Ordinarily in mathematics, when one introduces a new concept one defines it. For example, if this were a book on “blobs” I would begin with a definition of this new predicate: is a blob in case is a topological space such that no uncountable subset is Hausdorff. Then we would be all set to study blobs. Fortunately, this is not a book about blobs, and I want to do something different. I want to begin by introducing a new predicate “standard” to ordinary mathematics without defining it.
The reason for not defining “standard” is that it plays a syntactical, rather than a semantic, role in the theory. It is similar to the use of “fixed” in informal mathematical discourse. One does not define this notion, nor consider the set of all fixed natural numbers. The statement “there is a natural number bigger than any fixed natural number” does not appear paradoxical. The predicate “standard” will be used in much the same way, so that we shall assert “there is a natural number bigger than any standard natural number.” But the predicate “standard”— unlike “fixed”—will be part of the formal language of our theory, and this will allow us to take the further step of saying, “call such a natural number, one that is bigger than any standard natural number, unlimited.”
We shall introduce axioms for handling this new predicate “standard” in a consistent way. In doing so, we do not enlarge the world of mathematical objects in any way, we merely construct a richer language to discuss the same objects as before. In this way we construct a theory extending ordinary mathematics, called Internal Set Theory that axiomatizes a portion of Abraham Robinson’s nonstandard analysis. In this construction, nothing in ordinary mathematics is changed.
Nelson appears to claim that his approach allows one to avoid reification of abstract quantities. Anti-reification stance is deeply personal for him and has spiritual roots, as explained in his paper Mathematics and Faith:
I must relate how I lost my faith in Pythagorean numbers. One morning at the 1976 Summer Meeting of the American Mathematical Society in Toronto, I woke early. As I lay meditating about numbers, I felt the momentary overwhelming presence of one who convicted me of arrogance for my belief in the real existence of an infinite world of numbers, leaving me like an infant in a crib reduced to counting on my fingers. Now I live in a world in which there are no numbers save those that human beings on occasion construct.
During my first stay in Rome I used to play chess with Ernesto Buonaiuti. In his writings and in his life, Buonaiuti with passionate eloquence opposed the reification of human abstractions. I close by quoting one sentence from his Pellegrino di Roma “For [St. Paul] abstract truth, absolute laws, do not exist, because all of our thinking is subordinated to the construction of this holy temple of the Spirit, whose manifestations are not abstract ideas, but fruits of goodness, of peace, of charity and forgiveness.”
This is really remarkable because the the Idealisation Axiom of Nelson's Internal Set Theory is a mathematical reformulation of the process of reification:
Let be a classical relation.
In order to be able to find an with for all standard ,
a necessary and sufficient condition is
for each standard finite part , it is possible to find an such that holds for all .
Nelson hides reified objects in the non-standard part of his universe!