For reference, here are the axioms of Z set theory — TonesInDeepFreeze
the answer is that a calculus book is about setting forth the most basic mathematics that is put to use in different fields of study; it's not about very much understanding the foundations of that mathematics or its broader mathematical context. — TonesInDeepFreeze
However, my primary focus is on the objects themselves, such as the Cartesian coordinate system. I believe this system needs a parts-from-whole, continuum-based reinterpretation, as the current understanding relies heavily on the notion of actual infinity. — keystone
It seems that a k-continuum is a certain kind of finite(?), undirected(?), loopless(?) graph whose k-vertices are either k-points or k-curves. And (I surmise) no k-vertex is connected to itself. — TonesInDeepFreeze
Is there a natural number n>1 such that there is no k-continuum such that there occurs n number of k-curves connected to one another? (In other words do you disallow that a k-continuum may have arbitrarily finitely many connections of curves from one to another?) — TonesInDeepFreeze
I'm not sure that exhausts all possible configurations. You should figure it out to define 'is a k-continuum'. — TonesInDeepFreeze
But I see trouble ahead anyway with your personal notion of partitioning. I pessimistically predict that ultimately you won't be able to get past handwaving with it. — TonesInDeepFreeze
Not sure what you mean by actual infinity. Are you speaking of infinity as a sort of number that can be arithmetically manipulated, or infinity as unboundedness? I have always used the concept of the latter rather than the former. But set theorists use both I think. Please provide an instance of "actual infinity" in the Euclidean plane. A projection onto a sphere is not allowed. — jgill
The biggest hurdle for an intelligent but amateur mathematician is rediscovering a result established some time ago. Hence, my words of caution. — jgill
I have two continua described by Graph 1 and Graph 2, respectively. — keystone
I seem to lack your insight in this example. — jgill
Define a continuum as an abstract entity and not in terms of the real line. As a matter of fact, use another word for your creation. State the properties of the continuum, again not referencing the real line or numbers. This is a tall order. Metric spaces and topological functions are perhaps inappropriate in this regard. I don't know. You will be going into unexplored territory. — jgill
As a constructivist, what are you constructing other than a few line segments? — jgill
Why don't you jump right into calculus concepts in 2D instead of dwelling on the trivial, incredibly boring 1D case. Either that or make the 1D case something interesting, to capture the attention of a reader. Just a suggestion. — jgill
But I see trouble ahead anyway with your personal notion of partitioning. I pessimistically predict that ultimately you won't be able to get past handwaving with it. — TonesInDeepFreeze
All graphs are 1D drawable (in that each can be embedded in a circle without any of its edges crossing) — keystone
BaDbE can be unified into BaC because DbE can be treated as a whole that captures all the structure of C. In other words — keystone
You seem not to understand how the mathematical method of handwaving works — TonesInDeepFreeze
Thank you. You saved me a lot of time and effort. Because my prediction that you would resort to half-baked handwaving is confirmed, so I am done with trying to help you formulate your stuff into mathematics. A circle is an infinite set. — TonesInDeepFreeze
What structure? A line segment has structure? One line segment has the same "structure" as another? You must see something there that eludes me. But I am old and a lot gets past me. — jgill
You seem not to understand how the mathematical method of handwaving works. It's not ZF or PA or one of those; it's the theory BS. — TonesInDeepFreeze
A good idea often begins with some handwaving as it's being formed, but through refinement and rigorous thought, it can mature into a precise and well-supported explanation. — keystone
True. I hope there is something of interest coming from this discussion. But we've been through metric spaces and topology and now are venturing into graph theory with some sort of hope of connecting that with calculus. I have my doubts, but am trying to keep an open mind. — jgill
What structure? A line segment has structure? One line segment has the same "structure" as another? You must see something there that eludes me. But I am old and a lot gets past me. — jgill
Sorry, it looks like you are taking a line segment and dividing it into two smaller segments. Then comparing. If you think there is something significant here you had better present a philosophical argument supporting it. — jgill
There is virtually no mathematics so far. — jgill
..the first step is to accept that k-curves are indivisible. k-vertices in these graphs cannot be partitioned — keystone
Why resort to graph theory and call a simple line an edge? — jgill
Is this an effort to enhance an almost trivial concept of line and point? — jgill
Again, why not go to 2D? Maybe your ideas will make more sense in that context. — jgill
Your second figure is bewildering. Maybe go back to 1D and explain the real numbers as you see them. Expressions like k-vertex instead of point are confusing. — jgill
How all this simplifies normal calculus is questionable. — jgill
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