Nothing takes place in the world whose meaning is not that of some maximum or minimum.

Logic is the foundation of the certainty of all the knowledge we acquire.

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Madam, I have just come from a country where people are hanged if they talk.

e^(iπ)+1 = 0

The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error.

Quand mon cerveau excite dans mon ame la sensation d'un arbre ou d'une maison, je prononce hardiment, qu'il existe réellement hors de moi un arbre ou une maison, dont je connois même le lieu, la grandeur ou d'autres propriétés. Ainsi ne trouve-t-on ni homme ni bête qui doutent de cette vérité. Si un paysan en vouloit douter ; s'il disoit, par exemple, qu'il ne croyait pas que son baillif existe, quoiqu'il fut devant lui, on le pretendroit pour un fou et cela avec raison : mais dès qu'un philosophe avance de tels sentimens, il veut qu'on admire son esprit et ses lumières, qui surpassent infiniment celles du peuple.

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Die Mathematik ist es, die uns vor dem Trug der Sinne schützt und uns den Unterschied zwischen Schein und Wahrheit kennen lehrt.

Quanquam nobis in intima naturae mysteria penetrare, indeque veras caussas Phaenomenorum agnoscere neutiquam est concessum: tamen evenire potest, ut hypothesis quaedam ficta pluribus phaenomenis explicandis aeque satisfaciat, ac si vera caussa nobis esset perspecta.

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Till now the mathematicians tried in vain to discover some order in the sequence of the prime numbers and we have every reason to believe that there is some mystery which the human mind shall never penetrate. To convince oneself, one has only to glance at the tables of primes which some people took the trouble to compute beyond a hundred thousand, and one perceives that there is no order and no rule. This is so much more surprising as the arithmetic gives us definite rules with the help of which we can continue the sequence of the primes as far as we please, without noticing, however, the least trace of order.

It will seem a little paradoxical to ascribe a great importance to observations even in that part of the mathematical sciences which is usually called Pure Mathematics, since the current opinion is that observations are restricted to physical objects that make impression on the senses. As we must refer the numbers to the pure intellect alone, we can hardly understand how observations and quasi-experiments can be of use in investigating the nature of numbers. Yet, in fact, as I shall show here with very good reasons, the properties of the numbers known today have been mostly discovered by observation, and discovered long before their truth has been confirmed by rigid demonstrations. There are many properties of the numbers with which we are well acquainted, but which we are not yet able to prove; only observations have led us to their knowledge. Hence we see that in the theory of numbers, which is still very imperfect, we can place our highest hopes in observations; they will lead us continually to new properties which we shall endeavor to prove afterwards. The kind of knowledge which is supported only by observations and is not yet proved must be carefully distinguished from the truth; it is gained by induction, as we usually say. Yet we have seen cases in which mere induction led to error. Therefore, we should take great care not to accept as true such properties of the numbers which we have discovered by observation and which are supported by induction alone. Indeed, we should use such discovery as an opportunity to investigate more exactly the properties discovered and to prove or disprove them; in both cases we may learn something useful.