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  "body": "\n## Summary\n\nThe modality of a critical point of a function counts continuous parameters, called moduli.\nIt is the number of moduli needed to describe all nearby critical points up to a change of coordinates.\nGerms of modality $0$ are the simple singularities: only finitely many types occur near them.\nArnold introduced the notion and classified the germs of modality $0$, $1$ and $2$.\nModality measures how far the classification of singularities is from being a finite list.\n\n## Explanations\n\n### L1 human\n\nNear some critical points there are only finitely many shapes of critical point, like a short catalogue.\nNear others the shapes vary continuously, like the angles in a family of triangles, and no finite catalogue suffices.\nThe modality is the number of such continuous dials.\nFor example, the function $x^4 + y^4$ on $\\C^2$ vanishes on four lines through $0$.\nFour lines through a point have a cross-ratio, which no change of coordinates can alter.\nSo nearby functions $x^4 + y^4 + a x^2 y^2$ give infinitely many different types, and the modality is $1$.\n",
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