I referred on the podcast to Goodman's 1947 article "Steps Toward a Constructive Nominalism." You can look at it here. The philosophical content is in the first couple of chapters; in fact, I'll just give you the first half of the first chapter here:
We do not believe in abstract entities. No one supposes that abstract entities -- classes, relations, properties, etc. -- exist in space-time; but we mean more than this. We renounce them altogether. We shall not forego all use of predicates and other words that are often taken to name abstract objects. We may still write "x is a dog", or "x is between y and z"; for here "is a dog" and "is between . . . and" can be construed as syncategorematic: significant in context but naming nothing. But we cannot use variables that call for abstract objects as values.1 In "x is a dog", only concrete objects are appropriate values of the variable. In contrast, the variable in "x is a zoological species" calls for abstract objects as values (unless of course, we can somehow identify the various zoological species with certain concrete objects). Any system that countenances abstract entities we deem unsatisfactory as a final philosophy.
Now, I support this on general principles; I don't like ontologies with weird Platonic forms or numbers as real or possible worlds (another target of Goodman). What's less clear to me as someone who hasn't read much philosophy of mathematics is the rationale:
What seems to be the most natural principle for abstracting classes or properties leads to paradoxes. Escape from the paradoxes can apparently be effected only by recourse to alternative rules whose artificiality and arbitrariness arouse suspicion that we are lost in a world of make-believe.
If anyone reading this who has read Quine's Mathematical Logicwants to fill us in, post a comment!