A small error in the former will produce an enormous error in the latter. |

To doubt everything, or, to believe everything, are two equally convenient solutions; both dispense with the necessity of reflection. |

If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of the same universe at a succeeding moment. |

The mathematical facts worthy of being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a physical law. |

To invent is to discern, to choose. |

If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living. |

If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by the laws. |

It has adopted the geometry most advantageous to the species or, in other words, the most convenient. |

A sane mind should not be guilty of a logical fallacy, yet there are very fine minds incapable of following mathematical demonstrations. |

No more than these machines need the mathematician know what he does. |

Thought is only a flash between two long nights, but this flash is everything. |

It is through science that we prove, but through intuition that we discover. |

It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introduces order, all that gives unity, that permits us to see clearly and to comprehend at once both the ensemble and the details. |

One would have to have completely forgotten the history of science so as to not remember that the desire to know nature has had the most constant and the happiest influence on the development of mathematics. |

Geometry is not true, it is advantageous. |

Thus, they are free to replace some objects by others so long as the relations remain unchanged. |

Mathematicians are born, not made. |

Invention consists in avoiding the constructing of useless contraptions and in constructing the useful combinations which are in infinite minority. |

A very small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say that the effect is due to chance. |

Science is facts. **Topics: **Short |