Jonathan Cender's List: Companion to the paper What If There Were A Different Zero

• But that this form of introduction into the differential calculus
  can make no claim to being scientific, no one will deny. For myself this feeling of dissatisfaction was so overpowering that I made the fixed resolve to keep meditating on the question till I should find a purely arithmetic and perfectly rigorous foundation for the principles of infinitesimal analysis. - Jonathan Cender on 2008-10-27
• ESSAYS ON THE THEORY OF NUMBERS
  I. CONTINUITY AND IRRATIONAL NUMBERS
  II. THE NATURE AND MEANING OF NUMBERS
  BY
  RICHARD DEDEKIND
  AUTHORISED TRANSLATION BY WOOSTER WOODRUFF BEMAN
  PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF MICHIGAN - Jonathan Cender on 2008-10-27
• I regard the whole of arithmetic as a necessary, or at least natural,
  consequence of the simplest arithmetic act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding; the simplest act is the passing from an already-formed individual to the consecutive new one to be formed. The chain of these numbers forms in itself an exceedingly useful instrument for the human mind; it presents an inexhaustible
  wealth of remarkable laws obtained by the introduction of the four fundamental operations of arithmetic. - Jonathan Cender on 2008-10-27
• Addition is the combination of any arbitrary repetitions of the above-mentioned simplest act into a single act; from it in a similar way arises multiplication. While the performance of these two operations is
  always possible, that of the inverse operations, subtraction and division, proves to be limited. Whatever the immediate occasion may have been, whatever comparisons or analogies with experience, or intuition, may have led thereto; it is certainly true that just this limitation in performing the indirect operations has in each case been the real motive for a new creative act; thus negative and fractional numbers have been created by the human mind; and in the system of
  all rational numbers there has been gained an instrument of infinitely greater perfection. This system, which I shall denote by R, possesses first of all a completeness and self-containedness which I have designated in another place1 as characteristic of a body of numbers [Zahlk¨orper] and which consists in this that the four fundamental operations are always performable with any two individuals in R, i. e., the result is always an individual of R, the single case of division by the number zero being excepted. - Jonathan Cender on 2008-10-27
• Instead of this I demand that arithmetic shall be developed out of itself.
  That such comparisons with non-arithmetic notions have furnished the immediate occasion for the extension of the number-concept may, in a general way, be granted (though this was certainly not the case in the introduction of complex numbers); but this surely is no sufficient ground for introducing these foreign notions into arithmetic, the science of numbers. Just as negative and fractional rational numbers are formed by a new creation, and as the laws of
  operating with these numbers must and can be reduced to the laws of operating with positive integers, so we must endeavor completely to define irrational numbers by means of the rational numbers alone. The question only remains how to do this. - Jonathan Cender on 2008-10-27
• “If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions.” As already said I think I shall not err in assuming that every one will at once grant the truth of this statement; the majority of my readers will be very much disappointed in learning that by this commonplace remark the secret of continuity is to be revealed. To this I may say that I am glad if every one finds the above principle so obvious and so in harmony with his own ideas of a line; for I am utterly unable to adduce any proof of its correctness, nor has
  any one the power. The assumption of this property of the line is nothing else than an axiom by which we attribute to the line its continuity, by which we find continuity in the line. - Jonathan Cender on 2008-10-27
• In science nothing capable of proof ought to be accepted without proof. Though this demand seems so reasonable yet I cannot regard it as having been met even in the most recent methods of laying the foundations of the simplest science; viz., that part of logic which deals with the theory of numbers.4 In speaking of arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number-concept entirely independent of the notions or intuitions of space and time, that I consider it an immediate result from the laws of thought. - Jonathan Cender on 2008-10-27
• My answer to the problems propounded in the title of this paper is, then, briefly this: numbers are free creations of the human mind; they serve as a means of apprehending more easily and more sharply the difference of things. It is only through the purely logical process of building up the science of numbers and by thus acquiring the continuous number-domain that we are prepared accurately
  to investigate our notions of space and time by bringing them into relation with this number-domain created in our mind.5
  
  5See Section III. of my memoir, Continuity and Irrational Numbers (Braunschweig, 1872), translated at pages 4 et seq. of the present volume. - Jonathan Cender on 2008-10-27
• In what way the gradual extension of the number-concept, the creation of zero, negative, fractional, irrational and complex numbers are to be accomplished by reduction to the earlier notions and that without any introduction of foreign conceptions (such as that of
  measurable magnitudes, which according to my view can attain perfect clearness only through the science of numbers), this I have shown at least for irrational numbers in my former memoir on Continuity (1872); in a way wholly similar, as I have already shown in Section III. of that memoir,7 may the other extensions be treated, and I propose sometime to present this whole subject in systematic form. From just this point of view it appears as something self-evident and not new that every theorem of algebra and higher analysis, no matter how remote, can be expressed as a theorem about natural numbers,—a declaration I have heard repeatedly from the lips of Dirichlet. - Jonathan Cender on 2008-10-27
• But I see nothing meritorious–and this was just as far from Dirichlet’s thought—in actually performing this wearisome circumlocution and insisting on the use and recognition of no other than rational numbers. On the contrary, the greatest and most fruitful advances in
  mathematics and other sciences have invariably been made by the creation and introduction of new concepts, rendered necessary by the frequent recurrence of complex phenomena which could be controlled by the old notions only with difficulty. - Jonathan Cender on 2008-10-27
• On the other hand, we intend here for certain reasons wholly to
  exclude the empty system which contains no element at all, although for other investigations it may be appropriate to imagine such a system. - Jonathan Cender on 2008-10-27
• TRANSFORMATION OF A SYSTEM.
  21. Definition.14 By a transformation [Abbildung]  of a system S we
  understand a law according to which to every determinate element s of S there belongs a determinate thing which is called the transform of s and denoted by (s); we say also that (s) corresponds to the element s, that (s) results or is produced from s by the transformation , that s is transformed into (s) by the transformation . If now T is any part of S, then in the transformation  of S is likewise contained a determinate transformation of T, which for the sake of simplicity may be denoted by the same symbol  and consists in this that to every element t of the system T there corresponds the same transform
  (t), which t possesses as element of S; at the same time the system consisting of all transforms (t) shall be called the transform of T and be denoted by (T), by which also the significance of (S) is defined. As an example of a transformation of a system we may regard the mere assignment of determinate symbols or names to its elements. The simplest transformation of a system is that by which each of its elements is transformed into itself; it will be called the identical transformation of the system. For convenience, in the following
  theorems (22), (23), (24), which deal with an arbitrary transformation  of an arbitrary system S, we shall denote the transforms of elements s and parts T respectively by s0 and T0; in addition we agree that small and capital italics without accent shall always signify elements and parts of this system S. - Jonathan Cender on 2008-11-07
• THE FINITE AND INFINITE.
  64. Definition.20 A system S is said to be infinite when it is similar to a proper part of itself (32); in the contrary case S is said to be a finite system.
  65. Theorem. Every system consisting of a single element is finite.
  Proof. For such a system possesses no proper part (2), (6).
  66. Theorem. There exist infinite systems.
  Proof.21 My own realm of thoughts, i. e., the totality S of all things, which can be objects of my thought, is infinite. For if s signifies an element of S, then is the thought s0, that s can be object of my thought, itself an element of S. If we regard this as transform (s) of the element s then has the transformation of S, thus determined, the property that the transform S0 is part of S; and S0 is certainly proper part of S, because there are elements in S (e. g., my own ego) which are different from such thought s0 and therefore are not contained in
  S0. Finally it is clear that if a, b are different elements of S, their transforms a0, b0 are also different, that therefore the transformation  is a distinct (similar) transformation (26). Hence S is infinite, which was to be proved. - Jonathan Cender on 2008-11-07
• Footnotes to THE FINITE AND INFINITE. [previous comment]
  20 If one does not care to employ the notion of similar systems (32) he must say: S is said to be infinite, when there is a proper part of S (6) in which S can be distinctly (similarly) transformed (26), (36). In this form I submitted the definition of the infinite which forms the
  core of my whole investigation in September, 1882, to G. Cantor and several years earlier to Schwarz and Weber. All other attempts that have come to my knowledge to distinguish the infinite from the finite seem to me to have met with so little success that I think I may be
  permitted to forego any criticism of them.
  21 A similar consideration is found in § 13 of the Paradoxien des Unendlichen by Bolzano (Leipzig, 1851). - Jonathan Cender on 2008-11-07
• 73. Definition. If in the consideration of a simply infinite system N set
  in order by a transformation [Greek letter phi?] we entirely neglect the special character of the elements; simply retaining their distinguishability and taking into account only the relations to one another in which they are placed by the order-setting transformation, [Greek letter phi?] then are these elements called natural numbers or ordinal numbers or simply numbers, and the base-element 1 is called the base-number of the number-series N. With reference to this freeing the elements from every other content (abstraction) we are justified in calling numbers a free creation of the human mind. The relations or laws which are derived entirely from the conditions alpha, beta, gamma, delta in (71) and therefore are always the same in all ordered simply infinite systems, whatever names may happen to be given to the individual elements (compare 134), form the first object of the science of numbers or arithmetic. From the general notions and theorems of IV. about the transformation of a system in itself we obtain immediately the following fundamental laws where
  a, b, . . . m, n, . . . always denote elements of N, therefore numbers, A, B, C, . . .parts of N, a', b', . . . m', n', . . . A', B', C' . . . the corresponding transforms, which are produced by the order-setting transformation [Greek letter phi?] and are always elements or parts of N; the transform n' of a number n is also called the number following n. - Jonathan Cender on 2008-11-07
• See (126) for function - leads to basis of addition, etc - Jonathan Cender on 2008-11-07

• Dedekind

• "Of all the tools created up until now by the human spirit for making his life, that is, the labour of thinking, easier, none has been as full of consequences and inseparably connected with his innermost nature as the concept of number. Arithmetic, whose only subject this concept is, has already become a science of immense proportions, and there is no doubt that its further development will have no bounds; equally immense is the field of its application, because every thinking human being, even if he is not clearly aware of this, is a creature of numbers, an arithmetican."
• We emphasize three basic themes in the mathematics of Dedekind: his view of numbers and mathematical concepts as free creations of human thought, the formation of concepts as a central moment of mathematical research, and the formation of sets, 'system formation' in the words of Dedekind, as a method for forming new concepts.

• It is known that geometry assumes, as things given, both the notion of space and the first principles of constructions in space. She gives definitions of them which are merely nominal, while the true determinations appear in the form of axioms. The relation of these assumptions remains consequently in darkness; we neither perceive whether and how far their connection is necessary, nor a priori, whether it is possible.
• I have in the first place, therefore, set myself the task of constructing the notion of a multiply extended magnitude out of general notions of magnitude.
• It will follow from this that a multiply extended magnitude is capable of different measure-relations, and consequently that space is only a particular case of a triply extended magnitude. But hence flows as a necessary consequence that the propositions of geometry cannot be derived from general notions of magnitude, but that the properties which distinguish space from other conceivable triply extended magnitudes are only to be deduced from experience. Thus arises the problem, to discover the simplest matters of fact from which the measure-relations of space may be determined; a problem which from the nature of the case is not completely determinate, since there may be several systems of matters of fact which suffice to determine the measure-relations of space - the most important system for our present purpose being that which Euclid has laid down as a foundation. These matters of fact are - like all matters of fact - not necessary, but only of empirical certainty; they are hypotheses. We may therefore investigate their probability, which within the limits of observation is of course very great, and inquire about the justice of their extension beyond the limits of observation, on the side both of the infinitely great and of the infinitely small.
• In this way, to come to solid ground, we cannot, it is true, avoid abstract considerations in our formulæ, but at least the results of calculation may subsequently be presented in a geometric form. The foundations of these two parts of the question are established in the celebrated memoir of Gauss, Disqusitiones generales circa superficies curvas.
• • Gauss's work on curves used by Riemann.
• • infinitesimals as used by Riemann
• The theory of surfaces of constant curvature will serve for a geometric illustration. It is easy to see that surface whose curvature is positive may always be rolled on a sphere whose radius is unity divided by the square root of the curvature; but to review the entire manifoldness of these surfaces, let one of them have the form of a sphere and the rest the form of surfaces of revolution touching it at the equator. The surfaces with greater curvature than this sphere will then touch the sphere internally, and take a form like the outer portion (from the axis) of the surface of a ring; they may be rolled upon zones of spheres having new radii, but will go round more than once. The surfaces with less positive curvature are obtained from spheres of larger radii, by cutting out the lune bounded by two great half-circles and bringing the section-lines together. The surface with curvature zero will be a cylinder standing on the equator; the surfaces with negative curvature will touch the cylinder externally and be formed like the inner portion (towards the axis) of the surface of a ring. If we regard these surfaces as locus in quo for surface-regions moving in them, as Space is locus in quo for bodies, the surface-regions can be moved in all these surfaces without stretching. The surfaces with positive curvature can always be so formed that surface-regions may also be moved arbitrarily about upon them without bending, namely (they may be formed) into sphere-surfaces; but not those with negative-curvature. Besides this independence of surface-regions from position there is in surfaces of zero curvature also an independence of direction from position, which in the former surfaces does not exist.
• • Riemann sphere ?

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• This collection contains mathematical papers of Bernhard Riemann, transcribed and edited by David R. Wilkins. These texts are based on the second edition of the Gesammelte Mathematische Werke, and, in the case of some of the papers, the original printed text in the Journal für die reine und angewandte Mathematik, Annalen der Physik und Chemie and Annali di Matematica. Included here are all papers published in Riemann's lifetime, papers and correspondence published after Riemann's death by Dirichlet and others prior to the publication of the first edition of the Gesammelte Mathematische Werke (with the exception of the fragment Mechanik des Ohres which is non-mathematical in character), and one of the papers from his Nachlass, first published in the Gesammelte Mathematische Werke. There is also a translation by W. K. Clifford of Riemann's inaugural lecture on the foundations of geometry, and a biographical sketch by Richard Dedekind that was included in the Gesammelte Mathematische Werke.
• Ueber die Hypothesen, welche der Geometrie zu Grunde liegen.
  (Habilitationsschrift, 1854, Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 13 (1868))

• Translations

    
   
  20. On the Hypotheses which lie at the Bases of Geometry(Bernhard Riemann, translated by William Kingdon Clifford, Nature, 8 (1873), 14-17, 36-37)
      

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• The first thing to say about zero is that there are two uses of zero which are both extremely important but are somewhat different. One use is as an empty place indicator in our place-value number system. Hence in a number like 2106 the zero is used so that the positions of the 2 and 1 are correct. Clearly 216 means something quite different. The second use of zero is as a number itself in the form we use it as 0. There are also different aspects of zero within these two uses, namely the concept, the notation, and the name. (Our name "zero" derives ultimately from the Arabic sifr which also gives us the word "cipher".)

• Bhaskara wrote over 500 years after Brahmagupta. Despite the passage of time he is still struggling to explain division by zero. He writes:- 

   A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth. 

   So Bhaskara tried to solve the problem by writing n/0 = ∞. At first sight we might be tempted to believe that Bhaskara has it correct, but of course he does not. If this were true then 0 times ∞ must be equal to every number n, so all numbers are equal. The Indian mathematicians could not bring themselves to the point of admitting that one could not divide by zero. Bhaskara did correctly state other properties of zero, however, such as 0² = 0, and √0 = 0. 

• emptiness (sunyata), a concept which does not mean "non-existence" or   "nihility" (abhava), but rather the lack of autonomous existence (nihsvabhava)
• Denial of autonomy   according to Nagarjuna does not leave us with a sense of metaphysical or   existential privation, a loss of some hoped-for independence and freedom, but   instead offers us a sense of liberation through demonstrating the   interconnectedness of all things, including human beings and the manner in   which human   life unfolds in the natural and social worlds.
• Nagarjuna counters by reminding the Buddhist philosophers that, just as Gautama Sakyamuni had rejected both metaphysical and empirical substantialism through the teaching of "no-soul" (anatman) and causal interdependence (pratityasamputpada), so Scholastic Buddhism had to remain faithful to this non-substantialist stance through a rejection of the causal theories which necessitated notions of fixed nature (svabhava), theories which metaphysically reified the difference between samsara and nirvana. This later rejection could be based on Nagarjuna's newly coined notion of the "emptiness," "zeroness" or "voidness" (sunyata) of all things.
• Are we to infer that Nagarjuna's philosophy boils down to some strange paradoxical mysticism in which there is some ambiguous sense in which things should be considered causally interdependent but interdependent in some utterly unexplainable and inscrutable way? Not at all! Nagarjuna has not refuted all available theories of cause and effect, he has only rejected all substantialist theories of cause and effect.
• are built not upon essence, as both the Universal Existence and True Doctrine schools posit, but upon the fact that nothing (sunya) ever defines and characterizes them eternally and unconditionally. It is not that things are in themselves nothing, nor that things possess a positive absence (abhava) of essence. Change is possible because a radical indeterminancy (sunyata) permeates all forms.
• "If my statement (about emptiness) were a philosophical thesis, then it would indeed be flawed; but I assert no thesis, and so the flaw is not mine."
• "nothing will prevail for those in whom emptiness will not prevail, while everything will prevail for whom emptiness prevails." This is a reiteration of Nagarjuna's commitment that theory and praxis are not a partnership in which only through the former's justification is the latter redeemed. The goal of practice is after all transformation, not fixity, and so if one insists on marrying philosophy to practice, philosophical reflection cannot be beholden to the unchanging, eternal essences of customary epistemology and metaphysics
• the practice of Buddhism is a new sort of human vehicle, a vehicle meant not to carry people from one realm to another realm, but a vehicle which could make people anew in the only realm where they have always lived.
• Interdependent causality and the emptiness which change depends on mean that things can always go either way, and so which way they in fact go depends intimately on one's own deeds. And this leads one to grasp that the proper site of practice for the Buddhist cannot be just the monastery, removed as it tries to be from the machinations of state, economy, social class and the other tumultuous and sundry affairs of suffering beings. As there is no difference between samsara and nirvana owing to the emptiness and constantly changing nature of both, so the change which a Buddhist effects upon herself and those around her is a change in the world, and this constant and purposeful change is the rightful mission of Buddhism.
• Mulamadhyamakakarika, (Fundamental Verses on the Middle Way) translated as The Philosophy of the Middle Way by David J. Kalapuhana, SUNY Press, Albany, 1986

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• Indian mathematicians worked on introducing zero into their number system over a period of 500 years beginning with Brahmagupta in the 7^th Century. The problem they struggled with was how to make zero respect the usual operations of arithmetic. Bhaskara II wrote in Bijaganita:- 

   A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth. 

   It was an attempt to bring infinity, as well as zero, into the number system. Of course it does not work since if it were introduced as Bhaskara II suggests then 0 times infinity must be equal to every number n, so all numbers are equal.  
• The symbol  ∞   which we use for infinity today, was first used by John Wallis who used it in De sectionibus conicis in 1655 and again in Arithmetica infinitorum in 1656. He chose it to represent the fact that one could traverse the curve infinitely often.

• Little progress was being made on the question of the actual infinite. The same arguments kept on appearing without any definite progress towards a better understanding. Gauss, in a letter to Schumacher in 1831, argued against the actual infinite:- 

   I protest against the use of infinite magnitude as something completed, which in mathematics is never permissible. Infinity is merely a facon de parler, the real meaning being a limit which certain ratios approach indefinitely near, while others are permitted to increase without restriction.

• Perhaps one of the most significant events in the development of the concept of infinity was Bernard Bolzano's Paradoxes of the infinite which was published in 1840. He argues that the infinite does exist and his argument involves the idea of a set which he defined for the first time:- 

   I call a set a collection where the order of its parts is irrelevant and where nothing essential is changed if only the order is changed. 

   Why does defining a set make the actual infinite a reality? The answer is simple. Once one thinks of the integers as a set then there is a single entity which must be actually infinite. Aristotle would look at the integers from the point of view that one can find arbitrarily large finite subsets. But once one has the set concept then these are seen as subsets of the set of integers which must itself be actually infinite.

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