Below I copy and paste wikipedia segments as well as a quote from Richard Dedekind. We concentrate here on just the nineteenth century, but see the last section where we pay tribute to Gottfried Wilhelm Leibniz.
I don't agree with the author. Indeed, a couple of axioms needed to be pinned down, and set theory provided the glue, but logical and rigorous exposition of mathematical reasoning took a quantum leap during all of the nineteenth century.
He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors.
Soundness of calculus
Weierstrass was interested in the soundness of calculus, and at the time, there were somewhat ambiguous definitions regarding the foundations of calculus, and hence important theorems could not be proven with sufficient rigour. While Bolzano had developed a reasonably rigorous definition of a limit as early as 1817 (and possibly even earlier) his work remained unknown to most of the mathematical community until years later, and many mathematicians had only vague definitions of limits and continuity of functions.
Delta-epsilon proofs are first found in the works of Cauchy in the 1820s. Cauchy did not clearly distinguish between continuity and uniform continuity on an interval. Notably, in his 1821 Cours d'analyse, Cauchy argued that the (pointwise) limit of (pointwise) continuous functions was itself (pointwise) continuous, a statement interpreted as being incorrect by many scholars. The correct statement is rather that the uniform limit of continuous functions is continuous (also, the uniform limit of uniformly continuous functions is uniformly continuous). This required the concept of uniform convergence, which was first observed by Weierstrass's advisor, Christoph Gudermann, in an 1838 paper, where Gudermann noted the phenomenon but did not define it or elaborate on it. Weierstrass saw the importance of the concept, and both formalized it and applied it widely throughout the foundations of calculus.
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory), axiomatic foundation for the natural numbers, algebraic number theory and the definition of the real numbers.
That which is provable, ought not to be believed in science without
— Richard Dedekind (1888)
REVOLUTION or CONSOLIDATION?
Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers.
Gottfried Wilhelm Leibniz:
We find this quote from Benoit B. Mandelbrot discussing and quoting his hero, Leibniz:
In Euclidis Prota ..., which is an attempt to tighten Euclid's axioms,
he states ...: "I have diverse definitions for the straight line. The
straight line is a curve, any part of which is similar to the whole,
and it alone has this property, not only among curves but among sets."
This claim can be proved today.
So Leibniz was attempting to revamp the axioms found in Euclid's Elements.
Sometimes you can pack a considerable amount of mathematics and logic into notation, but the Leibniz's notation is truly a remarkable achievement. Amazingly, wikipedia states
In the modern rigorous treatment of non-standard calculus,
justification can be found to again consider the notation as
representing an actual quotient.
I get the impression that Leibniz was the first person OF THIS WORLD interested in meta-mathematics; try a google search on keywords Gödel Leibniz.
He is still in the news today:
The Philosopher Who Helped Create the Information Age
Gottfried Wilhelm Leibniz isn’t a household name—but he should be.
Google doodle: Who was Gottfried Wilhelm Leibniz? What does the doodle mean?
Today’s Google doodle celebrates the birth date of German polymath and philosopher Gottfried Wilhelm Leibniz. So who was he?