The Continuum: A Critical Examination of the Foundation of Analysis

by Hermann Weyl

"The hard won power ... to assess correctly the continuum of the natural numbers grew out of titanic struggles in the realm of mathematical logic in which Hermann Weyl took a leading part." — John Archibald Wheeler
Hermann Weyl (1885–1955) ranks among the most important mathematicians and physicists of this century. Though Weyl was not primarily a philosopher, his wide-ranging philosophical reflections on the formal and empirical sciences remain extremely valuable. Besides indicating clearly which results of classical analysis are invalidated by an important family of "non-circular" (predicative) theories, The Continuum wrestles with the problem of applying constructive mathematical models to cases of concrete physical and perceptual continuity. This new English edition features a personal reminiscence of Weyl written by John Archibald Wheeler.
Originally published in German in 1918, the book consists of two chapters. Chapter One, entitled Set and Function, deals with property, relation and existence, the principles of the combination of judgments, logical inference, natural numbers, iteration of the mathematical process, and other topics. The main ideas are developed in this chapter in such a way that it forms a self-contained whole.
In Chapter Two, The Concept of Numbers & The Continuum, Weyl systematically begins the construction of analysis and carries through its initial stages, taking up such matters as natural numbers and cardinalities, fractions and rational numbers, real numbers, continuous functions, curves and surfaces, and more.
Written with Weyl's characteristic passion, lucidity, and wisdom, this advanced-level volume is a mathematical and philosophical landmark that will be welcomed by mathematicians, physicists, philosophers, and anyone interested in foundational analysis.

Genres: Mathematics, Philosophy, Science, Nonfiction, History

176 pages, Paperback

First published January 1, 1932

About the author

Hermann Klaus Hugo Weyl (9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski. His research has had major significance for theoretical physics as well as purely mathematical disciplines including number theory. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study during its early years.

Weyl published technical and some general works on space, time, matter, philosophy, logic, symmetry and the history of mathematics. He was one of the first to conceive of combining general relativity with the laws of electromagnetism. While no mathematician of his generation aspired to the 'universalism' of Henri Poincaré or Hilbert, Weyl came as close as anyone. Michael Atiyah, in particular, has commented that whenever he examined a mathematical topic, he found that Weyl had preceded him (The Mathematical Intelligencer (1984), vol.6 no.1).

Source: http://en.wikipedia.org/wiki/Hermann_...

Reviews

[William Bies · Rating 3 out of 5]

For one steeped in the history of mathematics ever since Thales, the continuum presents the reflective mind with a very great mystery indeed—not so for the glib logical formalist, who vainly imagines Dedekind and Russell to have tidied everything up by the first decade of the twentieth century with their arithmetization of it! But to take the longer view, and the great German mathematician and colleague of Einstein, Hermann Weyl seeks to do so in the present little work, the question as to the nature of the infinite implicit in the concept of the continuum has always exercised mathematicians’ imagination, and many of the celebrated way stations along the great tradition must be viewed as apt responses to this question (thus, Aristotle, the Neoplatonists, Newton, Cantor). We could hardly ask for a more authoritative source; in his legendary Raum, Zeit, Materie, Weyl spells out the epistemological foundations of the special and general theories of relativity better than Einstein himself ever did, and he distinguished himself early on as an original mind by inventing the concept of a gauge theory in his abortive attempt to unify gravity and electrodynamics in 1918; later, he would return to theoretical physics and make pioneering contributions to our understanding of the role of symmetry and groups in quantum mechanics, along with Eugene Wigner.

Starting point: the Pythagoreans thought of numbers as obtained by a process of running through (one may wish to consult any text in the history of mathematics); e.g., the tetractys. Think of the representation of small whole numbers on dominoes via a collection of dots. Run through all the dots and one gets the corresponding number. As far as this reviewer can tell, when the Pythagoreans say that everything in the world is reducible to number, they imagine it as broken down into its constituent dots; count up the dots and one will arrive at the number of the thing. Such a manner of viewing things is not necessarily as naïve as it might seem, if we take the standpoint of modern theoretical physics, where of course we have to operate with a broader concept of number than the usual one.

By the way, it is already contentious to take a view such as this; Aristotle refuses to countenance it in his discussion of the continuum (which he regards as potentially divisible without limit, but one can never complete the process in order to arrive at atomic constituents). The Neoplatonist doctrine of the mathematicals as intermediaries between the sensible world of physics and the intelligible world of the ideas implicitly goes beyond a denumerable infinity. The mathematicals are concerned with forms of existing things; but, as a moment’s reflection should convince one of, spatial form implies a rule for coordinating any discrete collection of points; that is to say, it draws a boundary such that points lying on one side belong and those on the other do not (these could be seen as generalizing Dedekind’s cuts [Durchschnitte] of the rational number line to collections of greater than one dimension). Thus, the mathematicals inherently correspond to a non-denumerable infinity. All through the middle ages—despite what superficial accounts might lead one to suppose—, the reception of Aristotle was colored by Neoplatonic tendencies originating in the later church fathers, who were very much at home in the thought-world of pagan late antiquity. For several hundred years, inquiry into true nature of infinity furnished a topic of discussion at a serious and advanced level by the medieval scholastics, but this reviewer must disclaim any expertise and skip ahead to the modern period. Perhaps someday somebody will unearth what the technically quite proficient medievals were really talking about all along and phrase it in an idiom congenial to modern ears, but not yet, by a long shot! Any gifted and contrarian beginning graduate students out there? Excellent material for a ground-breaking dissertation here.

But we lowly ones must, however, limit ourselves to what a modern training in mathematics equips us to understand. As we just saw above, a vague concept of motion in mathematics can be attributed to the Pythagoreans and formed an ever-present backdrop to the classical Greek static view of ideas existing in a Platonic intelligible realm. But, in the early modern period, it would reemerge to play an instrumental role in stimulating the development of the differential and integral calculus. Perhaps, the early quadratures carried out by the indivisiblists (such as Cavalieri) led one to find it natural to conceive of a mathematical quantity to be arrived at by successive addition of infinitely many infinitely small parts. In any event, Newton’s guiding idea of fluents and fluxions underlay the discovery of the integral and of the derivative. When one thinks of fluents and fluxions in Newton’s informal terms, the reciprocity between the operations of integration and differentiation expressed by the fundamental theorem of calculus becomes intuitively compelling, although a strict proof of this relation had to wait two hundred years until an unexceptionable definition of the real number continuum became available. Let us remark on why the continuum matters in this context: it supplies us with a non-trivial notion of infinity. Of course, one could take any finite set and duplicate it transfinitely many times (as one in fact does in many places in higher algebra), but a procedure such as this would yield only a disaggregated heap of elements bearing no relation to one another. But, with the continuum, its elements are not merely disparate but are interrelated, in the sense that, entering into collectives defined with the aid of an order relation, they satisfy topological properties of nearness or distance, etc. without which modern real analysis would be inconceivable. For only then can one speak of functions having the properties of continuity, measurability etc.

Where matters stood as of the time of Weyl’s writing in 1917 was that, on the basis of Cantor’s set theory and the rigorous arithmetization of the continuum at the hands of Weierstrass and Dedekind, real analysis was experiencing its golden age. But, simultaneously, the crisis in the logical foundations of axiomatic set theory must have caused any reflective observer to wonder whether, or to what extent, the new perspectives in analysis were altogether sound and credible. Indeed, the Dutch mathematician Brouwer contested just this, and, with his so-called intuitionism suggested that one must content oneself with something more modest, yet, at the same time, more carefully wrought. Weyl himself, at this stage, was tending towards Brouwer’s position, although later in his career he would nuance his partisanship with the intuitionist school. What’s in Weyl’s present idiosyncratic little contribution to the debate: he parcels his exposition into two parts, 1) on an analysis of the construction of the mathematical concepts of set and function; 2) on the concept of number in relation to that of the continuum, which Weyl styles as a foundation of a calculus of infinitesimals.

Weyl proceeds from elementary definitions. His goal in Part I is to show what he considers to be the vicious circle in the customary concepts of set and function, as they were being employed in the analysis of his time, and to propose a means of getting around it. Roughly speaking, Weyl’s approach is to start with a ground category of elements that can be defined by immediately perceivable properties and relations, what he will call original relations. From this procedure, one arrives at sets of the first stage. Second-stage sets will then be defined via the introduction of ideal elements by means of the principles of substitution and iteration. The key point is that one cannot make use of the concepts of set equality or existence when defining relations at the second stage; otherwise, one would fall into the contradiction of trying to define relations in terms of themselves. Part II revolves around the distinction between the intuitive versus the mathematical continuum, where by the latter, Weyl seems to have in mind the arithmetical continuum of Dedekind’s with which we all are familiar. The suggestion Weyl wants to put forward is that the neat and tidy picture of the foundations of analysis one pretends to have gained via the mooted arithmetization of the continuum is deceptive, in that it fails to make adequate allowance for the transcendental conditions of intuition, upon which follows an appeal to Bergson’s philosophy: ‘Bereits dem Begriff des Punktes im Kontinuum mangelt es dazu an der nötigen Stütze in der Anschauung. Es ist ein Verdienst der Bergsonschen Philosophie, mit Nachdruck auf diese tiefe Fremdheit der mathematischen Begriffswelt gegenüber der immittelbar erlebten Kontinuität der phänomenalen Zeit (»la durée«) hingewiesen zu haben’ (pp. 68-69). The analytical clarity of the conventional approach hides the fact that the operations it stipulates are purely formal.

What, then, is Weyl’s counterproposal in Part II? First, he redefines what we should mean by a real number, and then sketches how the theory of functions, curves and surfaces in space will look different under the new definitions. We cannot recapitulate the entire argument here; suffice it to say that Weyl’s idea is tantamount to a reintroduction of the concept of motion in mathematics at an even more radical level than what Newton and Leibniz ever entertained. For (let those who are experts pardon this reviewer if what follows is misleadingly simplistic), Weyl’s revised concept of real number seems to be something that results from a sequence of subsets of pairs of integers corresponding to sets of rationals obeying some relation; as the relation is specified more precisely, one approaches a definite real number in the limit. Weyl’s proposal here recalls Brouwer’s choice sequences, which—at the risk of drastic simplification—one could view as specifying a real number through a choice of the successive digits in its decimal expansion. From an intuitionistic point of view, one can never think of the totality of such choices as having been realized all at once; all one can have at one’s disposal would be a finite expansion, up to an indefinite number of decimal places.

Now, it seems to this reviewer that what we are dealing with here, fundamentally considered, is a poor man’s substitute for Hegelian dialectics. There is a gap between intensive and extensive definition, which the conventional analytical philosophy of mathematics seeks to elide. For a very complicated extensive definition of a set will correspond to intensive definition via a property that no one would regard as belonging naturally to intuition. Weyl is aware of this problem. So, what he really wants to do is to define the sets in his intuitionistic schema via a sequence of increasingly refined properties. But what will this sequence be, in any given real case? Here is where dialectics comes in. As Aristotle points out in the Topics, dialectics has to do with the debate over concepts that do not admit precise analytical definitions upon which everyone can agree (as, for him, supposedly is the case in the sciences). Hence, we could suggest that Weyl’s choice sequences will be the outcome of a dialectical process in which intersubjective agreement as to the relevant properties will be reached by convergence. But, with a statement such as this, we are going far beyond what we would be entitled to draw from anything contained expressis verbis in Weyl’s succinct text. Nevertheless, something like this must be what he really intends! For isn’t what we have just intimated the logical consequence of Weyl’s statement in his Vorrede to the effect that:

Unsere Betrachtung des Kontinuumsproblems liefert einen Beitrag zu der erkenntniskritischen Frage nach den Beziehungen zwischen dem unmittelbar (anschaulich) Gegebenen und den formalen Begriffen (der mathematischen Sphäre), durch welche wir in Geometrie und Physik jenes Gegebene zu konstruieren suchen (p. iv)?

Weyl recognizes that constructivism à la Brouwer entails the need for a revision to our inherited ideas of classical logic; the principle of the excluded middle has to go by the wayside, for instance. But, unfortunately, he lacks the imagination to sketch anything like a systematic alternative, and we are left high and dry. What if Hegel had actually been in possession of a technically refined command of mathematics? But first one must master his science of logic, in which he strives heroically to make good on the promise of the phenomenology of spirit of 1807 that marks his point of departure as a thinker of stature. This reviewer must register his regret that, under the prevailing circumstances, an intellectually satisfactory treatment of the profound issues raised by Hegel and by Weyl can scarcely be expected to be forthcoming anytime soon. For what has hitherto passed under the name of philosophy of mathematics for the past century or so stands too much in the grip of Anglo-American analytic methodology and remains stifling in its preoccupation with trivialities, hence too boring to attract any really original minds who could compete in a class with Hegel himself. The present, incomplete sketch must be rated at 3 stars: unfortunately, Weyl’s prescient aim exceeds even his masterful grasp, but he has set the agenda for us, or for anyone who wishes seriously to contemplate the deeper significance of the continuum (so maybe good old Weyl deserves a half-point in extra credit!).

What is primordially at stake when we speak of motion in mathematics? Something like wielding a rudimentary enveloping framework, perceived intuitively, that has to be condensed into clear and distinct concepts and definitions in order to become fruitful in applications. When one declines this last step, one remains speculative in the pejorative sense [as captured by that untranslatable German term of abuse, ‘Schwärmerei’]; but when out of false modesty one leaves off speculation altogether, one winds up a mere derivative epigone. Clearly, one ought to balance the two and in the present work on the continuum Weyl points us modern mathematicians in a restorative direction, if only we would listen!

[David Olmsted · Rating 4 out of 5]

Herman Wyle (1885-1955) was a renowned and creative mathematician and theoretical physicist. His mathematical work was instrumental in forming the foundations for modern quantum physics. This 1917 book is an essay against the set theory foundations for mathematics that was starting to become widely accepted. His view was that set theory leads to contradictions and requires just as much human intuition as does working directly with geometry and natural numbers:

“A set-theoretic treatment of the natural numbers such as that offered in Dedekind (1888) may indeed contribute to the systematization of mathematics; but it must not be allowed to obscure the fact that our grasp of the basic concepts of set theory depends on a prior intuition of iteration and of the sequence of natural numbers.” (p 24)

His alternate approach was based upon objects and iteration but in the end he was not satisfied with the result. An “object” is a form of super-variable in which spaces for various sorts of data are reserved for use by operations inherent to the object itself. His conceptual use of objects in this book may be one of the first uses of an idea that has come to dominate computer science.

“The foundation consists of, first, one or more individual categories of object (the ‘basic categories’) and, second, certain individual immediately exhibited properties of and relations between objects of the basic categories (the ‘primitive relations’).” ( p 41)

His preface to a 1932 reprint sums up his view of the state of mathematical foundations:

“The point of view adopted in this monograph continues to strike me as a natural transitional stage in the development of foundational research. However, in the period since its appearance, my work has been superseded by two trends identified by the catchwords Intuitionism and Formalism. Still, this deeper grounding of the foundation has not lead to an even moderately satisfying or defensible conclusion; things remain in a state of flux.” (p 3)

[Qiang Wang · Rating 1 out of 5]

Totally do not agree with this book, It did not solve any problem, did not solve CH at all.
Please see the book :

https://www.amazon.com/Principles-Lar...

and



It cracks the CH thoroughly.
CH is only simplest problem in .
All problems and Puzzles solved in .are much more difficult than the CH. The CH is nothing.

Humans should step forward instead of staying in old dreams.