A model for the Smarandache anti‐geometry

The Smarandache anti‐geometry is a non‐euclidean geometry that denies all Hilbert’s 20 axioms, each axiom being denied in many ways in the same space. In this paper, one finds an economics model to this geometry by making the following correlations: a point is the balance in a particular checking account, expressed in US currency (points are denoted by capital letters); a line is a person, who can be a human being (lines are denoted by lower case italics); and a plane is a US bank, affiliated to the FDIC (planes are denoted by lower case boldface letters).


Introduction
David Hilbert's (1899)``Foundations of geometry'' contain 19 statements, labeled axioms, from which every theorem in Euclid's Elements can be derived by deductive inference, according to the classical rules of logic. The axioms use three property words``point'',``straight'' and``plane'' and three relation words`i ncident'',``between'' and``congruent'' for which no definition is given. These words have, of course, a so-called intuitive meaning in English (as the German equivalents actually used by Hilbert have in his language). But Hilbert believed they ought to be understood in whatever sense was compatible with the constraints prescribed by the axioms themselves [1]. To show that some of his axioms were not logical consequences of the others he unhesitatingly bestowed unorthodox meanings on the undefined terms. This enabled him to produce models that satisfied all the axioms but one, plus the negation of the excluded axiom.
The mathematician-philosopher, Gottlob Frege, showed little understanding for Hilbert's procedure. Frege thought that the undefined terms stood for properties and relations that Hilbert assumed to be well-known and that the axioms were intended as true statements about them. Hilbert disabused him: Smarandach anti-geometry 877 infinitely many systems of basic elements. One needs only to apply an invertible one-one transformation and to stipulate that the axioms for the transformed things are respectively the same. [. . .] This feature of theories can never be a shortcoming and is in any case inevitable (Hilbert to Frege, 29 December 1899;in Frege 1967, pp. 412-13).
Hilbert's reply has continued to sound artificial to those unwilling or unable to follow him in his leap to abstraction, because it is not possible to find a set of familiar relations among chimney-sweeps, laws and states of being in love which, when equated with Hilbert's relations of incidence, betweenness and congruence, would make his axioms to be true. But Hilbert's point can now be made crystal-clear thanks to Florentin Smarandache's anti-geometry [2].
Anti-geometry rests on a system of 19 axioms, each one of which is the negation of one of Hilbert's 19 axioms [3]. Such wholesale negation brings about a complete collapse of the constraints imposed by Hilbert's axioms on its conceivable models. The immediate consequence of this is that models of antigeometry can be readily found in all walks of life [4]. On the other hand, and for the same reason, the truths concerning these models that can be obtained from Smarandache's axioms by deductive inference are somewhat uninteresting, to say the least.
I shall now state my interpretation of the undefined terms in Smarandache's (and Hilbert's) axioms and show, thereupon, that Smarandache's 19 axioms come out true under this interpretation. Following Chimienti and Bencze (nd) [2] I say``line'' where Hilbert says``straight'' (gerade) [5]. Points lying on one and the same line are said to be collinear; points or lines lying on one and the same plane are said to be coplanar. Two lines are said to meet or intersect each other if they have a point in common.
In my interpretation the geometrical terms employed in the axioms are made to stand for ordinary, non-geometric objects and relations, with which I assume the reader is familiar. As a matter of fact, Smarandache's system, despite its vaunted vanguardistic libertarianism, still imposes a few existential constraints on admissible models; for example, his Axiom III presupposes the existence of infinitely many of the objects called``lines''. This has forced me to introduce three existence postulates which my model is required to comply with, at least one of which is plainly unnatural (EP3).
I list below the meaning I bestow on Hilbert's property words: .
A point is the balance in a particular checking account, expressed in US currency. (Points are denoted by capital letters). Two points A and B may be distinct, because they are balances from different accounts, which may or may not belong to different persons, and yet be equal in amount, in which case we shall say that A equals B (symbolized A = B). If A and B are the same point, we say that A and B are identical. Of course, in current mathematical parlance,``equal'', signified by``='', means``identical'', but, like Humpty Dumpty and David Hilbert, I feel free to use words any way I wish, provided that I explain their meaning clearly. I use the standard symbol < to express that a given balance is smaller than another. The above items take care of betweenness and the three kinds of incidence we find in Hilbert and Smarandache. Hilbert's relation of congruence does not apply, however, to points, lines or planes, but to two sorts of figures constructed from points and lines, namely segments and angles. I must therefore define these figures in terms of my points and lines: . Def. I. If two balances A and B belong to the same person x, the collection formed by A, B and all balances Y belonging to x and such that A is less than Y and Y is less than B is called the segment AB. By our definition of``betweenness'', the points belonging to segment AB but not identical with A or B do not lie between A and B. However, the Smarandache axioms are stated in such a way that none of them contradicts this surprising theorem.
. Def. II. If a balance O is owned in common by persons h and k, the set formed by h, k and O is called the angle hO k. h and k could be the same person, in which case the qualification``in common'' is trivial. If h and k are distinct persons, such that h besides O owns a balance P, not shared with k, and k, besides O, owns a balance Q, not shared with h, angle hOk may be called``the angle POQ''. In other words, the expression``angle POQ'' has a referent if and only if there exist persons h and k who respectively own balance P and balance Q separately from one another, and share the balance O; otherwise, this expression has no referent.
. Def. III. Person a acquired balance A partly from person b if and only if a part of balance A was electronically transferred from funds owned by b to the account owned by a which shows balance A. Instead of``a acquired A partly from b'' we write %(a,A,b).
I am now in a position to define Hilbert's two sorts of congruence: (1) Segment AB is congruent with segment CD if and only if there is a person x such that %(h,A,x) and %(h,B,x) and %(k,C,x) and %(k,D,x), where h denotes the owner of balances A and B, and k denotes the owner of balances C and D. (2) Angle (h,P,k) is congruent with angle (f,Q,g) if there is a person x such that %(h,P,x) and %(k,P,x) and %(f,Q,x) and %(g,Q,x).
We shall also need the following definitions: .
Def. IV. Two distinct lines a and b are said to be parallel if and only if persons a and b have accounts with the same bank a but do not own any balance in common.
. Def. V. Let A be a balance belonging to a person h. Any other balances owned by h can be divided into three classes: (i) those that are less than A; (ii) those that are greater than A; and (iii) those that are equal to A. Balances of class (i) and (ii) which are held by h in other accounts with the same bank where he has A will be said to lie, respectively, on one and on the other side of A (on h).
As I said, the fairly weak but nevertheless inescapable constraints implicit in some of Smarandache's axioms force me to adopt three existence postulates. The first of these is highly plausible; the second is, as far as I know, false in fact, but not implausible; while the third is quite unnatural, though not more so than the supposition, involved in Smarandache's Axiom III, that there are infinitely many distinct objects in any model of his system. The existence postulates are: . EP1. Mr John Dee has four checking accounts, with balances of 5,000, 5,000, 5,000 and 8,000 dollars, respectively. EP1 ensures the truth of Smarandache's Axiom II.3. .

EP2.
There are some checking accounts for whose balance two different banks are held responsible. I shall refer to such accounts as two-bank accounts. EP2 is needed to ensure the truth of Smarandache's Axiom I.4; it is also presupposed by his Axiom I.6. We could be more specific and stipulate that checks drawn against such accounts will be cashed at the branches of either bank, that the banks share the maintenance costs and monthly service charges, etc. But all such details are irrelevant for the stated purpose. .

EP3.
There exist infinitely many supernatural persons who may secretly own bank accounts, usually in common. EP3 is needed to take care of the last of the four situations contemplated in Smarandache's Axiom III (the Axiom of Parallels), which involves a point that is intersected by infinitely many lines. In our model, this amounts to a balance in current account that is owned in common by infinitely many International Journal of Social Economics 29,11 880 persons. EP3 is certainly weird, but not more so than say, the postulation of points, lines and a plane at infinity in projective geometry. As in the latter case, we may regard talk of supernatural persons as a fac Ëon de parler. EP3 will perhaps sound less unlikely if the banks of our model are Swiss instead of American.
I shall now show that, with one partial exception (I.7), all of the axioms of Smarandache's anti-geometry hold in our model. As we shall see, the said exception is due to an inconsistency in Smarandache's axiom system.

Group I. Anti-axioms of connection
The anti-axioms of connection are: .  4. There is at least one plane a and at least three points A, B, C, which lie on a but not on the same line, such that A, B, C do not completely determine the plane a. Three points A, B and C on plane a completely determine a if and only if any fourth point D, coplanar with A, B and C, also lies on a. However, according to EP2, the balances A, B and C may pertain to three two-bank accounts held, say, with bank a and bank b. In that case, D could belong to b and not to a. . The third condition, however, cannot be fulfilled, for EP1 demands the existence of at least four points. However, EP1 was solely introduced to secure the truth of Axiom II.3, which actually requires the existence of four distinct points. Therefore Axiom II.3 cannot be satisfied in a model that satisfies the last disjunct of Axiom I.7. Thus, the Smarandache axioms of anti-geometry are inconsistent as stated. I propose to delete the last disjunct of I.7. By the way,``space'' is not a term used in Hilbert's axioms. Indeed, since``space'' stands for the entire domain of application of Smarandache's system it ought not to occur in it, either.

Group II. Anti-axioms of order
The anti-axioms of order are: . Axiom II.1. Let A, B, and C be three collinear points, such that B lies between A and C. This does not entail that B lies also between C and A. Obviously, if A = B < C, CB. Thus, in fact, our model satisfies also the stronger axiom:``If B lies between A and C then B does not lie between C and A''. .

Axiom II.2. Let A and C be two collinear points. Then, there does not always exist a point B lying between A and C, nor a point D such that C lies between A and D.
Obviously, if a given person owns A and C there is no reason why she or he should own a third checking account, let alone one with a balance that is either equal to A and less than C, or greater than both C and A.
. Axiom II.3. There exist at least three collinear points such that one point lies between the other two, and another point lies also between the other two. This is so, of course, if the line is Mr. John Dee (by EP1).
. Axiom II.4. Four collinear points A, B, C, D cannot always be ordered so that (i) B lies between A and C and also between A and D, and (ii) C lies between A and D and also between B and D. In fact, under our definition of betweenness, four collinear points can never be ordered in this way. Condition (i) means than B equals A and is less than C and D; condition (ii) means that C equals A and B and is less than D. These two conditions are plainly incompatible.
. Axiom II.5. Let A, B, and C be three non-collinear points, and h a line which lies on the same plane as points A, B, and C but does not pass through any of these points. Then, the line h may well pass through a point of segment AB, and yet not pass through a point of segment AC, nor through a point of segment BC. Suppose that h does not pass through A, B or C but passes nevertheless through a point of segment AB. This entails that person h owns in common with the owner of both A and B a checking account whose balance X is greater than A and less than B. Obviously, h need not own any balances in common with the International Journal of Social Economics 29,11 882 owner of both B and C, nor with the owner of both A and C, let alone one that meets the requirements imposed by our definition of segment, viz., that the balance in question be greater than B and less than C, or greater than A and less than C.
Group III. Anti-axiom of parallels Let h be a line on a plane a and A a point on a but not on h. On plane a there can be drawn through point A either (i) no line, or (ii) only one line, or (iii) a finite number of lines, or (iv) an infinite number of lines which do(es) not intersect the line h. The line(s) is (are) called the parallel(s) to h through the given point A [6]. Let A be the balance of a checking account with bank a and h a client of bank a who does not own that account. The account in question may belong to a person who shares another balance with h (case i), or to a person h, or to finitely many persons c1,. . .,c n, none of whom shares a checking account with h (cases ii and iii). According to EP3, A may also be owned secretly by infinitely many supernatural persons who do not share an account with h (case iv). By Def. IV, the lines comprised in cases (ii), (iii) and (iv) all meet the requirements for being parallel with h. In Chimienti and Bencze's (nd) article, Axiom III includes the following supplementary condition, enclosed in parentheses:``(At least two of these situations should occur)'', where``these situations'' are cases (i) through (iv). Since I do not understand what this condition means, I did not consider it in the preceding discussion. Anyway, the following is clear: No matter how you interpret the terms``point'' and``line'' and the predicates`c oplanar'' and``intersect'', case (i) excludes cases (ii) and (iv). However, (i) implies (iii) and therefore can occur together with it, if by``finite number'' you mean``any natural number'' in Peano's sense, i.e. any integer equal to or greater than zero. In contemporary mathematical jargon, this would be the usual meaning of the term in this context. By the same token, (ii) implies (iii), for``one'' is a finite number. Finally, (iv) certainly implies (iii), for any infinite set includes a finite subset. In the light of this, the condition in parenthesis is obvious and trivial and few would think of mentioning it. Therefore, the fact that it is mentioned suggests to me that it is being given some other meaning, which eludes me.

Group IV. Anti-axioms of congruence
The anti-axioms of congruence are: . Axiom IV.1. If A, B are two points on a line h, and A 0 is a point on the same line or on another line h 0 , then, on a given side of A 0 on line h 0 , we cannot always find a unique B so that the segment AB is congruent to the segment A 0 B 0 . If balances A and B belong to person h, and A 0 belongs to h 0 (who may or may not be the same person as h), there is no reason at all why there should exist a unique balance B 0 such that segments AB and A 0 B 0 meet the condition of congruence, viz., that there exists a person x such that %(h,A,x) and %(h,B,x) and %(h 0 ,A 0 ,x) and %(h 0 ,B 0 ,x). For the expression``on a given side of A'', see Def. V. . Axiom IV.2. If segment AB is congruent with segment A 0 B 0 and also with segment A 00 B 00 , then segment A 0 B 0 is not always congruent with segment A 00 B 00 . Assume that (i) the owner of A and B got the monies in the respective accounts partly from a person x and partly from a person y; (ii) the owner of A 0 and B 0 got these monies partly from x but not from y; (iii) the owner of A 00 and B 00 got these monies partly from y but not from x. If these three conditions are met, Axiom IV.2 is satisfied.
. Axiom IV.3. If AB and BC are two segments of the same line h which have no points in common besides the point B, and A 0 B 0 and B 0 C 0 are two segments of h or of another line h 0 which have no points in common besides B 0 , and segment AB is congruent with segment A 0 B 0 and segment BC is congruent with B 0 C 0 , then it is not always the case that segment AC is congruent with segment A 0 C 0 . Again, let B and B 0 be acquired by h and h 0 respectively, partly from x and partly from y; A and A 0 from x but not from y; C and C 0 from y but not from x. Then segment AB is congruent with segment A 0 B 0 ; segment BC is congruent with segment B 0 C 0 , but segment AC is not congruent with segment A 0 C 0 . This axiom is not easy to apply, for it contains the terms``half-line'',`i nterior points (of an angle)'' and``side (of a line on a plane)'' which have not been defined and are not used anywhere else in the axioms. I shall take the half-line k issuing from a point O to mean a person k who owns O and owns another bank balance less than O in a different account with the same bank, but does not own a bank balance greater than O in a different account with the same bank. As for the other two expressions, since they are otherwise idle, we could simply ignore them. But if the readers do not like this expedient, they may equally well use the following one: let angle aPb be an angle, such that P is the balance held in common by a and b in their checking account with a particular branch of bank a; the interior points of angle a Pb are the cashiers of that particular branch. We say that the cashiers who are younger than a, lie on one side of a (on a), and that the cashiers who are older than a, lie on the other side of a (on a). The condition on interior points in axiom IV.4 will obviously be met if the line (i.e. bank client) h 0 is so chosen that the branch of bank b where h 0 holds the balance O 0 in common with k 0 has cashiers who are both younger and older than h 0 . Surely this requirement is not hard to meet, if b ranges freely over all banks in the USA.
If the axiom is understood in this way, its meaning is clear enough. It is so weak that there is no difficulty in satisfying it. Take the arbitrarily Group V. Anti-axiom of continuity (anti-archimedean axiom) Let A, B be two points. Take the points A1 , A2, A3, A4, so that A1 lies between A and A2, A2 lies between A 1 and A3, A3 lies between A2 and A4,. . ., and the segments AA 1, A1A2, A2A3, A3A4, are congruent to one another. Then, among this series of points, there does not always exist a certain point An such that B lies between A and An.
Let A and B be two checking account balances. Consider a series of n checking account balances A1, A2,. . ., An, such that all of them belong to the owner of A, and all except An amount to the same sum as A. Suppose that An is greater than A. Now, the condition denied in the apodosis, viz., that B lies between A and An, can hold if and only if B belongs to the owner of both A and An, and B is equal to A. Obviously this is not implied by the initial condition on B, viz., that B is a point, i.e. a checking account balance.

Conclusion
There is a simple moral to be drawn from this exercise. Because Smarandache anti-geometry has removed the stringent constraints on points, lines and planes prescribed by the Hilbert axioms, it is child's play to find uninteresting applications for it, like the one proposed above. When first confronted with this model, Dr Minh L. Perez wrote me that he had the impression that Smarandache's message was directed against axiomatization. Such an attack would be justified only if we take an equalitarian view of axiom systems. To my mind, equalitarianism in the matter of mathematical axiom systems ± though favored by some early twentieth century philosophers ± is like placing all games of wit and skill on an equal footing. The clever Indian who invented chess is said to have demanded 2 64 corn grains minus 1 for his creation. Who would have the chutzpah to charge even a trillionth of that for tic-tac-toe? But Smarandache's Anti-Euclidean geometry does not derogate Hilbert's axiom system for Euclidean geometry. Indeed this system, as well as Hilbert's axiom system for the real number field (1900a), deserve much more not less attention and praise in view of the fact that one can also propose consistent yet vapid axiom systems.