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Posts Tagged ‘**Significant mathematics**’

### Efficiency has limits within cultural biases, and within mathematicians…

Posted December 11, 2009

on:**Efficiency has limits within cultural bias**; (Dec. 10, 2009)

Sciences that progressed so far have relied on mathematicians: many mathematical theories have proven to be efficacious in predicting, classifying, and explaining phenomena.

In general, fields of sciences that failed to interest mathematicians stagnated or were shelved for periods; maybe with the exception of psychology.

People wonder how a set of abstract symbols that are linked by precise game rules (called formal language) ends up predicting and explaining “reality” in many cases.

Biology has received recently a new invigorating shot: a few mathematicians got interested working for example on patterns of butterfly wings and mammalian furs using partial derivatives, but nothing of real value is expected to further interest in biology.

Economy, mainly for market equilibrium, applied methods adapted to dynamic systems, games, and topology. Computer sciences is catching up some interest.

“**Significant mathematics**” or those theories that offer **classes of invariant** relative to operations, transformations, and relationship almost always find applications in the real world: they generate new methods and tools such as theories of group and functions of a complex variable.

For example, the **theory of knot** was connected to many applied domains because of its rich manipulation of “mathematical objects” (such as numbers, functions, or structures) that remain invariant when the knot is deformed.

**What is the main activity of a modern mathematician? **

First of all, they do systematic organization of classes of “mathematical objects” that are equivalent to transformations. For example, surfaces to a homeomorphisms or plastic transformation and invariant in deterministic transformations.

There are several **philosophical groups within mathematicians**.

1. The **Pythagorean mathematicians** admit that natural numbers are the foundations of the material reality that is represented in geometric figures and forms. Their modern counterparts affirm that real physical structure (particles, fields, and space-time…) is identically mathematical. Math is the expression of reality and its symbolic language describes reality.

2. The “**empirical mathematicians**” construct models of empirical (experimental) results. They know in advance that their theories are linked to real phenomena.

3. The “**Platonist mathematicians**” conceive the universe of their ideas and concepts as independent of the world of phenomena. At best, the sensed world is but a pale reflection of their ideas. Their ideas were not invented but are as real though not directly sensed or perceived. Thus, a priori harmony between the sensed world and their world of ideas is their guiding rod in discovering significant theories.

4. There is a newer group of mathematicians who are not worried to getting “dirty” by experimenting (analyses methods), crunching numbers, and adapting to new tools such as computer and performing surgery on geometric forms.

This new brand of mathematicians do not care to be limited within the “Greek” cultural bias of doing mathematics: they are ready to try the **Babylonian and Egyptian cultural way** of doing math by computation, pacing lands, and experimenting with various branches in mathematics (for example, Pelerman who proved the conjecture of Poincaré with “unorthodox” techniques and Gromov who gave geometry a new life and believe computer to be a great tool for theories that do not involve probability).

Explaining phenomena leads to generalization (reducing a diversity of phenomena, even in disparate fields of sciences, to a few fundamental principles). Mathematics extend new concepts or strategies to resolving difficult problems that require collaboration of various branches in the discipline.

For example, the theory elaborated by **Hermann Weyl** in 1918 to unifying gravity and electromagnetism led to the **theory of “jauge”** (which is the cornerstone theory for quantum mechanics), though the initial theory failed to predict experimental results.

The cord and **non-commutative geometry theo**ries generated new horizons even before they verified empirical results.

Axioms and propositions used in different branches of mathematics can be combined to developing new concepts of sets, numbers, or spaces.

Historically, mathematics was never “empirically neutral”: theories required significant work of translation and adaptation of the theories so that formal descriptions of phenomena are validated.

Thus, mathematical formalism was acquired by bits and pieces from the empirical world. For example, the theory of general relativity was effective because it relied on the formal description of the **invariant tensor calculus** combined with the fundamental equation that is related to **Poisson’s equations in classical potential**.

The same process of adaptation was applied to quantum mechanics that relied on algebra of operators combined with **Hilbert’s theory of space** and then the atomic spectrum.

In order to comprehend the efficiency of mathematics, it is important to master the production of mental representations such as ideas, concepts, images, analogies, and metaphors that are susceptible to lending rich invariant.

Thus, the **discovery of the empirical world** is done both ways:

First, the learning processes of the senses and

Second, the acquisition processes of mathematical modeling.

Mathematical activities are extensions to our perception power and written in a symbolic formal language.

Natural sciences grabbed the interest of mathematicians because they managed to extract invariant in natural phenomena.

So far, mathematicians are wary to look into the invariant of the much complex human and social sciences. Maybe if they try to find analogies of invariant among the natural and human worlds then a great first incentive would enrich new theories applicable to fields of vast variability.

It appears that “significant mathematics” basically decodes how the brain perceives invariant in what the senses transmit to it as signals of the “real world”. For example, **Stanislas Dehaene **opened the way to comprehending how elementary mathematical capacity are generated from neuronal substrate.

I conjecture that, since individual experiences are what generate intuitive concepts, analogies, and various perspectives to viewing and solving problems, most of the useful mathematical theories were essentially founded on the vision and auditory perceptions.

New breakthrough in significant theories will emerge when mathematicians start mining the processes of the brain of the other senses (they are far more than the regular six senses). Obviously, the interested mathematician must have witnessed his developed senses and experimented with them as hobbies to work on decoding their valuable processes in order to construct this seemingly “coherent world”.

A wide range of interesting discoveries on human capabilities can be uncovered if mathematicians dare direct scientists and experimenters to additional varieties of parameters and variables in cognitive processes and brain perception that their theories predict.

Mathematicians have to expand their horizon: the cultural bias to what is Greek has its limits. It is time to take serious attempts at number crunching, complex computations, complex set of equations, and adapting to newer available tools.

**Note**: I am inclined to associate algebra to the deductive processes and generalization on the macro-level, while viewing analytic solutions in the realm of inferring by manipulating and controlling possibilities (singularities); it is sort of experimenting with rare events.

### Who is towing sciences?

Posted December 11, 2009

on:**Who is towing sciences? (Dec. 6, 2009)**

In the previous 2 centuries it was **mathematics that was towing sciences** and especially physics in its theoretical aspects. Actually, most theories in the sciences were founded on mathematical abstract theorems that were demonstrated many decades ago by mathematicians.

It appears that this century is witnessing a different trend: **sciences are offering opportunities for mathematicians to expand their fields of interests, away from the internal problems of solving conjectures** (axioms or hypotheses) that were enunciated a century ago.

**Cedric Villani,** professor of mathematics at the Institute of Henry Poincaré in Paris, thinks that physics still remains the main engine for mathematicians to opening new fields of study. For example, **equations of fluid mechanics are not yet resolved** (those related to Navier-Stokes and Euler); compressed fluids; **Bose-Einstein condensation**; rarefied gas environment. We even **cannot explain why water boils.**

There is also the study of how the borders that separate **two phases of equilibrium** among chaotic, random, and unstable physical systems behave. The mathematician **Wendelin Werner** (Fields Prize) has been interested in that problem and I will publish a special post on how he resolved these phenomena.

There is an encouraging tendency among a few mathematicians to dealing with new emerging fields in sciences. The most promising venue is in **computer sciences** or “informatique”. In computer sciences the **problems of verification** (for example, theory of verifying proofs) is capturing interest: mathematical tools for validating theorems in the realm of logic or **exhuming errors are challenging**.

**Biology is an exciting field but it didn’t capture the interest of mathematicians**: the big illusion that mathematicians will approach biology faded away simply because fields related to the living world is too variable in complexity to attract mathematicians. The **number of “variability” is “absurdly” numerous** and does not lend simple and clean-cut laws that prove that the world is well structured and “mathematically” ordered.

This has been the case in the 70’s for **cognitive sciences and artificial intelligence**: scientists in those two fields hoped that mathematicians would get interested and drive them to innovative results, but nothing much happened.

The good trend is this kind of social re-organization within the mathematical community for deeper cooperative undertaking in solving problems. The web or internet has kind of revolutionized cooperation; it opened up this great highway for sharing ideas instantly and cooperating among several researchers.

For example, **Terry Tao and Tim Gowens** propose problems on their blog (**Polymath project**) and then the names of contributors are disseminated after a problem comes to fruition.

Still, individual initiatives are the norm; for example, Tao and Gowens ended up solving the problems most of the time. **Mikhail Gromov** (Abel Prize) has given **geometry a new life line in mathematics**.

It appears that “Significant mathematics” basically decoded how the **brain perceives “invariants”** in what the senses transmit to it. I conjecture that since individual experiences are what generate the intuitive concepts, analogies, and various perspectives to viewing a problem. Most of the mathematical theories were essentially founded on the stored vision and auditory perceptions.