Letter Letter to the Editor YAROSLAV

T he Editor-in-Chief of The Mathematical Intelligencer, Prof. Marjorie Senechal, has invited me to answer a letter [1] concerning my recent note The Olympic Medals Ranks, Lexicographic Ordering, and Numerical Infinities [2]. I am always happy to communicate with readers and thank Prof. Senechal for giving me this opportunity to reply. The reader writes that results presented in [2] can be obtained ‘‘in many different ways (say, with positional notation for ordinals, or using non-Archimedean fields and nonstandard models for reals),’’ but he does not suggest that this can be done numerically. I have explicitly stated (see [2], page 5, before formula (5)) that these computations can be executed symbolically and, in fact, all the ways the reader lists of dealing with infinity are symbolic, whereas numerical infinities are the most important words in my title. Numerical computations work with approximate floating point numbers,whereas symbolic computations are the exact manipulations with mathematical expressions containing variables with no given value and are thus manipulated as symbols. Let me explain this point in more detail, emphasizing also some differences between the -based methodology and traditional approaches to infinity and infinitesimals. Let us start with analyzing numerical computations with finite quantities. When we execute these computations, the same numerals are used for different purposes (e.g., 10 can express the number of elements of a set, indicate the position of an element in a sequence, or execute practical numerical computations). In contrast, when we face the necessity of working with infinities and/or infinitesimals, the situation changes drastically and we face a number of distinctions and complications. First, different numerals are used to work with infinities and infinitesimals in different situations. For example we use the symbol 1 in standard analysis, symbol x for working with ordinals, symbols @0;@1; ::: for dealing with cardinalities, etc. Second, traditionally theories dealing with infinite and infinitesimal quantities have a symbolic (not numerical) character and only algebraic manipulations can be done. For instance, nonstandard models and non-Archimedean fields use either a generic infinite number or a generic infinitesimal in their constructions (e.g., Levi-Civita numbers are built using a generic infinitesimal e), whereas our numerical computations with finite quantities are concrete and not generic. If we consider a finite n, then different values can be assigned to it, for example, we can use the numeral 34 and write n 1⁄4 34. Clearly, any other numeral used to express finite quantities and consisting of a finite number of symbols can be taken for this purpose. The finiteness of the number of symbols is necessary for executing practical computations, because we should be able to write down (and/or store) values with which we execute operations. In contrast, if we consider a nonstandard infinitem then it is not clear which numerals consisting of a finite number of symbols can be used to assign a concrete value tom. Again, it is not clear which numerals can be used to assign a value to the generic infinitesimal e and to write e 1⁄4 :::. Moreover, approaches of this kind leave unclear such issues as, for example, whether the infinite 1=e is an integer or not, and whether 1=e is the number of elements of a concrete infinite set. If one wishes to consider two infinitesimals (or infinities) h1 and h2, where h2 is not expressed in terms of h1, then it is not clear how to compare them because numeral systems that can express different values of infinities and infinitesimals are not provided by this kind of technique. In contrast, when we work with finite quantities, then we can compare n and k if they assume numerical values, for example, if k 1⁄4 25 and n 1⁄4 78, then, by using rules of the numeral system the symbols 25 and 78 belong to, we can compute that n[ k. Third, many arithmetics used to deal with infinities not only should be used for specific purposes only, but in addition are quite different with respect to the way we execute computations with finite quantities. Let me give some examples: