What do the @physicists think of this? There are fundamental problems with true infinities in the real world, but real numbers can be infinite. So maybe they do not really exist?
Over the past year, the Swiss physicist Nicolas Gisin has published four papers that attempt to dispel the fog surrounding time in physics. As Gisin sees it, the problem all along has been mathematical. Gisin argues that time in general and the time we call the present are easily expressed in a century-old mathematical language called intuitionist mathematics, which rejects the existence of numbers with infinitely many digits. When intuitionist math is used to describe the evolution of physical systems, it makes clear, according to Gisin, that ātime really passes and new information is created.ā Moreover, with this formalism, the strict determinism implied by Einsteinās equations gives way to a quantum-like unpredictability. If numbers are finite and limited in their precision, then nature itself is inherently imprecise, and thus unpredictable.
Physicists are still digesting Gisinās work ā itās not often that someone tries to reformulate the laws of physics in a new mathematical language ā but many of those who have engaged with his arguments think they could potentially bridge the conceptual divide between the determinism of general relativity and the inherent randomness at the quantum scale.
CortĆŖs called Gisinās approach āextremely interestingā and āshocking and provocativeā in its implications. āItās really a very interesting formalism that is addressing this problem of finite precision in nature,ā she said.
As a mathematical fictionalist, I agree that real numbers do not exist. But then integers donāt exist either.
I welcome the attempts to use intuitionist mathematics in physics. But I am doubtful that it will work out.
Intuitionism comes from looking at mathematics as derived from logic. And the intuitionist concern with ācomplete infinitiesā arises from the apparent paradoxes (such as the Russell paradox) that derive from naive set theory. Iām a mathematical traditionalist, so I accept the traditions of set theory, as carefully managed with axiom systems such as ZFC.
As I see it, mathematics is not really derived from logic. It uses logic. But mathematics really comes from the needs of physics for modeling reality. And much of what the intuitionists donāt like about traditional mathematics is there in order to support the needs of science. Thatās why I doubt that Gisinās program will work out.
Seems to me itās a bit absurd to say that if physical quantities take on real number values, that means thereās infinite information. Maybe it requires infinite information to represent it with infinite precision, but attributing that information to the physical quantity itself is to confuse the representation for reality.
Some of the people who do know what it means donāt need it either.
Whenever a YEC asks me to state my assumptions I like to answer āZFCā. Then I have to explain it, but it it denies the claim that science is assuming any conclusions.
This is part of the distinction between theory and practice.
Our theories presume infinite information. But, in practice, we only have access to measurements that we make, so that is finite information.
If you try to work with a mathematics of actual measurements, it gets very messy. So our theories idealize, in order to have more workable mathematics.
Why is that absurd? That is in fact the case. It requires infinite digits to represent a real number with infinite precision. This is a fact made us of in compression algorithms: Arithmetic coding - Wikipedia.
As I said in the next sentence of my post: sure, it takes infinite information to fully represent a real number. But attributing that information to the physical quantity itself (and even further, assuming this entails physical properties associated with physical representations of that information, such as having enough energy density to form a black hole) is to confuse the representation for the reality.
