Paradoxes of the Infinite (Routledge Revivals)

Bernard Bolzano

• Publisher: Routledge
• Year: 2014
• ISBN:
  9781317748571
• Edition: 1
• Topic: Philosophie
• Subtopic: Logik in der Philosophie

Paradoxes of the Infinite presents one of the most insightful, yet strangely unacknowledged, mathematical treatises of the 19th century: Dr Bernard Bolzano's Paradoxien. This volume contains an adept translation of the work itself by Donald A. Steele S.J., and in addition an historical introduction, which includes a brief biography as well as an evaluation of Bolzano the mathematician, logician and physicist.

PARADOXES OF THE INFINITE

by Dr. Bernard Bolzano

EDITED POSTHUMOUSLY
FROM THE AUTHOR’S MANUSCRIPTS
BY DR. FR. PŘIHONSKÝ

Je suis tellement pour l’infini actuel, qu’au lieu d’admettre que la nature l’abhorre, comme l’on dit vulgairement, je tiens qu’elle l’affecte partout, pour mieux marquer les prefections de son auteur. (Leibniz, Opera omnia studio Ludov. Dutens, Tom. II part I. p. 243.

LEIPZING

C. H. RECLAM SEN.

1851

EDITOR’S PREFACE^*

THE author of this remarkable treatise on the Paradoxes of the Infinite began writing it as early as 1847, during his residence in the company of the Editor at the charming country house in Liboch, near Melnik, but was interrupted by tasks of a different nature and did not complete it until the summer months of the following year, the last of his life. The work not only gave evidence that his intellectual powers (despite his advancing age of sixty-seven years and the visible decline of his bodily strength) had lost nothing of their vigour and alertness, but also furnished the learned world with fresh proofs of the uncommon insight he enjoyed into the most abstract depths of mathematics, natural science and metaphysics. Indeed, had Bolzano written and bequeathed to us nothing else than this one treatise, it is our firm belief that on its account alone he would have to be numbered among the most distinguished minds of our century. He knows how to solve with admirable ease the most interesting and complicated of the problems which are raised by the idea of the infinite and have at all times engrossed the attention of workers in these aprioristic sciences; and he can. disentangle them before the very eyes of a reader with such clearness that anyone not a complete stranger to the subject, even though he may have hitherto grasped but a few of the matters in question, can still follow the author’s exposition, and find at least the majority of his propositions easy to understand. The experts on their part (if only they devote some attention to the treatise, and may we not expect this much from every scholar whatever?) are also bound to notice before long how important are the ideas which Bolzano suggests in this work and develops more circumstantially in others, especially in his Logic and in his Athanasia, and how he aims at nothing less than a complete transformation of all the modes of scientific exposition hitherto adopted.

Upon receiving the manuscript of this treatise from the heirs of the author, the Editor undertook to have it printed as soon as possible, accepting this obligation all the more gladly in that it harmonised with his own inmost feelings; for Bolzano was his unforgettable teacher and friend. He would, moreover, have willingly fulfilled it sooner had his path not been barred by serious obstacles which he was only able to overcome in the course of the current year. He found himself at length in a position to correct his long-standing copy from the sometimes rather illegible and occasionally even incorrect manuscript, to facilitate the use of the book by preparing a full analytical table of contents and to select a satisfactory place of publication. His choice falls on Leipzig, one reason being his hope of a wider dissemination of the treatise, and another being his purpose of doing honour to the celebrated City of Books, the pride and ornament of his adopted Fatherland—for he is by birth a Bohemian. He is confident in fact that once the high genius of Bolzano is generally recognised, it will not be Leipzig’s least title to fame to have contributed to the appearance in print of these Paradoxes.

Budissin, 10 July 1850.

* By Dr. Přihonský.

SYNOPSIS OF CONTENTS

The left-hand column is a literal version of the summary drawn up for the posthumous first edition in 1851. Since Přihonský fails to notice many topics whose importance has since been better realised, his summary is inadequate for the modern reader, and does imperfect justice to Bolzano. The right hand column therefore attempts a more modern analysis.

Editor’s Analysis	Translator’s Analysis
§1. Why the author intends to deal with no other paradoxes than those of the infinite.	§1. Infinity is the source of most if not all mathematical and allied paradoxes, and the first necessity is to define it.
§§2–10. The idea of the infinite as conceived by mathematicians, and a discussion of that idea.	§2. Approaches to a definition: criteria for finite and infinite sets will cast light on the infinite as such.
	§3. The logical constant ‘and,’ together with the notion of an ‘aggregate’ (Inbegriff) as derived from logical conjunction.
	§4. ‘Sets’ (Mengen) are aggregates considered as indifferent to permutation. ‘Multitudes’ (Vielheiten) are sets whose members are co-specific, and ‘unities’ (Einheiten) are cospecific entities considered as members of a set.
	§5. Sets can be main sets composed of subsets, and are called ‘sums’ (Summen) when subset members can equally well be regarded as mainset members. Associativity enters into the very definition of a sum.
§§2–10. The idea of the infinite as conceived by mathematicians and a discussion of that idea.	§6. ‘Quantities’ (Grössen) are members of sets such that each two of their members are either identical, or else stand to one another in the relation of logical sum to logical summand.
	§7. When a fixed one-one relation exists, such that each member of an aggregate is either a prerelatum with a unique fellow-member as postrelatum, or else a postrelatum with a unique fellow-member as prerelatum: then the aggregate is called a ‘sequence’ (Reihe) and its members are called the ‘terms’ (Glieder) of the sequence. The fixed relation is called the ‘law of construction’ (Bildungsgesetz) while the pre- and postrelatum are called the ‘antecedent’ and ‘consequent’ (vorderes und hinteres Glied). Terms having both an antecedent and a consequent are called ‘inner’ (innere Glieder) and terms, if any, which lack the one or the other are called ‘outer’ (äussere Glieder) being either the ‘first’ if any or the ‘last’ if any.
	§8. Let the first term of a sequence be a unity of given species, and let each following term be a multitude formed by adjoining to the antecedent term some yet unadjoined but cospecific unity. Then each such following term is called a ‘finite’ (endlich) or ‘countable’ (zählbar) multitude. The name ‘number,’ or more definitely ‘whole number’ (Zahl, ganze Zahl), shall apply both to each such following term and also to the initial unity. Aliter: whole numbers are finite if they can be reached from 1 by terminated mathematical induction.
	§9. Multitudes would be called ‘infinite’ (unendlich) if every finite multitude were equipotent with a proper part of them. Their existence will be proved in the sequel.
	§10. The infinite as such, being infinite only in virtue of some latent infinite multitude, is hereby incidentally determined.
§11. How Hegel and other philosophers conceive the infinite.	§11. Hegel and others rightly reject the infinite as an unbounded variable, but wrongly allow it in cases other than those of latent infinite multitude.
§12. Other definitions of the infinite and the judgement to be passed upon them.	§12. Even mathematicians err in taking the infinite as the limit of unlimited increase, as the incapacity of further increase, as the absence of termination, or as that which exceeds any assignable magnitude.
§13. The objectivity of the concept set up by the author is evinced by examples from the realm of the non-actual. The set of all ‘absolute propositions and truths’ is infinite.^*	§13. The existence of infinite sets is evinced by an example from logic in parallel with the above notion of number.
§14. The refutation of some objections urged against this concept.	§14. Answers to the objections that infinite sets can never be completed wholes, and that aggregates arise solely through our actual or possible mental acts. The logical analysis of impossibility.
§15. The set of all natural numbers is infinite.	§15. Answers to the objection that the set of all natural numbers is enumerated by its last member and thus is finite instead of infinite.
§16. The set of all quantities whatever is an infinite set.	§16. The set of all quantities is with still greater reason an infinite set—Bolzano counting our ‘positive real numbers’ (rational and irrational) as quantities but not as numbers.
§17. The set of all the simple parts constituting either space or time at large, or any closed spatial or temporal interval, is an infinite set.	§17. Further examples of infinites are: the points or instants in space or time as a whole, or in any closed spatial or temporal interval. These examples are non-actual, being neither substances nor qualities.
§18. Not every quantity envisaged by us as the sum of infinitely many other quantities, each of which is finite, is itself an infinite quantity.	§18. The ‘sum’ of infinitely many individual finite terms need not be infinite. Proposals for a more rigorous summation of convergent geometrical series.
§19. There exist infinite sets which are greater or less than other infinite sets.	§19. One infinite set can exceed another as to multitude, namely by being a whole of which the other is a proper part.
§20. A remarkable relation between two infinite sets, consisting in the possibility that each member of the one set can be so coupled with a member of the other that no member in either set remains uncoupled, and no member in either set occurs in two or more of the couples.	§20. Biunivocal correspondence can obtain between the members of an infinite main set and those of an infinite proper subset, with two examples.^*
§21. Despite their equinumerosity in members (in Hinsicht auf die Vielheit ihrer Teile gleich) infinite sets can still be unequal as multitudes (Ungleichheit ihrer Vielheiten) in that the one turns out to be a proper part of the other.	§21. Conversely, infinite sets in holomeric correspondence are not necessarily equal as multitudes, but can also stand to one another in the relation of whole and part.
§§22–23. Why the case is different with finite sets, and why the grounds of difference cease with infinite ones.	§22. This apparent paradox arises only because holom...