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Sometimes, in a math paper, we read something like "Although a priori this statement could be true, it is in fact false, as proven by this theorem". But aren't all mathematical statements a priori? So why include that misleading bit of language in the beginning?

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  • "A priori" is relative to one's background knowledge, what is meant is, "as far as we know prior to proving the theorem". The stronger rationalist sense of "by reason alone", "prior to any sense experience", etc., is not in much use these days (mostly because math is not "prior" to humans who invented it under modern philosophies), so it is not particularly misleading. – Conifold May 1 '20 at 23:48
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    IMO, the statement "Although a priori this statement could be true, it is in fact false, as proven by this theorem" is quite meaningless... Maybe the author (if any) want to say that the statement seems plausible, but in fact it has been proven false. – Mauro ALLEGRANZA May 2 '20 at 7:40
  • Now the new post has an accepted answer and we can close this one as duplicate. – Mauro ALLEGRANZA May 5 '20 at 12:48
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In my experience as a mathematician, the mathematical usage of a priori corresponds more to the traditional definition of prima facie: it is applied for “conclusions we can draw fairly easily based on our current definitions/assumptions (together with previously acquired background knowledge)”, typically contrasted with “things that take some new non-trivial work to deduce”. A couple of examples:

Constructively this approach does not work, because it is not a priori evident that such a function exists. Once the mapping function is known to exist, it can easily be shown to be such a function.

Foundations of Constructive Analysis, Errett Bishop, 1967

…we don't know how to embed H*SF(n+1) as a sub-algebra of an algebra which we know a priori is commutative (if n is odd)…

The homology of iterated loop spaces, Peter May, 1976

This works if we can specify the degree a priori. If we cannot, or do not wish to, then it is more convenient to use binary forms.

Combinatorics the Rota way, Kung, Rota, Yan, 2009

As the question points out, this does not quite match the accepted philosophical sense(s) of a priori. I strongly doubt this is due to any intentional distinction — rather, it’s just the usual divergence between colloquial and technical usages of a term.

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