Being a constructive automatically implies empirical testability of any assumption and prediction of a given theory (see for more detail arXiv:1509.06801). Meanwhile, in at least some physical theories, and especially in quantum mechanics and string theory, one can find examples of non-constructive statements and proofs (e.g., the existence of universal wave function or a particular vacuum of string theory). So, the question is, should we abandon development a physical theory if chances are that this theory will never be testable?
http://arxiv.org/abs/1509.06801
But if we pursue an untestable physical theory, then what will be gains of it? How will such a theory be differ from a pure mathematical theory whose notion of truth is not empirical but abstract?
No, George, I’m not talking about our technical inability to test some theoretical predictions. We can, of course, work at a theoretical model even if we would be able to empirically verify it only in the (distant) future.
What I mean is much more 'dramatic': It is mathematical (not ‘technical’!) non-constructivism of certain physical theories. For example, in some interpretations of quantum mechanics, the existence of a wave function (or a quantum state vector) of the entire universe is assumed. However, to find such a wavefunction (that is, calculate it) and in this way to prove its existence would demand an infinite amount of steps (meaning an infinite time).
Likewise, even if we were to find compelling evidence that some vacuum of string theory describes our universe, we might never be able to find that vacuum explicitly and therefore to prove string theory.
Thank you, dear Mohamed, for bringing those interesting references to my attention.
Dear George,
Mathematically speaking, a theory can be both correct and non-constructive. An example of this will be classical analysis: If a real function F(n) defined over the set of natural numbers n is bounded from bellow F(n)>0, then, conforming to classical analysis, the global minimum F(n) does exist. However, the fact that there exists a global minimum of this function (and hence, in principle, it is possible to find it) does not mean that one can actually find it!
Obviously, the problem is that even if an actual non–zero global minimum of F(n) has been just reached, there is no way (i.e., an effective algorithm) to recognize it as such since defining the property of being the global minimum min F< F(n) involves quantification over the entire set of the natural numbers and thus undecidability.
But if non-constructivism may live well in mathematics, then what’s about physics?
Dear Mohamed,
I think you’ve nailed the very problem…
Unfortunately (or at least I feel it this way) many parts of modern physics look more and more like some branch of mathematics. And the matter is not a high degree of arduous mathematical machinery those parts use but rather their complete non-constructivism...
Dear George,
One thing is for sure: The foundations of mathematics must play a crucial role in resolution of many ‘stubborn’ physical problems.
For example, in 1991, a Chinese student, Don Wang, published a beautiful paper, in which he provided a convergent power series solution of the N-body problem – i.e., the problem of the motion of a group of celestial objects that interact with each other gravitationally. But did that paper mean the end of probably the oldest problem of classical physics unsuccessfully attacked by the greatest mathematicians of the last three centuries? Does this work imply that now we know everything about gravitating bodies, about the motion of planets and stars?
Unfortunately, the answer is no, we do not. The trouble with this ‘final solution’ is that it is non-constructive. This solution is expanded in power series and – though principally convergent on the whole real axis – has a very slow convergence. One would have to sum up millions of terms to determine the motion of the particles for insignificantly short intervals of time. The round-off errors make these series unusable in numerical work. Thus, from the physical point of view, such a solution adds nothing to what was previously known about the N-body problem.
What’s more, there are other examples in physics that have the form of the non-constructive power series. So, what to do with them? Eliminate power series solutions from physics as totally useless solutions?
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