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How Materialism Proves Unbounded Scientific Ignorance

There is an infinite number of things that are true that we cannot prove scientifically and never will

Science is based on a glut of laws from physics, chemistry, mathematics, and other areas. The assumption of scientific materialism, as I understand it, is that science has explained or will explain everything. The final conclusion of scientific materialism, also known as scientism, is nicely captured in a question chemist Peter Atkins asked philosopher William Lane Craig in a debate: “Do you deny that science can account for everything?”

Scientism’s assumption that science can establish everything is self-refuting. Careful analysis shows that there is an infinite number of things that are true that we cannot prove scientifically and never will.

Stephen Hawking saw the tip of the iceberg of this truth when he said, “Up to now, most people have implicitly assumed that there is an ultimate theory, that we will eventually discover.” This Theory of Everything, as it is often called, would link together all physical aspects of the universe under one vast umbrella theory. Some are still searching. But Hawking abandoned the search. In defending his switch of position, Hawking invoked Kurt Gödel (1906–1978):

Some people will be very disappointed if there is not an ultimate theory, that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind. I’m now glad that our search for understanding will never come to an end, and that we will always have the challenge of new discovery. Without it, we would stagnate. Gödel’s theorem ensured there would always be a job for mathematicians.

Stephen Hawking, “Gödel and the End of Physics” at Texas A&M University (public lecture, March 8, 2002)

Just as it is hard to talk about evolution without mentioning Charles Darwin, it’s hard to talk about metamathematics without mentioning Kurt Gödel. Hawking’s invocation of Gödel’s (first) incompleteness theorem has even deeper implications. Hawking is right in saying there are possibly many more discoveries in physics remaining. But Gödel also showed that there are unprovable truths and, more dramatically, there is an infinite number of unprovable truths forever beyond the reach of scientific proof.

Here is a result of Gödel’s incompleteness theorem supporting this. Given a finite set of assumptions (axioms), there will be revealed truths (theorems) that are unprovable. It sounds like an oxymoron, but these truths in Gödel’s proof are provably unprovable. The axioms can be thought of as inflating to a Gödelian bubble filled with provable proofs. There are also unprovable truths revealed within the bubble whose proof may lie outside the bubble.

One solution is to take an unprovable truth in Gödel’s bubble and add to the list of axioms as an assumed truth. We can’t prove it, so let’s assume it. If the bubble of truth was spawned from a billion axioms, the augmented axiom list now has a billion and one entries. This new list will birth an even bigger bubble. The bigger bubble subsumes the smaller. The larger bubble contains another truth that cannot be proved within the larger bubble. So, as before, we add this new unprovable truth to the set of axioms and now have a billion and two axioms. The bubble gets bigger. Continuing, we can add an unbounded number of axioms to the original list. The bubble will continue to expand with each new assumed axiom but will always contain new unprovable truths. There will always be a new unprovable truth to be added to the collection of axiomatic assumptions.

The first bubble contains unprovable truths in our universe. In talking about Gödel’s theorem in physics, Hawking was referring only to the first Gödelian bubble generated by the laws of our universe. Proving something is true but unprovable is difficult. But there are many candidates. In mathematics, Goldbach’s conjecture that every even number is the sum of two primes is almost certainly true but escapes rigorous proof. Evidence grows there is no proof of Goldbach’s conjecture as mathematician after mathematician has historically tried to crack the problem. This is similar to other open math problems including the Collatz Conjecture, the Riemann Hypothesis, and the Twin Prime Problem.

But what about science? String theory provides a TOE. But there is to date no empirical proof that the theory is either right or wrong. I was once told by a physicist that string theory is so mathematically beautiful, if it isn’t true, it should be. If string theory is true, it may be unprovable. The same thing goes for parallel universes.

A common claim is that the existence of God cannot be proven. Yet, as with Goldbach’s Conjecture, the evidence for the existence of a Creator is overwhelming. Many theists believe God is a truth outside our universe’s bubble of provability.

Scientism’s claim is built on a foundation of physical laws expressed mathematically. The list of these axiomatic laws can be written on a single sheet of paper. They include Schrodinger’s equation for quantum mechanics, Maxwell’s equation for electromagnetics, Einstein’s general relativity, Newtonian mechanics, and even string theory. These equations are built on the foundations of mathematics like calculus, differential equations, and probability. If you can think of some axioms not considered here, add them to the list. The final axiom tally will be finite and that is all that is required to build the first bubble of truths in our universe.

Some may question whether Gödel’s theorem is applicable here. It is. All measurements in our universe are of finite precision and therefore can be massaged into the umbra of the natural numbers considered by Gödel in his analysis of formal systems. For this reason, Hawking’s application of Gödel’s theorem to the search for the TOE is spot on, as is our claim of the existence of an unbounded number of unprovable truths.

Scientific materialists believe that all truth can be proved by science. Famously, this claim of scientism itself cannot be proved by science. The claim is philosophical rather than scientific. The scientific materialists must add this nonmaterialist belief to their list of materialist truths in order to have a consistent discussion. Even then, ironically, the assumptions of scientism augmented by this additional axiom leads to the inescapable conclusion that there is an unbounded number of unprovable truths.

So when an adherent of scientism prophesies that this or that scientific breakthrough will occur in the future, there is a chance they could be flat out wrong.

Acknowledgement: I appreciate my conversation with my good friend Daniel Andrés Díaz Pachón on this topic.

You may also wish to read: Gregory Chaitin’s “almost” meeting with Kurt Gödel. This hard-to-find anecdote gives some sense of the encouraging but eccentric math genius.

Robert J. Marks II

Director, Senior Fellow, Walter Bradley Center for Natural & Artificial Intelligence
Besides serving as Director, Robert J. Marks Ph.D. hosts the Mind Matters podcast for the Bradley Center. He is Distinguished Professor of Electrical and Computer Engineering at Baylor University. Marks is a Fellow of both the Institute of Electrical and Electronic Engineers (IEEE) and the Optical Society of America. He was Charter President of the IEEE Neural Networks Council and served as Editor-in-Chief of the IEEE Transactions on Neural Networks. He is coauthor of the books Neural Smithing: Supervised Learning in Feedforward Artificial Neural Networks (MIT Press) and Introduction to Evolutionary Informatics (World Scientific). For more information, see Dr. Marks’s expanded bio.

How Materialism Proves Unbounded Scientific Ignorance