5.154 In an urn there are equal numbers of white and black balls (and no others). I draw one ball after another and lay them back again in the urn. Then by this experiment I can determine that the number of black balls drawn and the number of white balls drawn get nearer to one another as the drawing goes on.

That is therefore not a mathematical fact.

If I now say: It is as probable that I will draw a white ball as a black, then that means: All circumstances known to me (including laws of nature assumed as hypotheses) give to the occurrence of the one event no more probability than to the occurrence of the other. That is, they give – as can be easily gathered from the explanations above – to each the probability ½.

What I confirm by the experiment is that the occurrence of both events is independent of the circumstances with which I am no closer acquainted.

Since the first paragraph is entirely a priori, why is this not a mathematical fact? Is it logical? Empirical? And what circumstances is he talking about at the end? Obscure to me.