What is an infinitesimal?

In mathematics, an infinitesimal is a quantity that is less than any finite positive quantity but is not zero. While such quantities do not exist within the standard real number system, they were historically crucial in the development of calculus . Early mathematicians like Isaac Newton used infinitesimals to explain calculus procedures before the concept of a limit was formally established. Modern mathematics, through nonstandard analysis , has developed rigorous ways to incorporate infinitesimals. This allows for mathematical objects that are smaller than any positive real number, yet not zero, without leading to paradoxes. The term "infinitesimal" itself derives from the Latin word for "infinite," reflecting the idea of endless divisibility.

How are infinitesimals used in mathematics?

Infinitesimals, quantities smaller than any finite positive number yet not zero, were historically crucial in the development of calculus . Early mathematicians like Isaac Newton used them to explain derivatives and integrals, leading to the term "infinitesimal calculus". While infinitesimals do not exist within the standard real number system, Abraham Robinson developed nonstandard analysis in the 1960s. This framework allows for the rigorous use of infinitesimals and infinities, providing a different perspective on concepts like limits. This approach, though outside the mathematical mainstream, offers powerful new methods for mathematical problems.

Why did mathematicians invent the idea of infinitesimals?

Mathematicians introduced infinitesimals to provide a foundation for calculus before the concept of limits was rigorously defined. Early developers like Isaac Newton and Gottfried Wilhelm Leibniz used infinitesimals to explain derivatives and integrals, treating them as quantities smaller than any positive real number but not zero. This approach, while powerful for solving complex problems, lacked logical justification and was criticized by philosophers like George Berkeley, who famously questioned their existence as "ghosts of departed quantities". The unease with these nebulous ideas ultimately spurred the development of the rigorous concept of limits in the 19th century, which provided a solid logical basis for calculus. While infinitesimals do not exist within the real number system, modern mathematical logic has shown ways to construct them as consistent mathematical objects.

How do infinitesimals relate to calculating slopes and areas?

Infinitesimals, though not existing within the real number system, were foundational to early calculus, often referred to as infinitesimal calculus . They were used to explain how derivatives, representing slopes, could be found as the ultimate ratios of infinitesimals. Similarly, integrals, used for calculating areas, were conceptualized by summing rectangles of infinitesimal width. While the rigorous concept of limits has largely replaced the direct use of infinitesimals, mathematical logicians have since constructed systems where objects behave like infinitesimals, rehabilitating early calculus arguments.

What are some problems that can occur when using infinitesimals?

When using infinitesimals, a primary problem is their lack of existence within the real number system. Early mathematicians like Isaac Newton used infinitesimals heuristically to explain calculus, treating them as numbers smaller than any positive real number but not zero. This nebulous concept led to unease and spurred the development of the more rigorous concept of limits. Philosophers such as George Berkeley criticized these early calculus methods, questioning the logical justification for infinitesimals, famously calling them "the ghosts of departed quantities". It wasn't until the mid-20th century that Abraham Robinson demonstrated that infinitesimals could be logically consistent and introduced as a new type of number through nonstandard analysis. While this approach offers a rigorous way to work with infinitesimals, it is considered outside the mathematical mainstream.

How are infinitesimals understood or replaced in modern mathematics?

In modern mathematics, infinitesimals, which are quantities smaller than any finite positive number yet not zero, are understood and replaced through rigorous concepts like limits. Early calculus, known as infinitesimal calculus, relied on these nebulous ideas, but unease with their lack of clarity led to the development of the limit concept. While infinitesimals do not exist within the standard real number system, mathematical logicians have constructed systems where they do exist. Abraham Robinson developed nonstandard analysis , a variant of real analysis where infinitesimals and infinities are genuine mathematical objects, allowing for rigorous use of arguments that were previously considered nonrigorous.

Infinitesimals

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Infinitesimals

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John Colin Stillwell

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Top Questions

What is an infinitesimal?
In mathematics, an infinitesimal is a quantity that is less than any finite positive quantity but is not zero. While such quantities do not exist within the standard real number system, they were historically crucial in the development of calculus. Early mathematicians like Isaac Newton used infinitesimals to explain calculus procedures before the concept of a limit was formally established.

Modern mathematics, through nonstandard analysis, has developed rigorous ways to incorporate infinitesimals. This allows for mathematical objects that are smaller than any positive real number, yet not zero, without leading to paradoxes. The term "infinitesimal" itself derives from the Latin word for "infinite," reflecting the idea of endless divisibility.
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Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
How are infinitesimals used in mathematics?
Infinitesimals, quantities smaller than any finite positive number yet not zero, were historically crucial in the development of calculus. Early mathematicians like Isaac Newton used them to explain derivatives and integrals, leading to the term "infinitesimal calculus".

While infinitesimals do not exist within the standard real number system, Abraham Robinson developed nonstandard analysis in the 1960s. This framework allows for the rigorous use of infinitesimals and infinities, providing a different perspective on concepts like limits. This approach, though outside the mathematical mainstream, offers powerful new methods for mathematical problems.
4 Britannica Sources
Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
Why did mathematicians invent the idea of infinitesimals?
Mathematicians introduced infinitesimals to provide a foundation for calculus before the concept of limits was rigorously defined. Early developers like Isaac Newton and Gottfried Wilhelm Leibniz used infinitesimals to explain derivatives and integrals, treating them as quantities smaller than any positive real number but not zero. This approach, while powerful for solving complex problems, lacked logical justification and was criticized by philosophers like George Berkeley, who famously questioned their existence as "ghosts of departed quantities". The unease with these nebulous ideas ultimately spurred the development of the rigorous concept of limits in the 19th century, which provided a solid logical basis for calculus. While infinitesimals do not exist within the real number system, modern mathematical logic has shown ways to construct them as consistent mathematical objects.
4 Britannica Sources
Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
How do infinitesimals relate to calculating slopes and areas?
Infinitesimals, though not existing within the real number system, were foundational to early calculus, often referred to as infinitesimal calculus. They were used to explain how derivatives, representing slopes, could be found as the ultimate ratios of infinitesimals. Similarly, integrals, used for calculating areas, were conceptualized by summing rectangles of infinitesimal width. While the rigorous concept of limits has largely replaced the direct use of infinitesimals, mathematical logicians have since constructed systems where objects behave like infinitesimals, rehabilitating early calculus arguments.
3 Britannica Sources
Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
What are some problems that can occur when using infinitesimals?
When using infinitesimals, a primary problem is their lack of existence within the real number system. Early mathematicians like Isaac Newton used infinitesimals heuristically to explain calculus, treating them as numbers smaller than any positive real number but not zero. This nebulous concept led to unease and spurred the development of the more rigorous concept of limits.

Philosophers such as George Berkeley criticized these early calculus methods, questioning the logical justification for infinitesimals, famously calling them "the ghosts of departed quantities". It wasn't until the mid-20th century that Abraham Robinson demonstrated that infinitesimals could be logically consistent and introduced as a new type of number through nonstandard analysis. While this approach offers a rigorous way to work with infinitesimals, it is considered outside the mathematical mainstream.
4 Britannica Sources
Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
How are infinitesimals understood or replaced in modern mathematics?
In modern mathematics, infinitesimals, which are quantities smaller than any finite positive number yet not zero, are understood and replaced through rigorous concepts like limits. Early calculus, known as infinitesimal calculus, relied on these nebulous ideas, but unease with their lack of clarity led to the development of the limit concept.

While infinitesimals do not exist within the standard real number system, mathematical logicians have constructed systems where they do exist. Abraham Robinson developed nonstandard analysis, a variant of real analysis where infinitesimals and infinities are genuine mathematical objects, allowing for rigorous use of arguments that were previously considered nonrigorous.
5 Britannica Sources
Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

Infinitesimals were introduced by Isaac Newton as a means of “explaining” his procedures in calculus. Before the concept of a limit had been formally introduced and understood, it was not clear how to explain why calculus worked. In essence, Newton treated an infinitesimal as a positive number that was smaller, somehow, than any positive real number. In fact, it was the unease of mathematicians with such a nebulous idea that led them to develop the concept of the limit.

The status of infinitesimals decreased further as a result of Richard Dedekind’s definition of real numbers as “cuts.” A cut splits the real number line into two sets. If there exists a greatest element of one set or a least element of the other set, then the cut defines a rational number; otherwise the cut defines an irrational number. As a logical consequence of this definition, it follows that there is a rational number between zero and any nonzero number. Hence, infinitesimals do not exist among the real numbers.

This does not prevent other mathematical objects from behaving like infinitesimals, and mathematical logicians of the 1920s and ’30s actually showed how such objects could be constructed. One way to do this is to use a theorem about predicate logic proved by Kurt Gödel in 1930. All of mathematics can be expressed in predicate logic, and Gödel showed that this logic has the following remarkable property:

A set Σ of sentences has a model [that is, an interpretation that makes it true] if any finite subset of Σ has a model.

This theorem may be used to construct infinitesimals as follows. First, consider the axioms of arithmetic, together with the following infinite set of sentences (expressible in predicate logic) that say “ι is an infinitesimal”: ι > 0, ι < 1/2, ι < 1/3, ι < 1/4, ι < 1/5, ….

Any finite subset of these sentences has a model. For example, say the last sentence in the subset is “ι < 1/n”; then the subset can be satisfied by interpreting ι as 1/(n + 1). It then follows from Gödel’s property that the whole set has a model; that is, ι is an actual mathematical object.

The infinitesimal ι cannot be a real number, of course, but it can be something like an infinite decreasing sequence. In 1934 the Norwegian Thoralf Skolem gave an explicit construction of what is now called a nonstandard model of arithmetic, containing “infinite numbers” and infinitesimals, each of which is a certain class of infinite sequences.

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In the 1960s the German-born American Abraham Robinson similarly used nonstandard models of analysis to create a setting where the nonrigorous infinitesimal arguments of early calculus could be rehabilitated. He found that the old arguments could always be justified, usually with less trouble than the standard justifications with limits. He also found infinitesimals useful in modern analysis and proved some new results with their help. Quite a few mathematicians have converted to Robinson’s infinitesimals, but for the majority they remain “nonstandard.” Their advantages are offset by their entanglement with mathematical logic, which discourages many analysts.

John Colin Stillwell

Infinitesimals

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