The Unreasonable Effectiveness of Number Theory
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You may print 0 more time s before then. You may print a maximum of 0 pages at a time. NOTE: Printing will start on the current page. Firefox users may need to click "Back" when printing completes. Print 0 Pages. Go to book description Print multiple pages Multi-Page Printing You have printed 0 times in the last 24 hours. This paper samples applications of number theory in the real world, ranging from concert hall acoustics to general relativity. Introduction Number theory has been considered since time immemorial to be the very paradigm of "pure" some would say useless mathematics.
According to Carl Friedrich Gauss, "mathematics is the queen of sciences - and number theory is the queen of mathematics.
How the unreasonable effectiveness of mathematics can work elsewhere
One is almost tempted to call them wholesome numbers. The principal emphasis is on the invention of concepts. Wigner , , W. However, he makes a distinction. On the other hand, Most more advanced mathematical concepts, such as complex numbers, algebras, linear operators, Borel sets — and this list could be continued almost indefinitely — were so devised that they are apt subjects on which the mathematician can demonstrate his ingenuity and sense of formal beauty.
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Wigner , Ingenuity, inventiveness, the skill of the virtuoso to develop interesting connections, guided by a sense of formal beauty and a basic concern for logical coherence are what drive pure mathematics. Strict formalism interprets mathematical systems as a game of symbols. The network of relations thus codified in a formal calculus restricts possible applications or interpretations of the system. But the question is whether his way of presenting the matter — in particular, his conception of mathematics — does justice to the new framework, or perhaps adds unnecessary complications, raising the specter of non-existent problems and creating weighty miracles out of thin air.
We find a brief reference to Hilbert on foundations , another passing reference to M. The case of Walter Dubislav is particularly interesting. He was a member of the Berlin Association for Empirical Philosophy and one of the signers of the famous Vienna Circle manifesto.
See also Esfeld The brief presentation of the philosophy of mathematics offered in his textbook is very clear, emphasizing mathematical logic, axiomatic thinking, and a form of empiricism in the case of applied mathematics. The imprint of the Hilbert school is undeniable here. Put grossly: pure logic and pure mathematics, taken in themselves, are games of formulas [Formelspiele] and nothing else. Dubislav This was not yet a familiar point of view. As it turns out, Dubislav was a Privatdozent at the Technische Hochschule Berlin from and a colleague of Wigner there.
First, logical empiricism would continue to be prominent in the philosophical context around Wigner and is visible in his But the second point is more directly interesting for my purposes. We have seen that Dubislav was a strict formalist, and that Wigner himself still defended a kind of formalism in his remarks about mathematics. The previous generation of physicists and mathematicians were not formalists and it was only the generation that matured in the s that understood the new ideas about axiomatics, structures, logic and foundations in a radical way.
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Hilbert was not a formalist at the level of epistemology. His celebrated formalism was a method adopted in the context of studies of the foundations of mathematics, for the goals of metamathematics consistency proofs, decision procedures. Using the axiomatic method, one may begin by considering a particular field of work with concrete ideas. But there is much to gain methodologically by disregarding the particular meaning of the concepts, considering the axioms as schematic conditions and adopting full freedom of interpretation.
In foundational research, this attitude can be amplified to achieve strict formalization, which makes it possible to study and solve precise questions, such as the incompleteness of Peano Arithmetic and other axiom systems; or the relative consistency of AC to the other Zermelo-Fraenkel axioms. Mathematics is not like a game in which the problems are determined by rules invented arbitrarily, but a conceptual system [endowed] with inner necessity, that can only be this, and not any other way.
In this case, other more general sources have to be found, coming largely from the intellectual context. As we discussed above, the young intellectuals in Berlin, during the s, were living a rather chaotic, rapidly changing, and heated cultural atmosphere. Such a setting promoted forms of modernism in the sciences, modernistic tendencies that presented themselves as a radical break with the past. The interwar period was a particularly 17 D. Hilbert, Natur und mathematisches Erkennen, lectures delivered in , ed. The point has been made repeatedly by experts such as Corry ; Rowe, ; Mancosu , Hilbert was basically right: formalism is very good as a method of studying foundations, but philosophical questions about the epistemic nature of mathematical knowledge require more sophisticated answers.
Moreover, the examples Wigner gives from advanced mathematics do not support his formalist views. His list included complex numbers, algebras, linear operators, and Borel sets. His idea was that the elaboration of such concepts is guided by intra-mathematical considerations, disregarding considerations of the potential for application to natural phenomena.
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The square root of — 1 was introduced by Italian mathematicians in the 16th century in order to manipulate numbers and expressions in algebraic equations. Even so, the early results and procedures did not establish a secure position for imaginary numbers in the world of mathematics.
From Wikipedia, the free encyclopedia
For Gauss and Riemann, considering the system of complex numbers as the natural general framework for number in general was a basic commitment and fundamental principle of pure mathematics. This is why theoretical physicists, such as Wigner and Penrose, have placed great emphasis on the complex number structure.
Wigner himself stressed that the real number system was devised so as to mirror the properties of measurable quantities, and the topology of C is essentially inherited from R. Given that Wigner pioneered group-theoretic methods in quantum mechanics, it is noteworthy that group theory is not among his examples. The reasoning behind this statement is that group structures are implicit behind all kinds of ancient concepts and practices — symmetry considerations, operations of translation and congruence in basic geometry, measuring operations, and so on.
Probably Wigner chose the example of Borel sets as one of the central concepts of set theory in the first third of the 20th century — what better example of the purest in pure math? Yet this case goes rather against his thesis. Borel sets are strongly linked with the function concept and their study was motivated by a desire to restrain the most general and arbitrary possibilities opened by set theory, to focus on concrete ideas closer to classical math.
But the study of functions has constantly been promoted by extra-mathematical considerations, mostly physical. What are the Borel sets? You start with open sets and build more complex ones by two operations — taking complement, or forming countable unions. You obtain a hierarchy of 20 Thus Grassmann in coming from geometry, and Dedekind in from algebraic number theory, were among the first to clearly articulate modern ideas about the subject Kleiner , Let us look at it from another perspective.
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Consider point-sets in a Euclidean space such as R3. General set theory aims to study any point-set, including the highly arbitrary ones that one may obtain by use of the Axiom of Choice AC. Thus we know that there are point-sets in R3 that are not Lebesgue measurable, and so on. In contrast to this, the study of Borel set represents a great restriction: one focuses only on sets obtained from well-understood basic cases, via simple operations; the whole thing has strong geometric underpinnings.
And, indeed, all the Borel sets are Lebesgue measurable. The hierarchy of Borel sets has a natural counterpart in the hierarchy of Baire functions. Here again, the inspiration was to focus on relatively simple functions, instead of loosing oneself in the generality of arbitrary functions.
The Baire functions are obtained starting from continuous functions, introducing new fs as pointwise limits of sequences of functions; this gives rise to a hierarchy indexed by countable ordinals. Lebesgue measurability , and set theorists were hard at 20 work studying how far beyond those properties applied. Also, both scientists were inclined to general philosophical reflection, and it is interesting that both emphasized the importance of the aesthetic element in guiding pure mathematics.
Thus in The Value of Science he writes: physics has not only forced us to choose among problems which came in a crowd; it has imposed upon us problems such as we should without it never have dreamed of. In an interesting paper for the general public, published in , he writes: The most vitally characteristic fact about mathematics is, in my opinion, its quite peculiar relationship to the natural sciences, or, more generally, to any science which interprets experience on a higher than purely descriptive level.
I am led to mention this because Wigner, like his Hungarian friends Szilard, Teller and von Neumann, was strongly anti-communist — and it may be the case that his political views colored his philosophical ideas. Many of the great abstractions introduced in mathematics from the midth century have strong roots in the physically motivated mathematics of functions, analysis, the real-number continuum, and geometry.
And this increase in flexibility provides a very rational explanation of the applicability of mathematics! It is hardly surprising that a much more general and flexible theory of geometrical structures e. Riemannian differential geometry can be applied in many contexts where the rigid structures of Euclidean geometry would not be applicable.
Yet perhaps the empirical success of mathematical laws in physics requires something else. References: U. Dahan Dalmedico Changing Images in Mathematics: From the French revolution to the new millenium. Corry Modern Algebra and the Rise of Mathematical Structures. Hilbert, Methoden der mathematischen Physik, vol.