Calculus Quotes - page 3
Images and pictures... have played a key role in shaping our scientific picture of the world. ...Carefully constructed families of pictures can act as a calculus all their own. Like any successful systems of symbols, with an appropriate grammar they enlarge the number of things that we can do without consciously thinking.
John D. Barrow
James Buchanan ... produced several seminal contributions to the literature prior to the appearance of The Calculus, and a continual stream of books and articles afterward. I view Buchanan's Nobel Prize as a reward for this great body of work, and not just limited to The Calculus. Pursuing this line of reasoning, Gordon Tullock's prodigious research output following the publication of The Calculus also justifies a Nobel Prize. Indeed, Tullock's (1967c) seminal analysis of rent seeking, and his subsequent work on this topic might be regarded as sufficient achievements for the awarding of a prize.
Dennis Mueller
When looking at the two streams of research these men produced, it is tempting to characterize Buchanan as a somewhat romantic optimist holding the belief that the pursuits of selfish individuals can be channeled for the common good, if the right set of constitutional constraints are adopted, and believing that individuals are capable of designing constitutions that achieve this goal. Thus, some 35 years after the publication of The Calculus, we find Buchanan, in joint work with Roger Congleton (1998), accepting the fact that legislatures always will adopt the simple majority rule, and trying to design constitutional constraints that will lead to outcomes benefiting all members of the community under that rule.
Dennis Mueller
The minimum principle that unified the knowledge of light, gravitation, and electricity of Hamilton's time no longer suffices to relate these fundamental branches of physics. Within fifty years of its creation, the belief that Hamilton's principle would outlive all other physical laws of physics was shattered. Minimum principles have since been created for separate branches of physics... but these are not only restricted... but seem to be contrived...
A single minimum principle, a universal law governing all processes in nature, is still the direction in which the search for simplicity is headed, with the price of simplicity now raised from a mastery of differential equations to a mastery of the calculus of variations.
Morris Kline
Brook Taylor... in his Methodus Incrementorum Directa et Inversa (1715), sought to clarify the ideas of the calculus but limited himself to algebraic functions and algebraic differential equations. ...Taylor's exposition, based on what we would call finite differences, failed to obtain many backers because it was arithmetical in nature when the British were trying to tie the calculus to geometry or to the physical notion of velocity.
Morris Kline
As is known, scientific physics dates its existence from the discovery of the differential calculus. Only when it was learned how to follow continuously the course of natural events, attempts, to construct by means of abstract conceptions the connection between phenomena, met with success. To do this two things are necessary: First, simple fundamental concepts with which to construct; second, some method by which to deduce, from the simple fundamental laws of the construction which relate to instants of time and points in space, laws for finite intervals and distances, which alone are accessible to observation.
Bernhard Riemann
With demands for special education or standardized test prep being shouted in their ears, public schools can't always hear a parent when he says: 'I want my child to be able to write contracts in Spanish,' or, 'I want my child to shake hands firmly,' or, 'I want my child to study statistics and accounting, not calculus.
Amity Shlaes
The statement is so frequently made that the differential calculus deals with continuous magnitude, and yet an explanation of this continuity is nowhere given; even the most rigorous expositions of the differential calculus do not base their proofs upon continuity but, with more or less consciousness of the fact, they either appeal to geometric notions or those suggested by geometry, or depend upon theorems which are never established in a purely arithmetic manner. Among these, for example, belongs the above mentioned theorem, and a more careful investigation convinced me that this theorem, or any one equivalent to it, can be regarded in some way as a sufficient basis for infinitesimal analysis. It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity. I succeeded Nov. 24, 1858.
Richard Dedekind
A nonlinear differential equation of the Riccati type is derived for the covariance matrix of the optimal filtering error. The solution of this 'variance equation' completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or non-stationary statistics.
The variance equation is closely related to the Hamiltonian (canonical) differential equations of the calculus of variations. Analytic solutions are available in some cases. The significance of the variance equation is illustrated by examples which duplicate, simplify, or extend earlier results in this field.
The duality principle relating stochastic estimation and deterministic control problems plays an important role in the proof of theoretical results. In several examples, the estimation problem and its dual are discussed side-by-side.
Properties of the variance equation are of great interest in the theory of adaptive systems. Some aspects of this are considered briefly.
Rudolf E. Kálmán
To make psychology into experimental epistemology is to attempt to understand the embodiment of mind. Here we are confronted by what seem to be three questions, although they may ultimately be only one. The three exist as categorically disperate desiderata.
The first is at the logical level: We lack an adequate, appropriate calculus for triadic relations.
The second is the psychological level: We do not know how we generate hypotheses that are natural and simple.
The third is that the physiological level: we have no circuit theory for the reticular formation that marshals our abductions.
Logically, the problem is far from simple. To be exact, no proposed theory of relations yields a calculus to handle our problem. When I was growing up, only the Aristotelian logic of classes was ever taught, and that badly. The Organon itself contains only a clumsy description of the apagoge - perhaps from the notes of some students who have not understand his master...
Warren S. McCulloch
That was the danger of being old and a politician. Habits outlived the situations that created them. Policies remained in place after the situations that inspired them had changed. The calculus of all human power was changing, and the models she used to make sense of it shifted with them, and she had to keep reminding herself that the past was a different place. She didn't live there anymore.
Daniel Abraham
Our First Amendment expresses a far different calculus for regulating speech than for regulating nonexpressive conduct and that is as it should be. The right to swing your fist should end at the tip of my nose, but your right to express your ideas should not necessarily end at the lobes of my ears."
Alan Dershowitz