History and philosophy of mathematics

Mathematics of the 19th–21st centuries

Rigorization of analysis with the $\forall\varepsilon>0\,\exists\delta>0:0<|x-a|<\delta\Rightarrow|f(x)-L|<\varepsilon$ definition, Galois' criterion linking solvability by radicals to group structure, Riemann's manifolds $ds^2=g_{ij}\,dx^i dx^j$ and zeta function $\zeta(s)=\sum_{n=1}^{\infty} n^{-s}$ , Cantor's set theory, and a path through Hilbert's program, Grothendieck's schemes, and today's open problems.

IntuitionA journey from rigor to the research frontier

By 1800, calculus had worked wonders for two centuries but rested on shaky ground: nobody could say precisely what an "infinitesimal" was. The 19th century closed that gap with hard logic; the 20th and 21st centuries opened doors that Newton and Leibniz never imagined — abstract structures, spaces with more than three dimensions, and machine-checked proofs. This topic follows that arc: from the $\varepsilon$ - $\delta$ definition of a limit, through Galois' group theory and Riemann's geometry, to Grothendieck's schemes, Perelman's proof of the Poincaré conjecture, and today's AI-assisted formal verification.

Domain coloring of a complex function: 19th-century analysts (Cauchy, Riemann) studied functions on the complex plane, where zeros and poles reveal deep structure — the still-unproven Riemann Hypothesis is a statement about exactly where the zeta function's zeros lie.

SchoolRigorizing analysis: the $\varepsilon$ - $\delta$ definition of a limit

Definition: Cauchy–Weierstrass limit

We say $\lim_{x\to a}f(x)=L$ when: $\forall\varepsilon>0\,\exists\delta>0:0<|x-a|<\delta\Rightarrow|f(x)-L|<\varepsilon$ . In words: no matter how small a tolerance $\varepsilon$ you demand on the output, there is a window of radius $\delta$ around $a$ on the input side that guarantees it. Cauchy (1821) and later Weierstrass gave this precise, quantifier-based meaning to a concept Newton and Leibniz had only described intuitively as a quantity "approaching" a limit.

\forall\varepsilon>0\,\exists\delta>0:0<|x-a|<\delta\Rightarrow|f(x)-L|<\varepsilon

Each symbol carries weight: $\forall\varepsilon>0$ says "for every positive tolerance," $\exists\delta>0$ says "there is a positive input radius," and the implication $0<|x-a|<\delta\Rightarrow|f(x)-L|<\varepsilon$ says that being within $\delta$ of $a$ (but not equal to $a$ ) forces $f(x)$ within $\varepsilon$ of $L$ . This single line replaced two centuries of hand-waving about "vanishing quantities" and let mathematicians prove — not just believe — statements about continuity, derivatives, and convergence.

UndergraduateGalois theory: solvability by radicals as a group property

Definition: Galois' solvability criterion

Évariste Galois attached to every polynomial equation a group — its Galois group, the symmetries of its roots that preserve every rational relation among them — and proved: the equation is solvable by radicals (by a formula using $+,-,\times,\div$ and $n$ -th roots) if and only if this group is a solvable group, meaning it has a chain of subgroups $1=G_0\triangleleft G_1\triangleleft\cdots\triangleleft G_k=G$ with each quotient $G_{i+1}/G_i$ abelian. For example, $x^3-2$ over $\mathbb{Q}$ has Galois group $S_3$ , which is solvable via $1\triangleleft A_3\triangleleft S_3$ (both quotients, $A_3/1\cong\mathbb{Z}/3\mathbb{Z}$ and $S_3/A_3\cong\mathbb{Z}/2\mathbb{Z}$ , are abelian).

1\triangleleft A_3\triangleleft S_3

By contrast, a generic quintic such as $x^5-x-1$ has Galois group $S_5$ , and $S_5$ is not solvable — its only normal subgroups are $1$ , $A_5$ , and $S_5$ itself, and $A_5$ is simple and non-abelian, so no chain of abelian quotients exists. This is the Abel–Ruffini theorem in Galois' language: there is no general radical formula for degree-5 (or higher) equations, even though specific quintics like $x^5-1$ (whose roots are the 5th roots of unity, with abelian, hence solvable, Galois group) can still be solved by radicals.

The 19th century also freed geometry from Euclidean flatness. Bernhard Riemann proposed measuring distance intrinsically, on any smooth space, by a metric $ds^2=g_{ij}\,dx^i dx^j$ (repeated indices summed): distances and angles are read off the tensor $g_{ij}$ point by point, with no need to embed the space in a larger flat one. In the same 1859 line of work on prime-counting, Riemann extended the series $\zeta(s)=\sum_{n=1}^{\infty} n^{-s}$ to a function of a complex variable $s$ and conjectured that its non-trivial zeros all have real part $1/2$ — the Riemann Hypothesis, still open today and central to understanding the distribution of primes.

ds^2=g_{ij}\,dx^i dx^j

\zeta(s)=\sum_{n=1}^{\infty} n^{-s}

Meanwhile Georg Cantor built a rigorous theory of infinite sets and showed (Theorem 1 below) that some infinities are strictly larger than others — the real numbers $\mathbb{R}$ cannot be listed in a sequence the way the natural numbers can. At the 1900 International Congress of Mathematicians in Paris, David Hilbert crowned this century of rigor with a program: a list of 23 open problems, and a broader ambition to formalize all of mathematics on secure logical foundations. Three decades later, Kurt Gödel's incompleteness theorems (see "the incompleteness theorems" in Related Topics) showed that ambition could never be fully realized — any consistent formal system rich enough for arithmetic contains true statements it cannot prove.

Three centuries of mathematics, at a glance
Century	Key mathematicians	Landmark results	Key dates
19th	Cauchy, Riemann, Galois, Cantor	Rigorous $\forall\varepsilon>0\,\exists\delta>0:0<\|x-a\|<\delta\Rightarrow\|f(x)-L\|<\varepsilon$ limits; Galois' solvability criterion; Riemannian geometry $ds^2=g_{ij}\,dx^i dx^j$ and the zeta function $\zeta(s)=\sum_{n=1}^{\infty} n^{-s}$ ; set theory and uncountable infinities	1821–1874
20th	Hilbert, Gödel, Grothendieck, Poincaré	Hilbert's 23 problems; Gödel's incompleteness theorems; Grothendieck's schemes $\mathrm{Spec}(R)$ ; the Langlands program	1900–1970s
21st	Perelman, Maynard, Viazovska	Proof of the Poincaré conjecture; bounded gaps between primes; optimal sphere packing in dimensions 8 and 24; AI-assisted formal proof verification	2002–2024

Theorem: Cantor's theorem: the reals are uncountable

The interval $[0,1]$ (and hence $\mathbb{R}$ ) is uncountable: there is no way to list all of its elements as a sequence $x_1, x_2, x_3, \ldots$ indexed by the natural numbers.

Why is it true?

This is the first proof that infinite sets come in different sizes: the natural numbers and the reals are both infinite, but one infinity is strictly bigger. It underlies why most real numbers cannot be described by any finite formula, and it is the ancestor of the diagonal arguments used throughout logic and computer science (e.g. the halting problem).

Proof

Suppose, for contradiction, that $[0,1]$ were countable: every real number in it appears exactly once in some enumeration $x_1, x_2, x_3, \ldots$ . Write each in decimal form as $x_n = 0.d_{n1}d_{n2}d_{n3}\ldots$ , choosing the expansion that does not end in an infinite string of 9's when a number has two representations.

Now build a new number $y$ digit by digit, one digit at a time, by looking down the diagonal of this list: $y = 0.e_1e_2e_3\ldots$ , where the $n$ -th digit is defined by $e_n=\begin{cases}5 & d_{nn}\neq5\\6 & d_{nn}=5\end{cases}$ . Restricting the choice to $\{5,6\}$ guarantees $y$ never ends in all 0's or all 9's, so its decimal expansion is unique and unambiguous.

For every index $n$ , the number $y$ differs from $x_n$ at the $n$ -th decimal digit by construction ( $e_n\neq d_{nn}$ ), so $y\neq x_n\ \forall n$ . Since $y$ 's digits are all $5$ or $6$ , $y\in[0,1]$ .

But then $y$ is a real number in $[0,1]$ that is not equal to any $x_n$ in the supposedly complete list — a contradiction, since the list was assumed to contain every element of $[0,1]$ . Therefore no enumeration of $[0,1]$ can exist, and $[0,1]$ (hence the larger set $\mathbb{R}$ ) is uncountable.

Theorem: Bolzano–Weierstrass theorem (bisection proof)

Every bounded sequence $(x_n)$ of real numbers has a convergent subsequence $(x_{n_k})$ .

Why is it true?

This is the key compactness fact that makes real analysis work: it guarantees that a bounded process cannot wander forever without accumulating somewhere, and it underlies proofs of the extreme value theorem, existence of minimizers in optimization, and completeness arguments throughout analysis.

Proof

Let $(x_n)$ be bounded, so there exist $a<b$ with $x_n\in[a,b]$ for every $n$ . We build a nested sequence of intervals by repeated bisection. Split $[a,b]$ into its two halves $[a,\tfrac{a+b}{2}]$ and $[\tfrac{a+b}{2},b]$ . Since the sequence has infinitely many terms (counted with index) and only two halves are available, by the pigeonhole principle at least one half must contain $x_n$ for infinitely many indices $n$ ; call that half $[a_k,b_k]$ with $k=1$ .

Repeat the same bisection on $[a_1,b_1]$ : split it in two, and again by pigeonhole at least one half contains $x_n$ for infinitely many $n$ ; call it $[a_2,b_2]$ . Continuing forever produces a nested chain $[a,b]\supset[a_1,b_1]\supset[a_2,b_2]\supset\cdots$ , each containing $x_n$ for infinitely many indices, and each half the width of the previous one, so the width of $[a_k,b_k]$ is exactly $(b-a)/2^{k}$ , which tends to $0$ as $k\to\infty$ .

Now build the subsequence: since $[a_1,b_1]$ contains infinitely many terms of the sequence, pick any index $n_1$ with $x_{n_1}\in[a_1,b_1]$ . Since $[a_2,b_2]$ also contains infinitely many terms (all but finitely many indices remain available), pick $n_2>n_1$ with $x_{n_2}\in[a_2,b_2]$ . Continuing inductively, at each step $k$ choose $n_k>n_{k-1}$ with $x_{n_k}\in[a_k,b_k]$ ; this is always possible because $[a_k,b_k]$ contains infinitely many terms, so infinitely many indices beyond $n_{k-1}$ remain.

By the nested interval property of the real numbers (each $[a_k,b_k]$ is closed, nested, and their widths shrink to $0$ ), the intersection is a single point: $\bigcap_{k=1}^{\infty} [a_k,b_k] = \{L\}$ for some $L\in[a,b]$ . Since both $x_{n_k}$ and $L$ lie in $[a_k,b_k]$ , whose width is $(b-a)/2^{k}$ , we get $|x_{n_k}-L|\le (b-a)/2^{k}\to 0$ . As $k\to\infty$ , the right-hand side tends to $0$ , forcing $x_{n_k}\to L$ . Thus $(x_{n_k})$ is a convergent subsequence of $(x_n)$ , as required.

AdvancedThe 20th century: schemes and the Langlands program

Classical algebraic geometry studied the solution sets of polynomial equations over the complex or real numbers. In the 1960s Alexander Grothendieck rebuilt the subject from its foundations: to any commutative ring $R$ he attached a geometric space $\mathrm{Spec}(R)$ (its "spectrum," whose points are the prime ideals of $R$ ), turning every ring — including rings with nilpotents, or rings of integers modulo $p$ — into a genuine geometric object called a scheme. This let the same geometric intuition and machinery (dimension, smoothness, cohomology) apply uniformly to number theory and geometry at once, and it was essential to Wiles' proof of Fermat's Last Theorem and to Deligne's proof of the Weil conjectures. In parallel, the Langlands program, proposed by Robert Langlands from 1967 onward, conjectures a deep correspondence between Galois representations (symmetries of solutions to polynomial equations, generalizing Galois groups) and automorphic forms (highly symmetric functions from harmonic analysis) — a unifying dictionary that has already produced major theorems and still drives much of modern number theory.

ResearchThe 21st century: today's research frontier

UndergraduateReal-World Applications and Worked Examples

These abstractions are not idle. Galois theory of finite fields is the algebraic backbone of modern cryptography: AES encrypts data using arithmetic in the finite field $\mathrm{GF}(2^8)$ , and elliptic-curve cryptography (ECC), which secures most web traffic (TLS) and cryptocurrencies, relies on the group structure of points on an elliptic curve over a finite field — a direct descendant of 19th-century field and group theory. Riemann's intrinsic geometry $ds^2=g_{ij}\,dx^i dx^j$ is the mathematical language of Einstein's general relativity, where spacetime curvature (encoded in $g_{ij}$ ) replaces Newtonian gravity. And in machine learning, both directions of this history matter today: formal verification tools built on the same logical foundations Hilbert sought (Lean, Isabelle) are now used to machine-check mathematical proofs, while representation theory — the modern descendant of Galois' symmetry groups — informs the design of equivariant neural network architectures that respect physical or geometric symmetries.

Example: Proving a limit rigorously with $\varepsilon$ - $\delta$

Prove rigorously, using the $\forall\varepsilon>0\,\exists\delta>0:0<|x-a|<\delta\Rightarrow|f(x)-L|<\varepsilon$ definition, that $\lim_{x\to2}(3x-1)=5$ .

Solution

Let $\varepsilon>0$ be given. We must produce a $\delta>0$ such that $0<|x-2|<\delta$ forces $|(3x-1)-5|<\varepsilon$ . Start from the target inequality and simplify: $|(3x-1)-5|=3|x-2|<3\delta=\varepsilon$ — that is, $|(3x-1)-5|=|3x-6|=3|x-2|$ .

So we need $3|x-2|<\varepsilon$ , which is equivalent to $|x-2|<\varepsilon/3$ . This tells us exactly what $\delta$ to choose: take $\delta=\varepsilon/3$ .

Now verify it works. Suppose $0<|x-2|<\delta=\varepsilon/3$ . Multiplying both sides of $|x-2|<\varepsilon/3$ by $3$ gives $3|x-2|<\varepsilon$ , i.e. $|(3x-1)-5|<\varepsilon$ , exactly the conclusion required. Since $\varepsilon>0$ was arbitrary and we exhibited a valid $\delta$ for each one, the $\varepsilon$ - $\delta$ definition is satisfied, so $\lim_{x\to2}(3x-1)=5$ is proven rigorously — not just plausible from a graph, but a logical certainty.

Example: The Galois group of $x^3-2$ is $S_3$ , and $S_3$ is solvable

Determine the Galois group of $x^3-2$ over $\mathbb{Q}$ , and confirm that it is solvable.

Solution

The three roots of $x^3-2$ in $\mathbb{C}$ are $\sqrt[3]{2}$ , $\omega\sqrt[3]{2}$ , and $\omega^2\sqrt[3]{2}$ , where $\omega=e^{2\pi i/3}$ is a primitive cube root of unity. The splitting field — the smallest field containing all three roots — is $\mathbb{Q}(\sqrt[3]{2},\omega)$ .

Build this field in two steps. First adjoin the real root: $[\mathbb{Q}(\sqrt[3]{2}):\mathbb{Q}]=3$ , since $x^3-2$ is irreducible over $\mathbb{Q}$ (by Eisenstein's criterion at $p=2$ ) and is the minimal polynomial of $\sqrt[3]{2}$ . This extension alone is not yet a splitting field, because $\mathbb{Q}(\sqrt[3]{2})\subset\mathbb{R}$ contains no complex root like $\omega\sqrt[3]{2}$ . Then adjoin $\omega$ : since $\omega$ satisfies $x^2+x+1=0$ , which stays irreducible over $\mathbb{Q}(\sqrt[3]{2})$ (that field is real, but $\omega$ is not), this second step has degree 2. Multiplying, $[\mathbb{Q}(\sqrt[3]{2},\omega):\mathbb{Q}]=6$ .

The Galois group has order equal to this degree, so $|\mathrm{Gal}|=6$ , and it acts faithfully by permuting the 3 roots (any automorphism is determined by where it sends the roots, since they generate the splitting field). The only group of order 6 that can act as all permutations of 3 objects is $S_3$ itself (order $3!=6$ ), and indeed one can exhibit generators explicitly: an order-3 automorphism $\sigma$ cycling the three roots (fixing $\omega$ , sending $\sqrt[3]{2}\mapsto\omega\sqrt[3]{2}\mapsto\omega^2\sqrt[3]{2}\mapsto\sqrt[3]{2}$ ) and an order-2 automorphism $\tau$ swapping two roots by complex conjugation (fixing $\sqrt[3]{2}$ , sending $\omega\mapsto\omega^2$ ). Together $\sigma$ and $\tau$ generate a group of order 6 acting as the full symmetric group on the roots, so $\mathrm{Gal}(\mathbb{Q}(\sqrt[3]{2},\omega)/\mathbb{Q})\cong S_3$ .

Finally, solvability: $S_3$ has the normal series $1\triangleleft A_3\triangleleft S_3$ , where $A_3=\{1,\sigma,\sigma^2\}\cong\mathbb{Z}/3\mathbb{Z}$ is a normal subgroup of index 2. Both quotients $A_3/1\cong\mathbb{Z}/3\mathbb{Z}$ and $S_3/A_3\cong\mathbb{Z}/2\mathbb{Z}$ are abelian (in fact cyclic), so $S_3$ is solvable — confirming, via Galois' criterion, that $x^3-2$ is solvable by radicals, exactly as the explicit cube-root formula for its roots already showed.

Common mistake. A common confusion: students remember "Abel–Ruffini says quintics can't be solved by radicals" and wrongly generalize this to "degree

\ge 5

means unsolvable." But Galois' criterion is about the group, not the degree. Some quintics are perfectly solvable:

x^5-1

has roots that are the 5th roots of unity, forming a cyclic (hence abelian, hence solvable) Galois group

\mathbb{Z}/5\mathbb{Z}

. What Abel–Ruffini really says is that there is no single radical formula that works for every quintic, because a generic quintic like

x^5-x-1

has Galois group

S_5

, which is not solvable. A second, related trap: confusing "countably infinite" with "uncountably infinite" — the rationals

\mathbb{Q}

are countable (the same size as

\mathbb{N}

) despite being dense in

\mathbb{R}

, while

\mathbb{R}

itself is strictly larger, as Theorem 1 above proves; density has nothing to do with cardinality.

Research frontier as of 2026

Three landmark results anchor the 21st century so far. In 2002–2003, Grigori Perelman posted preprints proving the Poincaré conjecture (every simply connected closed 3-manifold is topologically a sphere) via Richard Hamilton's Ricci flow program, one of the deepest results in topology and geometry to date (he declined both the Fields Medal and the Clay Millennium Prize money). In 2013–2014, James Maynard and Yitang Zhang independently produced major breakthroughs on bounded gaps between primes, showing there are infinitely many prime pairs within a bounded distance of each other, and Maynard's later refinements pushed results on primes in arithmetic progressions and on gaps between consecutive primes. In 2016, Maryna Viazovska solved the sphere-packing problem in dimension 8 using modular forms, and shortly after, with collaborators, in dimension 24 — the first optimal sphere-packing proofs in dimensions above 3 since Kepler's conjecture in dimension 3. A fourth, ongoing shift is methodological rather than a single theorem: large-scale formalization projects (the Liquid Tensor Experiment, formalizing deep results of Peter Scholze in the Lean proof assistant; growing Mathlib and Isabelle/HOL libraries) and AI systems — DeepMind's AlphaProof and AlphaGeometry reached silver-medal-equivalent performance at the 2024 International Mathematical Olympiad — are starting to make machine-checked proof a practical tool for research mathematics, not just an aspiration Hilbert could only sketch in 1900.

Using the $\forall\varepsilon>0\,\exists\delta>0:0<|x-a|<\delta\Rightarrow|f(x)-L|<\varepsilon$ definition, what value of $\delta$ (in terms of $\varepsilon$ ) proves $\lim_{x\to2}(3x-1)=5$ ?

The polynomial $x^3-2$ over $\mathbb{Q}$ has Galois group isomorphic to which group?

Which 19th-century algebraic structure underlies modern elliptic-curve cryptography (ECC) that secures most web traffic (TLS)?

Riemann's intrinsic geometry, introduced in the 19th century, provides the mathematical language for which 20th-century physical theory?

References

Morris Kline (1980). Mathematics: The Loss of Certainty
Maryna Viazovska (2016). The sphere packing problem in dimension 8 · arXiv:1603.04246
DeepMind (2024). AI solves IMO problems at silver medal level