For a smart material to be able to send out a more complex signal it needs to be nonlinear. If you hit a tuning fork twice as hard it will ring twice as loud but still at the same frequency. Thats a linear response. If you hit a person twice as hard theyre unlikely just to shout twice as loud. That property lets you learn more about the person than the tuning fork.
The fraction of life can be increased in value not so much By increasing your numerator as by lessening your denominator. Nay unless my Algebra deceives me unity itself divided by zero will give infinity.
The mathematical life of a mathematician is short. Work rarely improves after the age of twenty-five or thirty. If little has been accomplished by then little will ever be accomplished.
Mathematics is not a careful march down a well-cleared highway but a journey into a strange wilderness where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made and the real explorers have gone elsewhere.
Mathematics the non-empirical science par excellence … the science of sciences delivering the key to those laws of nature and the universe which are concealed by appearances.
About Thomas Hobbes: He was 40 years old before he looked on geometry; which happened accidentally. Being in a gentlemans library Euclids Elements lay open and twas the 47 El. libri I [Pythagoras Theorem]. He read the proposition By God sayd he this is impossible: So he reads the demonstration of it which referred him back to such a proposition; which proposition he read. That referred him back to another which he also read. Et sic deinceps that at last he was demonstratively convinced of that truth. This made him in love with geometry.
How happy the lot of the mathematician. He is judged solely by his peers and the standard is so high that no colleague or rival can ever win a reputation he does not deserve.
The good Christian should beware of mathematicians and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.
Einstein is an analytical mathematician seeking to give a physical interpretation to the conclusions of his mathematical process. In this he is hampered by a load of contradictory and absurd assumptions of the school that he follows which throws him in to all manner of difficulty. Einstein has such a faculty for embracing both sides of a contradiction that one would have to be of the same frame of mind to follow his thought it is so peculiarly his own. The whole Relativity theory is as easy to follow as the path of a bat in the air at night.
I think youre begging the question said Haydock and I can see looming ahead one of those terrible exercises in probability where six men have white hats and six men have black hats and you have to work it out by mathematics how likely it is that the hats will get mixed up and in what proportion. If you start thinking about things like that you would go round the bend. Let me assure you of that!
I had a feeling once about Mathematics that I saw it all. Depth beyond depth was revealed to me the Byss and Abyss. I saw as one might see the transit of Venus or even the Lord Mayors Show a quantity passing through infinity and changing its sign from plus to minus. I saw exactly why it happened and why the tergiversation was inevitable but it was after dinner and I let it go.
Dont talk to me of your Archimedes lever. He was an absentminded person with a mathematical imagination. Mathematics commands all my respect but I have no use for engines. Give me the right word and the right accent and I will move the world.
… it is interesting to note that the original problem that started my research is still outstanding – namely the problem of planning or scheduling dynamically over time particularly planning dynamically under uncertainty. If such a problem could be successfully solved it could eventually through better planning contribute to the well-being and stability of the world.
When I am violently beset with temptations or cannot rid myself of evil thoughts [I resolve] to do some Arithmetic or Geometry or some other study which necessarily engages all my thoughts and unavoidably keeps them from wandering.
Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers and we have reason to believe that it is a mystery into which the human mind will never penetrate.
One is hard pressed to think of universal customs that man has successfully established on earth. There is one however of which he can boast the universal adoption of the Hindu-Arabic numerals to record numbers. In this we perhaps have mans unique worldwide victory of an idea.
If the entire Mandelbrot set were placed on an ordinary sheet of paper the tiny sections of boundary we examine would not fill the width of a hydrogen atom. Physicists think about such tiny objects; only mathematicians have microscopes fine enough to actually observe them.
Since you are now studying geometry and trigonometry I will give you a problem. A ship sails the ocean. It left Boston with a cargo of wool. It grosses 200 tons. It is bound for Le Havre. The mainmast is broken the cabin boy is on deck there are 12 passengers aboard the wind is blowing East-North-East the clock points to a quarter past three in the afternoon. It is the month of May. How old is the captain?
Mathematicians are like lovers. Grant a mathematician the least principle and he will draw from it a consequence which you must also grant him and from this consequence another.
When you have mastered numbers you will in fact no longer be reading numbers any more than you read words when reading books. You will be reading meanings.
On Ramanujan: I remember once going to see him when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one and that I hoped it was not an unfavorable omen. No he replied it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.
[The works of Archimedes] are without exception monuments of mathematical exposition; the gradual revelation of the plan of attack the masterly ordering of the propositions the stern elimination of everything not immediately relevant to the purpose the finish of the whole are so impressive in their perfection as to create a feeling akin to awe in the mind of the reader.
Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes bathe and not make messes in the house.
The errors of definitions multiply themselves according as the reckoning proceeds; and lead men into absurdities which at last they see but cannot avoid without reckoning anew from the beginning.
This seems to be one of the many cases in which the admitted accuracy of mathematical processes is allowed to throw a wholly inadmissible appearance of authority over the results obtained by them. Mathematics may be compared to a mill of exquisite workmanship which grinds your stuff to any degree of fineness; but nevertheless what you get out depends on what you put in; and as the grandest mill in the world will not extract wheat flour from peascods so pages of formulae will not get a definite result out of loose data.
… the science of calculation also is indispensable as far as the extraction of the square and cube roots: Algebra as far as the quadratic equation and the use of logarithms are often of value in ordinary cases: but all beyond these is but a luxury; a delicious luxury indeed; but not to be in indulged in by one who is to have a profession to follow for his subsistence.
Most of the arts as painting sculpture and music have emotional appeal to the general public. This is because these arts can be experienced by some one or more of our senses. Such is not true of the art of mathematics; this art can be appreciated only by mathematicians and to become a mathematician requires a long period of intensive training. The community of mathematicians is similar to an imaginary community of musical composers whose only satisfaction is obtained by the interchange among themselves of the musical scores they compose.
It is India that gave us the ingenious method of expressing all numbers by means of ten symbols each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius two of the greatest men produced by antiquity.
In my opinion a mathematician in so far as he is a mathematician need not preoccupy himself with philosophy — an opinion moreover which has been expressed by many philosophers.
We come finally however to the relation of the ideal theory to real world or real probability. If he is consistent a man of the mathematical school washes his hands of applications. To someone who wants them he would say that the ideal system runs parallel to the usual theory: If this is what you want try it: it is not my business to justify application of the system; that can only be done by philosophizing; I am a mathematician. In practice he is apt to say: try this; if it works that will justify it. But now he is not merely philosophizing; he is committing the characteristic fallacy. Inductive experience that the system works is not evidence.
We lay down a fundamental principle of generalization by abstraction: The existence of analogies between central features of various theories implies the existence of a general theory which underlies the particular theories and unifies them with respect to those central features….
A mathematician of the first rank Laplace quickly revealed himself as only a mediocre administrator; from his first work we saw that we had been deceived. Laplace saw no question from its true point of view; he sought subtleties everywhere; had only doubtful ideas and finally carried the spirit of the infinitely small into administration.
… Newton was an unquestioning believer in an all-wise creator of the universe and in his own inability like the boy on the seashore to fathom the entire ocean in all its depths. He therefore believed that there were not only many things in heaven beyond his philosophy but plenty on earth as well and he made it his business to understand for himself what the majority of intelligent men of his time accepted without dispute (to them it was as natural as common sense) the traditional account of the creation.
What does this desire and this inability of ours proclaim to us but that there was once in man a genuine happiness of which nothing now survives but the mark and the empty outline; and this he vainly tries to fill from everything that lies around him seeking from things that are not there the help that he does not get from those that are present? Yet they are quite incapable of filling the gap because this infinite gulf can only be filled by an infinite and immutable object – that is God Himself. He alone is mans veritable good and since man has deserted Him it is a strange thing that there is nothing in nature that has not been capable of taking His place for man: stars sky earth elements plants cabbages leeks animals insects calves serpents fever plague war famine vices adultery incest. And since he has lost the true good everything can equally appear to him as such – even his own destruction though that is so contrary at once to God to reason and to nature.
Mathematicians do not study objects but relations between objects. Thus they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant: they are interested in form only.
A mathematician is a device for turning coffee into theorems.
For a smart material to be able to send out a more complex signal it needs to be nonlinear. If you hit a tuning fork twice as hard it will ring twice as loud but still at the same frequency. Thats a linear response. If you hit a person twice as hard theyre unlikely just to shout twice as loud. That property lets you learn more about the person than the tuning fork.
The fraction of life can be increased in value not so much By increasing your numerator as by lessening your denominator. Nay unless my Algebra deceives me unity itself divided by zero will give infinity.
The mathematical life of a mathematician is short. Work rarely improves after the age of twenty-five or thirty. If little has been accomplished by then little will ever be accomplished.
Mathematics is not a careful march down a well-cleared highway but a journey into a strange wilderness where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made and the real explorers have gone elsewhere.
Mathematics the non-empirical science par excellence … the science of sciences delivering the key to those laws of nature and the universe which are concealed by appearances.
About Thomas Hobbes: He was 40 years old before he looked on geometry; which happened accidentally. Being in a gentlemans library Euclids Elements lay open and twas the 47 El. libri I [Pythagoras Theorem]. He read the proposition By God sayd he this is impossible: So he reads the demonstration of it which referred him back to such a proposition; which proposition he read. That referred him back to another which he also read. Et sic deinceps that at last he was demonstratively convinced of that truth. This made him in love with geometry.
How happy the lot of the mathematician. He is judged solely by his peers and the standard is so high that no colleague or rival can ever win a reputation he does not deserve.
The good Christian should beware of mathematicians and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell.
For the things of this world cannot be made known without a knowledge of mathematics.
What is algebra exactly; is it those three-cornered things?
Einstein is an analytical mathematician seeking to give a physical interpretation to the conclusions of his mathematical process. In this he is hampered by a load of contradictory and absurd assumptions of the school that he follows which throws him in to all manner of difficulty. Einstein has such a faculty for embracing both sides of a contradiction that one would have to be of the same frame of mind to follow his thought it is so peculiarly his own. The whole Relativity theory is as easy to follow as the path of a bat in the air at night.
The different branches of Arithmetic Ambition Distraction Uglification and Derision.
I think youre begging the question said Haydock and I can see looming ahead one of those terrible exercises in probability where six men have white hats and six men have black hats and you have to work it out by mathematics how likely it is that the hats will get mixed up and in what proportion. If you start thinking about things like that you would go round the bend. Let me assure you of that!
I had a feeling once about Mathematics that I saw it all. Depth beyond depth was revealed to me the Byss and Abyss. I saw as one might see the transit of Venus or even the Lord Mayors Show a quantity passing through infinity and changing its sign from plus to minus. I saw exactly why it happened and why the tergiversation was inevitable but it was after dinner and I let it go.
Dont talk to me of your Archimedes lever. He was an absentminded person with a mathematical imagination. Mathematics commands all my respect but I have no use for engines. Give me the right word and the right accent and I will move the world.
… it is interesting to note that the original problem that started my research is still outstanding – namely the problem of planning or scheduling dynamically over time particularly planning dynamically under uncertainty. If such a problem could be successfully solved it could eventually through better planning contribute to the well-being and stability of the world.
Mathematics seems to endow one with something like a new sense.
Proof is the idol before whom the pure mathematician tortures himself.
When I am violently beset with temptations or cannot rid myself of evil thoughts [I resolve] to do some Arithmetic or Geometry or some other study which necessarily engages all my thoughts and unavoidably keeps them from wandering.
I dont believe in mathematics.
Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers and we have reason to believe that it is a mystery into which the human mind will never penetrate.
One is hard pressed to think of universal customs that man has successfully established on earth. There is one however of which he can boast the universal adoption of the Hindu-Arabic numerals to record numbers. In this we perhaps have mans unique worldwide victory of an idea.
If the entire Mandelbrot set were placed on an ordinary sheet of paper the tiny sections of boundary we examine would not fill the width of a hydrogen atom. Physicists think about such tiny objects; only mathematicians have microscopes fine enough to actually observe them.
Since you are now studying geometry and trigonometry I will give you a problem. A ship sails the ocean. It left Boston with a cargo of wool. It grosses 200 tons. It is bound for Le Havre. The mainmast is broken the cabin boy is on deck there are 12 passengers aboard the wind is blowing East-North-East the clock points to a quarter past three in the afternoon. It is the month of May. How old is the captain?
Mathematicians are like lovers. Grant a mathematician the least principle and he will draw from it a consequence which you must also grant him and from this consequence another.
If others would but reflect on mathematical truths as deeply and as continuously as I have they would make my discoveries.
When you have mastered numbers you will in fact no longer be reading numbers any more than you read words when reading books. You will be reading meanings.
To be a scholar of mathematics you must be born with talent insight concentration taste luck drive and the ability to visualize and guess.
On Ramanujan: I remember once going to see him when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one and that I hoped it was not an unfavorable omen. No he replied it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.
[The works of Archimedes] are without exception monuments of mathematical exposition; the gradual revelation of the plan of attack the masterly ordering of the propositions the stern elimination of everything not immediately relevant to the purpose the finish of the whole are so impressive in their perfection as to create a feeling akin to awe in the mind of the reader.
Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes bathe and not make messes in the house.
The errors of definitions multiply themselves according as the reckoning proceeds; and lead men into absurdities which at last they see but cannot avoid without reckoning anew from the beginning.
This seems to be one of the many cases in which the admitted accuracy of mathematical processes is allowed to throw a wholly inadmissible appearance of authority over the results obtained by them. Mathematics may be compared to a mill of exquisite workmanship which grinds your stuff to any degree of fineness; but nevertheless what you get out depends on what you put in; and as the grandest mill in the world will not extract wheat flour from peascods so pages of formulae will not get a definite result out of loose data.
The union of the mathematician with the poet fervor with measure passion with correctness this surely is the ideal.
… the science of calculation also is indispensable as far as the extraction of the square and cube roots: Algebra as far as the quadratic equation and the use of logarithms are often of value in ordinary cases: but all beyond these is but a luxury; a delicious luxury indeed; but not to be in indulged in by one who is to have a profession to follow for his subsistence.
Most of the arts as painting sculpture and music have emotional appeal to the general public. This is because these arts can be experienced by some one or more of our senses. Such is not true of the art of mathematics; this art can be appreciated only by mathematicians and to become a mathematician requires a long period of intensive training. The community of mathematicians is similar to an imaginary community of musical composers whose only satisfaction is obtained by the interchange among themselves of the musical scores they compose.
It is India that gave us the ingenious method of expressing all numbers by means of ten symbols each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius two of the greatest men produced by antiquity.
In my opinion a mathematician in so far as he is a mathematician need not preoccupy himself with philosophy — an opinion moreover which has been expressed by many philosophers.
We come finally however to the relation of the ideal theory to real world or real probability. If he is consistent a man of the mathematical school washes his hands of applications. To someone who wants them he would say that the ideal system runs parallel to the usual theory: If this is what you want try it: it is not my business to justify application of the system; that can only be done by philosophizing; I am a mathematician. In practice he is apt to say: try this; if it works that will justify it. But now he is not merely philosophizing; he is committing the characteristic fallacy. Inductive experience that the system works is not evidence.
Medicine makes people ill mathematics makes them sad and theology makes them sinful.
We lay down a fundamental principle of generalization by abstraction: The existence of analogies between central features of various theories implies the existence of a general theory which underlies the particular theories and unifies them with respect to those central features….
A mathematician of the first rank Laplace quickly revealed himself as only a mediocre administrator; from his first work we saw that we had been deceived. Laplace saw no question from its true point of view; he sought subtleties everywhere; had only doubtful ideas and finally carried the spirit of the infinitely small into administration.
In mathematics you dont understand things. You just get used to them.
The most painful thing about mathematics is how far away you are from being able to use it after you have learned it.
… Newton was an unquestioning believer in an all-wise creator of the universe and in his own inability like the boy on the seashore to fathom the entire ocean in all its depths. He therefore believed that there were not only many things in heaven beyond his philosophy but plenty on earth as well and he made it his business to understand for himself what the majority of intelligent men of his time accepted without dispute (to them it was as natural as common sense) the traditional account of the creation.
Mathematicians boast of their exacting achievements but in reality they are absorbed in mental acrobatics and contribute nothing to society.
What does this desire and this inability of ours proclaim to us but that there was once in man a genuine happiness of which nothing now survives but the mark and the empty outline; and this he vainly tries to fill from everything that lies around him seeking from things that are not there the help that he does not get from those that are present? Yet they are quite incapable of filling the gap because this infinite gulf can only be filled by an infinite and immutable object – that is God Himself. He alone is mans veritable good and since man has deserted Him it is a strange thing that there is nothing in nature that has not been capable of taking His place for man: stars sky earth elements plants cabbages leeks animals insects calves serpents fever plague war famine vices adultery incest. And since he has lost the true good everything can equally appear to him as such – even his own destruction though that is so contrary at once to God to reason and to nature.
I have hardly ever known a mathematician who was capable of reasoning.
Mathematicians do not study objects but relations between objects. Thus they are free to replace some objects by others so long as the relations remain unchanged. Content to them is irrelevant: they are interested in form only.