???
Distance “exists” in the sense that it has affect, although isn’t an object but a property of space. Segments are merely portions of distance. They too are not objects, yet still imbue affect.
The overlooked idea, perhaps is, that segments in a convergent sequence reduce to smaller and smaller segments, and are measurable by their size between lines(points)
A
t minimal convergence, the size is no longer measurable because it is news urged between 2 points which are indescribable.
That doesen’t mean they become identical, therefore the differential of the function applies to unbounded progressions, as well.
Therefore the function is no linger differentiable, but that does not mean that the progressive reduction
has transferred the function into an identity.
Here lies the paradox, where spatial derivatives (may act like real mathematical integers.
There is, according to this a tangential relationship between mathematical and spacial determinants, where the act like both.
I am not sure of the idea of affectance, but given the little exposure I had with it, I am sure there is some sort of tie in.
At critical points affects and their objects may not be differentiable, on extremely small sizes.
It is always and forever merely up to what standard of minimal measure one chooses - what degree of infinitesimal is going to represent “1”. If none is chosen, then a physical ontology cannot be formed because every infinitesimal distance could be infinitely divided such as to have no means to sum up any distance at all.
Ontology is a Choice. One must choose how many infinitesimals are going to exist between 0 and 1.
And a necessary choice if any understanding is to be formed or maintained. Ontological understanding is not reality itself, which has no limits for infinity. Thus the limits for infinity and infinitesimal must be set by choosing a standard independent of the fact that reality has no limit. It really isn’t any different than choosing the length of a meter. Reality has no such thing as a meter, so someone must choose a length, else length measurements are impossible to logically handle.
That insight reminds me of the intuitionist real line. I don’t know too much about intuitionism. What I do know is that it’s related to constructivism, so that every mathematical object has to be constructed, not just posited to exist. It’s much more restrictive than standard math. In fact we can describe the three great philosophies of the continuum as:
intuitionist line —>> standard line —>> hyperreal line
fewer points ------->> points ----------->> more points
I thought of this because you used the word “indescribable”. Although there are uncountably many points in a mathematical interval; only countably many of them are describable or constructible in the sense of being expressible by finite strings of symbols, or generated by algorithms. [There are subtle distinctions between “definable” and “computable” that don’t concern us here].
There’s also a psychological and subjective aspect to intuitionist philosophy. As I understand it (please don’t quote me) the limit of a sequence doesn’t exist until an active observer chooses it. To me this sounds interesting. I dislike the fact that the real line is full of holes where all the nonconstructable numbers live. That bothers me.
Intuitionism is making a comeback these days through computability theory from computer science; and alternate modern foundations like intuitionist type theory.
Mathematicians will have a different view of foundations a hundred years from now than they do today; just as foundations changed dramatically from a century ago, and a century before that. Mathematics is a historically contingent human activity.
Yes that is true. But the object and property of space, may be conflated by the fact, that perception of an object, predicates it’s property as an objective criteria, regardless of it’s supposed distinction.
At the point of maximum and minimum the
conceptual object as a property as objective criteria is not distinguishable from its existential object.
For these reasons , regardless of later discernment, where this idea cannot be argued reversely, or backwardly, the idea of the objective criteria which
space represents, cannot be explained other then the
objects affect it. Therefore, affectance of object-ness of space is dependent of objects which determine it.
Space is as much an object of affectance, as it affects the objects which determine it. The objects determining it, are spatial determinants, primarily
ones of extension, boundedness, usually consisting of
elliptically bound spaces. The unbounded spaces they occupy, and are surrounded by, are merely shells of varying closed curved surfaces, whose
object-ness is less determined by the materiality of the
content of the shells, but by the nature and interplay of forces generated by the enclosed curvature with the unbounded curvature surrounding it.
The force fields generated may be the result of affectance of forcing such differentiation into inner
and outer space.
In fact matter may very well be affected by such antagonistic force fields.
So the i materializations of space counter indicates such earlier ideas, as the existence of ether, or some
force field within what appears to be only a property
of space.
May be, a time will come when Einstein’s GR will show relations to quantum mechanical behavior, until then it is conjecture.
My bet is, that math will develop to satisfy both criteria.
That is such a strange remark.
Based on the content I’ve been posting here, I imagine people would say I’m the kind of person likely to have passed through a class or two on logic along the way.
Secondly, your statement that one would learn about limits by studying logic, is flat out falsified by objective facts.
Nowhere in the study of logic as it is taught in universities is either the history, the philosophy, or the ontology of limits. Not in basic logic, not in upper division, not in grad school.
This is called a fact because it may be objectively verified by any observer who consults books or flips through some random Wiki pages. The theory of limits; their history, their philosophy, their ontology; are not taught in logic class.
[Of course the mathematical formalism of limits – the epsilon/delta arguments, the multiple quantifiers in just the right order, etc. – IS studied in logic; but as you reject that formalism with anger, I assume it’s not of interest to you].
So what I really don’t get is this: Of ALL the things you could hurl at me that would be true, or have the ring of truth, or would make people think or laugh … why would you toss off a non sequitur that is
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Manifestly false about me; and
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Manifestly false about the world outside your head?
Many of us are pluralistic, and seek only to understand others’ points of view and share our own. In so doing we gain deeper understanding of our own ideas; and sometimes find virtue in the ideas of others.
False. And you are the proof of it.
Therein you display your ignorance of logic and ontology.
The very most fundamental construct in any ontology or understanding as well as any logical syllogism is not actually the axioms, as is often believed, but rather the definitions. In a sense, a definition is an axiom, but of a different kind. A definition is an axiom of communication. And when building an ontology, communicating the understanding is the entire task, as it also is with logic.
Truth is found through definitions within ontologies. Logic is merely keeping the language straight. There can be no “facts” nor empirical observations without first having ontological definitions.
In this topic, the issue is one of defining exactly what “infinitely” or the ellipsis, “…”, means. Does it stop at a first order infinity or infinitesimal? Where does it stop? The definition of those words implies that it does NOT STOP. And that leads to the inescapable conclusion that there is no final sum.
The limit is another matter. The limit is what is endlessly, infinitely, approached without ever reaching. Again, the fact that it is never reached (by definition of “infinitely”) implies the inescapable FACT that the limit is not the final sum.
It really isn’t up to ordained mathematicians to choose otherwise.
It is not a case of choosing how many infinitesimals are going to exist between 0 and 1 but utilising which ever ones are necessary with regard to explaining their function. And the analogy with the length of a metre is somewhat misplaced as objects or distances that require measurement are real whereas the number line
is abstract like all of mathematics is. It is further misplaced if you accept that mathematics is a non human discipline which was discovered rather than invented Whereas the metre like all measurements is purely of human origin
I think that is what I said with:
???
Math is a discovered “non-human” discipline???
The proof was posted by JSS himself. It’s on wikipedia. It’s well-known.
What isn’t well-known is that fact of its flaw.
The symbols are human of course but mathematics itself might not be
Its perfection is what makes its apparent human invention susceptible
Well, I guess you could be right. It is a little dubious for me to discern between inventing and discovering. The numbering systems are definitely invented, but the consequential logic that follows is usually discovered. But when someone comes up with a new way to resolve a complex equation, has he invented it or discovered it? I’m not really sure how to call such things.
Location points and segments of space are different how ??
You do realize that infinitesimal 1 floods it’s dimension in a single line right???
You never get to the 1!!
An ontology begins with segments of distance in 3 dimensions, not location points. The segments physically exist. The location points do not. So what you have to do is decide on the size of segments for your ontology. No matter how infinitesimal that is, the inverse gets you back to exactly 1.
If you choose to not have a limit for infinity (“absolute infinity”), then the inverse is as close to zero as you can get. Thus absolute infinity times 1/(absolute infinity) still accomplishes 1.
So you say…
I think you’re conceptualizing a point without dimension as being different than a point with dimension… And then you say you are right because you have a point with dimension ???
Am I not correct???
The question is… Can a point be without dimension in the first place!!!
If it can’t, then there’s no difference between your two terms!!
You have convinced me to change my mind. I now think that 0.999 … does not equal 1. I only accepted the alternative
since it is generally regarded as being true. However I can no longer accept this since it makes zero logical sense to me
As I cannot accept something as being objectively true [ even if it is ] if I do not understand why it is supposed to be so
The proof has been posted several times already. There is no question that .999… = 1.
You said something the other day that is actually the solution to the mystery:
Exactly. .999… = 1 is an exercise in formal mathematics. It’s a category error to try to ask if it’s “true in the real world.” The question is meaningless. The statement .999… = 1 can only be approached mathematically; and in that context, it’s a valid logical deduction, with an extremely simple proof.
“The number line is abstract like all of mathematics is.” I could not have said it better and wish I’d said it earlier. It’s like the rules of chess. Nobody asks if the rules of chess are “true.” Yet one can determine from first principles whether a given position is a legal one.
The statement “.999… = 1” is simply a legal position in the game of mathematics. You can no more ask whether it’s true in the real world than you can ask if a bishop only moves diagonally in the real world. The question is a category error. A formal game is neither true nor false about the world, but can only be valid or invalid within its own context.
If a mathematical theory is useful, the physical sciences use it. Utility is the standard by which we apply abstractions to reality. There is no question of truth. There’s always a better physical theory coming along, and a mathematical abstraction to support it.
.999… = 1 is a fact of mathematics.
What it means “in the real world” is a matter of philosophy; but that is a completely different question. It’s actually meaningless in the real world, since the real world is not made up of mathematical points, nor do the real numbers exist in the real world.
There is nothing real about the real numbers.