Keywords

1 Background and Overview

This chapter refers to the famous book “The Structure of Scientific Revolutions”, which Thomas Kuhn published the first time in 1962 (see [1]). The book describes the life cycle of so-called paradigms, which starts with the introduction of new scientific theories, an agenda of problems to be solved, and methods that are rapidly accepted by the scientific community at the expense of a previously used paradigm. This revolutionary stage is followed by a period of “normal science”, where the problems — i.e. the “puzzles” in Kuhn’s terminology — of the new paradigm are treated and successfully solved by means of its specific methods. At the end, the limitations of the paradigm become more and more visible since many of its remaining “puzzles” turn out to be unsolvable. This is the time, when the scientific community is ready for another scientific revolution by abandoning the existing paradigm in favour of a new one. Thus, science according to Kuhn is a sequence of paradigms, which in the stage of normal science monopolistically dominate the activities of a scientific community.

In spite of its excellent reputation, Kuhn’s book has two major shortcomings: First, it is mainly based on historical examples and thus neglects the institutional framework of contemporary science like peer-reviewing or the publish-or-perish rule for academic careers. Second, Kuhn’s book mainly deals with sequences of mutually exclusive paradigms and thus does not really come to grips with multi-paradigmatic situations, which are so typical for the humanities and social sciences (see [2]). In order to tackle these difficulties, we present in the following sections a simulation model, which is based on game theoretical premises. As proposed by [3], it takes the institutional settings of modern science better into account and offers the possibility to reproduce the coexistence of paradigms.

Obviously there are other simulation models of scientific revolutions (see [4]), the most prominent ones being developed by Sterman [5] and Sterman/Wittenberg [6], who directly refer to Kuhn’s work. Whereas these two authors used a systems dynamics approach (see [7]: Chap. 3), the present chapter is based on evolutionary game theory (see e.g. [8]), which we consider as much more appropriate to the study of competition between paradigms. Moreover, the cited works [5] and [6] of Sterman and Wittenberg are purely theoretical, whereas this paper attempts to corroborate the theoretical simulations with empirical data. The respective analyses in Sect. 4 demonstrate that our model is able to grasp not only the complete replacement of successive paradigms, as described by Kuhn [1], but also the more complex reality of multi-paradigmatic scientific communities. This again is a major advantage over the older model of Sterman/Wittenberg, which seems to explain only the total replacement of an exhausted paradigm by a new one (see [6]: 329, Fig. 7a).

2 A Game Theoretical Model of the Competition Between Paradigms

In its simplest form, evolutionary game theory (see [8], [9]: Chap. 8, [10]) departs from the idea of two randomly interacting species and an associated matrix of 2 × 2 = 4 pairs of possible payoffs, which determine the so-called fitness of the two species as well as their reproduction and death rates: the higher the mentioned fitness of the first species as compared to the second, the higher its population growth at the expense of the other.

These basic ideas from theoretical biology have successfully been used for the analysis of dynamic social processes (see e.g. [11–14]). Thus we are going to tackle the modelling of Kuhn’s scientific revolutions on the basis of evolutionary game theory. Obviously, the two interacting species are in this case the supporters of the old and the new paradigm, which we describe by:

$$ \mathrm{S}_{n} = \mathrm{Share}\ \mathrm{of}\ \mathrm{the}\ \mathrm{supporters}\ \mathrm{of}\ \mathrm{the}\ \underline{\mathrm{n}}\mathrm{ew}\ \mathrm{paradigm}, \vspace*{-7pt}$$
(1a)
$$ \mathrm{S}_{o}=1-\mathrm{S}_{n} = \mathrm{Share}\ \mathrm{of}\ \mathrm{the}\ \mathrm{supporters}\ \mathrm{of}\ \mathrm{the}\ \underline{\mathrm{o}}\mathrm{ld}\ \mathrm{paradigm}. $$
(1b)

The arenas where these two “species” encounter are editorial boards of scientific journals, search committees for filling academic posts, or institutions for funding research projects. In each of these arenas academics appear in the role as suppliers and requesters of publication space, posts at universities, or research money. Due to the exclusiveness of their paradigms, interactions in the mentioned arenas are rather hostile for encounters of different paradigms and relatively friendly between representatives of the same paradigm. This has consequences for the academic careers of the requesters, which we are going to analyse in the following paragraph for the case of the submission of articles to scientific journals.

If we assume that the composition of reviewers of journals by paradigm corresponds to the paradigm-orientation of the general population of scientists, it is possible to calculate the total acceptance rates A o of the old paradigm and a respective value A n for the new paradigm. Both are the sums of the acceptance rates A i of intra-paradigmatic and A e of extra-paradigmatic works, weighted by the population shares S n and S o . For reviewers supporting the old paradigm, intra-paradigmatic works are authored by members of the old paradigm and extra-paradigmatic works by supporters of the new one. For reviewers representing the new paradigm, the definitions of intra- and extra-paradigmatic works are just the reverse. Thus, according to Table 1:

Table 1 The acceptance rates of the old and the new paradigm
Full size table
$$ \mathrm{A}_{n} = \mathrm{S}_{o}{\raisebox{2pt}{$\scriptstyle*$}}\, \mathrm{A}_{e} + \mathrm{S}_{n}{\raisebox{2pt}{$\scriptstyle*$}}\, \mathrm{A}_{i} \vspace*{-7pt}$$
(2a)
$$ \mathrm{A}_{o} = \mathrm{S}_{o}{\raisebox{2pt}{$\scriptstyle*$}}\, \mathrm{A}_{i} + \mathrm{S}_{n}{\raisebox{2pt}{$\scriptstyle*$}}\, \mathrm{A}_{e} $$
(2b)

Implicitly we are postulating in the formulas (2a) and (2b) that the result of the reviewing process is influenced by the randomness of the assignment of reviewers to manuscripts, as demonstrated by [15]. If we assume in addition that there is a publication bias against new ideas (see [16]: p. 71 and [17]: Chap. 3) such that

$$ \mathrm{A}_{e} < \mathrm{A}_{i} $$
(3)

the Eqs. (2a) and (2b) imply that new paradigms have at the beginning of their existence a lower acceptance rate A n than the dominating old paradigm, since in this situation S o  ≈ 1 and S n  ≈ 0.

At the beginning, however, new paradigms have the advantage of offering to ambitious scientists a lot of easy-to-solve new puzzles such that the ease of discovery E n is at this stage for the new paradigm much higher than the ease of discovery E o of the old paradigm. As mentioned by Kuhn (see [1]: Chap. 7), the latter is in its final stage often confronted with insurmountable difficulties in solving its own scientific puzzles. Thus the ease of discovery obviously has consequences for the scientific productivity F o of the supporters of the old and F n of the new paradigm, which modify the effects of the initial non-acceptance of the new paradigm in the following way:

$$ \mathrm{F}_{n} = \mathrm{E}_{n}{\raisebox{2pt}{$\scriptstyle*$}}\, \mathrm{A}_{n} = \mathrm{E}_{n}{\raisebox{2pt}{$\scriptstyle*$}}\, \left(\mathrm{S}_{o}{\raisebox{2pt}{$\scriptstyle*$}}\, \mathrm{A}_{e} + \mathrm{S}_{n}{\raisebox{2pt}{$\scriptstyle*$}}\, \mathrm{A}_{i}\right) \vspace*{-7pt}$$
(4a)
$$ \mathrm{F}_{o} = \mathrm{E}_{o}{\raisebox{2pt}{$\scriptstyle*$}}\, \mathrm{A}_{o} = \mathrm{E}_{o}{\raisebox{2pt}{$\scriptstyle*$}}\, \left(\mathrm{S}_{o}{\raisebox{2pt}{$\scriptstyle*$}}\, \mathrm{A}_{i} + \mathrm{S}_{n}{\raisebox{2pt}{$\scriptstyle*$}}\, \mathrm{A}_{e}\right) $$
(4b)

Since the above-mentioned productivity in terms of accepted and published papers determines the careers of the respective scientists, we are using in (4a) and (4b) as left-hand-terms the letters F n and F o , which stand for the fitness of the two groups of scientists. Hence we hypothesise in accordance with the general assumptions of evolutionary game theory (see e.g. [10]: Chap. 3) that the growth of the supporters of the new paradigm is

$$ \varDelta \mathrm{S}_{n}=\updelta\ {\raisebox{2pt}{$\scriptstyle*$}}\, \left(\mathrm{F}_{n}-\mathrm{F}_{o}\right)\!,\ \mathrm{if}\ 0 < \mathrm{S}_{n} < 1,\ \mathrm{else}\ \varDelta \mathrm{S}_{n} = 0, $$
(5a)

where δ is a constant laps of time. Similarly we assume that the growth of the supporters of the old paradigm equals

$$ \varDelta \mathrm{S}_{o}=\updelta\ {\raisebox{2pt}{$\scriptstyle*$}}\, \left(\mathrm{F}_{o}-\mathrm{F}_{n}\right)\!,\ \mathrm{if}\ 0 < \mathrm{S}_{o} < 1,\ \mathrm{else}\ \varDelta \mathrm{S}_{o} = 0. $$
(5b)

Both equations are conceptualised in such a way that the shares S o and S n do not leave their definition interval [0,1] and always sum up to 1.Footnote 1 The changes which they describe are partly due to the transitions of established scholars between paradigms and partly to the rational choice of young scientists, who start their careers with the paradigm that promises the more successful professional future.

The advantage of the new paradigm in terms of a higher ease of discovery E n tends to decrease by the number of newly solved scientific puzzles, which is proportionate to F n \( \raisebox{2pt}{$\scriptstyle*$} \)S n , i.e. the product of the relative size S n of the population of scientists and its productivity F n . Similar things hold true for the dynamics of the ease of discovery with the old paradigm. Consequently we postulate:

$$ \varDelta \mathrm{E}_{n}=-\updelta\, {\raisebox{2pt}{$\scriptstyle*$}}\, \mathrm{F}_{n}\, {\raisebox{2pt}{$\scriptstyle*$}}\, \mathrm{S}_{n},\ \mathrm{with}\ \mathrm{initial}\ \mathrm{value}\ \mathrm{E}_{n} = 1\ \mathrm{and}\ \updelta = \mathrm{constant}\ \mathrm{laps}\ \mathrm{of}\ \mathrm{time}. $$
(6a)
$$ \varDelta \mathrm{E}_{o}=-\updelta\, {\raisebox{2pt}{$\scriptstyle*$}}\, \mathrm{F}_{o}\, {\raisebox{2pt}{$\scriptstyle*$}}\, \mathrm{S}_{o},\ \mathrm{with}\ \mathrm{initial}\ \mathrm{value}\ \mathrm{E}_{o}\le 1\ \mathrm{and}\ \updelta = \mathrm{constant}\ \mathrm{laps}\ \mathrm{o}\mathrm{f}\ \mathrm{time}. $$
(6b)

Hence, after some time, both paradigms are depleted and may be replaced by a third paradigm, which is however not considered in the simulations that follow.

3 Model Simulation

3.1 Introductory Remarks

This section pursues two related goals:

  1. (i)

    We want to look for an inventory of the different types of population dynamics that can be reproduced by the model. Of special interest are on the one hand the empirically observed coexistence of two paradigms and on the other the complete replacement of the old paradigm by a new one, as described by Kuhn [1].

  2. (ii)

    We attempt to analyse the determinants of the mentioned patterns of population-dynamics. Given the limited number of exogenous model parameters, we focus on the acceptance rates of extra-paradigmatic works A e and the initial ease of discovery E o by the old paradigm. For reasons of standardisation we set for the start of the simulations the ease of discovery of the new paradigm E n  = 1 and the acceptance rate of intra-paradigmatic works A i  = 1. This way E o and A e become relative values, i.e. fractions of the former ones.

In view of the complexity of our model we tackled the goals (i) and (ii) by simulation experiments: they allowed to study the effects of parameter changes on the population dynamics of the supporters of the old and the new paradigm in a rather easy way. This method obviously required the translation of the model into a computer program. We used for this purpose an EXCEL spread-sheet with columns being defined as time-dependent variables, like e.g. S o and S n and rows representing subsequent time-points with a laps of time δ = 0.1. The rows are linked in such a way that changes of variable-values on one line are propagated to the next, as described by the difference equations (5a, 5b) and (6a, 6b). This process always started under the assumption that between t = 0 and t = 1 a new paradigm showed up and lowered the share of the supporters of the old paradigm from an initial de facto monopole S o  = 1.0 to S o  = 0.95. By simulation of the subsequent population dynamics it was e.g. possible to analyse, under which conditions the new paradigm is crowded out by the old or alternatively further spreads and finally becomes the mainstream of the scientific community.

3.2 Simulated Types of Population Dynamics of Scientists

As a matter of fact, a relatively small number simulation experiments with randomly selected parameter values E o and A e show the population dynamics that correspond to the classical revolutions described by Kuhn [1]. Figure 1a is an example for these rather rare situations, where the new paradigm immediately attracts a growing number of scientists until it completely replaces the old one.

Fig. 1
figure 1

(a) Kuhn’s classical revolution: immediate and complete change from the old to a new paradigm (initial parameter values: E o  = 0.2, A e  = 0.2). (b) An incomplete revolution: coexistence of the old and a new paradigm (initial parameter values: E o  = 0.6, A e  = 1.0)

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Much more frequent than the classical “perfect” revolutions are in our simulation experiments the incomplete ones, as exemplified by Fig. 1b: the new paradigm immediately starts to grow at the expense of the old. The latter however recovers after some time and leads to a multi-paradigmatic situation, which is often observed in the social sciences. A closer look at Fig. 1b explains this fluctuation in the support for the two paradigms: the rapid start of the new paradigm leads to its early exhaustion and soon lowers its ease of discovery E n . Between time t=30 and t=100, the E n of the new paradigm is already smaller than the E o of the old, which this way gets a chance for a revival (see Fig. 1b). This dynamic of E o and E n is in sharp contrast to the classical scientific revolution, depicted in Fig. 1a: here the ease of discovery of the new paradigm is for a much longer time, i.e. until t = 50, above the old one and thus leads to its complete victory.

In about half of all simulation experiments with randomly selected parameter values E o and A e , the change of paradigm is delayed: the new paradigm is available, but for some years the old is still vigorous enough to exert monopolistic control of the scientific community. Only after a latent period of further depletion, the old paradigm breaks down and triggers either a complete (Fig. 2a) or an incomplete revolution (Fig. 2b).

Fig. 2
figure 2

(a) A delayed classical revolution: complete change to a new paradigm (initial parameter values: E o  = 0.8, A e  = 0.2). (b) A delayed incomplete revolution: transition to the coexistence of the old and a new paradigm (initial parameter values: E o  = 0.9, A e  = 0.5; source: [22]: Fig. 1)

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Last but not least there is the rather rare possibility that the outbreak of a scientific revolution is not only temporarily but even infinitely delayed and consequently ends in a failed revolution. Hence, from the perspective of evolutionary game theory there are particular conditions (see Sect. 3.3), under which paradigms can be evolutionarily stable.

In sum, this model is able to reproduce not only the scientific revolutions of Kuhn [1] but many other phenomena of scientific change like delayed revolutions, where new ideas come too early to be accepted by the scientific community, or incomplete revolutions that lead to multi-paradigmatic science. Thus in view of the last-mentioned category of changes, the model fulfils one of the major goals of this chapter (see Sect. 1).

3.3 The Determinants of the Stability and Long-Term Dominance of Paradigms

The previously encountered types of paradigm-changes differentiate mainly with regard to the following two dimensions:

  1. (i)

    The stability of the old paradigm in the case of the arrival of a new one: it may be immediately unstable, temporarily stable, or permanently stable. In the first case we expect a classical or an incomplete revolution, in the second delayed changes, and in the third a failed revolution.

  2. (ii)

    The paradigm, which finally dominates after the changes induced by the arrival of a new paradigm have fully developed. In the long run the dominating model of science may be the new paradigm, the old paradigm, or both paradigms. The first case corresponds to the effect of a classical revolution, the second of a failed revolution, and the third of an incomplete revolution.

    Fig. 3
    figure 3

    (a) The stability of the old paradigm, by values of A e and E o (source: [22]: Fig. 2a). (b)~Long-term dominance of different paradigms, by values of A e und E o (time horizon: 400 units of time; source: [22]: Fig. 2b)

    Full size image

The outcome of the model with regard to both dimensions (i) and (ii) depends on the ease of discovery of the old paradigm E o and the acceptance of extra-paradigmatic works A e . Thus we ran the simulation model over a span of 400 units of time with δ = 0.1 and for varying values E o  = 0., 0.2, 0.4, …, 1 and A e  = 0., 0.2, 0.4, …, 1.Footnote 2 The results of these simulations are presented in Fig. 3a, b.

As Fig. 3a demonstrates, scientific revolutions immediately break out if A e ≥ E o . Thus, if the reviewers of the old paradigm, who initially have full control of the editorial boards, are too indulgent to new extra-paradigmatic ideas, a change of paradigms is very likely. If A e > 0.2 the revolution remains incomplete and leads to a multi-paradigmatic compromise (see Fig. 3b). If A e ≤ 0.2, the revolution ends with the dominance of the new paradigm, as described by Kuhn [1] (see Fig. 3b). It is important to note that this kind of a complete classical revolution only occurs for a small minority of randomly selected parameter values of A e and E o .

Alternatively, if A e  < E o , but A e  > 0, the ease of discovery with the old paradigm is too high for an immediate swing from the old to the new paradigm (see Fig. 3a). Nevertheless, after some time the old paradigm is sufficiently depleted and a delayed revolution breaks out. For A e  > 0.2 it ends again with a compromise between the old and the new paradigm (see Fig. 3b). If A e  ≤ 0.2, the growth of the supporters of the new paradigm is slower but finally leads to a classical revolution, where the dominance of the scientific field completely shifts from the old to the new paradigm (see Fig. 3b).

Finally, if A e  = 0, the supporters of the old paradigm use their initial control of the scientific production to exert a perfect “censorship”: no extra-paradigmatic work from the new paradigm is accepted for publication. As Fig. 3a demonstrates, this kind of censorship is an evolutionary stable strategy, which turns the invasion of the field by supporters of the new paradigm into a failure, at least as long as there are any puzzles from the old paradigm left that can be solved by its representatives. In the very moment when E o reaches the level 0, a classical revolution is immediately triggered, which ends with a complete victory of the new paradigm, as Fig. 3a, b show for E o  = 0.

4 An Empirical Test of the Model

4.1 The Explanandum and Its Operationalization

This paper aims at an explanation of the rise and fall of two paradigms of social simulation: systems dynamics simulation (see [7]: Chap. 3) and agent based modelling (ABM) (see [18], [7]: Chap. 8). The former was introduced by Forrester [19] and dominated the simulation literature of the 1970s and 1980s. The latter has its roots in the work of Schelling [20] and spread after 1990 at the expense of the older systems dynamics approach. However, agent based modelling was never able to crowd the other competitive approaches out. According to Table A1 (see Data Appendix), this rise of ABM was rather an incomplete than a complete revolution, and thus resembles Fig. 1b more than Fig. 1a. Hence the case under consideration is an interesting test on whether our model is able to reproduce also the quantitative aspects of an incomplete scientific revolution.

Unfortunately, the shares of scientists S o and S n adhering to the old and the new paradigm are much more difficult to measure than the shares of their respective publications P o and P n , which can easily be extracted from bibliographies. Therefore we tested our model by explaining the publication shares P o and P n , which we hypothesise to be the standardised products of the shares of scientists and their fitness related productivity:

$$ \mathrm{P}_{n} = \left(\mathrm{F}_{n}{\raisebox{2pt}{$\scriptstyle*$}}\,\mathrm{S}_{n}\right)\ /\ \left(\mathrm{F}_{n}{\raisebox{2pt}{$\scriptstyle*$}}\,\mathrm{S}_{n} + \mathrm{F}_{o}{\raisebox{2pt}{$\scriptstyle*$}}\,\mathrm{S}_{o}\right) \vspace*{-12pt}$$
(7a)
$$ \mathrm{P}_{o} = \left(\mathrm{F}_{o}{\raisebox{2pt}{$\scriptstyle*$}}\, \mathrm{S}_{o}\right)\ /\ \left(\mathrm{F}_{n}{\raisebox{2pt}{$\scriptstyle*$}}\, \mathrm{S}_{n} + \mathrm{F}_{o}{\raisebox{2pt}{$\scriptstyle*$}}\, \mathrm{S}_{o}\right) $$
(7b)

Obviously the shares of the old- and the new-paradigm publications P o and P n sum up to 1.Footnote 3

In order to measure P n , we used the electronic bibliography of Scholar Google [21] as a basic resource that allowed us to count for each year between 1993 and 2012 the absolute number of articles with the keyword “agent based” in the title. For measuring P o we utilised the same bibliography and determined the number of articles with the title-words “system dynamics” or alternatively “systems dynamics”. Subsequently we calculated the relative shares P o and P n by dividing the number of articles in the old, respectively in the new paradigm through the number of both types of articles. The intermediate and final results of this procedure are presented in the annex in Table A1 (see Data Appendix). The figures are obviously only a rough approximation to reality, with many erroneous omissions and inclusions of articles. Its also important to keep in mind that the data refer not only the social sciences but to any scientific activity covered by Scholar Google, thus e.g. including engineering.

4.2 An Empirical Tests with Preliminary Results

As shown in the previous Figs. 1a and 2b, the dynamics of the model depend very much on the values of its “free” parameters like e.g. the acceptance of extra-paradigmatic works or the ease of discovery with different paradigms. Thus, these parameters have the advantage that they can be used in order to fit the model to the data. Ideally they should be determined with regression-like statistical methods. However, this is for the present model rather difficult, among others because of the missing time series data for the ease of discovery. Hence the author changed the values of the parameters δ, E o , and A e by trial and error, until there was for the whole analysed period between 1993 and 2012 a good correspondence between the outcome of the respective simulation run and the observed shares P n of publications, referring to the new agent based modelling paradigm. This ad hoc method has yielded the following results:

$$ \updelta = 0.0191\vspace*{-7pt} $$
(8a)
$$ \mathrm{A}_{e} = 1 \vspace*{-7pt}$$
(8b)
$$ \mathrm{S}_{o} = 0.872\ \mathrm{at}\ \mathrm{time}\ \mathrm{t}=1 \vspace*{-7pt}$$
(8c)
$$ \mathrm{E}_{o} = 0.892\ \mathrm{at}\ \mathrm{time}\ \mathrm{t}=1, $$
(8d)

where for reasons of standardisation all simulation-experiments started at time t=1 with the parameter values A i  = 1 and E n  = 1. The resulting model-fit,Footnote 4 defined as the mean difference between the observed and the simulated share of publications equals 0.0041 and thus appears to be quite ok: the simulated trajectory of the publication share P n is on the average less than half of a per cent away from true share of these publications. This positive evaluation of the model is further corroborated by Fig. 4, which shows a good correspondence between the real and the simulated temporary evolution of P n , especially with regard to the geometrical properties of the two curves, like e.g. the peaks or the phases of acceleration. However, it has to be kept in mind that a more profound assessment of the model is only possible on the basis of additional examples of paradigm changes, preferably with other types of revolutionary dynamics.

Fig. 4
figure 4

The temporary evolution of the observed and the simulated share P n of publications based on agent based modelling (ABM) (Time = 1 ≈ 1993, Time = 11 ≈ 1994, …, Time = 181 ≈ 2011, Time = 191 ≈ 2012)

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The parameter-estimates (8a) to (8d) are not only useful for a good model fit but also help to understand the modelled processes of science: especially striking in this respect is the estimate A e  = 1 (see (8b)),Footnote 5 which means that the extra-paradigmatic papers are treated by the journal reviewers in a very similar way as the intra-paradigmatic papers with the same value A i  = 1. This is probably due to the fact that the representatives of the new ABM-paradigm had even at the beginning of the simulated period enough opportunities to publish in journals, which were not under control of the older systems dynamics paradigm. Of similar interest as A e  = 1 is the strikingly high ease of discovery E o  = 0.892 of the old paradigm at the initial time-point t=1. This probably reflects the fact that the systems dynamics paradigm was not really in crisis, when agent based modelling entered the scientific scene. Obviously this is a different situation from the one described by Kuhn [1] at the outbreak of a scientific revolution. Consequently we cannot expect the disappearance of the old paradigm, as suggested by Kuhn, and the coexistence of two paradigms seems to be an intuitively plausible result of the model.

5 Summary and Conclusions

In this chapter we present a model that allows to simulate the classical scientific revolution of Th. Kuhn [1] as well as many other forms of paradigmatic changes like the stable coexistence of an old and a new paradigm. According to Fig. 3b, Kuhn’s revolution seems to be a possible but rather special event that can only occur if the acceptance A e of external paradigms is rather low. Given the large number of scientific journals, the difference between the intra- and extra-paradigmatic acceptance A i  = 1 and A e is probably often only small. Thus, A e too is for many cases close to 1 such that Kuhn’s revolution becomes according to Fig. 3b a rare event. Moreover, due to the mentioned high values of A e , the old paradigm need not really be depleted in order to enable the immediate start a new one: as shown in Fig. 3a, the triggering of this kind of paradigmatic change simply requires that the old paradigm has an ease of discovery E o  < A e . The high acceptance rate A e of new external paradigms makes this a likely event, which leads according to Fig. 3b to multi-paradigmatic science — in reality not only with two, but often several paradigms coexisting in parallel. Obviously, the model presented in this paper is not made for situations with more than two simultaneous paradigms and thus requires in the future an additional modification.