SOLSTICE:  AN ELECTRONIC JOURNAL OF GEOGRAPHY AND MATHEMATICS.
Volume II, Number 2.  Winter, 1991.
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\centerline{\big SOLSTICE:}
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\centerline{\bf  AN ELECTRONIC JOURNAL OF GEOGRAPHY AND MATHEMATICS}
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\centerline{\bf WINTER, 1991}
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\centerline{\bf Volume II, Number 2}
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\centerline{\bf Institute of Mathematical Geography}
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\centerline{\bf Ann Arbor, Michigan}
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\centerline{\bf SOLSTICE}
\line{Founding Editor--in--Chief:  {\bf Sandra Lach Arlinghaus}. \hfil}
\smallskip
\centerline{\bf EDITORIAL BOARD}
\smallskip
\line{{\bf Geography} \hfil}
\line{{\bf Michael Goodchild}, University of California, Santa Barbara.
\hfil}
\line{{\bf Daniel A. Griffith}, Syracuse University. \hfil}
\line{{\bf Jonathan D. Mayer}, University of Washington; joint appointment
in School of Medicine.\hfil}
\line{{\bf John D. Nystuen}, University of Michigan (College of
Architecture and Urban Planning).}
\smallskip
\line{{\bf Mathematics} \hfil}
\line{{\bf William C. Arlinghaus}, Lawrence Technological University. \hfil}
\line{{\bf Neal Brand}, University of North Texas. \hfil}
\line{{\bf Kenneth H. Rosen}, A. T. \& T. Bell Laboratories.
\hfil}
\smallskip
\line{{\bf Robert F. Austin, Ph.D.} \hfil}
\line{President, Austin Communications Education Services \hfil}
\line{Past-president, AM/FM International \hfil}

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The purpose of {\sl Solstice\/} is to promote  interaction
between geography and mathematics.   Articles in which  elements
of   one  discipline  are used to shed light on  the  other  are
particularly sought.   Also welcome,  are original contributions
that are purely geographical or purely mathematical.   These may
be  prefaced  (by editor or author) with  commentary  suggesting
directions  that  might  lead toward  the  desired  interaction.
Individuals  wishing to submit articles,  either short or full--
length,  as well as contributions for regular  features,  should
send  them,  in triplicate,  directly to the  Editor--in--Chief.
Contributed  articles  will  be refereed by  geographers  and/or
mathematicians.   Invited articles will be screened by  suitable
members of the editorial board.  IMaGe is open to having authors
suggest, and furnish material for, new regular features.
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\noindent {\bf Send all correspondence to:}
\vskip.1cm
\centerline{\bf Institute of Mathematical Geography}
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\centerline{\bf Ann Arbor, MI 48105-1429}
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\centerline{\bf (313) 761-1231}
\centerline{\bf IMaGe@UMICHUM}
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{\nn  Copyright  will  be taken out in  the  name  of  the
Institute of Mathematical Geography, and authors are required to
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There are no page charges; authors will be given  permission  to
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of \$15.95 per year (plus shipping and handling; hard copy is issued once yearly, in the Monograph series of the Institute of Mathematical Geography. Order directly from IMaGe. It is the desire of IMaGe to offer electronic copies to interested parties for free. Whether or not it will be feasible to continue distributing complimentary electronic files remains to be seen. Presently {\sl Solstice\/} is funded by IMaGe and by a generous donation of computer time from a member of the Editorial Board. Thank you for participating in this project focusing on environmentally-sensitive publishing.} \vskip.5cm Copyright, December, 1991, Institute of Mathematical Geography. All rights reserved. \vskip1cm {\bf ISBN: 1-877751-53-7} {\bf ISSN: 1059-5325} \vfill\eject \centerline{\bf SUMMARY OF CONTENT} \smallskip \noindent{\bf 1. REPRINT.} \smallskip {\bf Saunders Mac Lane}. {\bf Proof, Truth, and Confusion}. Given as the Nora and Edward Ryerson Lecture at The University of Chicago in 1982. Republished with permission of The University of Chicago and of the author. \smallskip \noindent I. The Fit of Ideas. II. Truth and Proof. III. Ideas and Theorems. IV. Sets and Functions. V. Confusion via Surveys. VI. Cost-benefit and Regression. VII. Projection, Extrapolation, and Risk. VIII. Fuzzy Sets and Fuzzy Thoughts IX. Compromise is Confusing. \smallskip \noindent{\bf 2. ARTICLE } \smallskip\noindent {\bf Robert F. Austin}. {\bf Digital Maps and Data Bases: Aesthetics versus Accuracy$^{\bf \star }$} \smallskip\noindent I. Introduction. II. Basic Issues. III. Map Production. IV. Digital Maps. V. Computerized Data Bases. VI. User Community. \smallskip \noindent{\bf 3. FEATURES} \smallskip \item{i.} {\bf Press Clipping} --- reprint of Briefing" from {\sl Science\/} mentioning {\sl Solstice\/} --- Online Journals," by Joseph Palca. \item{ii.} {\bf Educational feature} --- Word search puzzle promoting mathematical and geographical awareness. \item{iii.} {\bf Software Briefs} --- Brief descriptions of software provided by the creator. Look for reviews of the software in subsequent issues of {\sl Solstice \/}. a. RangeMapper$^{\hbox{TM}}$--- version 1.4. Created by Kenelm W. Philip, Tundra Vole Software, Fairbanks, Alaska. Program and Manual by Kenelm W. Philip; commentary from the Manual. A utility for biological species range mapping, and similar mapping tasks in other fields." b. XYNIMAP" --- created by David H. Douglas, University of Ottawa; a comprehensive system for computer cartography and geo-spatial analysis." Preliminary Version. \item{iv.} Index to Volumes I (1990) and II (1991) of {\sl Solstice}. \item{v.} Other publications of the Institute of Mathematical Geography. \item{vi.} Solution to Word Search Puzzle. \vfill\eject \centerline{\bf ACKNOWLEDGMENTS} \vskip.5cm The 1982 Ryerson Lecture, Proof, Truth, and Confusion," is republished in {\sl Solstice\/} with the permission of The University of Chicago and with the permission of Professor Saunders Mac Lane. The column Online Journals" is republished in {\sl Solstice \/} with the permission of the American Association for the Advancement of Science. It appeared originally in {\sl Science \/}, Briefings," Vol. 254, No. 5036 (Nov. 29, 1991), p. 1291. Copyright is held by the AAAS. \vfill\eject \noindent 1. REPRINT \smallskip \centerline{\bf PROOF, TRUTH, AND CONFUSION} \vskip.2cm \centerline{Saunders Mac Lane} \centerline{Max Mason Distinguished Service Professor of Mathematics} \centerline{The University of Chicago} \smallskip \centerline{The 1982 Nora and Edward Ryerson Lecture} \centerline{at The University of Chicago.} \smallskip Copyright, 1982 by The University of Chicago, republished with permission of that University and of the author. \smallskip \centerline{\bf Saunders Mac Lane } %photograph of Mac Lane, c. 1982. \vfill\eject \centerline{\bf Introduction} \centerline{by} \centerline{Hanna H. Gray} \centerline{President of the University} Each year, one of the annual events at the University is the selection of the Nora and Edward Ryerson Lecturer. The selection is made by a committee of faculty which receives nominations from their faculty colleagues. Each year, this committee comes up with an absolutely superb selection for the Ryerson Lecturer, and this year is triumphant confirmation of that generalization. The selection emanates from faculty nomination and discussion, and it is analogous to the process of the selection of faculty in this University, representing a selection based on the work and contribution, on the high esteem for the intellectual imagination and breadth of a colleague. The peer review process, in this as in faculty appointments, stresses scholarship and research, stresses the contribution of a member of the faculty to the progress of knowledge. In addition, of course, the faculty appointment process looks also to teaching and to institutional citizenship. If I had the nerve to fill out an A-21 form for Professor Saunders Mac Lane, I would, I think, be creating a new mathematics because I would award him 100 percent for research, 100 percent for teaching, and 100 percent for contribution or citizenship. Even I can add that up to 300 percent. Now, of course, in the evaluation of younger scholars for junior appointments similar judgments are made. They are based on the same three categories, and they are judgments about the promise of continuing creativity, continuing growth, continuing intellectual contribution. That judgment of the young Saunders Mac Lane was made a very long time ago in Montclair, New Jersey. Montclair, New Jersey is the home of the Yale Club of Montclair. I once had the enormous privilege of being invited to the Yale Club of Montclair where I was given something called the Yale Bowl, which had on it an inscription testifying that I had earned my Y" in the Big Game of Life." In 1929, the young Saunders Mac Lane went to the Yale Club of Montclair, I was told---he was then finishing his senior year at Yale---and there was a young dean of the Yale Law school who was leaving in order to go to the University of Chicago as president. And the Yale Club of Montclair, which usually gave its awards to football players, decided on this occasion to give recognition to a young mathematician and to a young law school dean, and that was where Mr. Hutchins and Mr. Mac Lane met. As Saunders Mac Lane graduated from Yale, Mr. Hutchins encouraged him personally to come to Chicago. And Saunders came. He had, however, neglected to take steps that are usually taken when one travels to enter another university, and the chairman of the Department of Mathematics, Mr. Bliss, had to say to him rather directly, Young man, you've got to apply first." He did apply, and fortunately he was accepted. Within a year, he had received his M.A. from Chicago and had come into contact with lots of extraordinary people, but two very extraordinary people in particular. One was the great mathematician E. H. Moore, and the other was a graduate student in economics named Dorothy Jones, who was in 1933 to become Mrs. Mac Lane. Now, those of us who know Saunders think of him as a Hyde Parker, and indeed as a Hyde Parker forever. And, of course, he is a Hyde Parker, and a Hyde Parker forever, but he had a period in his life, I have to tell you, after he had taken his M.A., when he became something of an academic traveler. We really ought to have been able to trace those travels when we think about Saunders' choice of costume. Now, that's not easy to figure out today because I think that necktie came out of a safe this morning. But if you think about the flaming reds, for example, that Saunders affects, you are perhaps reminded of Cambridge, Massachusetts. If you think of the Scottish plaids which he affects, that's a harder one, because I would say that that has to do with the great tradition which took him to New England and made him for a time a resident of Connecticut. And then, of course, there is the Alpine hat which could only have come from Ithaca, New York. Saunders received his doctorate from the University of G\"ottingen in 1934. He had spent the years 1933-34 again at Yale as a Sterling Fellow. He then spent two years at Harvard. He then spent a year at Cornell. Then he came to the University of Chicago for a year. And then again he moved, called back to Harvard as an assistant professor, and there he rapidly went through the ranks. Fortunately, in 1947, he returned to the University of Chicago and, in 1963, became the Max Mason Distinguished Service Professor. Between 1952 and 1958, he succeeded Marshall Stone as chairman of the Department of Mathematics for two three-year terms, and he has served the University as he has his department with total dedication. Saunders has extended his role beyond our University, serving primarily and prominently in a number of national scholarly organizations and institutions devoted to large questions of the relationship of learning to policy. He was president of the Mathematical Association of America and received its Distinguished Service Award in 1975 in recognition of his sustained and active concern for the advancement of undergraduate mathematical teaching and undergraduate mathematics. He was also president of the American Mathematical Society in 1973-74. He has been a member of the National Science Board and vice-president of the National Academy of Sciences. His work in mathematics, of course, has been widely recognized. Alfred Putnam, who studied with Saunders at Harvard, had this to say of Saunders in a biographical sketch that he has published. He wrote, Beginning as a graduate student with a brief exposure to group extensions, I've watched the development of Saunders Mac Lane's mathematics through homological algebra to category theory. Saunders Mac Lane belongs in a category by himself." And so he does. So he does as a mathematican, as an academic citizen, as a spokesman for the fundamental values and principles of the University, and, of course, in sartorial wonder. Now, it is to this category that we look for the Nora and Edward Ryerson Lecturers. When the Trustees established the lectureship in 1973, they sought a way to celebrate the relationship that the Ryersons and their family have had with our University---a relationship of shared values and a commitment to learning at the most advanced level. Mr. Ryerson was elected to the Board in 1923 and became Chairman of the Board in 1953. Nora Butler Ryerson was a founding member, if not {\sl the\/} founder, of the University's Women's Board. Both embraced a civic trust that left few institutions in our city untouched, and they passed to future generations of their family the sense of engagement and participation. Saunders Mac Lane, through his staunch loyalty to our University, his broad interest in the community of scholars and their work, his distinguished scholarly career, represents these values for us in a special way, and, of course, he is entirely uncompromising also in his commitment to them. It is a pleasure to introduce this year's Ryerson Lecturer, Saunders Mac Lane. \vfill\eject \centerline{\bf Proof, Truth, and Confusion} \smallskip \centerline{\bf Saunders Mac Lane} \smallskip \centerline{\bf I. The Fit of Ideas} It is an honor for a mathematician to stand here. Let me first say how much I appreciate the initiative taken by the trustees on behalf of the Ryerson family in providing for this series of lectures, which afford opportunity for a few fortunate faculty members to present aspects of their scholarly work which might be of interest to the whole university community. In my own case, though the detailed development of mathematics tends to be highly technical, I find that there are some underlying notions from mathematics and its usage which can and will be of general interest. I will try to disentangle these and to relate them to the general interest. This intent accounts for my title. Mathematicians are concerned to find truth, or, more modestly, to find a few new truths. In reality, the best that I and my colleagues in mathematics can do is to find proofs which perhaps establish some truths. We try to find the right proofs. However, some of these proofs and the techniques and numbers which embody them have turned out to be so popular that they are applied where they do not belong---with results which produce confusion. For this, I will try to cite examples and to draw conclusions. This involves a thesis as to the nature of mathematics: I contend that this venerable subject is one which does reach for truth, but by way of proof, and does get proof, by way of the concatenation of the right ideas. The ideas which are involved in mathematics are those ideas which are formal or can be formalized. However, they are not purely formal; they arise from aspects of human activity or from problems arising in the advance of scientific knowledge. The ideas of mathematics may not always lead to truth; for this reason it is important that good ideas not be confused by needless compromise. In brief, the ideas which matter are the ideas that fit. However, the fit may be problematical. A friend of mine with a vacation home in Vermont wanted to suitably decorate his barn, and so asked the local painter to put on the door the biggest number which can be written on the broad side of a barn door." The painter complied, painting on the barn door a digit 9 followed by as many further such digits as could be squeezed onto the door (Figure 1.a). A competitor then claimed he could do better by painting smaller 9's and so a bigger number. A second competitor then rubbed out the first line and wrote instead: The square of the number 9,$\ldots$(Figure 1.b). Even that didn't last, because another young fellow proposed the paradoxical words, One plus the biggest number that can be written on the broad side of this barn door" (Figure 1.c). At each moment, this produces a bigger number than anything before. We may conclude that there is no such biggest number. This may illustrate the point that it is not easy to get the ideas that fit---on barn doors or otherwise. \midinsert \vskip 6in \noindent {\bf Figure 1}. a. The biggest number on that barn door. b. A bigger number on that barn door. c. An even bigger number on that barn door.\endinsert \smallskip \centerline{\bf II. Truth and Proof} \smallskip I return to the truth" of my title. When I was young I believed in RMH--which sometimes stands for Robert Maynard Hutchins, who to my great profit first encouraged me to come to Chicago---and which sometimes stands for the slogan, Reach Much Higher." At any rate, when young I thought that mathematics could reach very much higher so as to achieve absolute truth. At that time, {\sl Principia Mathematica\/} by Whitehead and Russell seemed to model this reach; it claimed to provide all of mathematics firmly founded on the truths of logic. The logic in {\sl Principia\/} was elaborate, symbolic, and hard to follow. As a result, it took me some years to discover that {\sl Principia Mathematica\/} was not a {\sl Practica Mathematica\/}---much of mathematics, in particular most of geometry, simply wasn't there in {\sl Principia\/}. For that matter, what {\sl was\/} there didn't come exclusively from logic. Logic could provide a framework and a symbolism for mathematics, but it could not provide guidelines for a direction in which to develop. This limitation was a shocking discovery. Logic, even the best symbolic logic, did not provide all of absolute truth. What did it provide instead? It provided proof---the rigorous proof of one formal statement from another prior statement; that is, the deduction of theorems from axioms. For such a deduction, one needed logic to provide the rules of inference. In addition, one needed the subject matter handled in the deductions: the ideas used in the formulation of the axioms of geometry and number theory, as well as the suggestions from outside mathematics as to what theorems might usefully be proved from these axioms. Deductive logic is important not because it can produce absolute truth but because it can settle controversy. It has settled many. One notable example arose in topology, a branch of mathematics which studies qualitative properties of geometric objects such as spheres. From this perspective, a smooth sphere and a crinkly sphere would have the {\sl same\/} qualitative properties---and we would consider not just the ordinary spheres---two-dimensional, since the surface has two dimensions---but also the spheres of dimensions 3, 4, and higher (Figure 2). For these spheres, topologists wished to calculate a certain number which measures the connectivity---a measure two dimensions up" from the dimension of the sphere. The Soviet topologist L. Pontrjagin in 1938 stated that this desired measure was one. Others thought instead that the measure was two. In a related connection, the American reviewer of another paper by Pontrjagin wrote, Both theorems (of Pontrjagin) contradict a previous statement of the reviewer. It is not easy to see who is wrong here." Fortunately, it was possible to see. With careful analysis of the proof, Pontrjagin did see who was wrong---and in 1950 published a statement correcting his 1938 error: that the measure of connectivity two dimensions up is {\sl not\/} one, but two. \midinsert \vskip 4in \centerline{\bf Figure 2} \endinsert A few years ago, the {\sl New York Times\/} carried an item about a similar fundamental disagreement between a Japanese topologist and one of our own recent graduate students, Raphael Zahler. Analysis of the deductions showed that Zahler was right. There lies the real role of logic: it provides a formal canon designed to disentangle such controversies. Truth may be difficult to capture, but proof can be described with complete accuracy. Each mathematical statement can be written as a word or sentence in a fixed alphabet---using one letter for each primitive mathematical notion and one letter for each logical connective. A proof of a theorem is a sequence of such statements. The initial statement must be one of the axioms. Each subsequent statement not an axiom must be a consequence of prior statements in the sequence. Here consequence" means consequence according to one of the specified rules of inference"---rules specified in advance. A typical such rule is that of {\sl modus ponens\/}: Given statments S" and S implies T," one may infer the statement T." This description gives a firm standard of proof. Actual proofs may cut a few corners or leave out some obvious steps, to be filled in if and when needed. Actual proofs may even be wrong. However, the formal description of a proof is complete and definitive. It provides a formal standard of rigor, not necessarily for absolute truth, but for absolute proof. There is a surprising consequence: no one formal system suffices to establish all of mathematics. Precisely because there is such a rigorous description of a proof" in a formal system," Kurt G\"odel was able to show that, in each such system with calculable rules of inference, one could formulate in the system a sentence which was not decidable in the system---that is, a sentence$G$which can neither be proved nor disproved according to the specified rules of inference. More exactly, this is the case for any system which contains the numbers and the rules of arithmetic, and in which the rules of inference can be explicitly listed or numbered in the fashion called recursive." In such a system, all statements are formal and are constructed from a fixed alphabet. Hence we can number {\sl all\/} the possible proofs. Moreover, we can formulate within the system a sentence which reads, $n$is the number of the proof of the statement with the number$k$." On this basis, and adapting ideas illustrated by the paradox of the barn door, one then constructs another sentence$G=G(p)$(with number$p$) which reads, There is no number which is the proof of the sentence number$p$." This means in particular that this very sentence$G$cannot be proved in the system. This is because$G$itself states that there is no proof in the system for me"---hence$G$is true (Figure 3). Hence, unless the system is inconsistent, it can contain no refutation of$G$. Thus in such a formal system we can write one statement (and hence many) which, though true, is simply undecidable, yes or no, within the system. \midinsert \vskip 3in \centerline{\bf Figure 3} \endinsert This result is startling. It may seem catastrophic---but it turns out to be not quite so disastrous. It shows that there is an intrinsic limitation on what can be proved within {\sl any one formal\/} system; thus proof within one such system cannot give all of truth. Very well then, as we shall see, there can be more than one formal system and hence more than one way in which to reach by proof for the truth. \smallskip \centerline{\bf III. Ideas and Theorems} \smallskip Some observers have claimed that mathematics is just formalism. They are wrong. A mathematical proof in a given formal system must be {\sl about\/} something, but it is not about the outside world. I say it is about ideas. Thus the formal system of Euclidean geometry is about certain pictorial" ideas: point, line, triangle, and congruence; in their turn, these ideas arose as means of formulating our spatial experiences of shape, size, and extent and our attempts to analyze motion and symmetry. Each branch of formal mathematics has a comparable origin in some human activities or in some branch of scientific knowledge. In each such case, the formal mathematical system can be understood as the realization of a few central ideas. Mathematics is built upon a considerable variety of such ideas---in the calculus, ideas about rate of change, summation, and limit; in geometry, ideas of proximity, smoothness, and curvature. To further illustrate what I mean here by idea," I choose a small sample: The related ideas of connect," compose," and compare." To Connect" means to join. There are different ways in which mathematicians have defined what it means for a piece of space to be connected. One definition says that a piece of space is connected if it does not fall apart into two (or more) suitably disjoint pieces. Another definition says that a piece of space is {\sl path-connected\/} if any two points in the piece can be joined {\sl within\/} the piece by a path---that is, by a continuous curve lying wholly in the piece. These two formal explications of the idea of connected" are not identical; a piece of space which is path-connected is always connected in the first sense, but not necessarily vice-versa. This simple case of divergence illustrates the observation that the same underlying pre-formal idea can have different formalizations. Compose" is the next idea. To compose two numbers$x$and$y$by addition is to take their sum$x+y$; to compose them by multiplication is to take their product$xy$. To compose one motion$L$with a second motion$M$is to follow$L$by$M$to get the composite" motion which we write as$L \circ M$. Thus to rotate a wheel first by$25^{\circ}$and then by$45^{\circ}$will yield after composition a rotation by$70^{\circ}$. To compose a path$L$connecting a point$p$to a point$q$with a path$M$connecting$q$to a third point$s$is to form the longer path$L \circ M$which follows first$L$and then$M$, as in the top of figure 4. In all such cases of composition, the result of a composition$L \circ M \circ N$of three things in succession depends on the factors composed and the sequence or order in which they were taken---but {\sl not\/} on the position of the parenthesis. Thus arises one of the formal laws of composition, the associative law: $$L \circ (M \circ N) = (L \circ M) \circ N.$$ However,$L \circ M$may very well differ from$M \circ L$! The order matters. \midinsert \vskip3in \centerline{\bf Figure 4.} \endinsert The third sample idea is Compare." One may compare one triangle with another as to size, so as to study congruent triangles. One may compare one triangle with another as to shape, and so study more generally similar triangles. Another comparison is that by deformation: Two paths in a piece of space may be compared by trying to deform the first path in a continuous way into the second---as in Figure 4, the composite path$L  \circ M$from$p$to$r$can be deformed smoothly and continuously into the path$K$also joining$p$to$r$. Ideas such as these will function effectively in mathematics only after they have been formalized, because then explicit theorems about the ideas can be proved. The idea of composition is formalized by the concept of a group, which applies to those compositions in which each thing$L$being composed has an inverse" thing or operation$L^{-1}$so that$L \circ L^{-1} =  1$. One readily sets down axioms for a group of things" with such composition. The axioms are quite simple, but the concept has proven to be extraordinarily fruitful. There are very many examples of groups: Groups of rotations, groups of symmetry, crystallographic groups, groups permuting the roots of equations, the gauge groups of physics, and many others. There is a sense (analyzed by Eilenberg-Mac Lane in a series of papers) in which any group can be built up by successive extensions from certain basic pieces, called the simple" groups. Specifically, a group is said to be {\sl simple\/} when it cannot be collapsed into a smaller group except in a trivial way. A long-standing conjecture suggested that the number of elements in a finite simple group was necessarily either an even number or a prime number. About twenty years ago, here at Chicago, Thompson and Feit succeeded in proving this to be true (and I could take pleasure in the fact that Thompson, one of my students, had achieved such a penetrating result). The Thompson-Feit method turned out to be so suggestive and powerful that others have now been able to go on to explicitly determine {\sl all\/} the finite simple groups. For example, the biggest sporadic one has$2^{46}\cdot
3^{20}\cdot   5^9\cdot  7^6\cdot  11^2\cdot  13^3\cdot    17\cdot
19\cdot  23\cdot  29\cdot 41\cdot 47\cdot 59\cdot  71$elements (that number is approximately 8 followed by 53 zeros). This simple group is called the Monster" (Figure 5). Another one of our former students has been able to make a high dimensional geometric picture which shows that this monster really exists. He needed a space of dimension 196,884. \midinsert \vskip3in \centerline{\bf Figure 5} \endinsert Groups also serve to measure the connectivity of spaces. In particular, there are certain homology groups which count the presence of higher dimensional holes in space. To start with, a piece$\chi $of space is said to be {\sl simply connected\/} if any closed path in the space$\chi$can be deformed into a point. For example, the surface of a sphere is simply connected, so its first homology group is zero; however, it has a non-zero second homology group---meaning the hole" represented by the inside of the sphere. These properties characterize the two-dimensional sphere. Long ago, the French mathematician Poincar\'e said that the same should hold for a three-dimenional sphere (Figure 6). This famous conjecture has not yet been settled---but some years ago, Smale showed that the characterization was true for a sphere of dimension 5 or higher. Just during the last year, the Californian Michael Friedman, in a long proof, showed that it is also true for a sphere of dimension 4. Except for a solution which was announced on April 1, nobody yet knows the answer for a three-dimensional sphere. Proof advances, but slowly. \midinsert \vskip3in \centerline{\bf Figure 6} \endinsert \vfill\eject \centerline{\bf IV. Sets and Functions} \smallskip As already indicated, Whitehead and Russell, by {\sl Principia Mathematica\/}, had suggested that all mathematical truth could be subsumed in one monster formal system. Their system, corrupted as it was with types," was too complicated--- but others proposed a system based on the idea of a {\sl set\/}. A set is just a collection of things---nothing more. Mathematics does involve sets, such as the set of all prime numbers or the set of all rational numbers between 1 and 2. Mathematical objects can be defined in terms of sets. For example, a circle is the set of all points in the plane at a fixed distance from the center, while a line can be described as the set of all its points. Numbers can be defined as sets---the number two is the set of all pairs; an irrational number is the set of all smaller rational numbers. In this way numbers, spatial figures, and everything else mathematical can be defined in terms of sets (Figure 7). All that matters about a set$S$is the list of those things$x$which are members of$S$. When this is so, we write$x\,  \epsilon \,  S$, and call this the membership relation." \midinsert \vskip3in \centerline{\bf Figure 7} \endinsert There are axioms (due to Zermelo and Fraenkel) which adequately formalize the properties of this membership relation. These axioms claim to provide a formal foundation--I call this the grand set-theoretic doctrine---for all of mathematics. By 1940 or so this grand set-theoretic foundation had become so prominent in advanced mathematics that it was courageously taught to freshmen right here in the Hutchins college. This teaching practice spread nationally to become the keystone of the New Math." As a result, twenty years later sets came to be taught in the kindergarten. There is even that story about the fond parents inquiring as to little Johnny's progress. Yes said the teacher, he is doing well in math except that he can't manage to write the symbol$\epsilon$when$x$is a member of the set$S$. Johnny was not the only one in trouble. The grand doctrine of the new math: Everything is a set" came at the cost of making artificial and clumsy definitions. Moreover, putting everything in one formal system of axioms for set theory ran squarely into the difficulties presented by G\"odel's undecidable propositions. Fortunately, just about the time when sets reached down to the kindergarten, an alternative approach to a system of all" (better most") of mathematics turned up. This used again the idea of composition for functions$f:  S  \longrightarrow  T$sending the elements of a set$S$to some of those of another set$T$. Another function$g:  T \longrightarrow U$can then be composed with$f$to give a new function$g \circ f$(Figure 8). It sends an element of$S$first by$f$into$T$and then by$g$into$U$. The prevalence of many such compositions led Eilenberg and Mac Lane in 1945 to define the formal axioms for such composition. With no apologies to Aristotle, they called such a system a category"---because many types of mathematical objects did form such categories, and these properties were useful in the organization of mathematics. Note especially that the intuitive idea of composition" has several different formalizations: category and group. \midinsert \vskip4in \centerline{\bf Figure 8} \endinsert Then in 1970 Lawvere and Tierney made a surprising discovery: that in treating a function$f:  S \longrightarrow T$one could forget all about the elements in$S$and$T$, and write enough axioms on composition alone to do almost everything otherwise done with sets and elements. This formal system is called an elementary topos"---to suggest some of its connections to geometry and Top"-ology. Their success in discovering this wholly new view of mathematics emphasizes my fundamental observation: That the ideas of mathematics are various and can be encapsulated in different formal systems. Waiting to be developed, there must be still other formal systems for the foundation and organization of mathematics. \vfill\eject \centerline{\bf V. Confusion via Surveys} \smallskip The crux of any search for the right alternative to set theory is the search for the right concatenation of ideas---in the same way in which leading ideas in mathematics have been combined in the past to solve problems (the Poincar\'e conjecture on spheres) and to give new insights. Thus it was with our example, where the related ideas of connection, composition, and comparison came together in group theory, in the application of groups to geometry, and in category theory. But sometimes the wrong ideas are brought together, or the right ideas are used in the wrong way. Today the use of numbers and of quantitative methods is so pervasive that many arrays of numbers and of other mathematical techniques are deployed in ways which do not fit. This I will illustrate by some examples. Recently, in connection with my membership on the National Science Board, I came across the work of one prominent social scientist who was promoting (perhaps with reason) the use of computer-aided instruction in courses for college students. However, the vehicle he chose for such instruction was the formal manipulation of the elementary consequences of the Zermelo-Fraenkel axioms for set theory---and the result was an emphasis on superficial formalism with no attention to ideas or meaning. It was, in short, computer-aided pedantry. Opinion Surveys" provide another example of the confusion of ideas. For some social and behavioral research, the necessary data can be obtained only by survey methods, and responsible scientists have developed careful techniques to help formulate the survey questions used to probe for facts. Unfortunately these techniques are often used carelessly---both because of commercial abuse, statistical malpractice, or poor formulation of survey questions. First the malpractice: On many surveys the percentage of response is uncomfortably low, with the result that the data acquired are incomplete. This situation has led the statisticains into very elaborate studies of means for approximately completing such incomplete data." One recent and extensive such publication (by the National Research Council) seemed to me technically correct but very elaborate---perhaps overdone, and in any event, open to the misuse of too much massaging of data that are fatally incomplete. In opinion surveys touching directly on the academic profession some of the worst excesses are those exhibited by the so-called Survey of the American Professoriate." Successive versions of this survey are replete with tendentious and misleading questions, often such likely to create" opinion rather than to measure actual existing opinions. Despite heroic attempts by others to suggest improvements, the authors of this particular survey have continued in their mistaken practices in new such surveys---as has been set forth with righteous indignation by Serge Lang in his publication {\sl The File: A Case Study in Correction\/}. That otherwise useful publication {\sl Science Indicators\/} from the National Science Board makes excessive use of opinion surveys. The most recent report of the series ({\sl Science Indicators 1980\/}) coupled results from a new, more carefully constructed opinion survey with a simple continuation of poorly formulated questions taken from previous and less careful surveys. The main new opinion survey commissioned for this NSB report used an elaborate design---but this design still involved some basic misconceptions about science and some questions about science so formulated as to distort the opinions which were to be surveyed. For example, its Question 71 first observes that Science and technology can be directed toward solving problems in many different areas"---while I would claim that science cannot be directed" in the fashion intended by government bureaucrats. The question then lists fourteen areas and asks Which three areas on the list would you {\sl most\/} like to receive science and technology funding from your tax money?" Of the fourteen areas, some had little to do with science or technology and much to do with the political and economic structure of society (for example, controlling pollution, reducing crime, and conserving energy). Only one of the fourteen dealt with basic knowledge. With an unbalanced list of questions like this, the report goes on to claim that the answers suggest that the public interest tends to focus on the practical and immediate rather than on results that are remote from daily life." This may be so, but it cannot be demonstrated by answers to a survey questionnaire which itself is so constructed as to focus on the practical and immediate." To get comparisons of opinions across time, new surveys try to continue questions which have been used before---and so often use older questions of a clearly misleading character. In {\sl Science Indicators\/}, a typical such previous question is the hopelessly general one, Do you feel that science and technology have changed life for the better or the worse?" The current version of this question does still more to lead the respondent to a negative answer. It reads, Is future scientific research more likely to cause problems than to find solutions to our problems?" It is no wonder that this latter slanted question, in the 1979 survey, had only 60\% answers favorable to science, while the earlier one had 75\% favorable in 1974 and 71\% in 1976. Surveys also may pose questions which the respondents are in no position to answer. For instance, one question in this survey probed the respondents' expectations of scientific and technological achievements: During the next 25 years or so, would you say it is very likely, possible but not too likely, or not likely at all that researchers will discover a way to predict when and where earthquakes will occur?" How can the general public have a useful or informed opinion on this highly technical and speculative question? The question brought answers of 57\% very likely," 34\% possible," and 7\% not likely." After giving these figures, the text obscures the careful tripartite posture of the question as stated by lumping the first two categories together in the following summary: About 9 out of 10 consider it possible or very likely$\ldots$" The other five questions asking for similar 25-year predictions (for example, a cure for the common forms of cancer) are not much better. In sum, the public opinion surveys currently used in {\sl Science Indicators\/} are poorly constructed and carelessly reported. By emphasizing remote and speculative uses of science, the thrust of the questions misrepresents the very nature of scientific method. (There are worse misrepresentations, for example, in a report for GAO (General Accounting Office), mistitled {\sl Science Indicators: Improvements Needed in Design, Construction, and Interpretation\/}). To summarize: Opinion surveys may attempt to reduce to numbers both nebulous opinions and other qualities not easily so reducible. It would be wiser if their use were restricted to those things which are properly numerical. My own chief experience with other unhappy attempts to use mathematical ideas where they do not fit comes from studying many of the reports of the National Research Council (in brief, the NRC). I recently served for eight years as chairman of the Report Review Committee for this Council. This Council operates under the auspices of the National Academy of Sciences, which by its charter from the government is required to provide, on request, advice on questions of science or art. There are many such requests. Each year, to this end, the NRC publishes several hundred reports, aimed to apply scientific knowledge to various questions of public policy. Some of these policy questions are hard or even impossible of solution, so it may not be surprising that the desire to get a solution and to make it precise may lead to the use of quantitative methods which do not fit. This lack of fit can be better understood at the hand of some examples. \smallskip \centerline{\bf VI. Cost-Benefit and Regression} \smallskip Before making a difficult decision, it may be helpful to list off the advantages and the disadvantages of each possible course of action, trying to weigh the one against the other. Since a purely qualitative weighing of plus against minus may not be objective (or at any rate can't be done on a computer), there has grown up a quantitative cost-benefit analysis, in which both the costs and the benefits of the action are reduced to a common unit---to dollars or to some other such numeraire." The comparison of different actions and thus perhaps a decision between them can then be made in terms of a number, such as the ratio of cost to benefit. In simple cases or for isolated actions this may work well; I am told that it did so function in some of its initial uses in decisions about plans for water resources. However, the types of decisions considered in NRC reports were usually not so straight-forward. I studied many such reports which did attempt to use cost-benefit analysis. In every such case which came to my attention in eight years, these attempts at quantitative cost-benefit analysis were failures. In most cases, these failures could have been anticipated. Sometimes the intended cost-benefit analysis was not an actual numerical analysis but just a pious hope. For instance, one study tried to describe ways to keep clean air somewhere way out west." In this case, there weren't enough dependable data to arrive at any numbers for either the costs or the benefits of that clean air. Hence the report initially included a long chapter describing how these costs and benefits {\sl might\/} be calculated---although it really seemed more likely that there never would be data good enough to get dependable numbers for such a calculation. There are also cost-benefit calculations which must factor in the value of the human lives which might be saved by making (or not making) this or that decision. In such cases, the value ascribed to one human life can vary by a factor of 10, ranging from one hundred thousand to one million dollars. Much of the variation depends on whether one gets the value of that life in terms of discounted future earnings or by something called implicit self-valuation of future satisfaction. However, I strongly suspect that whatever the method, there isn't {\sl any\/} one number which can adequately represent the value of human life for such cost-benefit purposes. Our lives and our leisures are too various and their value (to us or to others) is not monetary. The consequence is that decisions which deal substantially with actions looking to the potential saving of lives cannot be based in any satisfactory way on cost-benefit analysis. Another aspect of cost-benefit methods came to my attention just yesterday, in the course of a thesis defense. Cost-benefit methods attend only to gross measures, in a strictly utilitarian way, and give no real weight to the distribution of benefits (or of costs) between individuals. Another striking example of the problems attending the use of cost-benefit analysis in policy studies is provided by a 1974 NRC study, Air Quality and Automobile Emission Control," prepared for the Committee on Public Works, U.S. Senate. That committee was considering the imposition of various levels of emission controls on automobiles; it requested advice on the merits of such controls, and in particular wanted a comparison of the costs and the benefits of such control. Some benefits of the control of automobile emissions are to be found in cleaner air and some in better health (less exposure to irritating smog). A number of studies of such health effects had been done; the NRC committee examined them all and considered all but one of them inadequate. The one adequate study was for students in a nursing school in the Los Angeles area. Each student carefully recorded daily discomforts and illnesses; these records were then correlated with the observed level of smog in Los Angeles. The results of this one study were then extrapolated by the NRC committee to the whole of the United States in order to estimate the health benefits of decreasing smog! It was never clear to me why Los Angeles is typical or how such a wide extrapolation can be dependable. Just as in the case of saving lives, the benefits of good health can hardly be reduced to numbers. Some other difficulties with this particular study concern the use of regression, a mathematical topic with a considerable history. Mathematics deals repeatedly with the way in which one quantity$y$may depend upon one or more other quantities$x$. When such a$y$is an explicitly given function of$x$, the differential calculus has made extraordinarily effective use of the concept of a derivative$dy/dx = y'$; in the first instance, the use of the derivative amounts to approximating$y$by a linear function, such as$y=ax + c$, choosing$a$to be a value of the derivative$y'$. The number$a$then is units of$y$per unit of$x$and measures the number of units change in$y$due (at$x$) to a one-unit change in$x$. For certain purposes these linear approximations work very well, but in other cases, the calculus goes on to use higher stages of approximation --- quadratic, cubic, and even an infinite series of successive powers of$x$. But a variable quantity$y$involved in a policy question is likely to depend not just on one$x$, but on a whole string of other quantities$x$,$z$, and so on. Moreover, the fashion of this dependence can be quite complex. One approximation is to again try to express$y$as a constant$a$times$x$plus a constant$b$times$z$and so on---in brief to express$y$as a linear function $$y=ax + bz + \cdots$$ with coefficients$a$,$b$,$\ldots $which are not yet known. Given enough data, the famous method of least squares" will provide the best" values of the constants$a$,$b$,$\ldots $to make the formula fit the given data. In particular, the coefficient$a$estimates the number of units change in$y$per unit change in$x$---holding the other quantities constant (if one can). \midinsert \vskip 3in \centerline{\bf Figure 9} \endinsert This process is called a multiple regression" of$y$on$x$,$z$,$\ldots $. This curious choice of a word has an explanation. It was first used by Galton in his studies of inheritance. He noted that tall fathers had sons not quite so tall---thus height had regressed on the mean." This technique of regression has been amply developed by statisticians and others; it is now popular in some cost-benefit analyses. For example, with the control of auto emission, how does one determine the benefit of the resulting clean air? Clean air cannot be purchased on the market, so the benefits of cleaner air might be measured by shadow" prices found from property values, on the grounds that homes in a region where the air is clear should command higher prices than comparable homes where the air is thick. In the NRC study, the prices of houses in various subregions of greater Boston were noted and then expressed as a (linear) function of some thirteen different measured variables thought to influence these prices: Clean air, proximity to schools, good transportation, proximity to the Charles River, and so on. The constants in this linear expression of house prices were then determined by regression. In this equation, the coefficient$a$for the variable representing clean air" (of units of dollars per measure of smog-free air) was then held to give the shadow price" for clean air. The resulting shadow price from this and one other such regression was then extrapolated to the whole U.S.A. to give a measure of the benefit of cleaner air to be provided by the proposed auto emission control. \vskip.1cm \smallskip \hrule \smallskip \quad In Boston: House \$ = a (Smog) + b (Charles River)
+ $\cdots$ one dozen more
\smallskip
\hrule
\smallskip
\vskip.1cm

This is surely a brash attempt to get a number, cost what it
may.  In my considered judgment, the result is nonsense.   It  is
not clear that buyers of houses monitor the clean air before they
sign the mortgage.  A nebulous (or even an airy) quantity said to
depend on thirteen other  variables  is  not  likely  to  be well
grasped  by  any  linear  function  of  those  variables.    Some
variables may have quadratic effects, and there  could  be  cross
effects  between  different  variables.   That  list  of thirteen
variables  may  have  duplicates  or  may  very  well  miss  some
variables which  should  be  there.   Moreover, the  coefficients
in that function are likely to be still more uncertain  than  the
known costs the equation estimates.   These coefficients  are not
regard.  They employ a mathematics which does not fit.

The  difficulties  which have been noted in interpreting the
coefficients  in  some  regressions  are  by  no  means new.  For
example, you can find them discussed with vigor and clarity in  a
text   by   Mosteller   and   Tukey,   {\sl   Data  Analysis  and
Regression\/},  kept  here  on  permanent  reserve in the Eckhart
Library.  I trust that such reserve has not kept it from the eyes
of  economists or other  users  of  regression.  What with canned
formulas from  other  sources  and fast computers, any big set of
data  can  be analyzed by regression---but that doesn't guarantee
that the  results will fit!

I  have  not  studied  the  extensive  academic literature on
cost-benefit analysis, but these and other  flagrant  examples  of
the misuse of these analyses in  NCR  reports leave me disquieted.
Current political dogma may create pressure for more  cost-benefit
analysis. In Congress, the House is now considering a Regulatory
Reform  Bill"  which  requires  that   independent  and  executive
agencies of the government  make a  cost-benefit  analysis  before
issuing  any new regulation (except for those  health  and  safety
regulations required by law).  It is high time  that  academicians
and politicians give  more  serious thought to the limitations  of
such methods of analysis.

The future is inscrutable.  However, people  are  curious, so
fashion usually provides some  method  for  its  scrutiny.   These
methods may range from consultation with the Oracle  at  Delphi to
opinion polls to the examination of the entrails of a  sacrificial
animal.  Now,  thanks  to  the  existence  of fast computers, some
economists  can  scrutinize  the  future  without  entrailing such
sacrifice.  The  short-term  predictions by econometric models can
be sold at high prices, though I am told that some of these models
deliver more dependable short-term predictions  when  the original
modeler is at hand to suitably massage the output figures.

At the NRC, my chief contact with  projection  was  on a very
much longer time scale---econometric  projections  of  the  energy
future of the United States going forward for fifty years or more.
This was done in connection with a massive NRC study called CONAES
(for the Committee on Nuclear and Alternative Energy Systems). For
this  study,  there  was  not  just  one econometric projection of
energy needs,  but  a half  dozen  such  models,  with  a  variety
of time horizons.  Now projections for a span of  forty  or  fifty
years cannot possibly take account of  unexpected  events  such as
wars, oil cartels,  depressions,  or even the discovery of new oil
fields. Since the present differs drastically from the past, there
is  little  or no hope of checking a fifty-year projection against
fifty  years  of actual  past  development.    Consequently,  this
particular NRC study  did not check theory against fact, but  just
theory against theory---by asking  just how  much  agreement there
was between  the half-dozen models.  It hardly  seemed  reasonable
to me to  conclude that agreement---even a  perfect  agreement ---
in  the  results   of  several  fictive  models   can  be  of  any
predictive  value.    In  the case of the CONAES report, there was
even a  proposal  to  use the thirty-five-year projection of those
models to assess the future economic value of the breeder reactor.
Such  assessment breeds total futility.  All told, despite the use
of  fast  computers  and  multiple models,  the ambiguities of the
models being computed still leave the future dark and inscrutable.

Projections over time into an unknown future are not the only
examples of policy-promoted projection of the unknown.  Many other
types  of   extrapolation   can   be   stimulated---for   example,
extrapolation designed to estimate risks.   Since  it  is  claimed
society has become more risk-averse, there is great demand to make
studies of future risks, as in the reports of the NRC Committee on
the Biological Effects of Ionizing  Radiation  (BEIR  for  short).
The third report of this committee,  a report  commonly  known  as
BEIR III," dealt with extrapolation, another kind of projection.
Data available from Hiroshima and  Nagasaki  give  the numbers of
cancers caused by high dosages of radiation.  For present puropses
one wants rather the effect of  low  doses,  on  which  there  are
little or no data.  To estimate this effect, one may  assume  that
the effect $E$ is proportional to dosage $D$---so that $E=kD$  for
some constant $k$.  Alternatively, on may assume that  the  effect
is quadratic so that $E$ depends both on $D$ and  on  $D^2$.  Then
the curve giving $E$ as a  function of  $D$ is  parabolic  (Figure
10).  The constants involved---such as  the proportionality factor
$k$---are then  chosen  to get  the  best  fit  of the line or the
parabola to the high dosage data.  The resulting  formula  is then
used to calculate the effect at low dosage.  Quite naturally,  the
linear  formula and the quadratic one give substantially different
results by this extrapolation; this is the cause  of  considerable
controversy.  Is the  linear formula right?   Does  the  choice of
formula  depend  on  the  type  of cancer considered?  There is no
secure and scientific answer to these  pressing  policy questions.
In particular, the mathematical methods themselves cannot possibly
produce  an  answer.  Mathematical  models  such as these  may  be
internally consistent, but  that  doesn't imply that they must fit
the facts.  Here, as in the case  of  regression,  the  assumption
that the variables of interest are connected by a  linear equation
\midinsert \vskip4in
\centerline{\bf Figure 10}
\endinsert

That BEIR III report deals with  just  one  of many different
kinds of risks that plague mankind.  There  are  many  others that
might be estimated, by extrapolation or otherwise.  From all these
cases there has arisen some hope that  there  might  be  effective
general principles underlying such cases--and  so  constituting  a
general subject of risk analysis."  The  hope  to  get  at  such
generality   may   resemble   the  process  of  generalization  so
successful  in  mathematics,  where  properties  of  numbers  have
been  widely  extended  to form the subject of number  theory  and
properties  of  specific  groups have led to general group theory.
However,  I  am doubtful  that there can yet be a generalized such
risk analysis" --- and this I judge  from  another  current  NRC
report.

This report arose as follows:  The  various  public  concerns
about risks were reflected in Congress,  so  a  committee  of  the
Congress  instructed  the  National  Science  Foundation  (NSF) to
establish  a  program  supporting  research on risk analysis.  The
NSF,  in its turn,  did not know how to go about choosing projects
in  such  a  speculative field---so it asked the National Research
Council for advice on how to do this.  The NRC, again in turn, set
up a committee of experts on risk analysis. This committee, in its
turn,  prepared  a  descriptive  report  on   risk  analysis  in
general."  The report also commented  on  specific  cases  of risk
analysis.  For example,  there were extensive comments on the BEIR
III report---but  these  comments  did  not  illuminate  the  BEIR
III  problem  of  extrapolation  and   made   no   other  specific
suggestions.  The  report had little of positive value to help the
NSF decide which projects in risk analysis to fund. Such a general
study of  risk  analysis  is clearly interdisciplinary, but I must
conclude that it is not yet disciplined.

These and other examples of unsatisfactory reports may  serve
to illustrate the confusion resulting  from  questionable  uses of
quantitative methods or of mathematical models.  But why are there
so many cases of such confusion?  Perhaps the troubled history  of
that report on risk analysis  is  typical.   A  practical  problem
appears; many people are concerned, and so is the Congress or  the
Administration. Since the problem is intractable, but does involve
some science, it is passed on to the scientists, perhaps to  those
at the NRC.  Some of these problems  can be---and are---adequately
treated.  For others there is not yet any adequate technique---and
so those techniques which happen to be available (opinion surveys,
cost-benefit  analysis,  regression,  projection,   extrapolation,
decision analysis, and others) get applied to contexts where  they
do not fit.  Confusion arises when the wrong idea is used, whether
for political reasons or otherwise.

There  are also  political  reasons for  such confusion.  Our
representatives, meeting  in  that  exclusively  political city of
Washington, represent a variety of sharply different interests and
constituencies.  To get  something  done,  a  compromise  must  be
struck.  This happens in many ways.  One which I have seen, to  my
sorrow,  is  the  adjustment  of  the  onerous  and   bureaucratic
regulations of the OMB (Office of  Management  and  Budget)  about
cost  principles  for  universities.   Their  Circular  A-21   now
requires faculty members to report the percentage distribution  of
their various university activities, with results  to  add  up  to
100\%, on a Personnel Activity Report Form" (PAR!). Such numbers
are meaningless; they are fictions fostered by accountants. Use of
such numbers makes for extra  paperwork---but  it  also  tends  to
relocate some control of scientific research from universities  to
the government bureaucrats.  For  A-21,  there was recently a vast
attempt at improvement,  combining  all  parties:  the  government
bureaucrats, their accountants, university financial officers, and
a  few  faculty.   What  resuted?   A  compromise,  and not a very
brilliant one.

Thus government policy, when it requires scientific advice on
matters that are intrinsically uncertain, is likely to  fall  into
the government mold: compromise.  And that, I believe, is a source
of confusion.
\smallskip
\centerline{\bf VIII.  Fuzzy Sets and Fuzzy Thoughts}
\smallskip

The misuse of numbers and equations to project  the future  or
to extrapolate  risks  is  by  no  means  limited  to the National
Research Council.  Within the academic community itself there  can
be similar fads and fancies.  Recently I have been reminded of one
curious such case:  The doctrine of fuzzy" sets.

How can a set be fuzzy?  Recall that a set $S$ is  completely
determined  by  knowing  what  things  $x$  belong to $S$ (thus $x \, \epsilon \, S$)  and  what  things  do  not  so belong.  But
sometimes,  it  is  said,  one  may  not know whether or not $x \, \epsilon \, S$.   So  for  a  fuzzy  set  $F$  one  knows only the
likelihood (call it  $\lambda (x)$)  that  the  thing  $x$  is  in
the fuzzy set $F$.  This  measure  of  likelihood may range from 0
($x$ is certainly not in $F$)  all  the way to 1 ($x$ is certainly
in $F$).  Now  I  might  have  said  that  $\lambda (x)$  is  the
probability that $x$ is in $F$, to make  this definition a part of
the  well-established  mathematical  theory  of  probability.  The
proponents  do  not  so  formulate  it, because their intention is
different  and  much more ambitious:   Replace  sets everywhere by
fuzzy sets!

By the grand set-theoretic doctrine, every mathematical concept
can be defined in terms of sets, hence this  replacement  is  very
extensive.  It even turns out that many mathematical concepts  can
be fuzzed up in several ways, say, by varying the fuzzy meaning to
be    attached    to   the   standard   set-theoretic   operations
(intersection, union, etc.)  of the usual Boolean algebra of sets.
And  so  this  replacement  doctrine  has   already   produced   a
considerable literature:  on  fuzzy  logic,  fuzzy  graphs,  fuzzy
pattern  recognition, fuzzy systems theory, and the like.  Much of
this  work  carries  large  claims for  applications of this fuzzy
theory.   In  those  cases  which  I  have  studied,  none  of the
applications  seem  to  be  real; they  do not answer any standing
problems  or  provide  any  new  techniques for specific practical
situations.   For  example, one  recent  book  is   entitled  {\sl
Applications  of Fuzzy Sets to  Systems  Analysis\/}.  The  actual
content  of  the  book  is a sequence of formal fuzzy restatements
of standard mathematical formulations of materials on programming,
automata,  algorithms,  and  (even!)  categories,  but there is no
example of  specific  use of such fuzzy restatement.  One reviewer
(in {\sl  Mathematical Reviews\/}) noted a minimal use  or  lack
of   instructive  examples---the  title   of  the   book  purports
applications."  Another more recent book on fuzzy decision  theory
states as one of  its six conclusions, It  is  a great pity that
there  exist  only  very  few  practical  applications   of  fuzzy
decision theories, and even practical  examples  to illustrate the
theories are scarce."  This leads me to suspect that the initially
ingenious  idea  of  a  fuzzy  set  has  been  overdeveloped  in a
confusing  outpouring  of  words  coupled  with spurious claims to
importance.

There are other examples---cybernetics, catastrophe theory---
where  an  originally  ingenious  new  idea  has   been   expanded
uncritically to lead to meaningless confusion.
\smallskip
\centerline{\bf IX.  Compromise Is Confusing}
\smallskip

But enough  of  such troubling examples of confusion.  Let me
summarize where we have come.  As with any branch of learning, the
real substance of mathematics resides in the ideas.  The ideas  of
mathematics are those which can be formalized and which have  been
developed  to  fit issues arising in science or in human activity.
Truth in mathematics is approached by way of proof  in  formalized
systems.   However,  because  of  the  paradoxical  kinds of self-
reference exhibited by the barn door and Kurt G\"odel,  there  can
be no  single formal system which subsumes all mathematical proof.
To  boot,  the  older  dogmas  that  everything  is  logic"   or
everything is a set"  now  have  competition--everything  is a
function."  However, such questions of foundation are  but  a very
small part of mathematical activity,  which  continues  to  try to
combine the right ideas to attack substantive problems.   Of these
I have touched on only a few examples:  Finding all simple groups,
putting groups together by extension, and  characterizing  spheres
by their connectivity.  In such cases,  subtle  ideas,  fitted  by
hand to the problem, can lead to triumph.

Numerical and mathematical methods can be used for  practical
problems.  However, because of political pressures, the desire for
compromise, or the simple  desire  for  more  publication,  formal
ideas may be applied in practical cases where the ideas simply  do
not  fit.   Then  confusion  arises --- whether  from   misleading
formulation  of  questions  in  opinion  surveys,   from  nebulous
calculations  of  airy  benefits, by regression, by extrapolation,
or otherwise.  As the case of fuzzy sets indicates, such confusion
is  not  fundamentally  a  trouble  caused  by  the  organizations
issuing reports, but is occasioned by academicians making careless
use of good ideas where they do not fit.

As Francis Bacon once said, Truth ariseth more readily from
error  than  from  confusion."   There  remains  to  us, then, the
pursuit  of  truth,  by  way  of proof, the concatenation of those
ideas which fit, and the beauty which results when they do fit.
\vskip.2cm

\midinsert

If only Longfellow were here to do justice to the situation:
\smallskip
\centerline{\bf Tell Me Not in Fuzzy Numbers}
\smallskip
\centerline{In the time of Ronald Reagan}
\centerline{Calculations reigned supreme}
\centerline{With a quantitative measure}
\centerline{Of each qualitative dream}
\smallskip
\centerline{With opinion polls, regressions}
\centerline{No nuances can be lost}
\centerline{As we calculate those numbers}
\centerline{For each benefit and cost}
\smallskip
\centerline{Though his budget will not balance}
\centerline{You must keep percents of time}
\centerline{If they won't sum to one hundred}
\centerline{He will disallow each dime.}
\endinsert
\vfill\eject
\centerline{\bf References}
\smallskip
\ref Air Quality and Automobile Emission Control."  A report
prepared by the Commission on Natural Resources, National Academy
of Sciences for the Committee on Public Works, U. S. Senate.
September 1974.  Volume 4.  {\sl The Costs and Benefits of
Automobile Emission Control\/}.

\ref Eilenberg, S. and Mac Lane, S.  Group Extensions and Homology,"
{\sl Annals of Math.\/} 43 (1941), 758-831.

\ref Eilenberg, S. and Mac Lane, S.  General Theory of Natural
Equivalences" (category theory), {\sl Trans. Am. Math. Soc.\/}
vol. 28 (1945), 231-294.

\ref Energy Modeling for an Uncertain Future:  Study of Nuclear and
Alternative Energy Systems.  A series:  Supporting Paper \#1,

\ref Feit, Walter and Thompson, John G. Solvability of Groups
of Odd Order," {\sl Pacific J. Math.\/} 13 (1968), 775-1029.

\ref Kickert, Walter J. M.  Fuzzy Theories in Decision-making:
A Critical Review.  Martinus Nijhoff, Social Sciences Division,
Leyden, 1978.  182 pp.  Reviewed in {\sl Mathematical Reviews\/}
vol. 81f (June 1981), \#90006.

\ref Ladd, Everett C. and Lipset, Seymour M.  The 1977 Survey of the
American Professoriate.

\ref Lang, Serge.  The File:  A Case Study in Correction.  712 pp.
Springer Verlag, New York, 1981.

\ref Lawvere, R. W. Toposes, Algebraic Geometry, and Logic,"
{\sl Springer Lecture Notes in Math.\/} No. 274 (1972).

\ref Mosteller, F. and Tukey, John W. Data Analysis and Regression.

\ref Negoita, C. V. and Ralescu, D. A.  Applications of Fuzzy Sets
to Systems Analysis, 1975.  John Wiley and Sons, New York.  191pp.
Reviewed in {\sl Mathematical Reviews\/} vol. 58 (1979),
\#9442a.

\ref Science Indicators:  Improvements Needed in Design, Construction,
and Interpretation.  Report by the Comptroller General of the
United States.  PAD 79-35, September 25, 1979.

\ref Whitehead, A. N. and Russell, Bertrand.  {\sl Principia
Mathematica\/}, vol.1, 2nd edition.  Cambridge University Press,
1925.  674 pp.
\vfill\eject
The Ryerson Lecture was given April 20, 1982 in the Glen A. Lloyd
Auditorium of the Laird Bell Law Quadrangle.
\smallskip
\smallskip
{\nn  The Nora and Edward Ryerson Lectures were established by the
trustees of the University in December 1972.  They are intended to
give a  member of the faculty the opportunity each year to lecture
to an  audience from the entire University on a significant aspect
of his or her research and study.  The president of the University
appoints the lecturer on the recommendation of a faculty committee
which  solicits  individual  nominations  from  each member of the
faculty  during the winter quarter preceding the academic year for
\smallskip
{\sl The Ryerson Lecturers have been:}
\smallskip
\line{1973-74:
John Hope Franklin, The Historian and Public Policy" \hfil}
\line{1974-75:
S. Chandrasekhar, Shakespeare, Newton, and Beethoven: \hfil}
\line{1975-76:
Philip B. Kurland, The Private I: Some Reflections on \hfil}
\line{1976-77:
Robert E. Streeter, WASPs and Other Endangered Species" \hfil}
\line{1977-78:
Dr. Albert Dorfman, Answers Without Questions and \hfil}
\line{1978-79:
Stephen Toulmin, The Inwardness of Mental Life" \hfil}
\line{1979-80:
Erica Reiner,  Thirty Pieces of Silver" \hfil}
\line{1980-81:
James M. Gustafson, Say Something Theological!" \hfil}
\vfill\eject
\noindent{\bf ARTICLE}
\vskip.5cm
\centerline{\bf DIGITAL MAPS AND DATA BASES:}
\centerline{\bf AESTHETICS VERSUS ACCURACY $^{\bf \star }$}
\vskip.5cm
\centerline{Robert F. Austin, Ph.D.}
\centerline{President}
\centerline{Austin Communications Education Services}
\centerline{28 Booth Boulevard}
\centerline{Safety Harbor, FL  34695-5242}
\centerline{Past-president, AM-FM International}
\smallskip
\centerline{\bf I. Introduction}

Of the many courses lectured by Immanuel Kant  at  the  University  of
K\"onigsberg, legend has it that one of the  most  frequently  offered
was a course on natural  philosophy  (that  is,  physical  geography).
It was argued  that individuals  could  acquire understanding  through
three  distinct  perspectives:    the  perspective of formal logic and
mathematics, the perspective of time (history),  and  the  perspective
of  space (geography).  The last of these --- the perspective of space
--- acknowledges the importance of distance, site characteristics, and
relative  location  in  describing  the  relationships  among  several
objects or facilities.

Maps are the primary means of representing such relationships.    Maps
are  analytical  tools  which depict spatial relationships and portray
objects from the perspective of space.   The power of  maps  rests  on
their synoptic representation of complex phenomena.  To paraphrase the
Confucian wisdom, a map is worth a thousand words.

{\narrower\smallskip\noindent
Recently  it  has become common to convert spatial phenomena
to digital form and store the data on tapes or discs.   These
data  can then be manipulated by a computer to supply answers
to  questions that formerly required  a  drawn  map $\ldots$
This  stored  geographic  information  is  referred  to  as a
[data base]." [1] \smallskip}

The  maps  produced  from  such  data  bases  are termed digital maps.
Computerized data bases, which may  be  queried and  used  by  several
people  simultaneously,  and  digital  maps  are  of immense value  to
engineers, comptrollers, planners,  and managers.   The combination of
a digital map and data base is worth a thousand mega-words."

The  advantages  of  digital  maps over manually drafted maps are most
apparent in situations of  frequent  growth  or  change.   Among these
advantages  are  the  ease  and speed of revision and  the  fact  that
special  purpose  maps  can  be produced in small volume at reasonable
cost. Moreover, digital maps offer greater precision in representation
and  analysis.   As  more  governmental bodies  [and other agencies]
expend the necessary one-time capital investment,  and begin  to  reap
the vast rewards of computer-assisted record and  map keeping,  others
\smallskip
\centerline{\bf II. Basic Issues}

After  an  agency or firm has decided to convert its manual records to
digital map and data base form, several issues must be addressed.

The  first,  and  most  important  question  the agency must answer is
related  to  the  source  documents to be used by the mapping firm. In
general  terms  the  choice  is  between   cartographic   sources  and
mechanically drafted sources.

Cartography is defined as:

{\narrower\smallskip\noindent
The  art,  science  and technology of making maps,  together
with their study as scientific documents and  works  of  art.
In  this  context maps may be regarded as including all types
of  maps,  plans,  charts,  and  sections,  three-dimensional
models  and  globes  representing  the Earth or any celestial
body at any scale."[3] \smallskip}

In  particular,   cartography  is  concerned  with  the  accurate  and
consistent  depiction  on  a flat surface of activities occurring on a
sphere.

It  is  not possible to duplicate, without distortion, the features on
the  surface of a sphere on any object other than a sphere.  A surface
of constant positive curvature may be represented on a surface of zero
curvature only if distortion is introduced in the representation.   As
a  simple  illustration  of  this  fact,   consider   the  problem  of
flattening" an orange peel: it will tear.  If the orange was made of
rubber,  it would be possible to flatten it without tearing,  but  not
without distortion of another kind --- a topological transformation.

The  methods by which cartographers represent the surface of the earth
on  a  flat  piece  of  paper  are known as map projections.   For any
particular  purpose,   the  selection   of  a   particular  projection
(transformation) is based on the  properties  of  a  sphere  that  the
projection  loses or retains.   Every method of mapping large areas is
affected,  whether it is continuous mapping  or  facet  mapping.    No
coherent,   distortion-free  transformation  exists,  nor,  given  the
theorems  of  mathematics,  can  it  ever  exist.  [4,  5]    However,
cartographers can identify projections that suit a client's particular
purpose.

Quite often appropriate  cartographic  source  documents  already  are
available  to  a public utility and mapping firm team.   Indeed,  such
sources may have served as the base for the construction  of  existing
records.   In other cases, it may be necessary for the mapping firm to
perform  an  aerial  photographic  survey  and  to   translate   these
photographs   into  cartographic  documents  ---  a process  known  as
photogrammetry.

It is also possible to produce maps from non-cartographic sources such
as tax assessor sheets.   Certainly the most  common  non-cartographic
sources  are  mechanically  drafted  cadastral  maps  and  engineering
drawings  or  plans.   These  documents  have  been  defined   by  the
International Association of Assessing Officers:

{\narrower\smallskip\noindent
map,  cadastral  ---  A  map  showing  the boundaries of
subdivisions  of land,  usually  with  the  bearings and
lengths  thereof  and  the  areas  of individual tracts,
for purposes of describing and recording ownership.
\smallskip}
\smallskip

{\narrower\smallskip\noindent
map, engineering --- A  map  showing information that is
essential  for   planning  an  engineering   project  or
development and for estimating its cost.  An engineering
map  is  usually  a  large-scale  map of a comparatively
small area or of a route. [6]
\smallskip}

Although  such  drawings  have  some  value  for  small  area  design,
engineering,  and planning purposes,  there are  a number  of problems
associated with their use as source documents for large area  mapping.
The most critical of these is related to accuracy; tax assessor sheets
in the United States,  for example, are designed to be used as indices
only and are subordinate to  actual  legal  descriptions.    They  are
highly  stylized  and,  despite  their  appearance  and  name,  highly
inaccurate in terms of geographic placement.

Small plans look" correct primarily because they correspond  to  the
limited  range  of  vision of human beings at ground level.   However,
these  documents  also  suffer  from   the   transformation   problem.
Non-uniform, interpretive, subjective  corrections by a draftsman make
this  problem  appear  to  vanish  on  individual  sheets.   But  such
corrections preclude accurately merging sheets for a large area.

In the language of the philosophy of science,  the distinction is  one
between  an  iconic model and a symbolic model.   An iconic model (the
mechanically drafted plan) is designed to look,  in some  metaphorical
fashion, like the object of study. Often, the closer the similarity in
appearance,  the  less  valuable the model for analytical purposes.  A
symbolic  model  (the  map)  is  designed  to  facilitate quantitative
measurements of characteristics of interest to analysts, managers, and
engineers.

A second issue that must be considered by public utilities is the  use
to  which the digital maps will be put.   This will determine the type
of output the mapping firm will generate.   This also  will  determine
the accuracy levels needed. [7]  In general terms, the types of output
products  correspond  to  the types of input products: maps and plans.
[8]    In  our  experience,  public  utility  clients  generally  have
expressed  a  preference  for  digital   maps   related   to   spatial
relationship  data  bases  because  they  facilitate the more accurate
analysis  of  physical  plant  attributes  and  distributions  over  a
large area in a geographic information system.

The primary purpose of most digital map and data base conversion  work
is  to  provide a means to manage corporate assets.   Often the actual
maps produced are used for  index  only,  not  for  scaled  or  direct
measurement.  This is in part a function of the distortion inherent to
any mechanical production or reproduction process,  in part a function
of the  demonstrated  superiority  of  a  fully  digital,  displayable
linked-attribute  data base management system (see Section 5),  and in
part a function of the distortion inherent to all map projections (the
transformation problem previously discussed.)

In some cases,  an agency may wish to construct a geographic data base
that  will  support  a  computerized  plan  generation  and facilities
management system.   As an example,  consider the case where  facility
data  will  be superimposed on a merged cadastral and land base.   The
data base must guarantee the  geographic  locations  of  features  and
their  connectivity,  relationships,  and other characteristics.   The
final digital plan and data base may include  information  on  street,
road,  and highway names,  centerlines, and rights-of-ways; political,
legal,  and natural boundaries;  township,  range,  and section lines;
river,  stream,  and  creek  centerlines  and names; and legal lot and
parcel lines and numbers, among other data. [9]
\smallskip
\centerline{\bf III. Map Production}

Regardless  of the type of source document or output product,  several
stages in cartographic production remain relatively constant.    These
are  considered  first  in  a general manner and then as they apply to
digital map production per se.

First,  we must define the purpose and accuracy standards of the  map.
For example, will the map be used for scaled measurement?  Or will the
map  be used as an index?   Second,  we must identify the features and
activities to be mapped.   The nature of these features will influence
the  amount  of  detail  appropriate for the base map and the finished
map.   The strength of a map may be diminished by displaying too  much
detail.

Third,  we  must  prepare or obtain a land base or base map.   In this
regard, it is important to consider the variety of map projections and
coordinate  systems  available  for  particular tasks.  Using a widely
accepted  system  such  as  the  UTM grid or  latitude  and  longitude
coordinates  has  a  number  of  advantages,  including  ease  of data
exchange and reduced production time and cost.

The next step is to collect and compile the data to be  mapped.    The
basic  rule  is  to  compile  data  at  the  most  detailed  level  of
measurement  possible  and  to  aggregate   the  data  only  at  later
analytical  stages.   Finally,  we  must design and construct the map.
This  is  a  two  step  process  that  involves:  (a)  the  design  of
symbols,  patterns,  legends, and other cartographic devices,  and (b)
the location and actual placement of the features and activities.
\smallskip
\centerline{\bf IV. Digital Maps }

As in traditional cartography the first step in constructing a digital
map  is to establish accuracy levels and to determine which attributes
should be displayed and which should simply be stored.

A  displayable  linked-attribute  data  base  system (discussed below)
allows for the  construction of a fully digital geographic information
system for data management,  as well as for the construction of  index
and general route maps. For such maps, placing items of plant on the
right side of the street" generally is adequate: in the real world,  a
utility pole 60 feet tall is clearly visible at an intersection.   The
critical  attributes  of each item of plant appear as numbers or words
on the map and also as  manipulable  information in the data base.  On
the other hand,  if the map  is  to  be  used  for scaled measurement,
accurate placement of items of plant is paramount.  It should be noted
that this second approach implies  considerable  supplementary  manual
adjustment and therefore substantially higher production costs.
The next question is the method of land base construction. Land bases,
or  base  maps,  may  be  constructed in a variety of ways.  It may be
possible  to  develop an accurate land  base  by  digitizing  existing
source documents.  The information may be captured by board digitizing
vectors  (line strings and endpoints) or by raster scanning.   If  the
quality  of the source documents is high,  these methods are extremely
cost effective.

If the quality  of  the  available  source  documents  is  unknown  or
suspect,  it is common to conduct an aerial photographic survey and to
compile the photographs into a model" of the region (the air  photos
are  rectified  to  generate  a  plane view of the photographed region
similar to a map projection).  These models are then  stereo digitized
and  used   as  highly   accurate   source  documents  for  land  base
construction  using  the  digitizing  or scanning methods noted above.
(+ 10 feet accuracy is standard,  but + 1 foot accuracy is  possible.)
Although  more  expensive than working from existing source documents,
such  an  approach  guarantees  extreme  accuracy.   Moreover, the end
product  often  has  resale  value  which will offset the initial cost
incurred by the end user.

Data  sources  also  may  be combined to form a hybrid land base.  For
example, individual assessor sheets at a wide variety of scales can be
overlaid  on  a  stereo  digitized  base.  The stereo digitized street
centerline network can be modified to agree with cadastral maps so the
assessor data will scale properly and satisfy aesthetic criteria.

Because such a hybrid is,  by definition, unique, it is appropriate to
discuss in detail at the outset of the project the problems likely  to
be  encountered  in  production.   Disadvantages  of  such an approach
include the substantial cost to make the map look"  like the  source
documents, many difficult production problems (e.g., warp of cadastral
data and fitting cadastral information to an accurate land base),  and
the volume of source documents required.

Nevertheless,  some utilities wish to use a hybrid approach because it
gives  them  a product that is internally acceptable from an aesthetic
viewpoint.   We have encountered many situations where end  users  are
uncomfortable  with  the computer-plotted version of a geographic data
base because it does not look" like the product they have  used  for
many years.  The issue of user acceptance is of critical importance to
the  ultimate success of a conversion project.   The cost and problems
associated  with  a  hybrid  approach may be justified in the long run
because the end user is comfortable with the look" of  the  product.
However,  care  must  be  taken to avoid simply computerizing existing
problems.

The process of generating a hybrid combines land base construction and
data compilation.   When a data base  management  system  approach  is
used,  the  distinction between these two production processes is much
clearer.  After a land base is assembled, a decision is made to define
some  selection  of  attributes"  of facilities or items of plant as
displayable. Displayable attributes are those attributes that actually
will be plotted on a map.   Other attributes may be stored in the data
base,  but  not,  as  a  matter  of  course,  be  displayed  on  maps.
(Information  of  interest  to  comptrollers  may  be of little use to
engineers.   Conversely,   attributes   that   facilitate  engineering
procedures may be unimportant to comptrollers.)

Once agreement on the attributes to be displayed is reached,  the data
are coded and laid out on the source document. Some of the information
will  be  recorded interactively at a board digitizing station.  Other
information  will  be  keypunched  and bulk loaded at the construction
phase.   After   construction,    updating   normally   is   performed
interactively.   One  significant  advantage of a data base management
system approach is  its ability  to  grow  or  change  with  technical

{\narrower\smallskip\noindent
Maps  today  are  strongly  functional  in  that  they   are
designed,  like  a  bridge or a house,  for a purpose.  Their
primary purpose is to convey information or to get across' a
geographical  concept or relationship $\ldots$  The mapmaker
is  essentially  a  faithful  recorder  of  given  facts, and
geographical integrity cannot be  compromised  to  any  great
degree.  Nevertheless,  the   range   of  creativity  through
scale,  generalization,  and  graphic manipulation  available
to the cartographer is comparatively great."[10]
\smallskip}

Given its pragmatic character,  it may be surprising to learn that the
physical appearance of a map is a common point of disagreement.   Most
frequently  such  disagreement  arises   because   different  sets  of
aesthetic principles have been applied by  the  client and the mapping
firm.

The general question of aesthetics is not at all simple;  as  the  art
historian  Ivins  has  argued,   the  notion   of   simple   geometric
relationships is not invariant in aesthetic assessments.  [11] Indeed,
aesthetic  issues  are  involved  in  both   the   creation   and  the
appreciation or perception of a work of art.  Within cartography,  the
term aesthetics" is reserved for consideration of  the  placement of
elements such as compass rose,  legends,  and scales;  the  balance of
these elements vis-a-vis the map object;  the selection  of type faces
from the range  of  standard  or customary fonts; and similar elements
of visual display.  To equate correct"  with cartographic aesthetics
is inappropriate, except in the limited sense that some  features  are
required  by cartographic convention  (e.g.,  italic  type  fonts  for
bodies  of  water).   Styles  as to what is correct" from a creative
standpoint  also  vary  from  one academic discipline to another:  the
geographer prefers a fine-line drawing  while  the  urban planner uses
heavy  lines  to  focus  attention and the landscape architect employs
pictorial  symbols.   Differences  in  styles  of map creation help to
condition the manner in which maps are appreciated  by consumers; that
is, aesthetics is more  than  simply  giving  the consumer  what he is
used to.  For example, many public utilities have developed a sense of
aesthetics  conditioned   by   experience   with   manually   drafted,
subjective,  and  highly  symbolic  plans.   Unless  discussed  at the
beginning  of  a  conversion  project  this  point  can  become   most
difficult,  because a mapping  firm  must  assume  that  map  accuracy
takes  precedence  over  map  symbology,  visual  appearance,  and the
superficial aesthetics of the perception of appearance. Often in cases
of this  sort,  the  aesthetics  of  appreciation of the consumer give
way to concerns of accuracy on the part of the mapping  firm  which in
turn may rest on the aesthetics of cartographic creation.

\centerline{\bf V. Computerized Data Bases}

{\narrower\smallskip\noindent
Computer  systems  are  increasingly  used  to aid in the
management of  information,  and  as a result,  new  kinds
of data-oriented  software  and  hardware  are  needed  to
enhance  the  ability  of  the  computer to carry out this
task.   [Data base  systems are]  computer systems devoted
to  the  management  of relatively  persistent data.   The
computer software employed in a data base system is called
a data base management system (DBMS)."  [12]
\smallskip}

Of  the  several  methods  of  classifying data bases used by software
engineers, one  most  important  dichotomy  is  that  between  network
(hierarchical) and relational data bases.   [13]  Certainly  the  most
useful  approach  for many users is the relational data base,  because
this approach permits a larger variety of queries.   Regardless of the
approach,  the method of manipulating the data base remains a critical
issue.

As  noted  in  the quotation of Blasgen [12], the software used with a
data  base  is  known  generically  as  a  data base management system
(DBMS).  Such  a  system  generally  will  have  provisions  for  data
structure  definition as well as for data base creation,  maintenance,
query, and  verification.  Blasgen  observed that in 1981 an estimated
50 companies were marketing 54 different DBMS packages.  [14]

In our experience,  as noted  earlier,  most  public  utility  clients
prefer working with a displayable linked-attribute" DBMS.  This term
describes a system in which selected attributes or characteristics  of
the  company's physical plant are stored in the data base,  where they
may be manipulated by users and also displayed  on  the  digital  map.
For example,  the age,  length,  size, and identification number for a
piece of cable may be stored and displayed.   Because  the  length  of
cable  in  a  given  span is known from installation and stored in the
data base, scaled measurement is unnecessary.

A second approach to building and maintaining a data  base  is  termed
the hybrid" approach.  Consider the following example.

Analytics  technicians  select  National Geodetic Survey monuments and
photo identifiable points which provide a network for accurate control
of the area.   On-site field  survey  crews  accurately  survey  these
points  and  target  them  for aerial photography.   After the flight,
analytics technicians assemble these points into an  accurate  control
network  and  place  them in a digital file.   Using the network file,
features defined in contract specifications are stereo digitized  from
the photography in a digital format.  The major features are portrayed
in  the  form of a detailed centerline network of interstate highways,
public roads and private roads.   The file then is divided into facets
(corresponding to individual maps) and plotted at a scale of 1:100'.

Tax  assessor  sheets and the 1:100' stereo digitized centerline plots
are joined at the next production phase.   The 1:100' plot is overlaid
on individual assessor sheets, intersections are held for control, and
as  rights-of-ways,   lot  lines,   boundaries,  and  text  are  board
digitized.

The product is the best of the digital and mechanical worlds.   It  is
accomplished  in  digital  format  so future modifications can be made
easily,  and  it  is  aesthetically  pleasing  ---  it  looks  like an
engineering  drawing  or cadastral map.   The information is accurate,
the data can be scaled, and bearings can be extracted.

The success of endeavors of this type depends on an  excellent  vendor
and client relationship.  For such a project, it is recommended that a
test or pilot study in a pre-selected area be completed prior to final
contract  negotiations.    It  is  the responsibility of the client to
define  his  needs  as  accurately  as  possible  and   convey   these
requirements  in  understandable  language   to  the  vendor.    Vague
terminology  and  ambiguous  specifications  can  compound  production
problems.   The vendor,  based  on his background and experience, must
make meaningful suggestions  to  the  client as early in production as
possible as  alternative options may be considered.
\smallskip
\centerline{\bf VI. User Community }

In  summary,  a  comprehensive  digital  map  and data system has many
consider and resolve several  questions  related  to actual needs  and
aesthetic conditioning.  Users should understand that digital maps and
data bases  constructed  using  highly  accurate  aerial  photographic
source  documents  will  not,  as  a rule,  look like familiar graphic
products.  They must consider the  distinctions  between  mechanically
drafted products and cartographic products and decide at the outset of
the  project  whether they are comfortable with the graphic
specifications.

Close communication between the utility company and the  mapping  firm
is  critical.    The  more  carefully  specified the project is at the
beginning,  the fewer the changes that will be required.   Changes  in
specifications  made  during  production,  no  matter how trivial they
appear, generally affect production schedules and costs adversely.

Thus, the creation of a digital map involves not only the mastery
of current technology, in order to produce an accurate" map,  but it
also  involves  an  awareness  of  aesthetics,  as well.  Attention to
aesthetics,  as appreciation  of the map by the consumer will ensure a
satisfied  client;  to this end,  considerable education of the client
with  attention  to  close communication is appropriate.  At the other
end, the mapping firm needs to consider the aesthetics of map creation.
When the aesthetics of creation help to guide the choice of technology,
an accurate and satisfying digital map is generally the end product.
\vfill\eject
\centerline{\bf References}

\ref 1. Robinson,  A., R. Sale and J. Morrison (1978),
{\sl Elements of Cartography\/} (4th edition),
New York: John Wiley, p. 4.

\ref 2.  Robinson, A., et al., p. 272.

\ref 3.  International Cartographic Association
(Commission II, E. Meynen, Chairman) (1973),
{\sl Multilingual Dictionary of Technical Terms in Cartography\/},

\ref 4.  P. W.  McDonnell, Jr. (1979),
{\sl Introduction to Map Projections\/},
New York and Basel: Marcel Dekker, Inc.

\ref 5.  H. S. M. Coxeter,  (1974),
{\sl Projective Geometry\/}, (second edition),
Toronto; University of Toronto Press.

\ref 6.    American Congress on Surveying and
Mapping and American Society of Civil Engineers,
Joint Committee (1978),
{\sl Definitions of Surveying and Associates Terms\/} (rev.),
p. 101.

\ref 7.  Symposium on the National Map Accuracy Standard,"
{\sl Surveying and Mapping\/}, 1960, v.20,  n.4,  pp.  427-457,
and M. M.  Thompson and G. H. Rosenfield,  On  Map Accuracy
Specifications," {\sl Surveying and Mapping\/},
1971, v.31, n.1, pp. 57-64.

\ref 8.  Cuff, D.J. and M.T.  Mattson (1982),
{\sl Thematic Maps: Their Design and Production\/},
New York and London: Methuen.

\ref 9.    For  additional discussion of this approach,
see Easton,  C.H. (1975),
The Land Records Information System in Forsyth County,
North Carolina," pp.  261-267  in
Chicago: International  Association  of  Assessing Officers.

\ref 10.  Robinson, A., et al., p.7.
See also: Amheim, R. (1976), The Perception of Maps,"
{\sl The American Cartographer\/}, 1976, v.3, n.1, pp. 5-10.

\ref 11.  Irvins, W. Jr. (1964),
{\sl Art and Geometry: A Study in Space Intuitions\/},
New  York:  Dover
(reprint  of  1949 Harvard University Press edition).
\ref 12.  Blasgen, M. W.,
Data Base Systems,"
{\sl Science\/}, 1982,  v.215,  12 February.

\ref 13.   Codd,  E.,
Relational Data Base: A Practical Foundation for Productivity,"
{\sl Communications\/} of the ACM, 1982, v.25, n.2.

\ref 14.  Blasgen [12].
\vfill\eject
\centerline{\bf $^{\bf \star }$ Acknowledgments}

An earlier version of this essay appeared as a Chicago  Aerial  Survey
Production Report.   Comments  on  the previous version by Mr.  Harold
Flynn are gratefully acknowledged,  as  is  the  support  provided  by
Geonex Corporation during its preparation.  The author wishes to thank
an  anonymous  referee  for  comments  useful in preparing the current
version.
\vfill\eject
\noindent{\bf FEATURES}
\vskip.5cm
\noindent{\bf i.  Press Clipping}

{\sl Science\/}, November 29, 1991, Vol.  254,  No. 5036,  copyright,
the   American   Association  for  the  Advancement of Science.  Many
thanks to Joseph Palca at {\sl Science\/} for his continuing interest
in  online  journals.   The  citation  appeared  in  Briefings" and
is entitled {\bf  Online Journals}," by Joseph Palca.

\centerline{\bf Online Journals}

When the AAAS and OCLC Online Computer Library Center
announced  the  scheduled  debut next year of their new
journal---{\sl The On\-line Journal of Current Clinical
Trials\/} --- they said it would be the  world's  first
peer-reviewed, on\-line science journal ({\sl Science\/},
27 September, p. 1480).   Since  then,  two other  such
journals   have   made    their   presence   known   to
{\sl Science\/}.  They are {\sl Solstice: An Electronic
Journal of Geography and Mathematics\/},  published  by
Sandra Lach Arlinghaus of the Institute of Mathematical
Geography in Ann Arbor, Michigan, and {\sl Flora Online
at the Buffalo Museum of Science. Both have been around
for about 2 years and are available  free  over several
popular research computer networks."

NOTE:   Readers  wishing  to contact Richard Zander can do so at

VISBMS@UBVMS
\vfill\eject
\noindent{\bf ii. Word Search Puzzle}

The  point of this puzzle is to develop familiarity  (dispelling fear)
with  a  selection of  words possibly not familiar to student readers.
The words are embedded in the jumble of letters below; not all letters
in  this  array  are  part  of  a word in the list and other words may
appear in the puzzle.  Words from the list may be written from left to
right, from right to left, from top to bottom,  from bottom to top, or
diagonally (in any direction).  Solution is on the last  page  of this
issue.

\centerline{\bf WORDS IN THE PUZZLE}

$$\vbox{\settabs 3 \columns \+ Algorithm &Gnomonic &Parallel \cr \+ Asymptote &Graticule &Polyconic \cr \+ Azimuthal &Integral &Projection \cr \+ Circumpolar &Inverse &Rotation \cr \+ Circumscribe &Jacobian &Solstice \cr \+ Conic &Lambert &Stereographic \cr \+ Converge &Latitude &Tangent \cr \+ Curvature &Logarithm &Translation \cr \+ Cylindrical &Longitude &Vector \cr \+ Divergent &Matrix & \cr \+ Equatorial &Mercator & \cr \+ Equidistant &Meridian & \cr \+ Equinox &Norm & \cr \+ Exponent &Oblique & \cr \+ Fractal &Orthogonal & \cr}$$
$$\vbox{\settabs 26 \columns \+ P&O&R&T&H&P&O&L&Y&C&O&N&I&C&I&H&P&A&R&G&O&E&R&E&T&S \cr \+ E&L&O&N&G&I&T&U&D&E&E&Q&U&I&D&I&S&T&A&N&T&N&E&B&C&C \cr \+ Q&A&L&G&O&R&I&T&H&M&L&O&D&I&V&E&R&G&E&N&T&G&G&I&U&V \cr \+ U&H&S&L&G&H&T&M&I&H&P&R&C&Y&M&S&A&N&E&G&R&R&P&R&E&R \cr \+ A&T&E&Y&R&M&E&T&O&T&P&M&Y&S&A&P&L&N&U&O&A&R&V&C&E&O \cr \+ T&U&L&A&M&B&E&R&T&I&A&C&I&N&O&M&O&N&G&T&T&A&T&S&A&T \cr \+ O&M&R&O&N&P&T&R&B&R&E&N&N&R&O&P&P&X&I&C&T&O&I&M&L&A \cr \+ A&I&X&M&R&T&T&P&M&A&S&C&G&C&X&C&M&C&A&U&R&R&T&U&N&T \cr \+ L&Z&R&S&A&T&Z&O&L&G&C&M&D&E&N&U&U&C&R&A&U&N&E&C&O&I \cr \+ R&A&T&F&E&T&R&S&T&O&M&U&Q&Q&N&L&C&E&R&F&E&G&L&R&A&O \cr \+ I&R&C&V&R&E&R&G&N&L&E&U&V&U&E&T&R&A&N&S&L&A&T&I&O&N \cr \+ P&O&O&I&T&A&R&I&S&T&I&L&C&A&T&O&I&N&M&U&T&H&A&C&Z&I \cr \+ M&T&N&O&R&E&C&X&X&N&W&F&A&T&B&N&C&A&J&I&O&C&A&R&N&N \cr \+ E&A&V&E&R&D&A&T&O&F&J&A&C&O&B&I&A&N&T&G&E&R&G&T&A&V \cr \+ T&C&E&T&S&A&N&X&A&R&S&X&A&R&R&T&G&U&O&A&N&R&E&V&B&E \cr \+ G&R&R&A&M&R&S&I&A&L&X&S&E&I&P&R&D&N&D&G&E&G&A&R&V&R \cr \+ R&E&G&E&C&I&T&S&L&O&S&T&E&A&Z&E&A&M&U&T&R&H&A&L&S&S \cr \+ A&M&E&R&I&D&I&A&N&Y&L&A&A&L&E&L&L&A&R&A&P&R&A&L&T&E \cr \+ I&S&G&A&L&R&N&O&I&T&C&E&J&O&R&P&O&B&L&I&Q&U&E&C&A&L \cr}$$
\vfill\eject
\noindent{\bf iii. Software  Briefs} ---  Brief descriptions  of
software provided by the  creator.  Look for reviews
of    the   software   in   subsequent   issues   of
{\sl Solstice \/}.   The  Institute  of Mathematical
Geography (IMaGe) makes  no claim as to the accuracy
of  the  statements  made  by  the   creators;   the
appearance of their comments in {\sl Solstice\/}  is
{\bf not} an endorsement, either direct or indirect,
of the product by IMaGe or by anyone associated with
either  IMaGe or {\sl Solstice\/}.   These Briefs"
are simply  presented as a way for software creators
to share infomation,  in an  e-journal,  with  other
possibly interested parties.

a.  RangeMapper$^{\hbox{TM}}$  ---  version 1.4.
Created  by Kenelm W. Philip,  Tundra Vole Software,
Fairbanks, Alaska.  Program and Manual by  Kenelm W.
Philip; commentary below from the Manual.
A  utility  for  biological species range mapping,
and similar mapping tasks in other fields."

RangeMapper$^{\hbox{TM}}$  is a Macintosh mapping utility
program  designed specifically for the field or museum biologist
who wants  to be  able to produce,  rapidly and easily,  species
range maps for various organisms.  The program may also  be used
for  mapping  other  kinds  of  data,  in medical, sociological,
geological, geophysical, biological, etc. applications.

The program is aimed at people whose mapping needs cover sizable
areas.  The most accurate data files in the map base are derived
from  the  CIA  mapping files, which are suitable for displaying
regions  down  to  20-30  miles  or  so in linear extent without
showing a polygon' effect from data  point spacing.    The data
files  also include the  Micro World Data Bank files,  which are
suitable for  mapping on a whole-world scale and down to regions
perhaps 500 miles in linear extent.

On a Macintosh II or other machines in the Macintosh II family,
most  maps  will  be  plotted  in  well under one minute.  Once
plotted,  they  can  be  saved to disk and re-loaded as needed.
In either case, data from properly formatted latitude/longitude
textfiles can be read by the  program  and plotted to the maps.
If  your  species  data  (including  lat/long  coordinates  for
collecting  sites) are stored in a computer database, it should
be easy to arrange to export, for any given species, a textfile
of lat/long coordinates that RangeMapper can read directly ---

In  conjunction  with the word processor NISUS, RangeMapper may
be used as a map-based visual interface to a text database.

The  current  version  (1.4) of RangeMapper is set up for world
mapping   in   six   projections  ---  north  polar  azimuthal,
cylindrical,   Mercator,   orthographic,   stereographic,   and
Lambert azimuthal equal-area.   The north  polar azimuthal maps
are  quite  usable  down  to  the  southern limits of the lower
48  states  and  equivalent  latitudes  in  Eurasia,   and  are
excellent   for   higher-resolution  mapping  of  Alaska.   The
cylindrical and Mercator projections can show the entire  world
(barring the extreme polar regions  in  Mercator),  centered at
any  longitude.   The  last  three  projections  show up to one
hemisphere, which may be centered at any point  on  the earth's
surface.

The map data files from the  Micro  World  Data  Bank cover the
entire world.  The  only  CIA  file presently incorporated into
RangeMapper is the  Alaska  file  (approximately 150,000 points
for  coastlines,  islands,  rivers,  and  lakes).   The  entire
continent of North America will be added from CIA files  later,
permitting mapping of the U.S. and Canada to the same precision
as can be obtained using the current file for Alaska.

Other continents, and higher-precision files covering the U.S.,
\vfill\eject
b.         XYNIMAP"   ---  created  by David H. Douglas,
University of Ottawa;  a comprehensive system  for
computer cartography and geo-spatial analysis."
Preliminary Version.

XYNIMAP is a comprehensive system for computer cartography
and geo-spatial analysis, that does a lot of things, but not
everything, that other packages do. If you give  it a chance
you will find it does a number of things better  than  other
packages.  The diskette contains all  manuals and  operating
instructions. It is meant for PC computers.  PC-XT 286, 386,
486.

\centerline{COMPONENTS}

XYNITIZE:    An  interactive  map  digitizing system (with a
different way of interacting with the user).

BNDRYNET:  A program to convert a mass of intersecting lines
into a  topology to represent the polygons that are visually
evident. In other words BNDRYNET is a cartographic spaghetti
to polygon converter.

CONSURF: A contour to grid digital elevation model program.

POLYGRID: A polygon to grid converter.

XYBINASC \& XYASCBIN:   Programs  to  convert  a XYNIMAP {\sl
stream  feature\/} file back and forth from compressed binary
to readable (therefore editable) ASCII files.

GDEMIDRI:   A  program  to  convert  a  XYNIMAP  grid digital
elevation model to an IDRISI .img and .doc files.

XYTALLY:  A program to read a XYNIMAP stream feature file and
produce a printout of various measures: (lengths of lines and
areas of regions).

{\bf The following are tested workable and distributable programs }
but I am just not ready to put them out just yet.

XYNIDISP:  A comprehensive display system for the PC computer
with EGA or VGA graphics adapter cards.

XYNIDRAW:  A  comprehensive  display  system for line drawing
plotters

VUBLOK:   A  particularly robust perspective view map program
for   grid   digital  elevation  models.    It  produces  the

PILLAR:   A program to display a geographical distribution by
an image of  standing  vertical  pillars  on the surface of a
perspective  view  of  a  base  map.  The  program curves the
surface to a realistic projection.

PROCIR: A proportional circle display program."
\vfill\eject
\noindent{\bf iv.  Index to Volumes I (1990) and II (1991) of
{\sl Solstice}.}
\vskip.5cm
\noindent{\bf Volume I, Number 1, Summer, 1990}

\noindent 1.  REPRINT

William Kingdon Clifford, {\sl Postulates of the Science of Space\/}

This reprint of a portion of  Clifford's  lectures  to  the
Royal  Institution in the 1870's suggests many geographic topics
of concern in the last half of the twentieth century.   Look for
connections  to  boundary  issues,  to  scale problems, to self-
similarity and fractals, and to non-Euclidean  geometries  (from
those based on denial of Euclid's parallel  postulate  to  those
based on a sort of mechanical polishing").  What else did,  or

\noindent 2.  ARTICLES.

Sandra L. Arlinghaus, {\sl Beyond the Fractal.}

An original article.  The fractal notion of  self-similarity
is  useful  for  characterizing  change  in  scale;  the  reason
fractals are effective in the geometry of central  place  theory
is  because  that  geometry  is hierarchical in nature.  Thus, a
natural place to look for other connections of this  sort  is to
other geographical concepts that are also hierarchical.   Within
diffusion.

When the idea of diffusion is extended to see adopters" of
an innovation as attractors" of new adopters,  a  Julia set is
introduced as a possible axis against which to measure one class
of geographic phenomena.   Beyond the fractal  context,  fractal
concepts,  such  as  compression"  and  space-filling"   are
considered in a broader graph-theoretic setting.
\smallskip
\smallskip
William C. Arlinghaus, {\sl Groups, Graphs, and God}

An original article based on a talk given  before  a MIdwest
GrapH TheorY (MIGHTY) meeting.  The author,  an  algebraic  graph
theorist, ties his research interests to a broader  philosophical
realm,  suggesting  the  breadth  of  range  to  which  algebraic
structure might be applied.

The  fact  that  almost  all  graphs  are rigid (have trivial
automorphism groups) is exploited to argue probabilistically  for
the  existence  of  God.  This  is  presented  with the idea that
applications  of  mathematics  need  not be limited to scientific
ones.
\smallskip
\noindent 3.  FEATURES
\smallskip
\item{i.}  Theorem Museum --- Desargues's  Two  Triangle  Theorem
from projective geometry.
\item{ii.} Construction Zone --- a centrally symmetric hexagon is
derived from an arbitrary convex hexagon.
\item{iii.} Reference Corner --- Point set theory and topology.
\item{iv.}  Educational Feature --- Crossward puzzle on spices.
\item{v.}   Solution to crossword puzzle.
\smallskip
\smallskip
\noindent{\bf Volume I, Number 2, Winter, 1990}
\smallskip
\noindent 1.  REPRINT

John D. Nystuen (1974), {\sl A City of Strangers:  Spatial Aspects
of Alienation in the Detroit Metropolitan Region\/}.

This paper examines the urban shift from people space" to
machine space" (see R. Horvath,  {\sl Geographical Review\/},
April, 1974) in the Detroit metropolitan  region  of 1974.   As
with Clifford's {\sl Postulates\/}, reprinted in the last issue
of {\sl Solstice\/}, note  the  timely  quality  of many of the
observations.

\noindent 2.  ARTICLES

Sandra Lach Arlinghaus, {\sl Scale and Dimension: Their Logical
Harmony\/}.

Fallacy of Division and the Fallacy of Composition in a fractal
setting.
\smallskip
\smallskip
Sandra Lach Arlinghaus, {\sl Parallels between Parallels\/}.

The earth's sun introduces a symmetry in the perception of
its trajectory in the sky that naturally partitions the earth's
surface  into  zones  of  affine  and hyperbolic geometry.  The
affine zones, with  single  geometric  parallels,  are  located
north and south of the  geographic  parallels.   The hyperbolic
zone, with multiple geometric parallels, is located between the
geographic  tropical  parallels.   Evidence  of  this geometric
partition is suggested in the geographic environment --- in the
design of houses and of gameboards.
\smallskip
\smallskip
Sandra L. Arlinghaus, William C. Arlinghaus, and John D. Nystuen.
{\sl The Hedetniemi Matrix Sum:  A Real-world Application\/}.

In a recent paper, we presented an algorithm for finding the
shortest distance between any two nodes in a network of $n$ nodes
when  given  only  distances between adjacent nodes [Arlinghaus,
Arlinghaus, Nystuen,  {\sl Geographical  Analysis\/}, 1990].  In
that  previous   research,  we  applied  the  algorithm  to  the
generalized  road  network  graph surrounding San Francisco Bay.
Here,  we  examine consequent changes in matrix entires when the
the  1989  earthquake  that closed the San Francisco --- Oakland
Bay Bridge.
\smallskip
\smallskip
Sandra Lach Arlinghaus, {\sl Fractal Geometry  of Infinite Pixel
Sequences:  Su\-per\--def\-in\-i\-tion" Resolution\/}?

Comparison of space-filling qualities of square and hexagonal
pixels.
\noindent 3.  FEATURES
\item{i.}       Construction  Zone ---  Feigenbaum's  number;  a
triangular coordinatization of the Euclidean plane.
\item{ii.}  A three-axis coordinatization of the plane.
\smallskip
\noindent {\bf Volume II, Number 1, Summer, 1991}
\smallskip
\noindent 1.  ARTICLE

Sandra L. Arlinghaus, David Barr, John D. Nystuen.
{\sl The Spatial Shadow:  Light and Dark --- Whole and Part\/}

This account of some of the projects of sculptor David Barr
attempts to place them in a formal, systematic, spatial  setting
based  on  the  postulates  of  the  science of space of William
Kingdon Clifford (reprinted in {\sl Solstice\/}, Vol. I, No. 1.).
\smallskip
\smallskip
\noindent 2.  FEATURES

\item{i}  Construction Zone --- The logistic curve.
\item{ii.} Educational feature --- Lectures on Spatial Theory"
\vfill\eject
\noindent{\bf v. Other publications of }
\noindent{\bf the Institute of Mathematical Geography. }
\noindent{\bf 1991}
\smallskip
\centerline{\it INSTITUTE OF MATHEMATICAL GEOGRAPHY (IMaGe)}
\centerline{\it 2790 BRIARCLIFF}
\centerline{\it ANN ARBOR, MI 48105-1429; U.S.A.}
\centerline{(313) 761-1231; IMaGe@UMICHUM}
\vskip 0.2cm
\centerline{\it Imagination is more important than knowledge"}
\centerline{\it A. Einstein}
\vskip.2cm
\centerline{\bf MONOGRAPH SERIES}
\centerline{\sl Scholarly Monographs--Original Material}
\centerline{Prices on request, exclusive of shipping and handling;}
\centerline{payable in U.S. funds on a U.S. bank, only.}
\smallskip
Monographs are printed by {\bf Digicopy} on 100\% recycled paper
of achival quality; both hard and soft cover is available.
\vskip 0.2cm
1.  Sandra L. Arlinghaus and John D. Nystuen.  {\it Mathematical
Geography and Global Art:  the Mathematics of David Barr's Four
Corners Project\/},'' 1986.
\vskip 0.1cm

This monograph contains Nystuen's calculations, actually used
by Barr to position his abstract tetrahedral sculpture
within the earth.  Placement of the sculpture vertices in Easter
Island, South Africa, Greenland, and Indonesia was chronicled in
film by The Archives of American Art for The Smithsonian
Institution.  In addition to the archival material, this
monograph also contains Arlinghaus's solutions to broader theoretical
questions--was Barr's choice of a tetrahedron unique within his
initial constraints, and, within the set of Platonic solids?
\vskip 0.2cm
2.  Sandra L. Arlinghaus.  {\it Down the Mail Tubes:  the Pressured
Postal Era, 1853-1984\/}, 1986.
\vskip 0.1cm

The history of the pneumatic post, in Europe and in the
United States, is examined for the lessons it might offer to the
technological scenes of the late twentieth century.  As Sylvia L.
Thrupp, Alice Freeman Palmer Professor Emeritus of History, The
University of Michigan, commented in her review of this work
Such brief comment does far less than justice to the
intelligence and the stimulating quality of the author's writing,
or to the breadth of her reading.  The detail of her accounts of
the interest of American private enterprise, in New York and
other large cities on this continent, in pushing for construction
of large tubes in systems to be leased to the government, brings
out contrast between American and European views of how the new
technology should be managed.  This and many other sections of
the monograph will set readers on new tracks of thought.''
\vskip 0.2cm
3.  Sandra L. Arlinghaus.  {\it Essays on Mathematical Geography\/},
1986.
\vskip 0.1cm

A collection of essays intended to show the range of power
in applying pure mathematics to human systems.  There are two types of
essay:  those which employ traditional mathematical
proof, and those which do not.  As mathematical proof may itself
be regarded as art, the former style of essay might represent
traditional'' art, and the latter, surrealist'' art.  Essay
titles are:  The well-tempered map projection,'' Antipodal
graphs,'' Analogue clocks,'' Steiner transformations,'' Concavity
and urban settlement patterns,'' Measuring the vertical city,''
Fad and permanence in human systems,'' Topological exploration
in geography,'' A space for thought,'' and Chaos in human
systems--the Heine-Borel Theorem.''
\vskip 0.2cm
4.  Robert F. Austin, {\it A Historical Gazetteer of Southeast Asia\/},
1986.
\vskip 0.1cm
Dr. Austin's Gazetteer draws geographic coordinates of Southeast
Asian place-names together with references to these
place-names as they have appeared in historical and literary
documents.  This book is of obvious use to historians and to
historical geographers specializing in Southeast Asia.  At a
deeper level, it might serve as a valuable source in establishing
place-name linkages which have remained previously unnoticed, in
documents describing trade or other communications connections,
because of variation in place-name nomenclature.
\vskip 0.2cm
5.  Sandra L. Arlinghaus, {\it Essays on Mathematical Geography--II\/},
1987.
\vskip 0.1cm

Written in the same format as IMaGe Monograph \#3, that seeks to use
pure'' mathematics in real-world settings, this volume
contains the following material:  Frontispiece--the Atlantic
Drainage Tree,'' Getting a Handel on Water-Graphs,'' Terror in Transit:
A Graph Theoretic Approach to the Passive Defense of Urban Networks,''
Terrae Antipodum,'' Urban Inversion,''
Fractals:  Constructions, Speculations,
and Concepts,'' Solar Woks,'' A Pneumatic Postal Plan:  The
Chambered Interchange and ZIPPR Code,'' Endpiece.''
\vskip 0.2cm
6.  Pierre Hanjoul, Hubert Beguin, and Jean-Claude Thill, {\it Theoretical
Market Areas Under Euclidean Distance\/}, 1988.
(English language text; Abstracts written in French and in English.)
\vskip 0.1cm

Though already initiated by Rau in 1841, the economic theory of the
shape of
two-dimensional market areas has long remained concerned with a
representation of transportation costs as linear in distance.
In the general gravity model, to which
the theory also applies, this corresponds to a decreasing exponential
function of distance deterrence.  Other transportation cost and
distance deterrence functions also appear in the literature, however.
They have not always been considered from the viewpoint of the shape
of the market areas they generate, and their disparity asks the
question whether other types of functions would not be worth
being investigated.  There is thus a need for a general theory
of market areas:  the present work aims at filling this gap,
in the case of a duopoly competing inside the Euclidean plane
endowed with Euclidean distance. \vskip 0.1cm

(Bien qu'\'ebauch\'ee par Rau d\es 1841, la th\'eorie \'economique
de la forme des
aires de march\'e planaires s'est longtemps content\'ee de l'hypoth\ese
de co\^uts de transport proportionnels \a la distance.  Dans le mod\ele
gravitaire g\'en\'eralis\'e, auquel on peut \'etendre cette th\'eorie, ceci
correspond au choix d'une exponentielle d\'ecroissante comme fonction de
dissuasion de la distance.  D'autres fonctions de co\^ut de transport
ou de dissuasion de la distance apparaissent cependant dans la
litt\'erature.
La forme des aires de march\'e qu'elles engendrent n'a pas toujours \'et\'e
\'etudi\'ee ; par ailleurs, leur vari\'et\'e am\ene \a se demander
si d'autres fonctions encore ne m\'eriteraient pas d'\^etre examin\'ees.
Il para\^it donc utile
de disposer d'une th\'eorie g\'en\'erale des aires de march\'e : ce \a
quoi s'attache ce travail en cas de duopole, dans le cadre
du plan euclidien muni d'une distance euclidienne.)
\vskip 0.2cm
7.  Keith J. Tinkler, Editor, {\it Nystuen---Dacey Nodal Analysis\/}, 1988.
\vskip.1cm

Professor Tinkler's volume displays the use of this graph theoretical
tool in geography, from the original Nystuen---Dacey article, to a
bibliography of uses, to original uses by Tinkler.  Some reprinted
material is included, but by far the larger part is of previously
unpublished
material.  (Unless otherwise noted, all items listed below are previously
unpublished.)  Contents:   Foreward' " by Nystuen, 1988;
Preface" by Tinkler,
1988; Statistics for Nystuen---Dacey Nodal Analysis," by Tinkler, 1979;
Review of Nodal Analysis literature by Tinkler (pre--1979, reprinted
with permission;
post---1979,
new as of 1988); FORTRAN program listing for Nodal Analysis by Tinkler;
A graph theory
interpretation of nodal regions'' by John D. Nystuen and Michael F. Dacey,
reprinted with permission, 1961; Nystuen---Dacey data concerning
telephone flows in Washington and Missouri,
1958, 1959 with comment by Nystuen, 1988;
The expected distribution of nodality in random (p, q)
graphs and multigraphs,'' by Tinkler, 1976.
\vskip.2cm
8.  James W. Fonseca, {\it The Urban Rank--size Hierarchy:  A Mathematical
Interpretation\/}, 1989.
\vskip.1cm

The urban rank--size hierarchy can be characterized as an equiangular spiral
of the form $r=ae^{\theta \, \hbox{cot}\alpha}$.  An equiangular spiral
can also be constructed from a Fibonacci sequence.  The urban rank--size
hierarchy is thus shown to mirror the properties derived from Fibonacci
characteristics such as rank--additive properties.  A new method of
structuring the urban rank--size hierarchy is explored which essentially
parallels that of the traditional rank--size hierarchy below rank 11.
Above rank 11 this method may help explain the frequently noted
concavity of the rank--size distribution at the upper levels.  The
research suggests that the simple rank--size rule with the exponent equal
to 1 is not merely a special case, but rather a theoretically justified norm
against which deviant cases may be measured.  The spiral distribution model
allows conceptualization of a new view of the urban rank--size hierarchy in
which the three largest cities share functions in a Fibonacci hierarchy.
\vskip.2cm
9.  Sandra L. Arlinghaus, {\it An Atlas of Steiner Networks\/}, 1989.
\vskip.1cm

A Steiner network is a tree of minimum total length joining a prescribed,
finite, number of locations; often new locations are introduced into the
prescribed set to determine the minimum tree.  This Atlas explains the
mathematical detail behind the Steiner construction for prescribed sets
of n locations and displays the steps, visually, in a series of
Figures.  The
proof of the Steiner construction is by mathematical induction, and enough
steps in the early part of the induction are displayed completely that the
reader who is well--trained in Euclidean geometry, and familiar with
concepts from graph theory and elementary number theory, should be able to
replicate the constructions for full as well as for degenerate Steiner
trees.
\vskip.2cm
10.  Daniel A. Griffith, {\it Simulating $K=3$ Christaller Central Place
Structures:  An Algorithm Using A Constant Elasticity of Substitution
Consumption Function\/}, 1989.
\vskip.1cm

An algorithm is presented that uses BASICA or GWBASIC on IBM compatible
machines.  This algorithm simulates Christaller $K=3$ central place
structures, for a four--level hierarchy.  It is based upon earlier published
work by the author.  A description of the spatial theory, mathematics, and
sample output runs appears in the monograph.  A digital version is available
from the author, free of charge, upon request; this request must be
accompanied by a 5.5--inch formatted diskette.  This algorithm has been
developed for use in Social Science classroom laboratory situations,
and is designed to (a) cultivate a deeper understanding of central place
theory, (b) allow parameters of a central place system to be altered and
then graphic and tabular results attributable to these changes viewed,
without experiencing the tedium of massive calculations, and (c) help
promote a better comprehension of the complex role distance
plays in the space--economy.  The algorithm also should facilitate
intensive numerical research on central place structures; it is expected
that even the sample simulation results will reveal interesting insights
into
abstract central place theory.

The background spatial theory concerns demand and competition in the
space--economy; both linear and non--linear spatial demand functions are
discussed.  The mathematics is concerned with (a) integration of non--linear
spatial demand cones on a continuous demand surface, using a constant
elasticity of substitution consumption function, (b) solving for roots of
polynomials, (c) numerical approximations to integration and root
extraction, and (d) multinomial discriminant function classification of
commodities into central place hierarchy levels.  Sample
output is presented for contrived data sets, constructed from
artificial and empirical information, with the wide range of all possible
central place structures being generated.  These examples should facilitate
implementation testing.  Students are able to vary single or multiple
parameters of the problem, permitting a study of how certain changes
manifest
themselves within the context of a theoretical central place structure.
Hierarchical classification criteria may be changed, demand elasticities may
or may not vary and can take on a wide range of non--negative values, the
uniform transport cost may be set at any positive level, assorted fixed
costs and variable costs may be introduced, again within a rich range of
non--negative possibilities, and the number of commodities can be
altered.  Directions for algorithm execution are summarized.  An ASCII
version of the algorithm, written directly from GWBASIC, is included in
an appendix; hence, it is free of typing errors.
\vskip.2cm
11.  Sandra L. Arlinghaus and John D. Nystuen,
{\it Environmental Effects on Bus Durability\/}, 1990.
\vskip.1cm

This monograph draws on the authors'
previous publications on Climatic" and Terrain" effects on bus
durability.
Material on these two topics is selected, and reprinted, from
three published papers that appeared in the {\sl Transportation
Research Record\/} and in the {\sl Geographical Review\/}.  New
material concerning congestion" effects is examined at the
national level, to determine dense," intermediate," and
sparse" classes of congestion, and at the local level of
congestion in Ann Arbor (as suggestive of how one  might use local data).
This material is drawn together in a single volume, along with a summary
of the consequences of all three effects simultaneously, in order to suggest
direction for more highly automated studies that should follow naturally
with the release of the 1990 U. S. Census data.
\vskip.2cm
12.  Daniel A. Griffith, Editor.
{\sl Spatial Statistics:  Past, Present, and Future\/},  1990.
\vskip.1cm

Proceedings of a Symposium of the same name held at Syracuse
University in Summer, 1989.  Content includes a Preface by Griffith and
the
following papers:
\vskip.1cm

\line{{\sl Brian Ripley}, Gibbsian interaction models"; \hfil}
\line{{\sl J. Keith Ord}, Statistical methods for point pattern data";
\hfil}
\line{{\sl Luc Anselin}, What is special about spatial data"; \hfil}
\line{{\sl Robert P. Haining}, Models in human geography: \hfil}
estimating, and validating models for spatial data"; \hfil}
\line{{\sl R. J. Martin},
The role of spatial statistics in geographic modelling"; \hfil}
\line{{\sl Daniel Wartenberg}, \hfil }
\line{Exploratory spatial analyses:  outliers,
leverage points, and influence functions"; \hfil}
\line{{\sl J. H. P. Paelinck},
Some new estimators in spatial econometrics"; \hfil}
\line{{\sl Daniel A. Griffith}, \hfil }
\line{A numerical simplification for estimating parameters of
spatial autoregressive models"; \hfil}
\line{{\sl Kanti V. Mardia}
Maximum likelihood estimation for spatial models"; \hfil}
\line{{\sl Ashish Sen}, Distribution of spatial correlation
statistics";\hfil}
\line{{\sl Sylvia Richardson},
Some remarks on the testing of association between spatial
processes";\hfil}
\line{{\sl Graham J. G. Upton}, Information from regional data";\hfil}
\line{{\sl Patrick Doreian},
Network autocorrelation models:  problems and prospects." \hfil}

Each chapter is preceded by an Editor's Preface" and followed by a
Discussion and, in some cases, by an author's Rejoinder to the Discussion.
\vskip.2cm
13.  Sandra L. Arlinghaus, Editor.
{\sl Solstice---I\/},  1990.
\vfill\eject
\centerline{\it DISCUSSION PAPERS--ORIGINAL}
\centerline{\it Editor, Daniel A. Griffith}
\centerline{\it Professor of Geography}
\centerline{\it Syracuse University}
\centerline{Founder as an IMaGe series:  Sandra L. Arlinghaus}
\smallskip
\noindent 1.  {\sl Spatial Regression Analysis on the PC:
Spatial Statistics Using Minitab}.  1989.
\vfill\eject
\centerline{\it DISCUSSION PAPERS--REPRINTS}
\centerline{\it Editor of MICMG Series, John D. Nystuen}
\centerline{\it Professor of Geography and Urban Planning}
\centerline{\it The University of Michigan}
\smallskip
\noindent 1.  {\sl Reprint of the Papers of the Michigan InterUniversity
Community of Mathematical Geographers.}  Editor, John D. Nystuen.
\smallskip
Contents--original editor:  John D. Nystuen.
\smallskip
\noindent 1.  Arthur Getis, Temporal land use pattern analysis with the
use of nearest neighbor and quadrat methods."  July, 1963
\smallskip
\noindent 2.  Marc Anderson, A working bibliography of mathematical
geography."  September, 1963.
\smallskip
\noindent 3.  William Bunge, Patterns of location."  February, 1964.
\smallskip
\noindent 4.  Michael F. Dacey, Imperfections in the uniform plane."
June, 1964.
\smallskip
\noindent 5.  Robert S. Yuill, A simulation study of barrier effects
in spatial diffusion problems."  April, 1965.
\smallskip
\noindent 6.  William Warntz, A note on surfaces and paths and
applications to geographical problems."  May, 1965.
\smallskip
\noindent 7.  Stig Nordbeck, The law of allometric growth."
June, 1965.
\smallskip
\noindent 8.  Waldo R. Tobler, Numerical map generalization;"
and Waldo R. Tobler, Notes on the analysis of geographical
distributions."  January, 1966.
\smallskip
\noindent 9.  Peter R. Gould, On mental maps."  September, 1966.
\smallskip
\noindent 10.  John D. Nystuen, Effects of boundary shape and the
concept of local convexity;"  Julian Perkal, On the length of
empirical curves;" and Julian Perkal, An attempt at
objective generalization."  December, 1966.
\smallskip
\noindent 11. E. Casetti and R. K. Semple, A method for the
stepwise separation of spatial terends."  April, 1968.
\smallskip
\noindent 12.  W. Bunge, R. Guyot, A. Karlin, R. Martin, W. Pattison,
W. Tobler, S. Toulmin, and W. Warntz, The philosophy of maps."
June, 1968.
\vfill\eject
\centerline{\bf Reprints of out-of-print textbooks.}
\centerline{\bf Printer and obtainer of copyright permission:  Digicopy}
\centerline{Inquire for cost of reproduction---include class size}
\smallskip
1.  Allen K. Philbrick.  {\sl This Human World}.
\smallskip
\vfill\eject
Publications of the Institute of Mathematical Geography have
been reviewed in

\item{1.} {\sl The Professional Geographer\/} published
by the Association of American Geographers;
\item{2.}  {\sl The Urban Specialty Group Newsletter\/}
of the Association of American Geographers;
$$\vbox{\settabs 26 \columns \+ & & & & &P&O&L&Y&C&O&N&I&C&I&H&P&A&R&G&O&E&R&E&T&S \cr \+ &L&O&N&G&I&T&U&D&E&E&Q&U&I&D&I&S&T&A&N&T& & &B& &C \cr \+ &A&L&G&O&R&I&T&H&M& & &D&I&V&E&R&G&E&N&T& &G&I&U&V \cr \+ &H& & & & & & & &H& & & & & & &A& &E& & &R& &R&E&R \cr \+ &T& & & & &E&T&O&T&P&M&Y&S&A& &L&N& & &A& &V&C& &O \cr \+ &U&L&A&M&B&E&R&T&I&A&C&I&N&O&M&O&N&G&T& &A&T&S& &T \cr \+ &M&R&O&N& & & & &R& &N& & & &P&P& &I& &T&O& &M& &A \cr \+ &I& &M& & & & & &A& & &G& &X& &M&C& &U&R& & &U& &T \cr \+ L&Z& & &A& & & & &G&C& & &E& & &U& &R& & & & &C&O&I \cr \+ &A& &F& &T& & & &O& & &Q&Q&N&L&C&E& & & & &L&R& &O \cr \+ &R&C& &R& &R& &N&L& &U& &U&E&T&R&A&N&S&L&A&T&I&O&N \cr \+ &O&O&I& &A& &I& & &I& & &A& & &I& & & &T&H& &C& &I \cr \+ &T&N& &R& &C& &X&N& & & &T& & &C& & &I&O& & & &N&N \cr \+ &A&V& & &D& &T&O& &J&A&C&O&B&I&A&N&T&G& & & &T& &V \cr \+ &C&E& & & &N&X&A& & & & &R& & & &U&O& & & &E& & &E \cr \+ &R&R& & & & &I& &L& & & &I& & &D&N& & & &G& & & &R \cr \+ &E&G&E&C&I&T&S&L&O&S& & &A& &E&A& & & &R& & & & &S \cr \+ &M&E&R&I&D&I&A&N&Y& & & &L&E&L&L&A&R&A&P& & & & &E \cr \+ & & & & & &N&O&I&T&C&E&J&O&R&P&O&B&L&I&Q&U&E& & & \cr}$$