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from them take more kindly to the concrete facts of everyday life 
than to abstract concepts ; to particulars rather than to universals. 
Every child, when first it begins to learn algebra, asks in despair 
‘But what are x, y and z?’ and is satisfied when, and only when, 
it has been told that they are numbers of apples or pears or bananas 
or something such. In the same way, the old-fashioned physicist 
could not rest content with x, y and 2, but was always trying to ex- 
press them in terms of apples or pears or bananas. Yet a simple 
_ argument will show that he can never get beyond x, y and z. 
Physical science obtains its knowledge of the external world by 
a series of exact measurements, or, more precisely, by comparisons 
of measurements. ‘Typical of its knowledge is the statement that 
the line H« in the hydrogen spectrum has a wave-length of so many 
centimetres. ‘This is meaningless until we know what a centimetre 
is. ‘The moment we are told that it is a certain fraction of the earth’s 
radius, or of the length of a bar of platinum, or a certain multiple 
of the wave-length of a line in the cadmium spectrum, our know- 
ledge becomes real, but at that same moment it also becomes purely 
numerical. Our minds can only be acquainted with things inside 
themselves—never with things outside. ‘Thus we can never know 
the essential nature of anything, such as a centimetre or a wave- 
length, which exists in that mysterious world outside ourselves to 
which our minds can never penetrate; but we can know the 
numerical ratio of two quantities of similar nature, no matter how 
incomprehensible they may both be individually. 
For this reason, our knowledge of the external world must always 
consist of numbers, and our picture of the universe—the synthesis 
of our knowledge—must necessarily be mathematical in form. All 
the concrete details of the picture, the apples and pears and bananas, 
the ether and atoms and electrons, are mere clothing that we ourselves 
drape over our mathematical symbols—they do not belong to Nature, 
but to the parables by which we try to make Nature comprehensible. 
It was, I think, Kronecker who said that in arithmetic God made 
the integers and man made the rest; in the same spirit, we may 
add that in physics God made the mathematics and man made the 
rest. 
~The modern physicist does not use this language, but he accepts 
its implications, and divides the concepts of physics into obser- 
vablesand unobservables. In brief, the observables embody facts of 
observation, and so are purely numerical or mathematical in their 
content ; the unobservables are the pictorial details of the parables. 
The physicist wants to make his new edifice earthquake-proof— 
immune to the shock of new observations—and so builds only on 
the solid rock, and with the solid bricks, of ascertained fact. Thus 
he builds only with observables, and his whole edifice is one of 
