Finding the Antipodes: Mathematical Constructivism and the Changing Logic of Cartographic Objects, 1960-1975

Abstract of my AAG 2012 Paper

In mathematics everything is algorithm and nothing is meaning; even when it doesn't look like that because we seem to be using words to talk about mathematical things.
Even these words are used to construct an algorithm. ---Ludwig Wittgenstein

...a proof of the existence of a magnitude can only be seen as completely rigorous if it contains a method by which the magnitude whose existence is being claimed can really be found. ---Leopold Kronecker

We look upon maps not only as stores of spatially ordered information, but also as a means for the graphical solution of certain problems for which the mathematics proves to be intractable. --William Warntz

In the early years of computer cartography new levels of abstraction entered into the field of geographical analysis through the algorithmic development of theorems from pure mathematics. In an attempt to answer previously intractable geographical questions, concepts from pure mathematics, like existence theorems, whose basic logical structure contains statements that confirm or deny the existence of particular sets of mathematical objects, were employed in various computer mapping schemes. The development of these programs injected high levels of topological and algebraic abstraction into geographical analysis and changed the basic ontology of geographic objects. Existence theorems, although they provide logical proof for whatever mathematical entity they are claiming existence for, do not however, necessarily provide a way to find or calculate those objects. In the field of pure mathematics existence theorems had long been objects of controversy from both a practical and philosophical perspective and their use sparked debates among many mathematicians. Mathematicians and philosophers, like Leopold Kronecker and Ludwig Wittgenstein, questioned the utility of a mathematical proof that provided no algorithmic way to find the mathematical object whose existence is claimed, while others such as David Hilbert and Richard Dedekind, saw no conceptual or philosophical difficulties with their use. This debate among the so-called constructivists, like Wittgenstein, who believed that in mathematics “everything is algorithm”, and the formalists like Hilbert, has left a large body of philosophical literature that has deeply analyzed the ontology of mathematical objects. [1]

In the fields of geography and cartography, these theorems entered into early computer systems through the construction of practical algorithms that calculated particular sets of objects useful in geographic analysis. Two important papers that can be seen as case studies in the use of constructivist forms of existence theorems in early computer cartography were published in the series Harvard Papers in Theoretical Geography by William Warntz and his associates at the Harvard Lab for Computer Graphics and Spatial Analysis in the late 1960s and early 1970s. This series of papers developed algorithmic constructions of many existence theorems and two of the most interesting, because of the sheer complexity of the mathematics, the Borsuk-Ulam Theorem and the Ham Sandwich Theorem, were applied to real world geographic problems [2].
Besides using existence theorems, mathematical cartographers would also begin to re-conceptualize on a more general level questions about the use of pure mathematics and its role in defining the diagrammatic logic of maps. In an early lecture, later written as a discussion paper for the Michigan Inter-University Community of Mathematical Geographers, Warntz says that, "More than ever before geographers are using the tools of calculus, probability, topology, symbolic logic, the various algebras, geometries, for example, are being taken more literally than ever before." He elaborates on these comments by explaining to the reader that something as abstract and foreign to geography as Venn diagrams are being taken, "in a far more literal sense than they were originally intended and by substituting real space and attendent phenonema for ideal space and by insisting on the utilization of all geometric properties involved as well as just the topological ones, geographers can reinterpret, add to, and refine the conventional concepts in the methodology of uniform regional geography and provide it with a basis in logic." [3]

Many geographers at the time would push the concept logic form and notions from set theory further into geographic analysis and not just in the sense of a useful analogy. In a paper written for one of the classic compilations texts from early years mathematical geography called, The Philosophy of Maps, Warntz and others like Waldo Tobler, and William Bunge, would change not only the vocabulary used in analysis but would also alter the very form of its expression. In an article in the collection, called Some Elementary and Literal Notions About Geographical Analysis and Extended Venn Diagrams, Warntz would say that, "Maps showing regional classification can be regarded as logic diagrams. Mapping of sets is a general mathematical concept. Geographical mapping is merely a special case of this." [4] Warntz here sees almost a mereological or mereotopological relationship between the spatial extent of Venn diagrams and their isomorphic counterparts of geographic regions.

John Venn

It is quite remarkable that the two systems of logic that Warntz draws on in this paper, Venn diagrams and existential graphs, are both visual and not symbolic logical systems. Most of the work done in logic during the 20th century has focused on symbolic systems with little research, at least until quite recently, on the heterogeneous reasoning of the type Warntz is advocating. He says that, "It is part of our purpose here to extend the use of such diagrams to the mapping of geographical regions by making use of properties already inherent in Venn diagrams but as yet unutilized... We intend to apply spatial properties literally to real spatial distributions on the earth's surface..."


Venn diagrams can grow to extremely complex forms depending on the number of sets one is dealing with and recent research on the use of logical diagrams has shown that Warntz was ahead of his time in thinking that the spatial and geometrical component of logical diagrams would be useful analogs for spatial maps. [5]
As stated above, Warntz' paper calls on the work of Charles Sanders Peirce (above) and his existential logic diagrams, which he sees as mappings from non-spatial sets to geographical maps. Looking at the complexity of Peirce's systems, there is both an alpha and beta form depending on the required complexity, one wonders how deeply Warntz explored the subject of existential graphs. An important aspect of these graphs that Warntz thought useful for regional geographic analysis was the fact that a logic diagram can be drawn as a two-dimensional figure with spatial relations that are isomorphic with the structure of some logical statement. This is very important if one is going to try to apply set theory of the type Warntz is envisioning here, simply because these spatial relations are usually of a topographic nature.Logic diagrams, especially the type developed by Peirce (simple examples shown above with a page frm Peirce's notebook below), stand in the same relation to the various logical algebras as maps of areas stand in relation to their particular algebraic functions; they are simply other ways of symbolizing the same basic structure. [6]

In much of what Warntz has to say here we are reminded of the long way we have come when talking about set theory, topology and the formal properties of spatial structures and their relationship to cartography. One only has to look at books like Varzi and Casati's, Parts and Places: the Structures of Spatial Representations (MIT, 1999) [7] to get a feel for how our language and conceptual grasp of these topics has improved since Warntz and others involved in the early development of computer cartography were experimenting with what at the time were radically new ideas.
The current project envisioned here, which grew out of my research for the 20th century volume of the History of Cartography, will provide a mathematical and philosophical analysis of both of the Harvard papers mentioned above, along with others from this formative period that apply set theory and logical analysis, in an effort to show not only how constructivist methods migrated from mathematics to geography, but also to show how these new levels of abstraction changed the foundational ontology of geographic and cartographic objects. Using the philosophical debates that took place over things like existence theorems in the mathematical literature as a basis, this study will show that a foundational shift in the ontology of geographical objects opened the door to new conceptualizations of geographic space and formed the theoretical basis for the development of spatial logics and the current use of topological and abstract algebraic methods in geographical analysis.

[1] It is interesting to note that many early mathematical geographers had an interest in Wittgenstein. Waldo Tobler, in a private communication, told me recently that he was persuaded by Peter Gould (1932-2000) to take up the reading of Wittgenstein.
[2] The two papers are; Geography and an Existence Theorem: A Cartographic computer solution to the localization on a sphere of sets of equal-valued antipodal points for two-continuous distributions with practical applications to the real earth (1968) and The Sandwich Theorem: A basic one for geography (1971).
[3] A Note on Surfaces and Paths and Applications, William Warntz, Discussion Paper Number 6, Michigan Inter-University Community of Mathematical Geographers, 1965.
[4] The Philosophy of Maps, edited by John Nystuen, Michigan Inter-University Community of Mathematical Geographers Discussion Paper 12, 1968.
[5] For recent research on the logical status of Venn diagrams and the nature of spatial logic see, Eric Hammer (1995), Logic and Visual Information, Stanford CA: Center for the Study of Logic and Information; Nathaniel Miller (2007), Euclid and His Twentieth Century Rivals: Diagrams in the Logic of Euclidean Geometry, Studies in the Theory and Applications of Diagrams, Stanford CA: Center for the Study of Logic and Information and Sun-Joo Shin (1994), The Logical Status of Diagrams, New York: Cambridge University Press.
[6] For more on Peirce's Existential Graphs see Sun-Jo Shin's seminal study, The Iconic Logic of Peirce's Graphs, MIT Press, 2002.
[7] Achille Varzi and Roberto Casati, Part and Places: The Structure of Spatial Representations, MIT Press. 1999.

A Cartographic Commentary on the Henchir Mettich Inscription from Central Tunisa: Roman Surveying in the Medjerda Valley

The foundations of the science of land measurement lies in practical experience, since the truth about sites or area cannot be expressed without lines that can be geometrically measured.
--Frontinus, De arte mensoria

As noted in my previous post, the inscription from Henchir Mettich in the Bagradas valley of Central Tunisia is important as a window on the landscape archaeology of the region and the history of agriculture during the Roman empire, but it is also important for the history of cartography as it relates to Roman law and the running of imperial estates in the second century AD. As one can see in the example of face one below, this coming from a series of photographs that I took last year in Tunisia, the inscription is badly damaged.























The inscription was published as CIL 25902 and very different transciptions of it can be found in Kehoe's, The Economics of Agriculture on Roman imperial Estates in North Africa and in van Nostrand's, The Imperial Domains of Africa Proconsularis: an epigraphical study.


The most important part of the inscription from a cartographic perspective is to be found on side one of the column shown below as published in the CIL and as a lithograph from Toutain's L'Inscription D'Henchir Mettich: un noveau document sur la propriete agricole dans L'Afrique Romaine.










































The text on the first side of the column talks in some detail about the subject of subseciva or unallocated lands. The word is the subject of much discussion in the Corpus Agrimensorum and generally means lands unsuitable for allocation to settlers, either sirutated between the centuriae and the outer boundary of a communities territory or within centuriae. The word literally means "cut off" or "cut away below".























For example, Frontinus talks about the fact that he knows of fifteen different types of 'land dispute',
"...the position of boundary markers, a straight line boundary, boundary, site, area, ownership, possession, alluvial land, territorial juristdiction, subseciva, public places, places omitted and not enclosed, sacred and religious places, control of rain water and rights of way."

He continues later in his text on 'land disputes',
"A dispute over subseciva occurs when some or all of a centuria has not benn allocated and is possessed. Or if an adjacent landholder or someone else occupies any land from the edge of the allocated area, this also comes under disputes involving subseciva."

Hyginus also has much to say about subseciva. In his descritpion of categories of land he tells us that,
"Certain areas that protrude beyond the type of land which is curved or has angles, and are divided off by straight lines, are called subseciva, that is, pieces of land that remain when the boundary lines have cut them off and retain the character of peripheral areas."

The discussion of subseciva in the Henchir Mettich inscription begins after the dedication ends in line 6.

...qui eorum [i]ntra fundo Villae Mag-
[n](a)e Varian(a)e id est Mappalia Siga, eiseos agros qui su[b]-
[c]esiva sunt excolere permittitur lege Manciana
ita, ut eas qui excoluerit usum proprium habe-
at. [...]

The translation of this part of the column is not easy, but generally it says,

"To those coloni (who will have farmsteads) within the boundaries of the estate of Villae Magna or Mappalia Siga, who wish to cultivate more fields, permission is given to cultivate those fields which have not been alloted (subseciva) or have been classfied as unused, under the terms of the law of Mancia; namely that he who cultivates this lands shall have them for personal use."

One can infer from this that the land belonging to the Villa Magna was originally mapped and surveyed and then distributed to individuals, becoming some form of ager privatus. It certainly proves that this particluar area had boundaries drawn even though there are currently few physical remains of the Roman centuriation lines. Epigraphic evidence for Roman mapping has not been studied in a large scale fashion before and I hope to published my complete GIS of this information, at least from North Africa and Southern France, in the next year or so. For those who are interested I show the other four faces below...

Click on images to enlarge






Face II (above), Face III (below ), Face IV (below III)

Geometrical Problems from the Corpus Agrimensorum :
Mollweide meets Mommsen

A representation is made with a purpose or goal in mind, governed by criteria of adequacy pertaining to that goal, which guide its means, medium and selectivity.
--Bas Van Fraassen

The group of Roman surveying texts known as the Corpus Agrimensorum have provided a great deal of information regarding the actual practices of Roman surveyors in the field and given scholars insight into how the Romans allocated and measured land. My current project in locating and mapping the surviving remains of Roman surveying in North Africa takes its starting point from this 6th century compilation of surveying manuals. The texts themselves and the illustrations attached to them have attracted the attention of many scholars in past including historians of Roman law and agrarian practices such as Theodor Mommsen and Max Weber. In this sence the historiography associated with the texts is almost as interesting as the texts themselves.

What is less known about this historiography is the attraction the texts have held for more mathematically inclined historians of cartography such as C.B. Mollweide. Mollweide is best known as map projectionist, but also did fundamental research into some of the unsolved geometrical problems found in the Corpus.

One of the most interesting of these problems concerns the method for finding south. In the Corpus there are several methods given for this, but the one that interests us here is that given by the writer, Hyginus.

There is also another method of obtaining South, by marking three shadows. On level ground we shall set up a gnomon AB, and note any three of its shadows, CDE. These shadows we shall mark with the set square, to see their distances from each other. If we set them up before noon, the first shadow will be the longest; if after noon, the last. We shall then draw these shadows in proportion by a footrule... Let AB be a gnomon, B the ground. Let us take the longest shadow and mark it [i.e. its end opposite to B] on the ground as C; the second likewise D, the third E... Let us project hypotenuses from C on to A and from D on to A. Now with centre A and radius E let us draw a circle. Then let us project lines parallel to the base, i.e. ground, on to the perpendicular [AB] from the intersections of the hypotenuses and the circumference, from F on to G and from I on to K. Then we shall apply the longest line, GF, to the largest shadow, and from B we shall mark out [the length of] GF ; the second line to the second shadow, and we shall mark out [the length of] KI. Then from F and I we shall project a straight line, and likewise from C and D, the shadow ends. These two lines will meet at T. Join TE; this will be east-west.

The above text by Hyginus is accompanied in the Codex Arcerianus A by the figure shown below.

The figure as drawn by the scribe is however totally inadequate to explain the complexities of the method outlined by Hyginus and one has to question its purpose in the manuscript and whether it was in fact added to the text by a later copiest with less understanding of the method. Mollweide analyzes the text in an article 'Erlduterung einer in der Scriptoribus rei agrariae. . . gegebenen Vorschrift..' published in Zach's Monatliche Correspon-denz zur Beforderung der Erd- und Himmelkunde (Volume 28, 1813. p. 396-425).








Click on Figures to enlarge

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The method that is being described in the text by Hyginus is extremely complicated and there are open questions concerning the level of mathematics and solid geometry involved.














Mollweide's solution is a complex affair and according to Dilke the method as described by Hyginus probably goes back to Alexandrian mathematical scholarship that has been lost but that must have been dependent on Apollonius' Conics.

The modern solution that Dilke adapted from Mollweide can be condensed to the following along with the figure below:

ABC, ABD, ABE are right-angled triangles. The lines CA, DA, EA go towards the centre of the sun. The arc EIF is part of a circle forming the base of a regular cone, parallel to the sun's daily round and so to the equator, and FI is a chord of this circle. Since GF is equal and parallel to BL, FL will be equal and parallel to GB; similarly IM to KB. As FL and IM are parallel to AB and so to each other, they lie in a plane in which FI and LM lie. But FI is also in the plane ACD, and LM is in the horizontal plane BCD; so FI is the intersection of the plane FIML with the plane ACD, and LM the intersection of the plane FIML with the horizontal plane BCD. As the plane ACD is cut by the horizontal plane at CD, which when produced meets LM produced at T, it follows that T is in the plane ACD and also in the plane FIML, and so is a point on the common intersection of both planes, i.e. of FI produced. Since the latter lies entirely on the plane of the circle through FIE, T is also in this plane, but likewise in the horizontal plane BCD, and so is a point on the common intersection of both planes. Since E is also such a point, it follows that ET is the intersection of a plane parallel to the equator plane with the horizontal plane, and so parallel to the east-west line.

One can see how different the figure which displays the actual construction as it described by Hyginus is from the original manuscript illustration.