Mathematicians From The United States

"The theorem of Löwenheim–Skolem was the first truly important discovery about formal systems in general, and it remains probably the most basic. It is not a negative result at all, but plays an important role in many situations. For example, in Gödel's proof of the consistency of the Continuum Hypothesis, the fact that the hypothesis holds in the universe of constructible sets is essentially an application of the theorem."

"However, in all honesty, I must say that one must essentially forget that all proofs are transcribed in this formal language. In order to think productively, one must use all the intuitive and informal methods of reasoning at one's disposal. At the very end one must check that no errors have been committed; but in practice set theory is treated as any other branch of mathematics. The reason that we can do this is that we will never speak about proofs but only about models."

"It is now known that the truth or falsity of the continuum hypothesis and other related conjectures cannot be determined by set theory as we know it today. This state of affairs regarding a classical and presumably well-posed problem must certainly appear rather unsatisfactory to the average mathematician. One is tempted to look more closely and perhaps more critically at the foundations of mathematics. Although our present "Cantorian" mathematics is highly successful in its treatment of abstractions, one must not overlook the fact that from the very beginning the use of infinite processes was regarded with suspicion by many people."

"The object of mathematics is to discover "true" theorems. We shall use the term "valid" to describe statements formed according to certain rules and then shall discuss how this notion compares with the intuitive idea of "true"."

"The spinor genus is due to Eichler. His early results (1952) established the theory over the rational field and also, in certain special cases, over a number field. Kneser (1956) extended this to number fields in general. At about the same time Watson obtained Eichler's results by elementary methods over the rational field."

"Class field theory can be divided into two parts, local and global. In each part it is the study of all the abelian extensions of a certain base field. The underlying philosophy is to describe all abelian extensions in terms of objects residing within, or close to, the base field."

"It was not until my second year as a doctoral student that I began to understand that mathematics was an ever-expanding universe. My thesis advisor at Princeton was Emil Artin, one of the great algebraists of the century. Unfortunately, or perhaps fortunately, he offered me no advice in the selection of a thesis topic. I think it was a fluke that I got started at all. But once I did, a whole new world opened up, to which I would devote a vast amount of time and energy for over thirty years."

"[A]ccording to Weyl, complexity is essential in understanding the concept of a law of nature. If laws of nature may be arbitrarily complex, he argued, the very concept... becomes vacuous. What difference would remain... if the laws meant to explain them were as complex as the phenomena they are meant to explain? Laws of nature must be simple."

"Mathematicians are coming up with s that they use in their actual mathematical research. And these, like , I think is the name of one of them ... these are actually like s that have been engineered in a way that they can actually be used by working mathematicians to check the work they're doing."

"Problem: Name a book that combines mathematical history, philosophy, more than a whiff of theology, personal palaver, and brilliant insights, along with evidence of a Borges-like imagination, eyebrow-raising mathematical constructions, breathtaking excitement, grandiose ruminations, and some bosh. Solution: The book under review. ...Ω leads me to turn 180 degrees away from these classical philosophical questions—discussed ad nauseam and with diminishing profit—of why mathematics is true, whether its objects and constructs have ontological validity, whether it is the only mode of inference, what its limitations are, whether it is the unique language in which theoretical physics must be formulated. If we could focus instead on mathematical pragmatics—why mathematics throughout the millennia has been useful or deleterious to society—then I believe that we might be able to illuminate an aspect of mathematics that is often ignored: mathematics as a enterprise, for that is surely what it is."

"At first it might seem that quantum mechanics (QM), which began with Einstein's photon as the explanation for the photoelectric effect in 1905, goes further in the direction of discreteness. But the wave-particle duality discovered by de Broglie in 1925 is at the heart of QM, which means that this theory is profoundly ambiguous regarding the question of discreteness vs. continuity. QM can have its cake and eat it too, because discreteness is modeled via standing waves (eigenfunctions) in a continuous medium."

"Are there mathematical propositions for which there is a considerable amount of computational evidence, evidence that is so persuasive that a physicist would regard them as experimentally verified?"

"...Once you entomb mathematics in an artificial language à la Hilbert, once you set up a completely formal axiomatic system, then you can forget that it has any meaning and just look at it as a game that you play with marks on paper that enable you to deduce theorems from axioms. You can forget about the meaning of the game, the game of mathematical reasoning, it's just combinatorial play with symbols! There are certain rules, and you can study these rules and forget that they have any meaning!"

"Why do I think that Turing's paper "On computable numbers" is so important? Well, in my opinion it's a paper on epistemology, because we only understand something if we can program it, as I will explain in more detail later. And it's a paper on physics, because what we can actually compute depends on the laws of physics in our particular universe and distinguishes it from other possible universes. And it's a paper on ontology, because it shows that some real numbers are uncomputable, which I shall argue calls into question their very existence, their mathematical and physical existence."

"I'm interested in the computer as a new idea, a new and fundamental philosophical concept that changes mathematics, that solves old problems better and suggests new problems, that changes our way of thinking and helps us to understand things better, that gives us radically new insights..."

"Others have spoken and written about his solution of Hilbert's fifth problem, but perhaps not enough is said about his later work, especially his joint work with C. T. Yang. In a long series of papers written in the late 1960s and early 1970s, they used the study of group actions on homotopy 7-spheres to showcase and test the growing new techniques of differential topology, especially index theory and surgery theory. At a time when much work in topology consisted in building these machines, their papers demonstrated the beauty of applying this theory to unfurl complexities of symmetry and structure."

"One respectful question will be addressed to Jews who plan to continue to eat meat: In view of strong Jewish mandates to be compassionate to animals, preserve our health, help feed the hungry, preserve and protect the environment, conserve resources, and seek and pursue peace, and the very negative effects animal-centered diets have in each of these areas, will you now become a vegetarian, or at least sharply reduce your consumption of animal products?"

"THEOREM: if G is a locally euclidean, connected, simply connected topological group of dimension n greater than one, then G contains a closed proper subgroup of positive dimension."

"A group which has a simple structure may offer difficult questions when operating as a transformation group. For example, the ways in which a cyclic group of order 2 can operate on a manifold, even on E 3, are far from completely known."

"In 1917 Levi-Civita discovered his celebrated parallelism which is an infinitesimal transportation of tangent vectors preserving the scalar product and is the first example of a connection. The salient fact about the Levi-Civita parallelism is the result that it is the parallelism, and not the Riemannian metric, which accounts for most of the properties concerning curvature."

"Suppose you want to teach the "cat" concept to a very young child. Do you explain that a cat is a relatively small, primarily carnivorous mammal with retractible claws, a distinctive sonic output, etc.? I'll bet not. You probably show the kid a lot of different cats, saying "kitty" each time, until it gets the idea. To put it more generally, generalizations are best made by abstraction from experience. They should come one at a time; too many at once overload the circuits."

"The predecessors of Newton and Leibnitz knew perfectly well how to determine tangents and areas, but they had to approach each problem from first principles. The great contribution of Newton and Leibnitz was precisely to make the procedures for finding tangents, areas, etc. into a calculus, that is, a systematic way of calculating—a collection of algorithms, to use the currently fashionable word."

"The fact that N Bourbaki was not a real person and represented a group of mathematicians was known to very few, for example, Ralph Boas, but the world of mathematics had come to believe in his existence. In an article for the Encyclopaedia Britannica, Boas revealed the truth; he was severely reprimanded in a letter to him "From my ashram in the Himalayas", beginning with "You miserable worm, how dare you say that I do not exist?" and signed 'Nicolas Bourbaki'!"

"It is well known that in three-dimensional elliptic or spherical geometry the so-called Clifford's parallelism or parataxy has many interesting properties. A group-theoretical reason for the most important of these properties is the fact that the universal covering group of the proper orthogonal group in four variables is the direct product of the universal covering groups of two proper orthogonal groups in three variables. This last-mentioned property has no analogue for orthogonal groups in n (>4) variables. On the other hand, a knowledge of three-dimensional elliptic or spherical geometry is useful for the study of orientable Riemannian manifols of four dimensions, because their tangent spaces possess a geometry of this kind."

"S. S. Chern revolutionized differential geometry with the use of moving frames, the invention of characteristic classes, the modern concept of a connection and so much more, but he’ll probably always be most remembered for the yellowing University of Chicago mimeographed lecture notes from the 1950s. An entire generation of geometers learned the elements of differentiable manifolds from those notes."

"Recently, having refreshed my understanding of the mathematics of relativity theory, I called one of my old Berkeley professors to ask him some questions about the geometry of general relativity. S. S. Chern is arguably the greatest living geometer. We spoke on the phone for a long time, and he patiently answered all my questions. When I told him I was contemplating writing a book about relativity, cosmology, and geometry and how they interconnect to explain the universe, he said, "It's a wonderful idea for a book, but writing it will surely take too many years of your life ... I wouldn't do it." Then he hung up."

"I have no doubt that future historians of differential geometry will rank Chern as the worthly successor of Elie Cartan in that field."

"Integral geometry, started by the English geometer M. W. Crofton, has received recently important developments through the works of W. Blaschke, L. A. Santaló, and others. Generally speaking, its principal aim is to study the relations between the measures which can be attached to a given variety."

"Not all the geometrical structures are "equal". It would seem that the riemannian and complex structures, with their contacts with other fields of mathematics and with their richness in results, should occupy a central position in differential geometry. A unifying idea is the notion of a G-structure, which is the modern version of an equivalence problem first emphasized and exploited in its various special cases by Elie Cartan."

"The main object of study in differential geometry is, at least for the moment, the differential manifolds, structures on the manifolds (Riemannian, complex, or other), and their admissible mappings. On a manifold the coordinates are valid only locally and do not have a geometric meaning themselves."

"The treatises of Darboux (1842–1917) and Bianchi (1856–1928) on surface theory are among the great works in the mathematical literature. They are: G. Darboux, Théorie générale des surfaces, Tome 1 (1887), 2 (1888), 3 (1894), 4 (1896), and later editions and reprints. L. Bianchi. Lezioni di Geometria Differenziale, Pisa 1894; German translation by Lukat, Lehrbuch der Differentialgeometrie, 1899. The subject is basically local surface theory."

"Why do I paint? In my work as a mathematician, form, structure, and economy of expression are important. In music, add tone quality, balance and harmony. And now in painting, these aesthetic values are enhanced by color, contrast, and composition."

"Algebraic geometry has developed in waves, each with its own language and point of view. The late nineteenth century saw the function-theoretic approach of Riemann, the more geometric approach of Brill and Noether, and the purely algebraic approach of Kronecker, Dedekind, and Weber. The Italian school followed with Castelnuovo, Enriques, and Severi, culminating in the classification of algebraic surfaces. Then came the twentieth-century "American" school of Chow, Weil, and Zariski, which gave firm algebraic foundations to the Italian intuition. Most recently, Serre and Grothendieck initiated the French school, which has rewritten the foundations of algebraic geometry in terms of schemes and cohomology, and which has an impressive record of solving old problems with new techniques. Each of these schools has introduced new concepts and methods."

"Intellectual property is an oxymoron."

"The subject matter of mathematics is the expressions themselves together with the rules for manipulating them—nothing more."

"Nelson rejected the prevailing views to the effect that nonstandard analysis operates with some fictional elements that extend the standard world of mathematical entities. In his approach, nonstandard objects inhabit the realm of the most ordinary mathematical objects. Nelson emphasized the creative syntactic contribution of the new approach in the following terms: “Really new in nonstandard analysis are not theorems but the notions, i.e., external predicates.” (Nelson 1988 ...)"

"Maybe so, but something is going on with the primes."

"You can ask the question about these ancient topics, such as s and ... and ask, are these good problems... I'd like to give a small amount of evidence... that they are... [S]tudying them helped us develop all of elementary number theory and from elementary number theory we developed the rest of number theory, and also you can argue that from elementary number theory came algebra.."

"Howard Percy Robertson was a postdoctorate student at Göttingen and Munich from 1925 to 1927. While in Göttingen he completed an important paper on relativistic cosmology in which he derived a relation between the velocity of nebulae and their distances. For the radius of the observable world he calculated R = 2 \times 10^{25} m. Although he had a velocity-distance relation and referred to Slipher's s, he did not conclude that the universe was in a state of expansion."

"Robertson wrote an influential [1933] review of relativistic world models in which he specifically excluded those which have "arisen in finite time from the singular state R = 0." Although he included the Einstein-de Sitter paper in his bibliography, he did not mention it in the review. He also did not mention Lemaître's primeval-atom hypothesis."

"In 1928 Robertson found a non-static line element similar to the one Lemaître had found three years earlier. Also like Lemaître, he derived a linear relationship between apparent recessional velocities and distances, and he discussed it in relation to observation data. Within the same tradition was Tolman's 1929 derivation of a 'Hubble law', that is, a relationship of the form v = kr, with v = \frac{d\lambda}{\lambda}. Robertson and Tolman generalized the De Sitter model to an arbitrary scale factor F(t), but they remained within the static paradigm and did not realize the significance of F(t). In a paper of 1929, Robertson wrote the general line element of what would later be known as the Robertson-Walker models and he even referred to Friedmann's work. And yet, although he had evidently studied Friedmann, he 'misread' him and failed to realize the significance of the expanding metric."

"Robertson and Tolman developed much of the mathematics of the expanding universe, but without concluding or predicting prior to 1930 that the universe actually expands. They were not discoverers or codiscoverers of the expanding universe (and never claimed that they were)."

"Howard P. Robertson showed that the uncertainty relation follows from the commutation rule."

"Note.—The second part of this paper was considerably altered by me after the departure of Mr. Rosen for Russia since we had originally interpreted our results erroneously. I wish to thank my colleague Professor Robertson for his friendly in his assistance in the clarification of the original error."

"[I]n deriving the general line-element for the background geometry of FLRW [Friedmann–Lemaître–Robertson–Walker, sometimes called the Standard Model] cosmology, Robertson required four basic assumptions: i. a congruence of s, ii. , iii. homogeneity, and iv. . i. and ii. are required to satisfy of a causal coherence amongst s in the entire Universe, by which every single event in the bundle of fundamental world lines is associated with a well-defined three-dimensional set of others with which it ‘really’ occurs simultaneously. However, it seems that ii. is therefore mostly required to satisfy the concept that synchronous events in a given inertial frame should have occurred simultaneously, against which I’ve argued..."

"Important contributions to what would later appear as mainstream big bang cosmology were made by Americans Howard P. Robertson and Richard Tolman... Robertson (and independently, A. G. Walker in England) deduced in 1935 the most general form of the metric for a space-time satisfying the : the postulate that the universe is spatially homogeneous in its large scale appearance. This metric became generally known as the Robertson-Walker metric. Together with Tolman, Robertson pioneered the study of thermodynamics in the theory of the expanding universe."

"Many of the theoretical investigations of cosmology in the 1920s were examinations of De Sitter's model, which is particular because it can be understood both as a static model (as De Sitter did) and as an expanding model (as became the view after 1930). It is clearly problematic to read the pre-1930 literature in light of later knowledge. 'Expanding' versions of the De Sitter universe were found by in 1922, Hermann Weyl in 1923, and Lemaître in 1925, but were not conceived as expanding in any real sense. These works, as well as later works by Howard Percy Robertson and Richard Chase Tolman in the United States, consisted in transforming De Sitter's line element in such a way that it formally became static, that is, included a term F(t) referring to the time parameter. The metric, giving the distance in space-time between two neighboring points, would then be in the form ds^2 = c^2 dt^2 - F(t)(dx^2 = dy^2 +dz^2)."

"The basic cause and nature of cosmic expansion, along with its recently-observed acceleration, are significant problems of the standard model; so, condisering the evidence that the acceleration is best described by pure \Lambda, there is strong motivation to search for an alternative big bang model that would respect the pioneering concept of expansion, as a direct consequence of the ‘de Sitter effect’ in the modified . It is therefore worth investigating the axiomatic basis of the Robertson-Walker (RW) line-element."

"As Daniel Dennefink explains in his article on the story behind the writings of the Einstein-Rosen gravitational paper, the singularity that Einstein and Rosen had encountered was an apparent singularity introduced by their choice of coordinate system, similar to the singularity one encounters when attempting to find the longitude of the North Pole. In fact, in his referee report Robertson had indicated that the singularity was removed by a change to a cylindrical coordinate system."

"It was the work of... Friedmann, Robertson and Walker, which resulted in the general mathematical framework that is still used today when discussing relativistic cosmological models of a homogeneous and isotropic universe."