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The Trouble with Physics and mathematics

Paul Budnik

Two recent books, The Trouble with Physics by the physicist Lee Smolin and Not Even Wrong by the mathematician Peter Woit, are powerful indictments of foundations research in physics. For a quarter century research in particle physics has been dominated by a single approach, string theory, that has yet to make any testable experimental predictions. A theory that fails to make predictions is `not even wrong'. It is irrelevant to science. I hasten to add that both authors do not completely rule out the possibility that eventually this theory will lead to such predictions, but they are clearly pessimistic about this.

Something similar, albeit less dramatic, is happening in the foundations of mathematics. Solomon Feferman, the editor of Gödel's collected works, made the following comment in "Does mathematics need new axioms?".

I am convinced that the Continuum Hypothesis is an inherently vague problem that no new axiom will settle in a convincingly definite way. Moreover, I think the Platonistic philosophy of mathematics that is currently claimed to justify set theory and mathematics more generally is thoroughly unsatisfactory and that some other philosophy grounded in inter-subjective human conceptions will have to be sought to explain the apparent objectivity of mathematics.
This calls into question the entire theory of infinite cardinals which has been the focus of foundations research in mathematics for decades. Over thirty years ago, I told Professor Feferman that I did not think the Continuum Hypothesis was objectively true or false. I went on to explain a theory of mathematical truth that determined which mathematics statements were objectively determined. The idea is that only statements that are determined by a sequence of events that can be listed by a computer program (a recursively enumerable set) are objectively determined. This encompasses most of existing mathematics through generalizations of the halting problem. One can ask does a computer program have an infinite number of outputs. One can then ask does it have infinite number of outputs that can be interpreted as programs such that an infinite subset of these outputs are programs that themselves have an infinite number of outputs. Complex generalizations of this form can capture most of mathematics including questions that require quantification over the reals. At the time Feferman was not much interested in philosophical speculation by someone who had not created new mathematics although he was quite helpful in explaining my technical results were not original. You can learn more about my approach to mathematics here.

Fundamental research in the foundations of mathematics and physics may have been pursuing mostly dead ends for decades. I offer two explanations for this situation in mathematics and physics. First is technical. The old idea of infinity as a limit never reached makes more sense than the hierarchy of infinities in mathematics and the continuous space time structures on which the fundamental theories of physics are based. This approach to mathematics and physics is developed in my book: What is and what will be.

The other explanation is the narrow intellectual focus of academic mathematics and physics. My book argues that intuition needs to be coequal with intellect if the sciences are to realize their creative potential. Intellect is all too prone to try to fit problems to intellectually tractable models. Intuition, with its powerful pattern recognition capabilities, is capable of dealing with problems beyond intellects grasp albeit far less reliably. In both mathematics and physics, all to many academics have become overly enamored of their limited intellectual models. String theory is a sad example of taking mathematical models more seriously than the physical reality that those models are intended to describe. The theory of infinite cardinals is the symptom of a similar malady. The foundations of mathematics is perhaps the only major scientific discipline in which computer simulation is not widely used. Yet it is precisely the combinatorial complexity of possible computer programs that is at the core of mathematics.


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