CompSuppMathTexts.txt  ex mnA05

======== Computational support for mathematical writing

==== Introduction

In the following considerations, after noting the growing need
of making usable mathematical knowledge available for applications,
proposals are made as to how it would be appropriate to adopt for 
a production of texts for the presentation of mathematical notions 
to support innovative developments.

==== A01

Mathematics is becoming increasingly important in today's society.
There are more and more problems that raise serious concerns 
or high expectations in individuals and organizations that are judged to be 
sources of strong spin-offs and challenging and therefore such as to justify 
the employment of considerable resources and even specific tools.

These problems are believed to be addressed by procedures 
and by quantitative and structural methods to be exercised on data that must be 
as objective, accurate and complete as possible.

To solve or at least to contain one of these serious problems 
one must begin with the accurate modeling of the objects,
the situations and the processes it involves, and with the definition 
of the objectives he intend to achieve.

The models and data of the problem instances open the possibility of 
carry forward solution-oriented elaborations that may consist of 
numerical calculations, logical deductions, statistical predictions, 
simulations of behaviors, comparisons with previous experiences, and more.   

==== A02

It should be noted that solutions to major problems now can be 
made available easily enough through digital documents 
comprising numerical tables, diagrams and algorithmic schemes that 
allow one to carefully check the appropriateness of the solutions 
found on new and similar real-world situations and processes.

These tendentially critical reviews often lead to corrections, 
extensions and differentiations of the models and the procedures adopted 
and open the possibility of their further improvement.    %new
These possibilities for growth can be supported by further 
investigations and digital simulations held %?
with relative ease by the previous activities. %new

The improvements that can be achieved by the audits allow one to address 
further instances of the problems with greater expertise.
One can thus delineate a cyclical progression characterized by the 
possibility of self-correction, which in general presents itself as a promising 
source of innovation, a promise confirmed in many cases. 

This way of proceeding characterized by its potential circularity
can be considered a recent widespread manifestation of the scientific method.

==== A03

The possibility of success of these operational and cognitive  
circular procedures in these years is robustly supported by the growth
substantially continuous growth of availability by 
of many innovative technologies of tangible tools with high 
characteristics in the various directions of accuracy, range, 
speed, and adaptability;
contributions to this growth come in particular from advanced microscopy, 
quantum studies, electronic components, materials science 
and software engineering.

In particular, to the awareness of the effectiveness of projects
based on scientific-technological attitude can contribute to the
conviction of the possible synergy among the different disciplines called 
to cooperate, and to this conviction can greatly contribute 
the ability to proceed unitedly on multiple fronts 
that mathematics has been able to demonstrate with its ability to alternate 
between unifying abstraction and analytical specialization accompanied 
by computational efficiency.

==== A04

At the level of documentation, teaching and popularization one must 
intervene with the production of texts of different genres: books,
articles, Web pages, graphics and in perspective much more.
 
==== A05

Regarding the noticeable acceleration of scientific-technological progress 
found in recent years, two important factors must be acknowledged.

The first already evidently established is due to the advances 
of telecommunications and especially to the growth of the global Web 
with its ability to collect and convey data and with  
its ability to circulate ideas and support collaborations of many kinds.

The second, still at an early stage and with many ill-defined characters,
is the recent advances in artificial intelligence (AI) 
which have almost suddenly become widely evident 
with the performance of generative systems (ChatGPT, Bard, Copilot, ...)
and concretely verifiable in numerous specific applications.

Note, for example, that all 2024 Nobel laureates in Physics, Chemistry, and 
Medicine have connections to AI. %?

In biology and pharmacology, many researchers talk of a recent revolution 
induced by AI. Machine translation and various financial activities 
have been able to make clearly evident progress. 
Not inferior are the expectations in the industrial world. 
At this point we consider superfluous talking about military activities.  

==== A06

The above two growth factors also contribute to the prospect of 
a relevant growth in the demands for mathematical contributions, 
particularly from several manufacturing and administrative sectors. 

Recent surveys estimate that in the coming years 40 percent of activities  
will require precise computer and STEM  skills (hard sciences,
technology, economics and mathematics) predicting that 
in many countries (including Italy) these will be severely lacking.

==== B01

The above considerations therefore point to an increase in the
demands for mathematical achievements and skills.

These may involve different levels and many specializations;
in addition, the demands may also arise as urgencies, given the 
speed with which innovations in operational procedures 
and in the needs of society are occurring.  %?

It is also worth mentioning the close relations between mathematics and AI, also these
with characteristics of symbiosis: mathematics provides algorithms and methods
to AI and this promises to provide significant ideas for the resolution
of entire mathematical problems. [citations]

==== B02

A large part of the demand for mathematics by applicative activities
will also involve computer science, physics, engineering and even economics,
disciplines that evidently play important roles in a number
of applicative sectors.

In particular, there will be a need to enhance the production
of expositions of mathematical content at various levels and with different
focuses on both mathematical and applicative topics.

These are texts or documents that can either be dedicated
exclusively to mathematical notions, or present limited contents
(for example mathematical results that justify algorithms or methods)
within texts aimed at specific applications.

==== B03

These documents can be placed either at the level of wide-ranging
expositions in support of entire methodologies, or at the level
of teaching aimed at operators with defined tasks,
or at the level of updating methods required by applications
with multifaceted computational needs.
The mathematical concepts presented may be both general and
specific, their tone may range from the elementary to the specialist 
in-depth, and the expositions may differ in dependence 
of the various requirements they presuppose.

It may therefore be useful to think about the production on a relevant scale 
 of documents with mathematical content aimed 
to a rather wide range of applications.

==== B04

It would be desirable if these documents could be systematically   
linked in some workable way to a body of notions that can be shared.

This on the one hand would allow economies of scale, but on the other hand 
would pose the problem of consistency of extended sets of results.

There is no question that these documents must have a digital version:
only this allows for acceptable sharability by 
a thinkable group of authors and content managers;
furthermore, only a digital version allows for distributions and updates 
easy and timely.

Any demand for paper versions can be met without difficulty.

=== B05

Moreover, digital documents can naturally be provided with links
with other documents in an imaginable library and links to
other documents in the Web (now unavoidable citations).
---
Even the prospect of often needing to amend 
these documents, to expand their scope, to adapt them to changes in
technological, economic or administrative requires that we design 
and produce digital documents.

With regard to adapting to technological changes, we point out that 
the availability of certain innovations makes it very convenient that 
they can be implemented rapidly by adapting simultaneously 
the technical environment and documentation concerning the new.

It can also prove very important to have documents available 
updated in multiple languages, and this can be ensured by current 
automatic translators that can refer to lexicons 
with appropriately organized syntactic and semantic enrichments,
possibly made aware of the needs of particular areas of 
applications.

==== C01 

Any project for the systematic development of educational and 
popular scientific documents on mathematics 
(let us call them MatDD documents) must pose problems concerning 
languages and software products, problems no less than those 
posed by the editor/publisher of a series of paper-based texts.

What languages to use to express formulas?
What languages to use to present and possibly implement algorithms? 
What software systems to adopt to manage texts at different stages 
of editing and subsequent updates by multiple authors 
and coordinators, of subjectification, archiving and distribution?

Of course, all these issues must take into account the strategies 
that an editorial organisation is willing and is able to adopt to carry out 
such a project publishing; moreover the context requires careful attention.

Of course, in these few pages we can only point out
the existence of these problems and advocate the examination of their
many facets.

==== C02 %%%%

At this point let me indicate some characteristics that documents-MatDD 
in my opinion should have.

The overall imprint should be predominantly pragmatic:
it must be about mathematical notions clearly linked to applications
of relevance, both from the point of view of scientific consistency and from 
that of application utility.

Mathematical developments must be presented with clear links 
with computational tools and mathematical-physical models and more 
generally with the mathematical-applicative models that make mathematical 
results concretely usable in realistic environments.

It happens, however, that in many texts with limited destinations
it is necessary to limit the requirements of generality and rigor
of mathematics.

This need must be considered together with the fact that in order to be
easily readable, every mathematical exposition must use simplified terms 
(for example, not distinguishing between a structure such as a group 
and the set of its elements) and must resort to metaphors
and intuitive images.

==== C03

These texts with acknowledged compromises on generality, on the 
completeness of demonstrations and with appeals to intuition 
must be linked to texts or quotations from texts that do not give up 
demonstrative completeness, generality and even abstraction.

This coverage with ``rigorous mathematics'' demands it because of the rapidly 
change of the applications themselves with the frequent opportunity to 
devise more demanding solving procedures that require 
to re-observe more carefully the procedures to be improved and to 
analyze them according to viewpoints that may be more demanding
and therefore more general and more pprofound.

==== C04

As for abstraction, a feature of mathematics 
often hostile to people pressed by the need to obtain concrete 
specific concrete results, it should be emphasized that the ability to 
move with mastery between abstract notions and their particularizations 
more substantial can be decisive for the possibility of 
rapid adjustments to the procedures themselves. 

In general, it should be emphasized that many abstractions have made it possible to 
to arrive at systems of solving procedures adaptable to wide variations 
of applications with considerable economies of thought;
this also to the increasing availability of software tools 
with high adaptability qualities, also due to the high scalability 
of current hardware+software systems; furthermore, in this direction 
some developments in AI seem very promising.

==== C05

To return to the desirable features of MatDD documents. 
we simply add schematically the following requirements:

Need for understandable and 
meaningful ones capable of clarifying general results.

Opportunity to have available illustrative figures and animations.

Availability of bibliographies and sitographies and reporting of 
documents devoted to overviews, synopses, and taxonomies on the areas 
of mathematics and applications concerning the groups of procedures 
for which the greatest need for adjustment is anticipated.

Availability, alongside abbreviated and simplified developments, of documents
containing more complete and rigorous discourses linkable to the previous ones
with the greatest possible ease.

Availability of indexes of terms, symbols and notations that 
allow one to move over a large set of documents, 
that is, to have a network organization of the document set. 

Links to the best mathematical resources accessible on the Web.

==== C06

We conclude by pointing out an attempt at a fairly extensive presentation
of mathematical developments aimed at people interested not only 
episodically in the applications of mathematics.

This is the exposition called MATeXp available on the Web pages
of the undersigned.

These pages, still under development, despite their obvious 
imcompletenesses, contain considerations that I hope will be of 
some utility when confronted with some recent development.

=== C07

In addition, MATeXp can be considered as an experience for researching
of some ways in which texts of the kind 
desired above, having limited means.

Instead, it should be recognized that in a longer-term perspective 
to the systematic production of MatDD texts will have to be applied 
tools of far greater scope and commitment, inparticular AI tools
to be developed specifically.