Take, in illustration, some simple proposition of arithmetic, say: "The sum of the first odd numbers is ." Suppose we wish to interpret this proposition as applying to the progression , , ,... , ... In this progression, let be the relation of each term to its successor. Then "odd numbers" will mean "terms having to a relation which is a power of ," where is the relation of an to the next but one.[2] We can now define as meaning that power of which relates to , and we can further define as meaning[Pg 4] that to which has the relation . This decides the interpretation of "the sum of the first odd numbers." To define it will be best to define multiplication. We have defined ; consider the relation formed by the relative product of the converse of together with . This relation relates to ; its square relates to ; its cube relates to , etc. Any power of this relation can be shown to be equivalent to a certain power of the converse of multiplied relatively by a certain power of . There is thus one power of this relation which is equivalent to moving backward from to , and then forward; the term to which the forward movement takes us is defined as . Thus we can now interpret . It will be found that the proposition from which we started is true with this interpretation. It follows from the above that, if we start from Peano's undefined ideas and initial propositions, arithmetic and analysis are not concerned with definite logical objects called numbers, but with the terms of any progression. We may call the terms of any progression 0, 1, 2, 3,..., in which case, with a suitable interpretation of + and , all the propositions of arithmetic will be true of these terms. Thus 0, 1, 2, 3,..., become "variables." To make them constants, we must choose some one definite progression; the natural one to choose is the progression of finite cardinal numbers as defined by Frege. What were, in Peano's methods, primitive terms are thus replaced by logical structures, concerning which it is necessary to prove that they satisfy Peano's five primitive propositions. This process is essential in connecting arithmetic with pure logic. We shall find that a process similar in some respects, though very different in others, is required for connecting physics with perception.