68 
PYTHAGOREAN ARITHMETIC 
they supposed the elements of numbers to be the elements 
of all things, and the whole heaven to be a musical scale and 
a number/ 1 
This passage, .with its assertion of ‘ resemblances ’ and 
£ assimilation suggests numbers as affections, states, or rela 
tions rather than as substances, and the same is implied by 
the remark that existing things exist by virtue of their 
imitation of numbers. 2 But again we are told that the 
numbers are not separable from the things, but that existing 
things, even perceptible substances, are made up of numbers; 
that the substance of all things is number, that things are 
numbers, that numbers are made up from the unit, and that the 
whole heaven is numbers. 3 Still more definite is the statement 
that the Pythagoreans ‘ construct the whole heaven out of 
numbers, but not of monadic numbers, since they suppose the 
units to have magnitude ’, and that, ‘ as we have said before, 
the Pythagoreans assume the numbers to have magnitude’. 4 
Aristotle points out certain obvious difficulties. On the one 
hand the Pythagoreans speak of ‘ this number of which the 
heaven is composed ’; on the other hand they speak of ‘ attri 
butes of numbers ’ and of numbers as ‘ the causes of the things 
which exist and take place in the heaven both from the begin 
ning and now ’. Again, according to them, abstractions and 
immaterial things are also numbers, and they place them in 
different regions; for example, in one region they place 
opinion and opportunity, and in another, a little higher up or 
lower down, such things as injustice, sifting, or mixing. 
Is i t this same ‘ number in the heaven ’ which we must 
assume each of these things to be, or a number other than 
this number 1 ? 5 
May we not infer from these scattered remarks of Aristotle 
about the Pythagorean doctrine that ‘ the number in the 
heaven ’ is the number of the visible stars, made up of 
units which are material points'? And may this not be 
the origin of the theory that all things are numbers, a 
theory which of course would be confirmed when the further 
1 Metaph. A. 5, 985 b 27-986 a 2. 2 lb. A. 5, 987 b 11. 
:! Ih. N. 8, 1090 a 22-23; M. 7, 1080 b 17 ; A, 5, 987 a 19, 987 b 27, 
986 a 20. 
4 lb. M. 7, 1080 b 18, 32. 5 lb. A. 8, 990 a 18-29. 
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