Jointly published by Akadémiai Kiadó, Budapest and Kluwer Academic Publishers, Dordrecht

Scientometrics, Vol. 61, No. 3 (2004) 301–321

Reconsidering Price’s model of scientific growth: An overview ANTONIO FERNÁNDEZ-CANO,a MANUEL TORRALBO,b MÓNICA VALLEJOa a Facultad

de Ciencias de la Educación, Universidad de Granada (Spain) de Matemática, Universidad de Córdoba (Spain)

b Departamento

This paper presents an overview of the general model of scientific growth proposed by D. J. de Solla Price. Firstly, the formulation of the model is examined using the seminal sources. Later, forerunners, offshoots and criticisms to the model are discussed. Finally, an integrative review using retrieved empirical studies exposes the complexity and diversity of models of scientific growth and the absence of consistent patterns.

Formulation of the model Measuring the size of the science was, from early on, a problematical task when this sort of mathematical Science of Science1 came into being in the 1920s and 1930s. Clearly, the chief difficulty of this undertaking lay in devising some means of measuring scientific effort or outputs based on a consistent metrological theory. In fact, the construct formulated under the general term size of science encompassed multiple operative definitions or variables. Price2 acknowledged such conceptual and operational difficulties when he said: Both considerations concern what one might well call the size of science [italics in the original] – the magnitude of the effort in terms of numbers of men working, papers written, discoveries made, financial outlay involved. (p. 163) In an effort to examine the highly cumulative nature of science, an arithmetic process was developed by pioneer scholars. Derek John de Solla Price, the first great precursor to do this, had the ability to summarize important generalizations about science into simple quantitative patterns. Undoubtedly, he represents a giant with big shoulders where we, the common people, seek support. Price, a foremost intellectual leader, offered us a paradigm for the study of science overall and its various disciplines or sciences in particular. In spite of historical-qualitative approaches to the study of Received April 19, 2004 Address for correspondence: ANTONIO FERNÁNDEZ-CANO Dpt. Métodos de Investigación en Educación, Facultad de Educación Universidad de Granada, Campus de Cartuja, Granada, 18071, Spain E-mail: [email protected] 0138–9130/2004/US $ 20.00 Copyright © 2004 Akadémiai Kiadó, Budapest All rights reserved

A. FERNÁNDEZ-CANO et al.: Price’s model of scientific growth

science, Price attempted to put some logical and mathematical order in an exuberant and hence unexplored field, traditionally focused on episodic narratives and anecdotal records. He was an ex-physicist who decided to turn the tools of science on science itself. When Price arrived in Singapore, the University had just received a complete set of the Philosophical Transactions of the Royal Society. He read them, and not only became aware of the evolutionary nature and the historical aspects of science and technology, but also developed his theory of the growth of scientific literature. Price has provided us with some remarkable insights, some on a par with law, for the quantitative study of science: the scientometrical via. Perhaps one of his most commented upon and impacting conclusions was his model of scientific growth connecting the size of science and time. This model is a simple functional form between one variable (related to the size of science) with another (a certain span of years) regarding a particular subject or discipline. Initially, the size of science was measured according to the number of journals founded (not surviving) between 1665, when the earliest journal, the Philosophical Transactions of the Royal Society of London was first published, and a future year 2000. The figure proposed in his books Science since Babylon2 and Little Science, Big Science3 has been widely reproduced in multiple languages and books: a figure represented by two exponential lines corresponding to the number of scientific journals and abstract journals during the 1665-2000 time interval. As Price points out:2 ..the number of journals has grown exponentially rather than linearly ….The constant involved is actually about fifteen years for a doubling, corresponding to a power of ten in fifty years and a factor of one thousand in a century and a half. (p. 169) He then goes on to make a social and biological analogy, comparing scientific growth to bacteria and rabbits:2 The number of journals has behaved just like a colony of rabbits breeding among themselves and reproducing every so often. (p. 169) … the growth of most organisms tends to be directly related to their size: the bigger they get, the faster they grow. (Price,4 p. 240) Making yet another comparison to human birth rate, Price2 describes the constant rate of annual growth as follows: “every year about one journal in twenty, about 5 per cent of the population, has a journal-child” (p. 170); or to put it another way: “the density of science in our culture is quadrupling during each generation” (p.177). Thus, a typical trait of science is its contemporariness, its undeniable youth. Yet Price proposes differing annual rates of growth in his papers. In the work cited above,2 he gives a figure of 5%, while in another paper 5 he states:

302

Scientometrics 61 (2004)

A. FERNÁNDEZ-CANO et al.: Price’s model of scientific growth

Since the body of literature has been growing exponentially for a few centuries, and probably will continue at its present rate of growth of about 7 percent per annum, there will be about 7 new papers each year for every 100 previously published papers in a given field. (pp. 510-511) Price was soon to acknowledge that a list of papers is likely to be somewhat more comprehensive and more selective that any list of journals which may, from time to time, publish scientific papers immersed in nonscientific material. As a result of his conceptual reasoning, the paper came to be the basic unit of analysis but above all because it is a more cumulative unit for obtaining great numbers, offering sound patterns over a great mass of such data collected for other purposes such as scientific bibliography. Price also considered other variables, namely the number of authors, number of universities founded in Europe, funds, scientific manpower or the population of scientists, dissertation production, or the number of scientific books and abstracts and citations; for a list of scientific growth indicators see Price and Gürsey 6 and Gilbert.7 Going beyond the bounds of absurdity, Price2 was aware, albeit solely at a rational but not yet empirical level, that “a process of growth so much more vigorous than any population explosion or economic inflation cannot continue indefinitely but must lead to an intrinsically larger catastrophe than either of these patently apparent dangers” (p. 182). As he states,4 “it follows that saturation must be reached sooner or later if such growth is maintained even approximately” (p. 241). The internal rationality of this pattern was explained by what is known as saturation hypothesis in the following way: at a certain time some new experimental opportunities appear and new ideas emerge, leading to the rapid growth of a new element. After the opportunities of the methods are exhausted, the growth begins to fade until a new trend emerges and gives rise to a new curve growth.2–4 Hence “the curve of growth is a sigmoid or logistic curve, S-shaped and above and below its middle”. Later, however, Prices goes on to say: “we have not reached that stage quite yet, but it is only a very short time before we will – less than a human generation”2 (p. 183). Thus science in age of saturation must begin to look rather different from its accustomed state. The following figure offered by Price2,4 (pp. 185 and 241, respectively) will be reproduced to the point of satiety in a great number of books, papers and languages over the next 25 years. Briefly, Price observed three stages in the growth of knowledge: a) a preliminary phase with small increments; b) a phase of exponential growth with an increasing growth; c) a period of inalterable development. Although it has been argued that there is a period during which both the rates of increase and the absolute increase decline and eventually approach the zero level, we have encountered nothing in Price’s writings to confirm that claim.

Scientometrics 61 (2004)

303

A. FERNÁNDEZ-CANO et al.: Price’s model of scientific growth

Figure 1. Price’s model of scientific growth

Finally, in an article published in 1971, Price8 empirically verifies his complete model using data on PhD production and on scientific and manpower funds to show that there is a transition period of almost linear growth extending for two or three decades between the free exponential growth and the onset of effective saturation. Forerunners to model When examining the antecedents to Price’s formulation, our attention turns to Malthus’s argument regarding the exponential growth of human population with linear increase of available resources. That is, the population tends to increase at a greater pace than its means of subsistence with a constant rate or factor of proportionality which is often called the Malthusian parameter. In the 19th century, Verhulst described a simple model of population growth, assuming that the probability of death depends on the size of population. Yet references to the subject at hand date back long before our era such as the following biblical passage from the Ecclesiastes (12, 12): And furthermore, by these, my son, be admonished: of making many books there is no end; and much study is a weariness of the flesh. Prior to Price’s systemic formulation, during 1917-1935, several pioneer exploratory studies along this line were conducted by Cole and Eales9 on the historical development of comparative anatomy over three centuries since 1550; Hulme10 calculated the For Price and for us here, a linear process or function only is a straight line, not a curve, while a nonlinear process is defined as one that exhibits sudden discontinuities.

304

Scientometrics 61 (2004)

A. FERNÁNDEZ-CANO et al.: Price’s model of scientific growth

average annual growth in terms of the number of scientific papers published in the XIX century using a catalogue of journals compiled by the Royal Society of London; Wilson and Fred11 studied the growth of research in the area of nitrogen fixation of plants. Meadows12 in 2001 verified that the overgrowth had previously been acknowledged in the XIX century, provoking exasperated reactions due to the declining readability of scientific literature. Heir to mathematical formalism and philosophical positivism, Price was concerned with transforming every phenomena of science into some sort of mathematical relation. The models, such as they were finally constructed, were certainly long in the making. Price’s first publication on this subject appeared in the form of a conference paper and dates from 1951,13 although it was soon published in a French journal.14 A more extensive text was later published in a more easily readable form in 1956.4 This text was to grow into a separate little book of its own: Little Science, Big Science,3 published in 1963. Price continued to produce a series of research papers,15 exploring different quantitative aspects and reformulating his original model. Although his three first papers4,13,14 were all unknown and it took a good 25 years for his general ideas to become fully articulated, Price16 was sure of his success when he wrote: It took only a few days to convince myself that the phenomenon was no artifact of this one particular journal. The exponentially and, almost precisely, the same simple parameter of the cumulative growth emerged wherever I looked. (p. 70) Expanding the model Price’s ideas would soon enjoy great popularity, attracting “a reasonable number of competent (co)-workers”2 (p. 194).On the one hand, application studies were carried out in an attempt to verify the model’s applicability to empirical data, as we shall see in grater detail below. On the other hand, methodological studies17–20 were done offering technical solutions to the problem involved in time series. Yablonsky21 rejected the idea that the regularity of distribution of scientific productivity follows a Gaussian model. Egghe and Rao22 classified and characterised the different growth models using summative (t to t + 1) and multiplicative (t to 2t) time rate functions, T1 and T2, respectively. Later, using the non-linear regression method available in the STAGRAPHICS – Statistical Package, they made a distinction between the exponential, logistic or S-shaped, power, Gumpertz and Ware models of scientific growth. Other authors, such as Gupta and Karisiddappa23 used the SYSTAT software package for calculating T1 and T2. Expanding upon Price’s model, SzydVowski and Krawiec24 applied differential equations to model the evolution of science including additional aspects such as the death of results, the time required to learn or to apply results to new discoveries.

Scientometrics 61 (2004)

305

A. FERNÁNDEZ-CANO et al.: Price’s model of scientific growth

The field of scientometrics, as it is now called, achieved a progressive theoretical base, in the Lakatosian25 sense of term, because such a model: – – –

–

–

– – –

Can help to explain notable phenomena with a more precise language (i.e. as internal repressive effects in a discipline). Can predict future growth although it cannot be fully explained due to “specification errors”. Concerns the saturation of unwieldy literature for review and is a reaction towards the increasingly higher level of organization of scientific production. The problem was not just one of keeping reading time within acceptable limits: it was also one of retrieving information from rapidly expanding range of sources. 12 Contemplates the heuristic derivation of new laws such as those testing the qualityquantity relation; for example, Rescher’s W-quality index, Rousseau’s law or May’s classification of literature, which can be found in Tague et al. 20 Justifies the phenomena of concomitant specialization and the swarming of innovations. Remember that swarming is a peculiar trend where a notable finding causes intellectual cascades that percolate across disciplines. Studies the effects of wars, political crises and social conflicts on the observable progress of science. Makes objective judgments on the health of certain fields of inquiry. Helps to explain the research fronts phenomenon as a cycle characterized by four ages or life cycles: birth, increase, maturity, and senility.

Nowadays, in an age of dwindling resources, Price’s ideas are chiefly being reassessed in the yet-to-be systematized social and educational sciences, which are prone to fighting paradigm wars. The massive incorporation of female researchers into these fields further underlines the rationality and relevance of Price’s ideas. A renewed interest in the matter is being taken in scientifically developing countries like Brazil, India or Spain where excessive growth could lead to a sort of “clay-footed” or “builton-sand” scientific development. Critics and replies to the model Price16,26 understood that his model of scientific growth vigorously opposed Thomas H. Kuhn’s27 central idea about the revolutionary nature of the science structures. As Price16 argues: I also feel certain reservations in accepting Kuhn’s idea of some major “paradigm shift” [quotation marks in original] as a major feature of the historical unfolding of the jigsaw puzzle solution. (p. 91)

306

Scientometrics 61 (2004)

A. FERNÁNDEZ-CANO et al.: Price’s model of scientific growth

Although Price16 criticizes “the simple methodologies based in the isolated numbers because to be meaningful a statistic must be somehow anticipatable from its internal structure or its relation to other data” (p. 72), he was even so guided by a primary positivism that sought to produce the best interlocking system with the minimum number of independent parameters. It was precisely this parsimonious aspect of Price’s model that motivated the criticism published in two editorials published by Nature.28,29 Price30 nonetheless provided a suitable reply. The ideological stance of the left (i.e. Kedrov and Spirkin31, Konfederatov32) rejected this model as it was considered too abstract and did not take into account socioeconomic, political and cultural factors, which obstructively condition scientific growth. The leftists uphold the idea put forward by Engels in his article Umrisse zu einer Kritik der Nationalökonomie33 in which he rejected the existence of a future point of saturation. Thus, the logistic curve model is simply a manifestation of the manifold causes that hinder the exponential model. Konfederatov,32 for example, was optimistic about the indefinite expansion of the science, comparing it to a nuclear fission chain reaction. In general, the official Soviet stance took for granted the optimist exponential model, rejecting the logical realist model proposed by Price. For more on the discussion of early considered models see the work by Brode,34 Kochen,35,36 or López-Piñero.37 Additional explanations for the phenomenon can be found in Nalimov53,54, and Ziman,38 who viewed the proliferation of scientific literature as a natural process. Concretely, Granovsky39 comments that Nalimov defined the transition from an exponential to a logistic curve as an “adaptive inhibition” because research yield increases in direct proportion to the increase in the number of researchers, and is in inverse proportion to the general number of researchers engaged in the solution of some problem. The criticisms are directed along several lines as shown below. Methodological inadequacy Specifically, the statistical method based on arithmetic counts at the aggregate-level is inadequate for two reasons: a quantitative bias omits relevant qualitative features and, due to its simplicity, the model is insensitive to interactions and contextual variations. For example, the growth ratio in the exponential curve is not a fixed constant but an averaged value and procedures for goodness-of-fit are only based on a graphic-visual examination whereas sophisticated functional adjustments22,40,41 are also available and could be more appropriate. Nowadays, another larger problem stems from ensuring that scientific products which seem to be new are really new. Reprints, translations, conference papers or the

This other giant of scientometrical studies, Vassily V. Nalimov was mistreated by the “little people”.

Scientometrics 61 (2004)

307

A. FERNÁNDEZ-CANO et al.: Price’s model of scientific growth

duplicate papers with only cosmetic changes are counted several times. When, on the other hand, some national science generated in underdeveloped countries would be measured with difficulty because it is not included in relevant database (i.e. Science Citation Index of Institute for Scientific Information, Philadelphia); we are speaking of the problem of limbic literature (Fernández-Cano and Bueno42). Nevertheless, Price4 proved himself to be a persuasive quantitative researcher when he said: Since it is natural to prefer a quantitative approach, even if inexact, to any purely qualitative analysis, it is necessary to seek any data that can be obtained by a process of “head-counting” [quotation marks in original] throughout a series of annual or other periods. (p. 240) The attractiveness of using these by-products of scientific activity lies in the fact that it entails a non-interfering and silent way of measurement, making it relatively easier than determining exactly what it is that has been measured, regardless of its meaning. Price15 considers that “one uses the numbers as one pleases ... only admitting that a number is a meaningful statement about what one should expect and why” (p. 101). According to Price,16 “a meaningful statistic must be somehow anticipatable from its internal structure or its relation to other data” (p. 72). Obviously, every measurement entails qualitative consideration relative to questioning “what”, “why” and so on. Bias towards the English language That there is a strong focus on publishing in English is unquestionable. This has a direct repercussion on the motivation of scientists as they are forced to write in the lingua franca (English ) and may therefore overlook local and national problems. In our opinion, this criticism is excessive as it attributes the forces of every culture and language to only one man. Despising the individuality of the scientist Disregard for the individuality of the scientist (i.e. Newton, Gauss or Einstein) and human factors in the realm of scientific enterprise is clearly discernible from

*

Helmut H. Abt, the open reviewer of this paper, considers “that one unstated advantage of English is that it is very specific and is therefore much more suitable for science…Of course other western languages may have the advantage of English in being very specific, but we have not had a universal language since the demise of Latin” (personal communication).

308

Scientometrics 61 (2004)

A. FERNÁNDEZ-CANO et al.: Price’s model of scientific growth

representative interviews. According to Boring’s43 formulations, Price rejects the assumption that truly important scientific development can be made solely by great geniuses. Non-frameworked model Beck’s44 criticism was directed at the fact that the models are proposed on a qualitative level much like a bureaucratic practice in nature, which could be defined as a rigid ability to work without enough regard for important differences and variations; and that sound reasonable explanations for these trends were not provided. Despite his criticisms of the models, Beck44.quickly acknowledged that “perhaps Solla Price’s proposal is useful and informative in practice”. However, Price45 argued that the idiosyncratic characteristics which deviate from his model curve are merely peaks of spasmodic activity. Omitting the quality of research Yet another criticism lies in the fact that the model does not consider the quality of research and the amount of knowledge since papers are undifferentiated and all authors are homogeneous. Thus every product or person is equivalent per se and each has a single value (1), making them easy to accumulate. Consequently, creative papers with new ideas and results are equivalent to trivial duplications. We must not forget the ballbeetle’s syndrome as exemplified by the methodological dictum: “Garbage plus garbage is garbage”. However, it is possible to establish ranks between scientific products such as axioms, theorem theoretical model, empirical facts, heuristical practice, and routine practice as Beck46 proposed. Only scientific achievements equal in epistemological rank might be admitted for statistical counts. Nevertheless, Price2 bears this shortcoming in mind when he argues: “of course there is some essential difficulty in counting Physical Review as a single unit of the same weight as any Annual Broadsheet of the Society of Leather Tanners of Bucharest” (pp. 164-165). He2 also remarks that thus far nothing has been said about the quality of research as opposed to its quantity, although he does formulate the following hypothesis: One may study the growth of only important discoveries, inventions, and scientific laws, rather than all such things, important and trivial; any count of this sort immediately shows that the growth, though still exponential, possesses a doubling time that is much longer than that of the gross growth of science. (pp. 187-188)

Scientometrics 61 (2004)

309

A. FERNÁNDEZ-CANO et al.: Price’s model of scientific growth

Clearly, for him,15 “papers differ enormously in their worth, the most important ones being a tiny fraction of the total” (p. 100). In an earlier work, he 4 argues: If one attempts to measure “high-level” advances in science rather than its crude size, it seems that the growth constant is considerable larger, taking about three times as long to double. (p. 242) Price4 regarded exponential growth as a disease of science that exercises a retarding effect on the growth of stable science, producing narrower and less flexible specialists. To double the usefulness of science involves multiplying by about eight the gross number of workers and the total expenditure of manpower and national income. He3 suggested that although the doubling time for all scientific work is ten years, the doubling time for “very high quality” work is about 20 years (p. 46). As Price 15 declared in 1969: “these measures of scientific production and manpower have all increased exponentially with a quite impressive and fatalistic regularity” (p. 102). The chapter 8 of his2 book Science since Babylon, where his models are finally put forward, is entitled Diseases of science (pp. 161-195). A central idea, in the line of Popperian epistemological view of falsifiability criterion”, which consider that an ample expansion of the science would be regarded as a deficiency. Concretely, Popper47 states: “the growth of knowledge … is not a repetitive or cumulative process, but one of error elimination” (p. 144). It would then follow that science must be subtractive, not additive. Consequently, this frenzied growth should stop and unsubstantial science ought to be blocked. Understanding difficulties Doubtless, Price’s model has been misunderstood and his terms and formulations were used improperly. For example, the exponential character of scientific production gave rise to the inappropriately used term ‘informative explosion’ when, in fact, growth was not a sudden phenomenon but the result of a long process of uninterrupted expansion throughout the last three centuries. Another frequent misunderstanding occurs when scientific growth across time is matched with the cumulative advantage function, which crossed production with the number of authors following the successbreeds-success pattern initially studied by Lotka48 and later also by Price.3,26,49 However, a distinction must also be made between the growth of size of science, and the growth of knowledge, which is a concept that is more complex and abstract, and hence more difficult to assess. In fact, the growth of knowledge has always been a major concern of philosophers of science. Thus, Karl R. Popper50 expanded the title of his central book, Conjectures and Refutations, to The Growth of Scientific Knowledge. Lakatos51 was also interested in this matter when he associated Criticism and the Growth of Knowledge.

310

Scientometrics 61 (2004)

A. FERNÁNDEZ-CANO et al.: Price’s model of scientific growth

Single-minded determinist model Cole and Meyer52 argued that Price ignored theoretically interesting deviations from these laws and the absence of data to support them, proposing a “blind” determinist model with little capacity to know what influences or causes lead to the rate of scientific advance. Equally, he has been accused of giving his laws a presumptuous scientific status as they predict occurrences without error. Actually, Price never spoke of laws –only models. It is we who speak of laws. Finally, other alternative approaches (epistemological, self-organizing systems, or diffusion of technology) could be complementary to and considered jointly with this socio-biological mixture proposed by Price.

Empirical studies of confirmation Price aimed to verify his exponential law of scientific growth by investigating several bibliographies of the sciences. About thirty such analyses came to his notice, all with similar results. As he 2 stated in 1978: It seems beyond reasonable doubt that the literature in any normal, growing field of science increases exponentially, with a doubling in an integral ranging from about ten to about fifteen years. (p. 171) This was soon corroborated from the other side of the “iron curtain” by Nalimov53, 54 who pointed out that the exponential growth of publications doubled every 10-15 years. Table 1 provides an overview of the study on this questions including: – – – –

– –

Author/s and year of the study. The particular scientific discipline studied. Type of basic units of analysis which comprise the distribution sample: i.e. journal, publication, abstract, paper, citation, literature and so on. Analytical techniques used to verify goodness-of-fit to the model: graphic-visual (direct or with smoothing), bi-functional, correlational (R2 and phi binomial coefficient), or ARIMA. Model verified: exponential, logistic, linear, irregular-nonlinear, power, or Gumpertz. Rate of growth per annum as a percentage or a functional constant.

Scientometrics 61 (2004)

311

A. FERNÁNDEZ-CANO et al.: Price’s model of scientific growth

312

Scientometrics 61 (2004)

A. FERNÁNDEZ-CANO et al.: Price’s model of scientific growth

Scientometrics 61 (2004)

313

A. FERNÁNDEZ-CANO et al.: Price’s model of scientific growth

314

Scientometrics 61 (2004)

A. FERNÁNDEZ-CANO et al.: Price’s model of scientific growth

Scientometrics 61 (2004)

315

A. FERNÁNDEZ-CANO et al.: Price’s model of scientific growth

The general findings of this review are manifold. First, as Tague et al.20 stated, most models of growth seem to determine scientific growth visually from the empirical plots of a graph. Second, the functions of saturation (logistic, power, or Gumpertz) are the prevalent model with approximately three defined periods having notable contextual and disciplinal variations as follows: •

•

•

The initial period represents the “little science age”; a period a small-scale “innocent” science where little funding is available and the scientific societies play a chief role. The intermediate phase constitute Price’s “Big Science” age of exponential growth in which the scientific system moves away from an academically driven society to a mixed commercial and social marketplace. A final phase that might be considered a period of logistic expansion, termed “scape-goat science” by Mabe and Amin.55 Corporations with strongly vested interests dominate this “soft” crisis period where overambitious expectations do not yield scientific output. Priorities and scarce resources produce a science with marked shortages.

There are notable variations in distributions with no consistent patterns. Generally, for short time periods and on former research fronts, the exponential curve is prevalent. It would seem, then, that for long time periods and highly consolidated disciplines, the saturation functions fit the model better. Open questions Price’s acute insights give rise to manifold considerations and questions that were not addressed when expounding his ideas. He was aware of the problem involved in managing the massive overload of newly incoming information. In his56 second to last paper, he acknowledged the enormous possibilities and capacities of computer applications to manage and develop databases, the advent of the Web to facilitate better access to those databases and new electronic journals giving rise to new considerations. Enormous concerns regarding the management of databases were thus awakened, as every study of this type is affected by the quality and quantity of the data retrieved. Price alerted us to the tactics and strategies to be employed when applying our scientific efforts. The diagnosis of the petty illnesses of science – its superabundance of literature, its manpower shortages, its increasing and excessive specialization offering a fragmented knowledge, its tendency to deteriorate in quality – are all but symptoms of a widespread disease that even nowadays manifests itself. To the best of our knowledge, his last paper57 was an article published in Scientometrics about square matrices for scientometric transactions. Unfortunately, he passed away on September, 3, 1983.

316

Scientometrics 61 (2004)

A. FERNÁNDEZ-CANO et al.: Price’s model of scientific growth

Controversial political and educational implications are still being challenged. At present in Spain a conservative government applies scientific policy in an attempt to reduce the size of science by lowering student admissions to the sciences and eliminating doctoral programs in weaker departments and universities, albeit without slowing down the rate of scientific advance. Encouraged by forecasting problems, a new generation of greater statistical sophistication has been born (Bayesian models, stochastic growth, ARIMA for timeseries and other multivariate prediction models). New decisions and choices will require great care, however. Interest should not be directed so much at the adjustment of data to one or another model as an exploratory function. Instead, trends and cycles should be sought out and future predictions made (prospective function). Obviously, reducing investigations to solely one model would weaken and impoverish the study of scientific growth. Even potential alternative models (i.e. the ecological model, Goffman’s epidemical model, and Sorokin’s theory of cyclical movements or systems dynamics to name but a few) could constitute worthwhile lines of theoretical inquiry. It is possible, for example, to model science as an ecological system in equilibrium with standard (logistic) growth that expands when higher levels of nutrients are added and returns to earlier levels when the consumed excess such as balance must be re-established. With web dynamics, new questions related to online information, links, web-pages and electronic journals could be considered to address these models of growth. As Kealey58 observes: Economists and governments lag decades behind Derek Price’ thinking. In fact, there are few topics that are of greater import than the economics of science. Yet we scientists have erred in entrusting it to economists. In ignoring Price, we have incubated 37 years of mistakes. (p. 279) Additional considerations such as libraries canceling subscriptions to periodicals owing to increased purchase costs are now of general concern. Forecasting demands timely action as inputs to policy analysis will aid in budgeting and staffing decisionmaking processes. The evaluative possibilities of scientometric indicators (Glänzel and Schöpflin59) and the scientific status of scientometric studies (Wouters and Leysdesdorff60) yet continue to be open questions. This was Price’s greatest contribution: the study of science is submitted to the inexorable demands of Time and Space. Evidently, he can just saw a big field of study and its method; a locus containing phenomena, events, institutions, problems, persons, and processes, which themselves constitute the raw material for inquires of many kinds using above all a quantitative methodology.

Scientometrics 61 (2004)

317

A. FERNÁNDEZ-CANO et al.: Price’s model of scientific growth

To paraphrase to Eugene Garfield and Robert Merton, who lauded him in a posthumous, expanded version of Little Science, Big Science,26 we can hardly doubt that Price is, truly, the Father of Scientometrics. * We are very grateful to Helmut H. Abt, who gives us his open refereeing, and to an anonymous reviewer for their helpful commentaries.

References 1. D. J. de S. PRICE, The science of science. In: M. GOLDSMITH, A. MACKAY (Eds), The Science of Science. Souvenir Press, New York, 1964. 2. D. J. de S. PRICE, Science since Babylon (3rd ed.). Yale University Press, New Haven, 1978. (First edition in Litho Crafters, Chelsea, MI, 1964). 3. D. J. de S. PRICE, Little Science, Big Science (1st ed.). Columbia University Press, New York, 1963. (Spanish translation as Hacia una ciencia de la ciencia. Ariel, Barcelona, 1973). 4. D. J. de S. PRICE, The exponential curve of science, Discovery, 17 (1956) 240–243. 5. D. J. de S. PRICE, Networks of scientific papers. The pattern of bibliographic references indicates the nature of the scientific front, Science, 149 (3683) (1965) 510–515. 6. D. J. de S. PRICE, S. GÜRSEY, Some statistical results for the numbers of authors in the states of the United States and the nations of the world. In: Who is Publishing in Science 1975 Annual. Institute for Scientific Information, Philadelphia, 1975, pp. 26–34. 7. G. N. GILBERT, Measuring the growth of science: A review of indicators of scientific growth, Scientometrics, 1 (1) (1978) 9–34. 8. D. J. de S. PRICE, Principles for projecting funding of academic science in the 1970s, Science Studies, 1 (1) (1971) 85–94. 9. F. J. COLE, N. B. EALES, The history of comparative anatomy. Part 1. A statistical analysis of literature, Science Progress, 11 (1917) 578–596. 10. E. W. HULME, Statistical Bibliography in Relation to the Growth of Modern Civilization, Grafton, London, 1923. 11. P. W. WILSON, E. B. FRED, The growth curve of a scientific literature: Nitrogen fixation by plant, Scientific Monthly, 41 (1935) 240–250. 12. J. MEADOWS, Early reactions to information growth, Scientometrics, 51 (3) (2001) 553–561. 13. D. J. de S. PRICE, Quantitative measures of the development of science. In: Actes du VIeme Congrès International d´Histoire des Sciences (Amsterdam). Hermann & Cie, Paris, 1951, pp. 413–421. 14. D. J. de S. PRICE, Quantitatives measures of the development of science, Archives Internationales d´Histoire des Sciences, 14 (1951) 85–93. 15. D. J. de S. PRICE, Measuring the size of the science, Proceedings of the Israel Academic of Science and Humanities, 4 (6) (1969) 98–111. 16. D. J. DE S. PRICE, Towards a model for science indicators. In: Y. ELKANA, J. LEDERBERG, R. K. MERTON, A. THACKRAY, H. ZUCKERMAN (Eds) Toward a Metric of Science: The Advent of Science Indicators. John Wiley & Sons, New York, 1978, pp. 69–95. 17. G. E. VLEDUTS, V. V. NALIMOV, N. I. STYAZHKIN, Scientific and technical information as one of the tasks of cybernetics, Uspekhi Fizicheskikh Nauk, 69 (1959) 13–56 (Original in Russian). 18. B. C. BROOKES, The growth, utility and obsolescence of scientific periodical literature, Journal of Documentation, 26 (4) (1970) 283–294. 19. M. KOCHEN, A. BLAIVAS, A model for the growth of mathematical specialities, Scientometrics, 3 (4) (1981) 265–274.

318

Scientometrics 61 (2004)

A. FERNÁNDEZ-CANO et al.: Price’s model of scientific growth

20. J. TAGUE, J. BEHESHTI, L. REES-POTTER, The laws of exponential growth: Evidence, implications, and forecasts, Library Trends, 30 (1) (1981) 125–149. 21. A. I. YABLONSKY, On the fundamental regularities of the distribution of scientific productivity, Scientometrics, 2 (1) (1980) 3–34. 22. L. EGGHE, I. K. R. RAO, Classification of growth models based in growth rates and its implications, Scientometrics, 25 (1) (1992) 5–46. 23. B. M. GUPTA, C. R. KARISIDDAPPA, Modelling the growth of literature in the area of theoretical population genetics, Scientometrics, 49 (2) (2000) 321–355. 24. M. SZYDPOWSKI, A. KRAWIEC, Scientific cycle model with delay, Scientometrics, 52 (1) (2001) 83–95. 25. I. LAKATOS, La metodología de los programas de investigación científica. Alianza Universidad, Madrid, 1983. Spanish translation of Falsation and the Methodologies of Scientific Research Programmes. (1970). 26. D. J. de S. PRICE, Little Science, Big Science … and Beyond (Enlarged edition). Columbia University Press, New York, 1986. 27. T. T. KUHN, The Structure of Scientific Revolutions. Chicago University Press, Chicago, 1962. 28. EDITORIAL, The magic of numbers, Nature, 217 (1968) 793–794. 29. EDITORIAL, Can science afford scientist? Nature, 226 (1970) 10. 30. D. J. de S. PRICE, Similes at unobtrusive, Nature, 226 (1970) 985. 31. M. B. KEDROV, A. SPRIKIN, La ciencia [The science]. Grijalbo, México, 1967. (Original in Russian). 32. J. KONFEDERATOV, Exponential or logistic law of scientific development. In: Actes du XIIeme Congres International d’Histoire de Science. Tome II. Problemes généraux d’Historie des Science. Epistemologie. Blanchard, Paris, 1970, pp. 63–66. 33. F. ENGELS, Umrisse zu einer Kritik der Nationalökonomie. In Dialektik der Nature, 1884. Spanish translation as Dialéctica de la Naturaleza. Grijalbo, México, 1967. 34. W. R. BRODE, Manpower in science and engineering based on a saturation model, Science, 173 (3993) (1971) 206–213. 35. M. KOCHEN (Ed.), The Growth of Knowledge. Wiley, New York, 1967. 36. M. KOCHEN, Stability in the growth of knowledge, American Documentation, 20 (3) (1969) 186–197. 37. J. M. LÓPEZ-PIÑERO, El análisis estadísticos y sociométrico de la literatura científica [Statistical and sociometrical analysis of scientific literature]. Centro de Documentación e Informática Médica, Valencia, 1972. 38. J. M. ZIMAN, The proliferation of scientific literature: A natural process, Science, 208 (4442) (1980) 369–371. 39. Y. V. GRANOVSKY, Is it possible to measure science? V. V. Nalimov’s research in scientometrics, Scientometrics, 52 (2) (2001) 127–150. 40. R. A. FAIRTHORNE, Empirical hyperbolic distributions (Bradford-Zipf-Mandelbrot) for bibliometric description and prediction), Journal of Documentation, 25 (4) (1969) 319–343. 41. J. J. HUBERT, General bibliometric models, Library Trends, 30 (1) (1981) 65–81. 42. A. FERNÁNDEZ-CANO, A. BUENO, Multivariate evaluation of Spanish educational research journals, Scientometrics, 55 (1) (2002) 87–102. 43. E. G. BORING, Eponyms as placebo, Acta Psychologica, 23 (1964) 9–23. 44. I. M. BECK, A hypothetical sociological explanations of the Solla Price’s theory, Science of Science, 4 (13) (1984) 91–98. 45. D. J. de S. PRICE, Ups and downs in the pulse of science and technology, Sociological Inquiry, 48 (3-4) (1978) 162–172. 46. I. M. BECK, A method of measurement of scientific production, Science of Science, 4 (14) (1984) 183–195. 47. K. POPPER, Objective Knowledge: An Evolutionary Approach. Clarendon Press, Oxford, 1972. 48. A. F. LOTKA, The frequency distribution of scientific productivity, Journal of the Washington Academy of Science, 16 (1926) 317–323. 49. D. J. de S. PRICE, A general theory of bibliometric and other cumulative process, Journal of the American Society for Information Science, 27 (5) (1975) 292–306.

Scientometrics 61 (2004)

319

A. FERNÁNDEZ-CANO et al.: Price’s model of scientific growth

50. K. R. POPPER, Conjectures and Refutations: The Growth of Scientific Knowledge. Routledge and Kegan Paul, London, 1976. 51. I. LAKATOS, Criticism and the Growth of Knowledge: Proceedings of the International Colloquium in the Philosophy of Science, London, 1965 (3rd Ed.). Cambridge University Press, New York, 1994. 52. S. COLE, G. S. MEYER, Little science, big science revisited, Scientometrics, 7 (3-6) (1985) 443–458. 53. V. V. NALIMOV, Methods of cybernetics to the laboratory practice, The Industrial Laboratory, 28 (1962) 771-774. (Original in Russian). 54. V. V. NALIMOV, Y. P. ADLER, Y. V. GRANOVSKY, Information system for mathematical theory of experiment, In: Cybernetics and Documentation. Mechanization of Scientific Information Accumulation, Storage and Search. Nauka, Moscow, 1966. (Original in Russian). 55. M. MABE, M. AMIN, Growth dynamics of scholarly and scientific journals, Scientometrics, 51 (1) (2001) 147–162. 56. D. J. de S. PRICE, Terminal librarians and the ultimate invention. In: L. J. ANTHONY (Ed.), EURIM 4: A European Conference on Innovations in Primary Publication Impact on Producers and Users. Aslib, London, 1980, pp. 103–106. 57. D. J. de S. PRICE, The analysis of square matrices of scientometric transactions, Scientometrics, 3 (1) (1981) 55–63. 58. T. KEALEY, More is less. Economists and governments lag decades behind Derek Price’s thinking, Nature, 405 (6784) (2000) 279–279. 59. W. GLÄNZEL, U. SCHÖPLIN, Little scientometrics, big scientometrics and beyond?, Scientometrics, 30 (2-3) (1994) 375–384. 60. P. WOUTERS, L. LEYDESDORFF, Has Price’s dream come true: Is scientometrics a hard science?, Scientometrics, 31 (2) (1994) 193–221.

Studies revisited (in alphabetical order) G. ARCHIBALD, M. B. LINE, The size and growth of serial literature 1950-1987, in terms of the number of articles per serial, Scientometrics, 20 (1) (1991) 173–196. B. C. BENNION, L. A. NEUTON, The epidemiology of research on anomalous water, Journal of American Society for Information Science, 17 (1) (1976) 53–56. R. T. BOTTLE, M. K. REES, Liquid crystal literature: A novel growth pattern, Journal of Information of Science, 1 (2) (1979) 117–119. T. BRAUN, W. S. LYON, E. BUJDOSÓ, Literature growth and decay: An activation analysis résumé, Analytical Chemistry, 49 (1977) 161–164. R. R. BROOKS, L. E. SMYTHE, The progress of analytical chemistry 1910–1970, Talanta, 22 (1975) 495. G. G. BRUNK, Swarming of innovations, fractal patterns, and the historical time series of US patents, Scientometrics, 56 (1) (2003) 61–80. S. COLE, G. S. MEYER, Little science, big science revisited, Scientometrics, 7 (3-6) (1985) 443–458. D. CRANE, Invisible Colleges. University of Chicago Press, Chicago, 1972. F. DELEUS, M. M. VAN HULLE, Monitoring elasticity between science and technology domains and its visualization, Scientometrics, 56 (1) (2003) 147–160. E. N. EFTHIMIADIS, The growth of the OPAC literature, Journal of the American Society for Information Science, 41 (5) (1990) 342–347. L. EGGHE, I. K. R. RAO, Clasification of growth models based on growth rates and its applications, Scientometrics, 25 (1) (1992) 5–46. J, EXPÓSITO, A. FERNÁNDEZ-CANO, La productividad de la investigación sobre evaluación de programas educativos (1975–2000) [The productivity of the research in educational program evaluation], Revista de Investigación Educativa, 20 (1) (2002) 113–129.

320

Scientometrics 61 (2004)

A. FERNÁNDEZ-CANO et al.: Price’s model of scientific growth

A. FERNANDEZ-CANO, M. TORRALBO, L. RICO, P. GUTIERREZ, A. MAZ, Análisis cienciométrico de las tesis doctorales españolas en Educación Matemática (1976–1998) [Scientometrical analysis of Spanish doctoral thesis in Mathematics Education (1976-1998)], Revista Española de Documentación Científica, 26 (2) (2003) 162–176. J. D. FRAME, J. J. BAUM, M. CARD, An information approach to examining developments in an energy technology: Coal gasification, Journal of American Society for Information Science, 30 (4) (1979) 193–201. E. GARFIELD, In tribute to Derek John de Solla Price: A citation analysis of Little Science, Big Science, Scientometrics, 7 (3-6) (1985) 487–503. W. GOFFMAN, A mathematical approach to the spread of scientific ideas. The history of mast cell research, Nature, 212 (1966) 449–452. W. GOFFMAN, Coping with the biomedical literature explosion: A qualitative approach, Rockfeller Foundation, New York, 1978. W. GOFFMAN, G. HARMAN, Mathematical approach to prediction of scientific discovery, Nature, 229 (5280) (1971) 103–104. W. GOFFMAN, K. S. WARREN, The ecology of the medical literatures, Journal of the Medical Sciences, 263 (1972) 267–273. B. M. GUPTA, C. R. KARISIDDAPPA, Modelling the growth of literature in the area of theoretical population genetics, Scientometrics, 49 (2) (2000) 321–355. B. M. GUPTA, S. KUMAR, S. L. SANGAM, C. R. KARISIDDAPPA, Modeling the growth of world social science literature, Scientometrics, 53 (1) (2002) 161–164. J. C. HUBER, R. WAGNER-DÖBLER, Scientific production: A statistical analysis of authors in mathematical logic, Scientometrics, 50 (2) (2001) 323–337. M. MABE, M. AMIN, Growth dynamics of scholarly and scientific journals, Scientometrics, 51 (1) (2001) 147–162. B. M. MANZER, The Abstract Journal, 1790-1920: Origin, Development and Diffusion, Scarecrow Press, Metuchen, NJ, 1977. K. O. MAY, Quantitative growth of the mathematical literature, Science, 154 (1966) 1672–1673. H. W. MENARD, Science: Growth and Change. Harvard University Press, Cambridge, 1971. V. V. NALIMOV, Y. P. ADLER, Y. V. GRANOVSKY, Information system for mathematical theory of experiment, In: Cybernetics and Documentation. Mechanization of Scientific Information Accumulation, Storage and Search. Moscow, Nauka, 1966. (Original in Russian). D. J. de S. PRICE, Science since Babylon (1st Ed.). Litho Crafters, Chelsea, MI, 1964. (Third edition in Yale University Press, New Haven, 1978). D. J. de S. PRICE, Networks of scientific papers. The pattern of bibliographic references indicates the nature of the scientific front, Science, 149 (1965) 510–515. K. RODRIGUEZ, J. A. MOREIRO, The growth and development of research in the field of ecology as measured by dissertations title analysis, Scientometrics, 35 (1) (1996) 59–70. L. E. STRONG, O. T. BENFEY, Is chemical information growing exponentially?, Journal of Chemical Education, 37 (1960) 29. D. SULLIVAN, The state of science: Indicators in the specialty of weak interactions, Social Studies of Science, 7 (1977) 167–200. J. TAGUE, J. BEHESHTI, L. R. POTTER, The law of exponential growth: Evidence, implications and forecasts, Library Trends, 30 (1) (1981) 125–149. C. TENOPIR, D. W. KING, Electronic Publishing and the Economics of Information. Growth in Peer Reviewed Electronic-Journals. Available: http:// web.utk.edu/~tenopir/speeches/sla061101p1.ppt (Consulted on October 14, 2003). D. WOLFRAM, C. M. CHU, X. LU, Growth of knowledge: Bibliometric analysis using online database data. Informetrics 89/90, L. EGGHE, R. ROUSSEAU (Eds), 1990. Proceedings of the Second International Conference on Bibliometrics, Scientometrics and Informetrics, London (Canada), 1989, pp. 355–372.

Scientometrics 61 (2004)

321