As we continue to explore the mathematical domain of abstract reality and objective truth, we come to first the Godel point (as summarised by Nesher):
where, recall, the domain of facts starts with something like the surreal world of numbers:
and then also, we come to the world of Mathematical Platonism/ Realism.
So, let me continue by promoting a comment I just added to the objectivity of Mathematics thread:
Traditionally, mathematical platonism has referred to a collection of metaphysical accounts of mathematics, where a metaphysical account of mathematics is one that entails theses concerning the existence and fundamental nature of mathematical ontology. In particular, such an account of mathematics is a variety of (mathematical) platonism if and only if it entails some version of the following three Theses:
1] Existence: Some mathematical ontology exists. [–> RHK Webster’s College Dict: ONTOLOGY: “ the branch of metaphysics that studies the nature of existence or being as such.“]
2] Abstractness: Mathematical ontology is abstract.
3] Independence: Mathematical ontology is independent of all rational activities, that is, the activities of all rational beings . . . .
platonists have maintained that the items that are fundamental to mathematical ontology are objects, where an object is, roughly, any item that may fall within the range of the first-order bound variables of an appropriately formalized theory and for which identity conditions can be provided . . . object platonism is the conjunction of three theses: some mathematical objects exist, those mathematical objects are abstract, and those mathematical objects are independent of all rational activities. In the last hundred years or so, object platonisms have been defended by Gottlob Frege [1884, 1893, 1903], Crispin Wright and Bob Hale [Wright 1983], [Hale and Wright 2001], and Neil Tennant [1987, 1997].
Nearly all object platonists recognize that most mathematical objects naturally belong to collections (for example, the real numbers, the sets, the cyclical group of order 20). To borrow terminology from model theory, most mathematical objects are elements of mathematical domains. Consult Model-Theoretic Conceptions of Logical Consequence for details. It is well recognized that the objects in mathematical domains have certain properties and stand in certain relations to one another. These distinctively mathematical properties and relations are also acknowledged by object platonists to be items of mathematical ontology.
More recently, it has become popular to maintain that the items that are fundamental to mathematical ontology are structures rather than objects. Stewart Shapiro [1997, pp. 73-4], a prominent defender of this thesis, offers the following definition of a structure:
I define a system to be a collection of objects with certain relations. … A structure is the abstract form of a system, highlighting the interrelationships among the objects, and ignoring any features of them that do not affect how they relate to other objects in the system.
According to structuralists, mathematics’ subject matter is mathematical structures. Individual mathematical entities (for example, the complex number 1 + 2i) are positions or places in such structures . . .
These ideas of course arise from Mathematical practice, where we find ourselves dealing with abstracta and find ourselves constrained by facts tied to them and by logical relationships.
This sort of thing does not sit easily with a day and age incluned to evolutionary materialism and empiricism, while being haunted by Kant’s ugly gulch. A good example is how often we find ourselves tempted to reduce minded, responsible, rational contemplation to computation. Never mind the ugly gulch between blind GIGO limited mechanical and stochastic processes and the rational, responsible freedom required for mind to be coherent, and consequences of self-referentiality.
Another issue is, what is truth vs what is it that we have warranted as true. So, is there a string of ten or a hundred or a thousand zero’s or 1’s etc in the expansion of pi? That we may never know is different from there is or is not in some abstract sense. But many are inclined to think that until we have constructed a solution that manifests such a string, it is neither there nor not there.
That is, we here see a challenge to the law of the excluded middle, a key logical principle connected to distinct identity. Such then feeds into the view we have seen, where Mathematics is whatever it is Mathematicians as a circle of subjects do and accept.
I tend to see that we may err and have rather bounded rationality, but that does not affect what is or may be beyond the circle of what we know or may ever know. But of course, that cuts across the empiricist spirit of the age.
On the other hand, that such are forced to traffic in mathematical abstracta to practice science may well be a corrective message.
The plumb-line speaking to the crooked yardstick, in short.
So, I find myself drawn to Godel’s option C as was added to the OP; noting that there is a body of specific math facts and linked observable phenomena that beg for structuring through axiomatisation and systematic derivation of theorems and results, but also once a scheme is consistent such systems cannot be complete.
I also draw broad confidence in grand coherence from the way vast domains are locked together to infinite precision by our old friend,
0 = 1 + e^i*pi.
Where also, once a wider world of reality is, all that is must be consistent with all else that is. And any one thing is consistent within its own existence and core character. That is, there can be circles and squares but no circle squares.>>
I think we can begin to see a coherent picture. Of course, some will disagree. END