The Foundational Crisis of Mathematics
Disclaimer: I'm not a mathematician or historian of mathematics. This description is that of a layperson; corrections and pushback are welcome.
At the start of the 20th century, mathematicians became increasingly worried about the foundations of mathematics – the subjects and concepts that formed the basis for more advanced topics in the field. Various logicians had uncovered paradoxes arising from principles of logic and set theory (two of the fundamental areas of mathematical inquiry) and were concerned that these paradoxes would propagate to "infect" the whole superstructure of mathematics with contradiction and falsehood.
In an effort to eliminate such paradoxes once and for all, Alfred North Whitehead and Bertrand Russell (both mathematicians and philosophers) produced the Principia Mathematica (or PM), a work of herculean effort and rigor. (Famously, Principia Mathematica does not definitively arrive at the truth of the proposition "1 + 1 = 2" until page 86 of volume II – a full 752 pages into the text). This text set out to retrace mathematicians' steps, returning to the very first assumptions that mathematicians make and work back to arithmetic, number theory, etc., all the time eliminating any opportunity for paradox or contradiction to arise.
Gödel's Work
Though Whitehead and Russell had hoped to use PM to make mathematics unassailable and invulnerable to contradiction, logician Kurt Gödel showed that such iron-clad aspirations are not possible. Gödel's two Incompleteness Theorems ingeniously harnessed the power and rigor of Principia Mathematica, using the mathematical instrument that Whitehead and Russell had developed to create self-referential statements – propositions about PM written in the symbolic logic of PM. In this way, Gödel was able to re-introduce contradiction into Whitehead and Russell's project through the construction of statements similar to "this statement is false."
For a more thorough explanation of the Incompleteness Theorems (from people who actually know what they're talking about) see the following resources:
- Nagel & Newman: Gödel's Proof
- Hofstadter: Gödel, Escher, Bach- An Eternal Golden BraidGödel, Escher, Bach- An Eternal Golden Braid
Purchase a copy on IndieBound.
Read the Wikipedia article on GEB.
Background
I first encountered this book by Douglas Hofstadter in high school and became enraptured. The concepts from Gödel, Escher, Bach that have most influenced my thinking and artistic practice are summarized in the notes to "Creativity and Constraint: Queering the Formal System."
Gödel, Escher, Bach (also abbreviated as GEB) usually proves too difficult for me to adequately summarize in conversation; I'll try to be... - Veritasium: Math Has a Fatal Flaw
- Gödel: On Formally Undecidable Propositions of Principia Mathematica and Related Systems
Last modified on 01-28-2022.