Infinite and Geometry

“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)

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)

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 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)

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)

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)

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)

✓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)

"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 9: On Mistakes, Blunders and Errors I

"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 S Lewis, "The Great Divorce", 1945)

"Error does not carry any recognizable badge, for when we change our point of view, to focus on a different problem, what had been error may become information, and what had been information may become error." (Solomon Diamond, "Information and Error: An Introduction to Statistical Analysis", 1959) 

"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)

"There is the very real danger that a number of problems which could profitably be subjected to analysis, and so treated by simpler and more revealing techniques. will instead be routinely shunted to the computing machines [...] The role of computing machines as a mathematical tool is not that of a panacea for all computational ills." (Richard E Bellman & Paul Brock, "On the Concepts of a Problem and Problem-Solving", American Mathematical Monthly 67, 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)

"A proof is a story told to and dissected by people who have spent much of their life learning how to read such stories and find mistakes or inconsistencies: people whose main aim is to prove the storyteller wrong, and who possess the uncanny knack of spotting weaknesses and hammering away at them until they collapse in a cloud of dust." (Ian Stewart, "In Pursuit of the Unknown", 2012)

"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)

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💠On Problem Solving 8: On Science

"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)

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