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  "body": "\n## Summary\n\nSynthetic differential geometry develops differential calculus and differential geometry with genuine infinitesimals, inside a [[topos|6837RS7N82]] instead of the category of sets.\nIts basic object is a number line $R$ whose set $D$ of elements of square zero is not reduced to $0$.\nBy the Kock–Lawvere axiom, every function from $D$ to $R$ is affine.\nTangent vectors become maps from the infinitesimal set $D = \\set{d : d^2 = 0}$ into a space, and derivatives are defined without limits.\nModels built from [[C-infinity rings|4EJED0YZBB]] show that the theory is consistent and contains ordinary smooth manifolds.\n\n## Explanations\n\n### L1 human\n\nPhysicists and early analysts often computed with tiny quantities whose square is zero.\nSynthetic differential geometry makes this rigorous.\nIn it, $f(x + d) = f(x) + d\\, f'(x)$ holds exactly for every infinitesimal $d$, and this equation defines $f'(x)$.\nThe price is a different logic: one cannot assume that every number is either zero or not zero.\n",
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