Smooth infinitesimal analysis is a modern reformulation of the

calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...

in terms of

infinitesimal In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...

s. Based on the ideas of

F. W. Lawvere Francis William Lawvere (; February 9, 1937 – January 23, 2023) was an American mathematician known for his work in category theory, topos theory and the philosophy of mathematics. Biography Born in Muncie, Indiana, and raised on a farm outsi ...

and employing the methods of

category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...

, it views all functions as being continuous and incapable of being expressed in terms of

discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit * Discrete group, ...

entities. As a theory, it is a subset of synthetic differential geometry. Terence Tao has referred to this concept under the name "cheap nonstandard analysis." The ''nilsquare'' or ''

nilpotent In mathematics, an element x of a ring (mathematics), ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term, along with its sister Idempotent (ring theory), idem ...

'' infinitesimals are numbers ''ε'' where ''ε''² = 0 is true, but ''ε'' = 0 need not be true at the same time. ''

Calculus Made Easy ''Calculus Made Easy'' is a book on infinitesimal calculus originally published in 1910 by Silvanus P. Thompson. The original text continues to be available as of 2008 from Macmillan and Co., but a 1998 update by Martin Gardner is available fro ...

'' notably uses nilpotent infinitesimals.

Overview

This approach departs from the

classical logic Classical logic (or standard logic) or Frege–Russell logic is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this c ...

used in conventional mathematics by denying the

law of the excluded middle In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the three laws of thought, along with the law of noncontradiction and th ...

, e.g., ''NOT'' (''a'' ≠ ''b'') does not imply ''a'' = ''b''. In particular, in a theory of smooth infinitesimal analysis one can prove for all infinitesimals ''ε'', ''NOT'' (''ε'' ≠ 0); yet it is provably false that all infinitesimals are equal to zero. One can see that the law of excluded middle cannot hold from the following basic theorem (again, understood in the context of a theory of smooth infinitesimal analysis): :''Every function whose domain is'' R, ''the

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s, is continuous and infinitely differentiable.'' Despite this fact, one could attempt to define a discontinuous function ''f''(''x'') by specifying that ''f''(''x'') = 1 for ''x'' = 0, and ''f''(''x'') = 0 for ''x'' ≠ 0. If the law of the excluded middle held, then this would be a fully defined, discontinuous function. However, there are plenty of ''x'', namely the infinitesimals, such that neither ''x'' = 0 nor ''x'' ≠ 0 holds, so the function is not defined on the real numbers. In typical

models A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , . Models can be divided int ...

of smooth infinitesimal analysis, the infinitesimals are not invertible, and therefore the theory does not contain infinite numbers. However, there are also models that include invertible infinitesimals. Other mathematical systems exist which include infinitesimals, including

nonstandard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definitio ...

and the

surreal number In mathematics, the surreal number system is a total order, totally ordered proper class containing not only the real numbers but also Infinity, infinite and infinitesimal, infinitesimal numbers, respectively larger or smaller in absolute value th ...

s. Smooth infinitesimal analysis is like nonstandard analysis in that (1) it is meant to serve as a foundation for

analysis Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...

, and (2) the infinitesimal quantities do not have concrete sizes (as opposed to the surreals, in which a typical infinitesimal is , where ω is a von Neumann ordinal). However, smooth infinitesimal analysis differs from nonstandard analysis in its use of

nonclassical logic Non-classical logics (and sometimes alternative logics or non-Aristotelian logics) are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this ...

, and in lacking the

transfer principle In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure. One of the first examples was the Lefschetz principle, which states that any sentence in the firs ...

. Some theorems of standard and nonstandard analysis are false in smooth infinitesimal analysis, including the

intermediate value theorem In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two imp ...

and the

Banach–Tarski paradox The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then ...

. Statements in

can be translated into statements about

limits Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2009 ...

, but the same is not always true in smooth infinitesimal analysis. Intuitively, smooth infinitesimal analysis can be interpreted as describing a world in which lines are made out of infinitesimally small segments, not out of points. These segments can be thought of as being long enough to have a definite direction, but not long enough to be curved. The construction of discontinuous functions fails because a function is identified with a curve, and the curve cannot be constructed pointwise. We can imagine the intermediate value theorem's failure as resulting from the ability of an infinitesimal segment to straddle a line. Similarly, the Banach–Tarski paradox fails because a volume cannot be taken apart into points.

References

{{Reflist

External links

*Michael O'Connor
An Introduction to Smooth Infinitesimal Analysis
Nonstandard analysis Mathematics of infinitesimals

Overview

See also

References

Further reading

External links