Synge waits until well into his second volume (J. L. Synge, Relativity: The General Theory (1960), p. 104) to really let rip against Realism; and since he was himself very much a mathematical physicist, rather than an empirical experimenter, his testimony must be respected as coming from within the tent. He distinguishes “Natural Observations” (NO) from “Mathematical Observations”, and opines:
Between NO and MO there is a sharp and decisive break. Only the simplest MO (counting) can be regarded as being NO also … Generally MO involve infinity (irrational numbers, differential calculus, and so on) and so lie outside physics and outside nature.
This is exactly the position of Kronecker (“Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk”).
He then delivers himself of a curious passage, which Irrationalists would seize on with glee (fortunately none of them are reading this):
The peculiar fascination of theoretical physics lies in the art of forcing meaningful truth out of the meaningless equation NO = MO, which is a symbolic form of the assertion that natural phenomena obey exact mathematical laws. The true inequality NO =/= MO should not be spoken above a whisper, because it is extremely dangerous. If believed, it would sever mathematics from physics, and reduce both to sterility through lack of mutual fecundation. It is whispered here only as an apology to those readers who expect to see the mathematics of relativity [which he presents in great detail] tied to the physics of relativity by a strong chain of clear thought. It cannot be done.
These ring like the Night Thoughts of a relativistic physicist, on the eve of taking his own life.
Despite his conspiritorial tone in that passage, Synge was by no means alone in his reservations. Here is another anti-Realist view from the world of physics:
A. D’Abro, The Rise of the New Physics (1939), vol. II, p. 728:
A hyperspace is obviously a mathematical fiction; and waves that can be represented only in a fictitious space must themselves be unreal.
Now here indeed is a statement that has been overtaken by events. In the view of string theory, this hyperspace, far from a mathematical fiction, is a physical fact, the one we live in; indeed, we must beware lest those compactified but very real extra dimensions someday unfurl in our faces. -- The point here being, not to make any point whatsoever about cosmic geometry, let alone to proclaim the truth of string theory: but simply to counsel against that “obviously”, when dismissing the Realist picture.