P=NP remaining unproven means current complexity theory cannot rule out the existence of a polynomial algorithm, making 'unlikely' a sociological claim rather than a mathematical one.

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Complexity theory(where analogous issues might appear): A branch of computer science and mathematics that studies how difficult different problems are to solve and how much computational power they require.
P=NP(mathematics and computer science): A famous unsolved question in computer science asking whether two classes of problems (P and NP) are actually the same—if true, it would mean certain very hard problems can actually be solved quickly.
Polynomial algorithm(in computer science): A step-by-step procedure for solving a problem that doesn't take an absurdly long time—roughly, the time it takes grows at a reasonable rate as the problem gets bigger.
Sociological claim(as opposed to a logical claim): A statement about how society and social systems actually work in practice, based on how people behave together.

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