“The knowledge of which geometry aims is the knowledge of the eternal." (Plato)
"You say that just as space consists of an infinity of contiguous points, so time is but an infinite collection of contiguous instants? Good! Consider, then, an arrow in its flight. At any instant its extremity occupies a definite point in its path. Now, while occupying this position it must be at rest there. But how can a point be motionless and yet in motion at the same time?” (Zeno)
"Time and space are divided into the same and equal divisions. Wherefore also, Zeno’s argument, that it is impossible to go through an infinite collection or to touch an infinite collection one by one in a finite time, is fallacious. For there are two senses in which the term ‘infinte’ is applied both to length and to time and in fact to all continuous things: either in regard to divisibility or in regard to number. Now it is not possible to touch things infinite as to number in a finite time, but it is possible to touch things infinite in regard to divisibility; for time itself is also infinite in this sense." (Aristotle)
“Our account does not rob mathematicians of their science, by disproving the actual existence of the infinite in the direction of increase, in the sense of the untraceable. In point of fact they do not need the infinite and do not use it. They postulate any that the finite straight line may be produced as far as they wish.” (Aristotle, Physics)
"A finite straight line can be extended indefinitely to make an infinitely long straight line." (Euclid’s postulate)
"Given a straight line and any point off to the side of it, there is, through that point, one and only one line that is parallel to the given line." (Euclid’s postulate)
"We admit, in geometry, not only infinite magnitudes, that is to say, magnitudes greater than any assignable magnitude, but infinite magnitudes infinitely greater, the one than the other. This astonishes our dimension of brains, which is only about six inches long, five broad, and six in depth, in the largest heads." (Voltaire)
“Of late the speculations about Infinities have run so high, and grown to such strange notions, as have occasioned no small scruples and disputes among the geometers of the present age. Some there are of great note who, not contented with holding that finite lines may be divided into an infinite number of parts, do yet further maintain that each of these infinitesimals is itself subdivisible into an infinity of other parts or infinitesimals of a second order, and so on ad infinitum. These I say assert there are infinitesimals of infinitesimals, etc., without ever coming to an end; so that according to them an inch does not barely contain an infinite number of parts, but an infinity of an infinity of an infinity ad infinitum of parts.” (George Berkeley, “The Principles of Human Knowledge”, 1710)
“The introduction into geometrical work of conceptions such as the infinite, the imaginary, and the relations of hyperspace, none of which can be directly imagined, has a psychological significance well worthy of examination. It gives a deep insight into the resources and working of the human mind. We arrive at the borderland of mathematics and psychology.” (John Theodore Merz, “History of European Thought in the Nineteenth Century”, 1903)
Quotes and Resources Related to Mathematics, (Mathematical) Sciences and Mathematicians
29 August 2017
Infinite, Nature and Mathematics
“But Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end.” (Aristotle, “Generation of Animals”)
“There is a single general space, a single vast immensity which we may freely call Void: in it are innumerable globes like this on which we live and grow; this space we declare to be infinite, since neither reason, convenience, sense-perception nor nature assign it a limit.” (Giordano Bruno)
"Just as the stone thrown into the water becomes the centre and cause of various circles, [so] the sound made in the air spreads out in circles and fills the surrounding parts with an infinite number of images of itself." (Leonardo da Vinci)
"I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author." (Gottfried W Leibniz)
“What is man in nature? A Nothing in comparison with the Infinite, an All in comparison with the Nothing, a mean between nothing and everything. Since he is infinitely removed from comprehending the extremes, the end of things and their beginning are hopelessly hidden from him in an impenetrable secret; he is equally incapable of seeing the Nothing from which he was made, and the Infinite in which he is swallowed up.” (Blaise Pascal, "Pensées", 1670)
"Nature is an infinite sphere of which the center is everywhere and the circumference nowhere." (Blaise Pascal, "Pensées", 1670)
“In an infinite number universe, every point can be regarded as the center, because every point has an infinite of stars on each side of it.” (Stephen Hawking, "A Brief History of Time", 1988)
“There is a single general space, a single vast immensity which we may freely call Void: in it are innumerable globes like this on which we live and grow; this space we declare to be infinite, since neither reason, convenience, sense-perception nor nature assign it a limit.” (Giordano Bruno)
"Just as the stone thrown into the water becomes the centre and cause of various circles, [so] the sound made in the air spreads out in circles and fills the surrounding parts with an infinite number of images of itself." (Leonardo da Vinci)
"I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author." (Gottfried W Leibniz)
“What is man in nature? A Nothing in comparison with the Infinite, an All in comparison with the Nothing, a mean between nothing and everything. Since he is infinitely removed from comprehending the extremes, the end of things and their beginning are hopelessly hidden from him in an impenetrable secret; he is equally incapable of seeing the Nothing from which he was made, and the Infinite in which he is swallowed up.” (Blaise Pascal, "Pensées", 1670)
"Nature is an infinite sphere of which the center is everywhere and the circumference nowhere." (Blaise Pascal, "Pensées", 1670)
“In an infinite number universe, every point can be regarded as the center, because every point has an infinite of stars on each side of it.” (Stephen Hawking, "A Brief History of Time", 1988)
The Infinite and Mathematics
“The progress of mathematics can be viewed as progress from the infinite to the finite.” (Gian-Carlo Rota)
“The whole universe is one mathematical and harmonic expression, made up of finite representations of the infinite.” (Fritz L Kunz)
"Mathematics is, in many ways, the most precious response that the human spirit has made to the call of the infinite." (Cassius J. Keyser, “The Human Worth of Rigorous Thinking, 1925)
"Mathematics is the science of the infinite, its goal the symbolic comprehension of the infinite with human, that is, finite means.” (Hermann Weyl, “Mind and Nature”, 1934)
“Mathematics has been called the science of the infinite. Indeed, the mathematician invents finite constructions by which questions are decided that by their very nature refer to the infinite. This is his glory.” (Hermann Weyl, “Axiomatic versus constructive procedures in mathematics”, The Mathematical Intelligencer, 1985)
"Mathematics as we know it and as it has come to shape modern science could never have come into being without some disregard for the dangers of the infinite." (David Bressoud, “A radical approach to real analysis”, MAA, 2007)
“Mathematics are the result of mysterious powers which no one understands, and which the unconscious recognition of beauty must play an important part. Out of an infinity of designs a mathematician chooses one pattern for beauty's sake and pulls it down to earth.” (Marston Morse)
"Mathematics is the only infinite human activity. It is conceivable that humanity could eventually learn everything in physics or biology. But humanity certainly won't ever be able to find out everything in mathematics, because the subject is infinite. Numbers themselves are infinite." (Paul Erdos)
“The infinite more than anything else is what characterizes mathematics and defines its essence. […] To grapple with infinity is one of the bravest and extraordinary endeavors that human beings have ever undertaken.” (William Byers, “How Mathematicians Think”, 2007)
"Mathematics, in one view, is the science of infinity." (Phillip J Davis & Reuben Hersh, “The Mathematical Experience”, 1999)
“The whole universe is one mathematical and harmonic expression, made up of finite representations of the infinite.” (Fritz L Kunz)
"Mathematics is, in many ways, the most precious response that the human spirit has made to the call of the infinite." (Cassius J. Keyser, “The Human Worth of Rigorous Thinking, 1925)
"Mathematics is the science of the infinite, its goal the symbolic comprehension of the infinite with human, that is, finite means.” (Hermann Weyl, “Mind and Nature”, 1934)
“Mathematics has been called the science of the infinite. Indeed, the mathematician invents finite constructions by which questions are decided that by their very nature refer to the infinite. This is his glory.” (Hermann Weyl, “Axiomatic versus constructive procedures in mathematics”, The Mathematical Intelligencer, 1985)
"Mathematics as we know it and as it has come to shape modern science could never have come into being without some disregard for the dangers of the infinite." (David Bressoud, “A radical approach to real analysis”, MAA, 2007)
“Mathematics are the result of mysterious powers which no one understands, and which the unconscious recognition of beauty must play an important part. Out of an infinity of designs a mathematician chooses one pattern for beauty's sake and pulls it down to earth.” (Marston Morse)
"Mathematics is the only infinite human activity. It is conceivable that humanity could eventually learn everything in physics or biology. But humanity certainly won't ever be able to find out everything in mathematics, because the subject is infinite. Numbers themselves are infinite." (Paul Erdos)
“The infinite more than anything else is what characterizes mathematics and defines its essence. […] To grapple with infinity is one of the bravest and extraordinary endeavors that human beings have ever undertaken.” (William Byers, “How Mathematicians Think”, 2007)
"Mathematics, in one view, is the science of infinity." (Phillip J Davis & Reuben Hersh, “The Mathematical Experience”, 1999)
The Infinite and Its Difficulties I
"The infinite is imperfect, unfinished and therefore, unthinkable; it is formless and confused." (Aristotle)
“These are among the marvels that surpass the bounds of our imagination, and that must warn us how gravely one errs in trying to reason about infinites by using the same attributes that we apply to finites.” (Galileo Galilei)
"The eternal silence of these infinite spaces terrifies me." (Blaise Pascal)
“[…] the notion of the infinite […] forces itself upon our mind and yet is incomprehensible. When this notion takes possession of the understanding we have only to bow before it.” (Louis Pasteur)
“The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite.” (David Hilbert)
“The infinite has always stirred the emotions of mankind more deeply than any other question; the infinite has stimulated and fertilized reason as few other ideas have; but also the infinite, more than any other notion, is in need of clarification.” (David Hilbert)
"Our minds are finite, and yet even in those circumstances of finitude, we are surrounded by possibilities that are infinite, and the purpose of human life is to grasp as much as we can out of that infinitude." (Alfred N Whitehead)
“These are among the marvels that surpass the bounds of our imagination, and that must warn us how gravely one errs in trying to reason about infinites by using the same attributes that we apply to finites.” (Galileo Galilei)
"The eternal silence of these infinite spaces terrifies me." (Blaise Pascal)
“[…] the notion of the infinite […] forces itself upon our mind and yet is incomprehensible. When this notion takes possession of the understanding we have only to bow before it.” (Louis Pasteur)
“The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite.” (David Hilbert)
“The infinite has always stirred the emotions of mankind more deeply than any other question; the infinite has stimulated and fertilized reason as few other ideas have; but also the infinite, more than any other notion, is in need of clarification.” (David Hilbert)
"Our minds are finite, and yet even in those circumstances of finitude, we are surrounded by possibilities that are infinite, and the purpose of human life is to grasp as much as we can out of that infinitude." (Alfred N Whitehead)
The Infinite from Poet’s Pen
“Nature that framed us of four elements,
Warring within our breasts for regiment,
Doth teach us all to have aspiring minds:
Our souls, whose faculties can comprehend
The wondrous architecture of the world:
And measure every wand’ring planet’s course,
Still climbing after knowledge infinite,
And always moving as the restless spheres,
Wills us to wear ourselves and never rest,
Until we reach the ripest fruit of all,
That perfect bliss and sole felicity,
The sweet fruition of an earthly crown.”
(Christopher Marlowe, “Tamburlaine the Great”, 1590)
“I could be bounded in a nutshell, and count myself a king of infinite space.” (William Shakespeare, “Hamlet”, cca. 1600)
“There’s nothing of so infinite vexation
As man’s own thoughts.”
(John Webster, “The White Devil”, 1612)
“[Infinity is] […] a dark
llimitable ocean, without bound,
Without dimension; where length, breadth, and height,
And time, and place, are lost…”
(John Milton, “Paradise Lost”, 1667)
“As lines (so loves) oblique may well
Themselves in every angle greet:
But ours so truly parallel,
Though infinite, can never meet.”
(Andrew Marvell, “The Definition of Love”, 1681)
“Even as the finite encloses an infinite series
And in the unlimited limits appear,
So the soul of immensity dwells in minutia
And in narrowest limits no limit in here.
What joy to discern the minute in infinity!
The vast to perceive in the small, what divinity!”
(Jacques Bernoulli, “Ars Conjectandi”, 1713)
“As lines, so loves oblique, may well
Themselves in every angle greet
But ours, so truly parallel,
Though infinite, can never meet.”
(Andrew Marvell, “The Definition of Love”)
“So, Nat’ralists observe, a Flea
Hath smaller Fleas that on him prey,
And these have smaller Fleas to bite ’em
And so proceed, ad infinitum.”
(Jonathan Swift", “On Poetry: a Rhapsody”, 1733)
“If in the infinite you want to stride,
Just walk in the finite to every side.” (Johann Wolfgang von Goethe)
“To see the world in a grain of sand,
And a heaven in a wild flower;
Hold infinity in the palm of your hand,
And eternity in an hour.”
(William Blake, “Auguries of Innocence”, 1803)
“Action is transitory, - a step, a blow,
The motion of a muscle, this way or that -
’Tis done, and in the after-vacancy
We wonder at ourselves like men betrayed:
Suffering is permanent, obscure and dark,
And shares the nature of infinity.”
(William Wordsworth, “The Borderers”, 1842)
“God puts his finger in the other scale,
And up we bounce, a bubble. Nought is great
Nor small, with God; for none but he can make
The atom imperceptible, and none
But he can make a world; he counts the orbs,
He counts the atoms of the universe,
And makes both equal; both are infinite.”
(Philip James Bailey, “Festus", 1845)
“What miracle of weird transforming
Is this wild work of frost and light,
This glimpse of glory infinite!”
(John Greenleaf Whittier, “The Pageant", 1869)
“Great fleas have little fleas upon their backs to bite 'em,
And little fleas have lesser fleas, and so ad infinitum.
And the great fleas themselves, in turn have greater fleas to go on;
While these again have greater still, and greater still, and so on.”
(Augustus de Morgan, “A Budget of Paradoxes”, 1915)
“But the star-glistered salver of infinity,
The circle, blind crucible of endless space,
Is sluiced by motion,-subjugated never.”
(Hart Crane. “The Bridge”, 1930)
“Big whorls have little whorls
Which feed on their velocity,
And little whorls have lesser whorls,
And so on to viscosity.”
(Lewis Richardson)
“All finite things reveal infinitude:
The mountain with its singular bright shade
Like the blue shine on freshly frozen snow,
The after-light upon ice-burdened pines;
Odor of basswood on a mountain-slope,
A scent beloved of bees;
Silence of water above a sunken tree :
The pure serene of memory in one man, --
A ripple widening from a single stone
Winding around the waters of the world.”
(Theodore Roethke, “The Far Field” IV, 1964)
Warring within our breasts for regiment,
Doth teach us all to have aspiring minds:
Our souls, whose faculties can comprehend
The wondrous architecture of the world:
And measure every wand’ring planet’s course,
Still climbing after knowledge infinite,
And always moving as the restless spheres,
Wills us to wear ourselves and never rest,
Until we reach the ripest fruit of all,
That perfect bliss and sole felicity,
The sweet fruition of an earthly crown.”
(Christopher Marlowe, “Tamburlaine the Great”, 1590)
“I could be bounded in a nutshell, and count myself a king of infinite space.” (William Shakespeare, “Hamlet”, cca. 1600)
“There’s nothing of so infinite vexation
As man’s own thoughts.”
(John Webster, “The White Devil”, 1612)
“[Infinity is] […] a dark
llimitable ocean, without bound,
Without dimension; where length, breadth, and height,
And time, and place, are lost…”
(John Milton, “Paradise Lost”, 1667)
“As lines (so loves) oblique may well
Themselves in every angle greet:
But ours so truly parallel,
Though infinite, can never meet.”
(Andrew Marvell, “The Definition of Love”, 1681)
“Even as the finite encloses an infinite series
And in the unlimited limits appear,
So the soul of immensity dwells in minutia
And in narrowest limits no limit in here.
What joy to discern the minute in infinity!
The vast to perceive in the small, what divinity!”
(Jacques Bernoulli, “Ars Conjectandi”, 1713)
“As lines, so loves oblique, may well
Themselves in every angle greet
But ours, so truly parallel,
Though infinite, can never meet.”
(Andrew Marvell, “The Definition of Love”)
“So, Nat’ralists observe, a Flea
Hath smaller Fleas that on him prey,
And these have smaller Fleas to bite ’em
And so proceed, ad infinitum.”
(Jonathan Swift", “On Poetry: a Rhapsody”, 1733)
“If in the infinite you want to stride,
Just walk in the finite to every side.” (Johann Wolfgang von Goethe)
“To see the world in a grain of sand,
And a heaven in a wild flower;
Hold infinity in the palm of your hand,
And eternity in an hour.”
(William Blake, “Auguries of Innocence”, 1803)
“Action is transitory, - a step, a blow,
The motion of a muscle, this way or that -
’Tis done, and in the after-vacancy
We wonder at ourselves like men betrayed:
Suffering is permanent, obscure and dark,
And shares the nature of infinity.”
(William Wordsworth, “The Borderers”, 1842)
“God puts his finger in the other scale,
And up we bounce, a bubble. Nought is great
Nor small, with God; for none but he can make
The atom imperceptible, and none
But he can make a world; he counts the orbs,
He counts the atoms of the universe,
And makes both equal; both are infinite.”
(Philip James Bailey, “Festus", 1845)
“What miracle of weird transforming
Is this wild work of frost and light,
This glimpse of glory infinite!”
(John Greenleaf Whittier, “The Pageant", 1869)
“Great fleas have little fleas upon their backs to bite 'em,
And little fleas have lesser fleas, and so ad infinitum.
And the great fleas themselves, in turn have greater fleas to go on;
While these again have greater still, and greater still, and so on.”
(Augustus de Morgan, “A Budget of Paradoxes”, 1915)
“But the star-glistered salver of infinity,
The circle, blind crucible of endless space,
Is sluiced by motion,-subjugated never.”
(Hart Crane. “The Bridge”, 1930)
“Big whorls have little whorls
Which feed on their velocity,
And little whorls have lesser whorls,
And so on to viscosity.”
(Lewis Richardson)
“All finite things reveal infinitude:
The mountain with its singular bright shade
Like the blue shine on freshly frozen snow,
The after-light upon ice-burdened pines;
Odor of basswood on a mountain-slope,
A scent beloved of bees;
Silence of water above a sunken tree :
The pure serene of memory in one man, --
A ripple widening from a single stone
Winding around the waters of the world.”
(Theodore Roethke, “The Far Field” IV, 1964)
Defining the Infinite
“There is no smallest among the small and no largest among the large;
But always something still smaller and something still larger.” (Anaxagoras)
“A quantity is infinite if it is such that we can always take a part outside what has been already taken.” (Aristotle, Physics)
“For generally the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite […].” (Aristotle, Physics)
“What is that thing which does not give itself and which if it were to give itself would not exist? It is the infinite!” (Leonardo da Vinci)
“When we say anything is infinite, we signify only that we are not able to conceive the ends and bounds of the thing named.” (Thomas Hobbes)
“We call infinite that thing whose limits we have not perceived, and so by that word we do not signify what we understand about a thing, but rather what we do not understand.” (René Descartes)
“I protest against the use of infinite magnitude as something completed, which in mathematics is never permissible. Infinity is merely a facon de parler [manner of speaking], the real meaning being a limit which certain ratios approach indefinitely near, while others are permitted to increase without restriction.” (Carl F Gauss, 1831)
“An infinite set is one that can be put into a one-to-one correspondence with a proper subset of itself.” (Georg Cantor)
“The Infinite is often confounded with the Indefinite, but the two conceptions are diametrically opposed. Instead of being a quantity with unassigned yet assignable limits, the Infinite is not a quantity at all, since it neither admits of augmentation nor diminution, having no assignable limits; it is the operation of continuously withdrawing any limits that may have been assigned: the endless addition of new quantities to the old: the flux of continuity. The Infinite is no more a quantity than Zero is a quantity. If Zero is the sign of a vanished quantity, the Infinite is a sign of that continuity of Existence which has been ideally divided into discrete parts in the affixing of limits.” (George H Lewes, “Problems of Life and Mind”, Vol. 2, 1875)
“A collection of terms is infinite when it contains as parts other collections which have just as many terms in it as it has. If you can take away some of the terms of a collection, without diminishing the number of terms, then there is an infinite number of terms in the collection.” (Bertrand Russell. International Monthly, Vol. 4, 1901)
But always something still smaller and something still larger.” (Anaxagoras)
“A quantity is infinite if it is such that we can always take a part outside what has been already taken.” (Aristotle, Physics)
“For generally the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite […].” (Aristotle, Physics)
“What is that thing which does not give itself and which if it were to give itself would not exist? It is the infinite!” (Leonardo da Vinci)
“When we say anything is infinite, we signify only that we are not able to conceive the ends and bounds of the thing named.” (Thomas Hobbes)
“We call infinite that thing whose limits we have not perceived, and so by that word we do not signify what we understand about a thing, but rather what we do not understand.” (René Descartes)
“I protest against the use of infinite magnitude as something completed, which in mathematics is never permissible. Infinity is merely a facon de parler [manner of speaking], the real meaning being a limit which certain ratios approach indefinitely near, while others are permitted to increase without restriction.” (Carl F Gauss, 1831)
“An infinite set is one that can be put into a one-to-one correspondence with a proper subset of itself.” (Georg Cantor)
“The Infinite is often confounded with the Indefinite, but the two conceptions are diametrically opposed. Instead of being a quantity with unassigned yet assignable limits, the Infinite is not a quantity at all, since it neither admits of augmentation nor diminution, having no assignable limits; it is the operation of continuously withdrawing any limits that may have been assigned: the endless addition of new quantities to the old: the flux of continuity. The Infinite is no more a quantity than Zero is a quantity. If Zero is the sign of a vanished quantity, the Infinite is a sign of that continuity of Existence which has been ideally divided into discrete parts in the affixing of limits.” (George H Lewes, “Problems of Life and Mind”, Vol. 2, 1875)
“A collection of terms is infinite when it contains as parts other collections which have just as many terms in it as it has. If you can take away some of the terms of a collection, without diminishing the number of terms, then there is an infinite number of terms in the collection.” (Bertrand Russell. International Monthly, Vol. 4, 1901)
24 August 2017
Nature and Mathematics I
"The laws of nature are but the mathematical thoughts of God." (Euclid)
"I believe we can attach mathematically everything in nature and in the world of change." (Iambilichus)
"Whence is it that nature does nothing in vain; and whence arises all that order and beauty which we see in the world?" (Sir Isaac Newton)
"Nature is pleased with simplicity, and affects not the pomp of superfluous causes." (Sir Issac Newton)
"Although to penetrate into the intimate mysteries of nature and hence to learn the true causes of phenomena is not allowed to us, nevertheless it can happen that a certain fictive hypothesis may suffice for explaining many phenomena." (Leonhard Euler)
"For many parts of nature can neither be invented with sufficient subtlety, nor demonstrated with sufficient perspicuity, nor accommodated unto use with sufficient dexterity without the aid and
intervention of mathematics.” (Galileo Galilei)
"The great book of nature can be read only by those who know the language in which it was written. And this language is mathematics." (Galileo Galilei)
"Nature's great book is written in mathematical symbols." (Galileo Galilei)
"For many parts of nature can neither be invented with sufficient subtlety, nor demonstrated with sufficient perspicuity, nor accommodated unto use with sufficient dexterity without the aid and intervention of mathematics."(Morris Kline, "Mathematics and the Physical World, 1959)
"it is the most widely accepted axiom in the natural science that Nature makes use of the fewest possible means" (Johannes Kepler)
"Nature is pleased with simplicity, and affects not the pomp of superfluous causes." (Morris Kline, "Mathematics and the Physical World", 1959)
"I believe we can attach mathematically everything in nature and in the world of change." (Iambilichus)
"Whence is it that nature does nothing in vain; and whence arises all that order and beauty which we see in the world?" (Sir Isaac Newton)
"Nature is pleased with simplicity, and affects not the pomp of superfluous causes." (Sir Issac Newton)
"Although to penetrate into the intimate mysteries of nature and hence to learn the true causes of phenomena is not allowed to us, nevertheless it can happen that a certain fictive hypothesis may suffice for explaining many phenomena." (Leonhard Euler)
"For many parts of nature can neither be invented with sufficient subtlety, nor demonstrated with sufficient perspicuity, nor accommodated unto use with sufficient dexterity without the aid and
intervention of mathematics.” (Galileo Galilei)
"The great book of nature can be read only by those who know the language in which it was written. And this language is mathematics." (Galileo Galilei)
"Nature's great book is written in mathematical symbols." (Galileo Galilei)
"For many parts of nature can neither be invented with sufficient subtlety, nor demonstrated with sufficient perspicuity, nor accommodated unto use with sufficient dexterity without the aid and intervention of mathematics."(Morris Kline, "Mathematics and the Physical World, 1959)
"it is the most widely accepted axiom in the natural science that Nature makes use of the fewest possible means" (Johannes Kepler)
"Nature is pleased with simplicity, and affects not the pomp of superfluous causes." (Morris Kline, "Mathematics and the Physical World", 1959)
22 August 2017
On Proofs (Unsourced)
"A proof tells us where to concentrate our doubts." (Morris Kline)
"Any good theorem should have several proofs, the more the better. For two reasons: usually, different proofs have different strengths and weaknesses, and they generalise in different directions: they are not just repetitions of each other." (Sir Michael Atiyah)
"Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs." (Paul Halmos)
"I think some intuition leaks out in every step of an induction proof." (Jim Propp)
"In science nothing capable of proof ought to be accepted without proof." (Richard Dedekind)
"It always seems to me absurd to speak of a complete proof, or of a theorem being rigorously demonstrated. An incomplete proof is no proof, and a mathematical truth not rigorously demonstrated is not demonstrated at all." (James J Sylvester)
"Just give me the insights. I can always come up with the proofs!" (Bernhard Riemann)
"[…] mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs." (Felix Klein)
"Mathematics is the most exact science, and its conclusions are capable of absolute proof. But this is so only because mathematics does not attempt to draw absolute conclusions. All mathematical truths are relative, conditional." (Charles P Steinmetz)
"Mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict reasoning." (John Locke)
"Mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict reasoning." (John Locke)
"No human investigation can claim to be scientific if it doesn't pass the test of mathematical proof." (Leonardo Da Vinci)
"No mathematical exactness without explicit proof from assumed principles – such is the motto of the modern geometer." (George B Halsted)
"Proofs really aren't there to convince you that something is true they're there to show you why it is true." (Andrew Gleason)
"Simplification of modes of proof is not merely an indication of advance in our knowledge of a subject, but is also the surest guarantee of readiness for farther progress.” (Baron William Thomson Kelvin)
"Some facts can be seen more clearly by example than by proof." (Leonard Euler)
"The art of drawing conclusions from experiments and observations consists in evaluating probabilities and in estimating whether they are sufficiently great or numerous enough to constitute proofs." (Antoine Lavoisier)
"The essential quality of a proof is to compel belief." (Pierre de Fermat)
On Problem Solving IX: Errors
"Whenever there is a simple error that most laymen fall for, there is always a slightly more sophisticated version of the same problem that experts fall for." (Amos Tversky)
"It is better to do the right problem the wrong way than to do the wrong problem the right way." (Richard Hamming)
"A sum can be put right: but only by going back till you find the error and working it afresh from that point, never by simply going on." (Clive Staples Lewis, "The Great Divorce", 1945)
"Intuition implies the act of grasping the meaning or significance or structure of a problem without explicit reliance on the analytical apparatus of one’s craft. It is the intuitive mode that yields hypotheses quickly, that produces interesting combinations of ideas before their worth is known. It precedes proof: indeed, it is what the techniques of analysis and proof are designed to test and check. It is founded on a kind of combinatorial playfulness that is only possible when the consequences of error are not overpowering or sinful." (Jerome S Bruner, "On Learning Mathematics", Mathematics Teacher Vol. 53, 1960)
"I suppose it is tempting, if the only tool you have is a hammer, to treat everything as if it were a nail." (Abraham H Maslow, "Toward a Psychology of Being", 1962)
"Heuristics are rules of thumb that help constrain the problem in certain ways (in other words they help you to avoid falling back on blind trial and error), but they don't guarantee that you will find a solution. Heuristics are often contrasted with algorithms that will guarantee that you find a solution - it may take forever, but if the problem is algorithmic you will get there. However, heuristics are also algorithms." (S Ian Robertson, "Problem Solving", 2001)
"In specific cases, we think by applying mental rules, which are similar to rules in computer programs. In most of the cases, however, we reason by constructing, inspecting, and manipulating mental models. These models and the processes that manipulate them are the basis of our competence to reason. In general, it is believed that humans have the competence to perform such inferences error-free. Errors do occur, however, because reasoning performance is limited by capacities of the cognitive system, misunderstanding of the premises, ambiguity of problems, and motivational factors. Moreover, background knowledge can significantly influence our reasoning performance. This influence can either be facilitation or an impedance of the reasoning process." (Carsten Held et al, "Mental Models and the Mind", 2006)
"Swarm intelligence illustrates the complex and holistic way in which the world operates. Order is created from chaos; patterns are revealed; and systems are free to work out their errors and problems at their own level. What natural systems can teach humanity is truly amazing." (Lawrence K Samuels, "Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013)
On Problem Solving VIII: Sciences
"Banishing fundamental facts or problems from science merely because they cannot be dealt with by means of certain prescribed principles would be like forbidding the further extension of the theory of parallels in geometry because the axiom upon which this theory rests has been shown to be unprovable. Actually, principles must be judged from the point of view of science, and not science from the point of view of principles fixed once and for all." (Ernst Zermelo, "Neuer Beweis für die Möglichkeit einer Wohlordnung", Mathematische Annalen 65, 1908)
"Our science is like a store filled with the most subtle intellectual devices for solving the most complex problems, and yet we are almost incapable of applying the elementary principles of rational thought. In every sphere, we seem to have lost the very elements of intelligence: the ideas of limit, measure, degree, proportion, relation, comparison, contingency, interdependence, interrelation of means and ends." (Simone Weil, "The Power of Words", 1937)
"A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery." (George Polya, "How to solve it", 1944)
"The great progress in every science came when, in the study of problems which were modest as compared with ultimate aims, methods were developed that could be extended further and further." (John von Neumann & Oskar Morgenstern, "Theory of Games and Economic Behavior", 1944)
"The more the rate of change increases, the more the problems that face us change and the shorter is the life of the solutions we find to them. Therefore, by the time we find solutions to many of the problems that face us, usually the most important ones, the problems have so changed that our solutions to them are no longer relevant or effective; they are stillborn." (Russell Ackoff, 1981)
"Not only the mathematical way of thinking, but also simulations assisted by mathematical methods, is quite effective in solving problems. The latter is utilized in various fields, including detection of causes of troubles, optimization of expected performances, and best possible adjustments of usage conditions. Conversely, without the aid of mathematical methods, our problem-solving effort will get stuck most probably [...]" (Shiro Hiruta, "Mathematics Contributing to Innovation of Management", [in "What Mathematics Can Do for You"] 2013)
"As long as a branch of science offers an abundance of problems, so long is it alive." (David Hilbert)
"Nothing stimulates great minds to work on enriching knowledge with such force as the posing of difficult but simultaneously interesting problems." (John Bernoulli)
"Science is bound, by the everlasting vow of honour, to face fearlessly every problem which can be fairly presented to it." (Lord Kelvin)
"Science starts from problems, and not from observations." (Karl Popper)
On Problem Solving V: Approaches
"He who seeks for methods without having a definite problem in mind seeks for the most part in vain." (David Hilbert, 1902)
"The materials necessary for solving a mathematical problem are certain relevant items of our formerly acquired mathematical knowledge, as formerly solved problems, or formerly proved theorems. Thus, it is often appropriate to start the work with the question; Do you know a related problem?"
"We acquire any practical skill by imitation and practice. […] Trying to solve problems, you have to observe and to imitate what other people do when solving problems and, finally, you learn to do problems by doing them.
"We can scarcely imagine a problem absolutely new, unlike and unrelated to any formerly solved problem; but if such a problem could exist, it would be insoluble. In fact, when solving a problem, we should always profit from previously solved problems, using their result or their method, or the experience acquired in solving them." (George Polya, 1945)
"We have to find the connection between the data and the unknown. We may represent our unsolved problem as open space between the data and the unknown, as a gap across which we have to construct a bridge. We can start constructing our bridge from either side, from the unknown or from the data. Look at the unknown! And try to think of a familiar problem having the same or a similar unknown. This suggests starting the work from the unknown. Look at the data! Could you derive something useful from the data? This suggests starting the work from the data." (George Pólya, "How to solve it", 1945)
"We should give some consideration to the order in which we work out the details of our plan, especially if our problem is complex. We should not omit any detail, we should understand the relation of the detail before us to the whole problem, we should not lose sight of the connection of the major steps. Therefore, we should proceed in proper order." (George Pólya, "How to solve it", 1945)
"I believe, that the decisive idea which brings the solution of a problem is rather often connected with a well-turned word or sentence. The word or the sentence enlightens the situation, gives things, as you say, a physiognomy. It can precede by little the decisive idea or follow on it immediately; perhaps, it arises at the same time as the decisive idea. […] The right word, the subtly appropriate word, helps us to recall the mathematical idea, perhaps less completely and less objectively than a diagram or a mathematical notation, but in an analogous way. […] It may contribute to fix it in the mind." (George Pólya [in a letter to Jaque Hadamard, "The Psychology of Invention in the Mathematical Field", 1949])
"The problems are solved, not by giving new information, but
by arranging what we have known since long." (Ludwig Wittgenstein, "Philosophical
Investigations", 1953)
"In picking that problem be sure to analyze it carefully to see that it is worth the effort. It takes just as much effort to solve a useless problem as a useful one." (Charles F Kettering, 1955)
"A great many problems are easier to solve rigorously if you know in advance what the answer is." (Ian Stewart, "From Here to Infinity", 1987)
"There are many things you can do with problems besides solving them. First you must define them, pose them. But then of course you can also refine them, depose them, or expose them or even dissolve them! A given problem may send you looking for analogies, and some of these may lead you astray, suggesting new and different problems, related or not to the original. Ends and means can get reversed. You had a goal, but the means you found didn’t lead to it, so you found a new goal they did lead to. It’s called play. Creative mathematicians play a lot; around any problem really interesting they develop a whole cluster of analogies, of playthings." (David Hawkins, "The Spirit of Play", Los Alamos Science, 1987)
"An important symptom of an emerging understanding is the capacity to represent a problem in a number of different ways and to approach its solution from varied vantage points; a single, rigid representation is unlikely to suffice." (Howard Gardner, "The Unschooled Mind", 1991)
"Alternative models are neither right nor wrong, just more or less useful in allowing us to operate in the world and discover more and better options for solving problems." (Andrew Weil," The Natural Mind: A Revolutionary Approach to the Drug Problem", 2004)
"Mostly we rely on stories to put our ideas into context and give them meaning. It should be no surprise, then, that the human capacity for storytelling plays an important role in the intrinsically human-centered approach to problem solving, design thinking."
"A problem thoroughly understood is always fairly simple. Found your opinions on facts, not prejudices. We know too many things that are not true." (Charles F Kettering)
"Divide each problem that you examine into as many parts as you can and as you need to solve them more easily." (Descartes OEuvres, vol. VI)
"Each problem that I solved became a rule which served afterwards to solve other problems." (Descartes, Oeuvres, vol. VI)
"When a problem arises, we should be able to see soon whether it will be profitable to examine some other problems first, and which others, and in which order." (Descartes, OEuvres, vol. X)
"I do believe in simplicity. It is astonishing as well as sad, how many trivial affairs even the wisest thinks he must attend to in a day; how singular an affair he thinks he must omit. When the mathematician would solve a difficult problem, he first frees the equation of all encumbrances, and reduces it to its simplest terms. So simplify the problem of life, distinguish the necessary and the real. Probe the earth to see where your main roots run." (Henry D Thoreau)
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On Problem Solving VII: Mathematicians I
"An expert problem solver must be endowed with two incompatible qualities, a restless imagination and a patient pertinacity." (Howard W Eves)
"Finding the right answer is important, of course. But more important is developing the ability to see that problems have multiple solutions, that getting from X to Y demands basic skills and mental agility, imagination, persistence, patience." (Mary H Futrell)
"I knew nothing, except how to think, how to grapple with a problem and then go on grappling with it until you had solved it." (Sir Barnes Wallis)
"It’s not that I’m so smart, it’s just that I stay with problems longer." (Albert Einstein)
"Man is not born to solve the problems of the universe, but to find out where the problems begin, and then to take his stand within the limits of the intelligible." (Johann Wolfgang von Goethe)
"Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice." (George Polya)
"The life of a mathematician is dominated by an insatiable curiosity, a desire bordering on passion to solve the problems he is studying." (Jean Dieudonne)
"The measure of our intellectual capacity is the capacity to feel less and less satisfied with our answers to better and better problems." (Charles W Churchman)
"The real raison d’etre for the mathematician’s existence is simply to solve problems. So what mathematics really consists of is problems and solutions." (John Casti)
"When I am working on a problem I never think about beauty. I only think about how to solve the problem. But when I have finished, if the solution is not beautiful, I know it is wrong." (Buckminster Fuller)
21 August 2017
On Problem Solving IV: Solvability
"A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted. There remains always something to do; with sufficient study and penetration, we could improve any solution, and, in any case, we can always improve our understanding of the solution."
"We can scarcely imagine a problem absolutely new, unlike and unrelated to any formerly solved problem; but if such a problem could exist, it would be insoluble. In fact, when solving a problem, we should always profit from previously solved problems, using their result or their method, or the experience acquired in solving them." (George Polya, "How to Solve It", 1945)
"The answer to the question ‘Can there be a general method for solving all mathematical problems?’ is no! Perhaps, in a world of unsolved and apparently unsolvable problems, we would have thought that the desirable answer to this question from any point of view, would be yes. But from the point of view of mathematicians a yes would have been far less satisfying than a no is. […] Not only are the problems of mathematics infinite and hence inexhaustible, but mathematics itself is inexhaustible." (Constance Reid, "Introduction to Higher Mathematics for the General Reader", 1959)
"Some problems are just too complicated for rational logical solutions. They admit of insights, not answers." (Jerome B Wiesner, The New Yorker, 1963)
"A problem will be difficult if there are no procedures for generating possible solutions that are guaranteed (or at least likely) to generate the actual solution rather early in the game. But for such a procedure to exist, there must be some kind of structural relation, at least approximate, between the possible solutions as named by the solution-generating process and these same solutions as named in the language of the problem statement." (Herbert A Simon, "The Logic of Heuristic Decision Making", [in "The Logic of Decision and Action"], 1966)
"Deep in the human nature there is an almost irresistible tendency to concentrate physical and mental energy on attempts at solving problems that seem to be unsolvable." (Ragnar Frisch, "From Utopian Theory to Practical Applications", [Nobel lecture] 1970)
"In general, complexity and precision bear an inverse relation to one another in the sense that, as the complexity of a problem increases, the possibility of analysing it in precise terms diminishes. Thus 'fuzzy thinking' may not be deplorable, after all, if it makes possible the solution of problems which are much too complex for precise analysis." (Lotfi A Zadeh, "Fuzzy languages and their relation to human intelligence", 1972)
"A great many problems are easier to solve rigorously if you know in advance what the answer is." (Ian Stewart, "From Here to Infinity", 1987)
"A common mistake in problem solving is to encompass too much territory, which dilutes any solutions chance of success. [...] However, the opposite error occurs more frequently." (Terry Richey, "The Marketer's Visual Tool Kit", 1994)
"[…] the meaning of the word 'solve' has undergone a series of major changes. First that word meant 'find a formula'. Then its meaning changed to 'find approximate numbers'. Finally, it has in effect become 'tell me what the solutions look like'. In place of quantitative answers, we seek qualitative ones."
"Theorems are fun especially when you are the prover, but then the pleasure fades. What keeps us going are the unsolved problems." (Carl Pomerance, MAA, 2000)
"Heuristics are rules of thumb that help constrain the problem in certain ways (in other words they help you to avoid falling back on blind trial and error), but they don't guarantee that you will find a solution. Heuristics are often contrasted with algorithms that will guarantee that you find a solution - it may take forever, but if the problem is algorithmic you will get there. However, heuristics are also algorithms." (S Ian Robertson, "Problem Solving", 2001)
"Solving a problem for which you know there’s an answer is like climbing a mountain with a guide, along a trail someone else has laid. In mathematics, the truth is somewhere out there in a place no one knows, beyond all the beaten paths. And it’s not always at the top of the mountain. It might be in a crack on the smoothest cliff or somewhere deep in the valley." (Yōko Ogawa, "The Housekeeper and the Professor", 2003)
"Knowing a solution is at hand is a huge advantage; it’s like not having a 'none of the above' option. Anyone with reasonable competence and adequate resources can solve a puzzle when it is presented as something to be solved. We can skip the subtle evaluations and move directly to plugging in possible solutions until we hit upon a promising one. Uncertainty is far more challenging."
"Mathematical good taste, then, consists of using intelligently the concepts and results available in the ambient mathematical culture for the solution of new problems. And the culture evolves because its key concepts and results change, slowly or brutally, to be replaced by new mathematical beacons." (David Ruelle, "The Mathematician's Brain", 2007)
"The beauty of mathematics is that clever arguments give answers to problems for which brute force is hopeless, but there is no guarantee that a clever argument always exists!" (David Ruelle, "The Mathematician's Brain", 2007)
"Every problem has a solution; it may sometimes just need another perspective." (Rebecca Mallery et al, "NLP for Rookies", 2009)
"Don't mistake a solution method for a problem definition - especially if it’s your own solution method." (Donald C Gause & Gerald M Weinberg, "Are Your Lights On?", 2011)
"The really important thing in dealing with problems is to know that the question is never answered, but that it doesn't matter, as long as you keep asking. It's only when you fool yourself into thinking you have the final problem definition - the final, true answer - that you can be fooled into thinking you have the final solution. And if you think that, you're always wrong, because there is no such thing as a 'final solution'." (Donald C Gause & Gerald M Weinberg, "Are Your Lights On?", 2011)
"I have not seen any problem, however complicated, which, when you looked at it in the right way, did not become still more complicated." (Paul Anderson)
"It is efficient to look for beautiful solutions first and settle for ugly ones only as a last resort. [...] It is a good rule of thumb that the more beautiful the guess, the more likely it is to survive." (Timothy Gowers)
"One is always a long way from solving a problem until one actually has the answer." (Stephen Hawking)
"The best way to escape from a problem is to solve it." (Brendan Francis)
"The worst thing you can do to a problem is solve it completely." (Daniel Kleitman)
"This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by reason, for in mathematics there is no ignorabimus." (David Hilbert)
"We can not solve our problems with the same level of thinking that created them." (Albert Einstein)
"When the answer to a mathematical problem cannot be found, then the reason is frequently that we have not recognized the general idea from which the given problem only appears as a link in a chain of related problems." (David Hilbert)
On Problem Solving III: Understanding the Problem
"The correct solution to any problem depends principally on a true understanding of what the problem is." (Arthur M Wellington, "The Economic Theory of Railway Location", 1887)
"First, we have to understand the problem; we have to see clearly what is required. Second, we have to see how the various items are connected, how the unknown is linked to the data, in order to obtain the idea of the solution, to make a plan. Third, we carry out our plan. Fourth, we look back at the completed solution, we review and discuss it." (George Pólya, "How to solve it", 1945)
"The intelligent problem-solver tries first of all to understand the problem as fully and as clearly as he can. Yet understanding alone is not enough; he must concentrate upon the problem, he must desire earnestly to obtain its solution. If he cannot summon up real desire for solving the problem he would do better to leave it alone. The open secret of real success is to throw your whole personality into your problem." (George Pólya, "How to Solve It", 1945)
"Trying to find the solution, we may repeatedly change our point of view, our way of looking at the problem. We have to shift our position again and again. Our conception of the problem is likely to be rather incomplete when we start the work; our outlook is different when we have made some progress; it is again different when we have almost obtained the solution.
"An important symptom of an emerging understanding is the capacity to represent a problem in a number of different ways and to approach its solution from varied vantage points; a single, rigid representation is unlikely to suffice." (Howard Gardner, "The Unschooled Mind", 1991)
"[By understanding] I mean simply a sufficient grasp of concepts, principles, or skills so that one can bring them to bear on new problems and situations, deciding in which ways one’s present competencies can suffice and in which ways one may require new skills or knowledge." (Howard Gardner, "The Unschooled Mind", 1991)
"An internal model corresponds to a specific concrete situation in the external world and allows us to reason about the external situation. To do so you used information about the problem presented in the problem statement. The process of understanding, then, refers to constructing an initial mental representation of what the problem is, based on the information in the problem statement about the goal, the initial state, what you are not allowed to do, and what operator to apply, as well as your own personal past experience." (S Ian Robertson, "Problem Solving", 2001)
"Lack of understanding of a problem (or a concept, or a system of relations) can lead to superficial answers to problems, as well as to blindly following a procedure." (S Ian Robertson, "Problem Solving", 2001)
"Understanding a problem means building some kind of representation of the problem in one's mind, based on what the situation is or what the problem statement says and on one's prior knowledge. It is then possible to reason about the problem within this mental representation. Generating a useful mental representation is therefore the most important single factor for successful problem solving." (S Ian Robertson, "Problem Solving", 2001)
"If you can’t think of at least three things that might be wrong with your understanding of the problem, you don’t understand the problem."
"A problem thoroughly understood is always fairly simple." (Charles Kettering)
"One measure of our understanding is the number of independent ways we are able to get to the same result." (Richard P Feynman)
"Very often in mathematics the crucial problem is to recognize and discover what are the relevant concepts; once this is accomplished the job may be more than half done." (Israel N Herstein)
On Problem Solving II: What Makes a Problem Worthy?
"Moreover a mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock our efforts. It should be to us a guidepost on the mazy path to hidden truths, and ultimately a reminder of our pleasure in the successful solution." (David Hilbert [Paris International Congress], 1900)
"To find a new problem which is both interesting and accessible, is not so easy; we need experience, taste, and good luck. Yet we should not fail to look around for more good problems when we have succeeded in solving one. Good problems and mushrooms of certain kinds have something in common; they grow in clusters. Having found one, you should look around; there is a good chance that there are some more quite near." (George Pólya, "How to Solve It", 1945)
"The real raison d'etre for the mathematician's existence is simply to solve problems. So what mathematics really consists of is problems and solutions. And it is the 'good' problems, the ones that challenge the greatest minds for decades, if not centuries, that eventually become enshrined as mathematical mountaintops." (John L Casti, "Mathematical Mountaintops: The Five Most Famous Problems of All Time", 2001)
"A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery." (George Polya)
"I hope that seeing the excitement of solving this problem will make young mathematicians realize that there are lots and lots of other problems in mathematics which are going to be just as challenging in the future." (Andrew Wiles)
"No one should pick a problem, or make a resolution, unless he realizes that the ultimate value of it will offset the inevitable discomfort and trouble that always goes along with the accomplishment of anything worthwhile. So, let us not waste our time and effort on some trivial thing." (Charles F Kettering)
"Problems worthy of attack prove their worth by fighting back." (Piet Hein)
"The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver." (Israel N Herstein)
"There is no such thing as a problem without a gift. We seek problems because we need their gifts." (Richard Bach)
"You are never sure whether or not a problem is good unless you actually solve it." (Mikhail Gromov)
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On Problem Solving I (Stating the Problem)
"Difficulties in identifying problems have delayed statistics far more than difficulties in solving problems." (John W Tukey, Unsolved Problems of Experimental Statistics, 1954)
"We call a problem well-defined if there is a test which can be applied to a proposed solution. In case the proposed solution is a solution, the test must confirm this in a finite number of steps." (John McCarthy, "The Inversion of Functions Denned by Turing Machines", 1956)
"A problem that is located and identified is already half solved!" (Bror R Carlson, "Managing for Profit", 1961)
"No mathematical idea has ever been published in the way it was discovered. Techniques have been developed and are used, if a problem has been solved, to turn the solution procedure upside down, or if it is a larger complex of statements and theories, to turn definitions into propositions, and propositions into definitions, the hot invention into icy beauty. This then if it has affected teaching matter, is the didactical inversion, which as it happens may be anti-didactical." (Hans Freudenthal, "The Concept and the Role of the Model in Mathematics and Natural and Social Sciences", 1961)
"A problem will be difficult if there are no procedures for generating possible solutions that are guaranteed (or at least likely) to generate the actual solution rather early in the game. But for such a procedure to exist, there must be some kind of structural relation, at least approximate, between the possible solutions as named by the solution-generating process and these same solutions as named in the language of the problem statement." (Herbert A Simon, "The Logic of Heuristic Decision Making", [in "The Logic of Decision and Action"], 1966)
"Definitions, like questions and metaphors, are instruments for thinking. Their authority rests entirely on their usefulness, not their correctness. We use definitions in order to delineate problems we wish to investigate, or to further interests we wish to promote. In other words, we invent definitions and discard them as suits our purposes." (Neil Postman, "Language Education in a Knowledge Context", 1980)
"Define the problem before you pursue a solution." (John Williams, Inc. Magazine's Guide to Small Business Success, 1987)
"There are many things you can do with problems besides solving them. First you must define them, pose them. But then of course you can also refine them, depose them, or expose them or even dissolve them! A given problem may send you looking for analogies, and some of these may lead you astray, suggesting new and different problems, related or not to the original. Ends and means can get reversed. You had a goal, but the means you found didn’t lead to it, so you found a new goal they did lead to. It’s called play. Creative mathematicians play a lot; around any problem really interesting they develop a whole cluster of analogies, of playthings." (David Hawkins, "The Spirit of Play", Los Alamos Science, 1987)
"Most people would rush ahead and implement a solution before they know what the problem is." (Q T Wiles, Inc. Magazine, 1988)
"An internal model corresponds to a specific concrete situation in the external world and allows us to reason about the external situation. To do so you used information about the problem presented in the problem statement. The process of understanding, then, refers to constructing an initial mental representation of what the problem is, based on the information in the problem statement about the goal, the initial state, what you are not allowed to do, and what operator to apply, as well as your own personal past experience." (S Ian Robertson, "Problem Solving", 2001)
"Problem solving starts off from an initial given situation or statement of a problem (known as the initial state of the problem). Based on the problem situation and your prior knowledge you have to work towards a solution. When you reach it you are in the goal state of the problem. On the way from the initial state to the goal state you pass through a number of intermediate problem states."
"The way a problem is defined determines how we attempt to solve it. […] If the definition is wrong, you will develop the right solution to the wrong problem." (James P Lewis, "Project Planning, Scheduling, and Control" 3rd Ed., 2001)
"Understanding a problem means building some kind of representation of the problem in one's mind, based on what the situation is or what the problem statement says and on one's prior knowledge. It is then possible to reason about the problem within this mental representation. Generating a useful mental representation is therefore the most important single factor for successful problem solving." (S Ian Robertson, "Problem Solving", 2001)
"Don't mistake a solution method for a problem definition - especially if it’s your own solution method." (Donald C Gause & Gerald M Weinberg, "Are Your Lights On?", 2011)
"The fledgling problem solver invariably rushes in with
solutions before taking time to define the problem being solved. Even
experienced solvers, when subjected to social pressure, yield to this demand
for haste. When they do, many solutions are found, but not necessarily to the
problem at hand."
"Framing the right problem is equally or even more important than solving it." (Pearl Zhu, "Change, Creativity and Problem-Solving", 2017)
"A problem well-defined is a problem half solved." (John Dewey)
"The greatest challenge to any thinker is stating the problem in a way that will allow a solution." (Bertrand Russell)
"The mere formulation of a problem is often far more essential than its solution. To raise new questions, new possibilities, to regard old problems from a new angle, requires creative imagination and marks real advances in science." (Albert Einstein)
"To ask the right question is harder than to answer it." (Georg Cantor)
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20 August 2017
On Art: Poetry and Science I
"In science, reason is the guide; in poetry, taste. The object of the one is truth, which is uniform and indivisible; the object of the other is beauty, which is multiform and varied." (Charles C Colton, "Lacon: Many Things in Few Words", 1820)
"True poetry is truer than science, because it is synthetic, and seizes at once what the combination of all the sciences is able, at most, to attain as a final result." (Henri-Frédéric Amiel, 1852)
“[…] those who have never entered upon scientific pursuits know not a tithe of the poetry by which they are surrounded.” (Herbert Spencer, 1855)
"Without poetry our science will appear incomplete, and most of what now passes with us for religion and philosophy will be replaced by poetry." (Matthew Arnold, "The Study of Poetry", 1880)
“The story of scientific discovery has its own epic unity - a unity of purpose and endeavour - the single torch passing from hand to hand through the centuries; and the great moments of science when, after long labour, the pioneers saw their accumulated facts falling into a significant order - sometimes in the form of a law that revolutionised the whole world of thought - have an intense human interest, and belong essentially to the creative imagination of poetry.” (Alfred Noyes, "Watchers of the Sky", 1922)
"What the world needs is a fusion of the sciences and the humanities. The humanities express the symbolic, poetic, and prophetic qualities of the human spirit. Without them we would not be conscious of our history; we would lose our aspirations and the grace of expression that move men’s hearts. The sciences express the creative urge in man to construct a universe which is comprehensible in terms of the human intellect. Without them, mankind would find itself bewildered in a world of natural forces beyond comprehension, victims of ignorance, superstition and fear." (Isidor I Rabi, [address] 1954)
"In science, one tries to say what no one else has ever said before. In poetry, one tries to say what everyone else has already said, but better. This explains, in essence, why good poetry is as rare as good science." (Anthony Zee, "Fearful Symmetry: The Search for Beauty in Modern Physics", 1986)
“The poetry of science is in some sense embodied in its great equations, and these equations can also be peeled. But their layers represent their attributes and consequences, not their meanings.” (Graham Farmelo, 2002)
Poets and Mathematicians
"[…] it is impossible to be a mathematician without being a poet in soul […] imagination and invention are identical […] the poet has only to perceive that which others do not perceive, to look deeper than others look. And the mathematician must do the same thing." (Sophia Kovalevskaya)
“A mathematician who is not also something of a poet will never be a complete mathematician.” (Karl Weierstrass)
"A mathematician, like a painter or a poet, is a maker of patterns. [...]. The mathematician's patterns, like the painter's or the poet's, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics." (Godfrey Harold Hardy, “A Mathematician's Apology”, 1941)
"The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal." (William James)
“The difference between the poet and the mathematician is that the poet tries to get his head into the heavens while the mathematician tries to get the heavens into his head.” (Gilbert Keith Chesterton)
"Imagination does not breed insanity. Exactly what does breed insanity is reason. Poets do not go mad […] mathematicians go mad.” (Gilbert Keith Chesterton)
"The imagination in a mathematician who creates makes no less difference than in a poet who invents […]." (Jean Le Rond D'Alembert, Discours Preliminaire de L'Encyclopedie, 1967)
"[…] mathematicians and poets are people who believe in the power of words, of concepts and giving names to concepts" (Cédric Villani)
“A mathematician who is not also something of a poet will never be a complete mathematician.” (Karl Weierstrass)
"A mathematician, like a painter or a poet, is a maker of patterns. [...]. The mathematician's patterns, like the painter's or the poet's, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics." (Godfrey Harold Hardy, “A Mathematician's Apology”, 1941)
"The union of the mathematician with the poet, fervor with measure, passion with correctness, this surely is the ideal." (William James)
“The difference between the poet and the mathematician is that the poet tries to get his head into the heavens while the mathematician tries to get the heavens into his head.” (Gilbert Keith Chesterton)
"Imagination does not breed insanity. Exactly what does breed insanity is reason. Poets do not go mad […] mathematicians go mad.” (Gilbert Keith Chesterton)
"The imagination in a mathematician who creates makes no less difference than in a poet who invents […]." (Jean Le Rond D'Alembert, Discours Preliminaire de L'Encyclopedie, 1967)
"[…] mathematicians and poets are people who believe in the power of words, of concepts and giving names to concepts" (Cédric Villani)
On Art: Poetry and Mathematics I
“Poetry is a form of mathematics, a highly rigorous relationship with words.” (Tahar Ben Jelloun)
"Mathematics is pure poetry." (Immanuel Kant, "Opus Postumum")
“Here, where we reach the sphere of mathematics, we are among processes which seem to some the most inhuman of all human activities and the most remote from poetry. Yet it is here that the artist has the fullest scope of his imagination.” (Havelock Ellis)
“The reason why we do maths is because it's like poetry. It's about patterns, and that really turned me on. It made me feel that maths was in tune with the other things I liked doing.” (Marcus du Sautoy)
"Mathematics is the art of giving the same name to different things. [As opposed to the quotation: Poetry is the art of giving different names to the same thing]. (Henri Poincare)
"There is no getting out of it. Through and through the world is infected with quantity. To talk sense is to talk in quantities. […] You cannot evade quantity. You may fly to poetry and to music, and quantity and number will face you in your rhythms and your octaves. Elegant intellects which despise the theory of quantity are but half developed. They are more to be pitied than blamed." (Alfred N Whitehead)
“You cannot evade quantity. You may fly to poetry and music, and quantity and number will face you in your rhythms and your octaves.” (Alfred N Whitehead)
“What, after all, is mathematics but the poetry of the mind, and what is poetry but the mathematics of the heart?” (David E Smith)
"In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in poetry, it's the exact opposite." (Paul A M Dirac)
"Poetry is as exact a science as geometry." (Gustave Flaubert)
"Mathematics is pure poetry." (Immanuel Kant, "Opus Postumum")
“Here, where we reach the sphere of mathematics, we are among processes which seem to some the most inhuman of all human activities and the most remote from poetry. Yet it is here that the artist has the fullest scope of his imagination.” (Havelock Ellis)
“The reason why we do maths is because it's like poetry. It's about patterns, and that really turned me on. It made me feel that maths was in tune with the other things I liked doing.” (Marcus du Sautoy)
"Mathematics is the art of giving the same name to different things. [As opposed to the quotation: Poetry is the art of giving different names to the same thing]. (Henri Poincare)
"There is no getting out of it. Through and through the world is infected with quantity. To talk sense is to talk in quantities. […] You cannot evade quantity. You may fly to poetry and to music, and quantity and number will face you in your rhythms and your octaves. Elegant intellects which despise the theory of quantity are but half developed. They are more to be pitied than blamed." (Alfred N Whitehead)
“You cannot evade quantity. You may fly to poetry and music, and quantity and number will face you in your rhythms and your octaves.” (Alfred N Whitehead)
“What, after all, is mathematics but the poetry of the mind, and what is poetry but the mathematics of the heart?” (David E Smith)
"In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in poetry, it's the exact opposite." (Paul A M Dirac)
"Poetry is as exact a science as geometry." (Gustave Flaubert)
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Music and Mathematics I (Unsourced)
"Although some older art, music or wines may be better than the newer, it is rather unlikely that this would often apply to science or mathematics." (William F Lucas)
"Architecture is akin to music in that both should be based on the symmetry of mathematics." (Frank L Wright)
"Besides language and music, mathematics is one of the primary manifestations of the free creative power of the human mind." (Hermann Weyl)
"In music, mathematics appears formally, as revelation, as creative idealism. All enjoyment is musical, consequently mathematical. The highest life is mathematics." (Novalis)
"It is Proportion that beautifies everything, the whole Universe consists of it, and Musicke is measured by it." (Orlando Gibbons)
"Little comes to us through time as a complete monument; much comes as remnants; much as techniques, as practical manual; some things because of their close affinity to man, like mathematics; other things because they are always encouraged, like astronomy and geography; other things because of man’s needs, like medicine; and finally some things, because the human being, without wanting to, continues to produce them, like music and the other fine arts." (Johann Wolfgang von Goethe)
"[…] mathematics has liberated itself from language; and one who knows the tremendous labor put into this process and its ever-recurring surprising success, cannot help feeling that mathematics nowadays is more efficient in it particular sphere of the intellectual world than, say, the modern languages in their deplorable condition of decay or even music are on their fronts." (Andreas Speiser)
"Mathematics is music for the mind;
Music is mathematics for the soul." (Stanley Gudder)
"Mathematics is, as it were, a sensuous logic, and relates to philosophy as do the arts, music, and plastic art to poetry." (Friedrich von Schlegel)
"May not Music be described as the Mathematics of sense, and Mathematics as the Music of reason?" (James J Sylvester)
"Music is architecture translated or transposed from space into time; for in music, besides the deepest feeling, there reigns also a rigorous mathematical intelligence." (Georg W F Hegel)
"Music is the arithmetic of sounds as optics is the geometry of light." (Claude Debussy)
"Music is the hidden arithmetical exercise of a soul unconscious that it is calculating." (Gottfried W Leibniz)
"Music is the pleasure the human soul experiences from counting without being aware it is counting." (Gottfried W Leibniz)
"The pleasure we obtain from music comes from counting, but counting unconsciously. Music is nothing but unconscious arithmetic." (Gottfried W Leibniz)
"The Pythagoreans considered all mathematical science to be divided into four parts: one half they marked off as concerned with quantity, the other half with magnitude; and each of these they posited as twofold. A quantity can be considered in regard to its character by itself or in relation to another quantity, magnitudes as either stationary or in motion. Arithmetic, then, studies quantity as such, music the relations between quantities, geometry magnitude at rest, spherics magnitude inherently moving." (Diadochus Proclus)
"There is geometry in the humming of the strings; there is music in the spacing of the spheres." (Pythagoras, cca. 6th century BC)
See also:
Music and Mathematics II
Music and Mathematics III
The Music of Numbers
"Architecture is akin to music in that both should be based on the symmetry of mathematics." (Frank L Wright)
"Besides language and music, mathematics is one of the primary manifestations of the free creative power of the human mind." (Hermann Weyl)
"In music, mathematics appears formally, as revelation, as creative idealism. All enjoyment is musical, consequently mathematical. The highest life is mathematics." (Novalis)
"It is Proportion that beautifies everything, the whole Universe consists of it, and Musicke is measured by it." (Orlando Gibbons)
"Little comes to us through time as a complete monument; much comes as remnants; much as techniques, as practical manual; some things because of their close affinity to man, like mathematics; other things because they are always encouraged, like astronomy and geography; other things because of man’s needs, like medicine; and finally some things, because the human being, without wanting to, continues to produce them, like music and the other fine arts." (Johann Wolfgang von Goethe)
"[…] mathematics has liberated itself from language; and one who knows the tremendous labor put into this process and its ever-recurring surprising success, cannot help feeling that mathematics nowadays is more efficient in it particular sphere of the intellectual world than, say, the modern languages in their deplorable condition of decay or even music are on their fronts." (Andreas Speiser)
"Mathematics is music for the mind;
Music is mathematics for the soul." (Stanley Gudder)
"Mathematics is, as it were, a sensuous logic, and relates to philosophy as do the arts, music, and plastic art to poetry." (Friedrich von Schlegel)
"May not Music be described as the Mathematics of sense, and Mathematics as the Music of reason?" (James J Sylvester)
"Music is architecture translated or transposed from space into time; for in music, besides the deepest feeling, there reigns also a rigorous mathematical intelligence." (Georg W F Hegel)
"Music is the arithmetic of sounds as optics is the geometry of light." (Claude Debussy)
"Music is the hidden arithmetical exercise of a soul unconscious that it is calculating." (Gottfried W Leibniz)
"Music is the pleasure the human soul experiences from counting without being aware it is counting." (Gottfried W Leibniz)
"The pleasure we obtain from music comes from counting, but counting unconsciously. Music is nothing but unconscious arithmetic." (Gottfried W Leibniz)
"The Pythagoreans considered all mathematical science to be divided into four parts: one half they marked off as concerned with quantity, the other half with magnitude; and each of these they posited as twofold. A quantity can be considered in regard to its character by itself or in relation to another quantity, magnitudes as either stationary or in motion. Arithmetic, then, studies quantity as such, music the relations between quantities, geometry magnitude at rest, spherics magnitude inherently moving." (Diadochus Proclus)
"There is geometry in the humming of the strings; there is music in the spacing of the spheres." (Pythagoras, cca. 6th century BC)
See also:
Music and Mathematics II
Music and Mathematics III
The Music of Numbers
On Beauty: Beauty and Mathematics (-1899)
“The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree. And since these (e.g. order and definiteness) are obviously causes of many things, evidently these sciences must treat this sort of causative principle also (i.e. the beautiful) as in some sense a cause.” (Aristotle, "Metaphysica", cca. 350 BC)
“Thus, of all the honorable arts, which are carried out either naturally or proceed in imitation of nature, geometry takes the skill of reasoning as its field. It is hard at the beginning and difficult of access, delightful in its order, full of beauty, unsurpassable in its effect. For with its clear processes of reasoning it illuminates the field of rational thinking, so that it may be understood that geometry belongs to the arts or that the arts are from geometry.” (Agennius Urbicus, “Controversies about Fields”, cca. 4 century BC)
"Wherever there is number, there is beauty.” (Proclus)
“Mathematics make the mind attentive to the objects which it considers. This they do by entertaining it with a great variety of truths, which are delightful and evident, but not obvious. Truth is the same thing to the understanding as music to the ear and beauty to the eye. The pursuit of it does really as much gratify a natural faculty implanted in us by our wise Creator as the pleasing of our senses: only in the former case, as the object and faculty are more spiritual, the delight is more pure, free from regret, turpitude, lassitude, and intemperance that commonly attend sensual pleasures.” (John Arbuthnot, “An Essay on the Usefulness of Mathematical Learning”, 1701)
“By the word symmetry […] one thinks of an external relationship between pleasing parts of a whole; mostly the word is used to refer to parts arranged regularly against one another around a centre. We have […] observed [these parts] one after the other, not always like following like, but rather a raising up from below, a strength out of weakness, a beauty out of ordinariness.” (Johann Wolfgang von Goethe)
“The most distinct and beautiful statement of any truth [in science] must take at last the mathematical form. We might so simplify the rules of moral philosophy, as well as of arithmetic, that one formula would express them both.” (Henry Thoreau, “A Week on the Concord and Merrimack Rivers”, 1873)
"As for everything else, so for a mathematical theory: beauty can be perceived but not explained." (Arthur Cayley, [President’s address] 1883)
“Thus, of all the honorable arts, which are carried out either naturally or proceed in imitation of nature, geometry takes the skill of reasoning as its field. It is hard at the beginning and difficult of access, delightful in its order, full of beauty, unsurpassable in its effect. For with its clear processes of reasoning it illuminates the field of rational thinking, so that it may be understood that geometry belongs to the arts or that the arts are from geometry.” (Agennius Urbicus, “Controversies about Fields”, cca. 4 century BC)
"Wherever there is number, there is beauty.” (Proclus)
“Mathematics make the mind attentive to the objects which it considers. This they do by entertaining it with a great variety of truths, which are delightful and evident, but not obvious. Truth is the same thing to the understanding as music to the ear and beauty to the eye. The pursuit of it does really as much gratify a natural faculty implanted in us by our wise Creator as the pleasing of our senses: only in the former case, as the object and faculty are more spiritual, the delight is more pure, free from regret, turpitude, lassitude, and intemperance that commonly attend sensual pleasures.” (John Arbuthnot, “An Essay on the Usefulness of Mathematical Learning”, 1701)
“By the word symmetry […] one thinks of an external relationship between pleasing parts of a whole; mostly the word is used to refer to parts arranged regularly against one another around a centre. We have […] observed [these parts] one after the other, not always like following like, but rather a raising up from below, a strength out of weakness, a beauty out of ordinariness.” (Johann Wolfgang von Goethe)
“The most distinct and beautiful statement of any truth [in science] must take at last the mathematical form. We might so simplify the rules of moral philosophy, as well as of arithmetic, that one formula would express them both.” (Henry Thoreau, “A Week on the Concord and Merrimack Rivers”, 1873)
"As for everything else, so for a mathematical theory: beauty can be perceived but not explained." (Arthur Cayley, [President’s address] 1883)
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