"Things [...] are some of them continuous [...] which are properly and peculiarly called 'magnitudes'; others are discontinuous, in a side-by-side arrangement, and, as it were, in heaps, which are called 'multitudes,' a flock, for instance, a people, a heap, a chorus, and the like. Wisdom, then, must be considered to be the knowledge of these two forms. Since, however, all multitude and magnitude are by their own nature of necessity infinite - for multitude starts from a definite root and never ceases increasing; and magnitude, when division beginning with a limited whole is carried on, cannot bring the dividing process to an end [...] and since sciences are always sciences of limited things, and never of infinites, it is accordingly evident that a science dealing with magnitude [...] or with multitude [...] could never be formulated. […] A science, however, would arise to deal with something separated from each of them, with quantity, set off from multitude, and size, set off from magnitude." (Nicomachus, cca. 100 AD)
"A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth." (Bhaskara II, "Bijaganita", 12th century)
"This world is finite, the other infinite,
reality is blocked by form and image." (Jalaluddin Rumi, "Masnavi-ye Ma ‘navi" Vol. I ["Spritual Verses"], 1262-1264)
"The existence of an actual infinite multitude is impossible. For any set of things one considers must be a specific set. And sets of things are specified by the number of things in them. Now no number is infinite, for number results from counting through a set of units. So no set of things can actually be inherently unlimited, nor can it happen to be unlimited." (St. Thomas Aquinas, "Summa Theologica", cca. 1266-1273)
"The infinity of All ever bringing forth anew, and even as infinite space is around us, so is infinite potentiality, capacity, reception, malleability, matter." (Giordano Bruno, "De immenso", 1591)
Quotes and Resources Related to Mathematics, (Mathematical) Sciences and Mathematicians
30 April 2020
Anaxagoras of Clazomenae - Collected Quotes
"All things were together, infinite both in number and in smallness; for the small too was infinite." (Anaxagoras, cca. 5th century BC)
"And since the portions of the great and the small are equal in number, so too all things would be in everything. Nor is it possible that they should exist apart, but all things have a portion of everything." (Anaxagoras, cca. 5th century BC)
"Mind is infinite and self-ruled, and is mixed with nothing, but is alone itself by itself." (Anaxagoras, cca. 5th century BC)
"The contents of the cosmos are not separated from each other or cut of by an axe not the hot from the cold nor the cold from the hot." (Anaxagoras, cca. 5th century BC)
"There is no smallest among the small and no largest among the large, But always something still smaller and something still larger." (Anaxagoras, cca. 5th century BC)
"Thought is something limitless and independent, and has been mixed with no thing but is alone by itself. […] What was mingled with it would have prevented it from having power over anything in the way in which it does. […] For it is the finest of all things and the purest." (Anaxagoras, cca. 5th century BC)
"And since the portions of the great and the small are equal in number, so too all things would be in everything. Nor is it possible that they should exist apart, but all things have a portion of everything." (Anaxagoras, cca. 5th century BC)
"Mind is infinite and self-ruled, and is mixed with nothing, but is alone itself by itself." (Anaxagoras, cca. 5th century BC)
"The contents of the cosmos are not separated from each other or cut of by an axe not the hot from the cold nor the cold from the hot." (Anaxagoras, cca. 5th century BC)
"There is no smallest among the small and no largest among the large, But always something still smaller and something still larger." (Anaxagoras, cca. 5th century BC)
"Thought is something limitless and independent, and has been mixed with no thing but is alone by itself. […] What was mingled with it would have prevented it from having power over anything in the way in which it does. […] For it is the finest of all things and the purest." (Anaxagoras, cca. 5th century BC)
29 April 2020
On Infinite (1675-1699)
"Only geometry can hand us the thread [which will lead us through] the labyrinth of the continuum's composition, the maximum and the minimum, the infinitesimal and the infinite; and no one will arrive at a truly solid metaphysics except he who has passed through this [labyrinth]." (Gottfried W Leibniz, "Dissertatio Exoterica De Statu Praesenti et Incrementis Novissimis Deque Usu Geometriae", 1676)
"Where, by the way, we may observe a great difference between the proportion of Infinite to Finite, and, of Finite to Nothing. For 1/∞, that which is a part infinitely small, may, by infinite Multiplication, equal the whole: But 0/1 , that which is Nothing can by no Multiplication become equal to Something." (John Wallis, "Treatise of Algebra", 1685)
"Infinities and infinitely small quantities could be taken as fictions, similar to imaginary roots, except that it would make our calculations wrong, these fictions being useful and based in reality." (Gottfried W Leibniz, [letter to Johann Bernoulli] 1689)
"To measure motion, space is as necessary to be considered as time....[They] are made use of to denote the position of finite: real beings, in respect one to another, in those infinite uniform oceans of duration and space." (John Locke, "An Essay Concerning Human Understanding", 1689)
"Where, by the way, we may observe a great difference between the proportion of Infinite to Finite, and, of Finite to Nothing. For 1/∞, that which is a part infinitely small, may, by infinite Multiplication, equal the whole: But 0/1 , that which is Nothing can by no Multiplication become equal to Something." (John Wallis, "Treatise of Algebra", 1685)
"Infinities and infinitely small quantities could be taken as fictions, similar to imaginary roots, except that it would make our calculations wrong, these fictions being useful and based in reality." (Gottfried W Leibniz, [letter to Johann Bernoulli] 1689)
"To measure motion, space is as necessary to be considered as time....[They] are made use of to denote the position of finite: real beings, in respect one to another, in those infinite uniform oceans of duration and space." (John Locke, "An Essay Concerning Human Understanding", 1689)
On Infinite (1650-1674)
"I will prove that there are infinite worlds in an infinite world. Imagine the universe as a great animal, and the stars as worlds like other animals inside it. These stars serve in turn as worlds for other organisms, such as ourselves, horses and elephants. We in our turn are worlds for even smaller organisms such as cankers, lice, worms and mites. And they are earths for other, imperceptible beings. Just as we appear to be a huge world to these little organisms, perhaps our flesh, blood and bodily fluids are nothing more than a connected tissue of little animals that move and cause us to move. Even as they let themselves be led blindly by our will, which serves them as a vehicle, they animate us and combine to produce this action we call life." (Cyrano de Bergerac,"The Other World", 1657)
"Whatever we imagine is finite. Therefore, there is no idea or conception of anything we call finite. No man can have in his mind an image of infinite magnitude; nor conceive infinite swiftness, infinite time, or infinite force, or inmate power." (Thomas Hobbes, "Of Man", 1658)
"Man is equally incapable of seeing the nothingness from which he emerges and the infinity in which he is engulfed." (Blaise Pascal, "Pensées", 1670)
"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)
"We know that there is an infinite, and we know not its nature. As we know it to be false that numbers are finite, it is therefore true that there is a numerical infinity. But we know not of what kind; it is untrue that it is even, untrue that it is odd; for the addition of a unit does not change its nature; yet it is a number, and every number is odd or even (this certainly holds of every finite number). Thus, we may quite well know that there is a God without knowing what He is." (Blaise Pascal, "Pensées", 1670)
"And just as the advantage of decimals consists in this, that when all fractions and roots have been reduced to them they take on in a certain measure the nature of integers, so it is the advantage of infinite variable-sequences that classes of more complicated terms (such as fractions whose denominators are complex quantities, the roots of complex quantities and the roots of affected equations) may be reduced to the class of simple ones: that is, to infinite series of fractions having simple numerators and denominators and without the all but insuperable encumbrances which beset the others." (Isaac Newton, "De methodis serierum et fluxionum" ["The Method of Fluxions and Infinite Series"], 1671)
"In practical life we are compelled to follow what is most probable; in speculative thought we are compelled to follow truth. […] we must take care not to admit as true anything, which is only probable. For when one falsity has been let in, infinite others follow." (Baruch Spinoza, [letter to Hugo Boxel], 1674)
"Whatever we imagine is finite. Therefore, there is no idea or conception of anything we call finite. No man can have in his mind an image of infinite magnitude; nor conceive infinite swiftness, infinite time, or infinite force, or inmate power." (Thomas Hobbes, "Of Man", 1658)
"Man is equally incapable of seeing the nothingness from which he emerges and the infinity in which he is engulfed." (Blaise Pascal, "Pensées", 1670)
"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)
"We know that there is an infinite, and we know not its nature. As we know it to be false that numbers are finite, it is therefore true that there is a numerical infinity. But we know not of what kind; it is untrue that it is even, untrue that it is odd; for the addition of a unit does not change its nature; yet it is a number, and every number is odd or even (this certainly holds of every finite number). Thus, we may quite well know that there is a God without knowing what He is." (Blaise Pascal, "Pensées", 1670)
"And just as the advantage of decimals consists in this, that when all fractions and roots have been reduced to them they take on in a certain measure the nature of integers, so it is the advantage of infinite variable-sequences that classes of more complicated terms (such as fractions whose denominators are complex quantities, the roots of complex quantities and the roots of affected equations) may be reduced to the class of simple ones: that is, to infinite series of fractions having simple numerators and denominators and without the all but insuperable encumbrances which beset the others." (Isaac Newton, "De methodis serierum et fluxionum" ["The Method of Fluxions and Infinite Series"], 1671)
"In practical life we are compelled to follow what is most probable; in speculative thought we are compelled to follow truth. […] we must take care not to admit as true anything, which is only probable. For when one falsity has been let in, infinite others follow." (Baruch Spinoza, [letter to Hugo Boxel], 1674)
On Infinite (2010-2019)
"Science has revealed a universe that is vast, ancient, violent, strange, and beautiful, a universe of almost infinite variety and possibility one in which time can end in a black hole, and conscious beings can evolve from a soup of minerals." (Leonard Mlodinow, "War of the Worldviews: Where Science and Spirituality Meet - and Do Not", 2011)
"A bell cannot tell time, but it can be moved in just such a way as to say twelve o’clock - similarly, a man cannot calculate infinite numbers, but he can be moved in just such a way as to say pi." (Daniel Tammet, "Thinking in Numbers: How Maths Illuminates Our Lives", 2012)
"Most of the world is of great roughness and infinite complexity. However, the infinite sea of complexity includes two islands of simplicity: one of Euclidean simplicity and a second of relative simplicity in which roughness is present but is the same at all scales." (Benoît Mandelbrot, "The Fractalist", 2012)
"Order is not universal. In fact, many chaologists and physicists posit that universal laws are more flexible than first realized, and less rigid - operating in spurts, jumps, and leaps, instead of like clockwork. Chaos prevails over rules and systems because it has the freedom of infinite complexity over the known, unknown, and the unknowable." (Lawrence K Samuels, "Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013)
"Why do mathematicians care so much about pi? Is it some kind of weird circle fixation? Hardly. The beauty of pi, in part, is that it puts infinity within reach. Even young children get this. The digits of pi never end and never show a pattern. They go on forever, seemingly at random - except that they can’t possibly be random, because they embody the order inherent in a perfect circle. This tension between order and randomness is one of the most tantalizing aspects of pi." (Steven Strogatz, "Why PI Matters" 2015)
"Zero seems as diaphanous as a fairy’s wing, yet it is as powerful as a black hole. The obverse of infinity, it’s enthroned at the center of the number line - at least as the line is usually drawn - making it a natural center of attention. It has no effect when added to other numbers, as if it were no more substantial than a fleeting thought. But when multiplied times other numbers it seems to exert uncanny power, inexorably sucking them in and making them vanish into itself at the center of things. If you’re into stark simplicity, you can express any number (that is, any number that’s capable of being written out) with the use of zero and just one other number, one." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)
"A bell cannot tell time, but it can be moved in just such a way as to say twelve o’clock - similarly, a man cannot calculate infinite numbers, but he can be moved in just such a way as to say pi." (Daniel Tammet, "Thinking in Numbers: How Maths Illuminates Our Lives", 2012)
"Most of the world is of great roughness and infinite complexity. However, the infinite sea of complexity includes two islands of simplicity: one of Euclidean simplicity and a second of relative simplicity in which roughness is present but is the same at all scales." (Benoît Mandelbrot, "The Fractalist", 2012)
"Order is not universal. In fact, many chaologists and physicists posit that universal laws are more flexible than first realized, and less rigid - operating in spurts, jumps, and leaps, instead of like clockwork. Chaos prevails over rules and systems because it has the freedom of infinite complexity over the known, unknown, and the unknowable." (Lawrence K Samuels, "Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013)
"Why do mathematicians care so much about pi? Is it some kind of weird circle fixation? Hardly. The beauty of pi, in part, is that it puts infinity within reach. Even young children get this. The digits of pi never end and never show a pattern. They go on forever, seemingly at random - except that they can’t possibly be random, because they embody the order inherent in a perfect circle. This tension between order and randomness is one of the most tantalizing aspects of pi." (Steven Strogatz, "Why PI Matters" 2015)
"Calculus is the study of things that are changing. It is difficult to make theories about things that are always changing, and calculus accomplishes it by looking at infinitely small portions, and sticking together infinitely many of these infinitely small portions." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
"In category theory there is always a tension between the idealism and the logistics. There are many structures that naturally want to have infinite dimensions, but that is too impractical, so we try and think about them in the context of just a finite number of dimensions and struggle with the consequences of making these logistics workable."
"Infinity is a Loch Ness Monster, capturing the imagination with its awe-inspiring size but elusive nature. Infinity is a dream, a vast fantasy world of endless time and space. Infinity is a dark forest with unexpected creatures, tangled thickets and sudden rays of light breaking through. Infinity is a loop that springs open to reveal an endless spiral." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
"Mathematics is particularly good at making things out of itself, like how higher-dimensional spaces are built up from lower-dimensional spaces. This is because mathematics deals with abstract ideas like space and dimensions and infinity, and is itself an abstract idea. […] Mathematics is abstract enough that we can always make more mathematics out of mathematics." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)"The Axiom of Choice says that it is possible to make an infinite number of arbitrary choices. […] Mathematicians don’t exactly care whether or not the Axiom of Choice holds over all, but they do care whether you have to use it in any given situation or not." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
"Zero seems as diaphanous as a fairy’s wing, yet it is as powerful as a black hole. The obverse of infinity, it’s enthroned at the center of the number line - at least as the line is usually drawn - making it a natural center of attention. It has no effect when added to other numbers, as if it were no more substantial than a fleeting thought. But when multiplied times other numbers it seems to exert uncanny power, inexorably sucking them in and making them vanish into itself at the center of things. If you’re into stark simplicity, you can express any number (that is, any number that’s capable of being written out) with the use of zero and just one other number, one." (David Stipp, "A Most Elegant Equation: Euler's Formula and the Beauty of Mathematics", 2017)
On Infinite (2000-2009)
"If you look at zero you see nothing; but look through it and you will see the world. For zero brings into focus the great, organic sprawl of mathematics, and mathematics in turn the complex nature of things." (Robert Kaplan, "The Nothing that Is: A Natural History of Zero", 2000)
"Zero is powerful because it is infinity’s twin. They are equal and opposite, yin and yang. They are equally paradoxical and troubling. The biggest questions in science and religion are about nothingness and eternity, the void and the infinite, zero and infinity. The clashes over zero were the battles that shook the foundations of philosophy, of science, of mathematics, and of religion. Underneath every revolution lay a zero - and an infinity." (Charles Seife, "Zero: The Biography of a Dangerous Idea", 2000)
"Mathematical truth is not totally objective. If a mathematical statement is false, there will be no proofs, but if it is true, there will be an endless variety of proofs, not just one! Proofs are not impersonal, they express the personality of their creator/discoverer just as much as literary efforts do. If something important is true, there will be many reasons that it is true, many proofs of that fact. [...] each proof will emphasize different aspects of the problem, each proof will lead in a different direction. Each one will have different corollaries, different generalizations. [...] the world of mathematical truth has infinite complexity […]" (Gregory Chaitin, "Meta Math: The Quest for Omega", 2005)
"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! We just saw a clever argument to prove that there are infinitely many primes, but we don't know any argument to prove that there are infinitely many pairs of twin primes." (David Ruelle, "The Mathematician's Brain", 2007)
"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 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)
"The system is highly sensitive to some small changes and blows them up into major alterations in weather patterns. This is popularly known as the butterfly effect in that it is possible for a butterfly to flap its wings in São Paolo, so making a tiny change to air pressure there, and for this tiny change to escalate up into a hurricane over Miami. You would have to measure the flapping of every butterfly’s wings around the earth with infinite precision in order to be able to make long-term forecasts. The tiniest error made in these measurements could produce spurious forecasts. However, short-term forecasts are possible because it takes time for tiny differences to escalate." (Ralph D Stacey, "Strategic Management and Organisational Dynamics: The Challenge of Complexity" 5th Ed. , 2007)
"Zero is the mathematically defined numerical function of nothingness. It is used not for an evasion but for an apprehension of reality. Zero is by far the most interesting number among all the others: It is a symbol for what is not there. It is an emptiness that increases any number it's added to. Zero is an inexhaustible and indispensable paradox. Zero is the only number which can be divided by every other number. Zero is also only number which can divide no other number. It seems zero is also the most debated number in mathematics. We know that mathematicians are involved in heated philosophical and logical discussions around the issues of zero: Can we divide a number by zero? Is the result of this division infinity or not? Is zero a positive or a negative number? Is it even or is it odd?" (Fahri Karakas, "Reflections on zero and zero-centered spirituality in organizations", 2008)
"Zero is powerful because it is infinity’s twin. They are equal and opposite, yin and yang. They are equally paradoxical and troubling. The biggest questions in science and religion are about nothingness and eternity, the void and the infinite, zero and infinity. The clashes over zero were the battles that shook the foundations of philosophy, of science, of mathematics, and of religion. Underneath every revolution lay a zero - and an infinity." (Charles Seife, "Zero: The Biography of a Dangerous Idea", 2000)
"Mathematical truth is not totally objective. If a mathematical statement is false, there will be no proofs, but if it is true, there will be an endless variety of proofs, not just one! Proofs are not impersonal, they express the personality of their creator/discoverer just as much as literary efforts do. If something important is true, there will be many reasons that it is true, many proofs of that fact. [...] each proof will emphasize different aspects of the problem, each proof will lead in a different direction. Each one will have different corollaries, different generalizations. [...] the world of mathematical truth has infinite complexity […]" (Gregory Chaitin, "Meta Math: The Quest for Omega", 2005)
"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! We just saw a clever argument to prove that there are infinitely many primes, but we don't know any argument to prove that there are infinitely many pairs of twin primes." (David Ruelle, "The Mathematician's Brain", 2007)
"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 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)
"The system is highly sensitive to some small changes and blows them up into major alterations in weather patterns. This is popularly known as the butterfly effect in that it is possible for a butterfly to flap its wings in São Paolo, so making a tiny change to air pressure there, and for this tiny change to escalate up into a hurricane over Miami. You would have to measure the flapping of every butterfly’s wings around the earth with infinite precision in order to be able to make long-term forecasts. The tiniest error made in these measurements could produce spurious forecasts. However, short-term forecasts are possible because it takes time for tiny differences to escalate." (Ralph D Stacey, "Strategic Management and Organisational Dynamics: The Challenge of Complexity" 5th Ed. , 2007)
"Zero is the mathematically defined numerical function of nothingness. It is used not for an evasion but for an apprehension of reality. Zero is by far the most interesting number among all the others: It is a symbol for what is not there. It is an emptiness that increases any number it's added to. Zero is an inexhaustible and indispensable paradox. Zero is the only number which can be divided by every other number. Zero is also only number which can divide no other number. It seems zero is also the most debated number in mathematics. We know that mathematicians are involved in heated philosophical and logical discussions around the issues of zero: Can we divide a number by zero? Is the result of this division infinity or not? Is zero a positive or a negative number? Is it even or is it odd?" (Fahri Karakas, "Reflections on zero and zero-centered spirituality in organizations", 2008)
On Infinite (1990-1999)
"Probability does pervade the universe, and in this sense, the old chestnut about baseball imitating life really has validity. The statistics of streaks and slumps, properly understood, do teach an important lesson about epistemology, and life in general. The history of a species, or any natural phenomenon, that requires unbroken continuity in a world of trouble, works like a batting streak. All are games of a gambler playing with a limited stake against a house with infinite resources. The gambler must eventually go bust. His aim can only be to stick around as long as possible, to have some fun while he's at it, and, if he happens to be a moral agent as well, to worry about staying the course with honor!" (Stephen J Gould, 1991)
"The scope of Theories of Everything is infinite but bounded; they are necessary parts of a full understanding of things but they are far from sufficient to reveal everything about a Universe like ours. In the pages of this book, we have seen something of what a Theory of Everything might hope to teach us about the unity of the Universe and the way in which it may contain elements that transcend our present compartmentalized view of Nature's ingredients. But we have also learnt that there is more to Everything than meets the eye. Unlike many others that we can imagine, our world contains prospective elements. Theories of Everything can make no impression upon predicting these prospective attributes of reality; yet, strangely, many of these qualities will themselves be employed in the human selection and approval of an aesthetically acceptable Theory of Everything. There is no formula that can deliver all truth, all harmony, all simplicity. No Theory of Everything can ever provide total insight. For, to see through everything, would leave us seeing nothing at all." (John D Barrow, "New Theories of Everything", 1991)
"The digits of pi march to infinity in a predestined yet unfathomable code: they do not repeat periodically, seeming to pop up by blind chance, lacking any perceivable order, rule, reason, or design - ‘random’ integers, ad infinitum." (Richard Preston, "The Mountains of Pi", The New Yorker, March 2, 1992)
"Pi is not the solution to any equation built from a less than infinite series of whole numbers. If equations are trains threading the landscape of numbers, then no train stops at pi." (Richard Preston, "The Mountains of Pi", The New Yorker, March 2, 1992)
"Much of what the universe had been, was, and would be, Newton had disclosed, was the outcome of an infinity of material particles all pulling on one another simultaneously. If the result of all that gravitational tussling had appeared to the Greeks to be a cosmos, it was simply because the underlying equation describing their behavior had itself turned out to be every bit a cosmos-orderly, beautiful, and decent." (Michael Guillen," Five Equations That Changed the World", 1995)
"While the equations represent the discernment of eternal and universal truths, however, the manner in which they are written is strictly, provincially human. That is what makes them so much like poems, wonderfully artful attempts to make infinite realities comprehensible to finite beings." (Michael Guillen," Five Equations That Changed the World", 1995)
"In an infinite universe, every point in space-time is the center." (David Zindell, "War in Heaven", 1998)
"Mathematics, in one view, is the science of infinity." (Phillip J Davis & Reuben Hersh, "The Mathematical Experience", 1999)
"The scope of Theories of Everything is infinite but bounded; they are necessary parts of a full understanding of things but they are far from sufficient to reveal everything about a Universe like ours. In the pages of this book, we have seen something of what a Theory of Everything might hope to teach us about the unity of the Universe and the way in which it may contain elements that transcend our present compartmentalized view of Nature's ingredients. But we have also learnt that there is more to Everything than meets the eye. Unlike many others that we can imagine, our world contains prospective elements. Theories of Everything can make no impression upon predicting these prospective attributes of reality; yet, strangely, many of these qualities will themselves be employed in the human selection and approval of an aesthetically acceptable Theory of Everything. There is no formula that can deliver all truth, all harmony, all simplicity. No Theory of Everything can ever provide total insight. For, to see through everything, would leave us seeing nothing at all." (John D Barrow, "New Theories of Everything", 1991)
"The digits of pi march to infinity in a predestined yet unfathomable code: they do not repeat periodically, seeming to pop up by blind chance, lacking any perceivable order, rule, reason, or design - ‘random’ integers, ad infinitum." (Richard Preston, "The Mountains of Pi", The New Yorker, March 2, 1992)
"Pi is not the solution to any equation built from a less than infinite series of whole numbers. If equations are trains threading the landscape of numbers, then no train stops at pi." (Richard Preston, "The Mountains of Pi", The New Yorker, March 2, 1992)
"Much of what the universe had been, was, and would be, Newton had disclosed, was the outcome of an infinity of material particles all pulling on one another simultaneously. If the result of all that gravitational tussling had appeared to the Greeks to be a cosmos, it was simply because the underlying equation describing their behavior had itself turned out to be every bit a cosmos-orderly, beautiful, and decent." (Michael Guillen," Five Equations That Changed the World", 1995)
"While the equations represent the discernment of eternal and universal truths, however, the manner in which they are written is strictly, provincially human. That is what makes them so much like poems, wonderfully artful attempts to make infinite realities comprehensible to finite beings." (Michael Guillen," Five Equations That Changed the World", 1995)
"In an infinite universe, every point in space-time is the center." (David Zindell, "War in Heaven", 1998)
"Mathematics, in one view, is the science of infinity." (Phillip J Davis & Reuben Hersh, "The Mathematical Experience", 1999)
On Infinite (1980-1989)
"'Infinity' is not a phenomenon - it is only a word which enables us somehow to learn truths about finite things." (Yuri I Manin, "Mathematics and Physics", 1981)
"The ‘eyes of the mind’ must be able to see in the phase space of mechanics, in the space of elementary events of probability theory, in the curved four-dimensional space-time of general relativity, in the complex infinite dimensional projective space of quantum theory. To comprehend what is visible to the ‘actual eyes’, we must understand that it is only the projection of an infinite dimensional world on the retina." (Yuri I Manin, "Mathematics and Physics", 1981)
"In the initial stages of research, mathematicians do not seem to function like theorem-proving machines. Instead, they use some sort of mathematical intuition to ‘see’ the universe of mathematics and determine by a sort of empirical process what is true. This alone is not enough, of course. Once one has discovered a mathematical truth, one tries to find a proof for it." (Rudy Rucker, "Infinity and the Mind: The science and philosophy of the infinite", 1982)
"At present, no complete account can be given - one may as well ask for an inventory of the entire products of the human imagination - and indeed such an account would be premature, since mental models are supposed to be in people's heads, and their exact constitution is an empirical question. Nevertheless, there are three immediate constraints on possible models. […] 1. The principle of computability: Mental models, and the machinery for constructing and interpreting them, are computable. […] 2. The principle of finitism: A mental model must be finite in size and cannot directly represent an infinite domain. […] 3. The principle of constructivism: A mental model is constructed from tokens arranged in a particular structure to represent a state of affairs." (Philip Johnson-Laird, "Mental Models" 1983)
"[…] a mathematician's ultimate concern is that his or her inventions be logical, not realistic. This is not to say, however, that mathematical inventions do not correspond to real things. They do, in most, and possibly all, cases. The coincidence between mathematical ideas and natural reality is so extensive and well documented, in fact, that it requires an explanation. Keep in mind that the coincidence is not the outcome of mathematicians trying to be realistic - quite to the contrary, their ideas are often very abstract and do not initially appear to have any correspondence to the real world. Typically, however, mathematical ideas are eventually successfully applied to describe real phenomena […]"(Michael Guillen, "Bridges to Infinity: The Human Side of Mathematics", 1983)
"[…] mathematics is not a science – it is not capable of proving or disproving the existence of real things. In fact, a mathematician’s ultimate concern is that his or her inventions be logical, not realistic." (Michael Guillen,"Bridges to Infinity: The Human Side of Mathematics", 1983)
"The invention of the differential calculus was based on the recognition that an instantaneous rate is the asymptotic limit of averages in which the time interval involved is systematically shrunk. This is a concept that mathematicians recognized long before they had the skill to actually compute such an asymptotic limit." (Michael Guillen,"Bridges to Infinity: The Human Side of Mathematics", 1983)
"The progress of mathematics can be viewed as a movement from the infinite to the finite. At the start, the possibilities of a theory, for example, the theory of enumeration appear to be boundless. Rules for the enumeration of sets subject to various conditions, or combinatorial objects as they are often called, appear to obey an indefinite variety of and seem to lead to a welter of generating functions. We are at first led to suspect that the class of objects with a common property that may be enumerated is indeed infinite and unclassifiable." (Gian-Carlo Rota [Preface to (Ian P Goulden and David M Jackson, "Combinatorial Enumeration", 1983)])
"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, in one view, is the science of infinity." (Phillip J Davis & Reuben Hersh, "The Mathematical Experience", 1985)
"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 world of science lives fairly comfortably with paradox. We know that light is a wave and also that light is a particle. The discoveries made in the infinitely small world of particle physics indicate randomness and chance, and I do not find it any more difficult to live with the paradox of a universe of randomness and chance and a universe of pattern and purpose than I do with light as a wave and light as a particle. Living with contradiction is nothing new to the human being." (Madeline L'Engle, "Two-Part Invention: The Story of a Marriage", 1988)
"The ‘eyes of the mind’ must be able to see in the phase space of mechanics, in the space of elementary events of probability theory, in the curved four-dimensional space-time of general relativity, in the complex infinite dimensional projective space of quantum theory. To comprehend what is visible to the ‘actual eyes’, we must understand that it is only the projection of an infinite dimensional world on the retina." (Yuri I Manin, "Mathematics and Physics", 1981)
"In the initial stages of research, mathematicians do not seem to function like theorem-proving machines. Instead, they use some sort of mathematical intuition to ‘see’ the universe of mathematics and determine by a sort of empirical process what is true. This alone is not enough, of course. Once one has discovered a mathematical truth, one tries to find a proof for it." (Rudy Rucker, "Infinity and the Mind: The science and philosophy of the infinite", 1982)
"At present, no complete account can be given - one may as well ask for an inventory of the entire products of the human imagination - and indeed such an account would be premature, since mental models are supposed to be in people's heads, and their exact constitution is an empirical question. Nevertheless, there are three immediate constraints on possible models. […] 1. The principle of computability: Mental models, and the machinery for constructing and interpreting them, are computable. […] 2. The principle of finitism: A mental model must be finite in size and cannot directly represent an infinite domain. […] 3. The principle of constructivism: A mental model is constructed from tokens arranged in a particular structure to represent a state of affairs." (Philip Johnson-Laird, "Mental Models" 1983)
"[…] a mathematician's ultimate concern is that his or her inventions be logical, not realistic. This is not to say, however, that mathematical inventions do not correspond to real things. They do, in most, and possibly all, cases. The coincidence between mathematical ideas and natural reality is so extensive and well documented, in fact, that it requires an explanation. Keep in mind that the coincidence is not the outcome of mathematicians trying to be realistic - quite to the contrary, their ideas are often very abstract and do not initially appear to have any correspondence to the real world. Typically, however, mathematical ideas are eventually successfully applied to describe real phenomena […]"(Michael Guillen, "Bridges to Infinity: The Human Side of Mathematics", 1983)
"[…] mathematics is not a science – it is not capable of proving or disproving the existence of real things. In fact, a mathematician’s ultimate concern is that his or her inventions be logical, not realistic." (Michael Guillen,"Bridges to Infinity: The Human Side of Mathematics", 1983)
"The invention of the differential calculus was based on the recognition that an instantaneous rate is the asymptotic limit of averages in which the time interval involved is systematically shrunk. This is a concept that mathematicians recognized long before they had the skill to actually compute such an asymptotic limit." (Michael Guillen,"Bridges to Infinity: The Human Side of Mathematics", 1983)
"The progress of mathematics can be viewed as a movement from the infinite to the finite. At the start, the possibilities of a theory, for example, the theory of enumeration appear to be boundless. Rules for the enumeration of sets subject to various conditions, or combinatorial objects as they are often called, appear to obey an indefinite variety of and seem to lead to a welter of generating functions. We are at first led to suspect that the class of objects with a common property that may be enumerated is indeed infinite and unclassifiable." (Gian-Carlo Rota [Preface to (Ian P Goulden and David M Jackson, "Combinatorial Enumeration", 1983)])
"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, in one view, is the science of infinity." (Phillip J Davis & Reuben Hersh, "The Mathematical Experience", 1985)
"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 world of science lives fairly comfortably with paradox. We know that light is a wave and also that light is a particle. The discoveries made in the infinitely small world of particle physics indicate randomness and chance, and I do not find it any more difficult to live with the paradox of a universe of randomness and chance and a universe of pattern and purpose than I do with light as a wave and light as a particle. Living with contradiction is nothing new to the human being." (Madeline L'Engle, "Two-Part Invention: The Story of a Marriage", 1988)
On Infinite (1970-1979)
"I am incapable of conceiving infinity, and yet I do not accept finity." (Simone de Beauvoir, "La Vieillesse", 1970)
"We say the map is different from the territory. But what is the territory? Operationally, somebody went out with a retina or a measuring stick and made representations which were then put on paper. What is on the paper map is a representation of what was in the retinal representation of the man who made the map; and as you push the question back, what you find is an infinite regress, an infinite series of maps. The territory never gets in at all. […] Always, the process of representation will filter it out so that the mental world is only maps of maps, ad infinitum." (Gregory Bateson, "Steps to an Ecology of Mind", 1972)
"The conception of the mental construction which is the fully analysed proof as being an infinite structure must, of course, be interpreted in the light of the intuitionist view that all infinity is potential infinity: the mental construction consists of a grasp of general principles according to which any finite segment of the proof could be explicitly constructed." (Michael Dummett, "The philosophical basis of intuitionistic logic", 1975)
"There is an infinite regress in proofs; therefore proofs do not prove. You should realize that proving is a game, to be played while you enjoy it and stopped when you get tired of it." (Imre Lakatos, "Proofs and Refutations", 1976)
"Mathematical induction […] is an entirely different procedure. Although it, too, leaps from the knowledge of particular cases to knowledge about an infinite sequence of cases, the leap is purely deductive. It is as certain as any proof in mathematics, and an indispensable tool in almost every branch of mathematics." (Martin Gardner, "Aha! Insight", 1978)
"We say the map is different from the territory. But what is the territory? Operationally, somebody went out with a retina or a measuring stick and made representations which were then put on paper. What is on the paper map is a representation of what was in the retinal representation of the man who made the map; and as you push the question back, what you find is an infinite regress, an infinite series of maps. The territory never gets in at all. […] Always, the process of representation will filter it out so that the mental world is only maps of maps, ad infinitum." (Gregory Bateson, "Steps to an Ecology of Mind", 1972)
"The conception of the mental construction which is the fully analysed proof as being an infinite structure must, of course, be interpreted in the light of the intuitionist view that all infinity is potential infinity: the mental construction consists of a grasp of general principles according to which any finite segment of the proof could be explicitly constructed." (Michael Dummett, "The philosophical basis of intuitionistic logic", 1975)
"There is an infinite regress in proofs; therefore proofs do not prove. You should realize that proving is a game, to be played while you enjoy it and stopped when you get tired of it." (Imre Lakatos, "Proofs and Refutations", 1976)
"Mathematical induction […] is an entirely different procedure. Although it, too, leaps from the knowledge of particular cases to knowledge about an infinite sequence of cases, the leap is purely deductive. It is as certain as any proof in mathematics, and an indispensable tool in almost every branch of mathematics." (Martin Gardner, "Aha! Insight", 1978)
On Infinite (1960-1969)
"The eternal lesson is that Mathematics is not something static, closed, but living and developing. Try as we may to constrain it into a closed form, it finds an outlet somewhere and escapes alive." (Péter Rózsa, "Playing with Infinity", 1961)
"[A] sequence is random if it has every property that is shared by all infinite sequences of independent samples of random variables from the uniform distribution." (J. N. Franklin, 1962)
"Mathematics is not a question of calculation perforce but rather the presence of royalty: a law of infinite resonance, consonance and order." (Le Corbusier, "Architecture and the Mathematical Spirit", 1962)
"It is paradoxical that while mathematics has the reputation of being the one subject that brooks no contradictions, in reality it has a long history of successful living with contradictions. This is best seen in the extensions of the notion of number that have been made over a period of 2500 years. From limited sets of integers, to infinite sets of integers, to fractions, negative numbers, irrational numbers, complex numbers, transfinite numbers, each extension, in its way, overcame a contradictory set of demands." (Philip J Davis, "The Mathematics of Matrices", 1965)
"The real difficulty lies in the fact that only a finite number of angels can dance on the head of a pin, whereas the mathematician is more apt to be interested in the infinite angel problem only." (Henri Lebesgue, "Mechanized Mathematics", Bulletin of the American Mathematical Society Vol. 72 (5), 1966)
"Older mathematics appears static while the newer appears dynamic, so that the older mathematics compares to the still-picture stage of photography while the newer mathematics compares to the moving-picture stage. Again, the older mathematics is to the newer much as anatomy is to physiology, wherein the former studies the dead body and the latter studies the living body. Once more, the older mathematics concerned itself with the fixed and the finite while the newer mathematics embraces the changing and the infinite." (Howard W Eves, "In Mathematical Circles", 1969)
"[A] sequence is random if it has every property that is shared by all infinite sequences of independent samples of random variables from the uniform distribution." (J. N. Franklin, 1962)
"Mathematics is not a question of calculation perforce but rather the presence of royalty: a law of infinite resonance, consonance and order." (Le Corbusier, "Architecture and the Mathematical Spirit", 1962)
"It is paradoxical that while mathematics has the reputation of being the one subject that brooks no contradictions, in reality it has a long history of successful living with contradictions. This is best seen in the extensions of the notion of number that have been made over a period of 2500 years. From limited sets of integers, to infinite sets of integers, to fractions, negative numbers, irrational numbers, complex numbers, transfinite numbers, each extension, in its way, overcame a contradictory set of demands." (Philip J Davis, "The Mathematics of Matrices", 1965)
"The real difficulty lies in the fact that only a finite number of angels can dance on the head of a pin, whereas the mathematician is more apt to be interested in the infinite angel problem only." (Henri Lebesgue, "Mechanized Mathematics", Bulletin of the American Mathematical Society Vol. 72 (5), 1966)
"Older mathematics appears static while the newer appears dynamic, so that the older mathematics compares to the still-picture stage of photography while the newer mathematics compares to the moving-picture stage. Again, the older mathematics is to the newer much as anatomy is to physiology, wherein the former studies the dead body and the latter studies the living body. Once more, the older mathematics concerned itself with the fixed and the finite while the newer mathematics embraces the changing and the infinite." (Howard W Eves, "In Mathematical Circles", 1969)
Subscribe to:
Posts (Atom)
On Thresholds (From Fiction to Science-Ficttion)
"For many men that stumble at the threshold Are well foretold that danger lurks within." (William Shakespeare, "King Henry th...