Like Weierstrass and Dedekind, Cantor aimed to formulate an adequate definition of the real numbers which avoided the presupposition of their prior existence, and he follows them in basing his definition on the rational numbers. The domain B of such symbols may be considered an enlargement of the domain A of rational numbers. After imposing an arithmetical structure on the domain B , Cantor is emboldened to refer to its elements as real numbers.
Cantor then shows that each point on the line corresponds to a definite element of B. Conversely, each element of B should determine a definite point on the line. Realizing that the intuitive nature of the linear continuum precludes a rigorous proof of this property, Cantor simply assumes it as an axiom, just as Dedekind had done in regard to his principle of continuity. For Cantor, who began as a number-theorist, and throughout his career cleaved to the discrete, it was numbers, rather than geometric points, that possessed objective significance.
Indeed the isomorphism between the discrete numerical domain B and the linear continuum was regarded by Cantor essentially as a device for facilitating the manipulation of numbers. Cantor's arithmetization of the continuum had the following important consequence. It had long been recognized that the sets of points of any pair of line segments, even if one of them is infinite in length, can be placed in one-one correspondence. But Cantor's identification of the set of points on a linear continuum with a domain of numbers enabled the sizes of point sets to be compared in a definite way, using the well-grounded idea of one-one correspondence between sets of numbers.
Remarkable in their richness of ideas, these papers provide the first accounts of Cantor's revolutionary theory of infinite sets and its application to the classification of subsets of the linear continuum. In the fifth of these papers, the Grundlagen of , [ 33 ] are to be found some of Cantor's most searching observations on the nature of the continuum.
Cantor begins his examination of the continuum with a tart summary of the controversies that have traditionally surrounded the notion, remarking that the continuum has until recently been regarded as an essentially unanalyzable concept. This opens the way, he believes, to the formulation of an exact concept of the continuum. Repudiating any use of spatial or temporal intuition in an exact determination of the continuum, Cantor undertakes its precise arithmetical definition. Making reference to the definition of real number he has already provided i.
The distance between two such points is given by. After remarking that he has previously shown that all spaces G n have the same power as the set of real numbers in the interval 0,1 , and reiterating his conviction that any infinite point sets has either the power of the set of natural numbers or that of 0,1 , [ 34 ] Cantor turns to the definition of the general concept of a continuum within G n. For this he employs the concept of derivative or derived set of a point set introduced in a paper of on trigonometric series.
Cantor had defined the derived set of a point set P to be the set of limit points of P, where a limit point of P is a point of P with infinitely many points of P arbitrarily close to it. A point set is called perfect if it coincides with its derived set [ 35 ].
Accordingly an additional condition is needed to define a continuum. Cantor supplies this by introducing the concept of a connected set. Cantor now defines a continuum to be a perfect connected point set. Cantor has advanced beyond his predecessors in formulating what is in essence a topological definition of continuum, one that, while still dependent on metric notions, does not involve an order relation [ 37 ].
It is interesting to compare Cantor's definition with the definition of continuum in modern general topology.
In a well-known textbook see Hocking and Young  on the subject we find a continuum defined as a compact connected subset of a topological space. Now within any bounded region of Euclidean space it can be shown that Cantor's continua coincide with continua in the sense of the modern definition. Throughout Cantor's mathematical career he maintained an unwavering, even dogmatic opposition to infinitesimals, attacking the efforts of mathematicians such as du Bois-Reymond and Veronese [ 38 ] to formulate rigorous theories of actual infinitesimals.
Despite the great success of Weierstrass, Dedekind and Cantor in constructing the continuum from arithmetical materials, a number of thinkers of the late 19 th and early 20 th centuries remained opposed, in varying degrees, to the idea of explicating the continuum concept entirely in discrete terms. In his later years the Austrian philosopher Franz Brentano — became preoccupied with the nature of the continuous see Brentano . In its fundamentals Brentano's account of the continuous is akin to Aristotle's. Brentano regards continuity as something given in perception, primordial in nature, rather than a mathematical construction.
He held that the idea of the continuous is abstracted from sensible intuition. Brentano suggests that the continuous is brought to appearance by sensible intuition in three phases. First, sensation presents us with objects having parts that coincide. From such objects the concept of boundary is abstracted in turn, and then one grasps that these objects actually contain coincident boundaries.
Finally one sees that this is all that is required in order to have grasped the concept of a continuum. For Brentano the essential feature of a continuum is its inherent capacity to engender boundaries, and the fact that such boundaries can be grasped as coincident.
Plerosis is the measure of the number of directions in which the given boundary actually bounds. Thus, for example, within a temporal continuum the endpoint of a past episode or the starting point of a future one bounds in a single direction, while the point marking the end of one episode and the beginning of another may be said to bound doubly.
In the case of a spatial continuum there are numerous additional possibilities: here a boundary may bound in all the directions of which it is capable of bounding, or it may bound in only some of these directions. In the former case, the boundary is said to exist in full plerosis ; in the latter, in partial plerosis.
Brentano took a somewhat dim view of the efforts of mathematicians to construct the continuum from numbers. Brentano's analysis of the continuum centred on its phenomenological and qualitative aspects, which are by their very nature incapable of reduction to the discrete. Brentano's rejection of the mathematicians' attempts to construct it in discrete terms is thus hardly surprising. The American philosopher-mathematician Charles Sanders Peirce's — view of the continuum [ 40 ] was, in a sense, intermediate between that of Brentano and the arithmetizers. Like Brentano, he held that the cohesiveness of a continuum rules out the possibility of it being a mere collection of discrete individuals, or points, in the usual sense.
And even before Brouwer, Peirce seems to have been aware that a faithful account of the continuum will involve questioning the law of excluded middle. Peirce also held that any continuum harbours an unboundedly large collection of points—in his colourful terminology, a supermultitudinous collection—what we would today call a proper class. While accepting the arithmetic definition of the continuum, he questions the fact that as with Dedekind and Cantor's formulations the irrational numbers so produced are mere symbols, detached from their origins in intuition.
The Dutch mathematician L. Brouwer — is best known as the founder of the philosophy of neo intuitionism see Brouwer ; van Dalen . Brouwer's highly idealist views on mathematics bore some resemblance to Kant's. For Brouwer, mathematical concepts are admissible only if they are adequately grounded in intuition, mathematical theories are significant only if they concern entities which are constructed out of something given immediately in intuition, and mathematical demonstration is a form of construction in intuition.
Initially Brouwer held without qualification that the continuum is not constructible from discrete points, but was later to modify this doctrine. The resulting choice sequences cannot be conceived as finished, completed objects: at any moment only an initial segment is known.
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The mathematical continuum as conceived by Brouwer displays a number of features that seem bizarre to the classical eye. The failure of these seemingly unquestionable principles in turn vitiates the proofs of a number of basic results of classical analysis, for example the Bolzano-Weierstrass theorem, as well as the theorems of monotone convergence, intermediate value, least upper bound, and maximum value for continuous functions [ 42 ].
While the Brouwerian continuum may possess a number of negative features from the standpoint of the classical mathematician, it has the merit of corresponding more closely to the continuum of intuition than does its classical counterpart. Far from being bizarre, the failure of the law of excluded middle for points in the intuitionistic continuum may be seen as fitting in well with the character of the intuitive continuum. In Brouwer showed that every function defined on a closed interval of his continuum is uniformly continuous.
As a consequence the intuitionistic continuum is indecomposable, that is, cannot be split into two disjoint parts in any way whatsoever. In contrast with a discrete entity, the indecomposable Brouwerian continuum cannot be composed of its parts. Brouwer's vision of the continuum has in recent years become the subject of intensive mathematical investigation.
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Hermann Weyl — , one of most versatile mathematicians of the 20 th century, was preoccupied with the nature of the continuum see Bell . In his Das Kontinuum of he attempts to provide the continuum with an exact mathematical formulation free of the set-theoretic assumptions he had come to regard as objectionable.
As he saw it, there is an unbridgeable gap between intuitively given continua e. Rather, he believed that the mathematical continuum must be treated and, in the end, justified in the same way as a physical theory. However much he may have wished it, in Das Kontinuum Weyl did not aim to provide a mathematical formulation of the continuum as it is presented to intuition, which, as the quotations above show, he regarded as an impossibility at that time at least.
Rather, his goal was first to achieve consistency by putting the arithmetical notion of real number on a firm logical basis, and then to show that the resulting theory is reasonable by employing it as the foundation for a plausible account of continuous process in the objective physical world. Later Weyl came to repudiate atomistic theories of the continuum, including that of his own Das Kontinuum. In particular, he found compelling the fact that the Brouwerian continuum is not the union of two disjoint nonempty parts—that it is indecomposable.
Once the continuum had been provided with a set-theoretic foundation, the use of the infinitesimal in mathematical analysis was largely abandoned. And so the situation remained for a number of years. The first signs of a revival of the infinitesimal approach to analysis surfaced in with a paper by A. Laugwitz and C. Schmieden [ 43 ]. After Robinson's initial insight, a number of ways of presenting nonstandard analysis were developed. Here is a sketch of one of them.
The saturation property expresses the intuitive idea that the nonstandard universe is very rich in comparison to the standard one. Indeed, while there may exist, for each finite subcollection F of a given collection of properties P , an element of U satisfying the members of F in , there may not necessarily be an element of U satisfying all the members of P. Using the transfer principle, any function f between standard sets automatically extends to a function—also written f —between their inflates. From the transfer principle it follows that and have precisely the same first-order properties.
A standard hyperreal is then just a real, to which we shall refer for emphasis as a standard real. In that case the set I of infinitesimals contains not just 0 but a substantial number in fact, infinitely many other elements. The monad of a hyperreal a thus consists of all the hyperreals that are infinitesimally close to a : it may be thought of as a small cloud centred at a. Here are some examples of such translations: [ 46 ]. Now suppose that f is a real-valued function defined on some open interval a , b.
We have remarked above that f automatically extends to a function—also written f— on. The original motivation for the development of constructive mathematics was to put the idea of mathematical existence on a constructive or computable basis. While there are a number of varieties of constructive mathematics see Bridges and Richman  , here we shall focus on Bishop's constructive analysis see Bishop and Bridges ; Bridges , ; and Bridges and Richman  and Brouwer's intuitionistic analysis see Dummett . In constructive mathematics a problem is counted as solved only if an explicit solution can, in principle at least, be produced.
This fact led to the questioning of certain principles of classical logic, in particular, the law of excluded middle, and the creation of a new logic, intuitionistic logic see entry on intuitionistic logic. It also led to the introduction of a sharpened definition of real numbers—the constructive real numbers.
Continuity and Infinitesimals (Stanford Encyclopedia of Philosophy)
The set R of all constructive real numbers is the constructive real line. To say that two real numbers are equal is to say that they are equivalent in this sense. The real number line can be furnished with an axiomatic description. We begin by assuming the existence of a set R with. The elements of R are called real numbers. These relations and operations are subject to the following three groups of axioms, which, taken together, form the system CA of axioms for constructive analysis , or the constructive real numbers see Bridges .
In the second of these the notions bounded above, bounded below, and bounded are defined as in classical mathematics, and the least upper bound , if it exists, of a nonempty [ 47 ] set S of real numbers is the unique real number b such that. The least upper bound principle. Then S has a least upper bound. The constructive real line R as introduced above is a model of CA. Are there any other models, that is, models not isomorphic to R. If classical logic is assumed, CA is a categorical theory and so the answer is no.
But this is not the case within intuitionistic logic, for there it is possible for the Dedekind and Cantor reals to fail to be isomorphic, despite the fact that they are both models of CA. In constructive analysis, a real number is an infinite convergent sequence of rational numbers generated by an effective rule, so that the constructive real line is essentially just a restriction of its classical counterpart. Brouwerian intuitionism takes a more liberal view of the matter, resulting in a considerable enrichment of the arithmetical continuum over the version offered by strict constructivism.
As conceived by intutionism, the arithmetical continuum admits as real numbers not only infinite sequences determined in advance by an effective rule for computing their terms, but also ones in whose generation free selection plays a part. The latter are called free choice sequences. Without loss of generality we may and shall assume that the entries in choice sequences are natural numbers. While constructive analysis does not formally contradict classical analysis, and may in fact be regarded as a subtheory of the latter, a number of intuitionistically plausible principles have been proposed for the theory of choice sequences which render intuitionistic analysis divergent from its classical counterpart.
Another such principle is Bar Induction , a certain form of induction for well-founded sets of finite sequences [ 49 ]. Brouwer used Bar Induction and the Continuity Principle in proving his Continuity Theorem that every real-valued function defined on a closed interval is uniformly continuous, from which, as has already been observed, it follows that the intuitionistic continuum is indecomposable. Brouwer gave the intuitionistic conception of mathematics an explicitly subjective twist by introducing the creative subject.
The creative subject was conceived as a kind of idealized mathematician for whom time is divided into discrete sequential stages, during each of which he may test various propositions, attempt to construct proofs, and so on. In particular, it can always be determined whether or not at stage n the creative subject has a proof of a particular mathematical proposition p. While the theory of the creative subject remains controversial, its purely mathematical consequences can be obtained by a simple postulate which is entirely free of subjective and temporal elements.
Now if the construction of these sequences is the only use made of the creative subject, then references to the latter may be avoided by postulating the principle known as Kripke's Scheme. Taken together, these principles have been shown to have remarkable consequences for the indecomposability of subsets of the continuum. Not only is the intuitionistic continuum indecomposable that is, cannot be partitioned into two nonempty disjoint parts , but, assuming the Continuity Principle and Kripke's Scheme, it remains indecomposable even if one pricks it with a pin.
The intuitionistic continuum has, as it were, a syrupy nature, so that one cannot simply take away one point.
If in addition Bar Induction is assumed, then, still more surprisingly, indecomposability is maintained even when all the rational points are removed from the continuum. A major development in the refounding of the concept of infinitesimal took place in the nineteen seventies with the emergence of synthetic differential geometry , also known as smooth infinitesimal analysis SIA [ 50 ].
Since in SIA all functions are continuous, it embodies in a striking way Leibniz's principle of continuity Natura non facit saltus. In what follows, we use bold R to distinguish the real line in SIA from its counterparts in classical and constructive analysis. We shall call a quantity having the property that its square is zero a nilsquare infinitesimal or simply a microquantity. Of course 2 holds trivially in standard mathematical analysis because there 0 is the sole microquantity in this sense.
It is equation 4 that is taken as axiomatic in smooth infinitesimal analysis. It follows that R itself may be regarded as the space of ratios of microquantities. For this reason Lawvere has suggested that R be called the space of Euler reals. From the principle of microaffineness we deduce the important principle of microcancellation , viz. From the principle of microaffineness it also follows that all functions on R are continuous , that is, send neighbouring points to neighbouring points.
This is Fermat's rule. An important postulate concerning stationary points that we adopt in smooth infinitesimal analysis is the. It follows from this that two functions with identical derivatives differ by at most a constant. In ordinary analysis the continuum R is connected in the sense that it cannot be split into two non empty subsets neither of which contains a limit point of the other.
In smooth infinitesimal analysis it has the vastly stronger property of indecomposability : it cannot be split in any way whatsoever into two disjoint nonempty subsets. We claim that f is constant. For we have. Possibilities ii and iii may be ruled out because f is continuous. So f is locally, and hence globally, constant, that is, constantly 1 or 0. We observe that the postulates of smooth infinitesimal analysis are incompatible with the law of excluded middle of classical logic.
This incompatibility can be demonstrated in two ways, one informal and the other rigorous. First the informal argument. If the law of excluded middle held, each real number would then be either equal or unequal to 0, so that the function f would be defined on the whole of R.
But, considered as a function with domain R , f is clearly discontinuous. Since, as we know, in smooth infinitesimal analysis every function on R is continuous, f cannot have domain R there [ 53 ]. So the law of excluded middle fails in smooth infinitesimal analysis. To put it succinctly, universal continuity implies the failure of the law of excluded middle.
Here now is the rigorous argument. We show that the failure of the law of excluded middle can be derived from the principle of infinitesimal cancellation. This means that. Now suppose that the law of excluded middle were to hold. This may be written. So again the law of excluded middle must fail. It is, instead, intuitionistic logic, that is, the logic derived from the constructive interpretation of mathematical assertions. What are the algebraic and order structures on R in SIA? As far as the former is concerned, there is little difference from the classical situation: in SIA R is equipped with the usual addition and multiplication operations under which it is a field.
From a strictly algebraic standpoint, R in SIA differs from its classical counterpart only in being required to satisfy the principle of infinitesimal cancellation. The situation is different, however, as regards the order structure of R in SIA. Using these ideas we can identify three infinitesimal neighbourhoods of 0 on R in SIA, each of which is included in its successor.
These three may be thought of as the infinitesimal neighbourhoods of 0 defined algebraically, logically, and order-theoretically , respectively. In certain models of SIA the system of natural numbers possesses some subtle and intriguing features which make it possible to introduce another type of infinitesimal—the so-called invertible infinitesimals—resembling those of nonstandard analysis, whose presence engenders yet another infinitesimal neighbourhood of 0 properly containing all those introduced above.
In SIA the set N of natural numbers can be defined to be the smallest subset of R which contains 0 and is closed under the operation of adding 1. In some models of SIA, R satisfies the Archimedean principle that every real number is majorized by a natural number. Multiplicative inverses of nonstandard integers are infinitesimals, but, being themselves invertible, they are of a different type from the ones we have considered so far.
Finally, a brief word on the models of SIA.
These are the so-called smooth toposes, categories see entry on category theory of a certain kind in which all the usual mathematical operations can be performed but whose internal logic is intuitionistic and in which every map between spaces is smooth, that is, differentiable without limit. The construction of smooth toposes see Moerdijk and Reyes  guarantees the consistency of SIA with intuitionistic logic.
This is so despite the evident fact that SIA is not consistent with classical logic. For a comprehensive account of the evolution of the concepts of continuity and the infinitesimal, see Bell , on which the present article is based. Introduction: The Continuous, the Discrete, and the Infinitesimal 2.
The Continuum and the Infinitesimal in the Ancient Period 3. The Continuum and the Infinitesimal in the 17th and 18th Centuries 5.
Category: Smooth Infinitesimal Analysis
The Continuum and the Infinitesimal in the 19th Century 6. Critical Reactions to Arithmetization 7. Nonstandard Analysis 8.
Introduction: The Continuous, the Discrete, and the Infinitesimal We are all familiar with the idea of continuity. The Continuum and the Infinitesimal in the Ancient Period The opposition between Continuity and Discreteness played a significant role in ancient Greek philosophy. The Continuum and the Infinitesimal in the Medieval, Renaissance, and Early Modern Periods The scholastic philosophers of Medieval Europe, in thrall to the massive authority of Aristotle, mostly subscribed in one form or another to the thesis, argued with great effectiveness by the Master in Book VI of the Physics, that continua cannot be composed of indivisibles.
The Continuum and the Infinitesimal in the 17th and 18th Centuries Isaac Barrow [ 17 ] —77 was one of the first mathematicians to grasp the reciprocal relation between the problem of quadrature and that of finding tangents to curves—in modern parlance, between integration and differentiation. This is based on two definitions: Variable quantities are those that continually increase or decrease; and constant or standing quantities are those that continue the same while others vary.
The infinitely small part whereby a variable quantity is continually increased or decreased is called the differential of that quantity. And two postulates: Grant that two quantities, whose difference is an infinitely small quantity, may be taken or used indifferently for each other: or what is the same thing that a quantity, which is increased or decreased only by an infinitely small quantity, may be considered as remaining the same.
An Invitation to Smooth Infinitesimal Analysis
Grant that a curve line may be considered as the assemblage of an infinite number of infinitely small right lines: or what is the same thing as a polygon with an infinite number of sides, each of an infinitely small length, which determine the curvature of the line by the angles they make with each other.
The major differences between Nieuwentijdt's and Leibniz's calculi of infinitesimals are summed up in the following table: Leibniz Nieuwentijdt Infinitesimals are variables Infinitesimals are constants Higher-order infinitesimals exist Higher-order infinitesimals do not exist Products of infinitesimals are not absolute zeros Products of infinitesimals are absolute zeros Infinitesimals can be neglected when infinitely small with respect to other quantities First-order infinitesimals can never be neglected In responding to Nieuwentijdt's assertion that squares and higher powers of infinitesimals vanish, Leibniz objected that it is rather strange to posit that a segment dx is different from zero and at the same time that the area of a square with side dx is equal to zero Mancosu , As for fluxions and evanescent increments themselves, Berkeley has this to say: And what are these fluxions?
The velocities of evanescent increments? And what are these same evanescent increments? They are neither finite quantities nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities? Nor did the Leibnizian method of differentials escape Berkeley's strictures.
The Continuum and the Infinitesimal in the 19th Century The rapid development of mathematical analysis in the 18 th century had not concealed the fact that its underlying concepts not only lacked rigorous definition, but were even e. Critical Reactions to Arithmetization Despite the great success of Weierstrass, Dedekind and Cantor in constructing the continuum from arithmetical materials, a number of thinkers of the late 19 th and early 20 th centuries remained opposed, in varying degrees, to the idea of explicating the continuum concept entirely in discrete terms.
Nonstandard Analysis Once the continuum had been provided with a set-theoretic foundation, the use of the infinitesimal in mathematical analysis was largely abandoned. Cauchy's criterion for convergence. This is equivalent to saying that f maps the monad of x 0 into the monad of f x 0. Many other branches of mathematics admit neat and fruitful nonstandard formulations.
The Constructive Real Line and the Intuitionistic Continuum The original motivation for the development of constructive mathematics was to put the idea of mathematical existence on a constructive or computable basis. Archimedean axiom. Smooth Infinitesimal Analysis A major development in the refounding of the concept of infinitesimal took place in the nineteen seventies with the emergence of synthetic differential geometry , also known as smooth infinitesimal analysis SIA [ 50 ].
An important postulate concerning stationary points that we adopt in smooth infinitesimal analysis is the Constancy Principle. Bibliography Aristotle Physics , 2 volumes, trans. Cornford and Wickstead. Metaphysics, Oeconomica, Magna Moralia , 2 volumes, trans. Cooke and Tredinnick. On the Heavens , trans. Forster and Furley.
Arthur, R. Kurstad, M. Maeke and D. Snyder eds. Barnes, J. The Presocratic Philosophers , London: Routledge. Baron, M. Beeson, M. Bell, E. Men of Mathematics , 2 volumes, London: Penguin Books. Still, it's fun to be free of this law for awhile , knowing we can always find it again if we want to… Second, the explanations are so clear, so considerate ; the author must have taught the subject many times, since he anticipates virtually every potential question, concern, and misconception in a student's or reader's mind.
Third, the poet in me is partial to short books, which nowadays seem to be rather rare. Since this review is entirely complimentary, probably the best thing for me to do here is to offer a selection of enticing passages: Page 7: "Nonzero infinitesimals can, and will, exist only in a 'potential' sense. Basic features of smooth worlds; 2. Basic differential calculus; 3. First applications of the differential calculus; 4.
Applications to physics; 5. Multivariable calculus and applications; 6. The definite integral: Higher order infinitesimals; 7. Synthetic geometry; 8. Smooth infinitesimal analysis as an axiomatic system; Appendix; Models for smooth infinitesimal analysis. Nonstandard Analysis. This is what the other reviewer meant when he said the book was wrong - that it doesn't repeat the accepted mantras. Well, good - academic mathematics has in this regard been under some kind of 'evil spell' for a hundred years.
Time to snap out of it - that's why this book is so important, it has begun the process of waking everyone up. One final word - the approach advocated here is called smooth infinitesimal analysis SIA; which is different to non-standard analysis NSA. The only practical difference is that NSA neglects infinitesimals at the end of derivations whereas SIA neglects them during the process the higher power terms that is.
This stems from different views on LEM but both approaches have an 'algebraic' character, in contrast to the more obscurantist limit theory. Sep 11, Richard Houchin rated it liked it Shelves: hard-science. This book has the word 'primer' in the title so I thought it would be a good starting place for learning calculus. Turns out this is a primer for some advanced subset of calculus, and not at all for beginners.
Still, I learned a bit. A square-zero is a number so small it isn't zero, but its square and all higher powers are identical to zero. A continuum has no points, and is infinitely divisible. A point is indivisible, and cannot be part of a continuum. There is no such thing as a discrete particle, everything is wave, some waves just have particle like properties.
If points can't be part of continua, then points can't exist. At least, that's what it seems like to me. The author of this book noted that at one point in history the concept of zero was practically unacceptable. More recently the concept of infinity or infinitely large numbers was absurd, but we got over that.
Now, the concept of infinitely small numbers is going through the same process, and while infinitesimals are strange and not yet in the mainstream, they will get there just as zero and infinity did. Nov 27, Robert rated it did not like it. I prefer my math books to be correct. I was rather pissed that I wasted my time on this one. View 1 comment. About J.
John Bell b. In , he was elected a Fellow of the Royal Society of Canada.