Top 1200 Mathematical Analysis Quotes & Sayings - Page 2

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.

I think that if you don't do the full analysis of what the origin of the electrical power is, where it comes from, how you get batteries into these cars, what the cost is in terms of CO2 and the environment, I think the analysis that we are going to save the planet with electric cars is nonsense.

This common and unfortunate fact of the lack of adequate presentation of basic ideas and motivations of almost any mathematical theory is probably due to the binary nature of mathematical perception. Either you have no inkling of an idea, or, once you have understood it, the very idea appears so embarrassingly obvious that you feel reluctant to say it aloud.

We feel certain that the extraterrestrial message is a mathematical code of some kind. Probably a number code. Mathematics is the one language we might conceivably have in common with other forms of intelligent life in the universe. As I understand it, there is no reality more independent of our perception and more true to itself than mathematical reality.

The division of the psychical into what is conscious and what is unconscious is the fundamental premise of psycho-analysis; and it alone makes it possible for psycho-analysis to understand the pathological processes in mental life, which are as common as they are important, and to find a place for them in the framework of science.

A great advantage of X-ray analysis as a method of chemical structure analysis is its power to show some totally unexpected and surprising structure with, at the same time, complete certainty.

The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work-that is, correctly to describe phenomena from a reasonably wide area.

The classes of problems which are respectively known and not known to have good algorithms are of great theoretical interest. [...] I conjecture that there is no good algorithm for the traveling salesman problem. My reasons are the same as for any mathematical conjecture: (1) It is a legitimate mathematical possibility, and (2) I do not know.

Here is a quilted book about mathematical practice, each patch wonderfully prepared. Part invitation to number theory, part autobiography, part sociology of mathematical training, Mathematics without Apologies brings us into contemporary mathematics as a living, active inquiry by real people. Anyone wanting a varied, cultured, and penetrating view of today's mathematics could find no better place to engage.

Philosophy, like science, consists of theories or insights arrived at as a result of systemic reflection or reasoning in regard to the data of experience. It involves, therefore, the analysis of experience and the synthesis of the results of analysis into a comprehensive or unitary conception. Philosophy seeks a totality and harmony of reasoned insight into the nature and meaning of all the principal aspects of reality.

I try to keep in mind Oscar Wilde's comment that "saints always have a past and sinners always have a future," so no investment should be ruled out simply on the basis of past history. We focus on liquidation analysis and liquidation analysis alone.

Mathematical analysis is as extensive as nature itself; it defines all perceptible relations, measures times, spaces, forces, temperatures:;; this difficult science is formed slowly, but it preserves every principle which it has once acquired; it grows and strengthens itself incessantly in the midst of the many variations and errors of the human mind. It's chief attribute is clearness; it has no marks to express confused notations. It brings together phenomena the most diverse, and discovers the hidden analogies which unite them.

Buddhist epistemologists do argue that rational analysis leads to the conclusion that rational analysis cannot give us infallible access to truth, including that one. That's not self-defeating, though; it only induces an important kind of epistemic humility and a clearer view of what we do when we reason. We engage in one more fallible human activity among many.

Kant, discussing the various modes of perception by which the human mind apprehends nature, concluded that it is specially prone to see nature through mathematical spectacles. Just as a man wearing blue spectacles would see only a blue world, so Kant thought that, with our mental bias, we tend to see only a mathematical world.

The spider-mind acquires a faculty of memory, and, with it, a singular skill of analysis and synthesis, taking apart and putting together in different relations the meshes of its trap. Man had in the beginning no power of analysis or synthesis approaching that of the spider, or even of the honey-bee; but he had acute sensibility to the higher forces.

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.

It seems perfectly clear that Economy, if it is to be a science at all, must be a mathematical science. There exists much prejudice against attempts to introduce the methods and language of mathematics into any branch of the moral sciences. Most persons appear to hold that the physical sciences form the proper sphere of mathematical method, and that the moral sciences demand some other method-I know not what.

Paraphrase, in the sense of summary, is as indispensable to the novel-critic as close analysis is to the critic of lyric poetry. The natural deduction is that novels are paraphrasable whereas poems are not. But this is a false deduction because close analysis is itself a disguised form of paraphrase.

In fact, the science of thermodynamics began with an analysis, by the great engineer Sadi Carnot, of the problem of how to build the best and most efficient engine, and this constitutes one of the few famous cases in which engineering has contributed to fundamental physical theory. Another example that comes to mind is the more recent analysis of information theory by Claude Shannon. These two analyses, incidentally, turn out to be closely related.

That is, while we believe that cost-benefit analysis is an important tool to inform agency decision making, the results of the cost-benefit analysis do not trump existing law.

The sad truth is that it is precisely those who disagree most with the hypothesis of efficient market pricing of stocks, those who pooh-pooh beta analysis and all that, who are least able to understand the analysis needed to test that hypothesis.

The Reader may here observe the Force of Numbers, which can be successfully applied, even to those things, which one would imagine are subject to no Rules. There are very few things which we know, which are not capable of being reduc'd to a Mathematical Reasoning, and when they cannot, it's a sign our Knowledge of them is very small and confus'd; and where a mathematical reasoning can be had, it's as great folly to make use of any other, as to grope for a thing in the dark when you have a Candle standing by you.

This [Thelonious Monk: The Life And Times Of An American Original] is another one of those books with the perfect blend of anecdote and analysis. The analysis is built into the anecdote. It has that right feel about it. It's not too scholarly, either.

It can be shown that a mathematical web of some kind can be woven about any universe containing several objects. The fact that our universe lends itself to mathematical treatment is not a fact of any great philosophical significance.

You cannot get involved in debate on 'MOTD'. You can do it on Sky because they've got hours and hours. We've got a couple of minutes. It's a very disciplined show. Our primary purpose is to show the action, and the analysis is very secondary. We have lots of people who would prefer no analysis. We have lots of people who would prefer more analysis.

When a truth is necessary, the reason for it can be found by analysis, that is, by resolving it into simpler ideas and truths until the primary ones are reached. It is this way that in mathematics speculative theorems and practical canons are reduced by analysis to definitions, axioms and postulates.

From a mathematical standpoint it is possible to have infinite space. In a mathematical sense space is manifoldness, or combinations of numbers. Physical space is known as the 3-dimension system. There is the 4-dimension system, the 10-dimension system.

There will be, and should be, reams and reams of analysis, even praise, for our friend but also even larger measures of non-analysis and, certainly, condemnation for James Baldwin, the Negro writer.

Nature seems to take advantage of the simple mathematical representations of the symmetry laws. When one pauses to consider the elegance and the beautiful perfection of the mathematical reasoning involved and contrast it with the complex and far-reaching physical consequences, a deep sense of respect for the power of the symmetry laws never fails to develop.

On a motif such as was indicated by Reti one cannot build the plan of a whole well contested game; it is too meagre, too thin, too puny for such an end. Reti's explanations, wherever they are concerned with an analysis which covers a few moves, are correct and praiseworthy. But when he abandons the foundations of analysis in order to draw too bold, too general a conclusion, his arguments prove to be mistaken.

Cubism had been an analysis of the object and an attempt to put it before us in its totality; both as analysis and as synthesis, it was a criticism of appearance. Surrealism transmuted the object, and suddenly a canvas became an apparition: a new figuration, a real transfiguration.

... each of the 24 modes in the Ramanujan function corresponds to a physical vibration of a string. Whenever the string executes its complex motions in space-time by splitting and recombining, a large number of highly sophisticated mathematical identities must be satisfied. These are precisely the mathematical identities discovered by Ramanujan.

The mathematical question is "Why?" It's always why. And the only way we know how to answer such questions is to come up, from scratch, with these narrative arguments that explain it. So what I want to do with this book is open up this world of mathematical reality, the creatures that we build there, the questions that we ask there, the ways in which we poke and prod (known as problems), and how we can possibly craft these elegant reason-poems.

The great body of physical science, a great deal of the essential fact of financial science, and endless social and political problems are only accessible and only 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 worldwide 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.

[Before the time of Benjamin Peirce it never occurred to anyone that mathematical research] was one of the things for which a mathematical department existed. Today it is a commonplace in all the leading universities. Peirce stood alone-a mountain peak whose absolute height might be hard to measure, but which towered above all the surrounding country.

Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?

The research worker, in his efforts to express the fundamental laws of Nature in mathematical form, should strive mainly for mathematical beauty. He should take simplicity into consideration in a subordinate way to beauty ... It often happens that the requirements of simplicity and beauty are the same, but where they clash, the latter must take precedence.

Not only in geometry, but to a still more astonishing degree in physics, has it become more and more evident that as soon as we have succeeded in unraveling fully the natural laws which govern reality, we find them to be expressible by mathematical relations of surprising simplicity and architectonic perfection. It seems to me to be one of the chief objects of mathematical instruction to develop the faculty of perceiving this simplicity and harmony.

In the last analysis, even the best man is evil: in the last analysis, even the best woman is bad.

An absolute can only be given in an intuition, while all the rest has to do with analysis. We call intuition here the sympathy by which one is transported into the interior of an object in order to coincide with what there is unique andconsequently inexpressible in it. Analysis, on the contrary, is the operation which reduces the object to elements already known.

As in Mathematicks, so in Natural Philosophy, the Investigation of difficult Things by the Method of Analysis, ought ever to precede the Method of Composition. This Analysis consists in making Experiments and Observations, and in drawing general Conclusions from them by Induction, and admitting of no Objections against the Conclusions, but such as are taken from Experiments, or other certain Truths. For Hypotheses are not to be regarded in experimental Philosophy.

Accordingly, we find Euler and D'Alembert devoting their talent and their patience to the establishment of the laws of rotation of the solid bodies. Lagrange has incorporated his own analysis of the problem with his general treatment of mechanics, and since his time M. Poinsôt has brought the subject under the power of a more searching analysis than that of the calculus, in which ideas take the place of symbols, and intelligent propositions supersede equations.

We do liquidation analysis and liquidation analysis only.

Mathematical demonstrations being built upon the impregnable Foundations of Geometry and Arithmetick are the only truths that can sink into the Mind of Man, void of all Uncertainty; and all other Discourses participate more or less of Truth according as their Subjects are more or less capable of Mathematical Demonstration.

... mathematical knowledge ... is, in fact, merely verbal knowledge. "3" means "2+1", and "4" means "3+1". Hence it follows (though the proof is long) that "4" means the same as "2+2". Thus mathematical knowledge ceases to be mysterious.

The virtual suppression of ethical discussion after 1845 produces the semblance of purely descriptive analysis, dressed in the mantle of positivist objectivity, analysis which is, in fact, strung to a framework of crude, because unexplicated, moral assumptions.

String theory has had a long and wonderful history. It originated as a technique to try to understand the strong force. It was a calculational mechanism, a way of approaching a mathematical problem that was too difficult, and it was a promising way, but it was only a technique. It was a mathematical technique rather than a theory in itself.

One effect of benefit-cost analysis is to give any respectable engineer or economist a means for justifying almost any kind of project the national government wants to justify... Exclusive reliance on benefit-cost analysis has been one of the greatest threats to wise decisions in water development.

Conventions of generality and mathematical elegance may be just as much barriers to the attainment and diffusion of knowledge as may contentment with particularity and literary vagueness... It may well be that the slovenly and literary borderland between economics and sociology will be the most fruitful building ground during the years to come and that mathematical economics will remain too flawless in its perfection to be very fruitful.

That is, while we believe that cost-benefit analysis is an important tool to inform agency decision making, the results of the cost-benefit analysis do not trump existing law

What psycho-analysis reveals in the transference phenomena of neurotics can also be observed in the lives of some normal people. The impression they give is of being pursued by a malignant fate or possessed by some 'daemonic' power; but psycho-analysis has always taken the view that their fate is for the most part arranged by themselves and determined by early infantile influences.

We in the FBI have created a malware repository and analysis tool known as the Binary Analysis Characterization and Storage System, or BACSS, which provides near real-time investigative information. BACSS helps us link malware in different jurisdictions and paint a picture of cyber threats worldwide.

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.

For generations, field guides to plants and animals have sharpened the pleasure of seeing by opening our minds to understanding. Now John Adam has filled a gap in that venerable genre with his painstaking but simple mathematical descriptions of familiar, mundane physical phenomena. This is nothing less than a mathematical field guide to inanimate nature.

Although mathematical notation undoubtedly possesses parsing rules, they are rather loose, sometimes contradictory, and seldom clearly stated. [...] The proliferation of programming languages shows no more uniformity than mathematics. Nevertheless, programming languages do bring a different perspective. [...] Because of their application to a broad range of topics, their strict grammar, and their strict interpretation, programming languages can provide new insights into mathematical notation.

What Brad Bushman did is in 2010 he ran what's called a meta-analysis, which is an analysis that looks at a whole bunch of different studies. They concluded that, yes, there is a link between violent video games and aggression.

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.

As a man may be born with a mathematical faculty, and by training that faculty year after year may immensely increase his mathematical capacity, so may a man be born with certain faculties within him, faculties belonging to the soul, which he can develop by training and by discipline.

Respecting beings, places, and life ways would be a basis for a worthy systemic analysis. And such an analysis would be inherently conservative, assuming that technology - from the fire stick to the silicon chip - is apt to do more harm to the Whole than good.

There was a time when events that define the human condition were subject to very limited and time-delayed analysis. In the 21st Century and beyond, however, the human condition is subject to real-time, overwhelming analysis and dissection.