Mathematical Investigation Quotes

No human investigation can be called real knowledge if it does not pass through mathematical demonstrations; and if you say that the kinds of knowledge that begin and end in the mind have any value as truth, this cannot be conceded, but rather must be denied for many reasons, and first of all because in such mental discussions there is no experimentation, without which nothing provides certainty of itself.

One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution.

Universality is the distinguishing mark of genius. There is no such thing as a special genius, a genius for mathematics, or for music, or even for chess, but only a universal genius. … The theory of special genius, according to which for instance, it is supposed that a musical genius should be a fool at other subjects, confuses genius with talent. … There are many kinds of talent, but only one kind of genius, and that is able to choose any kind of talent and master it.

Johannes Kepler described his motivation thus: ‘The chief aim of all investigations of the external world should be to discover the rational order which has been imposed on it by God, and which he revealed to us in the language of mathematics.

A distinguished writer [Siméon Denis Poisson] has thus stated the fundamental definitions of the science: 'The probability of an event is the reason we have to believe that it has taken place, or that it will take place.' 'The measure of the probability of an event is the ratio of the number of cases favourable to that event, to the total number of cases favourable or contrary, and all equally possible' (equally like to happen). From these definitions it follows that the word probability, in its mathematical acceptation, has reference to the state of our knowledge of the circumstances under which an event may happen or fail. With the degree of information which we possess concerning the circumstances of an event, the reason we have to think that it will occur, or, to use a single term, our expectation of it, will vary. Probability is expectation founded upon partial knowledge. A perfect acquaintance with all the circumstances affecting the occurrence of an event would change expectation into certainty, and leave neither room nor demand for a theory of probabilities.

Science proceeds by inference, rather than by the deduction of mathematical proof. A series of observations is accumulated, forcing the deeper question: What must be true if we are to explain what is observed? What "big picture" of reality offers the best fit to what is actually observed in our experience? American scientist and philosopher Charles S. Peirce used the term "abduction" to refer to the way in which scientists generate theories that might offer the best explanation of things. The method is now more often referred to as "inference to the best explanation." It is now widely agreed to be the philosophy of investigation of the world characteristic of the natural sciences.

Q. Would you repeat, Dr. Seldon, your thoughts concerning the future of Trantor? A. I have said, and I say again, that Trantor will lie in ruins within the next three centuries. Q. You do not consider your statement a disloyal one? A. No, sir. Scientific truth is beyond loyalty and disloyalty." Q. You are sure that your statement represents scientific truth? A. I am. Q. On what basis? A. On the basis of the mathematics of psychohistory. Q. Can you prove that this mathematics is valid? A. Only to another mathematician. Q. ( with a smile) Your claim then is that your truth is of so esoteric a nature that it is beyond the understanding of a plain man. It seems to me that truth should be clearer than that, less mysterious, more open to the mind. A. It presents no difficulties to some minds. The physics of energy transfer, which we know as thermodynamics, has been clear and true through all the history of man since the mythical ages, yet there may be people present who would find it impossible to design a power engine. People of high intelligence, too. I doubt if the learned Commissioners— At this point, one of the Commissioners leaned toward the Advocate. His words were not heard but the hissing of the voice carried a certain asperity. The Advocate flushed and interrupted Seldon. Q. We are not here to listen to speeches, Dr. Seldon. Let us assume that you have made your point. Let me suggest to you that your predictions of disaster might be intended to destroy public confidence in the Imperial Government for purposes of your own! A. That is not so. Q. Let me suggest that you intend to claim that a period of time preceding the so-called ruin of Trantor will be filled with unrest of various types. A. That is correct. Q. And that by the mere prediction thereof, you hope to bring it about, and to have then an army of a hundred thousand available. A. In the first place, that is not so. And if it were, investigation will show you that barely ten thousand are men of military age, and none of these has training in arms. Q. Are you acting as an agent for another? A. I am not in the pay of any man, Mr. Advocate. Q. You are entirely disinterested? You are serving science? A. I am.

He will repeat it boldly (for it has been said before him), truths that form the basis of political and moral science are not to be discovered but by investigations as severe as mathematical ones, and beyond all comparison more intricate and extensive.

But what is the use of the humanities as such? Admittedly they are not practical, and admittedly they concern themselves with the past. Why, it may be asked, should we engage in impractical investigations, and why should we be interested in the past? The answer to the first question is: because we are interested in reality. Both the humanities and the natural sciences, as well as mathematics and philosophy, have the impractical outlook of what the ancients called vita contemplativa as opposed to vita activa. But is the contemplative life less real or, to be more precise, is its contribution to what we call reality less important, than that of the active life? The man who takes a paper dollar in exchange for twenty-five apples commits an act of faith, and subjects himself to a theoretical doctrine, as did the mediaeval man who paid for indulgence. The man who is run over by an automobile is run over by mathematics, physics and chemistry. For he who leads the contemplative life cannot help influencing the active, just as he cannot prevent the active life from influencing his thought. Philosophical and psychological theories, historical doctrines and all sorts of speculations and discoveries, have changed, and keep changing, the lives of countless millions. Even he who merely transmits knowledge or learning participates, in his modest way, in the process of shaping reality - of which fact the enemies of humanism are perhaps more keenly aware than its friends. It is impossible to conceive of our world in terms of action alone. Only in God is there a "Coincidence of Act and Thought" as the scholastics put it. Our reality can only be understood as an interpenetration of these two.

the age-structure of human populations is characterized by smoothly overlapping generations, without any clear-cut break points between them. We cannot simply impose generations on human social systems; we need to investigate mathematically whether they will arise naturally as a result of age-structured population and social dynamics.

Ohm found that the results could be summed up in such a simple law that he who runs may read it, and a schoolboy now can predict what a Faraday then could only guess at roughly. By Ohm's discovery a large part of the domain of electricity became annexed by Coulomb's discovery of the law of inverse squares, and completely annexed by Green's investigations. Poisson attacked the difficult problem of induced magnetisation, and his results, though differently expressed, are still the theory, as a most important first approximation. Ampere brought a multitude of phenomena into theory by his investigations of the mechanical forces between conductors supporting currents and magnets. Then there were the remarkable researches of Faraday, the prince of experimentalists, on electrostatics and electrodynamics and the induction of currents. These were rather long in being brought from the crude experimental state to a compact system, expressing the real essence. Unfortunately, in my opinion, Faraday was not a mathematician. It can scarcely be doubted that had he been one, he would have anticipated much later work. He would, for instance, knowing Ampere's theory, by his own results have readily been led to Neumann's theory, and the connected work of Helmholtz and Thomson. But it is perhaps too much to expect a man to be both the prince of experimentalists and a competent mathematician.

The genius of Laplace was a perfect sledge hammer in bursting purely mathematical obstacles; but, like that useful instrument, it gave neither finish nor beauty to the results. In truth, in truism if the reader please, Laplace was neither Lagrange nor Euler, as every student is made to feel. The second is power and symmetry, the third power and simplicity; the first is power without either symmetry or simplicity. But, nevertheless, Laplace never attempted investigation of a subject without leaving upon it the marks of difficulties conquered: sometimes clumsily, sometimes indirectly, always without minuteness of design or arrangement of detail; but still, his end is obtained and the difficulty is conquered.

I advise you to look for a chance to break away, to find a subject you can make your own. That is where the quickest advances are likely to occur, as measured by discoveries per investigator per year. Therein you have the best chance to become a leader and, as time passes, to gain growing freedom to set your own course. If a subject is already receiving a great deal of attention, if it has a glamorous aura, if its practitioners are prizewinners who receive large grants, stay away from that subject. Listen to the news coming from the hubbub, learn how and why the subject became prominent, but in making your own long-term plans be aware it is already crowded with talented people. You would be a newcomer, a private amid bemedaled first sergeants and generals. Take a subject instead that interests you and looks promising, and where established experts are not yet conspicuously competing with one another, where few if any prizes and academy memberships have been given, and where the annals of research are not yet layered with superfluous data and mathematical models.

Richard Charnin is an author and quantitative software developer with advanced degrees in applied mathematics and operations research. He paints a very clear portrait of the JFK witness deaths in the context of the mathematical landscape: I have proved mathematically what many have long suspected: The scores of convenient JFK unnatural witness deaths cannot be coincidental.15

The idea of considering the infinitely large not only in the form of the unlimitedly increasing magnitude and in the closely related form of convergent infinite series...but to also fix it mathematically by numbers in the definite form of the completed infinite was logically forced upon me, almost against my will since it was contrary to traditions which I had come to cherish in the course of many years of scientific effort and investigations.

If, as I suggested before, the ability to tell right from wrong should turn out to have anything to do with the ability to think, then we must be able to “demand” its exercise from every sane person, no matter how erudite or ignorant, intelligent or stupid, he may happen to be. Kant—in this respect almost alone among the philosophers—was much bothered by the common opinion that philosophy is only for the few, precisely because of its moral implications, and he once observed that “stupidity is caused by a wicked heart.”21 This is not true: absence of thought is not stupidity; it can be found in highly intelligent people, and a wicked heart is not its cause; it is probably the other way round, that wickedness may be caused by absence of thought. In any event, the matter can no longer be left to “specialists” as though thinking, like higher mathematics, were the monopoly of a specialized discipline.

In sum, the fruition of 50 years of research, and several hundred million dollars in government funds, has given us the following picture of sub-atomic matter. All matter consists of quarks and leptons, which interact by exchanging different types of quanta, described by the Maxwell and Yang-Mills fields. In one sentence, we have captured the essence of the past century of frustrating investigation into the subatomic realm, From this simple picture one can derive, from pure mathematics alone, all the myriad and baffling properties of matter. (Although it all seems so easy now, Nobel laureate Steven Weinberg, one of the creators of the Standard Model, once reflected on how tortuous the 50-year journey to discover the model had been. He wrote, "There's a long tradition of theoretical physics, which by no means affected everyone but certainly affected me, that said the strong interactions [were] too complicated for the human mind.")

That I might investigate the subject matter of this science with the same freedom of spirit we generally use in mathematics, I have labored carefully not to mock, lament, or execrate human actions, but to understand them; and to this end I have looked upon passions such as love, hatred, anger, envy, ambition, pity, and other perturbations of the mind, not in the light of vices of human nature, but as properties just as pertinent to it as are heat, cold, storm, thunder, and the like to the nature of the atmosphere.

When an official report in the UK was commissioned to examine the mathematics needed in the workplace, the investigator found that estimation was the most useful mathematical activity. Yet when children who have experienced traditional math classes are asked to estimate, they are often completely flummoxed and try to work out exact answers, then round them off to look like an estimate. This is because they have not developed a good feel for numbers, which would allow them to estimate instead of calculate, and also because they have learned, wrongly, that mathematics is all about precision, not about making estimates or guesses. Yet both are at the heart of mathematical problem solving.

(William) Hamilton recast the central ideas (of the evolutionary theory of aging) in mathematical form. Though this work tells us a good deal about why human lives take the course they do, Hamilton was a biologist whose great love was insects and their relatives, especially insects which make both our lives and an octopus’s life seem rather humdrum. Hamilton found mites in which the females hang suspended in the air with their swollen bodies packed with newly hatched young, and the males in the brood search out and copulate with their sisters there inside the mother. He found tiny beetles in which the males produce “and manhandle sperm cells longer than their whole bodies. Hamilton died in 2000, after catching malaria on a trip to Africa to investigate the origins of HIV. About a decade before his death, he wrote about how he would like his own burial to go. He wanted his body carried to the forests of Brazil and laid out to be eaten from the inside by an enormous winged Coprophanaeus beetle using his body to nurture its young, who would emerge from him and fly off. 'No worm for me nor sordid fly, I will buzz in the dusk like a huge bumble bee. I will be many, buzz even as a swarm of motorbikes, be borne, body by flying body out into the Brazilian wilderness beneath the stars, lofted under those beautiful and un-fused elytra [wing covers] which we will all hold over our “backs. So finally I too will shine like a violet ground beetle under a stone.

We must consider also whether soul is divisible or is without parts, and whether it is everywhere homogeneous or not; and if not homogeneous, whether its various forms are different specifically or generically; up to the present time those who have discussed and investigated soul seem to have confined themselves to the human soul. We must be careful not to ignore the question whether soul can be defined in a single account, as is the case with animal, or whether we must not give a separate account of each sort of it, as we do for horse, dog, man, god (in the latter case the universal, animal—and so too every other common predicate—is either nothing or posterior). Further, if what exists is not a plurality of souls, but a plurality of parts of one soul, which ought we to investigate first, the whole soul or its parts? It is also a difficult problem to decide which of these parts are in nature distinct from one another. Again, which ought we to investigate first, these parts or their functions, mind or thinking, the faculty or the act of sensation, and so on? If the investigation of the functions precedes that of the parts, the further question suggests itself: ought we not before either to consider the correlative objects, e.g. of sense or thought? It seems not only useful for the discovery of the causes of the incidental proprieties of substances to be acquainted with the essential nature of those substances (as in mathematics it is useful for the understanding of the property of the equality of the interior angles of a triangle to two right angles to know the essential nature of the straight and the curved or of the line and (the plane) but also conversely, for the knowledge of the essential nature of a substance is largely promoted by an acquaintance with its properties: for, when we are able to give an account conformable to experience of all or most of the properties of a substance, we shall be in the most favourable position to say something worth saying about the essential nature of that subject: in all demonstration a definition of the essence is required as a starting point, so that definitions which do not enable us to discover the incidental properties, or which fail to facilitate even a conjecture about them, must obviously, one and all, be dialectical and futile.

In 1935, three years before his death, Edmund Husserl gave his celebrated lectures in Vienna and Prague on the crisis of European humanity. For Husserl, the adjective "European" meant the spiritual identity that extends beyond geographical Europe (to America, for instance) and that was born with ancient Greek philosophy. In his view, this philosophy, for the first time in History, apprehended the world (the world as a whole) as a question to be answered. It interrogated the world not in order to satisfy this or that practical need but because "the passion to know had seized mankind." The crisis Husserl spoke of seemed to him so profound that he wondered whether Europe was still able to survive it. The roots of the crisis lay for him at the beginning of the Modern Era, in Galileo and Descartes, in the one-sided nature of the European sciences, which reduced the world to a mere object of technical and mathematical investigation and put the concrete world of life, die Lebenswelt as he called it, beyond their horizon. The rise of the sciences propelled man into the tunnels of the specialized disciplines. The more he advanced in knowledge, the less clearly could he see either the world as a whole or his own self, and he plunged further into what Husserl's pupil Heidegger called, in a beautiful and almost magical phrase, "the forgetting of being." Once elevated by Descartes to "master and proprietor of nature," man has now become a mere thing to the forces (of technology, of politics, of history) that bypass him, surpass him, possess him. To those forces, man's concrete being, his "world of life" (die Lebenswelt), has neither value nor interest: it is eclipsed, forgotten from the start.

From a number of such laws he hoped to reach laws of the second degree of generality, and so on. A suggested law should be tested by being applied in new circumstances; if it worked in these circumstances it was to that extent confirmed. Some instances are specially valuable because they enable us to decide between two theories, each possible as far as previous observations are concerned; such instances are called 'prerogative' instances. Bacon not only despised the syllogism, but undervalued mathematics, presumably as insufficiently experimental. He was virulently hostile to Aristotle, but thought very highly of Democritus. Although he did not deny that the course of nature exemplifies a divine purpose, he objected to any admixture of teleological explanation in the actual investigation of phenomena; everything, he held, should be explained as following necessarily from efficient causes.

J.W. Dunne was a distinguished man of science and professor of mathematics. [...] He embarked upon a lifetime study of precognition. In 1927 he published his basic conclusions in his bestselling book An Experiment with Time. [...] He argued that if time was a fourth dimension then the passage of time must itself take time. If therefore time takes time there must be a time outside of time. He called this "time 2". [...] Most of our life we live in "time 1", which is synonymous with the passing ordinary moments of everyday life. But during sleep a part of our personality (observer 2) can slip into this other dimension of time and experience events in the future which are communicated to our ordinary self (observer 1). Investigations led Dunne to conclude that under certain circumstances past, present and future events were accessible to consciousness and that during dreams we can enter this fourth dimension of space-time.

The legendary inscription above the Academy's door speaks loudly about Plato's attitude toward mathematics. In fact, most of the significant mathematical research of the fourth century BC was carried out by people associated in one way or another with the Academy. Yet Plato himself was not a mathematician of great technical dexterity, and his direct contributions to mathematical knowledge were probably minimal. Rather, he was an enthusiastic spectator, a motivating source of challenge, an intelligent critic, an an inspiring guide. The first century philosopher and historian Philodemus paints a clear picture: "At that time great progress was seen in mathematics, with Plato serving as the general architect setting out problems, and the mathematicians investigating them earnestly." To which the Neoplatonic philosopher and mathematician Proclus adds: "Plato...greatly advanced mathematics in general and geometry in particular because of his zeal for these studies. It is well known that his writings are thickly sprinkled with mathematical terms and that he everywhere tries to arouse admiration for mathematics among students of philosophy." In other words, Plato, whose mathematical knowledge was broadly up to date, could converse with the mathematicians as an equal and as a problem presenter, even though his personal mathematical achievements were not significant.

The part played by deduction in science is greater than Bacon supposed. Often, when a hypothesis has to be tested, there is a long deductive journey from the hypothesis to some consequence that can be tested by observation. Usually the deduction is mathematical, and in this respect Bacon underestimated the importance of mathematics in scientific investigation. The problem of induction by simple enumeration remains unsolved to this day. Bacon was quite right in rejecting simple enumeration where the details of scientific investigation are concerned, for in dealing with details we may assume general laws on the basis of which, so long as they are taken as valid, more or less cogent methods can be built up. John Stuart Mill framed four canons of inductive method, which can be usefully employed so long as the law of causality is assumed; but this law itself, he had to confess, is to be accepted solely on the basis of induction by simple enumeration. The thing that is achieved by the theoretical organization of science is the collection of all subordinate inductions into a few that are very comprehensive—perhaps only one. Such comprehensive inductions are confirmed by so many instances that it is thought legitimate to accept, as regards them, an induction by simple enumeration. This situation is profoundly unsatisfactory, but neither Bacon nor any of his successors have found a way out of it.

The concept of absolute time—meaning a time that exists in “reality” and tick-tocks along independent of any observations of it—had been a mainstay of physics ever since Newton had made it a premise of his Principia 216 years earlier. The same was true for absolute space and distance. “Absolute, true, and mathematical time, of itself and from its own nature, flows equably without relation to anything external,” he famously wrote in Book 1 of the Principia. “Absolute space, in its own nature, without relation to anything external, remains always similar and immovable.” But even Newton seemed discomforted by the fact that these concepts could not be directly observed. “Absolute time is not an object of perception,” he admitted. He resorted to relying on the presence of God to get him out of the dilemma. “The Deity endures forever and is everywhere present, and by existing always and everywhere, He constitutes duration and space.”45 Ernst Mach, whose books had influenced Einstein and his fellow members of the Olympia Academy, lambasted Newton’s notion of absolute time as a “useless metaphysical concept” that “cannot be produced in experience.” Newton, he charged, “acted contrary to his expressed intention only to investigate actual facts.”46 Henri Poincaré also pointed out the weakness of Newton’s concept of absolute time in his book Science and Hypothesis, another favorite of the Olympia Academy. “Not only do we have no direct intuition of the equality of two times, we do not even have one of the simultaneity of two events occurring in different places,” he wrote.

Philosophy is different from science and from mathematics. Unlike science it doesn't rely on experiments or observation, but only on thought. And unlike mathematics it has no formal methods of proof. It is done just by asking questions, arguing, trying out ideas and thinking of possible arguments against them, and wondering how our concepts really work. The main concern of philosophy is to question and understand common ideas that all of us use every day without thinking about them. A historian may ask what happened at some time in the past, but a philosopher will ask, "What is time?" A mathematician may investigate the relations among numbers, but a philosopher will ask, "What is a number?" A physicist will ask what atoms are made of or what explains gravity, but a philosopher will ask how we can know there is anything outside of our own minds. A psychologist may investigate how children learn a language, but a philosopher will ask, "What makes a word mean anything?" Anyone can ask whether it's wrong to sneak into a movie without paying, but a philosopher will ask, "What makes an action right or wrong?" We couldn't get along in life without taking the ideas of time, number, knowledge, language, right and wrong for granted most of the time; but in philosophy we investigate those things themselves. The aim is to push our understanding of the world and ourselves a bit deeper. Obviously, it isn't easy. The more basic the ideas you are trying to investigate, the fewer tools you have to work with. There isn't much you can assume or take for granted. So philosophy is a somewhat dizzying activity, and few of its results go unchallenged for long.

Those who are condemned to see the world in shadows will be forced to formulate a description of reality based on insufficient information and, therefore, contrivance. Human history verifies the axiom, “In the absence of knowledge, superstition prevails.” Impaired perception impedes comprehension and breeds fabrication. We find the gaps in our knowledge irritating and uncomfortable, like a road riddled with potholes, and so we fill them in with fabrications to make the ride smoother. The rutted highway of human knowledge is mended with all sorts of contrivances concerning the nature of the universe, a query that for many centuries was beyond investigation. The tools of modern science have enabled us to repave the road, in a manner of speaking, and to upend, one by one, the falsities of our former ignorance. But the road is long, and the work is slow. We must concede that our current conception of the universe is still infantile. Like a child staring bewilderedly at a blackboard chalked from end to end with the esoteric figures of a complex mathematical formula, we are able to recognize some of the numbers and symbols but cannot hope to comprehend the equation, much less solve it. But rather than accept the irreducible complexity before us, many Christians have endeavored to reduce what they cannot comprehend into facile religious concepts that they can. This “Sunday school reductionism” tends to transform profound truths into coloring book illustrations and connect-the-dot puzzles. Instead of illuminating the problem with the lamp of logic and admitting our ignorance, we tend to obscure the problem beneath a canopy of nebulous abstractions, commending ourselves with the false satisfaction of having “solved” it.

To prove to an indignant questioner on the spur of the moment that the work I do was useful seemed a thankless task and I gave it up. I turned to him with a smile and finished, 'To tell you the truth we don't do it because it is useful but because it's amusing.' The answer was thought of and given in a moment: it came from deep down in my mind, and the results were as admirable from my point of view as unexpected. My audience was clearly on my side. Prolonged and hearty applause greeted my confession. My questioner retired shaking his head over my wickedness and the newspapers next day, with obvious approval, came out with headlines 'Scientist Does It Because It's Amusing!' And if that is not the best reason why a scientist should do his work, I want to know what is. Would it be any good to ask a mother what practical use her baby is? That, as I say, was the first evening I ever spent in the United States and from that moment I felt at home. I realised that all talk about science purely for its practical and wealth-producing results is as idle in this country as in England. Practical results will follow right enough. No real knowledge is sterile. The most useless investigation may prove to have the most startling practical importance: Wireless telegraphy might not yet have come if Clerk Maxwell had been drawn away from his obviously 'useless' equations to do something of more practical importance. Large branches of chemistry would have remained obscure had Willard Gibbs not spent his time at mathematical calculations which only about two men of his generation could understand. With this trust in the ultimate usefulness of all real knowledge a man may proceed to devote himself to a study of first causes without apology, and without hope of immediate return.

Our mathematics is a combination of invention and discoveries. The axioms of Euclidean geometry as a concept were an invention, just as the rules of chess were an invention. The axioms were also supplemented by a variety of invented concepts, such as triangles, parallelograms, ellipses, the golden ratio, and so on. The theorems of Euclidean geometry, on the other hand, were by and large discoveries; they were the paths linking the different concepts. In some cases, the proofs generated the theorems-mathematicians examined what they could prove and from that they deduced the theorems. In others, as described by Archimedes in The Method, they first found the answer to a particular question they were interested in, and then they worked out the proof. Typically, the concepts were inventions. Prime numbers as a concept were an invention, but all the theorems about prime numbers were discoveries. The mathematicians of ancient Babylon, Egypt, and China never invented the concept of prime numbers, in spite of their advanced mathematics. Could we say instead that they just did not "discover" prime numbers? Not any more than we could say that the United Kingdom did not "discover" a single, codified, documentary constitution. Just as a country can survive without a constitution, elaborate mathematics could develop without the concept of prime numbers. And it did! Do we know why the Greeks invented such concepts as the axioms and prime numbers? We cannot be sure, but we could guess that this was part of their relentless efforts to investigate the most fundamental constituents of the universe. Prime numbers were the basic building blocks of matter. Similarly, the axioms were the fountain from which all geometrical truths were supposed to flow. The dodecahedron represented the entire cosmos and the golden ratio was the concept that brought that symbol into existence.

Two hundred years later, Piaget reformulated this question from a more psychological perspective. It is no coincidence that he referred to Kant as “the father of us all” (Piaget, 1965/1971, p. 220). Several researchers (e.g., Lourenço & Machado, 1996; Smith, 1993) consider the question about the origin of necessary truth one of the central issues of Piagetian epistemology. Indeed, Piaget's investigations into object permanence, conservation of quantities, and operational knowledge – concepts that form the bedrock of mathematics and natural sciences – are directly related to the Kantian question.

The doctrine of creation of the kind that the Abrahamic faiths profess is such that it encourages the expectation that there will be a deep order in the world, expressive of the Mind and Purpose of that world’s Creator. It also asserts that the character of this order has been freely chosen by God, since it was not determined beforehand by some kind of pre-existing blueprint (as, for example, Platonic thinking had supposed to be the case). As a consequence, the nature of cosmic order cannot be discovered just by taking thought, as if humans could themselves explore a noetic realm of rational constraint to whichthe Creator had had to submit, but the pattern of the world has to be discerned through the observations and experiments that are necessary in order to determine what form the divine choice has actually taken. What is needed, therefore, for successful science is the union of the mathematical expression of order with the empirical investigation of the actual properties of nature, a methodological synthesis of a kind that was pioneered with great skill and fruitfulness by Galileo.

...men are capable of perceiving the Pyramid in an astonishing number of ways. Some have thought the Pyramid was an astronomic and astrological observatory. Some have thought it functioned as the equivalent of a theodolite for surveyors in ancient times... Some think it performed as a giant sundial... Some think it records the mathematics and science of a civilization which vanished... Some think it is a huge water pump. Others have thought it was filled with fabulous treasures... One early investigator came away convinced it was the remains of a huge volcano. Another thought the pyramids were Joseph's granaries. Some thought they were heathen idols which should be destroyed. Some believe the Pyramid captures powerful cosmic energies... Some think it is a tomb. Some think it is a Bible in stone with prophecies built into the scheme of its internal passages... Some think it was a mammoth public works project which consolidated the position of the pharaoh and the unity of the nation. Some think it was built by beings from outer space. Some say it was a temple of initiation. Some hold that it was an instrument of science. Some believe it is an altar of Guild built through direct Divine Revelation. And today, judging by the uses to which it has been put, some apparently think it is an outhouse.

The fact that simultaneous discovery occurs in mathematics, as well as the sciences, points toward some objective element within their subject matter that is independent of the psyche of the investigator.

My wonderful PhD supervisor, Martin Hyland... instilled in me the idea of starting sentences with ‘there is a sense in which…’ because math isn’t about right and wrong, it’s not about absolute truth; it’s about different contexts in which different things can be true, and about different senses in which different things can be valid. Abstraction in mathematics is about making precise which sense we mean, so that instead of having divisive arguments... we can investigate more effectively what is causing certain outcomes to arise.

Most early scientists were compelled to study the natural world because of their Christian worldview. In Science and the Modern World, British mathematician and philosopher Alfred North Whitehead concludes that modern science developed primarily from “the medieval insistence on the rationality of God.” Modern science did not develop in a vacuum, but from forces largely propelled by Christianity. Not surprisingly, most early scientists were theists, including pioneers such as Francis Bacon (1561–1626), Johannes Kepler (1571– 1630), Blaise Pascal (1623–62), Robert Boyle (1627–91), Isaac Newton (1642–1727), and Louis Pasteur (1822–95). For many of them, belief in God was the prime motivation for their investigation of the natural world. Bacon believed the natural world was full of mysteries God intended for us to explore. Kepler described his motivation for science: “The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God, and which he revealed to us in the language of mathematics.

The main difficulty the student of groups meets is not that of following the argument, which is nearly always straightforward, but of grasping the purpose of the investigation.

its very nature, scientific investigation takes for granted such assumptions as that: there is a physical world existing independently of our minds; this world is characterized by various objective patterns and regularities; our senses are at least partially reliable sources of information about this world; there are objective laws of logic and mathematics that apply to the objective world outside our minds;

But what turns out to be true is that the more we investigate, the more laws we find, and the deeper we penetrate nature, the more this disease persists. Every one of our laws is a purely mathematical statement in rather complex and abstruse mathematics. Newton’s statement of the law of gravitation is relatively simple mathematics. It gets more and more abstruse and more and more difficult as we go on. Why? I have not the slightest idea.

neighborhood. So the school, being unable to utilize this everyday experience, sets painfully to work, on another tack and by a variety of means, to arouse in the child an interest in school studies. (Dewey, 1959, pp. 76–77) During Dewey’s tenure at the University of Chicago, he and his colleagues created a model of an educational process that sought to immerse children in those fundamental community activities from which the contemporary academic disciplines have emerged. Using such perennial vocations as gardening, cooking, carpentry, and clothing manufacture, students at the Laboratory School were drawn into the forms of problem-solving and investigation that led to the invention of biology, mathematics, chemistry,

The terminology "analytic-synthetic" was introduced by Kant. Although the distinction itself looks uncontroversial, it can be made to do real philosophical work. Here is one crucial piece of work the logical positivists saw for it: they claimed that all of mathematics and logic is analytic. This made it possible for them to deal with mathematical knowledge within an empiricist framework. For logical positivism, mathematical propositions do not describe the world; they merely record our conventional decision to use symbols in a particular way. Synthetic claims about the world can be expressed using mathematical language, such as when it is claimed that there are nine planets in the solar system. But proofs and investigations within mathematics itself are analytic.

The usefulness of mathematics allows us to build spaceships and investigate the geometry of our universe. Numbers may be our first means of communication with intelligent alien races. Some physicists have even speculated that an understanding of higher dimensions and of topology-the study of shapes and their interrelationships-may someday allow us to escape our universe, when it ends in either great heat or cold, and then we could call all of space-time our home.

You can believe in God, or you can believe in America as the greatest country on Earth, but you can’t believe in astrology. Astrology is like mathematics or physics or language. It’s a tool, an organized system of thought that can be used to investigate or discover something. Believing is an act of faith. I believe in God, but I don’t believe in astrology. I can only make observations that either uphold or refute astrological theory.

qualitative market research in Myanmar Address Ramakrishna Paramhans Ward, PO mangal nagar, Katni, [M.P.] 2nd Floor, Above KBZ Pay Centre, between 65 & 66 street, Manawhari Road Mandalay, Myanmar - Phone +95 9972107002 +91 7222997497 Subjective Statistical surveying in Myanmar: Revealing Bits of knowledge for Business Development In the present globalized commercial center, understanding buyer conduct and market elements is urgent for organizations to flourish. Myanmar, with its quickly developing economy, presents special open doors and difficulties for organizations hoping to lay out areas of strength for an in the district. As organizations look to acquire an upper hand, the meaning of subjective statistical surveying in Myanmar couldn't possibly be more significant. This article digs into the significance of qualitative market research in Myanmar and how it tends to be instrumental in driving business development in the powerful Myanmar market. Myanmar, previously known as Burma, has seen critical political and monetary changes as of late, prompting expanded unfamiliar speculation and development across different areas. This change has brought about shifts in buyer inclinations, buying power, and market patterns. To explore this advancing scene effectively, organizations should participate in thorough subjective statistical surveying to acquire nuanced experiences into buyer conduct, inclinations, and social impacts. qualitative market research in Myanmar centers around understanding the "whys" behind buyer conduct, digging into the basic inspirations, feelings, and insights that drive dynamic cycles. Dissimilar to quantitative exploration, which gives mathematical information and factual examination, subjective examination offers a more profound comprehension of customer perspectives and inclinations, making it priceless for organizations looking to fit their techniques to the Myanmar market. One of the critical benefits of subjective statistical surveying in Myanmar is its capacity to uncover social subtleties and context oriented factors that impact buyer conduct. Given Myanmar's different ethnic gatherings, dialects, and cultural standards, a nuanced comprehension of nearby traditions and customs is fundamental for organizations meaning to resound with the interest group. Subjective examination procedures, for example, inside and out interviews, center gatherings, and ethnographic investigations empower scientists to dive into these social complexities, giving organizations noteworthy bits of knowledge for item improvement, promoting methodologies, and brand situating. Also, subjective examination assumes a significant part in distinguishing arising patterns and market holes that may not be obvious through quantitative information alone. By connecting straightforwardly with buyers and key partners, organizations can acquire subjective experiences into advancing business sector elements, possible undiscovered portions, and moving customer inclinations. This, thus, enables organizations to adjust their contributions and methodologies proactively, remaining on the ball in Myanmar's quickly changing business sector scene. As well as illuminating vital business choices, subjective statistical surveying encourages a more profound association among organizations and the nearby local area. By effectively including Myanmar purchasers in the examination cycle, organizations show a promise to understanding and tending to their requirements, cultivating trust and brand reliability simultaneously. This human-driven approach is especially relevant in Myanmar, where individual connections and local area ties hold huge influence over customer conduct.

Mathematics is that portion of our intellectual activity which transcends our biology and our environment. The principles of biology as we know them may apply to life forms on other worlds, yet there is no necessity for this to be so. The principles of physics should be more universal, yet it is easy to imagine another universe governed by different physical laws. Mathematics, a creation of mind, is less arbitrary than biology or physics, creations of nature; the creatures we imagine inhabiting another world in another universe, with another biology and another physics, will develop a mathematics which in essence is the same as ours. In believing this we may be falling into a trap: Mathematics being a creation of our mind, it is, of course, difficult to imagine how mathematics could be otherwise without actually making it so, but perhaps we should not presume to predict the course of the mathematical activities of all possible types of intelligence. On the other hand, the pragmatic content of our belief in the transcendence of mathematics has nothing to do with alien forms of life. Rather it serves to give a direction to mathematical investigation, resulting from the insistence that mathematics be born of an inner necessity.

Regarding the fundamental investigations of mathematics, there is no final ending ... no first beginning.

Since Darwin’s day, science has been increasingly revealing an irreversible dramatic story-line in nature that mathematical analysis had previously failed to notice. Science has now exposed nature as a drama that makes it more open than ever to theological investigation.

By this time, Washington’s airports were buried in two feet of snow, so I boarded a train for Boston. During the long ride back I wondered how my research into the mathematical theory of a game might change my life. In the abstract, life is a mixture of chance and choice. Chance can be thought of as the cards you are dealt in life. Choice is how you play them. I chose to investigate blackjack. As a result, chance offered me a new set of unexpected opportunities.

Intuitionist mathematics is nothing more nor less than an investigation of the utmost limits which the intellect can attain in its self-unfolding.

Sometimes I wonder whether my whole life has been a singular quest for beauty. Beauty in mathematics, and beauty in literature and in music. I feel that creating mathematics and writing fiction are closely related. While authors are poets in the universe of language, mathematicians seek the poetry in the language of the universe. The German mathematician Karl Weierstrass once wrote that any great mathematician must also be a poet. When I was young, several people told me that I’d be a poet when I grew up. So in a way, it feels as if I’ve tried to investigate whether the reverse implication is true: whether every poet must also be a great mathematician. I still don’t know the answer, but I doubt that this is the case. Over the past few years, I’ve started to dream of writing a novel. I’ve marveled at how the enjoyment of hearing a piece of music often gets stronger the better you know the piece, while a novel rarely has the same impact on third reading. Is it because music relies on recognition, while literature relies on the unexpected? Or has it more to do with the structure of the music, how the themes reflect each other so that the listener discovers ever new connections? The way the interplay of colors in a painting can fluctuate in different light, so that the painting continually changes? If so, it must be possible to write a novel in the same way. A novel that gets richer every time you read it, because you discover new connections that were previously invisible. A novel that carries something of the eternal beauty of music and mathematics within it. One of the most alluring things about mathematics is perhaps the feeling of being able to uncover unshakeable truths. And that terms such as truth and beauty obtain a kind of objectivity, because mathematicians have a shared understanding of what constitutes a valid proof and what is aesthetically beautiful. The disadvantage is that the truths of mathematics don’t say anything about what is true in the world beyond mathematics itself.

Magic is a practical science,' he began quickly. He talked to the wall, as if dictating. 'There is all the difference in the world between a formula in physics and a formula in magic, although they have the same name. The former describes, in terse mathematical symbol, cause-effect relationships of wide generality. But a formula in magic is a way of getting or accomplishing something. It always takes into account the motivation or desire of the person invoking the formula—be it greed, love, revenge, or what not. Whereas the experiment in physics is essentially independent of the experimenter. In short, there has been little or no pure magic, comparable to pure science. 'This distinction between physics and magic is only an accident of history. Physics started out as a kind of magic, too—witness alchemy and the mystical mathematics of Pythagoras. And modern physics is ultimately as practical as magic, but it possesses a superstructure of theory that magic lacks. Magic could be given such a superstructure by research in pure magic and by the investigation and correlation of the magic formulas which could be expressed in mathematical symbols and which would have a wide application. Most persons practicing magic have been too interested in immediate results to bother about theory. But just as research in pure science has ultimately led, seemingly by accident, to results of vast practical importance, so research in pure magic might be expected to yield similar results.

Ebling Mis said unhurriedly, ‘You know what I’m doing these days?’ ‘I have your reports here,’ replied the mayor, with satisfaction, ‘together with authorized summaries of them. As I understand it, your investigations into the mathematics of psycho-history have been intended to duplicate Hari Seldon’s work and, eventually, trace the projected course of future history, for the use of the Foundation.

The discussion is designed to get students to engage with one another's sorts and deduce the attribute that defines each group.

There is, however, a completely satisfactory way of avoiding the paradoxes without betraying our science. The desires and attitudes which help us find this way and show us what direction to take are these: 1. Wherever there is any hope of salvage, we will carefully investigate fruitful definitions and deductive methods. We will nurse them, strengthen them, and make them useful. No one shall drive us out of the paradise which Cantor has created for us. 2. We must establish throughout mathematics the same certitude for our deductions as exists in ordinary elementary number theory, which no one doubts and where contradictions and paradoxes arise only through our own carelessness.

The hope originally fostered by the Old Formalists that mathematical science erected according to their principles would be crowned one day with a proof of non-contradictority, was never fulfilled, and, nowadays [1952], in view of the results of certain investigations of the last few decades, has, I think, been relinquished.

[Brouwer’s] construction of intuitionist mathematics is nothing more nor less than an investigation of the utmost limits which the intellect can attain in its self-unfolding.

The hope originally fostered by the Old Formalists that mathematical science erected according to their principles would be crowned one day with a proof of non-contradictority, was never fulfilled, and, nowadays, in view of the results of certain investigations of the last few decades, has, I think, been relinquished.

Christianity isn't anti-science, but it is anti-scientism. Scientism is the belief that science is the only way to know anything. But there are many things we know without the benefit of science at all, like logical and mathematical truths (which precede scientific investigations), metaphysical truths (which determine if the external world is real), moral and ethical truths (which set boundaries for our behavior), aesthetic truths (like determining beauty), and historical truths. Christians believe that science can tell us many important things but not all of the important things.

Dirac was investigating the movements of elementary particles. He was searching for the mathematical structure that connects their movements the way sinews connect our bones.

The definition of art is problematic, but, simplistically, it is the application of skills to the creation of aesthetic values. Science can be defined as the methodical pursuit of knowledge about the phenomena of the physical world on the basis of unbiased observation and systematic experimentation. Roughly speaking, the objects of art and science are beauty and truth, respectively. Yoga is an art because it evidently does not have the mathematical exactitude of the natural sciences. The British-American mathematician-philosopher Alfred North Whitehead once remarked: “Art flourishes when there is a sense of adventure, a sense of nothing having been done before, of complete freedom to experiment; but when caution comes in you get repetition, and repetition is the death of art.”2 These comments apply to Yoga quite well. It is an incredible adventure of the spirit, which seeks to create an altogether new destiny. Each time the practitioner applies the wisdom of Yoga to life’s many situations, he or she must engage the process as if it were the first time. Thus Yoga is continuous self-application but not merely repetition. The Sanskrit term abhyāsa, which literally means “repetition,” has the primary meaning of “practice” in the context of Yoga, and practice calls for what the Zen masters call “beginner’s mind.” Any efforts to squeeze Yoga into the much-celebrated scientific method is doomed to failure, which is not to say that Yoga cannot or should not be studied rigorously from a scientific perspective. In fact, since the 1920s various research organizations and individual researchers have conducted such research, especially medical investigations, with varying degrees of success, and their findings have definitely been helpful in appraising Yoga’s effectiveness.3 Yet, Yoga is not completely subjective and inexact either. It proceeds according to careful rules established over a long period of (repeatable) personal experimentation.

The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God, and which he revealed to us in the language of mathematics.

This set of four laws, taken conjointly, form a unified system by means of which virtually all macroscopic mechanical systems, from the swinging of a pendulum to the motion of the planets in their orbits around the sun, may be investigated, and their behavior predicted. Newton did not merely state these laws of mechanics; he himself, using the mathematical tools of the calculus, showed how these fundamental laws could be applied to the solution of actual problems.

...the strongholds of Fingerprints of the Gods lie in its analysis of mythology, in its exposure of a great worldwide spiritual system - older than history - encompassing astronomical, architectural, mathematical, and geodetic information, in opening up to wider view the extraordinary nature of ancient Egyptian civilisation & the ancient Egyptians monuments, in the case it makes for an inherited legacy of high knowledge from earlier times, in its investigation of the post-glacial cataclysms that shook the world...and in the correlation of these with myths of universal catastrophe...

In a real sense, the important question is never one of validity or truth. Truth exists in the realm of mathematics and in the philosophy of logic, not in perceptions of reality. For those who would understand the world about them, the question is not one of truth, but of utility. Do our investigations deepen our understanding, further our ability to ask more refined questions, and lead to better predictions of events? If so, then the research is justified. If not, it remains but sophistry.

Mathematical calculations cannot demonstrate the existence and career of Alexander the Great in the fourth century BCE. But converging historical evidence would make it absurd to deny that he lived and changed the political and cultural face of the Middle East.

This is manifestly false, as the unsolved problems of mathematics like Goldbach’s Conjecture, which is either necessarily true or necessarily false, though no one knows which, shows. By contrast I have tremendous certainty that George Washington was once the President of the United States, though this is a contingent historical truth. There is no reason a contingent truth which is known with confidence might not serve as evidence for a less obvious necessary truth.

Female college students performed better on a hard mathematics test when it included at the beginning the statement “You may have heard that women typically do less well at math tests than men, but this is not true for this particular test.”34 Conversely, white male math and engineering majors who received high scores on the math portion of the SAT (a group of people quite confident about their mathematical abilities) did worse on a math test when told the experiment was intended to investigate “why Asians appear to outperform other students on tests of math ability.

Like anything else in the delivery of justice, at the investigative or any other phase, the approach requires balance. There is no science, no mathematical formula, no precise scale on which you can balance these things, but they must be balanced nonetheless.

An Investigation of the Laws of Thought, On Which Are Founded the Mathematical Theories of Logic and Probability.

Thanks to her PhD research, Noether had become an expert in symmetry, and as she investigated Einstein’s theories, she spotted a deep truth about our universe. Specifically, in a mathematical statement now known as Noether’s theorem, she showed that for the laws of physics to be unvarying over time, energy must be conserved.

IN DISQUISITIONS of every kind, there are certain primary truths, or first principles, upon which all subsequent reasonings must depend. These contain an internal evidence which, antecedent to all reflection or combination, commands the assent of the mind. Where it produces not this effect, it must proceed either from some defect or disorder in the organs of perception, or from the influence of some strong interest, or passion, or prejudice. Of this nature are the maxims in geometry, that “the whole is greater than its part; things equal to the same are equal to one another; two straight lines cannot enclose a space; and all right angles are equal to each other.” Of the same nature are these other maxims in ethics and politics, that there cannot be an effect without a cause; that the means ought to be proportioned to the end; that every power ought to be commensurate with its object; that there ought to be no limitation of a power destined to effect a purpose which is itself incapable of limitation. And there are other truths in the two latter sciences which, if they cannot pretend to rank in the class of axioms, are yet such direct inferences from them, and so obvious in themselves, and so agreeable to the natural and unsophisticated dictates of common-sense, that they challenge the assent of a sound and unbiased mind, with a degree of force and conviction almost equally irresistible. The objects of geometrical inquiry are so entirely abstracted from those pursuits which stir up and put in motion the unruly passions of the human heart, that mankind, without difficulty, adopt not only the more simple theorems of the science, but even those abstruse paradoxes which, however they may appear susceptible of demonstration, are at variance with the natural conceptions which the mind, without the aid of philosophy, would be led to entertain upon the subject. The infinite divisibility of matter, or, in other words, the infinite divisibility of a finite thing, extending even to the minutest atom, is a point agreed among geometricians, though not less incomprehensible to common-sense than any of those mysteries in religion, against which the batteries of infidelity have been so industriously leveled. But in the sciences of morals and politics, men are found far less tractable. To a certain degree, it is right and useful that this should be the case. Caution and investigation are a necessary armor against error and imposition. But this untractableness may be carried too far, and may degenerate into obstinacy, perverseness, or disingenuity. Though it cannot be pretended that the principles of moral and political knowledge have, in general, the same degree of certainty with those of the mathematics, yet they have much better claims in this respect than, to judge from the conduct of men in particular situations, we should be disposed to allow them. The obscurity is much oftener in the passions and prejudices of the reasoner than in the subject. Men, upon too many occasions, do not give their own understandings fair play; but, yielding to some untoward bias, they entangle themselves in words and confound themselves in subtleties.

Pythagoras (550 BCE), with his theory of numbers, had been a source of inspiration for those who sought harmony in the Universe. His aim was to show in his philosophy that there was a high, structural, divine order to the Universe. This was a natural habitat for the souls. Mathematics was the tool to investigate this order.

Regarding the fundamental investigations of mathematics, there is no final ending... no first beginning.

Infinity earned its rightful place as a legitimate mathematical concept towards the end of the nineteenth century. The two big players behind this were German mathematicians Georg Cantor and his champion David Hilbert. No longer just accepting infinity as a general notion but actually investigating it rigorously did not go down well. Contemporary mathematicians described Cantor as a 'corrupter of youth'. This had a detrimental effect on Cantor, who already suffered from depression. Thankfully, Hilbert saw the power of what Cantor had done. He described Cantor's work as 'the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity', and famously said, 'no one shall expel us from the Paradise that Cantor has created'.

By apparently purging material reality of subjective experience, Galileo cleared the ground and Descartes laid the foundation for the construction of the objective or “disinterested” sciences, which by their feverish and forceful investigations have yielded so much of the knowledge and so many of the technologies that have today become commonplace in the West. The chemical table of the elements, automobiles, smallpox vaccines, “close-up” images of the outer planets—so much that we have come to assume and depend upon has emerged from the bold experimentalization of the world by the objective sciences. Yet these sciences consistently overlook our ordinary, everyday experience of the world around us. Our direct experience is necessarily subjective, necessarily relative to our own position or place in the midst of things, to our particular desires, tastes, and concerns. The everyday world in which we hunger and make love is hardly the mathematically determined “object” toward which the sciences direct themselves. Despite all the mechanical artifacts that now surround us, the world in which we find ourselves before we set out to calculate and measure it is not an inert or mechanical object but a living field, an open and dynamic landscape subject to its own moods and metamorphoses.