points which contradicts common sense as established by Georg Cantor through his idea of one-to-one correspondence. The limit notion further clarifies this idea there are no gaps in continuous space.
Philosophy Explorer wrote: ↑Sun Mar 25, 2018 3:38 am
Here's something some may take for a paradox.
Take a line that's an inch long with no gaps. Take a square whose sides are an inch long, again with no gaps.
I assert that both objects contain the same number of points and this can be proven geometrically or by Cantor's method of one-to-one correspondence using Cartesian coordinates.
IOW, the number of points is independent of distance. Any comments?
In reality there are not infinite points. The problem with math is it's completely fabricated and coincides with reality on occasion. Both the concept of a point and infinity are imaginary.
Philosophy Explorer wrote:Even though points are dimensionless, they can exist as parts of groups, i.e. two-dimensional or three-dimensional space which are composed of infinite numbers of points. ...
Do they? You'll have to explain to me how a dimensionless 'thing' exists?
The paradox is regardless of the size of space in question, they all have the same cardinal number of points which contradicts common sense as established by Georg Cantor through his idea of one-to-one correspondence. The limit notion further clarifies this idea there are no gaps in continuous space.PhilX
How is it a paradox? As a 'dimensionless thing' by definition is a contradiction of common sense?
Philosophy Explorer wrote: ↑Sat Apr 07, 2018 1:44 am
Here's a question for Arising. In solid space, how many points are there and where are the gaps?
PhilX
What's a solid space?
At a guess, if your 'points' exist then whether there are gaps or not will depend upon their shape, so what shape are these existing points in your space?
Philosophy Explorer wrote: ↑Sat Apr 07, 2018 1:44 am
Here's a question for Arising. In solid space, how many points are there and where are the gaps?
PhilX
What's a solid space?
At a guess, if your 'points' exist then whether there are gaps or not will depend upon their shape, so what shape are these existing points in your space?
You're going in circles as we've already established that the points are dimensionless so shapes are irrelevant.
Philosophy Explorer wrote:You're going in circles as we've already established that the points are dimensionless so shapes are irrelevant.
PhilX
But I'm still waiting for you to tell me how a dimensionless point can exist? If what you are saying is that they are not actually existent but are mathematical constructs then I'll leave the nuanced discussion of such stuff to you and wtf but would point that if you describe them in such a way then it appears obvious that they will have no relation to such a thing as distance as distance is a dimension I'd have thought, so your deduction that they have no relation based upon lines and squares is pointless as by definition they have no such relation.
Philosophy Explorer wrote:You're going in circles as we've already established that the points are dimensionless so shapes are irrelevant.
PhilX
But I'm still waiting for you to tell me how a dimensionless point can exist? If what you are saying is that they are not actually existent but are mathematical constructs then I'll leave the nuanced discussion of such stuff to you and wtf but would point that if you describe them in such a way then it appears obvious that they will have no relation to such a thing as distance as distance is a dimension I'd have thought, so your deduction that they have no relation based upon lines and squares is pointless as by definition they have no such relation.
If you mean the bit about how your 'deduction' about the relationship between your 'points' and distance then that was not an assumption but the logical conclusion of the fact that if you already define your 'points' as dimensionless so how can there be a relationship as distance is a dimension, so you have already assumed what you are attempting to prove.
Arising_uk wrote: ↑Sat Apr 07, 2018 1:54 pm
If you mean the bit about how your 'deduction' about the relationship between your 'points' and distance then that was not an assumption but the logical conclusion of the fact that if you already define your 'points' as dimensionless so how can there be a relationship as distance is a dimension, so you have already assumed what you are attempting to prove.
Not my assumption, but others would. The relationship I'm talking about is between size and number of points which you're not grasping (it's like trying to teach a blind person to see).
It doesn't matter if it's size or distance your points are DIMENSIONLESS and as such will have no relationship to size or distance.
The reason why you think you have a paradox that defies common-sense is that in your OP you are implicitly implying that points are not dimensionless and as such are in some way filling up the space so when you compare an inch line with an inch square common-sense would imply that there should be more points in the latter and if in reality points did work like this then they'd be dimensioned and it would be a paradox that there are the same amount of points in both but you justify your claim that there are by using these dimensionless points and by bloody definition these can have no relation to size or distance so the 'deduction' is pointless.
