Steven Smale - Collected Quotes

"A regular curve on a Riemannian manifold is a curve with a continuously turning nontrivial tangent vector. A regular homotopy is a homotopy which at every stage is a regular curve, keeps end points and directions fixed and such that the tangent vector moves continuously with the homotopy. A regular curve is closed if its initial point and tangent  coincides with its end point and tangent." (Steven Smale,"Regular Curves on Riemann Manifolds", 1956)

"The stable manifolds of the critical points of a nice function can be thought of as the cells of a complex while the unstable manifolds are the dual cells. This structure has the advantage over previous structures that both the cells and the duals are differentiably imbedded in M. We believe that nice functions will replace much of the use of С triangulations and combinatorial methods in differential topology." (Steven Smale, "The generalized Poincare conjecture in higher dimensions", Bull. Amer. Math. Soc. 66, 1960)

"Certainly, the problems of combinatorial manifolds and the relationships between combinatorial and differentiable manifolds are legitimate problems in their own right. An example is the questionof existence and uniqueness of differentiable structures on a combinatorial manifold. However, we don't believe such problems are the goal of differential topology itself. This view seems justified by the fact that today one can substantially develop differential topology most simply without any reference to the combinatorial manifolds." (Steven Smale, "A survey of some recent developments in differential topology", 1961)

"It has turned out that the main theorems in differential topology did not depend on developments in combinatorial topology. In fact, the contrary is the case; the main theorems in differential topology inspired corresponding ones in combinatorial topology, or else have no combinatorial counterpart as yet (but there are also combinatorial theorem whose differentiable analogues are false)."  (Steven Smale, "A survey of some recent developments in differential topology", 1961)

"[...] it is clear that differential geometry, analysis and physics prompted the early development of differential topology (it is this that explains our admitted bias toward differential topology, that it lies close to the main stream of  mathematics). On the other hand, the combinatorial approach to manifolds was started because it was believed that these means would afford a useful attack on the differentiable case." (Steven Smale, "A survey of some recent developments in differential topology", 1961)

"Of course, there are a number of important problems left in differential topology that do not reduce in any sense to homotopy theory and topologists can never rest until these are settled. But, on the other hand, it seems that differential topology has reached such a satisfactory stage that, for it to continue its exciting pace, it must look toward the problems of analysis, the sources that led Poincare to its early development." (Steven Smale, "A survey of some recent developments in differential topology", 1961)

"We consider differential topology to be the study of differentiable manifolds and differentiable maps. Then, naturally, manifolds are considered equivalent if they are diffeomorphic, i.e., there exists a differentiable map from one to the other with a differentiable inverse." (Steven Smale, "A survey of some recent developments in differential topology", 1961)

"My definition of global analysis is simply the study of differential equations, both ordinary and partial, on manifolds and vector space bundles. Thus one might consider global analysis as differential equations from a global, or topological point of view." (Steven Smale, "What is global analysis?", American Mathematical Monthly Vol. 76 (1), 1969)

"To abstract the qualitative features of a differential equation on M, the concept of a phase portrait become important. Usually the phase portrait means the picture of the solution curves of the differential equation. [...] Then two differential equations on M have the same phase portrait if they are topologically equivalent. A definition of phase portrait is thus a topological  equivalence class of differential equations on M. A main goal of the qualitative study of ordinary differential equations is to obtain information on the phase portrait of differential equations." (Steven Smale, "What is global analysis?", American Mathematical Monthly Vol. 76 (1), 1969)

"A fairly general procedure for mathematical study of a physical system with explication of the space of states of that system. Now this space of states could reasonably be one of a number of mathematical objects. However, in my mind, a principal candidate for the state space should be a differentiable manifold; and in case the has a finite number of degrees of freedom, then this will be a finite dimensional manifold. Usually associated with physical is the notion of how a state progresses in time. The corresponding object is a dynamical system or a first order ordinary differential equation on the manifold of states." (Stephen Smale, "Personal perspectives on mathematics and mechanics", 1971)

"Related is the idea of structural stability and certain variations. This kind of is a property of a dynamical system itself (not of a or orbit) and asserts that nearby dynamical systems have the same structure. The 'same structure' can be defined in several interesting ways, but the basic idea is that two dynamical systems have 'the same structure' if they have the same gross behavior, or the same qualitative behavior. For example, the original definition of 'same structure' of two dynamical systems was that there was an orbit preserving continuous transformation between them. This yields the definition of structural stability proper. It is a recent theorem that every compact manifold admits structurally stable systems, and almost all gradient dynamical systems are structurally stable. But while there exists a rich set of structurally stable systems, there are also important examples which are not stable, and have good but weaker stability properties." (Stephen Smale, "Personal perspectives on mathematics and mechanics", 1971)

"A criticism commonly made of economic theory is its failure to make predictions of crises in the country or to anticipate correctly unemployment or inflation. One must be cautious in the social sciences about looking toward physics for answers. However, some comparisons with the physical sciences seem profitable in connection with the above criticism. In those sciences, where theory itself is in a far more advanced state, limitations can be seen un a similar way." (Steven Smale, "Dynamics in General Equilibrium Theory", The American Economic Review Vol. 66 (2), 1976) 

"After all of this is said, equilibrium theory will eventually stand or fall, depending on its truth as an important idealization of actual economic systems or as a model with values of justice, of efficient distribution and of utilization of resources." (Steven Smale, "Dynamics in General Equilibrium Theory", The American Economic Review Vol. 66 (2), 1976) 

"In fact even the very best chess players don't analyze very many moves and certainly don't make future commitments. Their experience together with the environment at the moment (the position), some rules of thumb and some other considerations lead to decisions on the playing board." (Steven Smale, "Dynamics in General Equilibrium Theory", The American Economic Review Vol. 66 (2), 1976) 

"What is special about 'general equilibrium theory' as opposed to economic theory in general? For me the importance of general equilibrium theory lies in its traditions which are deeper than any other part of economic theory. These traditions, which of course derive from actual economic history, explain why equilibrium theory has played such a central role and possesses such depth in content. Many of the procedures and  mathematical methods used in other parts of economics grew out of developments in general equilibrium theory. Also  equilibrium theory plays an important role in communication within the economics profession." (Steven Smale, "Dynamics in General Equilibrium Theory", The American Economic Review Vol. 66 (2), 1976) 

"A diffeomorphism is a differentiable function between manifolds with a differentiable inverse. In particular Φ is one-to-one and onto; thus for example an allocation has a unique supporting price system, and this association is smooth over all optimal allocations." ((Steven Smale, "Global Analysis and Economics VI: Geometrical analysis of Pareto Optima and price equilibria under classical hypotheses", 1974)

"Dynamical systems have two kinds of classical attractors which persist under small perturbations of the differential equations. These are the stable equilibria and the stable nontrivial periodic solutions or oscillators. An important development of recent times is a new kind of attractor which is robust in the sense that its properties persist under perturbations of the differential equation (it is structurally stable). These new attractors are sometimes called strange attractors." (Steven Smale, "On the Problem of Reviving the Ergodic Hypothesis of Boltzmann and Birkhoff", [in "The Mathematics of Time"] 1980)

"Let me start by posing what I tike to call 'the fundamental problem of equilibrium theory': how is economic equilibrium attained? A dual question more commonly raised is: why is economic equilibrium stable? Behind these questions lie the problem of modeling economic processes and introducing dynamics into equilibrium theory. A successful attack here would give greater validity to equilibrium theory. It may be however that a resolution of this fundamental problem will require a recasting of the foundations of equilibrium theory." (Steven Smale, "Some Dynamics Questions in Mathematical Economics", 1980)

"Chaos is a characteristic of dynamics, that is, of time  evolution of a set of states of nature. Let me take time to be measured in discrete units. A state of nature will be idealized as a point in the two-dimensional plane." (Steven Smale, "What is chaos?", 1990)

"Here is the result of the horseshoe analysis that I found on that Copacabana beach. Consider all the points which, under the horseshoe mapping, stay in the square, i.e., don't drift out of our field of vision. These 'visual motions' correspond precisely to the set of all coin-flipping experiments! This discovery demonstrates the occurrence of unpredictability in fully non-linear motion and gives a mechanism of how determinism produces uncertainty."(Steven Smale, "What is chaos?", 1990)

"Instead of a state of nature evolving according to a mathematical fomula, the evolution is given geometrically. The full advantage of the geometrical point of view is beginning to appear. The more traditional way of dealing with dynamics was with the use of algebraic expressions. But a description given by formulae would be cumbersome. That form of description wouldn't have led me to insights or to perceptive analysis. My background as a topologist, trained to bend objects like squares, helped to make it possible to see the horseshoe." (Steven Smale, "What is chaos?", 1990)

"It is astounding how important scientific ideas can get lost, even when they are aired by leading mathematicians of the preceding decades." (Steven Smale, "What is chaos?", 1990)

"Let us tum to the question, 'What is chaos?' Certainly the typical answer, 'sensitive dependence on initial conditions', is reasonable. But to be relevant to physics, more is required: the property should be robust or preserved under perturbations of the system. A response &lat involves a mechanism and meets the criteria is: 'Chaos is the presence of a (transversal) homoclinic point'." (Steven Smale, "What is chaos?", 1990)

"Summarizing, we have indicated why homoclinic points imply chaos. The converse is less formalizable, but I believe it is basically true. However,  there is a stronger notion of chaos, which we might call full chaos, where the set of initial conditions with sensitive dependence has positive or even full measure. In this case the corresponding homoclinic point could well be associated with a chaotic attractor, such as the Lorenz attractor. " (Steven Smale, "What is chaos?", 1990)

"The horseshoe is a natural consequence of a geometrical way of looking at the equations of Cartwright-Littlewood and Levinson. It helps understand the mechanism of chaos, and explain the widespread unpredictability in dynamics." (Steven Smale, "What is chaos?", 1990)

"The laws of chance, with good reason, have traditionally been expressed in terms of flipping a coin. Guessing whether heads or tails is the outcome of a coin toss is the paradigm of pure chance. On the other hand it is a deterministic process that governs the whole motion of a real coin, and hence the result, heads or tails, depends only on very subtle factors of the initiation of the toss. This is 'sensitive dependence on initial conditions'." (Steven Smale, "What is chaos?", 1990)

"There are at least three (overlapping) ways that mathematics may contribute to science. The first, and perhaps the most important, is this: Since the mathematical universe of the mathematician is much larger than that of the physicist, mathematicians are able to go beyond existing frameworks and see geometrical or analytic structures unavailable to tie physicist. Instead of using the particular equations used previously to describe reality, a mathematician has at his disposal an unused world of differential equations, to be studied with no a priori constraints. New scientific phenomena, new discoveries, may thus generated. Understanding of the present knowledge may be deepened via the corresponding deductions. [...] The second way [...] has to do with the consolidation of new physical ideas. One may express this as the proof of consistency of physical theories. [...] mathematical foundations of quantum mechanics with Hilbert space, its operator theory, and corresponding differential equations. [...] The third way [...] is by describing reality in mathematical terms, or by simply constructing a mathematical model." (Steven Smale, "What is chaos?", 1990)

"This construction, the horseshoe, has some consequences. First, it yields the fact that homoclinic points do exist and gives a direct construction of them. Second, one obtains such a useful analysis of a general transversal homoclinic point that many properties follow, including sensitive dependence on initial conditions - 'a large number', anyway. Third, one can prove robustness of the horseshoe in a strong global sense structural stability)." (Steven Smale, "What is chaos?", 1990)

"At least in my own case, understanding mathematics doesn't come from reading or even listening. It comes from rethinking what I see or hear. I must redo the mathematics in the context of my particular background. And that background consists of many threads, some strong, some weak, some algebraic, some visual. My background is stronger in geometric analysis, but following a sequence of formulae gives me trouble. I tend to be slower than most mathematicians to understand an argument. The mathematical literature is useful in that it provides clues, and one can often use these clues to put together a cogent picture. When I have reorganized the mathematics in my own terms, then I feel an understanding, not before." (Stephen Smale, "Finding a Horseshoe on the Beaches of Rio", 2000)

"It is more a philosophy than mathematics, and even as a philosophy it doesn't explain the real world [...] as mathematics, it brings together two of the most basic ideas in modern math: the study of dynamic systems and the study of the singularities of maps. Together, they cover a very wide area - but catastrophe theory brings them together in an arbitrary and constrained way." (Stephen Smale)