On mathematical models of reality

                 _________________________________

 

                          Jorge Carrera



              Engienner Faculty, Graduate Programm

                 National University of Mexico

                   jorge@gauss.depfi.unam.mx



                   (received: April 30, 1996)

             (first printed - Vol. 2, No. 9 - June 1996)

 

1. Introduction

_______________

 

	With the development of cheaper and every time more powerful computers 

the use of computational models of physical, economical, ecological, etc. 

models has become a widespread phenomenon.  Now it is possible to run a 

program to calculate stresses in a piece of machinery in a desktop computer. 

Even a home computer can handle programs for the diffusion of an illness in a 

given population or to make prognoses about macroeconomical variables. From a 

pragmatic point of view it is a situation that can even get the adjective of 

"fantastic", with which we can only agree.



	But we can also think of Heiddeger's word in "Was heisst Denken" 

("What is thinking" or "What is the meaning of thinking"; the author 

apologizes because he does not have an English translation of Heiddeger's 

works, and has been forced to translate from German. Maybe the translations do 

not fit with the ones usually being employed, but surely it can be 

identified): "the most thinkable thing today is that we do not yet think". I 

am sorry to say so, but a great part of those employing computer models do not 

think about them. They take programs, or they program themselves, and use them 

with an immense faith, sometimes without even knowing the underlying 

mathematics.

 

	It has to be said that faith is a very common, nice and useful feature 

of science. I have faith in the value of the gravity constant, I have faith in 

the degree of purity of a substance that I am employing in my experiments, 

etc. But this faith is based on my theoretical capability to understand why 

and how this value or this substance was obtained. I have, at least 

theoretically,the possibility to check something or everything, if needed. 

That it is a rational faith, (no matter how contradictory it sounds, can be 

seen in fields like Economics or Sociology, where statistics about population, 

for example, are taken only within a certain degree of certitude, and, 

depending on the government or the government agency they are taken sometimes 

only as a doubtful reference.

 

	But what happens with a program that will help me, a civil engineer, 

to build a huge bridge? Bridges move, and, depending on their size, their 

deformations are so big as to render any linear approximation only a distant 

reference. That means that the mathematics employed in more or less any kind 

of realistic models are really complex, and clearly, any model used in these 

simulations has its drawbacks. Thus, a deep knowledge of the mathematical 

system, of the discretization process, of the resolution method (a non trivial 

thing if the system is nonlinear or even very big, in the sense of quantity of 

variables and single equations), and even of the computer implementation, due 

to limited length of the rational numbers used, due to rounding, etc. should 

be an unavoidable premise for its use.

 

	The engineer knows, almost per definition, that big uncertainty 

factors are involved, Therefore she or he, and the program, must include 

enough security factors, as we know they do, because bridges generally do not 

brake down. But this engineer thinks of many factors, like measurement 

processes, uncompletely knowledge of some variables, etc., but certainly 

she/he rarely gives a thought to the question: why do the mathematics work? 

Funny enough, because the foundation of all her calculus is precisely 

calculus, i.e. mathematics.

 

	Even worst, it is the author's unlucky experience that many practical 

engineers, and also people in the academic world, do not think it important to 

know to which functional space a solution of a partial differential equation 

belongs, for example. Here a most important ethical question arises: how much 

do I, as a woman or man working in practical situations, from building a 

bridge to analyzing the ecological dynamics in a given region, have to know 

about the deeper questions of the mathematics I am using? We do not handle 

this question here, it is only one more motivation to another question, this 

one belonging to knowledge theory: "why do mathematics working? why are 

mathematical models so successful?" A complete answer is, of course, not at 

hand. Here a partial answer is attempted, based on the concept of model.

 

2. Models 

__________

 

	A general concept of a model is also a complex thing. Two features of 

that concept are of interest here. First, in opposition to laws, theories or 

hypothesis, a model is something that gives information about concrete 

systems. Newton's laws tell us how mechanical systems behave, while an 

astronomer has a model of the solar system.

 

	More important for us here, a model is something different from the 

real system that it models. This is an elementary, not a trivial fact. If it 

is so, then the next question is: why, to study system `A', should I use 

another system `S'? The answers are as varied as the models.

 

	Sometimes I do not have direct or total access to system `A', for 

example in quantum physics, but this is also the case in very practical 

situations. If I need to study the dynamics of the components of a working 

turbine, any kind of instrumentation will disturb the form of the component, 

and thus its aerodynamics (today laser equipment  is available, allowing more 

faithful measured values, but this is not the point here). Other times 

economical considerations are involved. It is cheaper to work on a model than 

on the real system. And so on.

 

	Now, if I want to study system `A' using system `S', I have to 

minimize the differences and maximize the similarities between both systems. 

To do so two main points have to be considered:

 

	a) The limits or frontiers or borders of `A' have to be 

	perfectly defined.

 

	b) A subset of all features of `A' have to be chosen, 

	as the characteristics or properties to be modeled.

 

	Then, a relation between the limits or frontiers of `A' and `S' have 

to be established, and `S' should posses a subset of features that can be put 

in a delimited relation to those of the chosen subset of features of `A'.

 

	Because `S' is different from `A', even if it is possible to establish 

very precise relations between `A' and `S', a problem remains: what to do with 

the features of `S' that do not correspond to those of `A'? Ideally, `S' 

should possess only those features that can be put in correspondence with the 

chosen features of `A'.

 

3. Models and mathematics 

__________________________

 

	Two fundamental positions exist about the origin of mathematical 

concepts. Schematically expressed there are those who think they are 

discovered, and those who think they are invented. Everybody knows that 

mathematical concepts can be invented (we say: defined), naturally not in an 

arbitrary manner.

 

	On the other side it is clear that many mathematical concepts (and the 

author thinks the basic ones) evolved as abstractions of specific features of 

real systems, the most important one is the system of natural numbers (was 

Peano the one who said "God created the natural numbers, the rest is man's 

work"?).

 

	In a technical way, abstraction is a process for going deeper into 

reality. Something is analyzed leaving aside all its characteristics that are 

not wanted, or needed, or that are not accessible. Abstraction is a 

fundamental feature of science, but it is mathematics and philosophy which are 

champions of abstraction. If we deal with quantities,  mathematics isolates 

one feature only: a number (more general, a cardinality).

 

	By concentrating on a single, maybe the singlest, feature of a very 

concrete entity, like four apples, mathematics put a whole class, an immense 

aggregate of entities, in correspondence with those apples, like four chairs, 

four dinosaurs, four theorems, four stars. We touch the whole universe with 

that number (a very slight touch, but a touch nonetheless).

 

	Thus, extrapolating the mentioned situation, mathematics concentrate 

their effort in systems possessing as few features as possible (the author is 

aware that this extrapolation can give way to strong opposition by some 

theoreticians, but even they could agree on a weakened affirmation: in many 

important cases mathematics concentrate their effort in systems possessing so 

few features as only possible).

 

	Also, by their very nature, mathematical concepts have precise limits, 

borders and/or frontiers. For example, and leaving the paradoxes of set theory 

aside, by definition it should always be possible to know which elements 

belong to a set and which not, even in complex cases, like elements of a 

Sobolew space, to say nothing of geometrical entities.

 

	It is then clear why, under the presented assumptions, mathematics is 

an ideal field to build models. From all possible systems modeling `A', if a 

mathematical system `S' can be used, it will be the one nearest to the ideal, 

because it will have the least undesirable features.

 

	Physics is also a science with a great degree of abstraction, that is 

why its relation to mathematics has been a long and fruitful one.

 

4. Final remarks 

________________

 

	We have made a small trip to an important area of metaphysics. It has 

given us some hints about the  mathematics-reality relation. As any such trip, 

many more questions have been opened than answered. In this case a glance to 

an ethical question regarding mathematics and practical actions was made. The 

question about the nature of mathematical concepts was also touched. But we 

are convinced that the deepest questions are, like abstractions, intrinsically 

related to practical, real ones.

 

	It was said that mathematical models have "the least undesirable 

features". that means, even a mathematical model have something that is not 

possible to put in correspondence with some feature of the real system to be 

modeled. This feature can be a very hidden one. (?? For example, using Green's 

theorem we can incorporate boundary conditions to integral formulations; but, 

if this formulation is being used as a model of a physical system, it should 

be clear that, the physical entity has an specifical way to form a unity with 

its boundary, and the mathematical system another. Thus, even if subsystem `B' 

of system `A' is perfectly well modeled by feature `P' of `S', and subsystem 

`C' by `R', the relation between `B' and `C' is of a different nature that the 

relation between `P' and `R', no matter how similar both relations can be. 

this is an unavoidable consequence of `A' and `S' not being identical.

 

	In the practical work similarities are stressed. For example the 

geometry of a bridge has to be reproduced to a very small tolerance in paper 

or in the virtual 3-D space of a computer. A very exact method to calculate 

deformations has to be applied. Now, the relation between a given geometry in 

an affine space and the calculations over its associated vector space is 

similar, but not identical, to the relation between the involved forces in a 

bridge and its spatial delimitation. Account is taken of differences between 

model and system being modeled, but, do we understand why do these differences 

arise? Where the similarities have their origin?

 

	We need to contradict Heidegger, we have to think. Very often we only 

calculate, very often we accept for acceptance's sake. Philosophy, metaphysics 

are there to be used, to be a practical tool for building concepts, thought, 

but also better bridges and to gain deeper understanding of ecology.



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        The author thanks the editor of Metaphysical Review, Timothy Paul 

Smith, for his useful suggestions.