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Mathematical Quotes

Can the difficulty of an exam be measured by how many bits of information a student would need to pass it? This may not be so absurd in the encyclopedic subjects but in mathematics it doesn’t make any sense since things follow from each other and, in principle, whoever knows the bases knows everything. All of the results of a mathematical theorem are in the axioms of mathematics in embryonic form, aren’t they? Picture Quote #1

How then shall mathematical concepts be judged? They shall not be judged. Mathematics is the supreme arbiter. From its decisions there is no appeal. We cannot change the rules of the game, we cannot ascertain whether the game is fair. We can only study the player at his game; not, however, with the detached attitude of a bystander, for we are watching our own minds at play Picture Quote #1

Mechanics is the paradise of the mathematical sciences because by means of it one comes to the fruits of mathematics Picture Quote #1

The mathematical phenomenon always develops out of simple arithmetic, so useful in everyday life, out of numbers, those weapons of the gods: the gods are there, behind the wall, at play with numbers Picture Quote #1

Computer science is to biology what calculus is to physics. It’s the natural mathematical technique that best maps the character of the subject Picture Quote #1

We are not very pleased when we are forced to accept a mathematical truth by virtue of a complicated chain of formal conclusions and computations, which we traverse blindly, link by link, feeling our way by touch. We want first an overview of the aim and of the road; we want to understand the idea of the proof, the deeper context Picture Quote #1

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Deductivism in mathematical literature and inductivism in scientific papers are simply the postures we choose to be seen in when the curtain goes up and the public sees us. The theatrical illusion is shattered if we ask what goes on behind the scenes. In real life discovery and justification are almost always different processes Picture Quote #1

Chess will always be in the doldrums as a spectator sport while a draw is given equal mathematical value as a decisive result Picture Quote #1

And despite the fact that the basis of this mathematical way of thinking in art is in reason, its dynamic content is able to launch us on astral flights which soar into unknown and still uncharted regions of the imagination Picture Quote #1

It took me 1057 pages to describe the hundreds of mathematical equations, algorithms and programming techniques that I invented and used Picture Quote #1

Sometimes a line of mathematical research extending through decades can be thought of as one long conversation in which many mathematicians take part. This is fortunately true at present Picture Quote #1

You call a star a star, and say it is just a ball of matter moving on a mathematical course. But that is merely how you see it. By so naming things and describing them you are only inventing your own terms about them. And just as speech is invention about objects and ideas, so myth is invention about truth Picture Quote #1

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It is quite clear that beauty does depend on one’s culture and upbringing for certain kinds of beauty, pictures, literature, poetry and so on... But mathematical beauty is of a rather different kind. I should say perhaps it is of a completely different kind and transcends these personal factors. It is the same in all countries and at all periods of time Picture Quote #1

A theory with mathematical beauty is more likely to be correct than an ugly one that fits some experimental data Picture Quote #1

A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details Picture Quote #1

We often say that the earth is a sphere, but to be precise, the term sphere refers only to the surface. The correct mathematical term for the solid earth is a ball Picture Quote #1

Mathematics has beauty and romance. It’s not a boring place to be, the mathematical world. It’s an extraordinary place; it’s worth spending time there Picture Quote #1

Criticized for using formal mathematical manipulations, without understanding how they worked: Should I refuse a good dinner simply because I do not understand the process of digestion? Picture Quote #1

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If you attack a mathematical problem directly, very often you come to a dead end, nothing you do seems to work and you feel that if only you could peer round the corner there might be an easy solution. There is nothing like having somebody else beside you, because he can usually peer round the corner Picture Quote #1

The mathematical difficulties of the theory of rotation arise chiefly from the want of geometrical illustrations and sensible images, by which we might fix the results of analysis in our minds Picture Quote #1

More than sixty years ago, mathematical logicians, by defining precisely the concept of an algorithm, gave content to the ancient human idea of an effective calculation. Their definitions led to the creation of the digital computer, an interesting example of thought bending matter to its ends Picture Quote #1

How can a modern anthropologist embark upon a generalization with any hope of arriving at a satisfactory conclusion? By thinking of the organizational ideas that are present in any society as a mathematical pattern Picture Quote #1

Gauss replied, when asked how soon he expected to reach certain mathematical conclusions, that he had them long ago, all he was worrying about was how to reach them! Picture Quote #1

A book on the new physics, if not purely descriptive of experimental work, must essentially be mathematical Picture Quote #1

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Mathematicians can and do fill in gaps, correct errors, and supply more detail and more careful scholarship when they are called on or motivated to do so. Our system is quite good at producing reliable theorems that can be solidly backed up. It’s just that the reliability does not primarily come from mathematicians formally checking formal arguments; it comes from mathematicians thinking carefully and critically about mathematical ideas Picture Quote #1

He knew by heart every last minute crack on its surface. He had made maps of the ceiling and gone exploring on them; rivers, islands, and continents. He had made guessing games of it and discovered hidden objects; faces, birds, and fishes. He made mathematical calculations of it and rediscovered his childhood; theorems, angles, and triangles. There was practically nothing else he could do but look at it. He hated the sight of it Picture Quote #1

If theory is the role of the architect, then such beautiful proofs are the role of the craftsman. Of course, as with the great renaissance artists, such roles are not mutually exclusive. A great cathedral has both structural impressiveness and delicate detail. A great mathematical theory should similarly be beautiful on both large and small scales Picture Quote #1

Not only in geometry, but to a still more astonishing degree in physics, has it become more and more evident that as soon as we have succeeded in unraveling fully the natural laws which govern reality, we find them to be expressible by mathematical relations of surprising simplicity and architectonic perfection. It seems to me to be one of the chief objects of mathematical instruction to develop the faculty of perceiving this simplicity and harmony Picture Quote #1

This ambiguity is another example of a growing problem with mathematical notation: There aren’t enough squiggles to go around Picture Quote #1

Why you were born and why you are living depend entirely on what you are getting out of this world and what you are giving to it. I cannot prove that this is a balance of mathematical perfection, but my own observation of life leads me to the conclusion that there is a very real friendship, both quantitatively and qualitatively, between what you contribute and what you get out of life Picture Quote #1

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Can the difficulty of an exam be measured by how many bits of information a student would need to pass it? This may not be so absurd in the encyclopedic subjects but in mathematics it doesn’t make any sense since things follow from each other and, in principle, whoever knows the bases knows everything. All of the results of a mathematical theorem are in the axioms of mathematics in embryonic form, aren’t they? (Mathematical Quotes) How then shall mathematical concepts be judged? They shall not be judged. Mathematics is the supreme arbiter. From its decisions there is no appeal. We cannot change the rules of the game, we cannot ascertain whether the game is fair. We can only study the player at his game; not, however, with the detached attitude of a bystander, for we are watching our own minds at play (Mathematical Quotes) Mechanics is the paradise of the mathematical sciences because by means of it one comes to the fruits of mathematics (Mathematical Quotes) The mathematical phenomenon always develops out of simple arithmetic, so useful in everyday life, out of numbers, those weapons of the gods: the gods are there, behind the wall, at play with numbers (Mathematical Quotes) Computer science is to biology what calculus is to physics. It’s the natural mathematical technique that best maps the character of the subject (Mathematical Quotes) We are not very pleased when we are forced to accept a mathematical truth by virtue of a complicated chain of formal conclusions and computations, which we traverse blindly, link by link, feeling our way by touch. We want first an overview of the aim and of the road; we want to understand the idea of the proof, the deeper context (Mathematical Quotes) Deductivism in mathematical literature and inductivism in scientific papers are simply the postures we choose to be seen in when the curtain goes up and the public sees us. The theatrical illusion is shattered if we ask what goes on behind the scenes. In real life discovery and justification are almost always different processes (Mathematical Quotes) Chess will always be in the doldrums as a spectator sport while a draw is given equal mathematical value as a decisive result (Mathematical Quotes) And despite the fact that the basis of this mathematical way of thinking in art is in reason, its dynamic content is able to launch us on astral flights which soar into unknown and still uncharted regions of the imagination (Mathematical Quotes) It took me 1057 pages to describe the hundreds of mathematical equations, algorithms and programming techniques that I invented and used (Mathematical Quotes) Sometimes a line of mathematical research extending through decades can be thought of as one long conversation in which many mathematicians take part. This is fortunately true at present (Mathematical Quotes) You call a star a star, and say it is just a ball of matter moving on a mathematical course. But that is merely how you see it. By so naming things and describing them you are only inventing your own terms about them. And just as speech is invention about objects and ideas, so myth is invention about truth (Mathematical Quotes) It is quite clear that beauty does depend on one’s culture and upbringing for certain kinds of beauty, pictures, literature, poetry and so on... But mathematical beauty is of a rather different kind. I should say perhaps it is of a completely different kind and transcends these personal factors. It is the same in all countries and at all periods of time (Mathematical Quotes) A theory with mathematical beauty is more likely to be correct than an ugly one that fits some experimental data (Mathematical Quotes) A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details (Mathematical Quotes) We often say that the earth is a sphere, but to be precise, the term sphere refers only to the surface. The correct mathematical term for the solid earth is a ball (Mathematical Quotes) Mathematics has beauty and romance. It’s not a boring place to be, the mathematical world. It’s an extraordinary place; it’s worth spending time there (Mathematical Quotes) Criticized for using formal mathematical manipulations, without understanding how they worked: Should I refuse a good dinner simply because I do not understand the process of digestion? (Mathematical Quotes) If you attack a mathematical problem directly, very often you come to a dead end, nothing you do seems to work and you feel that if only you could peer round the corner there might be an easy solution. There is nothing like having somebody else beside you, because he can usually peer round the corner (Mathematical Quotes) The mathematical difficulties of the theory of rotation arise chiefly from the want of geometrical illustrations and sensible images, by which we might fix the results of analysis in our minds (Mathematical Quotes) More than sixty years ago, mathematical logicians, by defining precisely the concept of an algorithm, gave content to the ancient human idea of an effective calculation. Their definitions led to the creation of the digital computer, an interesting example of thought bending matter to its ends (Mathematical Quotes) How can a modern anthropologist embark upon a generalization with any hope of arriving at a satisfactory conclusion? By thinking of the organizational ideas that are present in any society as a mathematical pattern (Mathematical Quotes) Gauss replied, when asked how soon he expected to reach certain mathematical conclusions, that he had them long ago, all he was worrying about was how to reach them! (Mathematical Quotes) A book on the new physics, if not purely descriptive of experimental work, must essentially be mathematical (Mathematical Quotes) Mathematicians can and do fill in gaps, correct errors, and supply more detail and more careful scholarship when they are called on or motivated to do so. Our system is quite good at producing reliable theorems that can be solidly backed up. It’s just that the reliability does not primarily come from mathematicians formally checking formal arguments; it comes from mathematicians thinking carefully and critically about mathematical ideas (Mathematical Quotes) He knew by heart every last minute crack on its surface. He had made maps of the ceiling and gone exploring on them; rivers, islands, and continents. He had made guessing games of it and discovered hidden objects; faces, birds, and fishes. He made mathematical calculations of it and rediscovered his childhood; theorems, angles, and triangles. There was practically nothing else he could do but look at it. He hated the sight of it (Mathematical Quotes) If theory is the role of the architect, then such beautiful proofs are the role of the craftsman. Of course, as with the great renaissance artists, such roles are not mutually exclusive. A great cathedral has both structural impressiveness and delicate detail. A great mathematical theory should similarly be beautiful on both large and small scales (Mathematical Quotes) Not only in geometry, but to a still more astonishing degree in physics, has it become more and more evident that as soon as we have succeeded in unraveling fully the natural laws which govern reality, we find them to be expressible by mathematical relations of surprising simplicity and architectonic perfection. It seems to me to be one of the chief objects of mathematical instruction to develop the faculty of perceiving this simplicity and harmony (Mathematical Quotes) This ambiguity is another example of a growing problem with mathematical notation: There aren’t enough squiggles to go around (Mathematical Quotes) Why you were born and why you are living depend entirely on what you are getting out of this world and what you are giving to it. I cannot prove that this is a balance of mathematical perfection, but my own observation of life leads me to the conclusion that there is a very real friendship, both quantitatively and qualitatively, between what you contribute and what you get out of life (Mathematical Quotes)

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