Quotes & Sayings About Mathematical Analysis

Mathematical Analysis is as extensive as nature herself. — Joseph Fourier

Mathematical analysis and computer modelling are revealing to us that the shapes and processes we encounter in nature -the way that plants grow, the way that mountains erode or rivers flow, the way that snowflakes or islands achieve their shapes, the way that light plays on a surface, the way the milk folds and spins into your coffee as you stir it, the way that laughter sweeps through a crowd of people - all these things in their seemingly magical complexity can be described by the interaction of mathematical processes that are, if anything, even more magical in their simplicity. Shapes that we think of as random are in fact the products of complex shifting webs of numbers obeying simple rules. The very word "natural" that we have often taken to mean "unstructured" in fact describes shapes and processes that appear so unfathomably complex that we cannot consciously perceive the simple natural laws at work.They can all be described by numbers. — Douglas Adams

Now, the velocity of wave propagation can be seen, without the aid of any mathematical analysis, to depend on the elasticity of the medium and its density; for we can see that if a medium is highly elastic the disturbance would be propagated at a great speed. — Albert A. Michelson

Mathematical Analysis is ... the true rational basis of the whole system of our positive knowledge. — Auguste Comte

Whether game theory leads to clear-cut solutions, to vague solutions, or to impasses, it does achieve one thing. In bringing techniques of logical and mathematical analysis gives men an opportunity to bring conflicts up from the level of fights, where the intellect is beclouded by passions, to the level of games, where the intellect has a chance to operate. — Anatol Rapoport

This harmony that human intelligence believes it discovers in nature - does it exist apart from that intelligence? No, without doubt, a reality completely independent of the spirit which conceives it, sees it or feels it, is an impossibility. A world so exterior as that, even if it existed, would be forever inaccessible to us. But what we call objective reality is, in the last analysis, that which is common to several thinking beings, and could be common to all; this common part, we will see, can be nothing but the harmony expressed by mathematical laws. — Henri Poincare

The new mathematics is a sort of supplement to language, affording a means of thought about form and quantity and a means of expression,more exact,compact, and ready than ordinary language. The great body of physical science, a great deal of the essential facts of financial science, and endless social and political problems are only accessible and thinkable to those who have had a sound training in mathematical analysis, and the time may not be very remote when it will be understood that for complete initiation as an efficient citizen of one of the new great complex world wide states that are now developing, it is as necessary to be able to compute, to think in averages and maxima and minima, as it is now to be able to read and write. — H.G.Wells

So is not mathematical analysis then not just a vain game of the mind? To the physicist it can only give a convenient language; but isn't that a mediocre service, which after all we could have done without; and, it is not even to be feared that this artificial language be a veil, interposed between reality and the physicist's eye? Far from that, without this language most of the initimate analogies of things would forever have remained unknown to us; and we would never have had knowledge of the internal harmony of the world, which is, as we shall see, the only true objective reality. — Henri Poincare

The effects of heat are subject to constant laws which cannot be discovered without the aid of mathematical analysis. The object of the theory is to demonstrate these laws; it reduces all physical researches on the propagation of heat, to problems of the integral calculus, whose elements are given by experiment. No subject has more extensive relations with the progress of industry and the natural sciences; for the action of heat is always present, it influences the processes of the arts, and occurs in all the phenomena of the universe. — Joseph Fourier

In studying the action of the Analytical Engine, we find that the peculiar and independent nature of the considerations which in all mathematical analysis belong to operations, as distinguished from the objects operated upon and from the results of the operations performed upon those objects, is very strikingly defined and separated. — Ada Lovelace

The deep study of nature is the most fruitful source of mathematical discoveries. By offering to research a definite end, this study has the advantage of excluding vague questions and useless calculations; besides it is a sure means of forming analysis itself and of discovering the elements which it most concerns us to know, and which natural science ought always to conserve. — Joseph Fourier

A major triumph of mathematical imagination: the use of visual imagery to condense a large quantity of information into a single comprehensible picture ... Mathematicians are just beginning to understand these basic building blocks of change and to analyze how they combine. The methodology involved has a very different spirit from traditional modeling with differential equations: it is more like chemistry than calculus, requiring careful counterpoint between analysis and synthesis. — Ian Stewart

Analysis does not owe its really significant successes of the last century to any mysterious use of sqrt(-1), but to the quite natural circumstances that one has infinitely more freedom of mathematical movement if he lets quantities vary in a plane instead of only on a line. — Leopold Kronecker

Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science. — Richard Courant

I presume that few who have paid any attention to the history of the Mathematical Analysis, will doubt that it has been developed in a certain order, or that that order has been, to a great extent, necessary
being determined, either by steps of logical deduction, or by the successive introduction of new ideas and conceptions, when the time for their evolution had arrived. — George Boole

In engineering, as in other creative arts, we must learn to do analysis to support our efforts in synthesis. One cannot build a beautiful and functional bridge without a knowledge of steel and dirt, and a considerable mathematical technique for using this knowledge to compute the properties of structures. Similarly, one cannot build a beautiful computer system without a deep understanding of how to "previsualize" the process generated by the code one writes. — Gerald Jay Sussman

I was appalled to find that the mathematical notation on which I had been raised failed to fill the needs of the courses I was assigned, and I began work on extensions to notation that might serve. In particular, I adopted the matrix algebra used in my thesis work, the systematic use of matrices and higher-dimensional arrays (almost) learned in a course in Tensor Analysis rashly taken in my third year at Queen's, and (eventually) the notion of Operators in the sense introduced by Heaviside in his treatment of Maxwell's equations. — Kenneth E. Iverson

The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking. — John Von Neumann

The proof given by Wright, that non-adaptive differentiation will occur in small populations owing to "drift," or the chance fixation of some new mutation or recombination, is one of the most important results of mathematical analysis applied to the facts of neo-mendelism. It gives accident as well as adaptation a place in evolution, and at one stroke explains many facts which puzzled earlier selectionists, notably the much greater degree of divergence shown by island than mainland forms, by forms in isolated lakes than in continuous river-systems. — Julian Huxley

Probit analysis provides a mathematical foundation for the doctrine first established by the sixteenth-century physician Paracelsus: "Only the dose makes a thing not a poison." Under the Paracelsus doctrine, all things are potential poisons if given in a high enough dose, and all things are nonpoisonous if given in a low enough dose. To this doctrine, Bliss added the uncertainty associated with individual results. One reason why many foolish users of street drugs die or become very sick on cocaine or heroin or speed is that they see others using the drugs without being killed. They are like Bliss's insects. They look around and see some of their fellow insects still alive. However, knowing that some individuals are still living provides no assurance that a given individual will survive. There is no way of predicting the response of a single individual. — David Salsburg

In the eighteenth century, philosophers considered the whole of human knowledge, including science, to be their field and discussed questions such as: Did the universe have a beginning? However, in the nineteenth and twentieth centuries, science became too technical and mathematical for the philosophers, or anyone else except a few specialists. Philosophers reduced the scope of their inquiries so much that Wittgenstein, the most famous philosopher of this century, said, "The sole remaining task for philosophy is the analysis of language." What a comedown from the great tradition of philosophy from Aristotle to Kant! — Stephen Hawking

It is interesting thus to follow the intellectual truths of analysis in the phenomena of nature. This correspondence, of which the system of the world will offer us numerous examples, makes one of the greatest charms attached to mathematical speculations. — Pierre-Simon Laplace

Euclid ... manages to obtain a rigorous proof without ever dealing with infinity, by reducing the problem [of the infinitude of primes] to the study of finite numbers. This is exactly what contemporary mathematical analysis does. — Lucio Russo

It were much to be desired, that when mathematical processes pass through the human brain instead of through the medium of inanimate mechanism, it were equally a necessity of things that the reasonings connected with operations should hold the same just place as a clear and well-defined branch of the subject of analysis, a fundamental but yet independent ingredient in the science, which they must do in studying the engine. — Ada Lovelace

The further we analyse the manner in which such an engine performs its processes and attains its results, the more we perceive how distinctly it places in a true and just light the mutual relations and connexion of the various steps of mathematical analysis; how clearly it separates those things which are in reality distinct and independent, and unites those which are mutually dependent. — Ada Lovelace

Every attempt to employ mathematical methods in the study of chemical questions must be considered profoundly irrational and contrary to the spirit of chemistry ... if mathematical analysis should ever hold a prominent place in chemistry
an aberration which is happily almost impossible
it would occasion a rapid and widespread degeneration of that science. — Auguste Comte

Thus we can get the correct answer for the probability of partial reflection by imagining (falsely) that all reflection comes from only the front and back surfaces. In this intuitively easy analysis, the 'front surface' and 'back surface' arrows are mathematical constructions that give us the right answer, whereas ... a more accurate representation of what is really going on: partial reflection is the scattering of light by electrons inside the glass. — Richard P. Feynman

Now, in the development of our knowledge of the workings of Nature out of the tremendously complex assemblage of phenomena presented to the scientific inquirer, mathematics plays in some respects a very limited, in others a very important part. As regards the limitations, it is merely necessary to refer to the sciences connected with living matter, and to the ologies generally, to see that the facts and their connections are too indistinctly known to render mathematical analysis practicable, to say nothing of the complexity. — Oliver Heaviside

The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists of the analysis of Symbolic Logic itself. — Bertrand Russell

The analysis of variance is not a mathematical theorem, but rather a convenient method of arranging the arithmetic. — Ronald Fisher

In mathematical analysis we call x the undetermined part of line a: the rest we don't call y, as we do in common life, but a-x. Hence mathematical language has great advantages over the common language. — Georg C. Lichtenberg

The mathematical difficulties of the theory of rotation arise chiefly from the want of geometrical illustrations and sensible images, by which we might fix the results of analysis in our minds. — James Clerk Maxwell