Representation as preeminent

Ignacio Añon ianon at
Tue Mar 16 00:29:52 EDT 2021

Caballero wrote:

"M. Gromov,

My interpretation of this statement is that expressions are like
coordinate systems of a mathematical object. The same mathematical
object can be represented in several ways, using different
expressions. A trivial example is the number 2/3, which can be
represented as 2/3, 4/6, 6/9, etc. One of I. Gelfand's famous quotes

> all of mathematics is some kind of representation theory"

The idea behind Gromov's phrase, which you cite, is shared by an old
tradition in mathematics. You mention Gelfand, but Lagrange, Riemann and
Gauss, could have said the same thing; and yet Gromov expresses the
idea quite clumsily. It is dangerous to distinguish mathematical
spaces, from their representations: there is a tradition, enhanced by
Lie, and shared by invariant and string theory, which posits
representations as the "Whole" which determines the "Part". In this
metaphor, "part" is a symbol for the notion of  "mathematical
object", which in this vein is synonymous with "space". The problem is
that, as of 2021, we haven't been able yet to master all the layers of
these spaces, whose inner invariant topology, is determined, not by
its structure, but by its representation. It was this sphere of
problems, that led Stone to tackle the structure of boolean
representations, and Von Neumann to discover continuous geometry: in
this realm, representations, and the notion of a spatial mathematical
continuum on the interval (0, 1), are inseparable...

Caballero wrote:

"To define computation from mathematics is questionable from an
epistemological perspective since our direct experience is with
computation (including measurement as a computation in the physical
world), not with mathematics. Mathematics may be the result of
abstraction in a computation. A mathematical object may not be a
primitive concept but a concept derived from computation. The
following argument seems to be a counterexample of this claim:
from the premises:

> 1. Every language is an algebra
> 2. There are several languages with subgroups unrelated to computation

it can be possible to derive the conclusion

> L. Not every algebra is entirely for computation

but rather than a counterexample, it is a positive example. Indeed, an
algebra that is unrelated to computation is an abstraction from an
algebra that is related to computation: this abstraction is obtained
by ignoring the property of being computational."

It is quite important to free our notion of Representation, from
algebra, as you are beginning to do. E. Cartan, and recently S. K.
Donaldson, and originally Lie, were able to show that there exists a
compact real form, of any complex semi simple Lie algebra, which is
determined by the representation of its spatial geometry: the
algebraic structure, is dependant on it, and not viceversa. If you
enhance the given group representation, into a Magma(semigroup),
it is the magma that determines the spatial structure, in a more precise
sense, than the previous group.

Any representation, is an infinitesimal transformation, creating a space
within which computation is possible: all computations, if preserved by
a topological transposition map, can be shown to be symmetric to any
euclidean structure within a Hilbert space. In current math, the problem of
defining the different layers of these spaces, is still the horizon of
difficulty. The fact that there are "dimensionless" layers, where things
hausdorff gaps and nowhere differentiable functions appear, and others
where color variables exist linking the layer of harmonic
to topology, still makes of this field, a mystery.

When proof assistants, can illuminate certain aspects of these spaces
created by representations, they will cease to be mere proof assistants,
and become tools of discovery: but it seems to me that chip architecture
and software, are a long way from this goal....
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