5 Usage of *natural* by Mochizuki

Edit  As an example of the increasing prevalence of the notion of naturality in contemporary mathematics, it is notable that Shinichi Mochizuki’s four preprints asserting a proof of the ABC conjecture employ the word "natural" and its derivatives on more than six hundred occasions (for details and several related quotations, see this post on Gödel's Lost Letter and P=NP).

In accord with Daniel Miller's answer above, the study of "naturality" as a formal abstraction in mathematics can be traced back largely to two articles by Saunders Mac Lane and Samuel Eilenberg: Natural isomorphisms in group theory (1942) and General theory of natural equivalences (1945). Both articles are well worth reading.

How did these ideas arise? Roughly speaking, Eilenberg and MacLane began by recognizing that if an arbitrary choice of coordinates makes a difference to the quantities you are calculating and/or the theorems you are proving, then those quantities and theorems are not natural. Mac Lane has described this process in The development and prospects for category theory (1996) as follows:

I emphasize that the notions category and functor were not formulated or put in print until the idea of a natural transformation was also at hand.
Thus, one good start for students is to acquire a thorough practical grasp of naturality in the context of coordinate transformations in linear algebra and differential geometry.

A concrete example of a canonical usage of "naturality", which is accompanied by an extensive motivating discussion, is given in John Lee's Introduction to Smooth Manifolds as the following lemma:

Lemma 12.16: (Naturality of the Exterior Derivative)

If $G\colon M\to N$ is a smooth map, then the pullback map $G^\star\colon \mathcal{A}^k(N)\to \mathcal{A}^k(M)$ commutes with $d$. That is, for all $\omega \in \mathcal{A}^k(N)$, we have $G^\star(d\omega) = d(G^\star\omega)$.

As a commutative diagram the above lemma exhibits a canonical form:

$$\begin{array}{c@{}ccc@{}c} &&d&&\\ &\mathcal{A}^k(N)&\longrightarrow&\mathcal{A}^{k+1}(N)\\[2ex] G^\star\!\!\!\!\!\!&\big\downarrow&&\big\downarrow&\!\!\!\!\!\!\!G^\star\\[2ex] &\mathcal{A}^k(M)&\longrightarrow&\mathcal{A}^{k+1}(M)\\ &&d&& \end{array}$$

Being interested in practical applications of geometrically "natural" formalisms for quantum systems engineering, I've studied the usage in MacSciNet reviews of the words "natural*" (chiefly "natural" and "naturality") and "universal*" (chiefly "universal" and "universality").

Here are the numbers; their use is burgeoning!

• Year-Range (natural*, universal*)
• 2001-2005: (16788 uses, 05288 uses)
• 1996-2000: (14880, 04977)
• 1991-1995: (12550, 04432)
• 1986-1990: (10335, 03343)
• 1981-1985: (08775, 03013)
• 1976-1980: (07402, 02412)
• 1971-1975: (05668, 02040)
• 1966-1970: (03466, 01167)
• 1961-1965: (02211, 00610)
• 1956-1960: (01368, 00406)
• 1951-1955: (00880, 00253)
• 1946-1950: (00502, 00107)
• 1941-1945: (00251, 00060)

So to judge by the literature, it seems that we are entering into a Golden Era of mathematical "naturality" and "universality" ... we can hope so, anyway! :)

4 Added commutative diagram

In accord with Daniel Miller's answer above, the study of "naturality" as a formal abstraction in mathematics can be traced back largely to two articles by Saunders Mac Lane and Samuel Eilenberg: Natural isomorphisms in group theory (1942) and General theory of natural equivalences (1945). Both articles are well worth reading.

How did these ideas arise? Roughly speaking, Eilenberg and MacLane began by recognizing that if an arbitrary choice of coordinates makes a difference to the quantities you are calculating and/or the theorems you are proving, then those quantities and theorems are not natural. Mac Lane has described this process in The development and prospects for category theory (1996) as follows:

I emphasize that the notions category and functor were not formulated or put in print until the idea of a natural transformation was also at hand.
Thus, one good start for students is to acquire a thorough practical grasp of naturality in the context of coordinate transformations in linear algebra and differential geometry.

A concrete example of a canonical usage of "naturality", which is accompanied by an extensive motivating discussion, is given in John Lee's Introduction to Smooth Manifolds as the following lemma:

Lemma 12.16: (Naturality of the Exterior Derivative)

If $G\colon M\to N$ is a smooth map, then the pullback map $G^\star\colon \mathcal{A}^k(N)\to \mathcal{A}^k(M)$ commutes with $d$. That is, for all $\omega \in \mathcal{A}^k(N)$, we have $G^\star(d\omega) = d(G^\star\omega)$.

As a commutative diagram the above lemma exhibits a canonical form:

$$\begin{array}{c@{}ccc@{}c} &&d&&\\ &\mathcal{A}^k(N)&\longrightarrow&\mathcal{A}^{k+1}(N)\\[2ex] G^\star\!\!\!\!\!\!&\big\downarrow&&\big\downarrow&\!\!\!\!\!\!\!G^\star\\[2ex] &\mathcal{A}^k(M)&\longrightarrow&\mathcal{A}^{k+1}(M)\\ &&d&& \end{array}$$

Being interested in practical applications of geometrically "natural" formalisms for quantum systems engineering, I've studied the usage in MacSciNet reviews of the words "natural*" (chiefly "natural" and "naturality") and "universal*" (chiefly "universal" and "universality").

Here are the numbers; their use is burgeoning!

• Year-Range (natural*, universal*)
• 2001-2005: (16788 uses, 05288 uses)
• 1996-2000: (14880, 04977)
• 1991-1995: (12550, 04432)
• 1986-1990: (10335, 03343)
• 1981-1985: (08775, 03013)
• 1976-1980: (07402, 02412)
• 1971-1975: (05668, 02040)
• 1966-1970: (03466, 01167)
• 1961-1965: (02211, 00610)
• 1956-1960: (01368, 00406)
• 1951-1955: (00880, 00253)
• 1946-1950: (00502, 00107)
• 1941-1945: (00251, 00060)

So to judge by the literature, it seems that we are entering into a Golden Era of mathematical "naturality" and "universality" ... we can hope so, anyway! :)

3 Included the full lemma, with a link to Google books

In accord with Daniel Miller's answer above, the study of "naturality" as a formal abstraction in mathematics can be traced back largely to two articles by Saunders Mac Lane and Samuel Eilenberg: Natural isomorphisms in group theory (1942) and General theory of natural equivalences (1945). Both articles are well worth reading.

How did these ideas arise? Roughly speaking, Eilenberg and MacLane began by recognizing that if an arbitrary choice of coordinates makes a difference to the quantities you are calculating and/or the theorems you are proving, then those quantities and theorems are not natural. Mac Lane has described this process in The development and prospects for category theory (1996) as follows:

I emphasize that the notions category and functor were not formulated or put in print until the idea of a natural transformation was also at hand.
Thus, one good start for students is to acquire a thorough practical grasp of naturality in the context of coordinate transformations in linear algebra and differential geometry. For

A concrete example , of a canonical usage of "naturality", which is accompanied by an extensive motivating discussion, can be found is given in John Lee's Introduction to Smooth Manifolds as the following lemma:

Lemma 12.16: (Naturality of the Exterior Derivative)

If $G\colon M\to N$ is a smooth map, then the pullback map $G^\star\colon \mathcal{A}^k(N)\to \mathcal{A}^k(M)$ commutes with $d$. That is, for all $\omega \in \mathcal{A}^k(N)$, we have $G^\star(d\omega) = d(G^\star\omega)$.

Being interested in practical applications of geometrically "natural" formalisms for quantum systems engineering, I've studied the usage in MacSciNet reviews of the words "natural*" (chiefly "natural" and "naturality") and "universal*" (chiefly "universal" and "universality").

Here are the numbers; their use is burgeoning!

• Year-Range (natural*, universal*)
• 2001-2005: (16788 uses, 05288 uses)
• 1996-2000: (14880, 04977)
• 1991-1995: (12550, 04432)
• 1986-1990: (10335, 03343)
• 1981-1985: (08775, 03013)
• 1976-1980: (07402, 02412)
• 1971-1975: (05668, 02040)
• 1966-1970: (03466, 01167)
• 1961-1965: (02211, 00610)
• 1956-1960: (01368, 00406)
• 1951-1955: (00880, 00253)
• 1946-1950: (00502, 00107)
• 1941-1945: (00251, 00060)

So to judge by the literature, it seems that we are entering into a Golden Era of mathematical "naturality" and "universality" ... we can hope so, anyway! :)

2 Added example of "naturality" from John Lee's book
1