Characteristica universalis

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Characteristica Universalis from Latin is commonly interpreted as "Universal Character" in English. It is a recurring concept in the writings of Gottfried Leibniz. When writing in French, he sometimes employed the phrase specieuse generale to the same effect. The concept is sometimes paired with his notion of a calculus ratiocinator and with his plans for an encyclopeadia as a compendium of all human knowledge.

Contents 1 Leibniz's universal pictographic language 1.1 A universal language of logic and international communication 1.2 A universal language of science 1.2.1 Metaphysics 1.2.2 Science 1.3 The "general algebra" 1.4 His diagrammatic reasoning 1.5 Leibniz loses heart 2 Three criteria 3 Gödel alleges conspiracy 4 Related 17th century projects 5 More recent projects 6 See also 7 References 8 External links

[edit] Leibniz's universal pictographic language

"We have seen that the characteristic was inseparable in Leibniz's mind from the encyclopedia. The former presupposed the latter, or at least had to be established at the same time and developed in parallel with it. Conversely, once established, the characteristic would be the instrument of the encyclopedia and the key to all the sciences."

- Louis Couturat, The Logic of Leibniz in accordance with unpublished documents, Chpt 5: The Encyclopedia, §1 Translated by Donald Rutherford and R. Timothy Monroe, copyright (c) 1997-2002, all rights reserved.

Inconsistency, vagueness, and a lack of specifics in both English language translations and modern English language interpretations of Leibniz's writings render a clear exposition difficult. As with Leibniz's calculus ratiocinator two different schools of philosophical thought have come to emphasise two different aspects that can be found in Leibniz's writing. The first point of view emphasizes logic and language, and is associated with analytic philosophy and rationalism. The second point of view is more in tune with Couturat's views as expressed above, which emphasize science and engineering. This point of view is associated with synthetic philosophy and empiricism. Either or both of these aspects Leibniz hoped would guide human reasoning like Ariadne's thread and thereby suggest solutions to many of humanity's urgent problems.

[edit] A universal language of logic and international communication

Many Leibniz scholars writing in English seem to agree that he intended his characteristica universalis or "universal character" to be a form of pasigraphy, or ideographic language. This was to based on a rationalised version of the 'principles' of Chinese characters, as Europeans understood these characters in the seventeenth century. From this perspective it is common to find the characteristica universalis associated with contemporary Universal language projects like Esperanto, and formal logic projects like Frege's Begriffsschrift. The global expansion of European commerce in Leibniz's time provided mercantilist motivations for a universal language of trade so that traders could communicate with any natural language. The mercantile motivation informed some of the research into universal characters, and has also been present in the association of universal language projects with computer metalanguages like XML (Roy and Ramanujan 2000, Yen et. al 2002).

Others, such as Jaenecke, for example, have observed that Leibniz also had other intentions for the characteristica universalis, and these aspects appear to be a source of the aforementioned vagueness and inconsistency in modern interpretations. According to Jaenecke,

the Leibniz project is not a matter of logic but rather one of knowledge representation, a field largely unexploited in today's logic-oriented epistemology and philosophy of science. It is precisely this one-sided orientation of these disciplines, which is responsible for the distorted picture of Leibniz's work found in the literature. (Jaenecke 1996).

As Couturat wrote, Leibniz criticized the linguistic systems of George Dalgarno and John Wilkins for this reason since they focused on

...practical uses rather than scientific utility, that is, for being chiefly artificial languages intended for international communication and not philosophical languages that would express the logical relations of concepts. He favors, and opposes to them, the true "real characteristic," which would express the composition of concepts by the combination of signs representing their simple elements, such that the correspondence between composite ideas and their symbols would be natural and no longer conventional. (1901. chp 3)

[edit] A universal language of science

Leibniz said that his goal was an alphabet of human thought, a universal symbolic language (characteristic) for science, mathematics and metaphysics. According to Couturat, " In May 1676, he once again identified the universal language with the characteristic and dreamed of a language that would also be a calculus—a sort of algebra of thought." (1901, chp 3.). This characteristic was a universalisation of the various "real characteristics." Couturat wrote that Leibniz gave Egyptian and Chinese hieroglyphics and chemical signs as examples of real characteristics writing:

This shows that the real characteristic was for him an ideography, that is, a system of signs that directly represent things (or, rather, ideas) and not words, in such a way that each nation could read them and translate them into its own language. (1901, chpt. 3)

In a footnote Couturat added:

Elsewhere Leibniz even includes among the types of signs musical notes and astronomical signs (the signs of the zodiac and those of the planets, including the sun and the moon). It should be noted that Leibniz sometimes employs planetary signs in place of letters in his algebraic calculations. (ibid,)

The universal "representation" of knowledge would therefore combine lines and points with "a kind of pictures" (pictographs) to be manipulated by means of his calculus ratiocinator.

[edit] Metaphysics

Rogers emphasised the metaphysical aspect of the characteristica universalis by relating it to the "elementary theory of the ordering of the reals," defining it as "a precisely defineable system for making statements of science" (Rogers 1963: 934). Universal language projects like Esperanto, and formal logic projects like the Begriffsschrift are not commonly concerned with the epistemic synthesis of empirical science, mathematics, pictographs and metaphysics in the way Leibniz described (below). Hence scholars have had difficulty in showing how such projects embody Leibniz's full vision for his characteristica. Unfortunately there is no accessible explanation of this matter currently available in English language philosophy.

[edit] Science

In the domain of science, Leibniz aimed for his characteristica to form maps, diagrams or pictures, depicting any system at any scale, and understood by all regardless of native language. Leibniz wrote:

"And although learned men have long since thought of some kind of language or universal characteristic by which all concepts and things can be put into beautiful order, and with whose help different nations might communicate their thoughts and each read in his own language what another has written in his, yet no one has attempted a language or characteristic which includes at once both the arts of discovery and judgement, that is, one whose signs and characters serve the same purpose that arithmetical signs serve for numbers, and algebraic signs for quantities taken abstractly. Yet it does seem that since God has bestowed these two sciences on mankind, he has sought to notify us that a far greater secret lies hidden in our understanding, of which these are but the shadows." (Unless stated otherwise, all Leibniz quotations are from his On the General Characteristic as translated in Loemker 1969: 221-25. This passage is from p. 222.)

P.P.Weiner raised an example of a large scale application of Leibniz's characteristica to climatic science. A weather-forecaster invented by Athanasius Kirchner, "interested Leibniz in connection with his own attempts to invent a universal language" (1940).

Leibniz talked about his dream of a universal scientific language at the very dawn of his career, as follows:

"We have spoken of the art of complication of the sciences, i.e., of inventive logic... But when the tables of categories of our art of complication have been formed, something greater will emerge. For let the first terms, of the combination of which all others consist, be designated by signs; these signs will be a kind of alphabet. It will be convenient for the signs to be as natural as possible--e.g., for one, a point; for numbers, points; for the relations of one entity with another, lines; for the variation of angles and of extrmities in lines, kinds of relations. If these are correctly and ingeniously established, this universal writing will be as easy as it is common,and will be capable of being read without any dictionary; at the same time, a fundamental knowledge of all things will be obtained. The whole of such a writing will be made of geometrical figures, as it were, and of a kind of pictures -- just as the ancient Egyptians did, and the Chinese do today. Their pictures, however, are not reduced to a fixed alphabet... with the result that a tremendous strain on the memory is necessary, which is the contrary of what we propose." (On The Art of Combination, 1666, translated in Parkinson 1966: 10-11)

Rescher, reviewing Cohen's article, wrote that:

"Leibniz's program of a universal science (scientia universalis) for co-ordinating all human knowledge into a systematic whole comprises two parts: (1) a universal notation (characteristica universalis) by use of which any item of information whatever can be recorded in a natural and systematic way, and (2) a means of manipulating the knowledge thus recorded in a computational fashion, so as to reveal its logical interrelations and consequences (the calculus ratiocinator)."

Fearnley-Sander (1986) went one step further, defining Leibniz's characteristica as a combination of the algebra of logic (which Fearnley-Sander defined as the calculus ratiocinator) and the algebra of geometry (defined as the characteristica geometrica). Fearnley-Sander suggested that this combination had "come to pass" with the rise of universal algebra. It should be noted that "universal algebra," the study of the mathematical and logical properties of algebraic structures generally, has nothing to do with the subject of this entry.

[edit] The "general algebra"

Near the end of his life, he wrote that combining metaphysics with mathematics and science through a universal character would require creating what he called:

"... a kind of general algebra in which all truths of reason would be reduced to a kind of calculus. At the same time, this would be a kind of universal language or writing, though infinitely different from all such languages which have thus far been proposed; for the characters and the words themselves would direct the mind, and the errors -- excepting those of fact -- would only be calculation mistakes. It would be very difficult to form or invent this language or characteristic, but very easy to learn it without any dictionaries" (letter to Nicolas Remond, 10 January 1714, in Loemker 1969: 654. Translation revised.)

He hoped his pictorial algebra would advance the scientific treatment of qualitative phenomena, thereby constituting "that science in which are treated the forms or formulas of things in general, that is, quality in general." (On Universal Synthesis and Analysis, 1679, in Loemker 1969: 233).

[edit] His diagrammatic reasoning

Since the characteristica universalis is diagrammatic and employs pictograms, the diagrams in Leibniz's work warrant close study. On at least two occasions, Leibniz illustrated his philosophical reasoning with diagrams (reproduced in Loemker 1969: 83, 366). One diagram, the frontispiece to his 1666 De Arte Combinatoria (On the Art of Combinations), represents the Aristotelian theory of how all material things are formed from combinations of the elements earth, air, fire, and water. These four elements make up the four corners of a diamond. Opposing pairs of these are joined by a bar labeled "contraries" (earth-air, fire-water). At the four corners of the superimposed square are the four qualities defining the elements. Each adjacent pair of these is joined by a bar labeled "possible combination"; the diagonals joining them are labeled "impossible combination." Starting from the top, fire is formed from the combination of dryness and heat; air from wetness and heat; water from coldness and wetness; earth from coldness and dryness. This diagram is more clearly reproduced in Saemtliche Schriften und Briefe, Reihe VI, Band 1, p. 166.

[edit] Leibniz loses heart

Leibniz rightly saw that creating the characteristica would be difficult, fixing the time required for devising it as follows: "I think that some selected men could finish the matter in five years" (Loemker 1969: 224), later remarking: "And so I repeat, what I have often said, that a man who is neither a prophet nor a prince can ever undertake any thing of greater good to mankind of more fitting for divine glory". (Loemker 1969: 225). But later in life, a more sober note emerged. In a March 1706 letter to the Electress Sophia of Hanover, the spouse of his patron, he wrote:

"It is true that I once planned a new method of calculation proper to subjects having nothing in common with mathematics, and if this manner of Logic were put into practice, all reasoning, even analogical ones, would be carried out in a mathematical way. Then modest intellects could, with diligence and good will, not accompany but at least follow greater ones. For one could always say "let us calculate" and judge properly, insofar as reason and the data can furnish us the means to do so. But I do not know whether I will ever be able to execute such a project, one requiring more than one hand, and it would even seem that humanity is not yet sufficiently mature to pretend to the advantages to which this method could lead." (translation of a passage reproduced in Couturat 1901: 118, fn 2)

In an another 1714 letter to Nicholas Remond, he wrote:

"I have spoken to the Marquis de l'Hôpital and others about my general algebra, but they have paid no more attention to it than if I had told them about a dream of mine. I should have to support it too by some obvious application, but to achieve this it would be necessary to work out at least a part of my characteristic, a task which is not easy, especially in my present condition and without the advantage of discussions with men who could stimulate and help me in work of this nature." (Loemker 1969: 656)

[edit] Three criteria

Cohen (1954) set out three criteria which any project for a philosophical language would need to meet before it could be considered a version of the characteristica universalis. In setting out these criteria, Cohen made reference to the concept of "logistic". This concept is not the same as that used in statistical analysis. In 1918, Clarence Irving Lewis, the first English-speaking logician to translate and discuss some of Leibniz's logical writings, elaborated on "logistic" as follows:

Logistic may be defined as the science which deals with types of order as such. It is not so much a subject as a method. Although most logistic is either founded upon or makes large use of the principles of symbolic logic, still as science of order in general does not necessarily presuppose or begin with symbolic logic. (Lewis 1960: 3. Also see pp. 7-9)

Following from this Cohen stipulated that the universal character would have to serve as a:

• "International auxiliary language" enabling persons speaking different languages to communicate with one another;
• Symbolism for the exact and systematic expression of all present knowledge, making possible a "logistic" treatment of science in general. This symbolism could also be expanded to accommodate future knowledge;
• Instrument of discovery and demonstration.

These criteria together with the notion of logistic reveal that Cohen and Lewis both associated the characteristica with the methods and objectives of General systems theory.

[edit] Gödel alleges conspiracy

Because Leibniz never described the characteristica universalis in operational detail, many philosophers have deemed it an absurd fantasy. In this vein, Parkinson wrote:

"Leibniz's views about the systematic character of all knowledge are linked with his plans for a universal symbolism, a Characteristica Universalis. This was to be a calculus which would cover all thought, and replace controversy by calculation. The ideal now seems absurdly optimistic..." (Parkinson 1973: ix )

The logician Kurt Gödel, on the other hand, believed that the characteristica universalis was feasible, and that its development would revolutionize mathematical practice.(Dawson 1997) He noticed, however, that a detailed treatment of the characteristica was conspicuously absent from Leibniz's publications. It appears that Gödel assembled all of Leibniz's texts mentioning the characteristica, and convinced himself that some sort of systematic and conspiratorial censoring had taken place, a belief that became obsessional. Gödel apparently had failed to appreciate the magnitude of the task facing the editors of Leibniz's manuscripts, given that Leibniz left about 15000 letters and 40000 other manuscripts. Even now, most of this huge Nachlass remains unpublished.

[edit] Related 17th century projects

Others in the 17th century, such as George Dalgarno, attempted similar philosophical and linguistic projects, some under the heading of mathesis universalis. A notable example was John Wilkins, the author of An Essay towards a Real Character and a Philosophical Language, who wrote a thesaurus as a first step towards a universal language. He intended to add to his thesaurus an alphabet of human thought (an organisational scheme, similar to a thesaurus or the Dewey decimal system), and an "algebra of thought," allowing rule-based manipulation. The philosophers and linguists who undertook such projects often belonged to pansophical (universal knowledge) and scientific knowledge groups in London and Oxford, collectively known as the "Invisible College" and now seen as forerunners of the Royal Society.

[edit] More recent projects

A wide variety of constructed languages have emerged over the past 150 years which appear to support many of Leibniz's intuitions. If indeed they do support Leibniz's vision of unified science, then the remaining question is whether Ariadne's unifing thread can be discerned among these projects.

• Raymond F. Piper (1957; 432-433) claimed that O.L. Reiser's Unified Symbolism for World Understanding in Science (1955), an expansion of his A Philosophy for World Unification (1946), was inspired by Leibniz's Characteristica Universalis, and believed necessary for world understanding and unbiased communications so that "war may eventually be eliminated and that a worldwide organism of peaceful human beings may gradually be established." (Piper Ibid.).

• Boolean algebra and group theory in the 19th century proved correct Leibniz's intuition that algebraic methods could be used to reason about qualitative and non-numerical phenomena. Specifically, the members of the universal set of a Boolean algebra or group need not be numbers. Moreover, a fair bit of philosophy and theoretical science can be formalized as axiomatic theories embodying first-order logic and set theory. Note also how model theory has been employed to formalize and reason about such emphatically nonumerical subjects as semantics and pragmatics of natural languages. But these approaches have yet to result in any pictographic notations.

• Palko, Gy Bulcsu (1986) considered structured analysis for analyzing and designing hierarchic systems by using an iconic language, and suggested that such was an application of the universal characteristics Leibniz's project to the language of structured analysis and the formalization of an iconic control system.

• Kluge (1980) argued that Frege's landmark Begriffsschrift was consciously inspired by the characteristica universalis.

• Even though Charles Peirce, a founder of semiotics, believed that all reasoning was diagrammatic, the relation, if any, of the characteristica to his existential graphs and to semiotics has yet to be explored in the English literature.

• Several aspects of logical positivism, specifically:
  • The first-order theories of Rudolf Carnap's Aufbau (1928, English translation 1967) and of its successor, Goodman (1977), are Leibnizian in their sweep and ambition, although Leibniz would have taken strenuous exception to Carnap's resolute hostility to all metaphysics.
  • The unification of science movement of the 1930s, led by Otto Neurath, Rudolf Carnap, and Charles W. Morris, and later by Edward Haskell et al, bears comparison with the characteristica.
  • Otto Neurath's isotype pictogram system, and "international picture language." [1] Neurath wrote, "One of our slogans in Vienna was, 'Words divide: Pictures unite.'"

• The following attempts to recast parts of theoretical science as axiomatics first-order theories can be viewed as attempts to develop parts of the characteristica:
  • Special relativity, by Hans Reichenbach, Rudolf Carnap, and others during the 1920s (Carnap 1958: 197-212);
  • Biology, by Joseph Woodger (1937), also during the 1930s (Carnap 1958: 213-20):
  • Mechanics, by Suppes (1957: 291-305) and others during the 1950s.

• The objectives of the constructed language Toki Pona, and the 'Symbolator' or 'idea-computer' (Goppold 1994) resemble in some respects a less ambitious version of the characteristica universalis.

• Connections with the Jewish Cabbala, and the International auxiliary language policy of the Bahai faith have also been made.

• The characteristic has also been claimed as an ancestor of the pictographic Energy Systems Language and associated Emergy Synthesis of Odum's Systems Ecology (Cevolatti and Maud, 2004). The Energy Systems Language combines lines and points with "a kind of pictures" manipulated by means of digital computers and software packages like EXTEND(tm) (Odum, Odum, and Peterson 1995), and Valyi's Emergy Simulator. It was designed to provide a general systems language affording quantitative accounting and mathematical simulation of qualitative energy relationships between ecological entities: "that science in which are treated the forms or formulas of things in general, that is, quality in general." . A general algebra known as the emergy algebra emerged out of the repeated use of this language in modelling and simulating the energetic principles of ecological relations. In particular it afforded the discovery and demonstration of the maximum power principle, suggested as the fourth law of thermodynamics. If this ancestoral claim is granted, then simulation software like EXTEND(tm) and Valyi's Emergy Simulator can be seen as combining the characteristica and the Calculus ratiocinator, if and only if the digital computer is interpreted as a physical embodiment of the calculus ratiocinator.

• The work of Mario Bunge on the border of physics and metaphysics seems grounded in metaphysical presuppositions similar to those of Leibniz's characterisitica .

• Charles Bliss's Blissymbolics, presently used as an 'alternative and augmentative language' for disabled people but originally intended as an International 'Auxlang', is said to be in the mold of the Characteristica. See Bliss, Charles K. 1978. Semantography: Blissymbolics. 3rd enlarged edition. Sydney: Semantography-Blissymbolics Publications. ISBN 0-9595870-0-4 (Blissymbolics Communication International, http://www.blissymbolics.org).

[edit] See also

• Gottfried Leibniz
• Calculus ratiocinator
• Begriffsschrift
• existential graph
• alphabet of human thought
• mathesis universalis
• Universal language
• Unified Science
• Isotype
• Energy Systems Language
• Emergy Synthesis

[edit] References

On Leibniz's lifelong interest in the characteristica and the like, see the following texts in Loemker (1969): 165-66, 192-95, 221-28, 248-50, and 654-66. On the characteristica, see Rutherford (1995) and the still-classic discussion in Couturat (1901: chpts. 3,4). Also relevant to the characteristica is Mates's (1986: 183-88) discussion of what he called the lingua philosophica.

• Brown, M. T., 2004, "A picture is worth a thousand words: energy systems language and simulation," Ecological Modelling 178: 83-100.
• Bulcsu P. G., 1986, "Formalization of the Iconic Language for Structured Analysis and Symbolic Logic," [A STRUKTURALT ANALIZIS (SA) IKONIKUS NYELVENEK FORMALIZALASA ES A SZIMBOLIKUS LOGIKA.] Meres es Automatika 34: 315-22.
• Rudolf Carnap, 1958. Introduction to Symbolic Logic with Applications. Dover.
• -----, 1967. The Logical Structure of the World: Pseudoproblems in Philosophy. University of California Press. Translation of his 1928 Der Logische Aufbau der Welt. Leipzig: Felix Meiner Verlag.
• Cohen, C. Jonathan., 1954, "On the project of a universal character," Mind (New Series) 63: 249.
• Louis Couturat, 1901. La Logique de Leibniz. Paris: Felix Alcan. Donald Rutherford's English translation in progress.
• Dascal, M., 1987. Leibniz: Language, Signs and Thought, A Collection of Essays. Philadelphia: John Benjamins.
• Martin Davis, 2001. Engines of Logic: Mathematicians and the Origin of the Computer. W W Norton.
• Dawson, J. W. Jr., 1997. Logical Dilemmas: The Life and Work of Kurt Gödel. Wellesley MA: A. K. Peters.
• Fearnley-Sander, Desmond, 1982, "Hermann Grassmann and the Prehistory of Universal Algebra," The American Mathematical Monthly 89: 161-66.
• Nelson Goodman, 1977 (1951). The Structure of Appearance. Reidel.
• Jaakko Hintikka, 1997. Lingua Universalis vs. Calculus Ratiocinator. An ultimate presupposition of Twentieth-century philosophy. Kluwer.
• Jaenecke, P., 1996, "Elementary principles for representing knowledge," Knowledge Organization 23: 88-102.
• Kluge, E. H. W., 1980, "Frege, Leibniz and the notion of an ideal language," Studia Leibnitiana 12: 140-54.
• Clarence Irving Lewis, 1960 (1918). A Survey of Symbolic Logic. Dover.
• Loemker, Leroy, ed. and trans., 1969. Leibniz: Philosophical Papers and Letters. Synthese Historical Library. Dordrecht: D. Reidel.
• Mates, Benson, 1986. The Philosophy of Leibniz. Oxford Univ. Press.
• Odum, E.C., Odum, H.T. and Peterson, N.S., 1995, "Using Simulation to Introduce the Systems Approach in Education" in Hall, C.S., ed., Maximum Power: The Ideas and Applications of H. T. Odum. Colorado Univ. Press: 346-52.
• Parkinson, G. H. R., ed. and trans., 1966. Leibniz: Logical Papers. Oxford Univ. Press.
• ------, and Mary Morris, trans., 1973. Leibniz: Philosophical Writings. London: J M Dent.
• Piper, R. F. 1957, "Review of 'Unified Symbolism for World Understanding in Science' by Oliver L. Reiser," Philosophy and Phenomenological Research 17: 432-33.
• Rutherford, Donald, 1995, "Philosophy and language" in Jolley, N., ed., The Cambridge Companion to Leibniz. Cambridge Univ. Press.
• Reiser, Oliver Leslie 1946. A philosophy for world unification;: Scientific humanismas an ideology for cultural integration, Haldeman-Julius Publications.
• ------, 1940. The Promise of Scientific Humanism Toward a Unification of Scientific, Religious, Social and Economic Thought, Oskar Piest.
• ------, 1955. Unified symbolism for world understanding in science: Including Bliss symbols (semantography) and logic, cybernetics and semantics, Semantography Pub. Co.
• Nicholas Rescher, 1954, "Review of On the Project of a Universal Character by Jonathan Cohen," Journal of Symbolic Logic 19: 133.
• Hartley Rogers, Jr., 1963, "An Example in Mathematical Logic," The American Mathematical Monthly 70: 929-45.
• Roy, J., Ramanujan, A., 2000, "XML: Data's universal language," IT Professional 2: 32-36.
• Patrick Suppes, 1999 (1957). Introduction to Logic. Dover.
• Jean van Heijenoort , 1967, "Logic as calculus and logic as language," Synthese 17: 324-330. Reprinted in Hintikka (1997).
• Wiener, Philip P., 1940, "Leibniz's Project of a Public Exhibition of Scientific Inventions," Journal of the History of Ideas 1: 232-240.
• Woodger, Joseph H., 1937, The axiomatic method in biology, Cambridge University Press.
• Yen, D.C., Huang, S.-M., Ku, C.-Y. 2002, The impact and implementation of XML on business-to-business commerce, Computer Standards and Interfaces 24: 347-62.

[edit] External links

• Cevolatti, D., and Maud, S., 2004, "Realising the Enlightenment: H. T. Odum's Energy Systems Language qua G. W. v. Leibniz's Characteristica Universalis," Ecological Modelling 178: 279-92.
• Corazzon, Raul, 2005, "Language as Calculus vs. Language as Universal Medium". Includes bibliography, links to online papers, and passages from the writings of Jaakko Hintikka and Jean Van Heijenoort.
• Goppold, P. A., 1994, "Characteristica Universalis and the Origin of the Symbolator," 240 S., Projekt.
• Rotha, Paul, 1946, "From Hieroglyphs to Isotypes".
• Smith, Barry, 1978, "An Essay in Formal Ontology," Grazer Philosophische Studien 6: 39-62.
• --------, 1990, "Characteristica Universalis" in K. Mulligan, ed., Language, Truth and Ontology, (Philosophical Studies Series). Kluwer: 50-81.