Mathematics

The deilghts of science and mathematics--their revelations of natural beauty and harmony, their visions of things to come, and the joy of discovery in itself, the light and shadow it casts on the mystery dance of mind and nature--are too profound, and too important, to be left to scientists and mathematicians alone. They belong to the cultural heritage of the entire world, and to know something about them is to be acquainted with the finest new achievements of the human mind. Science and mathematics are to our technically inclined societies what the composition of epic poetry was to the Homeric Greeks, or shipbuilding to the ninth-century Norsemen, or landscape painting to the Sung Dynasty Chinese; they are what we do best . . . . [p. xi]** [to be continued . . . .]**

**Einstein's special theory of relativity** ** [ e = mc2 ] **

Begins with the assumption that the speed of light in a vacuum will always be measured at the same constant value regardless of the speed of the light source relative to the observer. [p. 185]

**Newton's law of gravitation** - That between any two particles of matter there is a force which is proportional to the product of their masses and inversely proportional to the square of their distance. That is to say, ignoring for the present the question of mass, if there is a certain attraction when the particles are a mile apart, there will be a quarter as much attraction when they are two miles apart, a ninth as much when they are three miles apart, and so on: the attraction diminishes much faster than the distance increases. [p. 194

**Interval is a Relationship** - We assume that there is a certain physical quantity called "interval," which is a relation between two events that are not widely separated. [p. 195]

**Continuous Relationship** - Every measurement is a physical process carried out with physical material; the result is certainly an experimental datum, but may not be susceptible of the simple interpretation which we ordinarily assign to it. We are, therefore, not going to assume to begin with that we know how to measure anything. We assume that there is a certain physical quantity called "interval," which is a relation between two events that are not widely separated; but we do not assume in advance that we know how to measure it, beyond taking it for granted that it is given by some generalization of the theorem of Pythagoras . . . .

We do assume, however, that events have an order, and that this order is four-dimensional. We assume, that is to say, that we know what we mean by sayin that a certain event is nearer to aother than a third, so that before making accurate measurements we can speak of the "neighborhood" of an event; and we assume that, in order to assign the positoin of an event in space-time, four quantities [coordinates] are necessary --e.g..... [perhaps] latitude, longitude, altitude, and time.... But we assume nothing about the way in which these coordinates are assigned, except that neigbuoring coordinates are adsigned to neighbouring events.

The way in which these numbers, called coordinates, are to be assigned is neither wholly abitrary nor a result of careful measurment--it lies in an intermediate region. While you are making any continuous journey, your coordinates must never alter by sudden jumps. In America one finds that houses between [say] fourteenth Street and Fifteenth Street are likely to have numbers between 1400 and 1500, while those between Fifteenth Street and Sixteenth Street have numbers between 1500 and 1600, even if the 1400s were not used up. This would not do for our purposes, because there is a sudden jump when we pass from one block to the next . . . . [p. 195] It is assumed that, independently of measurement, we know what a continuous journey is. And when your position in space-time changes continuously, each of your four coordinates must change continuously. One, two or three of them may not change at all, but whatever change does occur must be smooth, without sudden jumps. This explains what is not allowable in assigning coordiantes. [p. 196]

**Measurements of distances and times do not directly reveal properties of the things measured, but relations of the things to the measurer** - We are apt to think that, for really careful measurments, it is better to use a steel rod than a live eel. This is a mistake, not because teh eel tells us what teh steel rod was thought to tell, but because the steel rod really tells no more than the eel obviously does. The point is, not that eels are really rigid, but that steel rods really wriggle. To an observer in just one possible state of motion the eel would appear rigid, while the steel rod would seem to wriggle just as the eel does to us. For everybody moving differently both from this observer and ourselves, bott the eel and the rod would seem to wriggle. And there is no saying that one observer is right and another wrong. In such matters what is seen does not belong solely to the pysical process observed, but also to the standpoint of the observe . . . . What observation can tell us about the physical world is therefore more abstract than we have higherto believed . . . . Measurements of distances and times do not directly reveal properties of the things measured, but relations of the things to the measurer. [p. 197]

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