...a form of reasoning which transcends reasoning....
S. Bochner
The term “Boolean valued analysis” appeared within the realm of mathematical logic. It was Gaisi Takeuti, a renowned expert in proof theory, who introduced the term. Takeuti defined Boolean valued analysis as “an application of Scott–Solovay’s Boolean valued models of set theory to analysis.” Vopěnka invented similar models at the same time. That is how the question of the title receives an answer in zero approximation. However, it would be premature to finish at this stage. It stands to reason to discuss in more detail the following three questions.
1. Why should we know anything at all about Boolean valued analysis?Curiosity often leads us in science, and oftener we do what we can. However we appreciate that which makes us wiser. Boolean valued analysis has this value, expanding the limits of our knowledge and taking off blinds from the eyes of the perfect mathematician, mathematician par excellence. To substantiate this thesis is the main target of my talk.
2. What need the working mathematician know this for?Part of the answer was given above: to become wiser. There is another, possibly more important, circumstance. Boolean valued analysis not only is tied up with many topological and geometrical ideas but also provides a technology for expanding the content of the already available theorems. Every theorem, proven by the classical means, possesses some new and nonobvious content that relates to “variable sets.” Speaking more strictly, each of the available theorems generates a whole family of its next of kin in disguise which is enumerated by all complete Boolean algebras or, equivalently, nonhomeomorphic Stone spaces.
3. What do the Boolean valued models yield?The essential and technical parts of this talk are devoted to answering the question. We will focus on the general methods independent of the subtle intrinsic properties of the initial complete Boolean algebra. These methods are simple, visual, and easy to apply. Therefore they may be useful for the working mathematician.
Dana Scott foresaw the role of Boolean valued models in mathematics and wrote as far back as in 1969: “We must ask whether there is any interest in these nonstandard models aside from the independence proof; that is, do they have any mathematical interest? The answer must be yes, but we cannot yet give a really good argument.” Some impressive arguments are available today.
Furthermore, we must always keep in mind that the Boolean valued models were invented in order to simplify the exposition of Cohen’s forcing. Mathematics is impossible without proof. Nullius in Verba. Therefore, part of the talk will be allotted to the scheme of proving the consistency of the negation of the continuum hypothesis with the axioms of Zermelo–Fraenkel set theory (with choice) ZFC. Cohen was awarded a Fields medal in 1966 for this final step in settling Hilbert’s problem No. 1.
Logic and FreedomMathematics is the most ancient of sciences. However, in the beginning was the word. We must remember that the olden “logos” lives in logics and logistics rather than grammar. The order of mind and the order of store are the precious gifts of our ancestors.
The intellectual field resides beyond the grips of the law of diminishing returns. The more we know, the huger become the frontiers with the unbeknown, the oftener we meet the mysterious. The twentieth century enriched our geometrical views with the concepts of space-time and fractality. Each instance of knowledge is an event, a point in the Minkowski 4-space. The realm of our knowledge comprises a clearly bounded set of these instances. The frontiers of science produce the boundary between the known and the unknown which is undoubtedly fractal and we have no grounds to assume it rectifiable or measurable. It is worth noting in parentheses that rather smooth are the routes to the frontiers of science which are charted by teachers, professors, and all other kinds of educationalists. Pedagogics dislikes saltations and sharp changes of the prevailing paradigm. Possibly, these topological obstructions reflect some objective difficulties in modernizing education.
The proofs are uncountable of the fractality of the boundary between the known and the unbeknown. Among them we see such negative trends as the unleashed growth of pseudoscience, mysticism, and other forms of obscurantism which creep into all lacunas of the unbeknown. As revelations of fractality appear the most unexpected, beautiful, and stunning interrelations between seemingly distant areas and directions of science.
The revolutionary changes in mathematics in the vicinity of the turn of the twentieth century are connected not only with the new calculus of the infinite which was created by Cantor. Of similar import was the rise and development of mathematical logic which applied rigor and analysis to the very process of mathematical demonstration. The decidable and the undecidable, the provable and the improvable, the consistent and the inconsistent have entered the research lexicon of the perfect mathematician. Mathematics became a reflexive science that is engaged not only in search of truths but also in study of its own methods for attaining these truths.
Aristotle’s logic, the paradoxes of Zeno, the razor of William of Occam, the donkey of Buridan, the Lebnizian Calculemus, and Boolean algebras are the outstanding achievements of mankind which cast light on the road to the new stages of logical studies. Frege immortalized his name by inventing the calculus of predicates which underlies the modern logic.
The twentieth century is marked with deep penetration of the ideas of mathematical logic into many sections of science and technology. Logic is a tool that not only organizes and orders our ways of thinking but also liberates us from dogmatism in choosing the objects and methods of research. Logic of today is a major instrument and institution of mathematical freedom. Boolean valued analysis serves as a brilliant confirmation of this thesis.
The concept of continuum belongs to the most important general tools of science. The mathematical views of the continuum relate to the understanding of time and time-dependent processes in physics. It suffices to mention the great Newton and Leibniz who had different perceptions of the continuum.
The smooth and perpetual motion, as well as vision of the nascent and evanescent arguments producing continuous changes in the dependant variables, underlies Newton’s outlook, philosophy, and his method of prime and ultimate ratios. The principal difficulty of the views of Newton rests in the impossibility of imagining the immediately preceding moment of time nor the nearest neighbor of a given point of the continuum. As regards Leibniz, he viewed every varying quantity as piecewise constant to within higher order imperceptible infinitesimals. His continuum splits into a collection of disjoint monads, these immortal and mysterious ideal entities.
The views of Newton and Leibniz summarized the ideas that stem from the remote ages. The mathematicians of Hellas distinguished between points and monads, so explicating the dual nature of the objects of mathematics. The mystery of the structure of the continuum came to us from our ancestors through two millennia.
The set-theoretic stance revealed a new enigma of the continuum. Cantor demonstrated that the set of the naturals is not equipollent with the simplest mathematical continuum, the real axis. This gave an immediate rise to the problem of the continuum which consists in determining the cardinalities of the intermediate sets between the naturals and the reals. The continuum hypothesis reads that the intermediate subsets possess no new cardinalities.
The continuum problem was the first in the famous report by David Hilbert. An incontrovertible anti-ignorabimus, Hilbert was always inclined to the validity of the continuum hypothesis. It is curious that one of his most beautiful and appealing articles, which is dated as of 1925 and contains the famous phrase about Cantor’s paradise, was devoted in fact to an erroneous proof of the continuum hypothesis.
The Russian prophet Luzin viewed as implausible even the mere suggestion of the independence of the continuum hypothesis. He said in his famous talk “Real Function Theory: State of the Art” at the All-Russia Congress of Mathematicians in 1927: “The first idea that might leap to mind is that the determination of the cardinality of the continuum is a matter of a free axiom like the parallel postulate of geometry. However, when we vary the parallel postulate, keeping intact the rest of the axioms of Euclidean geometry, we in fact change the precise meanings of the words we write or utter, that is, ‘point,’ ‘straight line,’ etc. What words are to change their meanings if we attempt at making the cardinality of the continuum movable along the scale of alephs, while constantly proving consistency of this movement? The cardinality of the continuum, if only we imagine the latter as a set of points, is some unique entity that must reside in the scale of alephs at the place which the cardinality of the continuum belongs to; no matter whether the determination of this place is difficult or even ‘impossible for us, the human beings’ as J. Hadamard might comment.”
Gödel proved the consistency of the continuum hypothesis with the axioms of ZFC, by inventing the universe of constructible sets. Cohen demonstrated the consistency of the negation of the continuum hypothesis with the axioms of ZFC by forcing, the new method he invented for changing the properties of available or hypothetical models of set theory. Boolean valued models made Cohen’s difficult result simple, demonstrating to the working mathematician the independence of the continuum hypothesis with the same visuality as the Poincaré model for noneuclidean geometry. Those who get acquaintance with this technique are inclined to follow Cohen and view the continuum hypothesis as “obviously false.”
Returning to Takeuti’s original definition of Boolean valued analysis, we must acknowledge its extraordinary breadth. The Boolean valued model resting on the dilemma of “verum” and “falsum” is employed implicitly by the overwhelming majority of mathematicians. Our routine talks and discussions on seminars hardly deserve the qualification of articles of prose. By analogy, it seems pretentious to claim that Euler, Cauchy, and Abel exercised Boolean valued analysis.
Boolean valued analysis is a special mathematical technique based on validating truth by means of a nontrivial Boolean algebra.
From a category-theoretic viewpoint, Boolean valued analysis is the theory of Boolean toposes.
From a topological viewpoint, it is the theory of continuous polyverses over Stone spaces.
Ernest Mach taught us the economy of thought. It seems reasonable to apply his principle and to shorten the bulky term “Boolean valued analysis.” Mathematization of the laws of thought originated with George Boole and deserves the lapidary title “Boolean analysis.”
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