Formal language of math
Formal language vs. language built around separate sets of contexts and interpretations etc
Can the formal language of math be related to the existence which we perceive?
Answer: Basically, no.
Can the formal language of math be related to the existence which we perceive?
Answer: Basically, no, since there are too many limitations as to the patterns which a formal language can describe, and subsequently, a formal language can be virtually unrelated to any pattern of existence, but once formal language of math is entrenched in the professional math community, the formal language becomes both ever more complicated and relatable only to the patterns of other math literature, with nothing (or very little) to say about the existence we experience.
Science has also become equally formal and mathematical (using the same formal, complicated math language) with its math structures improperly interpreted in regard to the relation these math patterns have to observed patterns, ie if the nuclei cannot be described using the laws of particle-physics then this gap between math patterns and observed patterns needs to be taken as an indication that the formal math patterns are not relatable to the perceived properties of our existence.
That is, in both math and physics there is an endless stream of literature... , which only has a valid logical and formal relation to other math and science literature... , whose structures and laws cannot be used to accurately describe "what is observed" by applying, in a both a wide-ranging and sufficiently precise way, set of rules, which allows math descriptions to precisely describe the observed properties of general systems.
By the failing of math methods to describe the stable properties of physical systems, this means that the formal structures of math and science are inconsistent with what actually exists, ie the world's as well as life's containment set.
There now is an alternative containment set, interpretative context for "what it is that we observe" so that assumptions, axioms, contexts, interpretations, and containment sets, etc are re-invented so as to be able to describe the observed patterns of the physical world, but it is a description which transcends the idea of materialism. That is, it is an example of the intuitive method which works better than does the formal methods now used in math and science.
Is the context for physical description to be: differential equations defined for functions or functions spaces whose domain space is a metric-space,
Is the shape of a metric-space the fundamental property of the physical structure, where the metric-spaces (which have shapes) are, in turn, contained in a domain-space which is another (higher-dimensional) metric-space, which is also a shape; and so-on; and where derivatives are defined in a discrete manner on stable metric-space shapes, in a context in which there are different, and separate (because of the shapes of the metric-spaces), dimensional levels?
This is actually, the correct context which allows for stable physical properties to exist in a stable manner and to be described in a measurable context.
In general relativity, the question about the metric-space properties is central to describing physical properties, but it is framed in a non-linear context so that measuring is not reliable, and the context is quantitatively inconsistent, ie it is non-linear, so that no physical properties are describable, ie it is useless in regard to practical use.
Modern formal math is based on the idea that one can consider any context, which is vaguely related to the math structures of quantity and shape, and calculate with formal math patterns in a consistent manner, but this is not true.
Most contexts (of existence) are not consistent with the elementary properties of quantitative patterns, ie the formal math patterns are quantitatively inconsistent patterns and most shapes described by formal math patterns are not stable.
So the formalism of math allows for a great range for math literature but it is mostly quantitatively inconsistent nonsense, which can neither identify (stable) patterns, nor can the properties of formal math descriptions be consistently measured, so it is unrelated to a measurable context within which stable patterns exist.
Formal math does not have any (or it has very little) practical value.
Can math formalism be applied to any quantitative context? or Must math descriptions of math patterns always adhere closely to the properties of elementary arithmetic? The latter.
Must the patterns described in a math context, which is quantitatively consistent, be stable or can they be unstable? They must be stable.
Is a derivative a local linear model of measuring or can it be non-linear? (It must be linear.)
If many-dimensional then must the local matrix properties of a derivative, which are associated with the local linear vector structure, which the derivative gives to the local coordinates of the function's domain space, be continuously commutative, in order for the math description to remain quantitatively consistent, and so as to remain consistent with the properties of elementary arithmetic,
Is non-commutativeness a valid property of formal math descriptions? (Quantitatively consistent patterns require continuous commutative relations on matrices, as well as on function spaces.)
Are derivatives operators which act on function spaces so as to allow non-commutativiety between derivative operators,
Are derivatives about local measurable relations which exist in a discrete manner defined between stable metric-space shapes? (the latter is a better context for the description of existence)
In order for measuring to remain quantitatively consistent, must the domain space be metric-invariant, and be defined on a local linear context, so that all the local linear geometric measures are consistent with metric-invariance in a continuous manner, and so that the matrix associated to the metric-function's symmetric 2-tensor structure is continuously commutative? Yes.
Is a derivative best thought of as a local linear measure which is defined in a discrete manner in regard to the changes in the properties, such as position, to which its local measures are related, ie discrete changes associated to a derivative which is also defined discretely? Yes.
In this discrete definition there are also discrete relations which exist between different adjacent dimensional levels, and are related to discrete shapes of either the interaction structure, or discretely related to a stable shape for a metric-space.
Within a given metric-space the discrete structure of the derivative can be approximated so as to define the usual (usually second-order) sets of differential equations of physics, mostly classical physics, but the new description also accounts for the property of quantum randomness, and so differential equations associated to random properties do make sense, but they are very limited in regard to their practical value, since the stability of physical systems actually comes from their stable metric-space structures.
That is, formal math structure can exist but its relevance is very limited.
Are formal math structures to be fixed and proclaimed for all-time, and then applied in a formal way, where the validity of the math patterns is assumed to always be true? No.
This is a paper related to a book review in Bulletin of the American Math Society July 2013 concerning the book "Plato's Ghost:... , by J Gray, and Reviewed by D E Rowe. This is a review whose focus is on the conflict between the formalists (determine a fixed language for math) and the intuitionists (relate math patterns to the properties of existence so that math patterns are stable and quantitatively consistent) which occurred in math, in a time period around 1900.
Rowe's review apparently is more sympathetic to the intuitionists view (where the author of this paper is an intuitionist) than was Gray's outlook.
The main issue of the book is about how modern math has come to primarily be the domain of the formalists.
But formalism has led to some significant failures, since rigorous formalism can define a formal truth, but many, if not all, of the applications of the formal patterns of modern math fail to provide valid answers when applied to the real world, ie existence and the formal truths of mathematics are inconsistent with one another.
There exists a deeper criticism of the formal language of math, where the alternative to formal math is to continually re-building language so that it remain both valid and practically useful.
Where the formal language of math, though rigorous, (the language) is neither accurate (in regard to what, about the world, it is trying to describe) nor is the formal language of math practically useful, ie it is not the correct context in which to describe a physical system's measurable properties, however, formalism deals with extremely complicated patterns which delve into a very general context, but the math information about this general context seems to be un-relatable to practical uses or to actual (observed) properties.
Either, in probability
trying to define risk (or a quantum-system's spectral properties) based patterns
either defined by identifying vaguely distinguishable, but unstable, patterns
or on sets of spectral values (related to a function space) which the descriptive structures cannot identify,
Or, in geometry
Trying to describe an unstable geometric pattern by means of a quantitatively inconsistent descriptive (or containment) context, eg using a non-linear context to describe unstable shapes (or a chaotic context).
This is the realm of formal math.
That is, formal language claims to be capable of describing a system's measurable properties, by applying the formal language structures of math, based on formal properties of an assumed containment in regard to either coordinate domain spaces or function spaces, but it is either not the correct context, or the ideas are inconsistent with the actual patterns which (that) are being observed, yet the formal quantitative structure claims to be the correct containment space for the observed patterns of, for example,
eg trying to identify the probabilities of financial patterns, but the patterns can never be sufficiently defined, ie the patterns are unstable and thus not properly relatable to a probability containment context.
However, there are many examples where formal language is trying to relate patterns to a quantitative context but it is failing, and this is done "even though" there already exists an alternative set of: assumptions, contexts, and interpretations concerning pattern containment, within which one can choose many different quantitatively descriptive contexts,
there exists an already developed new "quantitatively based" language, which exists as an entire descriptive structure, whose logical basis is more intuitive and more consistent with the properties of existence, and which can account for the observed patterns, which the formal language is trying to describe.
An example of an alternative math language is based upon using the ideas of Thurston-Perelman geometrization in a many-dimensional context, in order to describe the stable spectral-orbital properties of physical systems which are observed at all size-scales.
Instead of defining derivatives on either general metric-spaces or general function spaces, rather define the descriptive context to be a discretely determined set of metric-spaces which posses stable shapes and are associated to specific dimensional levels and subspaces contained within an 11-dimensional hyperbolic metric-space.
This math structure, as well as the criticism of the accepted context of formal language used by the professional math and science communities, is presented in several books "A new Copernican Revolution" B Bash, P Coatimundi Trafford publishing, "The Authority of Material vs. the Spirit" D Hunter, Trafford publishing, and six books by M Concoyle published as e-books at: Scribd.com, search m concoyle.
In regard to social forces, formal math structures are about a formal power relation between math and the state, where the state wants a hierarchical structure imposed on the math community, and in this formalism the state can manipulate control of the subjects of math by placing obsessive competitive and aggressive types within such a community ie controlling a subject by the use of borderline autistic people. It is about control by requiring narrow vision in regard to a subject from which power, which can challenge their arbitrary social constructs of high-social-value, can emanate. This is about the cultural knowledge remaining consistent with the collective actions of a society which collectively supports the interests of the ruling class.
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