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Part 3. Mathematics and modeling

Let us try approaching to mathematics from one more direction - the mathematical modeling.

Of course, in science, modeling is playing an important role. By para-phrasing the famous Kant's statement*, I would say even more than that:

A particular branch of science is proper science only to the extent to which it is building models.

One can try also modeling of the process of modeling itself (cognitive science?).

I prefer here the most general notion of model (as it is used in computer science):

Model is a "structure" A used for some purpose instead of another "structure" B. The purpose may be: prediction, exploitation, destruction of B etc.


*) "Ich behaupte aber, dass in jeder besonderen Naturlehre nur soviel eigentliche Wissenschaft angetroffen werden koenne, als darin Mathematik anzutreffen ist."

In serious branches of science,

knowledge cannot be derived from "facts" alone...

As put by Albert Einstein: "... the human mind has first to construct forms independently, before we can find them in things. ... "

Full text:
"It seems that the human mind has first to construct forms independently, before we can find them in things. Kepler’s marvelous achievement is a particularly fine example of the truth that knowledge cannot spring from experience alone, but only from the comparison of the inventions of the intellect with observed fact."
(The Schiller Institute - ).

Einstein's original text in German:
"Es scheint, dass die menschliche Vernunft die Formen erst selbständig konstruieren muss, ehe wir sie in den Dingen nachweisen können. Aus Keplers wunderbarem Lebenswerk erkennen wir besonders schön, dass aus bloßer Empirie allein die Erkenntnis nicht erblühen kann, sondern nur aus dem Vergleich von Erdachtem mit dem Beobachteten."
(Albert Einstein über Kepler. Frankfurter Zeitung, 9. November 1930, see
online copy by Dr. Böttiger-Verlag-GmbH.)

See also Einstein's manuscript in Einstein Archives Online.

As a consequence:

George E.P. Box

Photo: University of Wisconsin-Madison

"All models are wrong, but some are useful."
George E. P. Box, statistician

Box, G. E. P. Robustness in the strategy of scientific model building. In R. L. Launer, & G. N. Wilkinson (Eds.), Robustness in statistics (pp. 201-236). New York: Academic Press, 1979

Mathematical modeling

Mathematicians are working with models in a very specific way.

Lobachevsky should have been aware of the idea of modeling when he tried using astronomical measurements to decide which of the two different geometries is better as a description of the physical space: the usual Euclidean one, or his "imaginary geometry"? For Kant, it was an impossible question to ask - he knew only about one "impeccable" kind of geometry.

In biology, the model of living cells is being improved steadily, covering increasingly the growing experimental data. But what, if we would stop this flow of new external information, and would declare that, from now, we will investigate our model of living cells "as it is" (despite of its limited adequacy)? And what, if we will do this for years? I would say that in this moment our model becomes a mathematical model.

Thus, for me, the true distinguishing feature of mathematical models is as follows: it is worthwhile to investigate these models without any reference to the modeled objects. And one can do this for years. For me, exactly this feature separates mathematical models from the non-mathematical ones.

Many people think that mathematical models are models built by means of the well-known mathematical stuctures (numbers, functions, spaces etc.).

I prefer the above-mentioned more general distinguishing feature. In other words: a model should be qualified as a mathematical model, if and only if it is both self-contained and stable. Self-contained - because external information flow to it is prohibited. Stable - because modification of a mathematical model is qualified as defining a new model.


Thus, mathematics is a specific branch of science not because of a specific subject of investigation, but because of a specific method of investigation: create and explore models distracted completely from the modeled objects.

The task of mathematics is developing methods of creating and exploring of this kind of models.

As put by Morris Kline: "More than anything else mathematics is a method."

Or, as put by Polonius: "Though this be madness, yet there is method in't." This "mathematical excourse" of William Shakespeare was first observed by Stanislaw Lem in his "Summa technologiae".

Mathematicans may modify their models arbitrary, even if this could reduce or destroy completely their (already restricted) similarity to the "originals". And explore such models for years... In exactly this way, the non-Euclidean geometries were created.

This kind of experimentation is put into mathematics "by definition" - because mathematical models are - "by definition" - completely distracted from the modeled objects.

Are databases "true" mathematical models?

The database of an enterprise information system can be regarded as a model of the enterprise. The more complete is the database - the better it can serve as a replacement of the enterprise itself - for analysis and decision support etc.

According to the above-mentioned definition, the database will become a mathematical model, if it will be "distracted" from the enterprise (i.e. we will stop updating it), and we will explore it as an independent structure.

Such an independent database can be modified arbitrary for various purposes. In this way, we can make the database completely diverse from any reality. Exactly like as mathematicians modify arbitrary their structures, axioms etc.!

Such independent databases - are their status essentially different from the models of the Solar system (which are, no doubt, recognized as "true" mathematical models)?

Formal models, or mathematical models?

Most people will agree that models that can be distracted from the modeled objects (i.e. self-contained and stable models) represent a significant class of models. Still, maybe, formal models would be a better term for this class of models? And mathematical models are more "subtle" structures?

For example, the database of an enterprise information system is, by no means, a formal model, but could it be qualified also as a mathematical model?

But, then, how should we call the scientific discipline investigating not a particular formal model, or a specific class of formal models, but formal models as such? Shouldn't we call this discipline "mathematics"?

Victor Mikhaylovich Glushkov (pictures) replied to this question as follows:

"If you think that mathematics should have bright future, then, perhaps, we must agree that the above-mentioned [formal - K. P.] methods also must be added to mathematics. Otherwise, mathematics will make its way to decline, and something new will arise instead." (Sorry, my own English translation.)

V. M. Glushkov. Gnoseological foundations of the mathematization of sciences. Preprint of the seminar "Methodological problems of cybernetics", Institute of Cybernetics, Ukrainian Academy of Science, 1965 (online copy).

This is why I insist that distraction from the modeled objects is the really significant distinguishing feature of mathematical models. For me, only with this thesis the really specific "orthogonal" nature of mathematics as a kind of science is revealed.

In mathematics, one may investigate any kinds of objects, processes, systems etc., without any restrictions. Specific is here only the method - creating of models that can be explored without any reference to the modeled "originals". The task of mathematics is development of methods allowing to create and investigate this kind of models.

This is now my favorite definition of mathematics.

From "numerical modeling" to mathematical ontologies

The natural number system arised as a model of processes of counting, the "real" number system - as a model of measurement processes. These two structures are used most frequently as means of building mathematical models. This is why many people, still, tend to thinking that the "numerical modeling" is the only "true" kind of mathematical modeling. But this is not true already since almost 200 years!

As the opposite extreme, in computer science, formal languages are used as a model of ontologies. In computer science, we are speaking about ontologies in plural, not about a single "true" ontology. It seems, in his above-mentioned speech of 1965, Glushkov already advanced towards this position...

For details, see the online text Ontology by John F. Sowa.


Mathematical models are distracted from the modeled objects, and can be explored (sometimes, for years) without any reference to these objects. Such models, by their very nature, allow modification "for modification's sake".

"Das Wesen der Mathematik liegt in ihrer Freiheit."
"The essence of mathematics resides in its freedom."
Georg Cantor,
in his 1883 paper about "Punktmannichfaltigkeiten" (English translation - one of the internet versions).

But this can destroy any similarity with "the rest of the world", and thus - lead to models that never will be applied for modeling of something useful. And even - to entire useless branches of mathematics...

George Box proposed to call this phenomenon "mathematistry":

"Mathematistry is characterized by development of theory for theory's sake, which since it seldom touches down with practice, has a tendency to redefine the problem rather than solve it."

G. E. P. Box. Science and statistics. Journal of the American Statistical Association, 71, 1976, pp. 791-799.

But it should be noted, once again, that this problem is caused by the very nature of mathematics: because mathematical models are distracted from the modeled objects! Without this kind of freedom there is no mathematics at all!

Why mathematicians do not agree with me...

The above-stated picture of the nature of mathematics (I would like to call it Advanced Formalism) is by far not commonly acknowledged. Where is the problem, why it is so hard to regard mathematics as the investigation of stable, self-contained models?

A personal communication by Svyatoslav Sergeevich Lavrov from 1988: " ... Theorems of any theory consist, as a rule, of two parts - the premise and the conclusion. Therefore, the conclusion of a theorem is derived not only from a fixed set of axioms, but also from a premise that is specific to this particular theorem. And this premise - is it not an extension of the fixed system of principles? ... Mathematical theories are open for new notions. Thus, in the Calculus after the notion of continuity the following connected notions were introduced: break points, uniform continuity, Lipschitz's conditions, etc. ... All this does not contradict the thesis about the fixed character of principles (axioms and rules of inference), yet it does not allow "working mathematicians" to regard mathematical theories as fixed ones."

About S. S. Lavrov - see also Kosmos. A Portrait of the Russian Space Age.

The first step only...

This is only the first step in understanding the nature of mathematics - concluding that distraction from the modeled objects is the distinguishing feature of mathematical models, and that the task of mathematics is development of methods allowing to create and investigate this kind of models. Without the next step, we would end up by representing mathematics as an unordered heap of mathematical models!

But, in fact, mathematics is a complicated system of interrelated theories each representing some significant mathematical structure (for example, natural numbers, real numbers, sets, groups, fields, algebras, all kinds of spaces, graphs, categories, computability, all kinds of logic, etc.).

Two hemispheres of human brain

It is well known that the human brain is operating by means of two distinct mechanisms:

a) The left-hemispherical mechanism - a kind of "computer", performing efficiently algorithmic activities, i.e. activities governed by pre-defined rules.

b) The right-hemispherical mechanism - an "artist", trespassing from time to time the barriers set by pre-defined rules.

Sergei Yu. Maslov perceived an analogy between these mechanisms and "some aspects of the evolution of mathematics":

Sergei Yu. Maslov

Photo: St. Petersburg Mathematical Society

See also:

Academician Andrei Ershov's Archive

Maslov S. Y. (1939-1982), human rights activist in ENCYCLOPAEDIA OF SAINT PETERSBURG.

"... in the most of applications, every particular process is modeled by a fixed system of some type, and one is exploring the properties of this deductive system. However, in more complicated situations the essence of the modeled process is a transition from one calculus to another one."

S. Yu. Maslov. Asymmetry of cognitive mechanisms and its consequences. Semiotics and information science, N20, pp.3-31, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1983 (in Russian).

S. Yu. Maslov. Theory of deductive systems and its applications. With a foreword by Nina B. Maslova. Translated from the Russian by Michael Gelfond and Vladimir Lifschitz. MIT Press Series in the Foundations of Computing. MIT Press, Cambridge, MA-London, 1987, xii+151 pp.

Two dimensions of mathematics

I would prefer a less careful position, ignoring "subtleties". I would extend this analogy not only to "some aspects of the evolution of mathematics", but to the entire mathematics.

In the world of mathematics, people are performing two kinds of activities:

a) "left-hemispherical" activities - working in a fixed formal theory (on a fixed mathematical structure),

b) "right-hemispherical" activities - changing a theory / structure (or, inventing a new one).

Thus, in a sense, we should think of mathematics as a "two-dimensional" activity. Sergei Yu. Maslov could have put it as follows: most of a mathematician's working time is spent along the first dimension (working in a fixed mathematical theory, on a fixed mathematical structure), but, sometimes, he/she needs also moving along the second dimension (changing his/her theories/structures or, inventing new ones).

Working along the first dimension, mathematicians have developed their ability to draw a maximum of conclusions from a given set of premises. I regard this as the true source of the "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" (Eugene Wigner).

For me, "Elements de Mathematique" by Nicolas Bourbaki can be regarded as an attempt of a systematic treatment of the second dimension of mathematics.

Now, this is my favorite model of mathematics. Do we need more than this, to understand the nature of mathematics?

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