I was reminded the other day of a conversation I had at the back-end of last year.

It seems there is a school of mathematical thinking whose slogan is: if you cannot give me explicit instructions on how to construct an object, I am under no obligation to accept its existence. This school is called constructivism.

Among other things this is to deny the Axiom of Choice, which is a part of the axiomatic set theory known as ZFC, and very widely regarded as the bed-rock of all modern mathematics. So let's see what this agenda might imply.

Off the top of my head, I would say that, by denying the AC, the well-ordering theorem - any set can be so ordered that each subset has a least element - need not apply in general; specifically, it may not apply to

. Hurrah!

It may also deny the validity of the Banach-Tariski paradox - any n-ball can be "dissected" and "reassembled" in such a way as to generate two n-balls, each with the same volume as the first. Hurrah again!

It may also mean that there are vector spaces with no basis. Boo! to that.

And in general, the proofs of many well-known theorems may need to be re-written, and some of these theorems may actually be unprovable under the constructivist agenda. This would be a Bad Thing.

It is often frequently said that constructivists also deny the Law of Excluded Middle - a statement that is not true is of necessity false, no deals. I am not sure they all did, but then, I am asking for your thoughts on this

It seems that constructivists are a bit of a fringe. But before we dismiss it as a lunatic fringe, note that some extremely big hitters have, and do, adhere to it. I name "Fixed point" Brouwer and "delta" Kroenecker, for two.

I am also aware that, even now, there is a form of mathematical etiquette that requires any proof implicitly using AC to say so. This seems to imply a certain ambivalence to the axiom.

What think you all?