what is science, modeling, scientific, model, Podnieks, Karlis, modelling, science, models, philosophy, scientific thinking, everything that is going on in human minds is modeling.
My personal page  click here.
Any comments are welcome  email to Karlis.Podnieks@mii.lu.lv
This lecture is continued in the next one: "On the Nature of Mathematics".
The ideas of this lecture are developed further in:
K.
Podnieks. The
Formalist Picture of Cognition. Towards a Total Demystification.
SciRePrints
Archive, ID
Code 157.
K.
Podnieks. Towards
a General Definition of Modeling. SciRePrints
Archive, ID
Code 155, 25 November 2010.
Princeton, 1950 
What is Science? Science is Modeling! Prof. Karlis Podnieks 65th Scientific Conference Picture from: MacTutor History of Mathematics archive  Godel Portraits 
This work is licensed under a Creative Commons License and is copyrighted © 20072008 by me, Karlis Podnieks. 

[Added September 16 and October 13, 2007 after googling for "science is modeling" (and "modelling")... The line of thought was initiated by David Hestenes in 1986!] "To the grand
philosophical question: "What
is a man?"
Aristotle answered: "Man
is a rational animal."
Modeling Theory offers a new answer: "Man
is a modeling animal!"
HOMO MODELUS!" "I will employ the
concept of the model to develop a particular view of science,
namely, science as
modeling (SAM).
This view is held by many, perhaps most, scientists themselves.
... I claim no originality in employing the idea that science is
modeling, but I do claim that it facilitates a perspicious
explication of the scientific enterprise." "Science is modeling; we
derive models of Nature for the purpose of understanding how
everything works." "... with quite
inspiring discussions about the future of science (raw data
sharing  is it ever going to be real?) and what are we actually
doing when modelling neurons (but hey, all science is modelling,
trying to find out the rules... ne?)..." "No science without
modeling or rather "science is modeling". With the help
of models we try to grasp our environment and to influence it for
our benefit. "... Screw that. Science
is modeling. Evolution is a successful model. Semantics doesn't
change that." "Why model the brain?
Science IS modeling." "Given that I am a
mathematical biologist, it is perhaps not surprising that I am in
favour of models. But it seems to me that all
science is modelling. It need not to
be a mathematical model, of course, ..." 

[Added January 31, 2009] "...
and that all philosophy (including science) is modelling,
abstraction, reduction of wholes into fundamentally unreal
(holistically abused) parts. It's not useless, because it can make
fridges and democracy, but one of the fundamental mistakes I see
in science, again and again, is that assumption that reduction 
or the approximated isolation of some feature from its
embeddedness in nature  will not cause distortion of
results." "Science is modeling,
not truth." "Actually, all of
science is modelling in one way or another. And it's continually
corrected. That describes two of the differences between science
and religion. Science is based on evidence, and science never
claims that an answer is final." "Because the truth of the matter is
that scientists have been VERY wrong MANY times before. We tend to
forget that. But that is what Science IS  modelling the world we
see around us as best we can." "All of modern science is computer
modeling. 

The human mind has first to
construct forms, independently, before we can find them in
things. Full text: Einstein's original text in German: See also Einstein's manuscript in Einstein Archives Online. By this quote, I would like to emphasize the active, essentially generating role of the human mind in cognition: some of the ideas and structures offered by scientists, are NOT copied or inspired from outside or from the above. 

All models are wrong, but some are
useful. Box, G. E. P. (1979). Robustness in the strategy of scientific model building. In R. L. Launer, & G. N. Wilkinson (Eds.), Robustness in statistics (pp. 201236). New York: Academic Press. Box, G. E. P. (1976). Science and statistics. Journal of the American Statistical Association, 71, 791799. MerriamWebster. By this quote, I would like to emphasize my limited appreciation of (any!) products of human activities. Many philosophers are searching for "truth" within themselves by means of introspection, "conceptual analysis" etc. But I do not trust our 2 liter "cognitive engines"! They represent, mainly, the thinking habits of ordinary "people from the street". Science has contributed very little to them. 

Neurophilosophy
Picture from Patricia's Churchland website. "She is associated with a school of thought called eliminativism or eliminative materialism, which argues that folk psychology concepts such as belief, free will, and consciousness will likely need to be revised as science understands more about the nature of brain function. She is also called a naturalist, because she thinks a priori methods alone cannot discover the nature of the mind. In this respect, she differs from philosophers who restrict themselves to conceptual analyses in hopes of saying something useful about the nature of the mind." (See Wikipedia  http://en.wikipedia.org/wiki/Patricia_Churchland). Patricia S. Churchland, Paul Churchland. Neural worlds and real worlds. Nature Reviews Neuroscience, November 2002, vol. 3, N 11, pp. 903907 (online copy). I like especially the terms used by Churchlands for the naive concepts: "folk physics" (Aristotle^{Wikipedia}), "folk thermodynamics" (caloric fluid^{Wikipedia}), and, of course, "folk cosmology" (...?), and now  even "folk semantics" (most theories of "meaning")! Of course, one cannot hope to obtain more than "folk science" by means of introspection and "conceptual analysis"... 

My main thesis: Any branch of human activities can be qualified as scientific only to the extent to which it is building models. If some set of concepts is pretending to be more than that, then it is... (what?) Of course, this formula is paraphrasing the famous statement of Immanuel Kant: "Ich behaupte, dass, in jeder besonderen Naturwissenschaft, nur soviel eigentliche Wissenschaft angetroffen werden kann, als darin Mathematik enthalten ist." (as quoted by David Hilbert in his Königsberg radioaddress of September 8, 1930 published by James T.Smith). "Ich behaupte sogar, dass in jeder besonderen Naturlehre nur soviel eigentliche Wissenschaft angetroffen werden könne, als darin Mathematik anzutreffen ist." (this may be closer to the original). The above thesis defines my favorite philosophy (let us call it "modeling approach"): if you would propose any set of concepts, any version of a philosophy, religion, ontology, theory etc., then I will try to understand it as a model of the world (or, of a part of it). [Or, as a framework for building such models  added later. K.P.] And I will try to find out, how could you arrive at your model (for example, which kind of analogies could have been used as a guide). Of course, one can try also modeling of the process of modeling itself (this could be the task of the scientific branch of philosophy  if such one exists). [Added July 1, 2008. Interestingly enough, the nonscientific branches of philosophy (such as "philosophy of language" and "philosophy of mind") also can be best analyzed from the modeling point of view. The models built in these branches are very simple, incomplete and based mainly on naive analogies. This explains why there are so many kinds of small "philosophies" ("isms"), and why all they are still existing and fighting each other. Many excellent expositions of the situation can be found in Wikipedia: Philosophy of Mind, Semantic Holism, Thought Experiment, Hilary Putnam and many other.] 

What is a model? A model is anything that is used instead of something other to predict the behavior of, to get on with, to master or to destruct this "something other". Thus, models are tiny fragments of the Universe used instead of other fragments (or, even instead of the entire Universe). [Added August 09, 2007. George Boolos about his
dissertation. [Answering the question about the difference between models and theories: some theories can serve as models, but most of them are frameworks or methods of building models. See below.] Of course, there must exist at least some analogy between the model and the corresponding "original" (referent, "modellee"). Some people are talking about "isomorphism" or "homomorphism" between the model and (some aspects of) its "modellee", thus creating a parody of the corresponding welldefined mathematical terms. I would propose using of a more general term "mapping" instead. Namely, there must be some mapping between the model and its "modellee". To which extent a tiny fragment of the Universe could serve as a model of the entire Universe? A good question... [Added July 1, 2008. Since anything can be used as a model, and the similarity of the model and the "modellee" is always only a restricted one, then, of course, there must be  as the extreme  also "models of nothing". These models either are not similar to anything other, or they are, but we do not know this in advance. An example  the first nonEuclidean geometry that was created by experiment as a "model of nothing", and only some time later many Euclidean interpretations of it were discovered. See below.] 

My first argument in favor of the "modeling approach" is The History of Modeling Modeling in its simplest (and very powerful) form  modeling by analogy  is as old as human thinking. All kinds of mythology were attempts of explaining the world (and making it predictable) by using analogies from human societies of the corresponding times (animals helping to "solve" people's problems etc.). By way of analogy, Democritus^{Wikipedia} produced the idea of atoms. The first still surviving mathematical model was created in the 6th century BC (at least, Pythagoras of Samos^{Wikipedia} was already aware of it)  the infinite sequence of natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, ... (for details, see my lecture "The Nature of Mathematics"). The second one was created a century later  now it is called the Euclidean geometry. The third model was created in the 17th century  the GalileanNewtonian mechanics. From it, the first serious cosmological model was derived (however, Giordano Bruno^{Wikipedia} anticipated essential features of the Newtonian cosmology). [Added August 07, 2007. Speaking very strictly, the above three theories are not models, but frameworks for building models. For example, the GalileanNewtonian mechanics (Newton's gravitation law included) is, in fact, a method of modeling an arbitrary system of particles (and other "masses"). One can try modeling in this way the Solar system^{Wikipedia}, our Milky Way galaxy^{Wikipedia}, or the entire Universe.] 

Plato As many models of that times, the abovementioned mathematical models were not comprehended as inventions. Beyond compare, they were qualified as "absolutely true" pictures of the world. And, the smartest among philosophers were puzzled by the question: how could this be possible? "The first one puzzled" among the smartest was Plato^{MacTutor} (427 BC  347 BC). Studying mathematics Plato came to his surprising philosophy of two worlds: the "world of ideas" (as perfect as the "world" of the Euclidean geometry) and the "world of things". According to Plato, each thing is only an imprecise, imperfect implementation of its "idea" (which does exist independently of the thing itself in the world of ideas). Completely fantastic was Plato's notion of the nature of mathematical investigation: before a mathematician is born, his soul is living in the world of ideas, and afterwards, doing mathematics, he simply remembers what his soul has learned in the world of ideas. For a materialist, this may seem an upsidedown notion of the real situation. Plato treats the end product of the evolution of mathematical concepts as an independent starting point of the evolution of the "world of things"? Still, it was the first brilliant attempt to explain the specific position of the mathematical knowledge among other branches of the human knowledge. Materialists tend to think of mathematics as "one of" the branches of science. It's true that the process of genesis of the first mathematical concepts (arithmetic and Euclidean geometry) was very similar to the process of genesis of the first concepts of "other" branches of science. But Plato realized that mathematical concepts have become stable, fixed, selfcontained, and hence, "independent forever", and tried explaining. Of course, for us, Plato's radical simple explanation of this independence seems completely fantastic... (more  in my paper "Platonism, Intuition and the Nature of Mathematics"). For me, Plato's fantasies are the first brilliant step towards understanding the nature of mathematics (and the human knowledge in general). 

Kant The second brilliant step was due to Immanuel Kant^{Wikipedia} (17241804). Fascinated (like as Plato) by the "precision" of the Euclidean geometry, and trying to explain, why we (i.e. people of his time) cannot imagine another kind of geometry, Kant declared the Euclidean geometry "a form of subjectivity", i.e. a builtin mechanism of the human mind used for arranging of sensations (his famous "synthetic apriori"). Thus, by the way, Kant concluded that Plato's invention of the "world of ideas" was not necessary. [Added August 02, 2007. Perhaps, Kant could arrive at the idea of modeling  if he would not think of "knowledge" and delusion as something existing separately in the human mind. In fact, in the human mind, "knowledge" and delusion are dissolved in each other, and even the proportion may be unknown. Models are exactly this kind of "compounds".] 

NonEuclidean geometries The third brilliant step was the invention of the nonEuclidean geometries^{Wikipedia} in the 19th century by Riemann, Lobachevsky, Gauss and Bolyai^{MacTutor} (mentioned in reverse alphabetical order). NonEuclidean geometries were invented, not discovered. People just tried experimenting with the axioms, by assuming that for a given line L and a point P, which is not on L, there are two different lines through P that do not intersect L. And when exploring the consequencies of this assumption they did not find any inconsistencies... Thus, a new kind of geometry was born! Trying to avoid the possible indignation of the general public of his time, Lobachevsky called this new geometry "imaginary". Thus, for the first time, the Euclidean geometry faced a competition. Which of the geometries is the best one as the description of the physical space? Kant could not arrive at such a question, but Lobachevsky did  and he even proposed to try finding the answer by using astronomical measurements. Shouldn't philosophers have concluded  in this very moment  that ALL products of the human mind are models of (fragments, aspects etc. of) the world, no more than that? [Or, frameworks for building such models  added later. K.P.] 

Philosophers are thinking not very fast... NonEuclidean geometries were followed by Maxwell's^{MacTutor} physics. Confronted with the Newtonian physics it produced Einstein' s special relativity theory (a direct competitor of the GalileanNewtonian mechanics) and Planck's^{MacTutor} first ideas of quantum physics. But what followed after that, demonstrated already a complete lack of respect for all the previous physics and philosophy: the general relativity theory, cosmological models assuming that the Universe is limited in size and age, starting with Big Bang etc., quantum physics... of particles that are not "flying decently"... etc. Thus, today, after all this humiliation, we have much more reason to conclude that ALL products of the human mind are models of (fragments, aspects etc. of) the world, no more than that! [Or, frameworks for building such models  added later. K.P.] [In principle, we could analyze in a similar way the work of Ptolemy, Copernicus and Kepler^{MacTutor}, and arrive at the same "modeling conclusion". Remarkably, Einstein concluded his article on the occasion of the 300th anniversary of Kepler's death with "...mind has first to construct forms..." (see above). ] 

My second argument in favor of the "modeling approach" is The Philosophy of Robots "It
turns out that many philosophical problems take new forms when
thought about in terms of how to design a robot." Imagine a universal autonomous robot. How could it proceed, if not by building models of its environment(s), deriving predictions from them, and acting according to these predictions? Of course, primitive specialized robots do not "derive" their behavior. They simply act according to specific reaction mechanisms built into them. In this way they can perform specific tasks and survive in specific environments. Similarly, reactions of animals to various situations are built into their organisms as reflexes (etc.), and are not derived from models. (See also: Behaviour Based Robotics & Deliberative Robotics by Tamie Salter.) In an unusually variable environment, primitive robots and animals hardly can survive on their own. But the survival capabilities can be improved by extending the builtin reaction mechanisms to more universal models. In such models, one is trying to reflect the "environment in general" (or even, the "world in general"), and not only the minimum of environment features that are essential for today's survival. From such models, one can derive action plans also for situations that have not yet occurred in practice. For me, this kind of universal modeling is the borderline between humans and animals. 

As the first experiment for testing the "modeling approach", I would propose Demystification of Gödel's Theorem In many texts, Gödel's Incompleteness Theorem is applied without any prior and proper definition of the context. But in fact, we are dealing here with a specific mathematical model of science. In this model, scientific theories are replaced by specific models of them  the socalled formal theories. Formal theories are formulated in an extremely precise way  so precise that reasoning in them can be simulated by a computer. And these formulations never change  if you change the formulation of your theory, then you are trying to propose a new theory! In this formal model of science Gödel's Theorem sounds as follows. Gödel's Incompleteness Theorem. If some formal theory is universal enough to represent the notion of natural numbers (1, 2, 3, ...), then, working in this theory, one of two things will happen inevitably: we will run either into contradictions, or into unsolvable problems. Kurt Gödel^{MacTutor} proved this theorem in Vienna, during the summer of 1930, and he first told about it to his colleagues on Tuesday, August 26 at the Cafe Reichsrat: 

What could this theorem mean for us? If we wish to derive from Gödel's Theorem some conclusions "for us", then (according to the "modeling approach") we must first determine, to which extent our situation corresponds to the formal model of science? Is our theory formulated precisely enough, i.e. can it be represented adequately by a formal theory? If not, then Gödel's Theorem is not applicable to it. But, then, simultaneously, we must confess, that our theory is not precisely formulated, i.e. at least in part, we do not understand what we are talking about. If our theory is formulated precisely enough, then it can be represented adequately by a formal theory. And then, Gödel's Theorem may be applicable to it. Thus: If our theory is universal enough, and we will try to formulate it precisely enough (and will not change this formulation in the future), then, working in this theory, one of two things will happen inevitably: we will run either into contradictions, or into unsolvable problems. As you may see now, the real significance of Gödel's Theorem differs completely from the funny things written about it in many texts and textbooks (about mathematical "truths" that cannot be proved formally, about minds and machines etc.). An example: set theory. Trying to formulate the principles of set theory precisely enough, we run either into contradictions (such as Russell's Paradox), or into unsolvable problems (such as the Continuum Problem). For details, see my text Axiomatic Set Theory. 

So what? If you think that the abovementioned perspective is unacceptable, then exactly three ways of stepping back can be noticed in the picture: 1) Let us lower our ambitions, i.e. let us give up building a universal theory. 2) Let us refuse formulating our theory precisely. 3) Let us formulate our theory (universal, or not) precisely, but let us change it in the future, if necessary. The first way is an individual choice of everyone. Wouldn't the second way expose us to ridicule? The third way is the one of the modern science: keep modeling, but don't make a fetish of your model. All models are wrong! 

Is philosophy of mathematics a kind of modeling? Notes taken January 11, 2006 during reading of "New Directions in the Philosophy of Mathematics", 1998, edited by Thomas Tymoczko^{Wikipedia}. Reuben Hersh^{Wikipedia} concludes his 1979 paper "Some Proposals for Reviving the Philosophy of Mathematics" as follows: "This account of mathematics contains nothing new. It is merely an attempt to describe what mathematicians actually are doing and have been doing for centuries. The novelty, if any, is the conscious attempt to avoid falsification and idealization." (p. 25 of Tymoczko's volume). Avoiding falsification may be acceptable, but why should idealization be avoided? It seems, Hersh is protesting here not against idealization as a method (if applied deliberately), but against unconscious replacing of the real process by its idealized version. If Galileo^{MacTutor} had tried avoiding idealization, then he never had arrived at the "obviously false" notion of the uniform linear motion (Newton's First Law^{Wikipedia}), and we would be staying with the "obviously true" principle "any motion stops, if one does not apply force". While being "false" (in the sense that exactly such thing does not exist in the real world), the notion of the uniform linear motion serves as a much better basis for physical modeling than the "empirically true" principle. Thus, why should idealization (as a method that is applied deliberately) be avoided when modeling mathematics? Or, is philosophy of mathematics not modeling? (It is  see other papers of Tymoczko's volume!) Thus, what, if we would think of formal theories as ideal models of mathematical theories  as a kind of "uniform linear" theories to which one does not "apply force"? When doing a serious mathematical reasoning, we never achieve this ideal standard, i.e. mainly (this model says!)  we are "applying force". In which sense? This model says: you are introducing new principles of reasoning implicitly. Indeed, if formal reasoning is the ideal standard, then the "delta" between it and real reasoning may be only this: implicit using of new principles that cannot be derived from the previously accepted principles. And if so, any such real reasoning must be analyzed to discover these implicit new principles. Otherwise (this model says), we would not behave as mathematicians! Which principles could serve as "Newton's Laws" in this model of mathematics? May we ignore formal theories, and still be able to analyze the nature of mathematics correctly? Shouldn't a purely empirical approach to analyzing mathematics lead us to representing it as a kind of a social ritual only? [Added August 15, 2007. Isn't the platonist reaction to formalism motivated similarly as the reaction of many people to Darwin's theory of natural evolution?] 

"Modeling" Immanuel Kant? Perhaps, serious philosophers will reject (as over simplification) my above reducing of Kant's philosophy of mathematics to a single idea of declaring the Euclidean geometry "a form of subjectivity", i.e. a builtin mechanism of the human mind used for arranging of sensations (his famous "synthetic apriori"). Yes, of course, Kant produced more than this one brilliant idea. Thus, in terms of the "modeling approach", instead of the Kant's authentic philosophy of mathematics, I'm using in this lecture a simplified "model of Kant". In this way, I'm trying to avoid the socalled overfitting^{Wikipedia}, known from statistics and data mining. When, modeling some phenomenon, we are trying to obtain a model "as precise as possible", and we do not restrict the complexity of the model, then it may represent not only essential features of the phenomenon, but also various kinds of "noise". A useful philosophy should be a robust one, oder? [Added July 18, 2008] The following modern subtlety was not known at the times of Kant: "The same point is to be made about perceptual space. The detailed experimental data ... argues from many different directions that the space of visual perception is not in any conclusive sense Euclidian in nature." (p. 390 of P. Suppes. RESPONSE TO MARIO VALENTINO BRAME. Epistemologia XXIX (2006), pp. 389396). Being aware of this, Kant could not be able to produce his brilliant idea! But it seems, Kant was interested in the notion of space taught at schools and used in engineering. "Perceptual space" was used in civil engineering for the last time well before the first pyramids were built... 

Ontologies In computer science, we are allowed to use this term in plural. See Wikipedia  http://en.wikipedia.org/wiki/Ontology_(computer_science). If our business is modeling, then we may try to promote "good practices" by collecting various constraints, rules, recommendations etc. And, if our set of constraints, rules etc. can be organized in a consistent system, then we can say that we have invented an ontology. Thus, any ontology is only a framework for building theories and models, no more than that. 

Is there such thing as "eternal truth"? [Added July 6, 2008] Many people think that the final goal of science and philosophy is "truth", and that there must be even such a kind of "truth", as "absolute truth", or even "eternal truth". Is it "eternally true" that atoms really exist? Or, atoms are an invention of Democritus^{Wikipedia} still used in modern physics, chemistry etc? Some people think, they have "seen" atoms on photos taken by using Wilson chamber^{Wikipedia}, hence, atoms do really exist. Thus, the following question will be much more interesting: is it "eternally true" that quarks^{Wikipedia} really exist? Or, quarks are a mere invention of overmathematized physicists? This question is more interesting because, according to the actual theory, quarks cannot be observed as isolated particles even in principle. I.e. everything we detect about quarks, can be only an inference. From the modeling point of view, some construct can be viewed as "eternally truly existing", if and only if it never will be dropped from the actual scientific models. Thus, "eternal truth" is simply a synonym for long living modeling invariants. "Universe really exists" is an "eternal truth", if we do not believe that this construct will ever be dropped from the actual scientific ontologies. BTW, by googling for "invariants as eternal truth" one finds the paper: Kunii, T.L. Science of computer graphics. Proceedings. Seventh Pacific Conference on Computer Graphics and Applications, 1999, pp. 2  3. 

Where are these ideas coming from? I don't think that my today's speech represents something radically new. Rather, it only formulates frankly a way of thinking preferred, in fact, by many people before me... When preparing my speech ''The Nature of Mathematics" for the IX Scientific Conference "Modern Logic: Problems of Theory, History and Applications in Science" (SaintPetersburg, Russia, June 2006), I considered three approches to defining mathematics: a) Mathematics as a social phenomenon (DavisHersh). Computerized mathematics against human mathematics (Zeilberger). b) Is mathematics an "ordinary" branch of science, or is it "something special"? Platonism and formalism. c) Mathematical models and mathematics. In this way I arrived at a general question about the role of modeling in science. Is modeling only a specific method? Or, it is the distinguishing feature of scientific thinking? 

I'm not an educated philosopher... In 1974, during a Soviet army training course, I discovered a simple (almost trivial) extension of Goedel's theorem  my double incompleteness theorem. Since that time, I'm an amateur philosopher of mathematics. My education up to Ph.D. in 1979 was purely mathematical, but I was elected Professor of Information Technologies (second class computer science). My philosophical credo stabilized in the middle of 1970s (I think, reading Ernst Mach^{Wikipedia} was the crucial event), and did not change significantly since that time. It was first published in 1981 as a small book for students "Vokrug teoremi Gedela" (in Russian, "Around Gödel's Theorem"). The first presentation "for the West"  in 1988 at the Heyting'88 Summer School & Conference on Mathematical Logic (view copy of Abstract). This was followed in 1992 by the 2nd extended edition of the book "Around Gödel's Theorem" (in Russian), and then in English: in 1994, on the QED mailing list (in 5 parts: #1, #2, #3, #4, #5), and in 1997  the 3rd extended Internet edition of the book  in English, and now renamed "What is Mathematics: Gödel's Theorem and Around". The general reaction of the Western public is indifference  no critics, no support. It seems that for "serious" mathematicians  I'm "not a mathematician" (this is true since 1980), and for "serious" philosophers  "not a philosopher" (this has been always true). However, reading all the funny things about mathematics written even by the most prominent philosophers and mathematicians, I'm feeling at least as their kind of person... 

What next? I would like to invite the audience to try analyzing your own "world views" (Weltanschauungen) as models of the world. Could your model arise in a natural way, for example, by way of analogy? Or, a different explanation is needed? 

A softened version of my main thesis There are two kinds of science: modeling sciences and nonmodeling sciences? Let us call for a "peaceful coexistence" between the two of them? [Added April 22, 2008.] Is the "nonmodeling part of science" really nonmodeling? 
This lecture is continued in the next one: "On the Nature of Mathematics".
what is science, modeling, scientific, model, Podnieks, Karlis, modelling, science, models, philosophy, scientific thinking, everything that is going on in human minds is modeling.