897: Remarks on Reverse Mathematics/4 -- a historical view

Sam Sanders sasander at me.com
Sun Oct 3 14:57:08 EDT 2021

Dear FOM,

Harvey Friedman wrote the following:

> The use of any functionals of higher type in actual mathematics is
> almost nil and certainly is not fundamental like the use of RCA0 which
> is pervasive.

I argue below that at least historically, third-order objects can be found in the 
the mainstream literature (Euler, Abel, Dirichlet, …), in the form of discontinuous functions.  

I do agree that there are few natural/elegant/explicit fourth-order objects to be found in mathematics.
A lone example is the Banach indicatrix, going back to Banach, which yields a characterisation of functions of bounded variation.  
Banach showed the case for continuous functions, while a theorem of Sierpinski generalises the results beyond that.  

However, discontinuous functions (say on R) are third-order and are can readily/explicitly be found in mainstream mathematics, 
and this has been true for hundreds of years.  My point is that I am not a historian, but have collected the following examples in 
a couple of years of looking through the old literature.  Hence, there must be many many examples in the “lesser” literature from 1700 to 1900.  

* Around 1767, Euler introduces a function that is zero everywhere, except for a finite non-zero value in one point ([1, SS39])

* Around 1823, Abel defines a number of functions via series, and notes that some are discontinuous ([2, p. 202])

* Around 1826, Abel defines a discontinuous function (via a series of sines) as a (purported) counterexample to Cauchy’s sum theorem.  Fourier had done the same by then.

* Around 1829, Dirichlet formulates conditions for the convergence of the Fourier series of a given function ([4]).  One of the conditions is that the function have only finitely many discontinuities.  As a counterexample to his conditions, he provides his “characteristic function of Q”.

* Around 1847, Seidel formulates a theorem concerning the (arbitrary slow) convergence of series approximating discontinuous functions ([5]). The paper was later republished in volume 116 of Ostwald’s Klas- siker der exakten Wissenschaften.

* Around 1854, Riemann defines a Riemann integrable function with countably many discontinuities via a series in his Habilitationsschrift ([6]).  The latter monograph is said to have 
really sparked the study of discontinuous functions (I would call this the consensus in the history of math).  

One more example relevant for my second email:

* Around 1875, Darboux proves a number of results about discontinuous functions (including examples).  This includes a supremum principle for any [0,1] -> [0,1] function.  




[1] L. Euler, E ́claircissements d ́etaill ́es sur la g ́en ́eration et la propagation du son, et sur la formation de l’ ́echo (Paper E340, 1767) (1926).

[2]  N H Abel  , Œuvres compl`etes, Tome II, Imprimerie de Grøndahl & Son, Christiania, 1981.

[3] ______ , Untersuchungen über die Reihe …. Journal für die Reine und Angewandte Mathematik, 1 (4) (1826), pp. 311-339

[4] L. P. G. Dirichlet, Sur la convergence des s ́eries trigonom ́etriques qui servent a` repr ́esenter une fonction arbitraire entre des limites donn ́ees (2008). https://arxiv.org/abs/0806 <https://arxiv.org/abs/0806>.


[5] Ph. L Seidel, Note u ̈ber eine Eigenschaft der Reihen, welche discontinuirliche Functio- nen darstellen, Abhandl. der Math. Phvs. Klasse der Kgl. Bayerischen Akademie der Wis- sensch ̈aften V (1847), no. 2, 381–394.

[6] Bernhard Riemann, Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe, Abhandlungen der Ko ̈niglichen Gesellschaft der Wissenschaften zu Go ̈ttingen, Volume 13. Habilitation thesis defended in 1854, published in 1867, pp. 47.

[7] Gaston Darboux, M ́emoire sur les fonctions discontinues, Annales scientifiques de l’E ́cole Normale Sup ́erieure 1875
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