/sci/ - Science & Math

>>11006776
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Now, to guess "why" more morally from an exotic (topos theoretic) standpoint, it's maybe worth pointing out that [math]\sum[/math] resp. [math]\prod[/math] (just like [math]\exists[/math] and [math]\forall[/math]) arise from looking at projections [math] p : E \to X [/math] which for discrete and countable objects end up counting volume of fibrations resp. it's section space. I.e. there [math]\sum_X[/math] ends up collecting all the volumes of the fibres and [math]\prod[/math] the amount of sections (from the base X into the individual fibres). And combinatorics as we do it is just not too much concerned with counting the amount of legal functions, but rather items in a space. If you want the amount of items in over each point in X, you just add them up by summing over X. But if we were to describe a space X and were interested in how many functions s on X there are that map into a space with certain conditions, then you'd naturally take products over X (you take one a in X, collect the legal values functions can take, then you take another one, b, and all the values, and by counting functions, you need to multiply - if that makes sense).

>>11006774
https://en.wikipedia.org/wiki/LogSumExp