A dimension is a time-zero unification of an infinite series from a baseline dimension upwards to the next higher dimension — ucarr

You'll have to be more specific. Put this in math terms.

Dimension is about the unsearchable fact of existence: transcendental being — ucarr

It looks like you’ve just restated Zeno’s paradoxes.

Physics observes without understanding. — ucarr

This is all arse backwards. Special relativity unified space and time under Poincare invariance. Dimensionality emerges under the 4D Minkowski signature where you have three directions of spatial symmetry and one of time~energy symmetry.

So what makes physics actually physics is that it creates the mathematical picture that answers to the observable facts. It doesn't do the naive Euclidean thing of building up from atomic points. It defines the global constraints that combine to zero the metric to a collection of points under a general covariant symmetry condition. The physics creates the metric from the top-down rather than builds it from the bottom-up.

You have to look at all the stuff – the 10 symmetries – that compose the holistic constraint on existence that is Poincare invariance. You have the three translational degrees of freedom, but also the three rotational dof, the three Lorentz boosts, and the time~energy dimension.

It all sounds spatialised, but it is also temporalised in being bounded by a "speed of causality" – the speed of light. The fundamental units are no longer metres and seconds but light-metres. Separations described in terms of the durations that are separating them.

Lorentz boosts deal with the frame-dependent contraction of length and dilation of time that then result from putting the speed of light at the centre of things as the Unit 1 measure of a 4D hypersphere where when you zoom in, the metric looks remarkably flat and Euclidean in its dimensionality, but zoom right out and you can see all the translations become instead rotations. Go far enough in a straight line and you wind up coming back at yourself.

In short, the point is that special relativity describes dimensionality not as points make lines and lines make planes, etc, but instead as a system of constraints which has a rich causal structure.

Dimensional flatness is only emergent in the limit. And dimensional curvature is then emergent as its complementary limit. The two tie together in Unit 1 hypersphere fashion so that each stands as the ratio of the other. This is why you wind up with three directions of translational symmetry complemented by three directions of rotational symmetry.

So rather than the usual mathematical story of a metaphysics of construction – start with the simplest thing and add complication – physics expresses a metaphysics of holistic constraint. Find a symmetry-breaking that can divide a potential against itself and develop that breaking towards its dichotomising limits.

If you start with a notion of dimensionality that is neither really curved nor straight, great. Now these can be the two directions that fly apart in a Big Bang. You can expand the hypersphere and find that the apparent distance to the curved horizon has stretched out ever flatter, allowing for Newtonian physics and its inertialised momentum, while simultaneously angular momentum has become also a thing. Objects can now rotate just as freely on the spot. They can rest in one place for a duration while spinning on their centre of mass.

The place to understand physical dimensionality is with the physics. The physics is about space, but also about the spatial dof of translation and rotation. And also about time or the way c closes the space in terms of its causal relations and ensures that localised entities can persist as they are underwritten by a global energy conservation principle.

Poincare invariance speaks to the package of stuff that composes the actual metric of a physical realm. And even then it is not the last word. It has only 10 symmetries. You can step up to the next level of the 15 symmetries encoded by twistor space – the de Sitter invariance which includes the further fact that the Big Bang happened and so the scale of things was as small and hot as it could be in the past, and will become as large and cold as it could be in the future. You add 5 extra "conformal" transformations that are there to preserve the angles between things but not necessarily the distances.

So again, in general, Euclidean geometry doesn't act as a good starting point for a leap into the metaphysics of physical reality. You instead need to work with a logic of constraints. You start with a state of "everythingness" – a vagueness or potential like the concept of a quantum foam. Then you start to consider the way it can break that symmetry into opposing tendencies – like the way that flatness can know it ain't curved and curvature can know it ain't flat as part of a mutual relativistically-closed spacetime relation.

Mathematically that is, you can place your geometry on a hypersphere. Poincare invariance placed 3D space on a 4D spacetime hypersphere. De Sitter invariance placed this 4D story on on a 5D conformal hypersphere.

So the maths is still bottom-up in that sense. The search was for a Unit 1 description which bound translations to rotations as a pair of local symmetry-breaking freedoms. And it started with 3D Galilean symmetry, then 4D Minkowski, then jumped another step to 5D de Sitter.

Maths is rather locked in to its way of building up complication from simplicity. And that can create a misleading impression about how to understand Nature in terms of its own holistic causality.

But physics came at the maths from the other direction. Special relativity stumbled into the importance of Poincare invariance as Lorentz boosts were something that got spat out of Maxwell's unification of electricity and magnetism. What worked at the level of local gauge symmetry suddenly had obvious implications for the global symmetries of spacetime as a unified 4D c-rate whole.

The same for de Sitter invariance as a mathematical next step on Minkowski space. That was originally an odd-ball way to rewrite Einstein's general relativity. A totally unphysical-seeming mathematical exercise where you could recover the GR equation from the symmetries of a Universe that was completely empty but also an exponentially expanding void driven by some "cosmological constant".

It seemed a joke back in the 1920s. Then eventually it did become evident the Big Bang happened and now dark energy has shown up as a physical thing. The future of the Big Bang will be a heat death state that arrives at this de Sitter story of a void that continues to grow at an ever-accelerating rate due to an inherent cosmological constant.

So if the OP is about metaphysics and how to relate its two offspring disciplines of maths and physics, a picture should be emerging.

The physics delivers the observations. The maths delivers the models. The metaphysics has to aim at articulating the holism of a structural logic that treats the Cosmos as a functional whole.

The problem is that when metaphysics got going in Ancient Greece, the maths of the time was pretty geometric and so that smuggled a good dose of physical realism into things. But the way maths has developed is that mathematicians want to break geometry down to algebra. It is just easier when wanting to earn a crust doing complicated calculations. The atomistic mindset came to dominate rather completely.

However maths also covers topology and symmetry. So it has its own holistic tendencies. And that is where the physicist would look for the broad metaphysical principles that might explain why everything has to be – at the structural level – exactly the way that we find them.

You'll have to be more specific. Put this in math terms. — jgill

Fibre bundle?

"Infinite series" caught my attention.

To the extent I understand you, your helpful cosmology primer on the math and physics of spacetime of the past three centuries or so, acting as a guide, enables me to see that my atomistic approach to dimensional expansion - in the mode of Zeno - in your view, employs the wrong mode going in the wrong direction.

The cosmos is not an accretion built up from infinitesimals. Instead, there are holistic cosmic symmetries that the transformations of topology "breaks" for analysis, then reassembles towards the general relativism of nodes of material existence.

My central global objective within my thesis is clarification of the relationship between analysis, i.e., science, and existence, i.e., being. At the center of my focus is the calculus, an analytic methodology of the infinitesimal as an approach to the curvilinear. When the n-gon parallelogram magically becomes the circle, we see what science makes a close approach to, but cannot attain to: dimensionally extended material things. You can smash up a thing towards understanding that its parts have a logical continuity, but you can't understand analytically the brute fact of the existence of dimensionally extended material things.

This gap separating analysis from existence, by my understanding, separates the series from the dimension. Algebraic geometry, topology, like all algebras, seeks to find the missing part via math operators governing the inter-relations of numbers. Well, dimensionally extended material things can be measured in accordance with the shuffling around of parts towards diagramming and memorizing the design for assemblage of the parts into a whole. However, the whole thing assembled dimensionally gestalts away from analysis to brute fact observable only. What science observes axiomatically, it cannot understand holistically.

Yes, math converts dimensionally extended wholes into logics and designs mathematically controllable. Once returned to its whole state of being, the dimensionally extended whole exerts its brute presence and science returns to its axiomatic observation without understanding.

Distance, the experience of logic and design, occupies the transcendental idealism of the mind as its own physics internalized as mind, per Kant.

We have a mental understanding of ourselves whilst not existentially understanding ourselves because 3-Space dimensional extension plus time is our native state as dimensionally extended beings. When I enter a room, I don't perceive myself penetrating an infinite series of planes as I traverse the cubic space of the room because parallelograms are a cognitive reality within the transcendental idealism of the mind.

Within an infinite series, you can keep changing the scale of your numerical progression so that you'll never exit a bounded infinity. Enlarge the scale and you immediately exit the bounded infinity. Might this be the way out of Zeno's Paradox?

In the calculus of the measurement of the area under the curve, however, the approach, by design, never scales up and beyond the bounded infinity. Irrational Pi tells us our math-controllable analysis never arrives at the circle. There are no circles outside of the mind, but spheres abound within our natural world. We can understand them only mentally as an infinite series of circles rotating around an axis; we can only observe spheres without understanding them existentially.

Within an infinite series, you can keep changing the scale of your numerical progression so that you'll never exit a bounded infinity. Enlarge the scale and you immediately exit the bounded infinity. Might this be the way out of Zeno's Paradox? — ucarr

Are you talking about a series or a sequence? What is a bounded infinity?

Are you talking about a series or a sequence? What is a bounded infinity? — jgill

I'm talking about a sequence. For a bounded infinity, you can configure a linear equation that extends between zero and one but never reaches either boundary.

On the other hand, if you change the scale of the same linear equation, it extends beyond both boundaries.

In both cases, the linear equation is configured to function within the realm of analysis, which is to say the plotting of the linear equation covers distance with time-positive. It is this role of time-positive that makes analysis possible.

Within the realm of dimension, distance is time-zero. You and I experience our navigation of the world as a whole person moving through distance time-positive. We don't say the infinite points, lines, and areas (that math articulates as the parts making up our native 3D extension) assemble and re-assemble as we move about. We move about as one whole person, not as our math-measurable parts continuously assembling and disassembling.

A crucially important question within math-physics is how time-positive and time-zero manage the realm of analysis and the realm of dimension.