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  "body": "\n## Summary\n\nLogicism is the view that mathematics, or at least arithmetic, is a part of logic.\nIts claim is that mathematical notions can be defined in purely logical terms and that mathematical truths can be proved from logical laws alone.\nGottlob Frege, and later Bertrand Russell and Alfred North Whitehead in Principia Mathematica, tried to carry out this program.\nIt is one of the classical answers in the [[foundations of mathematics|EFFRJFW67Y]].\n\n## Explanations\n\n### L1 human\n\nA logicist says that numbers are not new things beyond logic.\nFor example, the number $2$ can be described as what all two-element collections have in common.\nAnd \"two-element\" can be said in logic: \"there are $x$ and $y$ with $x \\ne y$, and nothing else\".\nThen a fact such as $1 + 1 = 2$ would be a truth of logic.\nFrege's system turned out to be inconsistent because of Russell's paradox.\nLater systems avoided the paradox, but they needed axioms, such as an axiom of infinity, that are hard to call purely logical.\n",
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