608 VIII. ON THE STRUCTURE OF MATHEMATICS |
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Not so, however, with temperature, or density, or many other derived magnitudes, as we call them. If we have a body of temperature of one degree and combine it with another body of equal temperature the synthesis will not have a temperature of two degrees, (as in the case of weight), but of one degree. This applies to density . , two bodies of density one each will not give us a body of density two, but of density one.
Before further analysis of the problems of linearity and additivity, it will be well to consider a few definitions.
If an entity u is changed into an entity v by some process, the change may be regarded as the result of an operation performed upon u, the operand, which has converted it into v. If we denote the operation by/, then the result might be written as v =fu. The symbol of the operation / is called the operator. We are familiar with many such; indeed the symbols for all mathematical operations may be treated as operators. So for instance the symbol
![]() ![]() ![]() ![]() the definite integral
![]() It is important to know that many of the rules of algebra and arithmetic when defined in this way, give rise to a calculus of operations. The fundamental notion in such a calculus is that of a product. If « is operated upon by / the result v is indicated by
![]() ![]() upon by g the result w is indicated by gv, or symbolically,
![]() the operation gf which converts w directly into w is called the product of / and g. If this operation is repeated several times in succession the usual notation of powers is used, for instance
![]() all, which we would denote by f°, leaves w unchanged, which we indicate symbolically by the equation
![]() ![]() plication by
![]() ![]() that the law of indices holds; namely, the
![]() For our purpose we will analyse only one special case; namely, where we have u, v and u +v as operands, and such an operator, /, that
![]() In terms of functions we would have
![]() called a functional equation. It has been proved that such a functional equation has one type of solutions; namely, when / is equivalent to a multiplication by a constant, or
![]() in science are stated in terms of variation. For purposes of analysis a statement |
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