Counterexamples to theorems published and proved in recent literature on
Clifford algebras, spinors, spin groups and the exterior algebra
If you are not familiar with Clifford algebras, see worked problems and preliminary discussions on R, C, H and O, the cross product, objects in 4D, rotations in 4D and the Maxwell equations in Cl_{3}. |
This web-page is an abridgement of my paper article on counterexamples, published in a peer-reviewed journal. The paper discusses the role of counterexamples in scientific progress, in evaluating validity of theories. Counterexamples falsify theories. In mathematics, theorems are provable statements, which cannot be falsified, by definition. However, it is possible to falsify statements published as theorems in mathematical literature.
I give a list of counterexamples, which satisfy all the assumptions of the alleged theorems without the conclusions being valid. I show that there are many false statements published as theorems, in the mathematical literature. In other words, I falsify purported theorems.
The job of a mathematician is to prove statements. Proofs are evaluated with a few colleagues and published in a refereed journal. The purpose of publication is to expose the theorems to public scrutiny. There follows a public debate, which might result in a revision of a theorem (or rather in a revision of a manifestation of the theorem in the literature).
Mathematics is universal and effectively applied to the real world. This often leads mathematicians to a cognitive illusion: when several members of a research group have accepted a new statement as a theorem, the statement becomes true and unfalsifiable (in the minds of the members of the research group). I show that such a collective position is untenable: I falsify statements, which groups of renowned mathematicians have labeled as theorems. Most of the renowned mathematicians are still living and can participate in the evaluation of possible validity of my counterexamples to their "theorems".
The mistakes occurred at frontiers of joint explorations of mathematicians, who still had inaccurate cognitive charts of the new domains they were exploring. Young scientists with recent PhD's are reluctant to admit their mistakes, because for them it is important to get a position. On the other hand, senior scientists also become reluctant to admit their mistakes toward the end of their careers, because they fear that their life works will collapse if mistakes are detected. In general, two groups are ready to admit their blunders: those far removed from the frontline of research and those who produce new knowledge. Producers of new knowledge know the factors of uncertainty: closer to the source of knowledge, the uncertainty increases. The transmitters and users of knowledge are more certain about their theories: university teachers and engineers are supposed to know the facts.
Some mistakes were confined to individuals, who could be easily convinced about their mistakes. Some mistakes were common to several mathematicians, or groups. Such groups of mathematicians had a collective cognitive illusion about the "mathematical reality", which they described and studied by means of a poor language. To break such a cognitive illusion, I often had to learn the poor language and culture of the mistaken groups. Sometimes such groups defended their cognitive bugs vigorously.
The falsified "theorems" were seldom used in subsequent deductions and did not have an impact on the works of other researchers, because I often presented my counterexamples soon after the first occurrence of the "theorems". Nevertheless, other researchers often repeated the same mistakes independently. From this observation, I come to the following main result of my findings:
Creative research mathematicians, exploring the frontiers of our common body of knowledge, tend to make similar mathematical mistakes.This leads to collective cognitive bugs, among advanced researchers. In the realm of such bugs, false statements collectively hold true, cannot be distinguished from the correct ones. In other words, there is no way to make a distinction between
1. a theorem, andIf there were, experts could just agree on classifying all statements labeled as theorems into the above two classes. Mathematicians have responded to this observation by the following comments:
2. a statement labeled as a theorem by all experts.
1. a theorem cannot be falsified, since it is a provable statement, andThe responses overlook cultural aspects of theorem-making: theorems held true today may well become false in the future, if language becomes more precise and resolution of the research topic enhances.
2. if a statement has a counterexample it cannot be a theorem.
I informed almost all of the mathematicians about their errors, prior to exhibition of this web-page. The mathematicians have mostly admitted their mistakes, after a reasoned dialogue, lasting for a few months or sometimes years. The course of events was usually as follows: I find that a theorem does not hold and work out the simplest non-trivial counterexample. I pay special attention to interpreting the text in the way the author has intended so as to make sure that my counterexample reveals an interior inconsistency. Then I send a letter to the author enclosing a detailed description of my counterexample. To avoid misinterepretation of my intentions, I also enclose a photocopy of the theorem in the envelope, underline false parts in red and mark in the margin the word WRONG, in red.
The authors usually defend their theorems, at first. After a few letters have been exchanged, most of the authors accept validity of my counterexamples and admit their mistakes. When the mistake has been understood, the author usually explains the error away as casual and insignificant.
Some of the counterexamples stem from the failure of the authors to list all the assumptions of their theorems or, more serioulsy, to check their statements for small numbers of indices or in low dimensions (typically 2,3). Quite a few of the counterexamples consist of exceptional cases in lower dimensions (typically 2,4,7,8). Counterexamples are also given in the cases where the author failed to notice a general pattern after some dimension (typically at and above 4,5,6). In some cases, the authors had just poorly designed or chosen concepts, which require exceptional cases, like the "inner product", a symmetrized contraction imitating a derivation.
Informing colleagues about their own errors is more subtle. It offers them an opportunity to learn more mathematics. It might also result in a feeling of insufficiency, a cognitive conflict, and instigate a learning process, and thus indirectly lead into the same final situation, namely cognitive growth. See Ginsburg & Opper 1988.
In order to benefit from mathematical arguments presented on this web-page, the viewer should have references at hand, and follow the reasoning of the counterexamples line by line. At first, we have to fix some preliminary notations, since some viewers might be unfamiliar with Clifford algebras.
(xe_{1}+ye_{2})^{2} = x^{2}+y^{2}.It follows that Cl_{2} has a basis (1,e_{1},e_{2},e_{12}), and the following multiplication rules are satisfied: the orthogonal unit vectors e_{1},e_{2} in R^{2} square up to 1
e_{1}^{2} = e_{2}^{2} = 1and anticommute
e_{1} e_{2} = -e_{2} e_{1}and their product equals e_{12} = e_{1}e_{2}. Computing the square of e_{12} we find
e_{12}^{2} = e_{1}e_{2}e_{1}e_{2} = -e_{1}^{2}e_{2}^{2} = -1.Thus, the basis element e_{12} cannot be a scalar (in R) nor a vector (in R^{2}), not even a linear combination of a scalar and a vector. In other words, it is a new kind of object, called a bivector. The basis elements
1span, respectively, the subspaces of scalars, vectors and bivectors (in Cl_{2}).
e_{1},e_{2}
e_{12}
The Clifford algebra Cl_{2} has a faithful matrix image, the matrix algebra Mat(2,R) of 2x2-matrices with entries in R. The basis elements 1,e_{1},e_{2},e_{12} of Cl_{2} can be represented by the matrices
In comparing Cl_{2} and Mat(2,R), which are isomorphic as associative algebras, it should be noted that Cl_{2} has more structure: in the Clifford algebra Cl_{2} there is a distinguished subspace, isometric to the Euclidean plane R^{2}. No such privileged subspace exist in Mat(2,R).
E_{0} = 1 0
0 1, E_{1} = 1 0
0 -1, E_{2} = 0 1
1 0, E_{12} = 0 1
-1 0.
1which span, respectively, the subspaces of scalars, vectors, bivectors, 3-vectors and 4-vectors. The vectors e_{1}, e_{2}, e_{3}, e_{4} satisfy the following multiplication rules: they are unit vectors with squares
e_{1}, e_{2}, e_{3}, e_{4}
e_{12}, e_{13}, e_{14}, e_{23}, e_{24}, e_{34}
e_{123}, e_{124}, e_{134}, e_{234}
e_{1234}
e_{1}^{2} = e_{2}^{2} = e_{3}^{2} = 1 and e_{4}^{2} = -1and they anticommute
e_{i} e_{j} = -e_{j} e_{i} for i not j.We denote
e_{ij} = e_{i}e_{j} for i not j, and for instance e_{1234} = e_{1}e_{2}e_{3}e_{4}.These rules and conventions already fix the computation rules of Cl_{3,1}.
Example. e_{1}e_{2}e_{1}e_{3} = -e_{1}^{2} e_{2}e_{3} = -e_{23}.
The Clifford algebra Cl_{3,1} of the Minkowski space-time R^{3,1} is isomorphic, as an associative algebra, to the real 4x4-matrix algebra Mat(4,R). This isomorphism allows us to view Cl_{3,1} through its faithful matrix image Mat(4,R).
For the convenience of viewers unfamiliar with Clifford algebras, I shall present the first counterexamples by means of a matrix algebra, namely Mat(4,R), and then translate the presentation into the corresponding Clifford algebra Cl_{3,1}.
satisfying the multiplication rules
E_{1} = 1 0 0 0
0 -1 0 0
0 0 -1 0
0 0 0 1, E_{2} = 0 1 0 0
1 0 0 0
0 0 0 1
0 0 1 0, E_{3} = 0 0 1 0
0 0 0 -1
1 0 0 0
0 -1 0 0, E_{4} = 0 -1 0 0
1 0 0 0
0 0 0 -1
0 0 1 0
E_{1}^{2} = E_{2}^{2} = E_{3}^{2} = I, E_{4}^{2} = -I and E_{i}E_{j} = -E_{j}E_{i} for i not j.Take an element a = (1+e_{1})(1+e_{234}) in Cl_{3,1}, represented by the matrix
The so-called Clifford-conjugation sending a in Cl_{3,1} to a^{-} corresponds in Mat(4,R) to the anti-automorphism sending A to
A = (I+E_{1}) (I+E_{2}E_{3}E_{4}) = 2 0 -2 0
0 0 0 0
0 0 0 0
0 -2 0 2.
Compute the products of A and A^{-} in different orders to find:
A^{-} = E_{4}A^{T}E_{4}^{-1} = 0 0 -2 0
0 2 0 0
0 0 2 0
0 -2 0 0.
In fact, AA^{-} is not even diagonal, that is, it is not a scalar multiple of I.
A^{-}A = 0 although AA^{-} = 0 0 -8 0
0 0 0 0
0 0 0 0
0 -8 0 0is not zero.
After this excursion into matrix algebras the viewer is hopefully prepared for Clifford algebras. Next, I will present some preliminary counterexamples by rewriting the above observation in terms of the Clifford algebra Cl_{3,1}.
a = (1+e_{1})(1+e_{234}) = 1+e_{1}+e_{234}+e_{1234},and apply Clifford-conjugation (the anti-automorphism of Cl_{3,1} extending the map x -> -x in R^{3,1})
a^{-} = (1+e_{234})(1-e_{1}) = 1-e_{1}+e_{234}+e_{1234}.Compute the products of a and a^{-} in different orders to find:
a^{-}a = 0 although aa^{-} = 4(e_{234}+e_{1234}) is not zero, not even a scalar in R.Harvey 1990 claims on p. 202, ll. 1, 4-5, in Lemma 10.45, that the following statements are equivalent: (c) aa^{-} e R, (d) a^{-}a e R. Compare the above result to the Lemma, claimed to have been proven by Harvey, and you have a counterexample to Harvey's lemma. In other words, my counterexample falsifies a result of Harvey 1990, Lemma 10.45, (c,d), p. 202, since a^{-}a = 0 is in R but aa^{-} = 4(e_{234}+e_{1234}) is nonzero and not in R. (Harvey introduces the Clifford-conjugation a^{-} on p. 183; he calls it a hat involution and denotes by a^{^}.)
Gilbert & Murray 1991 denote D(x) = x^{-}x and prove in Theorem 5.16 that for x such that D(x) is in R, it necessarily follows that D(x^{-}) = D(x) [p. 41, l. 19] and in particular that D(x) = 0 forces D(x^{-}) = 0 [p. 42, ll. 2-3]. Choose x = a to find D(a) = 0 in R, although D(a^{-}) = (a^{-})^{-}a^{-} = 4(e_{234}+e_{1234}) is not 0, and thereby not in R. Compare this result to Theorem 5.16, claimed to have been proven by Gilbert & Murray, and you have a counterexample to Gilbert & Murray's theorem. In other words, Gilbert & Murray's Theorem 5.16, stating that D(x^{-}) = D(x), has been falsified by my counterexample. (Gilbert & Murray's conjugation is the Clifford-conjugation, see p. 17.)
The element x = e_{1}+e_{23} in Cl_{3} serves as a counterexample to Knus 1991, p. 228, l. 13, since x^{-}x = -2e_{123} is not in Cl_{3}^{+}. (Knus introduces the Clifford-conjugation x^{-} on p. 195 and calls it the standard involution s(x). Knus could defend himself by arguing that x in m(x) = x^{-}x is homogeneous as in l. 3, p. 228.)
For x = e_{1}+e_{23} in Cl_{3}, x^{-}x = -2e_{123} is not in R, and we have a counterexample to Dabrowski 1988, p. 7, l. 12, who observed his error [see the errata sheet distributed along with his monograph]. In the Clifford algebra Cl_{3} of the Euclidean space R^{3} there are elements whose exponentials are vectors, like e_{3} = exp[(p/2)(e_{12}-e_{123})]. Therefore, for the multivalued inverse of the exponential,
log e_{3} = (p/2)(e_{12}-e_{123}).This shows that vectors can have logarithms in a Clifford algebra, and serves as a counterexample to Hestenes 1986, p. 75 [the error is corrected in Hestenes 1987].
All the above counterexamples are trivial, in the sense that an expert reader recognizes the mistakes at the first reading, except maybe the last one. The detection of the last mistake, concerning functions in Clifford algebras, requires knowledge of idempotents, nilpotents and minimal polynomials. A good place to start studying them is Sobczyk, 1997.
L_{p,q} = {s e Cl_{p,q}; for all x e R^{p,q}, sxs^{^-1} e R^{p,q}},Note the presence of the grade involution: s -> s^{^} (the automorphism of Cl_{p,q} extending the map x -> -x in R^{p,q}), and/or restriction to the even/odd parts Cl_{p,q}^{±}. The Lipschitz group L_{p,q} has a subgroup, normalized by the reversion: s -> s^{~} (the anti-automorphism of Cl_{p,q} extending the identity map x -> x in R^{p,q}),
L_{p,q} = {s e Cl_{p,q}^{+} \cup Cl_{p,q}^{-}; for all x e R^{p,q}, sxs^{-1} e R^{p,q}}.
Pin(p,q) = {s e L_{p,q}; ss^{~} = ±1},with an even subgroup
Spin(p,q) = Pin(p,q) \cap Cl_{p,q}^{+},which contains as a subgroup the two-fold cover
Spin_{+}(p,q) = {s e Spin(p,q); ss^{~} = 1}of the connected component SO_{+}(p,q) of SO(p,q) \subset O(p,q).
Although SO_{+}(p,q) is connected, its two-fold cover Spin_{+}(p,q) need not be connected. In particular,
Spin_{+}(1,1) = {x+ye_{12}; x,y e R, x^{2}-y^{2} = 1}has two components, two branches of a hyperbola [and so the group
Spin(1,1) = {x+ye_{12}; x,y e R, x^{2}-y^{2} = ±1}has four components]. This serves as a counterexample to Choquet-Bruhat et al. 1989, p. 37, ll. 2-3, p. 38, ll. 22-23 [see also p. 27, ll. 4-5]. Although the two-fold covers Spin(n) = Spin(n,0) ~ Spin(0,n), n > 2, and Spin_{+}(n-1,1) ~ Spin_{+}(1,n-1), n > 3, are simply connected, Spin_{+}(3,3) is not simply connected, and therefore not a universal cover of SO_{+}(3,3), since the maximal compact subgroup SO(3)xSO(3) of SO_{+}(3,3) has a four-fold universal cover Spin(3)xSpin(3). The two-fold cover Spin_{+}(3,3) of SO_{+}(3,3) is doubly connected, contrary to the claims of Lawson & Michelsohn 1989, p. 57, l. 22, and G\"ockeler & Sch\"ucker 1987, p. 190, l. 17.
Comment on Bourbaki 1959. The groups Pin(p,q) and Spin(p,q), obtained by normalizing the Lipschitz group L_{p,q}, are two-fold coverings of the orthogonal and special orthogonal groups, O(p,q) and SO(p,q), respectively. If one defines, instead of the Lipschitz group, a slightly different group
G_{p,q} = {s e Cl_{p,q}; for all x e R^{p,q}, sxs^{-1} e R^{p,q}},one obtains, only in even dimensions, a cover of O(p,q). Furthermore, for odd n=p+q, an element of G_{p,q} need not be even or odd, but might have an inhomogeneous central factor x+ye_{12...n} e R+/\^{n}R^{p,q}. Thus Bourbaki 1959, p. 151, Lemme 5, does not hold, as has been observed by Deheuvels 1981, p. 355, Moresi 1988, p. 621, and by Bourbaki himself [see Feuille d'Errata No. 10 distributed with Chapters 3,4 of Alg\`ebre Commutative 1961].
The confusion about proper covering of O(p,q) in Cl_{p,q} pops up frequently.
In the Lipschitz group every element s e L_{p,q} is of the form s = r g, where r e R\{0}, g e Pin(p,q). The group G_{p,q} does not have this property in odd dimensions. For instance, the central element z=x+ye_{123} e Cl_{3}, with non-zero x,y e R, satisfies z e G_{3}, but z \not= r g, g e Pin(3). This serves as a counterexample to Baum 1981, p. 57, l. -1. [Baum's C_{n,k} means Cl_{k,n-k}, see p. 51, and her Pin(n,k) means Pin(k,n-k), see p. 53. Note that the two-fold cover of O(3),
Pin(3) = Spin(3) \cup e_{123}Spin(3) ~ SU(2) \cup i SU(2),is a subgroup of G_{3}, but since the actions are defined differently, G_{3} does not cover O(3).]
For all s e G_{3}, ss^{~} > 0. Therefore, if we normalize G_{3} by the reversion, the central factor is not eliminated, but instead we get the group {s e G_{3}; ss^{~} = 1} ~ U(2), which does not cover O(3) but covers SO(3) with kernel {x+ye_{123}; x,y e R, x^{2}+y^{2} = 1} ~ U(1) \not~ {±1}. Compare this to Figueiredo 1994, p. 230, ll. -4.
Exponentials of bivectors. There are two possibilities to exponentiate a bivector B e /\^{2}R^{p,q}: the ordinary/Clifford exponential exp(B), and the exterior exponential exp^(B), where the product is the exterior product. If the exterior exponential exp^(B) is invertible with respect to the Clifford product, then it is in the Lipschitz group L_{p,q}. For the ordinary exponential we always have exp(B) e Spin_{+}(p,q).
All the elements of the compact spin groups Spin(n,0) ~ Spin(0,n) are exponentials of bivectors [when n > 1]. Among the other spin groups the same holds only for Spin_{+}(n-1,1) ~ Spin_{+}(1,n-1), n > 4, see M. Riesz 1958/1993 pp. 160, 172. In particular, the two-fold cover Spin_{+}(1,3) ~ SL(2,C) of the Lorentz group SO_{+}(1,3) contains elements which are not exponentials of bivectors: take (g_{0}+g_{1})g_{2} e /\^{2}R^{1,3}, [(g_{0}+g_{1})g_{2}]^{2} = 0, then -exp[(g_{0}+g_{1})g_{2}] = -1-(g_{0}+g_{1})g_{2} \not= exp(B) for any B e /\^{2}R^{1,3}.
Every element L of the Lorentz group SO_{+}(1,3) is an exponential of an antisymmetric matrix, L = exp(A), gA^{T}g^{-1} = -A; a similar property is not shared by SO_{+}(2,2), see M. Riesz 1958/1993, pp. 150-152, 170-171. There are elements in Spin_{+}(2,2) which cannot be written in the form ±exp(B), B e /\^{2}R^{2,2}; for instance ±e_{1234}exp(bB), B = e_{12}+2e_{14}+e_{34}, b > 0. This serves as a counterexample to Doran 1994, p. 41, l. 26, formula (3.16), and to Doran & Hestenes & Sommen 1993, p. 3650, ll. 16-18, formula (4.9).
Riesz also showed, by the same construction on pp. 170-171, that there are bivectors which cannot be written as sums of simple and completely orthogonal bivectors; for instance B = e_{12}+2e_{14}+e_{34} e /\^{2}R^{2,2}.
The above mistakes are not serious, in the sense that they could be rectified by stipulating the assertions, although such a correction is not obvious in the last examples. The above counterexamples should be easy to understand also for a non-expert, except maybe the last one by M. Riesz, which does require some knowledge of minimal polynomials of linear transformations. Good places to start studying minimal polynomials are Sobczyk, 1997, and M. Riesz 1958/1993, pp. 150-152, 170-171.
In mathematics, proving theorems, finding gaps and errors in the proofs, correcting the theorems, detecting errors in the corrected theorems, etc. is a normal activity. This is even more so in advanced mathematics because our cognitive charts are less accurate in new frontiers of knowledge. See Lakatos 1976.
In evaluating the validity of a mathematical theorem, one should either check every detail of its proof or point out a flaw in the chain of deductions or line of thoughts. After a counterexample has been presented, it is often easier to settle whether it fulfils all the assumptions than to check all the details of the proof. As in science, also in mathematics we are faced with the fact that a single counterexample can falsify a theorem or a whole theory. See Popper 1972.
The role of counterexamples in mathematics has been discussed by Lakatos 1976, Dubnov 1963 and Hauchecorne 1988. Lakatos focuses on the historical development of mathematics and Dubnov on various levels of abstraction. Both restrict themselves to a specific topic within mathematics (like this web-page). Hauchecorne gives counterexamples in almost all branches of mathematics. He also elaborates on virtues of counterexamples in teaching and in research: A theorem often necessitates several hypotheses -- to chart out its domain of applications it is important to become convinced about the relevancy of each hypothesis. This can be done by dropping one assumption at a time, and giving a counterexample to each new "theorem". Counterexamples cannot be ignored on the basis that "they do not treat the general case". Counterexamples are not "exceptions that confirm the rule". In mathematical research, the negation of a theorem, affirmation that it is false, is demonstrated by existence of a case, where all the hypotheses are verified without the conclusion being valid. The mathematical justification for the falsity of a theorem is completed by presenting a counterexample. After verification of the validity of a counterexample, further study in the same line, to rescue the "theorem", at whatever generality, is useless and erroneous activity. Falsification of theorems by verifying counterexamples opens new doors for cognitive growth and acts as an impetus of scientific progress .
There are several books listing counterexamples in various branches of mathematics: Capobianco & Molluzo 1978 (graph theory), Gelbaum & Olmsted 1964 (analysis), Forn\ae ss & Stens\o nes 1987 (several complex variables), Khaleelulla 1982 (vector spaces), Romano & Siegel 1986 (statistics), Steen & Seebach 1970 (topology), Stoyanov 1987 (probability) and Wise & Hall 1993 (real analysis). Similarly as Lakatos, Dubnov and Hauchecorne these authors do not point out errors of contemporary mathematicians. The present web-page differs from those studies in that respect: counterexamples are given to the works of living mathematicians, who can participate in a public debate about possible correctness of counterexamples presented in this web-page.
Some scientists refrain from participating in discussions about errors in published works, presumably because they anticipate a misinterpretation on the part of the author. Some scientists refrain from a public debate on errors, because of their mistaken belief that the peer refereeing system guarantees correctness of published works. Often scientists cannot come up with a suggestion on how to evaluate details of their works [other than the peer review system in abstract journals and academic appointments] -- in this web-page such a method has been suggested/revived: public scrutiny focusing on interior consistency of details [of publications available in scientific libraries -- thus guaranteeing everybody's access to a public debate].
H. Baum: Spin-Strukturen und Dirac-Operatoren \"uber pseudoriemannschen Mannigfaltigkeiten. Teubner, Leipzig, 1981.
R. Borasi: Reconceiving Mathematics Instruction: A Focus on Errors. Greenwood Publishing Group, Inc., 1996.
N. Bourbaki: Alg\`ebre, Chapitre 9, Formes sesquilin\'eaires et formes quadratiques. Hermann, Paris, 1959.
M. Capobianco, J. Molluzo: Examples and Counterexamples in Graph Theory. North Holland, Amsterdam, 1978.
C. Chevalley: Theory of Lie Groups. Princeton University Press, Princeton, 1946.
C. Chevalley: The Algebraic Theory of Spinors. Columbia University Press, New York, 1954. The Algebraic Theory of Spinors and Clifford Algebras. Springer, New York, 1997.
Y. Choquet-Bruhat, C. DeWitt-Morette: Analysis, Manifolds and Physics, Part II. North Holland, Amsterdam, 1989.
A. Crumeyrolle: Orthogonal and Symplectic Clifford Algebras, Spinor Structures. Kluwer Academic Publishers, Dordrecht, 1990.
L. Dabrowski: Group Actions on Spinors. Bibliopolis, Napoli, 1988.
R. Deheuvels: Formes quadratiques et groupes classiques. Presses Universitaires de France, Paris, 1981.
G.M. Dixon: Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics. Kluwer Academic Publishers, Dordrecht, 1994.
C. Doran: Geometric Algebra and its Applications to Mathematical Physics. Thesis, Univ. Cambridge, 1994.
C. Doran, D. Hestenes, F. Sommen, A. van Acker: Lie groups as spin groups. J. Math. Phys. 34 (8) (1993), 3642-3669.
Ya.S. Dubnov: Mistakes in Geometric Proofs. Heath, Boston, 1963.
V. Figueiredo: Clifford algebra approach to Cayley-Klein matrices; pp. 230-236 in P.S. Letelier, W.A. Rodrigues (eds.): Gravitation: The Space-Time Structure, SILARG VIII. Proc. 8th Latin American Symposium on Relativity and Gravitation (Brazil, 1993). World Scientific, Singapore, 1994.
J.E. Forn\ae ss, B. Stens\o nes: Lectures on Counterexamples in Several Complex Variables. Princeton University Press, Princeton, NJ, 1987.
B.R. Gelbaum, J.H.M. Olmsted: Counterexamples in Analysis. Holden-Day, San Francisco, 1964.
B.R. Gelbaum, J.H.M. Olmsted: Theorems and Counterexamples in Mathematics. Springer, New York, 1990.
J. Gilbert, M. Murray: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1991.
H.P. Ginsburg, S. Opper: Piaget's Theory of Intellectual Development. Prentice Hall, Englewood Cliffs, NJ, 1988.
M. G\"ockeler, Th. Sch\"ucker: Differential Geometry, Gauge Theories, and Gravity. Cambridge University Press, Cambridge, 1987.
F.R. Harvey: Spinors and Calibrations. Academic Press, San Diego, 1990.
B. Hauchecorne: Les contre-exemples en math\'ematiques. Ellipses, Paris, 1988.
D. Hestenes: New Foundations for Classical Mechanics. Reidel, Dordrecht, 1986, 1987, 1990.
S.M. Khaleelulla: Counterexamples in Topological Vector Spaces. Springer, Berlin, 1982.
M.-A. Knus: Quadratic and Hermitian Forms over Rings. Springer, Berlin, 1991.
I. Lakatos: Proofs and Refutations. The Logic of Mathematical Discovery. Cambridge University Press, Cambridge, 1976.
H.B. Lawson, M.-L. Michelsohn: Spin Geometry. Universidade Federal do Cear\'a, Brazil, 1983. Princeton University Press, Princeton, NJ, 1989.
R. Lipschitz: Principes d'un calcul alg\'ebrique qui contient comme esp\`eces particuli\`eres le calcul des quantit\'es imaginaires et des quaternions. C.R. Acad. Sci. Paris 91 (1880), 619-621, 660-664. Reprinted in Bull. Soc. Math. (2) 11 (1887), 115-120.
R. Lipschitz: Untersuchungen \"uber die Summen von Quadraten. Max Cohen und Sohn, Bonn, 1886, pp. 1-147. The first chapter of pp. 5-57 translated into French by J. Molk: Recherches sur la transformation, par des substitutions r\'eelles, d'une somme de deux ou troix carr\'es en elle-m\^eme. J. Math. Pures Appl. (4) 2 (1886), 373-439. French r\'esum\'e of all three chapters in Bull. Sci. Math. (2) 10 (1886), 163-183.
R. Lipschitz (purported author): Correspondence. Ann. of Math. 69 (1959), 247-251. Reprinted on pages 557-561 of A. Weil: Oeuvres Scientifiques, Collected Papers, Volume II. Springer, 1979.
P. Lounesto: Counterexamples in Clifford algebras with CLICAL, pp. 3-30 in R. Ablamowicz et al. (eds.): Clifford Algebras with Numeric and Symbolic Computations. Birkh\"auser, Boston, 1996.
P. Lounesto: Counterexamples in Clifford algebras. Advances in Applied Clifford Algebras 6 (1996), 69-104.
P. Lounesto: Clifford Algebras and Spinors. Cambridge University Press, Cambridge, 1997, 1998, (2nd ed.) 2001.
E.A. Maxwell: Fallacies in Mathematics. CUP, Cambridge, 1959.
R. Moresi: A remark on the Clifford group of a quadratic form; pp. 621-626 in Stochastic Processes, Physics and Geometry. Ascona/Locarno, 1988.
K. Popper: Objective Knowledge. Oxford University Press, Oxford, 1972.
I.R. Porteous: Topological Geometry. Van Nostrand Reinhold, London, 1969. Cambridge University Press, Cambridge, 1981.
I.R. Porteous: Clifford Algebras and the Classical Groups. Cambridge University Press, Cambridge, 1995.
J.M. Rassias: Counter-Examples in Differential Equations and Related Topics. Intercorporated, 1991.
M. Riesz: Clifford Numbers and Spinors. The Institute for Fluid Dynamics and Applied Mathematics, Lecture Series No. 38, University of Maryland, 1958. Reprinted as facsimile (eds.: E.F. Bolinder, P. Lounesto) by Kluwer Academic Publishers, 1993.
J.P. Romano, A.F. Siegel: Counterexamples in Probability and Statistics. Wadsworth & Brooks, Monterey, CA, 1986.
G. Sobczyk: Spectral integral domains in the classroom. Aportaciones Matematicas, Serie Comunicaciones 20 (1997), 169-188.
G. Sobczyk: The generalized spectral decomposition of a linear operator. The College Mathematics Journal 28:1 (1997), 27-38.
L.A. Steen, J.A. Seebach: Counterexamples in Topology. Holt, Rinehart and Winston, New York, 1970. Springer, New York, 1978.
J. Stoyanov: Counterexamples in Probability. John Wiley, Chichester, 1987, 1997.
G.L. Wise, E.B. Hall: Counterexamples in Probability and Real Analysis. Oxford University Press, New York, 1993.