Abstract
We analyze the connections of Lavoisier system of nomenclature with Leibniz’s philosophy, pointing out to the resemblance between what we call Leibnizian and Lavoisian programs. We argue that Lavoisier’s contribution to chemistry is something more subtle, in so doing we show that the system of nomenclature leads to an algebraic system of chemical sets. We show how Döbereiner and Mendeleev were able to develop this algebraic system and to find new interesting properties for it. We pointed out the resemblances between Leibniz program and Lavoisier legacy, particularly regarding the lingua philosophica for understanding and thinking Nature, in this particular case, chemistry. In the second part we discuss, from the linguistic viewpoint, how Lavoisian algebraic system may be taken further to build a language. We study the constituents of such a chemical language. Finally, we formalize some of the ideas here presented by using elements of network theory and discrete mathematics.
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Notes
This doctrine considered that celestial bodies could be perceived with and related to the intervals of strengthened strings, as in a musical harmonic play.
\( A \Gamma E \Omega M E T P H T O \Sigma \, M H \Delta E I\Sigma \, EI \Sigma I T \Omega \)
“SIMP: Io non dirò che questa vostra ragione non possa esser concludente, ma dirò bene con Aristotile che nelle cose naturali non si deve sempre ricercare una necessità di dimostrazion matematica. SAGR: Sí, forse, dove la non si può avere, ma se qui ella ci è, perché non la volete voi usare? Ma sarà bene non ispender piú parole in questo particolare, perché io credo che il signor Salviati ad Aristotile ed a voi senza altre dimostrazioni avrebbe conceduto, il mondo esser corpo, ed esser perfetto e perfettissimo, come opera massima di Dio.”(Galilei 1911)
“La filosofia é scritta in questo grandissimo libro che continuamente ci sta aperto innanzi agli occhi (io dico luniverso), ma non si pu intendere se prima non simpara a intender la lingua, e conoscer i caratteri, ne quali é scritto. Egli é scritto in lingua matematica, e i caratteri son triangoli, cerchi, ed altre figure geometriche, senza i quali mezi é impossibile a intenderne umanamente parola; senza questi é un aggirarsi vanamente per un oscuro laberinto.” (Galilei 1623)
There has been discussion about who came up first with Calculus ideas, whether Leibniz or Newton. An interesting summary is given in (Bardi 2006).
It is well known that mental structures and streams of thought are in continuous change. But the only invariant fact relating all thoughts and giving structure to them is their logic, which to exist needs to be expressed through a language. Our opinion is that the most suitable language for dealing with logic and relations in a more simple manner is mathematics. All sciences, in fact all aspects of the human activity when treated and analyzed in earnest are expressed with the assistance of such a language. Although mankind has struggled for centuries to understand this, leaving aside long periods of mysticism, it is surprising to find statements of highly recognized “contemporary thinkers” claiming that the “myth of the absolute power of mathematics” is over. This kind of assertion, found in a recent book by Laughlin (2006) (1998 Nobel prize in physics) is worrisome. What we think is that the time for science-as-usual, based primarily on Newtonian mathematics, is not the only possibility and perhaps not the best one. This does not mean, however, that it is the end of mathematics assistance for understanding the world; mathematics has a lot of branches and it is an extremely active field. Hence, what must be done is to look for different mathematical approaches to understand the phenomena we face. In this respect, Hankins recalls that “even philosophers like Diderot and Comte de Bouffon […], who during the 1750s accused their fellow scientists of an excessive reliance on mathematics, did not repudiate the application of reason, which they believed to be best exemplified in mathematics” (Hankins 1985). Rényi by his side says: “The new concepts invented by the mathematicians are like new ships which carry the discoverer farther on the great sea of thought” and “The main aim of the mathematician is to explore the secrets and riddles of the sea of human thought” (Rényi 1967).
Although Leibniz mooted his idea of a lingua philosophica in his Ars Combinatoria, he kept thinking on it througout his life, as Rutherford shows in (Rutherford 1995).
According to Leibniz, the lingua philosophica may include characters like “words, letters; chemical, astronomical, and Chinese figures; hieroglyphs; musical, cryptographic, mathematical, algebraic notations; and all other symbols which in our thoughts we use for the signified things” (Rutherford 1995). However, he states, not all of them are useful in the discovery of truths of reason. As Rutherford (1995) has pointed out, for Leibniz “alchemical and astrological symbols, Egyptian hieroglyphs, and Chinese characters are all deemed unsuitable for this purpose [discovery of truths of reason].” For Leibniz the most suitable characters ought to be “opposed to letters and Chinese characters which signify only conventionally, or ‘through the will of men’ ” (Rutherford 1995).
In Leibniz’s words: “It is obvious that if we could find characters or signs suited for expressing all our thoughts as clearly and exactly as arithmetic expresses numbers or geometrical analysis expresses lines, we could do in all matters insofar as they are subject to reasoning all that we can do in arithmetic and geometry. For all investigations which depend on reasoning would be carried out by the transposition of these characters and by species of calculus” (Rutherford 1995).
There is a Leibniz’s reply to Descartes’ assertion in (Couturat 1903).
Analysis refers to the philosophical method of resolving an idea or concept to its parts and it is opposed to synthesis, the method of composing concepts from their parts. Particularly, in the eighteenth century analysis was thought as the method of resolving problems by reducing them to equations and sometimes the very word analysis was used to name the proper scientific method (Hankins 1985). In fact, Condillac “insisted that all discovery must proceed by analysis, and that synthesis, which had been an important part of Newton’s method, could be employed only to demonstrate the validity of a mathematical proposition or experimental conjecture after the answer was known. Thus, he believed that the sciences, including mathematics, should be taught by analysis” (Hankins 1985). Isaac Barrow, Newton’s master, stated in 1665 that “Analysis […] seems to belong no more to Mathematics than to Physics, Ethics or any other Science. For this is only […] a certain Manner of using Reason in the Solution of Questions, and the Invention or Probation of Conclusions, which is often made use of in all Sciences” (Hankins 1985).
“Nous ne pensons qu’avec le secours des mots. Les langues sont de véritables méthods analytiques. L’algèbre la plus simple, la plus exacte et la mieux adaptée à son object de toutes les manières de s’énoncer, est à la fois une langue et une méthode analytique. L’art de raisonner se réduit à une langue bien faite. (Lavoisier 1789).
According to Leibniz: “I really believe that languages are the best mirror of the human mind, and that a precise analysis of the significations of words would tell us more than anything else about the operations of the understanding” (Rutherford 1995) and in a letter to Oldenburg (1673) he writes: “To anyone who wanted to speak or write about any topic, the genius of this language will supply not only the words but also the things. The very name of any thing will be the key to all that could reasonably be said or thought about it or done with it” (Rutherford 1995).
Elements of chemistry, in a new systematic order, containing all the modern discoveries. A scanned copy of the original manuscript may be obtained at http://moro.imss.fi.it/lavoisier/.
It is important to note that the new system of nomenclature, although mainly associated with Lavoisier, has its roots in the early work of Bergman (1772) (Crosland 1962).
In his Metaphysische Anfangsgründe der Naturwissenschaft of 1786, Kant states that “chemistry can become nothing more than a systematic art or experimental doctrine, but never science proper” for it uses no mathematics. However, as van Brakel has pointed out (Van Brakel 2006), Kant, in his later years (1796–1803), corrected his statement owing to Lavoisier’s influence in the Übergang von den Metaphysischen Anfangsgründen der Naturwissenschaft zur Physik. In fact, Kant changed to a big extent his position about chemistry to the point that he found it as a “paradigm for the method of critical philosophy” (Van Brakel 2006). However, the Übergang publication was delayed more than a century and its first abridged English edition appeared just until 1993. Therefore, it is not a surprise to find Kant’s first assertion on chemistry more long-lasting and influential than his corrected one (Van Brakel 2006).
As it was known at Lavoisier’s times (see Lavoisier 1965, p. 233).
About the classificatory power of a language, Lavoisier again refers to Condillac: “A child, is taught to give the name of tree to the first one which is pointed out to him. The next one he sees presents the same idea, and he gives it the same name. This he does likewise to a third and a fourth, till a last the word tree, which he first applied to an individual, comes to be employed by him as the name of a class or a genus, an abstract idea, which comprehends all trees in general. But, when he learns that all trees serve not the same purpose, that they do not all produce the same kind of fruit, he will soon learn to distinguish them by specific and particular names” (Lavoisier 1965). An interesting example of lacking a classificatory scheme for understanding and referring to the world is given by Borges in his tale “Funes el memorioso” (Borges 2008) (Funes, the memorious (Borges 1999)).
Lavoisier mentions this salt in (Lavoisier 1965, p. 231).
According to Lavoisier, besides meaningless, these names were sometimes even “improper, because they suggested false ideas.” As examples he called upon “oil of tartar per deliquium, oil of vitrol, butter of arsenic and of antimony, flowers of zinc, &c.” and he concluded by saying that those names are improper because “in the whole mineral kingdom, […] there exist no such thing as butter, oils, or flowers,” in fact they are “nothing less than rank poisons” (Lavoisier 1965).
It is interesting to see how Leibniz foresees the use of a lingua philosophica to gain understanding. In the following quote, Leibniz is highlighting the importance of giving more appropriate names to substances, in this case, to advance in the knowledge and understanding of substances themselves: “For my own part I admit (and the facts also proclaim) that certain chemical phenomena which time and opportunity reveal cannot now be derived from the name which we impose on (for example) gold, until we have obtained phenomena sufficient for determining the rest. It belongs to God alone in the first place to impose names of this intuitive sort of things. Nevertheless, the name which will be imposed on gold in this language will be the key to all those things which can be humanly, i.e. by reason and method, known about gold” (Rutherford 1995).
In so doing, he revived Boyle’s definition of chemical element, mooted more than a century before. In fact the idea of an element as the ultimate point of analysis has a long tradition that goes back to Aristotle’s times. For this philosopher an element was a body into which other bodies can be decomposed and which itself is not capable of being divided into others (Partington 1989).
An account of the substances eighteenth-century chemists used to analyse is given by U. Klein (2008).
“If we apply the term elements, […] to express […] the last point which analysis is capable of reaching, we must admit, as elements, all the substances into which we are capable, by any means, to reduce bodies by decomposition. Not that we are entitled to affirm, that these substances […] may not be compounded of two, or even of greater number of principles; but, since these principles cannot be separated, or rather since we have not hitherto discovered the means of separating them, they act with regard to us as simple substances, and we ought never supposed them compounded until experiment and observation has proved them to be so” (Lavoisier 1965).
U. Klein (2008) has shown the analytical methods eighteenth-century chemists used to obtain chemical elements. She also shows that the reverse process, namely the synthetic method based upon chemical elements was satisfactory to obtain binary substances belonging to the mineral kingdom. However, for more complex compounds, even inorganic ones, and mainly organic substances, their synthesis were rather complicated. Even though, Lavoisier was able to see that these experimental shortcomings would be overcome by future chemists, as it turned out to be in the nineteenth century with the successes of organic chemistry.
For transfermium elements eight criteria have to be met to claim that a new element has been obtained (Wilkinson et al. 1991).
See for example the “TABLE of the binary Combinations of Oxygen with simple Substances” in p. 184 of (Lavoisier 1965).
In Lavoisier’s times, acids were regarded as dissolved in water. H2SO4 used to be considered as sulphuric acid, SO3 dissolved in water.
In fact Lavoisier was not far from actually formulating algebraic equations. When dealing with the “acidification of metals,” Lavoisier wrote:
“Let any metallic substance be S. M.
Let any acid be \(\Upomega\)
Let water be ∇ \(\left[\ldots \right]\)
Then, we will have, for the general expression of any metallic solution
\(\left[\hbox{Metallic solution} =\right]\) S. M. (\(\nabla \Upomega\)).” After some additional steps Lavoisier ended up with more elaborated mathematical relations (Gillispie 1960). It is likely, according to Gillispie, that Lavoisier went toward a mathematization of chemistry owing to the influence several mathematicians had upon him, for example Laplace, with whom Lavoisier worked on the quantification of heat exchanges in chemical processes. Regarding algebra, Lavoisier stated: “Algebra is the analytic method per excellence: It has been contrived to facilitate the operations of the understanding, to make reasoning more concise, and to contract into a few lines what we would have needed many pages of discussion” (Swetz 1994). But the scientific interest was reciprocal between Lavoisier and his mathematical colleagues; according to Cajori “Through A. L. Lavoisier he [Lagrange] became interested in chemistry, which he found ‘as easy as algebra’ ” (Cajori 1991).
One may think that the algebra of sets discussed here missed a wealth of chemical information since what is presented does not take into account the stoichiometric relationships among the elements when forming compounds. However, the algebra can be refined in such a way that stoichiometry be regarded. In fact, there is a result where the patterns depicted by the periodic table are found by using binary compounds, their composition and stoichiometry. In such results, each chemical element is characterized by the list of elements which accompany the element in question in the different binary compounds studied. To take into account the stoichiometry of each combination, each chemical element of the mentioned list is weighted by the ratio of the stochiometric factors of the two elements in the binary compound. For further information the reader is referred to references (Leal and Restrepo 2009; Leal et al., under review).
Condillac criticized Newton for the use of the synthetic method in his research. However, it is noteworthy that Newton also used analysis (Newton 1671), for example when decomposing sunlight into colored rays to then by synthesis recombining them to obtain light again (Hankins 1985). A similar scientific method was applied by Lavoisier to chemistry more than a century later when finding chemical elements by analysis and generating chemical compounds by synthesis.
In 1854, George Boole came out with the book “An investigation of the laws of thought on which are founded the mathematical theories of logic and probabilities” (Boole 1958). There, he proposes to develop an algebraic system for modeling thought (Mazur 2005) that led to the development of symbolic logic, which is remarkably and algebra of sets, with rules and axioms similar to those of the arithmetic. It is also interesting to see that 188 years after Leibniz’s Ars Combinatoria, Boole was still thinking on a mathematization of thought, which shows that Leibniz ideas on a lingua philosophica were not abandoned. In fact, Chaitin, developer of the algorithmic information theory (AIT), acknowledges Leibniz’s influence in his work: “Some of the key ideas of AIT are clearly visible in embryonic form in his [Leibniz] 1686 Discourse on Methaphysics.” (Chaitin 2007)
The first triads Döbereiner studied (1817) were Ca–Sr–Ba, 12 years later he came up with Cl–Br–I. Afterwards, he found Li–Na–K and S–Se–Te (Scerri 2007).
That was the reason to refuse grouping nitrogen, carbon and oxygen, their lack of chemical resemblance. Same reason is behind the lack of acceptance of Kremer’s triads: O–Mg–S, S–Ca–Ti, and Ti–Fe–P (Scerri 2007), which although sound in numerical terms, lack of chemical resemblance.
This was a misinterpretation of the relation he had found: Chemical similarity-triadic numerical relationships. For he turned the relation into an implication, i.e. chemical similarity ⇔ triadic relationship. Hence, the fact of not having a numerical relationship made him conclude a lack of chemical similarity.
It is remarkable that De Chancourtois borned in mind a tendency to systematize the current knowledge, e.g. mineralogy, geology, geography, and, according to Scerri (Scerri 2007), was trying to produce a universal alphabet, which links him intellectually with Leibniz.
Meyer indicated that there should be an element of atomic weight greater than silicon’s by a difference of 44.55, which later was found out to be germanium. Mendeleev, by his side, predicted several properties for the afterwards known as germanium, gallium and scandium (Scerri 2007). His predictions for germanium were: atomic weight, density, molar volume, melting point, specific heat, valence, color, the method of isolation, reactivity with HCl and NaOH, reactivity with oxygen, empirical formula of the compound with oxygen (GeO2), GeO2 density, GeO2 solubility in acid, GeS2 solubility in water and ammonium sulfide, GeCl4 and GeEt4 boiling points (Greenwood and Earnshaw 2002).
Molecular structure is a particular kind of structure where X is a set of atoms and R a connectivity relation between pairs of them. Although this “structure” has allowed further development of chemical knowledge, the notion of “chemical structure” should not be reduced just to this particular case but should be regarded in its whole generality. Some examples of other chemical structures are the topology found for the chemical elements (Restrepo et al. 2004, 2006a), or the partially ordered sets that different kinds of chemicals form (Klein et al. 2008).
Klein has suggested that the periodic table is a multy-poset (Klein 1995), which means that the set of chemical elements is a partially ordered set where different partial order relations are intertwined one with each other (private communication).
In some cases the property of a substance can be successfully predicted by computing the average of the predecessors and successors of the substance in question (Klein and Bytautas 2000). This greatly resembles the averages used by Döbereiner in his triads; from a modern point of view Döbereiner dealt with total ordered sets of cardinality three. Klein has led forward the method by using averages on partially ordered sets. He also has used other interpolative/extrapolative methods such as the posetic cluster expansion (Ivanciuc et al. 2006) mooted by Rota in 1964 (Rota 1964) and the splinoid (Došlić and Klein 2005), based upon the splins used in numerical analysis.
A hyperdigraph is properly suited for representing chemical reactions wherein the reaction is complete and there is no room for chemical equilibria. In case of equilibria a double arc connecting two chemicals must be drawn, which indicates that those substances are doubly related, i.e. (R, P) and (P, R) are members of H (Definition 2).
Algorithmic complexity can be related to algorithmic information theory, which studies complexity on data structures. In few words a simple structure, say a string of characters, needs few instructions or words to be described or computed. In contrast, a complex structure needs more explanation to be described. An example of a simple structure is 11111111, which can be described by “repeat 1 eight times”; a more complex string is 10011101, which needs a further explanation “1, two 0s, then three 1s, a 0 and finally a 1”.
cf. footnote 19.
See footnote 21.
According to Schummer (2008), contemporary chemists, mainly concentrated on substances (stuff) with an associated “molecular structure”, are overlooking a wealth of raw material (stuff) proper of Chemistry. Hence, it seems that a big part of the reaction network is not being regarded.
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Acknowledgments
Special thanks are given to A. Bernal for giving us drafts of his work and for checking out the accuracy of the definitions shown in this manuscript. G. Restrepo thanks the Universidad de Pamplona for the financial support given during this research.
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Restrepo, G., Villaveces, J.L. Chemistry, a lingua philosophica. Found Chem 13, 233–249 (2011). https://doi.org/10.1007/s10698-011-9123-z
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DOI: https://doi.org/10.1007/s10698-011-9123-z