Where the ideas came from, and what mathematical truth even is.
10 topics · 17 curated works
No prior grounding assumed.
The Thirteen Books of Euclid's Elements
Euclid (trans. Thomas L. Heath) · 300 BCE
Builds the whole of classical geometry and much of number theory from five postulates and a handful of common notions, establishing the axiomatic…
+4 more at this level
Assumes you know the vocabulary.
Mathematical Problems
David Hilbert (trans. Mary Winston Newson) · 1900
Sets out twenty-three unsolved problems as a deliberate agenda for the coming century of mathematics, arguing that a field's health can be read from…
+5 more at this level
Primary sources and full treatments.
The Algebra of Mohammed Ben Musa
Muhammad ibn Musa al-Khwarizmi (trans. Frederic Rosen) · 820
Sets out systematic rules for solving equations by restoring and balancing terms, treating algebra as a general method for a class of problems rather…
+5 more at this level
12 of 17 works
Euclid (trans. Thomas L. Heath)
Builds the whole of classical geometry and much of number theory from five postulates and a handful of common notions, establishing the axiomatic method as the model for what a mathematical proof is.
link checked 17 Sept 2026David Brewster
Draws on Newton's papers and correspondence to argue that his scientific achievement and his private, often bitter disputes over priority came from the same intensely secretive and self-certain temperament.
about 7 hours to readlink checked 17 Sept 2026W.W. Rouse Ball
Traces mathematics from ancient Egypt through the calculus era as a continuous chain of individuals building on and correcting each other, arguing the history is best told through its practitioners rather than as an abstract progression of ideas.
link checked 17 Sept 2026Henri Poincare
Argues that mathematical discovery is driven by an unconscious aesthetic sense for which combinations of ideas are fruitful, against the view that proof is generated by mechanical logical deduction alone.
link checked 17 Sept 2026G.H. Hardy
Defends pure mathematics done for its own beauty rather than its use, arguing a mathematician's work is judged by seriousness and depth in the same way a poem or painting is, not by application.
56 pageslink checked 17 Sept 2026David Hilbert (trans. Mary Winston Newson)
Sets out twenty-three unsolved problems as a deliberate agenda for the coming century of mathematics, arguing that a field's health can be read from the difficulty and clarity of the problems it poses itself.
link checked 17 Sept 2026Richard Dedekind
Defines the real numbers as cuts in the rationals and the natural numbers by their order-generating properties alone, arguing that continuity and number themselves need logical construction rather than geometric intuition.
link checked 17 Sept 2026J. M. Child
Translates Leibniz's working notebooks from 1673 to 1677 and argues from their dated content that Leibniz arrived at the differential and integral calculus independently of, and partly before, Newton published his own method.
link checked 17 Sept 2026Richard Zach (Stanford Encyclopedia of Philosophy)
Explains Hilbert's proposal to secure all of mathematics by proving the consistency of formal systems using only finitary methods, and traces how Godel's incompleteness theorems undercut the program's original strong form.
link checked 17 Sept 2026Rosalie Iemhoff (Stanford Encyclopedia of Philosophy)
Presents Brouwer's view that mathematics is a mental construction rather than a discovery about an independent realm, and traces the consequence that a proof of existence must actually construct the object, so the law of excluded middle fails.
link checked 17 Sept 2026Oystein Linnebo (Stanford Encyclopedia of Philosophy)
Lays out the case that numbers and sets exist independently of us as abstract objects, and works through the main objection: how creatures like us could know anything about objects with no causal power at all.
link checked 17 Sept 2026Muhammad ibn Musa al-Khwarizmi (trans. Frederic Rosen)
Sets out systematic rules for solving equations by restoring and balancing terms, treating algebra as a general method for a class of problems rather than a collection of solved examples.
link checked 17 Sept 2026