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Two visions of ultrafinitism intricately intertwined
These two concepts of ultrafinitism—one positing a largest number, another asserting totality for addition and multiplication but not exponentiation—are…
Dec 29, 2025
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Joel David Hamkins
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Ultrafinitism with a largest number
According to the theory of finite arithmetic (FA), there is a largest natural number, and consequently addition and multiplication are only partially…
Dec 19, 2025
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Joel David Hamkins
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Ultrafinitism
Ultrafinitism is the philosophical view that only comparatively small or accessible numbers exist.
Dec 12, 2025
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Joel David Hamkins
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The surreal line is topologically compact—or is it?
Shocking instances of compactness in the surreal line
Nov 28, 2025
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Joel David Hamkins
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The surreal line is topologically disconnected—or is it?
The surreal line is topologically disconnected according to a natural conception of connectedness. Nevertheless, on another conception—attending to…
Nov 15, 2025
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Joel David Hamkins
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Most Popular
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Zeno's paradox
Jan 7, 2023
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Joel David Hamkins
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The surreal numbers
Jan 6, 2024
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Joel David Hamkins
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The Book of Numbers
Jan 2, 2023
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Joel David Hamkins
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Potential versus actual infinity
Apr 14, 2023
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Joel David Hamkins
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The Book of Infinity
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Take my Infinity final exam!
How well can you answer the questions on the final exam for my Infinity course?
Dec 21, 2024
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Joel David Hamkins
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Throwing Darts at the Real Line
Mathematical philosophers have proposed various thought experiments suggesting different resolutions of the continuum hypothesis. Can we hope…
Sep 25, 2024
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Joel David Hamkins
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What is the infinite?
What does it mean, exactly, to say that a set is infinite? Can we provide precise mathematical definitions of the finite and the infinite?
Sep 11, 2024
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Joel David Hamkins
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Infinite Games
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Tactics versus strategies—the case of chess
Does chess admit of winning or drawing tactics? Which information exactly do we need to include as part of the board position?
Aug 17, 2025
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Joel David Hamkins
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The tactical variation of the fundamental theorem
We prove the tactical variation of the fundamental theorem of finite games—for finite games with sufficiently rich board positions, one of the players…
Aug 10, 2025
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Joel David Hamkins
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Tactics versus strategies in the theory of games
How do tactics differ from strategies? Does the fundamental theorem of finite games hold for tactics? Must every finite game have a winning tactic for…
Aug 3, 2025
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Joel David Hamkins
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A Panorama of Logic
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Proof and the Art
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Philosophy of Mathematics
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The mathematics and philosophy of the infinite
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My Books
Lectures on the Philosophy of Mathematics, MIT Press 2021
Proof and the Art of Mathematics, MIT Press 2020
Proof and the Art of Mathematics: Examples and Extensions, MIT Press, 2021
A Mathematician's Year in Japan, Kindle KDP, 2015
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