Well, if you are saying that the textbook ought to have said "by the way, this particular result is based on hotly contested definitions, and who knows what might be wrong, but trust us this is a good thing to learn" I am agreeing to do that. The source just said that textbooks stated and proved that theorem, (I assumed without this addition).
I'm not saying that, as the whole of calculus was "hotly contested" as far as definitions were concerned. But telling students that is among the best ways to ensure they disregard the subject. If it weren't for textbooks using poor definitions, proofs which were actually only resembled proofs, and so on, we wouldn't have calculus as it is today. The fundamental advances in mathematics resulted from students being exposed to the mathematics of the time, and the caveat "by the way, the mathematical community hasn't really accepted any of this" is a sure way to prevent progress.
Or has it been proven? That's the point.
It has.
I don't know whether you consider Fermat's last theorem to be proved or not, but if you do (probably) either you have checked it yourself, or you are just accepting the consensus which has developed in its support.
I have read the proof (Wiles' proof and a number of works on it).
In the second case (which I suspect is true) it is not quite a satisfactory and absolutely sure thing for me.
Would you explain why?
My philosophical issue is that such a truth which is determine by consensus (in a clique) can never be treated as absolute and the chance remains (even a microscopic one) that it is wrong.
The chance that no reality exists and you are dreaming exists as well. Of course there is a chance. There are no absolutes with human knowledge. You have to trust that what you see, hear, etc. exists. You have to trust that if I say "if 2 is the smallest prime, then no number smaller than 2 is prime" that by accepting the antecent is true, the consequent must me. Science is plagued with consensus and similar problems. Mathematics, however, is much more of a closed system. It is not a matter of consensus the way the sciences can be. That's why the history of modern math contains so many examples of non-mathematicians changing mathematics forever. It's true that mathematics contains specialists of relatively small groups in particular areas. However, one of my areas of study is dynamical systems and fuzzy sets. A neighbor of mind is a math professor at Brown. His field is number theory. Yet he wrote a book on dynamical systems. How is this possible? Because so many different specialties related. Statistics, set theory, combinatorics, linear algebra, abstract algebra, graph theory, multivariate calculus, are all fundamentally related in many areas. So the "clique" argument is, I think, faulty.
