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Finitism and Physics – “A brief precis: Gravitational collapse limits the amount of energy present in any space-time region. This in turn limits the precision of any measurement or experimental process that takes place in the region. This implies that the class of models of physics which are discrete and finite (finitistic) cannot be excluded experimentally by any realistic process. Note any digital computer simulation of physical phenomena is a finitistic model. We conclude that physics (Nature) requires neither infinity nor the continuum. For instance, neither space-time nor the Hilbert space structure of quantum mechanics need be absolutely continuous. This has consequences for the finitist perspective in mathematics…” (previously)

re: finitism: “We experience the physical world directly, so the highest confidence belief we have is in its reality. Mathematics is an invention of our brains, and cannot help but be inspired by the objects we find in the physical world. Our idealizations (such as ‘infinity’) may or may not be well-founded. In fact, mathematics with infinity included may be very sick, as evidenced by Godel’s results, or paradoxes in set theory. There is no reason that infinity is needed (as far as we know) to do physics. It is entirely possible that there are only a (large but) finite number of degrees of freedom in the physical universe.”[1,2,3]

more here:

…the primacy of physical reality over mathematics (usually the opposite assumption is made!) — the parts of mathematics which are simply models or abstractions of “real” physical things are most likely to be free of contradiction or misleading intuition. Aspects of mathematics which have no physical analog (e.g., infinite sets) are prone to problems in formalization or mechanization. Physics (models which can to be compared to experimental observation; actual “effective procedures”) does not ever require infinity, although it may be of some conceptual convenience. Hence one suspects, along the lines above, that mathematics without something like the “axiom of infinity” might be well-defined. Is there some sort of finiteness restriction (e.g., upper bound on Godel number) that evades Godel’s theorem? If one only asks arithmetical questions about numbers below some upper bound, can’t one avoid undecidability?

@johncarlosbaez: “you *can* define truth for sentences with at most n quantifiers.”

I discuss fundamental limits placed on information and information processing by gravity. Such limits arise because both information and its processing require energy, while gravitational collapse (formation of a horizon or black hole) restricts the amount of energy allowed in a finite region. Specifically, I use a criterion for gravitational collapse called the hoop conjecture. Once the hoop conjecture is assumed a number of results can be obtained directly: the existence of a fundamental uncertainty in spatial distance of order the Planck length, bounds on information (entropy) in a finite region, and a bound on the rate of information processing in a finite region. In the final section I discuss some cosmological issues related to the total amount of information in the universe, and note that almost all detailed aspects of the late universe are determined by the randomness of quantum outcomes.

  • Minimum length and quantum gravity – “An implication of the result is that there may only be a finite number of degrees of freedom per unit volume in our universe – no true continuum of space or time. This means that there is only a finite amount of information or entropy in our universe (or at least in any finite patch of it).”
  • Feynman and Everett – “A couple of years ago I gave a talk at the Institute for Quantum Information at Caltech about the origin of probability — i.e., the Born rule — in many worlds (“no collapse”) quantum mechanics. It is often claimed that the Born rule is a consequence of many worlds — that it can be derived from, and is a prediction of, the no collapse assumption. However, this is only true in a particular (questionable) limit of infinite numbers of degrees of freedom — it is problematic when only a finite number of degrees of freedom are considered.”
  • Horizons of truth – “[Gregory Chaitin:] I think it’s reasonable to demand that set theory has to apply to our universe. In my opinion it’s a fantasy to talk about infinities or Cantorian cardinals that are larger than what you have in your physical universe. And what’s our universe actually like?”

a finite universe?

discrete but infinite universe (ℵ0)?

universe with continuity and real numbers (ℵ1)?

universe with higher-order cardinals (≥ ℵ2)?

Does it really make sense to postulate higher-order infinities than you have in your physical universe? Does it make sense to believe in real numbers if our world is actually discrete? Does it make sense to believe in the set {0, 1, 2, …} of all natural numbers if our world is really finite?

CC: Of course, we may never know if our universe is finite or not. And we may never know if at the bottom level the physical universe is discrete or continuous…

GC: Amazingly enough, Cris, there is some evidence that the world may be discrete, and even, in a way, two-dimensional. There’s something called the holographic principle, and something else called the Bekenstein bound. These ideas come from trying to understand black holes using thermodynamics. The tentative conclusion is that any physical system only contains a finite number of bits of information, which in fact grows as the surface area of the physical system, not as the volume of the system as you might expect, whence the term “holographic.”

also btw…
How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer. – “For 50 years, mathematicians have believed that the total number of real numbers is unknowable. A new proof suggests otherwise.”