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Feb 1, 2015 at 10:24 comment added Paul Taylor Wikipedia is not very informative on (Kolmogorov and) Arnold's work on this topic. Do you have references to (the original papers and) some "popular" account of the ideas that are involved? (The question should perhaps be tagged as real analysis.)
Feb 1, 2015 at 8:50 history edited David Spivak CC BY-SA 3.0
wanted to make the title question more searchable.
Feb 1, 2015 at 8:45 vote accept David Spivak
Feb 1, 2015 at 7:52 history edited David Spivak CC BY-SA 3.0
used Todd Trimble's comment to redesign my definition of truncated algebraic theories.
Feb 1, 2015 at 7:42 history edited David Spivak CC BY-SA 3.0
typo
Feb 1, 2015 at 1:32 answer added Eric Wofsey timeline score: 28
Feb 1, 2015 at 1:05 comment added Theo Johnson-Freyd My strong recollection is that you get all functions $\mathbb R^n \to \mathbb R$ by composing functions $\mathbb R \to \mathbb R$ and addition $+ : \mathbb R^2 \to \mathbb R$. My recollection is that this is true in both the continuous and smooth cases. But Qiaochu's comment makes me worried that perhaps I'm remembering Arnold's solution to Hilbert's problem for continuous functions, and perhaps my recollections are wrong for the smooth category.
Jan 31, 2015 at 23:38 comment added Todd Trimble Might you mean by $\mathcal{C}_{\leq n}$ the smallest Lawvere subtheory contained in $\mathcal{C}$ and with the same $k$-ary operations for $k = 0, \ldots, n$?
Jan 31, 2015 at 23:31 comment added Qiaochu Yuan Vladimir Arnold answered this question for continuous functions, it seems: en.wikipedia.org/wiki/Hilbert%27s_thirteenth_problem
Jan 31, 2015 at 23:17 history asked David Spivak CC BY-SA 3.0