Continuity and discontinuity of collaboration

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R. Wagner-DObler: Continuity and discontinuity of collaboration behaviour ..... of all coauthored papers) were excluded in the process of editing the database,.
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Scientometrics, Vol. 52, No. 3 (2001) 503-517

Continuity and discontinuity of collaboration behaviour since 1800 - from a bibliometric point of view R o l a n d W a g n e r -D o b l e r Institutfiir Philosophie, Universitat Augsburg, Augsburg (Germany) Time-series of collaboration trends indicated through co-authorships are examined from 1800 to presence in mathematics, logic, and physics. In physics, the share of co-authored papers expands in the second half of the 19th century, in mathematics in the first decades of the 20th century, in logic in the second half of the 20th century. Subdisciplines of mathematics, of physics, and areas of logic show large differences in their respective propensities to collaborate. None of the existing explanatory approaches meets this heterogeneity; the most salient feature is a propensitiy to collaborate in fields where theoretical and applied research is combined.

Necessities and riddles of collaboration That collaboration and teamwork are among the most important necessities of scientific and technological work today is beyond doubt. But it is also of no doubt in my eyes that both topics themselves deserve scientific reflection and interest as this workshop vividly demonstrates. I hope that it is no misperception if I say that those reflections since Beaver's and Rosen's investigations1'3 deliver a contradicting and theoretically not satisfying picture of the process. (Cf., e.g.. Ref. 4.) A satisfying theoretical approach should consider inter alia economic as well as sociological concepts to explain different forms of collaboration in terms of a cognitive economy to take up a concept of Rescher, and in terms of the allocation of cognitive resources. It should explain how the process of specialization, a secular trend of the last two centuries, is connected in the case of scientific work with increasing cooperation and collaboration and why this differs so much in different scientific disciplines and areas. Apart from those explanatory aims it has to be said that even the simple empirical historiography of collaboration is not established, despite several case studies, at least as far as I can see. Is it really correct that scientific collaboration began to expand significantly since World War II as it is suggested so often, is this true for all (natural) sciences, and what happens in the presence? The situation concerning empirical data is

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R. Wagner-DObler: Continuity and discontinuity of collaboration behaviour

characterized through a lack of comprehensiveness and a lack of retrospective validity and sensibility. Studies giving an empirical picture of collaboration in a certain scientific area in a certain time interval deserve genuine interest. Because thousands of scientific areas exist apart from different time intervals and geographic areas, those studies can augment beyond limits. However, to which extent is theoretical progress connected with that augmentation? In what follows I would like to propagandize a comparative synchronic approach to the study of collaboration, on the one hand, and a diachronic, historiographic approach, on the other hand. If I understood it correctly both approaches are characteristic for Donald deB. Beavers work, but too less scientometricians followed his steps. In addition, there is a scientific discipline which is far too less an object of scientometric interest, viz. mathematics. It is a commonspread prejudice concerning mathematics that collaboration plays no major role in this discipline the heart of which is purely theoretical. The outline of my sketch is as follows: First I would like to present representative data with regard to collaboration in mathematics from 1800 to presence. This will be contrasted with physics which is supposed to be a collaborating discipline par excellence. The results I would like to compare with a major subdiscipline of mathematics, namely mathematical logic. How similar is the time path of collaboration in this mathematical subdiscipline to the development of the whole? In addition, some subareas of logic itself will be compared to logic and mathematics as a whole. This gives a diachronic picture of collaboration. A synchronic comparison of the amount of collaboration in all about 60 major mathematical subdisciplines follows. Considered will be the world-wide output of mathematical papers with publication year 1985. This is compared with a similar investigation of the most important 19th-century mathematical areas. Again, this will be contrasted with physics. The empirical material will be the background for some considerations concerning the most commonspread explanatory concepts regarding collaboration. Before I come to the points I should make clear that I consider scientific collaboration as being indicated in coauthorships. Those coauthorships will embrace all forms of collaboration, from teamwork to loose forms of cooperation and contribution in all varieties. Despite coauthoring does not cover all cooperation, it will fulfill the purpose as a rough indicator.5

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Collaboration 1800 to presence I: mathematics and physics The expansion, institutionalization and professionalization of pure mathematics is a venture rooting in the 19th century. Were its literary output in the first years of the 19th century some dozens papers per year, this amounted to some hundreds of papers in the last years of the 19th century, growing to several ten thousands annual papers nowadays. Let us consider the amount of coauthoring in that abstract discipline. Data of 19th-century mathematics is based on the exploitation of the comprehensive Catalogue o f Scientific Papers delivered by the Royal Society of London.6 Data concerning 20th-century mathematics is based on the Zentralblatt fu r Mathematik und ihre Grenzgebiete which is a review journal covering mathematical literature in a world wide manner similar to the Mathematical Reviews, but beginning as early as 1931 and summing up to far over 1 million publications. I considered only journal papers, using the CD-ROM edition. As Figure 1 shows for the period of 1800 to 1998 in moving five-year averages, the level of coauthored papers remains with no more than two per cent in the first half and no more than one per cent in the second half of the 19th century extremely low. Not die slightest increasing trend can be observed (Figure 2). The period between 1900 and 1930 is not covered by the Catalogue or the Zentralblatt, respectively, but obviously exacdy in that period a distinct increase in the percentage of coauthored papers has to be supposed.

Year Figure 1. Percentage of papers with coauthors in mathematics, 1800-1998. Moving 5-year averages. Sum about 1.3 millions of papers. Sources: see text. Exact figures of all papers: See also Ref. 6,9.

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Figure 2. The same as in Figure 1, only for the 19th century

This strong growth continues in the next decades, and in the last years of the 20th century the threshold of 50 per cent is exceeded. We record that the time after World War II witnesses not a sudden increase of collaboration, at least not in mathematics, rather that the trend begins much earlier. And that modem mathematicians tend to work for their own or that mathematics is not suitable for collaboration seems to be a mere fable. Now I would like to compare these findings with the situation in another discipline which is, as I already mentioned, the paradigm for large-scale cooperative endevours in science, namely physics. To be exact I would like to detect whether the collaboration behaviour in 19th-century physics as reflected in coauthored publications shows early peculiarities and deviates from mathematics or whether both show similar low dispositions to collaborate. Again I exploited the Catalogue o f Scientific Papers, in this case the physics index which includes mechanics and embraces about 75,000 papers.7 Figure 3 shows the percentage of coauthored papers in moving five-year averages. There is no doubt that physics is a different case: Here a distinct growth of coauthoring begins in the second half of the 19th century. In the first half, however, there is, apart from some fluctuations, a more or less constant level of about 2 per cent of coauthored papers. I cannot present data for the 20th century here, but one can nevertheless conclude that the secular increase of collaboration in physics began not in the 20th century, not with World War II, but in the second half of the 19th century.

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Year Figure 3. Percentage of coauthored papers in 19th-century physics. Moving five-year averages. Sum: about 75.000 papers. Source: See text. Exact figures in Ref. 7

Collaboration 1800 to presence II: a mathematical subdiscipline Mathematics is a huge discipline with dozens of subdisciplines. How similar is the collaborative behaviour of a larger mathematical subdiscipline in the course of the last decades compared to the whole? Mathematical logic belongs to the most abstract and theoretical areas of a discipline which as such is rated as most abstract and theoretical. However, logic was a scientific nobody in the 19th century, amorphous stuff of some philosophizing mathematicians or mathematicizing philosophers, without any scientific or practical importance beyond those circles. It took several decades until logic reached institutionalization and an output of a hundred papers or so per year in the middle of our century. That was the time when logic suddenly gained unexpected usefulness and outraging importance: as a mathematical tool for constructing computing devices. Since when do the reasoners collaborate? At least long before World War II. Figure 4 shows the percentage of coauthored publications (including books in this case) of all logic publications from the beginning to 1990 in moving five-year averages. The data is based on an authoritative logic bibliography.8

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Year Figure 4. Percentage of coauthored publications (including books in this case) in mathematical logic from the beginning to 1990 in moving five-year averages. For comparison: Complete mathematics; constructive and intuitionistic logic. Exact figures in Ref. 8

As the graph reveals, coauthor publishing in logic began in the first years of the 20th century; before no coauthored publications at all appeared. For comparative purposes the graph for the total of mathematics is shown again. One can observe a roughly parallel development with logic on a lower level, following the overall trend in mathematics with a time lag of ten to fifteen years. Let us take a glance at an esoteric subarea inside the esoteric: constructive and intuitionistic logic. The graph shows an upspring in the pioneering years of that branch in the twenties, connected with the name of Brouwer. The activities seemed to have taken place in isolation since then until the sixties, where collaboration begins anew with a markedly increasing trend towards conformity to logic as a whole. One has to pay regard here to the interest of computer scientists in constructivistic and intuitionistic mathematical logic. Of course, after the comparison of time series of mathematics, logic, and intuitionism we cannot come to a general conclusion. The only general conclusion is that also the different parts of mathematics show an increasing trend to collaboration; and that the respective trends are similar, but not identical in their development. One can suppose that this might be transferred to all subdisciplines of mathematics. In the investigation of the following section, this idea will be supported indirectly.

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Collaboration in mathematical and physics subdisciplines: synchronic comparisons We now take a look at mathematics not from a diachronic, but from a synchronic perspective and want to compare the tendency to collaborate for all major mathematical subdisciplines. Here I prefer the Mathematical Reviews because its classification system is more suitable than the classification system of the Zentralblatt. All about 50,000 journal publications with publication year 1985 covered by the 1982-1987 CD-ROM version of the Mathematical Reviews are considered. The first two digits of the classification code lead to 60 subdisciplines. In Table 1, the names of the respective mathematical subdisciplines are listed together with the number of all papers of an area in publication year 1985; the second column gives the number of coauthored papers in per cent of all papers of the respective area. The areas are ranked according to their tendency to publish coauthored papers. Table 1 Papers with more than one author in mathematics, publishing year 1985 as percentage of all papers of that year (n = 31,600; bibliographical source; Mathematical Reviews, CD-ROM edition 1980-1987; only the main classification codes (“pc”) are considered) pu 894 117 423 737 1833 846 162 450 865 201 645 1001 232 809 50 1414 1662 1546 262 274 209 1200 501 181

% coau

cc

Field description

54.47 52.99 52.96 52.37 51.99 50.95 50.00 49.56 49.13 48.26 47.13 46.05 44.83 44.75 44.00 43.14 42.42 41.66 41.60 41.24 40.66 38.42 38.32 37.57

93 80 94 82 81 76 33 92 83 78 73 5 26 68 85 65 90 62 70 15 45 58 49 52

Systems, Control Classical Thermodynamcis, Heat Transfer Information and Communications, Circuits, Automata Statistical Physics, Structure of Matter Quantum Mechanics Fluid Mechanics Special Functions Biology and Behavioral Sciences Relativity Optics, Electromagnetic Theory Mechanics of Solids Combinatorics Real Functions Computer Science Astronomy and Astrophysics Numerical Analysis Economics, Operations Res., Programming, Games Statistics Mechanics of Particles and Systems Linear and Multilinear Algebra; Matrix Theoty Integral Equations Global Analysis, Analysis on Manifolds Calculus of Variation and Optimal Control Convex Sets and Geomtric Inequalities

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Table 1 (continued) pu 73 300 934 1572 250 678 87 1294 165 642 434 728 318 111 205 85 223 36 269 95 816 217 359 125 580 364 832 940 300 63 99 453 76 794 79 490

% coau 36.99 34.33 33.94 33.40 32.40 31.42 31.03 30.99 30.91 30.84 29.72 29.12 28.93 28.83 28.78 28.24 27.80 27.78 27.51 27.37 27.33 26.73 26.46 25.60 25.51 25.27 25.12 24.15 24.00 23.81 23.23 22.96 22.37 18.77 17.72 17.34

CC

Field description

44 22 34 35 6 53 86 60 55 30 16 47 41 18 28 8 13 4 51 12 46 17 14 39 54 42 20 11 57 40 31 32 0 3 43 1

Integral Transforms, Operational Calculus Topological Groups, Lie Groups Ordinary Differential Equations Partial Differential Equations Order, Lattices, Ordered Algebraic Structures Differential Geometry Geophysics Probability Theory and Stochastic Processes Algebraic Topology Functions of a Complex Varaible Associative Rings and Algebra Operator Theory Approximations and Expansions Category Theory, Homological Algebra Measure and Integration General Mathematical Systems Commutative Rings and Algebras Set Theory Geometry Field Theory, Polynomials, Algebraic Number Theory Functional Analysis (from MR year 1973 on) Nonassociative Rings and Algebras Algebraic Geometry Finite Differences and Functional Equations General Topology Fourier, Abstract Harmonic Analysis Group Theory Number Theory Topology, Geometry of Manifolds (nur MR-Jahre 1959-72) Sequences, Series, Summability Potential Theory Several Complex Variables and Analytic Spaces General Logic and Foundations Abstract Harmonic Analysis History and Biography

pu: number of publications; %coau: percentage of coauthored papers; cc: classification code

We can see that the first ranks are not hold by pure mathematics; e.g. (in the order of ranking): Systems and Control; Thermodynamics; Information and Communication, followed by a couple of areas of mathematical physics; then comes Real Functions; Computer Science and so on. Obviously, the upper ranks are occupied by subdisciplines of applied mathematics, whereas areas belonging to pure mathematics occupy lower ranks. However, the amount of collaboration in pure mathematics lies in a region between 35 per cent and under 20 per cent in the least collaborative areas.

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If one can conclude from the amount of publishing activities of areas to the amount of financial support - a conclusion that is not totally absurd at first sight small areas should reveal less collaboration. But it is obvious without computation that the correlation between the two rankings of Table 1 will be negligible. Let us go back to 19th-century mathematics and take a glance to a ranking list of 25 major areas of that blossoming science. We again take into account all journal articles based on the Catalogue o f Scientific Papers mentioned above. In Table 2 we can see that number one in collaboration - on an extremely low level, however - is the only area of those 25 fields amidst pure mathematics which could be called applied: Calculating Machines and Other [similar] Instruments.

Table 2 Papers with more than one author in 25 main areas of 19th-century mathematics (number of all 25 areas papers n = 36,565 i.e. more than 90% of all papers), as percentage of all contributions to that area (including coauthors contributions). Bibliographical source: mathematical index of the Catalogue o f Scientific Papers, 1800-1900. A hundred or so papers before 1800 are included, also some anonymous papers (minor differences to the database used in Ref. 6). To a small extent multiple classifications occurr. pu 418 2126 1776 520 1997 580 490 2236 3123 1777 2265 1059 363 618 1966 247 2839 1287 3258 2607 562 2289 857 1175 130

% coau 1.20 1.08 0.96 0.96 0.90 0.86 0.82 0.81 0.74 0.62 0.57 0.57 0.55 0.49 0.46 0.40 0.39 0.39 0.34 0.19 0.18 0.17 0.12 0.09 0.00

Field description with classification codes Calculating Machines and Other Instruments (0080) Transformations and Algebraic Configurations (8000-8100) Probability and Statistics (1630,1635) Foundations of Arithmetic (0400-0430) Geometry of Conics and Quadrics (7200-7260) Theory of Groups (1200-1230) Applications of Analysis to Physics (5600-5660) Algebraic Functions (4000-4070) Elementary Geometry (6800-6840) Linear Substitutions (2000-2070) Algebraic Curves and Surfaces of Degree > 2 (7600-7660) Foundations of Geometry (6400-6430) Differential Forms and Invariants (5200-5240) Theory of Functions of Complex Variables (3600-3640) Theory of Equations (2400-2470) Other Special Functions (4430-4470) Quadratic Forms (2830, 2840) Elements of Algebra (1600-1625,1640) Foundations of Analysis (3200-3280) Infinitesimal Geometry (8400-8490) Functions of Euler, Legendre, and Bessel (4410, 4420) Differential Equations (4800-4880) Universal Algebra (0800-0870) Differential Geometry (8800-8870) Philosophy (0000)

pu: number of publications; %coau: percentage of coauthored papers

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One should not think that in analogy to our present-day experience this area had some importance then; on the contrary, the level of publishing was extremely low with no increasing trend compared to other expanding areas of mathematics (cf. Ref. 6, p. 304), all this reflecting the negligible economic importance of calculating machines in the 19th century. Three areas bring up the rear in collaboration: The area of “Philosophy” of mathematics with no collaborative papers at all, next Differential Geometry and then Universal Algebra with just one collaborative paper each. Mathematical logic did not exist as an acknowledged field of its own as I already mentioned; rather, its ideas were partially hidden in the two areas of “Philosophy” and Universal Algebra. It took a hundred years until those isolated and seemingly anti­ cooperative occupations nourished the most important mathematical substance of computer science, which itself is the scientific substance of one of the leading technologies of our time, viz. calculating machines. Let us now inspect that logic subdiscipline again, now in a synchronic manner. How does a ranking list of all its major areas look like with concern to collaboration? In Table 3 one can see that number one in collaboration is an area with an enormous development since its invention some thirty years ago or so, namley fuzzy logic. It is followed by fields which seem to be strongly connected with computer science. Lowest collaboration seems to appear in logic fields with most distance to application. The difference between the most and the least collaborative fields is about five fold. Finally we want to investigate in a synchronic manner the collaboration intensity of 19th-century physics which we examined already in a diachronic analysis. In Table 4 all 53 major physics areas according to the corresponding index of the Catalogue o f Scientific Papers were listed again their tendency to publish coauthored papers. At first sight a tendency of application orientated areas to collaborate more intensively than theoretical areas seems to be visible. However, there are exemptions like the area of telescopes and microscopes. The clarification must be left to a physics historian.

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Table 3 Publications (articles and books) with more than one author in 64 main areas of mathematical logic 18741985 (n = 35,450 publications); multiple counts because of multiple classification codes; as percentage of all contributions of an area, including coauthors contributions. The percentages for the collaborative intensity would be somewhat higher if related to publications, not to contributions, which does not touch, however, the ranking between the areas. Bibliographical source: M(jller , Gert H. (Ed.), Omega-Bibliography of Mathematical Logic, Vol. 1-6, Berlin, Springer, 1987. Described in Ref. 8. pu 626 546 2248 1068 2591 1212 1788 1681 1058 277 351 888 1557 196 1182 3539 565 423 2300 660 369 173 1645 643 705 1064 347 1626 2301 795 1215 1332 715 628 1715 170 839 968 382 630 660

% coau 24.28 23.81 23.75 23.50 21.42 21.04 20.58 17.97 17.86 17.33 17.09 16.67 16.57 16.33 16.16 16.13 15.93 15.60 15.48 15.30 15.18 15.03 14.65 14.62 14.61 14.47 14.12 13.96 13.91 13.84 13.83 13.81 13.71 13.69 13.58 13.53 13.35 12.60 12.57 12.06 11.97

cc B52 D80 B75 E72 D05 B35 D15 E05 D10 CIO B51 E55 HXX F25 E50 GXX D45 C57 C05 D40 D03 B80 C60 C20 E45 B25 B46 D20 E75 C62 E35 E07 B40 C90 B50 D70 D35 D25 C25 C40 F35

Field description Fuzzy Logic Applications of Recursion Theory Logic of Algorithmic and Programming Languages Fuzzy Sets Automata and Formal Grammars Mechanization of Proofs and Logic Operations Complexity of Computation Combinatorial Set Theory Turing Machines Quantifier Elimination Quantum Logic Large Cardinals Nonstandard Models Relative Consistency and Interpretations Continuum Hypothesis Algebraic Logic Theory of Numerations Recursion-Theoretic Model Theory Equational Classes, Universal Algebra Word Problems Thue and Post Systems Other Applications of Logic Model-Theoretic Algebra Ultraproducts Constructibility Decidability Relevanz and Entailment Recursive Functions Applications of Set Theory Models of Arithmetic and Set Theory Consistency and Independence Results Relations and Orderings Combinatory Logic and Lambda-Calculus Nonclassical Models Many-Valued Logic Inductive Definability Undecidability and Degrees of Sets of Sentences Recursively Enumerable Sets Model-Theoretic Forcing Interpolation, Preservation, Definability Second- and Higher-Order Arithmetic

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Table 3 (continued) pu 531 643 293 1047 101 604 1113 973 147 700 1073 884 487 1344 778 2258 399 797 387 861 352 381 393

% coau 11.86 11.35 11.26 11.17 10.90 10.76 10.69 10.69 10.20 10.14 9.69 9.50 9.45 9.22 9.13 9.12 8.52 8.03 8.01 7.90 7.67 7.09 5.60

CC

B15 C35 D75 D55 F20 C75 B05 E70 C70 B10 F30 E25 B48 F50 B20 B45 F60 E30 B28 E10 B55 F05 F55

Field description Higher-Order Logic and Type Theory Categoricity and Completeness Abstract and Axiomatic Recursion Theory Hierarchies Complexity of Proofs Other Infinitary Logic Classical Propositional Logic Nonclassical Set Theories Logic on Admissible Sets Classical First-Order Logic First-Order Arithmetic and Fragments Axiom of Choice Probability and Inductive Logic Metamathematics of Constructive Systems Fragments of Classical Logic Modal and Tense Logic Constructive Recursive Analysis Axiomatics of Classical Set Theory Classical Foundations of Number Systems Ordinal and Cardinal Numbers Intermediate and Related Logics Cut Elimination Constructive and Ihtuitionistic Mathematics

pu: number of publications; %coau: percentage of coauthored papers; cc: classification code Table 4 Papers with mote than one author in 53 areas of 19th-century physics, as percentage of all contributions to that area (number of all 53 areas papers n = 90,109; cf. comments to Table 2; more multiple classifications in physics than in mathematics). About 80 cases pf papers with more than one author (about 2.3% of all coauthored papers) were excluded in the process of editing the database, containing the phrases “et alii” or “and other” pu 726 699 3907 2420 819 2212 1059 409 1151 5312 1993 1172

% coau

Field description and classification codes

13.36 10.01 9.70 9.46 7.44 7.14 7.08 6.84 6.42 6.28 6.27 6.05

Physics: 4240, Rontgen and Allied Radiation Physics: 1600-1695, Calometry and Specific Heat Physics: 4200-4275, Emission & Anal, of Radiation, Phosphoresc., Radioact., etc. Physics: 6800-6850, Electric Discharge Physics: 1200-1260, Thermometry Physics: 6200-6255, Electrophysics Physics: 1400-1450, Expansion and Stress Physics: 3400-3440, Velocity, Wave Length, etc., of Radiation Physics: 4225, Photochemistry and Photography Physics: 5600-5900, Electric Current and Conduction Physics: 5200-5270, Electrostatics Physics: 6600-6660, Electromagn. Waves, Gen. Electromagn. Theory of Light

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Table 4 (continued) pu 315 2110 2305 682 2057 4805 747 379 256 843 2829 3682 1353 1286 924 474 1822 2184 3538 792 2426 2895 1611 3933 4192 441 1172 1021 2397 3544 1403 1348 1573 2087 780 549 637 1217 194 1056 371

% coau 5.71 5.68 5.68 5.57 5.44 4.84 4.81 4.74 4.70 4.50 4.45 4.45 3.84 3.81 3.67 3.58 3.51 3.48 3.33 3.28 3.25 3.10 3.04 2.82 2.81 2.72 2.64 2.55 2.33 1.91 1.71 1.63 1.52 1.34 1.30 1.28 0.94 0.90 0.51 0.28 0.00

Field description and classification codes Physics: 1000-1014, Sources of Heat and Cold Physics: 1800-1940, Change of State Physics: 5400-5490, Magnetism Physics: 2000-2040, Thermal Conduction and Convention Physics: 3800-3875, Reflection, Refraction and Absorption of Radiation Physics: 6000-6090, Electrical Instruments and Apparatus Physics: 9400-9520, Physics of Music & the Sensation of Sound, Physiol. Acoust. Physics: 4230, Phosphorescence Physics: 3150-3165, Spectrum Analysis Physics: 9200-9340, Propagation of Sound, Methods of Analysis and Measurement Mechanics: 0100-0180, Measurement of Dynamical Quantities Physics: 0800-0845, Measurement of Dynamical & Mechanical Quantities, Elasticity Physics: 5610, Theory and Construction of Primaty Cells Physics: 2490, Theory of Heat Engines Physics: 4000-4050, Polarisation Physics: 4900-5000, Electricity & Magnetism, Gen. Dynamical Theory & Relations Mechanics: 3600-3670, Strength of Materials, Hardness, Friction, Viscosity, etc. Physics: 5600,6020,6450,6480,6485,6615, Telegraphy and Telephony Physics: 6400-6490, Electrodynamics, Special Phenomena Physics: 3600-3650, Interference and Diffraction Physics: 2400-2495, Thermodynamics Physics: 0100-0700, General Molecular Physics Mechanics: 3280, Principles of Construction, Resistance of Materials Mechanics: 3200-3290, Elasticity Physics: 3000-3100: Geometrical Optics Physics: 3200-3260, Optics of the Atmosphere Physics: 9000-9140, Vibrations Mechanics: 0180, 1220; Physics, 0700, Gravitation Mechanics: 2400-2540, Statics and Dynamics of Fluids Mechanics: 2790-2860, Hydraulics and Fluid Resistance Physics: 4400-4470, Physiological Optics Mechanics: 2820, Hydraulic Motors, Propellers, Pumps, Turbines Mechanics: 0400-0440, Geometry and Kinematics of Particles and Solid Bodies Mechanics: 1600-1650, Kinetics of Particles, Rigid Bodies, etc. Physics: 0600, 2990, 3400,6400,6410,6600, Ether Physics: 0500,0600,0700; Mechanics: 2450, Ultimate Physical Theories Mechanics: 2000-2100, General Analytical Mechanics Physics: 3080-3082, Telescopes and Microscopes Mechanics: 0000, Physics: 0000, Philosophy Mechanics: 1200-1270, Statics of Particles, Rigid Bodies, etc. Mechanics: 0800-0900, Principles of Rational Mechanics

pu: number of publications; %coau: percentage of coauthored papers

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Some concluding considerations If we compare different areas of mathematics of the 19th century, of 1985, of logic, and of physics with regard to collaboration we detect massive differences between the mathematical areas, between the logic areas, and between physics areas. It is hard to believe that the differences can be explained through differences in financial support which was suggested by some collaboration researchers. Even if we leave out the possibility that financial support of a field is reflected directly through the quantity of its publication output the influence seems not to be plausible. In pure logic, for example, with few exceptions no differences in financial support will exist; but marked differences in collaborativeness. Another concept often mentioned in connection with the increasing trend of collaboration is increasing specialization. But this remains only a slogan as long as it is not shown that, e.g., mathematicians working in Abstract Harmonic Analysis are far less specialized than mathematicians working in Lie Groups who collaborate twice as much as the former. Also the concept of professionalization seems not to be suitable here. A physics historian could better judge about the influence of large machines and other devices on collaboration in the 19th century. However, one trend at least was confirmed: If mathematicians approach reality through experiments they tend to collaborate; and in a similar manner, if logicians come into touch with working devices as computers, they also tend to collaborate. The heart of the often cited inter- or multidisciplinarity which leads to collaboration as it is maintained seems to lie in combinations of experimental or application oriented sciences, on the one hand, and theoretical sciences, on the other hand. It will be a long way until collaboration is understood in terms of the allocation of cognitive ressources and as concomitant of the cognitive division of labour. To be sure, there have to join further aspects: sociological, psychological, geographical, among others. Obviously, the development of such a systemic, comparative, time-sensible, and differential picture deserves collaborative efforts.

I thank Dipl.-Mathem. Dirk Kampjmeier for his invaluable help in building up bibliometric databases.

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References 1. D. deB. Beaver, R. Rosen, Studies in scientific collaboration, Part 1, The professional origins of scientific co-authorship, Scientometrics, 1 (1978) 65-84. 2. D. deB. Beaver, R. ROSEN, Studies in scientific collaboration, Part 2, Scientific co-authorship, research productivity and visibility in the French scientific elite, 1799-1830, Scientometrics, 1 (1979) 133-149. 3. D. deB. Beaver, R. Rosen, Studies in scientific collaboration, Part 3, Professionalization and the natural history of modem scientific co-authorship. Scientometrics, 1 (1979) 231-245. 4. M. A. H arsanyi, Multiple authors, multiple problems - bibliometrics and the study of scholarly collaboration: a literature review, Library & Information Science Research, 15 (1993) 325-354. 5. J. S. Katz, B. R. M artin, What is research collaboration? Research Policy, 26 (1997) 1-18. 6. R. W agner -DObler , J. B erg , Nineteenth-century mathematics in the mirror of its literature: A quantitative approach, Historia Mathematica, 23 (1996) 288-318. 7. R. W agner -DObler , J. B erg , Physics 1800-1900: A quantitative outline, Scientometrics, 46 (1999) 213-285. 8. R. WAGNER-DOBLER, J. BERG, Mathematische Logik von 1847 bis zur Gegenwart, Eine bibliometrische Untersuchung, de Gruyter, Berlin, New York, 1993. 9. R. WAGNER-DOBLER, Wachstumszyklen technisch-wissenschaftlicher Kreativitdt, Eine quantitative Studie unter besonderer Beachtung der Mathematik, Campus-Verl., Frankfurt/M., New York, 1997.

Received July 17,2001. Address fo r correspondence: R oland W agner -DObler

Institut fur Philosophie, Universitat Augsburg Universitatsstr. 10, D-86159 Augsburg, Germany E-mail: [email protected]

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