Infinitesimals. Non-standard Analysis
The early history of Calculus is the story of infinitesimals. Starting with Newton and Leibniz in the 17th century, practically all great mathematicians tried unsuccessfully to justify the employment of infinitesimals; till in the 19th century the infinitesimals were finally banished from mathematics and replaced with Weierstrass' ε-δ definition of the limit.
(As an aside, Weierstrass' definition is often singled out by mathematics educators as a piece of calculus especially difficult for the beginning students. For example, [Berlinghoff and Gouvêa, p. 49] wrote about Weierstrass, 'His clear, precise definitions removed any trace of mystery or geometric intuition from calculus, putting it all on a logical foundation that depended only on algebra and arithmetic. The new approach wasn't easy, though, as students who have had to learn his "epsilon-delta" approach to limits will still testify.' I believe that nothing is easy to an unprepared mind. Calculus is supposed to be a tool chest of methods for solving certain kinds of problems. This is how it has been used by mathematicians and physists even before, and certainly since, it was formulated as such by Newton and Leibniz. The place of "clear, precise definitions" is in Analysis, which is a foundational part of Calculus and is being studied by mathematically inclined students.)
Quite remarkably in the 20th century, Abraham Robinson from the Hebrew University of Jerusalem, Israel, while on leave at the Institute for Advanced Study in Princeton during the 1959-1960 academic year, managed to create a theory of infinitesimals on a superbly logical foundation that depended less on algebra and arithmetic but more on the understanding of that side of mathematics that makes use of formal languages to describe theories and their models. Robinson's construction of what became known as the Non-standard Analysis came on the heels of the work done earlier by the Norwegian Thoralf A. Skolem, the Russian Anatoli Maltsev, the American Leon A. Henkin, and the German/American Kurt Gödel [Davis and Hersh, p. 249].
Essential to the understanding of Robinson's approach is mathematical logic and theory of models. Mathematical logic [Marker, p. 635] is a study of formal languages that are used to describe mathematical structures. These concepts deserve a more extended exposition. Below is just a short, introductory outline of the ideas involved.
A formal language includes a set of rules for forming valid sentences. A theory (in a certain language) is a collection of sentences formed in that language. Mathematical structures are models or interpretations of such theories. For example, a language that includes some logical symbols along with variables and a symbol for a single binary operation may be used to describe Group Theory, for which various groups serve as models. For a theory and a sentence in a certain language, a meaningful question to ask is whether that sentence can be deduced in the theory following the rules of the language. In other words, it may be inquired whether a sentence has a proof in a theory. (A proof is a deduction in a finite number of steps.) In a model, a sentence may be true or not. Thus a sentence may or may not be a logical consequence of a theory and may or may not be true in one or more models of that theory.
Gödel's Completeness Theorem states that, for a certain kind (first-order, to be precise) of a language, a sentence can be proved in a theory if and only if it is true in every model of that theory.
A theory that has a model (i.e. a model in which all sentences of the theory are true) is said to be satisfiable. A theory in which it is impossible to derive a contradiction (i.e. a sentence and its negation) is consistent. A satisfiable system is necessarily consistent. If it is not satisfiable, then every sentence in the associated language is provable (since there is no model in which it might be false). In particular, in a not satisfiable theory a contradiction is provable, showing that the theory is inconsistent. This observation leads to the
Compactness Theorem which states that a theory is satisfiable if and only if every finite subtheory is satisfiable. This is so, because if a theory is inconsistent, i.e., if it is possible to derive a contradiction, then that contradiction is a consequence of a finite subtheory, meaning that the latter has no model. But if a subtheory has no model, the theory can't have one either, so that under such circumstances the theory is unsatisfiable.
Now is the time for a punch line: Compactness theorem directly implies the existence of infinitesimals and infinitely large numbers!
- W. P. Berlinghoff, F. Q. Gouvêa, Math Through the Ages, Oxton House Publishers, 2002
- P. J. Davis & R. Hersh, The Mathematical Experience, Houghton Mifflin Co, 1981, 237-254
- K. Ito, Nonstandard Analysis (293), in Encyclopedic Dictionary of Mathematics, v. 1, MIT Press, Press, 2000 (fourth printing), pp. 1100-1103
- D. Marker, Logic and Model Theory, in T. Gowers (ed.), The Princeton Companion to Mathematics, Princeton University Press, 2008, pp. 635-646
- A. Robinson, Non-standard Analysis, Princeton University Press (Rev Sub edition), 1996
- I. Stewart, Nonstandard Analysis, in R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996, pp. 518-524
- I. Stewart, Non-Standard Analysis, in From Here to Infinity: A Guide to Today's Mathematics, Oxford University Press, 1996, pp. 80-81.