Self-Boot-Strapping: Petard Hoisting by Another Name
It's not rocket science to realize that you have to use abstract objects to analyze and decide what they are. What do people think they're doing in the very process of assessing them?
That already assumes what they are: universally authoritative as components of evaluating and deciding questions about universals. Such as universals claiming what abstract objects are or must be, or assumptions about how to decide what abstract objects are or must be. Abstract objects are always already required, assumed, and used in the process of carrying out such an analysis in the first place.
Any discussion of abstract objects already assumes them and the role they play in thinking, among other equally basic assumptions and implications.
In terms of the ultimate criteria of general analysis, they are God Objects, and persons apply them personally to their lives in the structure of their own conceptualizing just about every waking moment of their lives. So the system of reason is necessarily theistic in the very definition of what it is.
But the approach of this book is a major fail and just blank about this stuff.
You have to use abstract objects to think at all, and specifically in this context to think about those abstract objects themselves. How do you decide what they are, or if that's not a problem, whether they're real or not? By using abstract objects as ontology-determining components in order to even construct claims and analyze them. In fact, however they're even defined, definitions require abstract objects, as do attempted reductions of abstract objects to useful fiction status or consensus or whatever.
It's a case of "by what authority do you say these things?" when saying anything already assumes a system of abstract objects. Questions about anything including questioning itself, assumes a system of inquiry made up of abstract objects.
A detailed look at all the definitions of abstract objects clearly shows that people are using them in order to think about them at all in the first place, but never mentioning that fact as they proclaim one abstract universal after another. And what that implies is a necessary theory of abstract objects, that cannot even be thought without assuming them in the process.
So to my mind, it's just a matter of collecting an inventory of them, maybe starting with the 1100 or so basic concepts of the latest Roget thesauruses, especially the mind/ideas section which contains about 250 or so of the most basic ones. And then see which ones can survive attempts to do without them.