This is a terrible view, that throws all of mathematics, and science out of the window. It is also a form of epistemological theft. The theft arises from the fact that the whole argument would be impossible without using the law of excluded middle, because it's the foundation of all reasoning. Without it would would have neither mathematics, neither computers, neither the Medium platform to publish this horrible idea.
Furthermore, basing something on presence or absence of proof steals the concept of "proof." Proof is based on the principle of excluded middle. Therefore, there is no difference in framing the condition in terms of proof that is stand in for evidence, vs. truth of falsity. If one wishes to use proof, he must accept the principle of excluded middle. Otherwise, he is a hypocrite.
The fallacy of the Stolen Concept was first formulated by Ayn Rand. In "Introduction to Objectivist Epistemology," she writes:
"When modern philosophers declare that axioms are a matter of arbitrary choice, and proceed to choose complex, derivative concepts as the alleged axioms of their alleged reasoning, one can observe that their statements imply and depend on 'existence,' 'consciousness,' 'identity,' which they profess to negate, but which are smuggled into their arguments in the form of unacknowledged, 'stolen' concepts."
All truth is derived from evidence, and also all proof is derived from evidence. The evidence is reality itself, and it is the reason we have invented mathematics in the first place: to help us count and measure real things.
But do I have an argument against the particular example in the article? Sure I do. If you define a mathematical theory and it contains statements that you can't prove, then either you still haven't found the a proof, or the whole mathematical model has a flaw, and you should redefine it.
Set theory is a model, it's not real. Even whole numbers are a model. What is real, then? Actual things that we count in numbers, or group in sets, or measure by comparing. And what about sets infinities, aren't they real? No. Infinities are not real, they are a model of something that can be repeated any arbitrary number of times. What about sets containing themselves, a la Bertrand Russell? They are invalid. See "How We Know" by Harry Binswanger.
In closing, the legitimation of this kind of faux logic by universities is an example of why government should not fund universities. Universities should be fully private, funded by investors seeking profit. But the proposed faux-logic is anti real world, and would not help earn a single cent. Mathematicians who are entertaining these ideas are waisting their time, and have a detrimental effect on all the undergrads that they teach.
