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In this essay it is shown how mathematical and computational approaches can be used to model the underlying mechanisms of historical processes, which transform the structure, dynamics and function of chemistry. By chemical knowledge, I refer to a complex dynamical system emerging from the interaction of the social, material and semiotic systems of chemistry. Besides instantiating some watershed events of the history of chemistry in this framework, the increasing availability of large datasets amenable to computational exploration is discussed, as well as the suitable mathematical theories to carry out these studies. I show how this framework allows for exploring possible alternative histories of chemistry by perturbing its past, leading to solving questions of the sort "what would have happened if." This not only sheds light on the past of chemistry, it rather allows modelling the future of the discipline, with its societal and pedagogical reaches. This approach complements conventional methodologies for the history of chemistry and becomes an interdisciplinary field of research for linguists, mathematicians, physicists, historians and chemists, to name but a few scholars and scientists.
Bull. Hist. Chem., VOLUME 47, Number 1: HIST Centennial (2022) 91
In this essay it is shown how mathematical and
computational approaches can be used to model
the underlying mechanisms of historical processes,
which transform the structure, dynamics and function
of chemistry. By chemical knowledge, I refer to a
complex dynamical system emerging from the inter-
action of the social, material and semiotic systems
of chemistry. Besides instantiating some watershed
events of the history of chemistry in this framework,
the increasing availability of large datasets amenable
to computational exploration is discussed, as well
as the suitable mathematical theories to carry out
these studies. I show how this framework allows for
exploring possible alternative histories of chemistry
by perturbing its past, leading to solving questions
of the sort “what would have happened if.” This not
only sheds light on the past of chemistry, it rather
allows modelling the future of the discipline, with
its societal and pedagogical reaches. This approach
complements conventional methodologies for the
history of chemistry and becomes an interdisciplin-
ary eld of research for linguists, mathematicians,
physicists, historians and chemists, to name but a
few scholars and scientists.
“The search for regularities in human history is becom-
ing a trie more respectable than it was formerly. That
could well portend some signicant improvement in
our ability to discuss the human future.”
—Murray Gell-Mann, 2017 (1)
Guillermo Restrepo, Max Planck Institute for Mathematics in the Sciences, Inselstraße 22,
04103 Leipzig, Germany;
1 Introduction
The charm of the history of chemistry lies in its
convincing, and often literary, narratives of past events
whose temporal paths have inuenced the evolution of
chemistry. History of chemistry shows how the backbone
of chemistry emerges from a multidimensional and noisy
dynamics of contingencies and certainties. Wonderfully
it teaches us about friendships, rivalries, cooperations,
academic-industrial alliances, professionalization and
technologies, which when embedded in changing social
and scientic contexts lead us to the chemistry of the
twenty rst-century.
History elaborates on the past but it should not be
forced to wait until events happen. Its niche lies in the
past, but it also includes the present and future, as well as
the possible pasts, presents and futures. History spans all
tenses. History of chemistry, therefore, ought not beguile
us only with the past of chemistry, but with its present
and future reaches. Moreover, given the key societal role
of chemistry, the history of chemistry endeavors with the
past, present and future of our civilization (2).
Tracing back the conditions leading to the discovery
of penicillin, or to the development of the Haber-Bosch
process is paramount, especially if we want to disen-
tangle the workings of innovation. Similarly, analyzing
the conditions facilitating the production and commer-
cialization of thalidomide or the formulation and use of
Napalm is extremely relevant to avoid repeating these
92 Bull. Hist. Chem., VOLUME 47, Number 1: HIST Centennial (2022)
disasters. These constitute a few examples of the enor-
mous contributions and responsibilities of the history of
chemistry. Addressing these questions entails that history
of chemistry, at its core, concerns itself with the search
for regularities leading to causal relationships (3).
However, most of the historical work, although il-
luminating, is far from the search for patterns. And this
has its own history, as expected. Carl von Clausewitz
(1780-1831) and Leo Tolstoy (1828-1910) believed that
historical processes were driven by some sort of law,
an idea supported by several nineteenth- and twentieth-
century historians (4, 5). Nevertheless, the concept of
history as a scientic enterprise in the search for patterns
lost popularity in the second half of the twentieth-century
(6); a movement epitomized by Karl Popper (1902-1994)
with his critics to search for historical regularities to
foretell the future. Currently, searching for regularities is
assumed as a task of the natural sciences, which deal with
far less complex systems than those of the humanities.
All in all, particles, atoms and genes lack free will, which
facilitates the detection of their patterns, while people
and their organizations are unpredictable and prone to
act upon contingencies. However, scientists are typically
busy working within their specialties and very seldom
venture beyond their disciplines to seek regularities.
Nevertheless, one would expect that chemists, having a
strong tradition of detecting patterns, could easily look
into their discipline and detect the relevant threads that
after being separated from noise would resolve into the
driving forces shaping chemistry. The fact of the matter
shows that historians are left alone in this central task.
In this essay I argue that Clausewitz and Tolstoy’s
belief in historical processes driven by patterns is not
only a reality but that the moment is ripe to undertake
these studies seizing upon present computational and
mathematical capacity to delve into the colossal corpus
of chemical information gathered to date.
2 History of Chemistry and the Search for
Historians are very good at devising narratives, often
involving causal relationships, where they weave the dif-
ferent dimensions of the historical subject under study. It
is gripping to nd how nineteenth-century organic chem-
istry beneted from the strong academe-industry relation-
ship in Germany (7); how professional rivalry hindered
the recognition of watershed chemical constructions or
theories, for example the frictions between Arrhenius
and Mendeleev or between the former and Nernst (8).
History contributes to understanding how the deluge of
organic chemicals in the rst quarter of the nineteenth-
century led to devising the molecular structural theory
(9), disciplinarily so rooted in our chemical minds. It is
fascinating to nd that Guyton de Morveau, Lavoisier,
Berthollet, de Fourcroy, Hassenfratz and Adet’s revolu-
tionary nomenclature (10) was motivated by philosophy
of language, which can be traced back to Leibniz (11).
These and several other remarkable developments in
the history of chemistry evidence how chemistry and
its knowledge have been driven by material, social and
semiotic aspects of the discipline (2).
Then, in its broadest sense, the history of chemistry
entails the temporal analysis of events leading to the
status of what has been regarded as chemistry in a given
time, this latter discipline understood as the science
devoted to transforming matter and theorizing upon it.
The events of interest for the history of chemistry involve
complex social, semiotic and material factors whose in-
teraction is driven by contingencies and regularities. The
question that arises is about the essence of these events.
What is the currency of the history of chemistry? I claim
it is chemical knowledge.
2.1 Chemical Knowledge, the Currency of History
of Chemistry
Jürgen Renn has framed the history of science within
a broader history of knowledge (12). This entails consid-
ering the history of science as the study of knowledge and
of its evolution. In this setting the different dimensions
of historical events are integrated into the new object
of study: knowledge. Following Renn, Jürgen Jost and
I posited that the history of chemistry entails analyzing
the evolution of chemical knowledge (2). But, what is
knowledge and what is specically chemical knowledge?
Humans accumulate experiences, which they cogni-
tively structure allowing for predictions of new experi-
ences (13). Knowledge entails developing those cogni-
tive structures and predicting, as well as the feedback
resulting from predictions (12, 14). When predictions
are realized, new experiences are added to the cognitive
structures and the predictive method is strengthened.
Otherwise, new experiences falling outside the scope
of the initial experiences are used to tune the predictive
model, while enlarging the cognitive structures. There-
fore, knowledge is a dynamical process and it depends on
social constraints, such as economic and political interest;
as well as cognitive frameworks, which involve theories
and meaning generation. Knowledge is stored, shared
Bull. Hist. Chem., VOLUME 47, Number 1: HIST Centennial (2022) 93
and transmitted across generations. Thus, it requires a
material system to be preserved and spread, for instance
through signs and text (2).
Chemical knowledge involves the cognitive struc-
tures generated to make sense of the experimentation
upon matter transformations. These structures are either
used by chemists to estimate future outcomes of their ex-
periments or to modify their cognitive structures in such
a manner that they span the new experimental ndings.
As chemists are embedded in social frameworks, which
vary over time and which determine particular ways of
thinking (15), their cultures and semiotic systems inu-
ence chemical knowledge, which also depends on the
materials and technologies available for exploring matter
transformations (2).
In our account of the evolution of chemical knowl-
edge we claim that it can be modelled as a complex dy-
namical system made of at least three interacting systems.
These are the semiotic, material and social systems (2).
As typical systems, they are made of objects and their
relationships (16, 17). The objects of the social system
include people, academic and scientic societies, com-
mittees, enterprises, industries and other forms of social
organization, plus computational objects such as robots
and articial intelligence technologies. These social ob-
jects are held together by economic, political, cultural,
academic and other relations (2). The semiotic system,
following Peirce’s distinction among objects, signs and
interpretants (18, 19), involves substances, reactions and
other concepts of chemistry along with their historical
representations (signs) that chemists (interpretants) have
associated to them (2). These semiotic objects are related
by the ternary relation object-sign-interpretant of Peirce’s
semiotics and by the high-order relations resulting from
their combinations (20). The material system is made
of substances, reactions, technologies and apparatus, as
well as the relationships they establish in the chemical
practice (2, 21). Our setting is that chemical knowledge
arises from the mutual interaction of the semiotic, social
and material systems of chemistry. Chemical knowledge
is an emerging property and history of chemistry involves
the analysis of the dynamics of this complex object.
So far this does not bring anything new, it sounds
just like the use of complex systems jargon to describe
what historians of chemistry have been doing for over a
century. Nonetheless, this setting brings new possibilities
for the history of chemistry. The theory of complex sys-
tems regards the emergence of macroscopic phenomena
as resulting from multiple relations caused by simple
dynamical rules (22), which are often detected through
the patterns they form (23). Hence, if by studying past
events of chemistry, we detect rules or patterns leading
to the rules, we will be in a good position to make pre-
dictions and even retrodictions, this latter understood as
“predictions” about the past.
The most realistic case of the two mentioned is that
of detecting patterns in a large corpus of chemical infor-
mation about the social, material and semiotic systems of
chemistry. If that happens, we have good reasons to think
there is an underlying rule driving the historical process.
The other case is a bit more difcult, as it involves the
direct detection of the rule driving the complexity of the
evolution of chemical knowledge. In any case the rule is
accepted or rejected insofar as it reproduces the patterns
of chemical knowledge.
The rst case, that is from patterns to rules, leads
to modeling. If we nd a pattern in the evolution of
chemical knowledge, we may devise a model and let it
evolve over time. If the resulting pattern obtained from
the model matches that of the historical evolution of
chemical knowledge, we have good reasons to take the
model as encoding the driving force of the pattern (24).
The second case, from model to pattern, is a derivation
of the rst one. This requires the direct evaluation of the
validity of the model by contrasting it with the historical
Thus, considering chemical knowledge as a com-
plex dynamical system leads to a model, which brings
the history of chemistry to new reaches. It allows for
estimating the future outcomes of the observed pattern.
Interestingly, it also allows for retrodictions by running
the model on arbitrary pasts and letting it predict events
occurring afterwards but still in the past (2). This is
particularly suitable to solve questions of the sort, “what
would have happened if.” One could ask, for instance,
what would have happened with the material and social
systems of chemistry, and with chemical knowledge
in general, if Berzelius had not devised its notation of
empirical formulae. What would have happened if the
pre World War I conditions of chemical knowledge had
been maintained for longer (25).
In short, the history of chemistry involves the detec-
tion of historical patterns in the evolution of chemical
knowledge, which we may model as a complex dy-
namical system arising from the mutual interaction of
the semiotic, material and social systems of chemistry.
The question that now arises is how to do it. I argue
that these patterns are to be detected by analyzing large
94 Bull. Hist. Chem., VOLUME 47, Number 1: HIST Centennial (2022)
corpora of historical chemical data through mathematical
and computational tools.
3 The Necessity of Mathematical and
Computational Methods for Studies in the
History of Chemistry
Computational approaches to the history of science
have been recently recognized as complementary to the
practice of history and sociology of science (26-29). As
recently noted by Abraham Gibson, Manfred D. Laubi-
chler and Jane Maienschein (30), history, as a discipline,
is currently undergoing a computational revolution. In
recent years Isis has published some papers on “the com-
putational turn,” “the computational revolution” and “the
electronic information revolution” (31-33) and in 2019
the journal dedicated a focus section to “Computational
history and philosophy of science” (30). In 2017 the an-
nual issue of Osiris was devoted to “Historicizing Big
Data” (34). In turn, in the American Historical Review
“the digital revolution” and “the digitized revolution”
has been discussed (35). Despite the pros and cons of
computation in the practice of history, addressed in the
above references, it is clear that “digital sources and
computational tools have transformed how we engage
with the historical record, including the history of sci-
ence” (30).
What does this computational turn offer for the
history of chemistry and for the search for historical pat-
terns? Computational approaches allow for processing
large amounts of historical data and, when coupled with
mathematical and statistical methods, for detecting the
sought-after historical patterns, if they actually exist. It
is important to note that these patterns are not observable
by traditional history of science methods, which are often
restricted to analyzing periods spanning decades and cov-
ering specic geographical regions. Furthermore, such
studies typically rely on fewer than hundreds of primary
and secondary sources, only some of which are digitized
(36). In contrast, datasets for computational studies
depend upon millions of digitized records spanning
centuries and often the whole globe (37). This change
of scale signals a change in the kinds of analysis offered
by computational approaches to the history of chemistry.
Furthermore, I propose that mathematical ap-
proaches to the history of chemistry promote insights
that are otherwise unsupported speculations, or simply
unavailable because they are out of reach. For example,
the proposal that chemical knowledge is a complex dy-
namical system involving social, material and semiotic
components can be elaborated by employing a suite of
mathematical theories for pattern formation, evolution
and adaptation, nonlinear dynamics, as well as systems,
network, game and collective behavior theories (38). I
discuss some particular instances of these theories in
section 5. But the central point of mathematical methods
for the history of chemistry goes beyond the use of math-
ematical theories as “canned” tools of straightforward
application (2). Rather, I posit that close interactions
between chemists, historians, and mathematicians gener-
ates fruitful interdisciplinary work while also stimulating
some surprising insights into the history of chemistry
(39), a topic further discussed in section 6, below. Some
of the aforementioned theories are instrumental for de-
tecting patterns and analyzing statistical properties. The
products of such combined mathematical methods allow
us to glimpse the future of chemical knowledge.
A case study of the mathematical and computational
approach to the history of chemistry follows.
4 The Evolution of the Chemical Space
Recent studies of the growth of the chemical space
provide a case study of patterns in the history of chem-
istry (40). By chemical space I mean the substances
reported over the history of chemistry, which have been
extracted or synthesized by chemists, apothecaries, phar-
macists, metallurgists and other chemistry practitioners
(2, 41). These substances are endowed with a notion of
nearness by chemical reactions and therefore constitute a
mathematical space (42). Hence, one talks of substances
that are closely related by very few synthetic steps, in
contrast to the majority of other substances which are
far apart in our current knowledge of possible synthesis
plans connecting them. Note that the notion of nearness
is arbitrary, as one could select other criteria of nearness.
For example, substances can be characterized by their
molecular structures and their nearness can be determined
by the resemblance of their structures (43).
In any case, the chemical space is an important con-
cept leading to new questions. For example: how does
it grow? How rapidly? How are its dynamics affected
by social perturbations such as wars or pandemics? Is it
perturbed by semiotic changes? Moreover, can we model
its evolution? Which are the rules driving its dynamics?
In 1963 de Solla Price briey discussed the annual
report of some chemical starting materials and the growth
of chemical elements (44). The rst complete account
Bull. Hist. Chem., VOLUME 47, Number 1: HIST Centennial (2022) 95
of the growth of the chemical space was reported by
Joachim Schummer in 1997, who analyzed the period
1800-1995 by manually screening the indexes of eight
printed sources, including handbooks of organic and in-
organic chemistry (45). He found an exponential growth
with an annual growth rate r = 5.5%, indicating a dou-
bling time of about 13 years. A further study analyzed
the growth of organic substances by computationally
treating the Beilstein database for the period 1850-2004
and an exponential growth was also found, with r = 8.3%
before 1900 and r = 4.4% afterwards (46).
In a more recent account, we analyzed 16,356,012
reactions and 14,341,955 substances published between
1800 and 2015 in chemical journals, gathered in the
Reaxys electronic database. We found that the chemical
space has historically grown at an exponential rate, with
a stable growth rate (r = 4.4%) (40). This indicates that
about each 16 years chemists have doubled the number
of new substances reported. The speed of this rapid
chemical production can be expressed in these terms:
the number of new chemicals reported by the chemical
community in 2015 roughly amounts to all substances
reported between 1800 and 1992. That is, in a single year
of contemporary chemistry, chemists produced the same
number of new substances as reported in 192 years of the
history of chemistry. This is the dramatic speed at which
the chemical space grows (47)!
In our model of chemical knowledge as a complex
dynamical system, we claim that chemical knowledge is
driven by the mutual interaction of the semiotic, social
and material systems of chemistry. Evidence of these
interactions, at least in their binary forms, have been
already reported and discussed in (2).
We found that the expansion of the chemical space
has been affected by social setbacks such as World Wars
(WWs) (40). We observed two drops in chemical produc-
tion around WWs and quantied the effect of these events
in the annual output of new chemicals. It was found that
WW I sent back chemical production 37 years, while WW
II 16 years. The dramatic effect of WW I follows from the
centralized structure of chemistry in the rst quarter of
the twentieth century, whose capital was Germany. WW I
actually motivated a restructuring of chemical industrial
and research production, prompting a decentralization
in which the USA began to take the lead. By the time of
WW II, the social system of chemistry had changed and
production was sufciently less centralized such that, to
a large extent, WW II did not affect the annual output
of new chemicals.
An interesting semiotic event leading to changes in
the material system of chemistry was the introduction of
the molecular structural theory (2). This theory solved a
semiotic crisis, mainly driven by the deluge of organic
substances brought about by the improvement of ana-
lytical methods of the early nineteenth-century (48, 49).
Before chemists gained control over organic substances,
their extractions and synthesis, chemical knowledge was
mainly driven by inorganic chemistry (49, 50). Therefore,
its semiotics and theoretical structure was tailored to
these substances. In this period, for instance, the Berze-
lian dualistic theory became a paper tool to understand
the chemical space and to keep expanding it (48).
At any rate, the 1830s and 1840s brought a grow-
ing number of organic substances, which challenged the
dualistic formulae of Berzelius and the way of thinking
these formulae incorporated in the chemistry of the
rst half of the nineteenth century (2, 48). Finding a
growing number of very different substances with the
same composition was totally unexpected, for instance.
The introduction of the molecular structure as a new
semiotic object of chemistry added a new dimension to
the Berzelian algebra of formulae and to the Lavoisian
concept of classes of substances (2). Chemists had a new
and powerful paper tool at their hands, a topological one
where elements hold relationships through chemical
bonds. Molecular structures added to the visual character
of chemistry and, as Berzelian formulae, became a way
of thinking for expanding the chemical space (2).
Our data-driven study of the evolution of the chemi-
cal space, besides revealing a stable historical growth,
one not halted by social setbacks such as WWs (51),
showed other patterns and sharp transitions. This is a
further instance of a result attainable only by applying
mathematical and computational tools to vast corpora
of chemical information. By analyzing the variability
of the annual output of new chemicals, we found three
clear statistical regimes of chemical production, the rst
one spanning the period 1800-1860, the second running
between 1860 and 1980 and the third and present one
beginning in 1980 (40).
Typically, growth studies involve analyzing growth
as the slope of the growth curves. However, in our in-
terdisciplinary work, we were fortunate enough to bring
mathematicians to our study, for example experts in time
series analyses, who indicated to us the importance of the
variability of the signal. Duc H. Luu found three statisti-
cal regimes where the variability of the annual output of
chemicals has been normally distributed (40). These are
96 Bull. Hist. Chem., VOLUME 47, Number 1: HIST Centennial (2022)
the three statistical regimes just mentioned, which are
characterized by an emphasis on inorganic chemistry
before 1860, where the variability of the annual output of
new chemicals was the highest in the history of chemistry.
This indicates an exploratory regime of the chemical
space, where some years brought several new substances
and some others not so many. It is clear that the size of
the chemical community also played a major role.
By 1860 there was a drastic reduction of the vari-
ability, which we argued was caused by the widespread
introduction of the structural theory that regularized
chemical production, assisted by an ever growing
chemical community, which was beneting from the
positive public image of chemistry and of its important
chemical industry (9). The interplay between a power-
ful chemical theory and the social conditions led to
regularize the annual output of chemicals, which were
mostly organic substances. This trend lasted more than
a century—impressively, with two WWs in between! It
is noteworthy that WWs did not delay chemical produc-
tion as observed in other disciplines according to their
bibliographic production (44), but rather caused a drop
in production followed by a rapid recovery after WWs,
leading chemical production to pre-WWs trends (40).
This is what Schummer has dubbed as a catching-up
phenomenon (45). Although he has explored the role of
the size of the chemical community in this effect, further
work is needed to gauge the dynamics of these postwar
A second sudden reduction of the variability of the
annual production of chemicals was observed around
1980, but it is still an open question as to the leading event
or collection of events causing this further regularization
of the variability. It could be caused by the increasing
computerization of the chemical practice or a delayed
effect of the widespread adoption of spectroscopic chemi-
cal instrumentation that took place in the 1950s. At any
rate, this third regime of chemistry was characterized by
a revival of compounds containing metals, specically
organometallics, followed by a surge of substances of
biological interest (40).
Having discussed the mathematical and computa-
tional analysis of the evolution of the chemical space, I
now turn to discuss the data and methods we have, and
we expect to have, to further extend the computational
history of chemistry.
5 Data and Methods
Chemistry is the science with the largest output
of publications (2) associated to its material practice.
Therefore, it is not short of data. Moreover, chemists have
developed a strong tradition of data curation, annotation,
storage and dissemination initiated by the encyclopedists
of the thirteenth century such as Batholomaeus Anglicus
(before 1203-1272), Vincent of Beauvais (c. 1190- c.
1264) and Albertus Magnus (1193-1280) (49) and
continued by towering nineteenth-century gures such
as Leopold Gmelin (1788-1853) and Friedrich Konrad
Beilstein (1838-1906). The efforts of these pioneers, plus
the colossal amount of new chemical data of the twen-
tieth and twenty-rst century are today at our ngertips
through electronic databases, which besides being a
source of chemical information, constitute an important
corpus of historical information.
A well organized electronic database of proven use
for history of chemistry is Reaxys, owned by Elsevier, and
a further one is SciFinder hosted by the American Chemi-
cal Society. These sources offer important information
about the material system of chemical knowledge, which
includes substances, reactions, substance properties and
reaction conditions. This data is also associated to authors
and to their afliations. Hence, these databases may be
used as a source of data for the social system of chemistry
as well. As most of the reactions stored in these sources
contain details about how the reactions were performed,
these databases contribute to the corpus needed to explore
the semiotic system of chemistry (2).
Despite the advantages of the chemical databases,
historians still lack a well organized and curated database
of information about the semiotic and social systems
of chemistry. In (2) we mention some reliable sources
which could be used to build up such historical data-
bases, which include other electronic databases such as
the ISI Web of Knowledge and Dimensions. However,
these databases do not convey a complete picture of the
social and semiotic systems. The methods of historians
are needed to collect, curate and digitize, for instance,
membership in chemical societies, chemical industries
and registration records of academic institutions, if the
aim is building up a complete database for the semiotic
and social systems (2).
Perhaps the most challenging system is the semi-
otic one, because gathering its data requires develop-
ing standards for selecting and curating information.
For example, it is important to dene what counts as a
Bull. Hist. Chem., VOLUME 47, Number 1: HIST Centennial (2022) 97
diagram, table or reaction scheme in historical records.
This entails going beyond a formal denition as it has
to account for the natural evolution of these terms and
concepts. A further question concerns the storage of these
data: should they be saved as images? Or is it better to
convert them into a machine readable format regardless
of whether it is human readable (52)?
Here we must briey examine the history of codi-
cation of molecular structures and reactions. In the 1950s
chemists started to ponder the problem of computational
encoding of molecular structures (53), which led to con-
nection tables, later on to SMILES and today to InChIs (2,
54). These frameworks were mainly adjusted to encode
organic molecules, therefore presenting difculties for
encoding organometallic or inorganic compounds (2).
Moreover, they are based on the “chemistry” of the mol-
ecules, as encoded in their molecular structures, which
disregards several dimensions of their semiotic load.
For instance, the intention of the chemist who draws
one particular shape, among many possible alterna-
tives to represent a molecule, is lost in these encodings.
Fortunately, the growing computational memory and
processing capacity is making it possible to store and
process molecular structures as images (55), which are
coupled with machine learning algorithms to advance
chemical knowledge, nevertheless mainly concentrated
on the material system. We consider this an opportunity
to use current computational power and algorithms to
collect chemical information of relevance for the semiotic
system of chemistry, and, in general, for the evolution of
chemical knowledge.
In (2) we discuss different computational and
mathematical methods we nd appropriate for stud-
ies on the history of chemistry. I have mentioned how
time series analysis becomes a powerful tool to analyze
historical data. The quality of results obtained through
these methods depends to a large extent on the temporal
resolution of the data. In section 4 we show the applica-
tion of these methods to data with annual resolution.
Current publishing speeds and editorial policies are
making it possible to think about continuous sources of
data, which will sharpen the possibilities of time series
analysis methods (56). A time series analysis study takes
a random temporal signal allowing for studying temporal
trends and forecasting (57), as well as detecting changes
and evolving behaviors (58). This technique has been
recently recognized by historians of science as a suit-
able method to assist their narratives and to nd causal
relationships (59).
A powerful collection of mathematical techniques
is found in statistical physics that involves the dynamic
behavior of mathematical structures, such as graphs and
hypergraphs (60-62), which I further discuss in section
6. In general these structures are suitable models for
chemical knowledge and its constitutive systems. The
general idea is to dene a set of objects of study and
some relations among them of interest for historical
study. Hence, for the material system one may think about
analyzing how substances relate to each other through
chemical reactions and how that structure of reactions
has evolved over time. Likewise, one may consider the
dynamics of the social structure of chemistry relating
people and institutions as well as the temporal behavior
of the connections between concepts and other semiotic
tokens of chemical knowledge. Graphs, and in general
hypergraphs, are perfectly suited to study the dynamics
of the relationships between the systems of chemistry
from which chemical knowledge emerges (2).
Another important tool is agent-based modelling,
which is used to model simple local interaction patterns
and to understand the emerging global complex dynam-
ics. Thus, these models are especially appropriate for
understanding the evolution of chemical knowledge.
Besides allowing for estimations, agent based models
become relevant to study retrodictions, for example,
of competing narratives in the history of chemistry (2).
Text analysis tools as well as natural language tech-
niques become important parts of the computational tools
for the history of chemistry. These techniques allow for
detecting concepts and topics of importance throughout
the evolution of chemical knowledge which can be ap-
plied, for instance, to treat large corpora of chemical
abstracts and reaction details. Therefore, these techniques
are applicable to each one of the systems of chemistry
(social, semiotic and material) as well as to our explora-
tion of their mutual interplay (2).
A further tool historians can prot from is machine
learning, which runs through large databases allowing for
predictions, for example through regression techniques,
or for classications. These algorithms also involve rein-
forcement learning, where the algorithm make decisions
based on rewards that it intends to maximize (2).
Bayesian networks constitute a powerful tool for
the practice of history of chemistry as they provide a
formal setting to explore causal relationships (63), of
utter importance for weaving historical narratives. In this
setting a time series signal is perturbed and the temporal
propagation of the perturbation is analyzed (2).
98 Bull. Hist. Chem., VOLUME 47, Number 1: HIST Centennial (2022)
Some other computational tools and mathematical
settings are discussed in (2), but nothing restricts the use
of others and the development of new ones motivated by
questions from the history of chemistry.
6 Mathematics Triggered by Research on the
History of Chemistry
Two questions that computational history of chem-
istry may address, which I have so far not discussed, are:
can we model the evolution of chemical space? What are
the rules regulating its dynamics? The answer to the rst
question is “yes, we can” and we are working on gauging
the right model for the evolution of the chemical space.
With information about the use of substances to expand
the space we can propose statistical models based on
the dynamics of chemical reaction networks, described
as hypergraphs. In (40) we analyzed whether there are
statistical patterns in the way chemists combine their
substrates in chemical reactions. We found that although
a large part of chemistry is exploratory, that is chemists
combine new chemicals with other new chemicals to
produce novel substances, there is also a high degree of
conservatism in the expansion of the chemical space.
We dubbed this trend as the x substrate approach and
it entails reactions of very well known substrates, such as
acetic anhydride, with new substances (64). Thus, as we
know the statistical patterns of the annual participation
of substrates in reactions, we can model the expansion of
the chemical space. The estimated space can therefore be
contrasted with the actual chemical space as recorded, for
instance in Reaxys. The mathematical setting to encode
participation of substances in chemical reactions is that
of hypergraphs, which may be regarded as belonging to
the eld of network science. What follows is a detailed
discussion of how these mathematical structures afford
this encoding and how history of chemistry questions
have triggered mathematical research in this area.
The success of network theory as a suitable model
for systems of an ample range of disciplines has not left
chemistry untouched. Collaboration networks and rela-
tionships among disciplines are but a few examples of
uses of networks to analyze cases of chemical interest. An
interesting case at the core of chemistry is Schummer’s
claim that the logical structure of chemical knowledge
corresponds to a network of substances related through
chemical reactions (65). This is what today is dubbed as
a chemical reaction network. Graph theory has become
the mathematical setting of choice for modeling networks
(66). In a graph (or network), the objects of the system
under study constitute the vertices (or nodes) of the
network and the different relationships among objects,
the edges of the network. A familiar graph-like chemi-
cal concept is that of molecular structure. In this setting,
atoms and bonds, respectively, correspond to vertices (V)
and edges (E) of the graph, which embodies the molecular
structure. A graph G is the couple (V, E). As in a typical
molecule, where pairs of atoms are bonded, in a graph,
edges connect pairs of vertices (67).
Not surprisingly, graphs have been used to model
networks of chemical reactions (46). However, Klamt,
Haus and Theis showed in 2009 that these structures miss
an important piece of chemical information when used
to model chemical reactions (68). As it is well known,
reactions typically entail sets of substances. That is, a
collection of substrates, solvents, catalysts, which are
typically heated and stirred to end up with a mixture
of products, which are later on diligently separated and
analyzed (69). Klamt, Haus and Theis’s argument is that
the essential information of a chemical reaction is the
AND connector for some substances. We say, A reacts
with B to produce C and D, for example, as the result of
an experiment (70). Here the important information is
that A AND B react together to produce C AND D. If we
model this reaction through a substrate-product graph,
where substrates are related with products in a directed
fashion, that is with arrows from every substrate to each
product (71), then we obtain the following relations:
A → C, A → D, B → C and B → D. Klamt, Haus and
Theis show that this model does not allow for deducing
that A AND B are the substrates of the reaction. This
occurs because there are further possible interpretations
of this model such as that A decomposes into C AND D.
The same can be said for the decomposition of B. We
can also infer that A AND B react together to produce C
through an addition reaction and that under other reac-
tion conditions they produce D. This lack of accuracy of
the model, or of several interpretations, multiply as the
network grows, that is as more and more reactions are
concatenated in the reaction network.
Klamt, Haus and Theis provide the solution to
this conundrum by highlighting the possibilities of hy-
pergraphs to gauge the essence of the AND relation of
chemical reactions. The basic idea is that reactions simply
relate sets of substrates with sets of products. Hence, we
talk of {A, B} → {C, D} as a suitable reaction model
emphasizing the direction from the left hand side set
of substrates to the right-hand side set of products. The
main difference between graphs and hypergaphs is that
while graphs depict only relations between pairs of ele-
Bull. Hist. Chem., VOLUME 47, Number 1: HIST Centennial (2022) 99
ments, hypergraphs do it for sets of elements of any size.
As seen, hypergraphs are a novel concept in chemistry
and it is also of recent appreciation in mathematics (72),
where the attention has been mostly on graphs and their
mathematical and statistical properties (73). Nevertheless
the further study of these properties for hypergraphs is
gaining momentum, in part, motivated by efforts to model
large networks of chemical reactions. Interestingly, hy-
pergraphs are a generalization of graphs in such a manner
that all that is known about graphs should be reduced to a
single case of a general theory of hypergraphs (74). This
is an example of mathematics triggered by the study of
the history of chemistry. In fact, the study of directed
hypergraphs, such as chemical reactions, is just beginning
and constitutes a niche of vibrant mathematical research
motivated by chemical questions (75).
These mathematical studies may shed light not only
on the very concept of chemical space but on the whole
of chemical knowledge, as we have argued that the most
suitable model for studying the interaction of the three
systems of chemical knowledge is a hypergraph. The
dynamics of hypergraphs allows for exploring arbitrary
moments in the temporal unfolding of chemical knowl-
edge, that is, they lead to predictions and retrodictions,
which I discuss in the next section.
7 Predictions and Retrodictions
Detecting statistical patterns in the history of chem-
istry enables the historical endeavor of the chemical
practice to extend beyond the past by allowing extrapo-
lations. It further allows perturbing the past to come up
with alternative pasts, presents and futures of chemistry.
All in all, statistical patterns in the history of chemistry
allow for simulations, predictions and retrodictions.
In 2005 Grzybowski and his team modelled chemi-
cal prices based on the use the chemical community made
of substances (46). They found that the price of a chemi-
cal rapidly decreases with both, the number of synthetic
ways of producing it (kin) and the number of uses the
substance has as starting material of other chemicals
(kout). In fact, the cost follows a power-law distribution
that is proportional to k−ν, with k being either kin or kout
(46). This research was conducted with a fraction of the
known chemical space, namely the organic chemical
space between 1850 and 2004. Hence, these trends apply
to prices of organic chemicals. The same team found that
chemists have traditionally used organic substrates of
150 g mol−1, on average, to produce substances with an
average of 250 g mol−1 (in 2004) (46). This pattern may
shed light on the large scale trends of organic chemistry.
However, further statistical tests need to be conducted to
analyze the reliability of possible estimations.
In section 4 we discussed an important pattern in
the evolution of the chemical space, namely that for
more than 200 years the growth rate of the chemical
space has been stable, with doubling times of about
16 years and with variabilities that have been reducing
over time. This trend allows us to predict an expansion
of chemical space at a similar pace (76). This leads
us to think that an extrapolation is possible and that
eventually we could know when, for instance, a specic
number of new chemicals would be afforded. However,
even if the present variability of the annual output of
new substances is the lowest, still the uncertainty of the
estimations is high (40). This result shows two important
things: i) that extrapolations must be supported by sound
statistical analyses to avoid oversimplications and ii)
that estimations of the expansion of the chemical space
are complex, presumably calling for information from
the social and semiotic systems of chemical knowledge.
This latter point highlights the relevance of building up
strong social and semiotic databases to foresee the future
of the discipline.
A further possibility for computational history of
chemistry lies in the retrodictions it permits. This basi-
cally entails perturbing past historical data to observe
the effects upon subsequent events of that arbitrary past.
We conducted a retrodictive study when analyzing the
inuence of the chemical space upon the evolution of the
periodic system (77). In that study we took the known
chemical space between 1800 and 1869, with annual reso-
lution, and by taking several systems of atomic weights in
vogue in that period, we perturbed the chemical space to
obtain the possible periodic systems that several leading
chemists such as Dalton and Berzelius may have devised
if they had attempted to do so with the chemical space at
their disposal (78). We also analyzed how the chemical
space by the time of Gmelin, Meyer, Odling, Hinrichs
and Mendeleev inuenced the systems they reported
(79). This allowed us to calculate the false positives and
true negatives rates for these chemists by contrasting the
periodic systems obtained from the chemical space and
those reported by them.
8 Conclusions and Outlook
The increasing amount of data and of computational
power are turning computational approaches into an in-
tegral part of the historians’ tools, which when combined
100 Bull. Hist. Chem., VOLUME 47, Number 1: HIST Centennial (2022)
with mathematical insight open new possibilities for the
practice of history of chemistry. Here I have shown how
computational history of chemistry provides new ways
to solve historical questions and allows for asking and
solving novel questions related to large scale historical
Arguing that the currency of the history of chemistry
is chemical knowledge, I show how a computational his-
tory of chemistry sheds light on the evolution of knowl-
edge by regarding it as a complex dynamical system
made of the mutual interaction of the social, semiotic
and material systems of chemistry.
Different sorts of data available to undertake studies
belonging to the setting here presented are analyzed, as
well as several methods and theoretical frameworks as-
sisting those studies. Likewise, I discuss the rich sources
of information for the material system of chemistry,
vastly documented in chemical databases, and I empha-
size the lack of databases for the social and semiotic
system. I argue that this is actually an opportunity for
the history of chemistry, rather than a limitation, where
historians can decide on the sorts of data relevant to
curate and to preserve as well as their formats. Clearly,
historians cannot create these databases alone; they will
require the support of computer scientists and chemists,
neither of which alone can come up with databases of
relevant use for the history of chemistry.
Computational history of chemistry therefore re-
quires above all interdisciplinary work and constitutes an
interesting research niche at least for historians of chem-
istry, chemists, physicists, computer scientists, mathema-
ticians, philosophers and semioticians. As discussed in
(2), the success of these approaches depends, more than
on data and methods, on the common and synchronized
work of different disciplines, which requires being able
to understand the language and jargon of other specialists
and even other ways of thinking. All in all, the success
of this setting for the history of chemistry is a matter of
interdisciplinary respect and of interdisciplinary thinking.
Besides interdisciplinary research, the approaches
here discussed also involve interdisciplinary research
assessment. That is, when refereeing a publication or
research proposal of this sort, a traditional historian, with
little knowledge of computation, data analysis or the
mathematical techniques discussed, will face problems,
as she/he cannot go beyond the validity of the historical
question and the kind of data suitable for the study. Like-
wise, computer scientists, mathematicians and chemists
cannot go beyond judging the methods, tools and logic
of the subject under study (80). A complete and holistic
evaluation of research in this eld requires the common
work of different experts, with coordinated communica-
tion among them during the assessment process.
The success of novel outcomes discussed in this
essay necessitate ready access to databases and adequate
computing capacity. Traditional history of science de-
partments do not count on computational resources and
when these are available, they normally do not offer the
minimal conditions of memory and processing capacity
to handle large corpora of chemical information. More-
over, they typically do not count on system administrators
in charge of machine and software updates. Therefore,
alliances between computer science departments and
history and chemistry ones are essential.
Based on an analysis by Daniel J. Cohen and Roy
Rosenzweig of the pros and cons of computational his-
tory (81), in general, the approach here presented solves
several issues of the traditional practices of history of
chemistry. It brings almost unlimited storage capacity,
as information is stored in electronic form. Traditional
storage capacities are driven by library and archive ca-
pacities of printed records such as books and letters. In
contrast, a colossal amount of information can be stored
in tiny computer memories. Another advantage is the ac-
cessibility to information. This is already evident when
consulting century-old printed records by searching,
for instance in the Internet Archive. Several users can
get direct and simultaneous access to the same records
without leaving their desks (82). Computational history
of chemistry also offers exibility, as digital information
may be converted into different formats, ranging from
text and image to audio and video. One can, for instance,
think of recorded interviews of leading chemists or even
politicians, which are stored as sound but can easily be
transformed into text. Graphical abstracts and videos of
current publications also enlarge the possibilities of the
data sources for the historical practices of the present
and the future.
Disciplinary diversity is a further advantage of the
mathematical and computational approaches. The very
fact of accessibility to the information makes it possible
that not only historians can access the archives, but also
practicing chemists and other scholars and scientists
with interests in the history of chemistry. Large reposi-
tories of information for historical studies contribute to
the manipulability and searchability of information we
have become accustomed to in the current information
era. It is almost unimaginable to search for a particular
string of characters in hundreds of handwritten records
Bull. Hist. Chem., VOLUME 47, Number 1: HIST Centennial (2022) 101
of the Hellenistic, Chinese or Arabic alchemy without
computational aid. Computation has turned this task into
a customary process, where segmentation and optical
character recognition techniques allow for rapid and ef-
cient queries. The approaches here presented also offer
the possibility of interactivity, where databases can be fed
from different sources, by different specialists (and even
laypeople as in the case of Wikipedia) and the different
users can interact and discuss particular subjects, leav-
ing digital records of their interactions, which constitute
sources of further historical inquiry.
Despite the advantages of computational history of
chemistry, there are challenges and problems to be faced.
One of these is data quality. Building up large repositories
of information relies on the common work of several an-
notators and curators. This practice may sound foreign
for traditional historians, which often look for primary
sources delving into the archives and rely basically on
their own interpretations. The approaches here presented
often rely on commercial databases which are regularly
updated. These updates may modify records of previous
entries, for example information dumped today cover-
ing the nineteenth century may vary from another dump
from the same database and spanning the same period
but performed months later. This occurs because annota-
tors and curators may include new sources, for instance
journals not considered before, or patents; or because
annotation errors are found and corrected. Although this
poses a problem for historical studies, as the historical
source may vary, it is actually a minor problem as long
as the bulk of the database is not affected. The reason is
statistical. If updates and corrections do not change the
historical trends, they may be considered as background
noise of the signal; otherwise, the previous observed
trends correspond to data artifacts, which could have
been detected running statistical tests over the data. In
any case, it is a good policy to perform regular dumps
of the data and run statistical tests to determine whether
variations are signicant or not.
A further issue of these approaches is the inacces-
sibility to some of the relevant data. Computational
history of chemistry requires accessing large databases,
some of them owned by private companies or involving
costly subscription licenses, which are normally granted
to institutions rather than to individuals. Moreover, math-
ematical and computational approaches require special
access to these databases, which go far beyond the limited
number of queries allowed to typical users and which
often imply dumping the entire database. This issue can
be overcome by signing collaboration agreements with
database owners. This has been my particular experience
with Reaxys, for instance. Providers of other relevant da-
tabases such as Clarivate and Digital Science & Research
Solutions Inc. offer a variety of similar opportunities for
academic partnerships. As mentioned, providing access
on individual basis is not the norm, rather, these com-
panies generally grant access to research groups who
already have the infrastructure to store and process the
information. Typically, a further condition is that access
is granted based on a guarantee of information security
to avoid data leaks, which may affect the commercial
interests of database owners.
A large part of this essay has been devoted to depict-
ing how methods of the natural sciences, especially those
devised to look for patterns, nd application in the his-
tory of chemistry. However, this relationship is far from
being unidirectional, and it should be actually taken as a
symmetric relation, where historians have a central role
in the advancement of the different sciences. As Robin
George Collingwood (1889-1943) claimed (83):
The methods of historical research have, no doubt,
been developed in application to the history of hu-
man affairs; but is that the limit of their applicability?
They have already before now undergone important
extensions: for example, at one time historians had
worked out their methods of critical interpretation
only as applied to written sources containing nar-
rative material, and it was a new thing when they
learnt to apply them to the unwritten data provided
by archaeology. Might not a similar but even more
revolutionary extension sweep into the historian’s net
the entire world of nature? In other words, are not
natural processes really historical processes, and is
not the being of nature an historical being?
I thank Michael Gordin, Evan Hepler-Smith, Ernst
Homburg, Jeffrey Johnson, Jürgen Jost, Ursula Klein,
Manfred Laubichler, Farzad Mahootian, Carsten Re-
inhardt, Jürgen Renn, Camilo Restrepo, Alan Rocke,
Jeff Seeman and Joachim Schummer, for interesting
and motivating discussions, over the years, about some
aspects discussed in this essay.
References and Notes
1. M. Gell-Mann, “Regularities in Human Affairs,” in D.
Krakauer, J., Gaddis and K. Pomeranz, Eds., History, Big
History, & Metahistory, ch. 4, pp 63-90, SFI Press, Santa
Fe, 2018.
102 Bull. Hist. Chem., VOLUME 47, Number 1: HIST Centennial (2022)
2. G. Restrepo and J. Jost, The Evolution of Chemical Knowl-
edge, a Formal Setting for its Analysis, Wissenschaft
und Philosophie—Science and Philosophy, Springer,
3. Note that I am not advocating either for a teleological or
for a whig history. I put forward a history of chemistry
advancing hand-in-hand with mathematics and statistics,
and as I will argue later, also with computer sciences.
4. P. Turchin, “Toward Cliodynamics—an Analytical,
Predictive Science of History,” Cliodynamics, 2011, 2,
5. W. McNeill, The Rise of the West: A History of the Human
Community, University of Chicago Press, 2009. https:// books?id=_RsPrzrsAvoC (accessed 12
Oct. 2021).
6. As Turchin has remarked, there were still some supporters
of the idea of history as the search for patterns such as S.
H. Williamson, “The History of Cliometrics,” Research in
Economic History, Suppl. 6, 1991, 15-31 and V. Bonnell
and L. Hunt, “Introduction,” in V. Bonnell and L. Hunt,
Eds., Beyond the Cultural Turn: New Directions in the
Study of Society and Culture, University of California
Press, Berkeley, 1999, pp 1-32.
7. J. A. Johnson, “Germany: Discipline—Industry—Profes-
sion. German Chemical Organizations, 1867-1914,” in
A. K. Nielsen and S. Štrbáňová, Creating Networks in
Chemistry: The Founding and Early History of Chemi-
cal Societies in Europe, Royal Society of Chemistry,
Cambridge, UK, 2008, ch. 6, pp 113-138.
8. R. M. Friedman, The Politics of Excellence: Behind the
Nobel Prize in Science, W. H. Freeman, New York, 2001.
9. A. Rocke, “The Theory of Chemical Structure and its
Applications,” in M. J. Nye, Ed., The Cambridge History
of Science, vol. 5, The Modern Physical and Mathemati-
cal Sciences, Publisher: Cambridge University Press,
Cambridge, UK, 2002, pp 255-271.
10. I am referring to Louis-Bernard Guyton de Morveau
(1737-1816), Antoine-Laurent de Lavoisier (1743-1794),
Claude Louis Berthollet (1748-1822), Antoine François,
comte de Fourcroy (1755-1809), Jean-Henri Hassenfratz
(1755-1827) and Pierre-Auguste Adet (1763-1834).
11. G. Restrepo and J. L. Villaveces, “Chemistry, a Lingua
Philosophica,” Found. Chem., 2011, 13, 233-249.
12. J. Renn, The Evolution of Knowledge: Rethinking Science
in the Anthropocene, Princeton Univ. Press, Princeton,
NJ, 2020.
13. A recent study models how we operate upon experiences
to build up such cognitive structures. See C. W. Lynn
and D. S. Bassett, “How Humans Learn and Represent
Networks,” Proc. Nat. Acad. Sci. USA, 2020, 117 , 29407-
14. J. Renn, “The Evolution of Knowledge: Rethinking Sci-
ence in the Anthropocene,” HoST—Journal of History
of Science and Technology, 2018, 12, 1-22, https://doi.
15. M. Foucault, Les mots et les choses, Gallimard, Paris,
16. L. v. Bertalanffy, “An Outline of General System The-
ory,” Br. J. Philos. Sci., 1950, 1, 134-165, https://doi.
17. A typical chemical system is a system of elements, where
chemical elements hold order and similarity relation-
ships (W. Leal and G. Restrepo, “Formal Structure of
Periodic System of Elements,” Proc. R. Soc. (London)
A, 2019, 475, 20180581,
rspa.2018.0581). I note in passing that systems of ele-
ments are not set in stone, but rather historical objects
emerging from the evolution of chemical knowledge.
See G. Restrepo, “Compounds Bring Back Chemistry
to the System of Chemical Elements,” Substantia,
2019, 3, 115-124 (2019),
Substantia-739 and G. Restrepo, Das periodische System
und die Evolution des chemischen Raums,” Nachrichten
aus der Chemie, 2020, 68, 12-15,
18. C. S. Peirce, The Essential Peirce, Volume 2, Indiana
University Press, Bloomington and Indianapolis, 1998.
19. T. L. Short, Peirce’s Theory of Signs, Cambridge Univer-
sity Press, Cambridge, UK, 2007.
20. An example of this ternary relation is the association of
the sign “water” to the colorless liquid wetting materials,
quenching our thirst, owing in rivers, boiling at 100 °C,
dissolving table salt, plus several other features; an asso-
ciation mediated by the interpretant (2). By high-order re-
lations I mean the further relations between ternary related
semiotic objects. For instance, a chemical reaction such as
A + B → C + D
corresponds to a high-order relation entailing the ternary
semiotic objects A, B, C and D. The ternary relation in
A is given by the sign the interpretant has given to this
substance, likewise occurs with B, C and D. As we have
shown elsewhere (2), a suitable high-order structure for
chemical reactions is a hypergraph, which I discuss in
section 6.
21. In Ref. 2, we have highlighted the strong relationship be-
tween the semiotic and the material systems, as there are
semiotic objects which, through experimental evidence,
have been introduced as objects of the material system.
This is the case of atoms and molecules, rst devised as
chemical entities lacking physical reality used to advance
chemical knowledge. Later experimental results led to the
adoption of Avogadro’s hypothesis and the kinetic theory
of gases, which prompted the introduction of atoms and
molecules as material entities of the chemical practice.
Bull. Hist. Chem., VOLUME 47, Number 1: HIST Centennial (2022) 103
See A. J. Rocke, Chemical Atomism in the Nineteenth
Century, Ohio State University Press, Columbus, 1984.
22. C. Hooker, “Introduction to Philosophy of Complex
Systems: A: Part a: Towards a Framework for Complex
Systems,” in C. Hooker, Ed., Philosophy of Complex
Systems, vol. 10 of Handbook of the Philosophy of Sci-
ence, North-Holland, Amsterdam, 2011, pp 3-90, https://
23. An example of simple rules leading to the emergence of a
pattern is the game of life by John Horton Conway (1937-
2020). The setting is as follows (A. Adamatzky, Game of
Life Cellular Automata, Springer, London, 2010, https:// imagine
a graph paper sheet, where every square can be either
black (dead) or white (live). The color of each square is
determined by the colors of an initial set of squares that
one selects at one’s will. Each square is surrounded by
eight other squares. The rules read that any live square
with two or three live neighboring squares survives. Any
dead square with three live neighboring squares becomes
a live square. All other live squares die in the next genera-
tion. Likewise, all other dead squares stay dead. The rst
generation of squares is created by simultaneously apply-
ing the aforementioned rules to every square. Births and
deaths occur simultaneously. Each generation depends
on the preceding one and the rules continue to be applied
repeatedly to create further generations. The Wikipedia
article, “Conway’s Game of Life (https://en.wikipedia.
org/wiki/Conway%27s_Game_of_Life, (accessed 12 Oct.
2021)) depicts interesting visualizations.
24. In practice, accepting a model requires some further steps,
such as validating it. This entails, for example, perturbing
(or deleting) part of the input data to observe the stability
of the model.
25. This was a period with one of the most rapid annual
outputs of new chemicals in the history of chemistry, as
shown in E. J. Llanos, et al., “Exploration of the Chemi-
cal Space and its Three Historical Regimes,” Proc. Nat.
Acad. Sci. USA, 2019, 116, 12660-12665, https://doi.
26. A. Buyalskaya, M. Gallo and C. F. Camerer, “The Golden
Age of Social Science,” Proc. Nat. Acad. Sci. USA, 2021,
118 ,
27. J. Li, Y. Yin, S. Fortunato, and D. Wang, “Scientic
Elite Revisited: Patterns of Productivity, Collaboration,
Authorship and Impact,” Journal of The Royal Society
Interface, 2020, 17, 20200135,
28. A. Edelmann, T. Wolff, D. Montagne and C. A. Bail,
“Computational Social Science and Sociology,” Annu.
Rev. Sociol., 2020, 46, 61-81,
29. J. Schummer, “The Impact of Instrumentation on Chemi-
cal Species Identity,” in P. Morris, Ed., From Classical to
Modern Chemistry: The Instrumental Revolution, Royal
Society of Chemistry, Cambridge, UK, 2002, pp 188-211.
30. A. Gibson, M. D. Laubichler and J. Maienschein,
“Introduction,” Isis, 2019, 11 0, 497-501, https://doi.
31. A.-L. Post and A. Weber, “Notes on the Reviewing of
Learned Websites, Digital Resources, and Tools,” Isis,
2018, 109, 796-800,
32. M. D. Laubichler, J. Maienschein and J. Renn, “Com-
putational Perspectives in the History of Science: To the
Memory of Peter Damerow, Isis, 2013, 104, 119-130,
33. S. P. Weldon, “Introduction,” Isis, 2013, 104, 537-539,
34. E. Aronova, C. v. Oertzen, and D. Sepkoski, “Introduc-
tion: Historicizing Big Data,” Osiris, 2017, 32, 1-17,
35. L. Putnam, “The Transnational and the Text-Searchable:
Digitized Sources and the Shadows They Cast,” Ameri-
can Historical Review, 2016, 121, 377-402, https://doi.
36. When I state that traditional historical studies concentrate
on specic regions and on periods of decades, I am mainly
referring to studies leading to journal publications. There
are, nevertheless, important publications, mainly books,
where historians present large analyses of longer periods
and often of global scope. These studies typically rely on
journal publications as well as on other secondary sources.
37. Here sizes are relative to the largest possible historical
records. Hence, one can take as a reference for the mate-
rial system all substances, reactions, instrumentation and
technologies used in the history of chemistry. Likewise,
the size of the records pertaining to the social system is
determined by the number of names and personal data of
all alchemists, apothecaries, metallurgists and chemists
in the history of chemistry as well as their organizations
and institutions. It would also require the inclusion of
novel technologies such as robots directly interacting
with humans in the practice of chemistry. The size of the
semiotic system is given by the collection of all semiotic
objects devised and used by practitioners of chemistry
over the evolution of chemistry. It is in comparison with
this scale that I argue that a traditional historical study
relies on small datasets, which are nevertheless a subset
of the datasets of computational studies.
38. Particular elds of mathematics in play include partial
differential equations, cellular automata, articial neural
networks, evolutionary computation, genetic algorithms,
machine learning, time series analysis, agent-based mod-
elling, as well as bifurcation, information, (hyper)graph
and complexity theories.
39. There exist examples of mathematics motivated by
chemistry. For instance, nineteenth-century settings of
104 Bull. Hist. Chem., VOLUME 47, Number 1: HIST Centennial (2022)
molecular structures as mathematical objects contributed
to graph theory with the works of James Joseph Sylves-
ter (1814-1897). Likewise, Arthur Cayley (1821-1895),
motivated by the question on the number of isomers for
some alkanes, contributed to enumerative mathematics,
which were expanded by George Pólya (1887-1985) in
the twentieth century. Further information on this subject
is found in D. J. Klein, “Mathematical Chemistry! Is It?
and if So, What Is It?” Hyle, 2013, 19, 35-85 (http://, accessed 12
Oct. 2021) and G. Restrepo, “Mathematical Chemistry,
a New Discipline,” in E. Scerri and G. Fisher, Eds., Es-
says in the Philosophy of Chemistry, Oxford University
Press, Oxford and New York, 2016, ch. 15, pp 332-351,
40. Llanos, et al. (Ref. 25).
41. Restrepo, “Das periodische System...” (Ref. 17).
42. Although there is no unique denition of mathematical
space, this concept encodes the idea of a set endowed with
a notion of nearness. For example, if nearness is associ-
ated with distance, then we talk about metric spaces. If
it is associated with relationships among elements, then
this remits to topological spaces.
43. This is, for example, the basis of the Quantitative Struc-
ture-Activity Relationship paradigm.
44. D. J. de Solla Price, Little Science, Big Science, Columbia
University Press, New York, 1963.
45. J. Schummer, “Scientometric Studies on Chemistry I:
The Exponential Growth of Chemical Substances, 1800-
1995,” Scientometrics, 1997, 39, 107-123.
46. M. Fialkowski, K. J. M. Bishop, V. A. Chubukov, C.
J. Campbell and B. A. Grzybowski, “Architecture and
Evolution of Organic Chemistry,” Angew. Chem. Int. Ed.,
2005, 44, 7263-7269, doi:10.1002/anie.200502272.
47. A natural question is how this speed compares to those
of other spaces, for instance biological spaces. In this
case, one has also to account for rates of extinction
and of mutation. See J. W. Bull and M. Maron, “How
Humans Drive Speciation as Well as Extinction,” Proc.
R. Soc. (London) B, 2016, 283, 20160600, https://doi.
48. U. Klein, Experiments, Models, Paper Tools: Cultures of
Organic Chemistry in the Nineteenth Century, Stanford
University Press, Stanford, CA, 2003.
49. H. Leicester, The Historical Background of Chemistry,
Dover Publications, 1971 (rst published Wiley, 1956).
50. W. H. Brock, The Norton History of Chemistry, W. W.
Norton and Company, New York, 1993.
51. See below in this section for a discussion on the recovery
after WWs.
52. Discussing these and other related questions was one of
the main reasons we organized the recent Computational
approaches to the history of chemistry meeting (Max
Planck Institute for Mathematics in the Sciences, Leipzig,
March 2021), where historians, chemists, computer sci-
entists and mathematicians analyzed different aspects
of data driven approaches for the practice of history of
chemistry. It was clear that more work is required to
advance the construction of electronic databases for the
social and semiotic systems of chemistry.
53. L. C. Ray and R. A. Kirsch, “Finding Chemical Records
by Digital Computers,” Science, 1957, 126, 814-819,
54. Connection tables are lists of atoms and bonds belonging
to a molecular structure with further information such
as coordinates in a two- or three-dimensional space.
SMILES (Simplied Molecular-Input Line-Entry Sys-
tem) are string representations of molecular structures.
International Chemical Identiers (InChIs) are strings
used to encode as much as possible information about
chemical substances, which include their associated
molecular structures.
55. J. Staker, K. Marshall, R. Abel and C. M. McQuaw, “Mo-
lecular Structure Extraction from Documents Using Deep
Learning,” J. Chem. Inf. Model., 2019, 59, 1017-1029,
56. Publications before the digital era had regular discrete
outputs ranging from annual to weekly, a periodization
that is becoming more continuous with the advent of
preprint servers and with the online publication of papers
right after acceptance, while proofs are edited (2).
57. Z. Wu, N. E. Huang, S. R. Long and C.-K. Peng, “On
the Trend, Detrending, and Variability of Nonlinear
and Nonstationary Time Series,” Proc. Nat. Acad. Sci.
USA, 2007, 104, 14889-14894,
58. D. Brillinger, “Time Series: General,” In N. J. Smelser
and P. B. Baltes, Eds., International Encyclopedia of the
Social and Behavioral Sciences, 15724-15731, Pergamon,
Oxford, 2001,
59. M. D. Laubichler, J. Maienschein and J. Renn, “Com-
putational History of Knowledge: Challenges and
Opportunities,” Isis, 2019, 110, 502-512, https://doi.
60. A. Aleta and Y. Moreno, “Multilayer Networks in a
Nutshell,” Annu. Rev. Condens. Matter Phys., 2019, 10,
61. M. M. Danziger, I. Bonamassa, S. Boccaletti and S.
Havlin, “Dynamic Interdependence and Competition in
Multilayer Networks,” Nature Physics, 2019, 15, 178-
62. P. Chodrow and A. Mellor, “Annotated Hypergraphs:
Models and Applications,” Applied Network Science,
2020, 5, 9.
Bull. Hist. Chem., VOLUME 47, Number 1: HIST Centennial (2022) 105
63. J. Pearl, Causality, Cambridge University Press, Cam-
bridge, UK, 2009.
64. This nding goes hand in hand with others indicating also
preferences for some particular reactions to explore the
chemical space, actually the space of substances of po-
tential pharmacological interest. In this case, for example,
Suzuki-Miyaura is one of the preferred reactions since
1980 (D. G. Brown and J. Boström, “Analysis of Past and
Present Synthetic Methodologies on Medicinal Chemis-
try: Where Have All the New Reactions Gone?” J. Med.
Chem., 2016, 59, 4443-4458,
65. J. Schummer, “The Chemical Core of Chemistry I: A
Conceptual Approach,” Hyle, 1998, 5, 129-162, http:// (accessed
12 Oct. 2021).
66. Sociologists were one of the rst to recognize the advan-
tages of graph theory. The same community further ex-
tended the mathematical possibilities of these structures,
which drew the attention of mathematicians. Graph theory
is today a vibrant area of research in discrete mathematics.
Although I highlight the important role of sociologists
for the expansion of graph theory, I do not disregard the
early works of mathematicians such as Leibniz and Euler
in presenting instances of graphs and their mathematics.
67. Interesting discussions on the development of chemical
graph theory are found in Klein (Ref. 39).
68. S. Klamt, U.-U. Haus and F. Theis, “Hypergraphs and
Cellular Networks,” PLOS Computational Biology, 2009,
5, 1-6.
69. This is a very broad depiction of a chemical reaction in
solution. One could claim that there are also solid state
reactions, or even one-pot reactions and others where
the separation process is automated, or others where
the evolution of the reaction is monitored in real time
through the direct insertion of a probe into the reaction
vessel (B. Wittkamp, In Situ Monitoring of Chemical
Reactions—a Molecular Video, https://www.chemeurope.
chemical-reactions-a-molecular-video.html, (accessed
12 Oct. 2021)). In any case the general idea of reactions
as relating sets of chemicals holds and it is an essential
part of what a chemical reaction is.
70. The development of the concept of chemical reaction is
of central historical and epistemological relevance for
understanding the evolution of chemical knowledge.
Much work in this direction is needed, where, for instance,
semioticians and philosophers of science may contribute
to a large extent. At any rate, Lavoisier initiated chemical
semiotics by using chemical equations to describe mate-
rial transformation in terms of substances undergoing
changes (11). Further discussions on alternative ways to
describe chemical transformations are found in (2)
71. In graph theory these arrows correspond to arcs.
72. Claude Berge (1926-2002) in the 1970s analyzed several
of their properties and discussed their delayed recogni-
tion in mathematics: C. Berge, Graphs and Hypergraphs,
North-Holland Mathematical Library, North-Holland,
Amsterdam, 1973.
73. W. Leal, G. Restrepo, P. F. Stadler and J. Jost, “Forman-
Ricci Curvature for Hypergraphs,” Advances in Complex
Systems, 2021, 24, 2150003,
74. In this setting a hypergraph H corresponds to the couple
(V, E), where V are the objects and E contains the subsets
of V that are related. For example, for the reaction A +
B → C + D, its corresponding hypergraph is H = (V, E ),
with V = {A, B, C, D} and E = {({A, B}, {C, D})}.
75. In (2) we discussed some of the mathematical properties
already studied for hypergraphs and some others still to
be explored.
76. Our results show no sign of saturation in this expansion,
as de Solla Price wrongly anticipated for the whole sci-
ence in the 1960s (44).
77. W. Leal, E. J. Llanos, A. Bernal, P. F. Stadler, J. Jost and G.
Restrepo, “Computational Data Analysis Shows that Key
Developments Towards the Periodic System Occurred in
the 1840s,” ChemRxiv, 2021.
78. The underlying hypothesis of this study is that the chemi-
cal space directly inuenced the ordering and similarities
of the chemical elements, therefore the periodic system.
See Restrepo “Compounds Bring Back…” (Ref. 17)
and G. Restrepo, “Challenges for the Periodic Systems
of Elements: Chemical, Historical and Mathematical
Perspectives,” Chem. Eur. J., 2019, 25, 15430-15440,
79. These are some of the formulators of periodic systems,
ranging from Leopold Gmelin (1788-1853) in the 1840s to
Julius Lothar Meyer (1830-1895), William Odling (1829-
1921), Gustavus Detlef Hinrichs (1836-1923) and Dmitri
Ivanovich Mendeleev (1834-1907) in the 1860s. See E.
Scerri, The Periodic Table: Its Story and Its Signicance,
Oxford University Press, New York, 2nd ed., 2019, https://
80. Sometimes, even for computer scientists it is difcult to
judge the value of a computational approach as reproduc-
ibility may become an issue.
81. D. Cohen and R. Rosenzweig, Digital History: A Guide
to Gathering, Preserving, and Presenting the Past on
the Web, University of Pennsylvania Press, Philadel-
phia, 2006, available online at
digitalhistory/ available online at
digitalhistory/ (accessed 12 Oct. 2021).
82. One is tempted to argue that this practice contributes to
lowering global warming emissions, as trips to consult
archives and other sources are reduced to a large extent.
106 Bull. Hist. Chem., VOLUME 47, Number 1: HIST Centennial (2022)
However, the environmental costs of computation are
far from being disregarded, as shown in E. Strubell, A.
Ganesh and A. McCallum, “Energy and Policy Consider-
ations for Deep Learning in NLP,” in Proceedings of the
57th Annual Meeting of the Association for Computational
Linguistics, 3645-3650, Association for Computational
Linguistics, Florence, Italy, 2019, https://www.aclweb.
org/anthology/P19-1355 (accessed 12 Oct. 2021). The
question that arises concerns the net environmental costs
of the approaches here presented and how they actually
compare with the traditional approaches to the history of
Born in the Same Year as HIST (1922)
Har Gobind Khorana: January 9, 1922, in Raipur, in the Punjab in (then British) India.
Robert William Holley: January 28, 1922, in Urbana, Illinois, USA.
George C. Pimentel: May 2, 1922, in Rolinda, California, USA.
John Goodenough: July 25, 1922 in Jena, Germany.
Khorana and Holley, born in the same month in widely separated places, shared the 1968 Nobel
Prize in Physiology or Medicine for their roles in deciphering the genetic code, the link between nucleic
acid and protein sequences. Khorana and his co-workers at the University of Wisconsin-Madison syn-
thesized short chains of RNA which were used to direct the synthesis of short protein fragments. Later
Khorana and co-workers made what is considered the rst articial gene. Holley and his co-workers
at Cornell University and the US Plant, Soil and Nutrition Laboratory on the Cornell campus, isolated
and sequenced alanine transfer RNA, which directs incorporation of alanine into proteins.
Pimentel’s research is well known to physical chemists and his service to chemical education
and the chemical profession is widely recognized in the American Chemical Society. Matrix isola-
tion methods and chemical lasers are his main legacies in physical chemistry. He received the highest
award of the American Chemical Society, the Priestley medal, in 1989, the year of his death. He had
served as ACS president in 1986. The ACS award in chemical education was renamed the George C.
Pimentel Award in Chemical Education in his honor.
Goodenough shared the 2019 Nobel Prize in Chemistry for the development of lithium ion batter-
ies, ubiquitous power-storage devices for mobile electronics. Goodenough and co-workers made their
key contributions to this technology at the University of Oxford in the 1970s and 1980s. The Materials
Chemistry division of the Royal Society of Chemistry have an award named for Goodenough.
83. R. G. Collingwood and W. J. van der Dussen, The Idea
of History, ACLS Humanities E-Book, Oxford University
Press, Oxford, UK, 1994,
About the Author
Guillermo Restrepo is a chemist working at the
Max Planck Institute for Mathematics in the Sciences
(Germany). His scientic interests include the evolution
of chemical knowledge, the relationship between chem-
istry and mathematics and the history and philosophy of
... 120 years. On the other hand, bibliographic analysis of the progress and sociology of chemistry has been used recently to great advantage (Seeman,Restrepo, 2020b, a;Llanos et al., 2019;Leal et al., 2022;Restrepo, 2022;Jost,Restrepo, 2023). This approach is certainly worth considering in the future. ...
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Despite decades of research and thought on the meaning and identification of revolutions in science, there is no generally accepted definition for this concept. This paper presents 13 different characteristics that have been used by philosophers and historians of science to characterize revolutions in science, in general, and in chemistry, in particular. These 13 characteristics were clustered into six independent factors. Suggestions are provided as to the use of these characteristics and factors to evaluate historical events as to their possible categorization as revolutions in chemistry. Challenges to the goal of creating a consensus definition of “revolutions in science” are also presented in this publication.
... We hope our results and methods contribute to the ongoing development of computational approaches to the history of science and the evolution of knowledge (8,46,47). ...
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The periodic system, which intertwines order and similarity among chemical elements, arose from knowledge about substances constituting the chemical space. Little is known, however, about how the expansion of the space contributed to the emergence of the system—formulated in the 1860s. Here, we show by analyzing the space between 1800 and 1869 that after an unstable period culminating around 1826, chemical space led the system to converge to a backbone structure clearly recognizable in the 1840s. Hence, the system was already encoded in the space for about two and half decades before its formulation. Chemical events in 1826 and in the 1840s were driven by the discovery of new forms of combination standing the test of time. Emphasis of the space upon organic chemicals after 1830 prompted the recognition of relationships among elements participating in the organic turn and obscured some of the relationships among transition metals. To account for the role of nineteenth century atomic weights upon the system, we introduced an algorithm to adjust the space according to different sets of weights, which allowed for estimating the resulting periodic systems of chemists using one or the other weights. By analyzing these systems, from Dalton up to Mendeleev, Gmelin’s atomic weights of 1843 produce systems remarkably similar to that of 1869, a similarity that was reinforced by the atomic weights on the years to come. Although our approach is computational rather than historical, we hope it can complement other tools of the history of chemistry.
Chemistry is concerned with a subject that is not static, but evolving in time, chemical space, that is, the collection of all substances and reactions reported over time. If we accept that premise, we can identify the path dependencies and self-reinforcing mechanisms that determined its current space and selected it across historical alternatives. In particular, data analysis allows us to identify two crucial turning points. One was the introduction of structural theory in 1860, the other a technological shift around 1980.
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Chemical space entails substances endowed with a notion of nearness that comes in two flavours: similarity and synthetic reachability. What is the maximum size for the chemical space? Is there an upper bound for its classes of similar substances? How many substances and reactions can it house? Can we store these features of the chemical space? Here I address these questions and show that the physical universe does not suffice to store the chemical one embodied in the chemical space. By analysing the historical evolution of the space as recorded by chemists over the centuries, I show that it has been mainly expanded by synthesis of organic compounds and unfolds at an exponential rate doubling its substances each 16 years. At the turn of the 20th century it left behind an expansion period driven by reactions and entered the current era ruled by substance discovery, which often relies on some few starting materials and reaction classes. Extrapolating from these historical trends, synthesising a large set of affordable chemicals in the foreseeable future would require trebling the historical stable speed rate of discovery of new chemicals. Likewise, creating a database of failed reactions accounting for 25% of the known chemical space to assist the artificial intelligence expansion of the space could be afforded if the synthetic efforts of the coming five years are entirely dedicated to this task. Finally, I discuss hypergraph reaction models to estimate the future shape of the network underlying the chemical space.
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Social science is entering a golden age, marked by the confluence of explosive growth in new data and analytic methods, interdisciplinary approaches, and a recognition that these ingredients are necessary to solve the more challenging problems facing our world. We discuss how developing a “lingua franca” can encourage more interdisciplinary research, providing two case studies (social networks and behavioral economics) to illustrate this theme. Several exemplar studies from the past 12 y are also provided. We conclude by addressing the challenges that accompany these positive trends, such as career incentives and the search for unifying frameworks, and associated best practices that can be employed in response.
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The integration of social science with computer science and engineering fields has produced a new area of study: computational social science. This field applies computational methods to novel sources of digital data such as social media, administrative records, and historical archives to develop theories of human behavior. We review the evolution of this field within sociology via bibliometric analysis and in-depth analysis of the following subfields where this new work is appearing most rapidly: ( a) social network analysis and group formation; ( b) collective behavior and political sociology; ( c) the sociology of knowledge; ( d) cultural sociology, social psychology, and emotions; ( e) the production of culture; ( f ) economic sociology and organizations; and ( g) demography and population studies. Our review reveals that sociologists are not only at the center of cutting-edge research that addresses longstanding questions about human behavior but also developing new lines of inquiry about digital spaces as well. We conclude by discussing challenging new obstacles in the field, calling for increased attention to sociological theory, and identifying new areas where computational social science might be further integrated into mainstream sociology. Expected final online publication date for the Annual Review of Sociology, Volume 46 is July 30, 2020. Please see for revised estimates.
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Throughout history, a relatively small number of individuals have made a profound and lasting impact on science and society. Despite long-standing, multi-disciplinary interests in understanding careers of elite scientists, there have been limited attempts for a quantitative, career-level analysis. Here, we leverage a comprehensive dataset we assembled, allowing us to trace the entire career histories of nearly all Nobel laureates in physics, chemistry, and physiology or medicine over the past century. We find that, although Nobel laureates were energetic producers from the outset, producing works that garner unusually high impact, their careers before winning the prize follow relatively similar patterns to those of ordinary scientists, being characterized by hot streaks and increasing reliance on collaborations. We also uncovered notable variations along their careers, often associated with the Nobel Prize, including shifting coauthorship structure in the prize-winning work, and a significant but temporary dip in the impact of work they produce after winning the Nobel Prize. Together, these results document quantitative patterns governing the careers of scientific elites, offering an empirical basis for a deeper understanding of the hallmarks of exceptional careers in science.
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Abstract Hypergraphs offer a natural modeling language for studying polyadic interactions between sets of entities. Many polyadic interactions are asymmetric, with nodes playing distinctive roles. In an academic collaboration network, for example, the order of authors on a paper often reflects the nature of their contributions to the completed work. To model these networks, we introduce annotated hypergraphs as natural polyadic generalizations of directed graphs. Annotated hypergraphs form a highly general framework for incorporating metadata into polyadic graph models. To facilitate data analysis with annotated hypergraphs, we construct a role-aware configuration null model for these structures and prove an efficient Markov Chain Monte Carlo scheme for sampling from it. We proceed to formulate several metrics and algorithms for the analysis of annotated hypergraphs. Several of these, such as assortativity and modularity, naturally generalize dyadic counterparts. Other metrics, such as local role densities, are unique to the setting of annotated hypergraphs. We illustrate our techniques on six digital social networks, and present a detailed case-study of the Enron email data set.
Chemistry shapes and creates the disposition of the world's resources and provides novel substances for the welfare and hazard of our civilisation at an exponential rate. Can we model the evolution of chemical knowledge? This book not only provides a positive answer to the question, it provides the formal models and available data to model chemical knowledge as a complex dynamical system based on the mutual interaction of the social, semiotic and material systems of chemistry. These systems, which have evolved over the history, include the scientists and institutions supporting chemical knowledge (social system); theories, concepts and forms of communication (semiotic system) and the substances, reactions and technologies (material system) central for the chemical practice. These three systems, which have traditionally been mostly studied in isolation, are brought together in this book in a grand historical narrative, on the basis of comprehensive data sets and supplemented by appropriate tools for their formal analysis. We thereby develop a comprehensive picture of the evolution of chemistry, needed for better understanding the past, present and future of chemistry as a discipline. The interdisciplinary character of this book and its non-technical language make it an ideal complement to more traditional material in undergraduate and graduate courses in chemistry, history of science and digital humanities.
Hypergraphs serve as models of complex networks that capture more general structures than binary relations. For graphs, a wide array of statistics has been devised to gauge different aspects of their structures. Hypergraphs lack behind in this respect. The Forman–Ricci curvature is a statistics for graphs based on Riemannian geometry, which stresses the relational character of vertices in a network by focusing on the edges rather than on the vertices. Despite many successful applications of this measure to graphs, Forman–Ricci curvature has not been introduced for hypergraphs. Here, we define the Forman–Ricci curvature for directed and undirected hypergraphs such that the curvature for graphs is recovered as a special case. It quantifies the trade-off between hyperedge (arc) size and the degree of participation of hyperedge (arc) vertices in other hyperedges (arcs). Here, we determine upper and lower bounds for Forman–Ricci curvature both for hypergraphs in general and for graphs in particular. The measure is then applied to two large networks: the Wikipedia vote network and the metabolic network of the bacterium Escherichia coli. In the first case, the curvature is governed by the size of the hyperedges, while in the second example, it is dominated by the hyperedge degree. We found that the number of users involved in Wikipedia elections goes hand-in-hand with the participation of experienced users. The curvature values of the metabolic network allowed detecting redundant and bottle neck reactions. It is found that ADP phosphorylation is the metabolic bottle neck reaction but that the reverse reaction is not similarly central for the metabolism. Furthermore, we show the utility of the Forman–Ricci curvature for quantification of assortativity in hypergraphs and illustrate the idea by investigating three metabolic networks.
We celebrate 150 years of periodic systems that reached their maturity in the 1860s. They began as pedagogical efforts to project corpuses of substances on the similarity and order relationships of the chemical elements. However, these elements are not the canned substances wrongly displayed in many periodic tables, but rather the abstract preserved entities in compound transformations. We celebrate the systems, rather than their tables or ultimate table. The periodic law, we argue, is not an all‐encompassing achievement, as it does not apply to every property of all elements and compounds. Periodic systems have been generalised as ordered hypergraphs, which solves the long‐lasting question on the mathematical structure of the systems. In this essay, it is shown that these hypergraphs may solve current issues such as order reversals in super‐heavy elements and lack of system predictive power. We discuss research in extending the limits of the systems in the super‐heavy‐atom region and draw attention to other limits: the antimatter region and the limit arising from compounds under extreme conditions. As systems depend on the known chemical substances (chemical space) and such a space grows exponentially, we wonder whether systems still aim at projecting knowledge of compounds on the relationships among the elements. We claim that systems are not based on compounds anymore, rather on 20th century projections of the 1860s systems of elements on systems of atoms. These projections bring about oversimplifications based on entities far from being related to compounds. A linked oversimplification is the myth of vertical group similarity, which raises questions on the approaches to locate new elements in the system. Finally, we propose bringing back chemistry to the systems by exploring similarity and order relationships of elements using the current information of the chemical space. We ponder whether 19th century periodic systems are still there or whether they have faded away, leaving us with an empty 150th celebration. Systems through time: This Essay discusses the Periodic Table and as it celebrates its 150th year. Discussions include its powers and limitations, inherent generalizations as well as the ordering and inter‐element relationships that exist in it currently and the previous periodic systems that have led to the Table as we know it today.