8 THE PRESIDENTIAL ADDRESS 
speed, they are very like the waves by which Maxwell described the 
flow of radiation through space, so that matter and radiation are much 
more like one another in the new physics than they were in the old. 
In other cases, ordinary time and space do not provide an adequate 
canvas for the wave-picture. ‘The wave-picture of two currents of 
electricity, or even of two electrons moving independently, needs 
a larger canvas—six dimensions of space and one of time. ‘There 
can be no logical justification for identifying any particular three of 
these six dimensions with ordinary space, so that we must regard 
the wave-picture as lying entirely outside space. ‘The whole picture, 
and the manifold dimensions of space in which it is drawn, become 
pure mental constructs—diagrams and frameworks we make for 
ourselves to help us understand phenomena. 
In this way we have the two co-existent pictures—the particle- 
picture for the materialist, and the wave-picture for the determinist. 
When the cartographer has to make two distinct maps to exhibit 
the geography of, say, North America, he is able to explain why two 
maps are necessary, and can also tell us the relation between the 
two—he can show us how to transform one into the other. He will 
tell us, for instance, that he needs two maps simply because he is 
restricted to flat surfaces—pieces of paper. Give him a sphere 
instead, and he can show us North America, perfectly and completely, 
on a single map. 
The physicist has not yet found anything corresponding to this 
sphere ; when, if ever, he does, the particle-picture and the wave- 
picture will be merged into a single new picture. At present some 
kink in our minds, or perhaps merely some ingrained habit of 
thought, prevents our understanding the universe as a consistent 
whole—just as the ingrained habits of thought of a ‘ flat-earther ’ 
prevent his understanding North America as a consistent whole. 
Yet, although physics has so far failed to explain why two pictures 
are necessary, it is, nevertheless, able to explain the relation between 
the particle-picture and the wave-picture in perfectly comprehensible 
terms. 
The central feature of the particle-picture is the atomicity which 
is found in the structure of matter. But this atomicity is only one 
expression of a fundamental coarse-grainedness which pervades the 
whole of nature. It crops up again in the fact that energy can only 
be transferred by whole quanta. Because of this, the tools with 
which we study nature are themselves coarse-grained ; we have only 
blunt probes at our disposal, and so can never acquire perfectly 
precise knowledge of nature. Just as, in astronomy, the grain of our 
photographic plates prevents our ever fixing the position of a star 
with absolute precision, so in physics we can never say that an electron 
is here, at this precise spot, and is moving at just such and such a 
