Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.
In common speech, an infinitesimal object is an object which is smaller than any feasible measurement, hence not zero size, but so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, "infinitesimal" in the vernacular means "extremely small".
Before the nineteenth century none of the mathematical concepts relating to infinitesimals as we know them today were formally defined, but many of these concepts were already used. The founders of calculus — Euler, Leibniz, Newton, Cauchy, and many others — calculated with infinitesimals and achieved correct results even though no formal definition was available. For comparison one should note that there was no formal definition of real numbers at the time, either.
The notion of infinitesimally small quantities was discussed by the Eleatic School. Archimedes, in The Method of Mechanical Theorems, was the first to propose a logically rigorous definition of infinitesimals. His Archimedean property defines a number x as infinite if it satisfies the conditions |x|>1, |x|>1+1, |x|>1+1+1, ..., and infinitesimal if x≠0 and the same set of conditions holds for 1/x. A number system is said to be Archimedean if it contains no infinite or infinitesimal members. In the ancient Greek system of mathematics, 1 represents the length of some line segment which has arbitrarily been picked as the unit of measurement. Thus the Archimedean definition really defines a notion of one number's being infinite or infinitesimal relative to another.
The Indian mathematician Bhāskara II (1114–1185) and Persian mathematician Sharaf al-Dīn al-Tūsī (1135–1213) made use of infinitesimals when they independently discovered the notion of a derivative. As an alternative to infinitesimals, the Kerala school of astronomy and mathematics in the 14th to 16th centuries made use of an intuitive notion of a limit to compute several series expansions.
When Newton and Leibniz developed calculus, they made use of infinitesimals. The use of infinitesimals was attacked as incorrect by Bishop Berkeley in his work The Analyst. Although mathematicians, scientists, and engineers continued to use infinitesimals to produce correct results, it was not until the second half of the nineteenth century that the calculus was given a formal mathematical foundation by Karl Weierstrass and others using the notion of a limit. While infinitesimals eventually disappeared from the calculus, their mathematical study continued through the work of Levi-Civita and others, throughout the late nineteenth and the twentieth centuries, as documented by Ehrlich (2006). In the 20th century, it was found that infinitesimals could also be treated rigorously. Neither formulation is wrong, and both give the same results if used correctly.
In extending the real numbers to include infinite and infinitesimal quantities, one typically wishes to be as conservative as possible by not changing any of their elementary properties. This guarantees that as many familiar results as possible will still be available. Typically elementary means that there is no quantification over sets, which would include statements of the form "for any number x..." For example, the axiom that states "for any number x, x + 0 = x" would still apply. The same is true for quantification over several numbers, e.g., "for any numbers x and y, xy = yx." However, statements of the form "for any set S of numbers ..." may not carry over. This limitation on quantification is referred to as first-order logic. It is clear that the extended system cannot agree with the reals on all properties that can be expressed by quantification over sets, because the goal is to construct a nonarchimedean system, and the Archimedean principle can be expressed by quantification over sets. Similarly, the completeness property cannot be expected to carry over, because the reals are the unique complete ordered field (up to isomorphism).
We can distinguish three levels at which a nonarchimedean number system could have properties compatible with those of the reals:
Systems in category 1, at the weak end of the spectrum, are relatively easy to construct, but do not allow a full treatment of classical analysis using infinitesimals in the spirit of Newton and Leibniz. For example, the transcendental functions are defined in terms of infinite limiting processes, and therefore there is typically no way to define them in first-order logic. Increasing the analytic strength of the system by passing to categories 2 and 3, we find that the flavor of the treatment tends to become less constructive, and it becomes more difficult to say anything concrete about the hierarchical structure of infinities and infinitesimals.
An example from category 1 above is the field of Laurent series with a finite number of negative-power terms. For example, the Laurent series consisting only of the constant term 1 is identified with the real number 1, and the series with only the linear term x is thought of as the simplest infinitesimal, from which the other infinitesimals are constructed. Dictionary ordering is used, which is equivalent to considering higher powers of x as negligible compared to lower powers. David O. Tall refers to this system as the super-reals, not to be confused with the superreal number system of Dales and Woodin. Since a Taylor series evaluated with a Laurent series as its argument is still a Laurent series, the system can be used to do calculus on transcendental functions if they are analytic. These infinitesimals have different first-order properties than the reals because, for example, the basic infinitesimal x does not have a square root.
The Levi-Civita field is similar to the Laurent series, but is algebraically closed. For example, the basic infinitesimal x has a square root. This field is rich enough to allow a significant amount of analysis to be done, but its elements can still be represented on a computer in the same sense that real numbers can be represented in floating point. It has applications to numerical differentiation in cases that are intractable by symbolic differentiation or finite-difference methods.
The field of transseries is larger than the Levi-Civita field. An example of a transseries is:
where for purposes of ordering x is considered to be infinite.
Conway's surreal numbers fall into category 2. They are a system that was designed to be as rich as possible in different sizes of numbers, but not necessarily for convenience in doing analysis. Certain transcendental functions can be carried over to the surreals, including logarithms and exponenentials, but most, e.g., the sine function, cannot. The existence of any particular surreal number, even one that has a direct counterpart in the reals, is not known a priori, and must be proved.
The most widespread technique for handling infinitesimals is the hyperreals, developed by Abraham Robinson in the middle of the 20th century. They fall into category 3 above, having been designed that way in order to allow all of classical analysis to be carried over from the reals. This property of being able to carry over all relations in a natural way is known as the transfer principle. For example, the transcendental function sin has a natural counterpart *sin that takes a hyperreal input and gives a hyperreal output, and similarly the set of natural numbers has a natural counterpart , which contains both finite and infinite integers. A proposition such as carries over to the hyperreals as .
The superreal number system of Dales and Woodin is a generalization of the hyperreals. It is different from the super-real system defined by Tall.
Alternatively, we can have synthetic differential geometry or smooth infinitesimal analysis with its roots in category theory. This approach departs from the classical logic used in conventional mathematics by denying the general applicability of the law of excluded middle — i.e., not (a ≠ b) does not have to mean a = b. A nilsquare or nilpotent infinitesimal can then be defined. This is a number x where x2 = 0 is true, but x = 0 need not be true at the same time.
Proving or disproving the existence of infinitesimals of the kind used in nonstandard analysis depends on the model and which collection of axioms are used. We consider here systems where infinitesimals can be shown to exist.
In 1936 Maltsev proved the compactness theorem. This theorem is fundamental for the existence of infinitesimals as it proves that it is possible to formalise them. A consequence of this theorem is that if there is a number system in which it is true that for any positive integer n there is a positive number x such that 0 < x < 1/n, then there exists an extension of that number system in which it is true that there exists a positive number x such that for any positive integer n we have 0 < x < 1/n. The possibility to switch "for any" and "there exists" is crucial. The first statement is true in the real numbers as given in ZFC set theory : for any positive integer n it is possible to find a real number between 1/n and zero, but this real number will depend on n. Here, one chooses n first, then one finds the corresponding x. In the second expression, the statement says that there is an x (at least one), chosen first, which is between 0 and 1/n for any n. In this case x is infinitesimal. This is not true in the real numbers (R) given by ZFC. Nonetheless, the theorem proves that there is a model (a number system) in which this will be true. The question is: what is this model? What are its properties? Is there only one such model?
In 1960, Abraham Robinson provided an answer following the first approach. The extended set is called the hyperreals and contains numbers less in absolute value than any positive real number. The method may be considered relatively complex but it does prove that infinitesimals exist in the universe of ZFC set theory. The real numbers are called standard numbers and the new non-real hyperreals are called nonstandard.
In 1977 Edward Nelson provided an answer following the second approach. The extended axioms are IST, which stands either for Internal Set Theory or for the initials of the three extra axioms: Idealization, Standardization, Transfer. In this system we consider that the language is extended in such a way that we can express facts about infinitesimals. The real numbers are either standard or nonstandard. An infinitesimal is a nonstandard real number which is less, in absolute value, than any positive standard real number.
In 2006 Karel Hrbacek developed an extension of Nelson's approach in which the real numbers are stratified in (infinitely) many levels i.e, in the coarsest level there are no infinitesimals nor unlimited numbers. Infinitesimals are in a finer level and there are also infinitesimals with respect to this new level and so on.