4 ;;Quote: proof verification is analytical and leads to nothing but the premises translated into another language

4+;;Quote: a real proof is fruitful, its conclusion is in a sense more general than its premises

4+;;Quote: there is no science but the science of the general

9 ;;Quote: proof by induction allows us to generalize, to pass from the finite to the infinite

20 ;;Quote: mathematicians study the relations between objects, not the objects themselves; i.e., they study form; matter does not engage their attention

21 ;;Quote: the square root of 2 is the cut between numbers whose square is less than and greater than 2; Kronecker

21+;;Quote: every rational and irrational number is symbol for a cut that divides the real numbers into two

22 ;;Quote: a physical continuum is not transitive (e.g., A=B, B=C, A<C); its contradictions lead to the mathematical continuum

31 ;;Quote: a line is a sequence of elements such that each cannot be distinguished from its predecessor

32 ;;Quote: a cut separates a continuum into multiple continuums by removing one or more elements

32+;;Quote: a continuum has two dimensions if it is cut by one-dimensional continuums

64 ;;Quote: construct a consistent, non-Euclidean world as a sphere whose temperature decreases to zero at its surface, and whose lengths are proportional to temperature

74 ;;Quote: a distance must be measured so there is no abstract distance; a property of straight lines includes the measurement

74+;;Quote: no experiment can contradict Euclid's postulate, nor can experiment contradict Lobatschewsky's postulate

75 ;;Quote: the law of relativity: a system depends on the state of its bodies and their mutual distances; it does not depend on absolute position and orientation

77 ;;Quote: the law of relativity applies to the entire universe

77+;;Quote: geometric empiricism and measurement do not have a rational meaning; only the relations between measurements

79 ;;Quote: experiments have reference to relations between bodies, not to space

90 ;;Quote: there is no absolute space, no absolute time, no simultaneity across distance, not even geometry; these concepts are no more than a language

104 ;;Quote: it is by definition that 'f=ma', that action and reaction are equal and opposite, that objects remain in uniform motion

110 ;;Quote: the laws of acceleration are conventions but they are not arbitrary; the laws were justified by experiment

140 ;;Quote: experiment is the sole source of truth; but every experiment is long and difficult

140+;;Quote: mathematical physics directs generalization; it allows scientists to select the important experiments

145 ;;Quote: every generalization supposes a belief in the unity and simplicity of Nature

145+;;Quote: any fact can be generalized in an infinite number of ways. The choice can only be guided by considerations of simplicity

169 ;;Quote: experiments have shown that ether, i.e., absolute space, does not exist

169+;;Quote: hypotheses are what we lack the least

173 ;;Quote: science shows opposite tendencies towards unity--simplicity and diversity--complication