New paper on arXiv today.
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 -- see excerpt below.
Fundamental Limit on Angular Measurements and Rotations from Quantum Mechanics and General Relativity
arXiv:2108.11990
Xavier Calmet, Stephen D.H. Hsu
We show that the precision of an angular measurement or rotation (e.g., on the orientation of a qubit or spin state) is limited by fundamental constraints arising from quantum mechanics and general relativity (gravitational collapse). The limiting precision is 1/r in Planck units, where r is the physical extent of the (possibly macroscopic) device used to manipulate the spin state. This fundamental limitation means that spin states S1 and S2 cannot be experimentally distinguished from each other if they differ by a sufficiently small rotation. Experiments cannot exclude the possibility that the space of quantum state vectors (i.e., Hilbert space) is fundamentally discrete, rather than continuous. We discuss the implications for finitism: does physics require infinity or a continuum?
From the conclusions:
IV. FINITISM: DOES PHYSICS REQUIRE A CONTINUUM?
Our intuitions about the existence and nature of a continuum arise from perceptions of space and time [21]. But the existence of a fundamental Planck length suggests that spacetime may not be a continuum. In that case, our intuitions originate from something (an idealization) that is not actually realized in Nature.
Quantum mechanics is formulated using continuous structures such as Hilbert space and a smoothly varying wavefunction, incorporating complex numbers of arbitrary precision. However beautiful these structures may be, it is possible that they are idealizations that do not exist in the physical world.
The introduction of gravity limits the precision necessary to formulate a model of fundamental quantum physics. Indeed, any potential structure smaller than the Planck length or the minimal angle considered here cannot be observed by any device subject to quantum mechanics, general relativity, and causality. Our results suggest that quantum mechanics combined with gravity does not require a continuum, nor any concept of infinity.
It may come as a surprise to physicists that infinity and the continuum are even today the subject of debate in mathematics and the philosophy of mathematics. Some mathematicians, called finitists, accept only finite mathematical objects and procedures [25]. The fact that physics does not require infinity or a continuum is an important empirical input to the debate over finitism. For example, a finitist might assert (contra the Platonist perspective adopted by many mathematicians) that human brains built from finite arrangements of atoms, and operating under natural laws (physics) that are finitistic, are unlikely to have trustworthy intuitions concerning abstract concepts such as the continuum. These facts about the brain and about physical laws stand in contrast to intuitive assumptions adopted by many mathematicians. For example, Weyl (Das Kontinuum [21, 22]) argues that our intuitions concerning the continuum originate in the mind’s perception of the continuity of space-time.
There was a concerted effort beginning in the 20th century to place infinity and the continuum on a rigorous foundation using logic and set theory. However, these efforts have not been successful. For example, the standard axioms of Zermelo-Fraenkel (ZFC) set theory applied to infinite sets lead to many counterintuitive results such as the Banach-Tarski Paradox: given any two solid objects, the cut pieces of either one can be reassembled into the other [23]. When examined closely all of the axioms of ZFC (e.g., Axiom of Choice) are intuitively obvious if applied to finite sets, with the exception of the Axiom of Infinity, which admits infinite sets. (Infinite sets are inexhaustible, so application of the Axiom of Choice leads to pathological results.) The Continuum Hypothesis, which proposes that there is no cardinality strictly between that of the integers and reals, has been shown to be independent (neither provable nor disprovable) in ZFC [24]. Finitists assert that this illustrates how little control rigorous mathematics has on even the most fundamental properties of the continuum.
David Deutsch [26]: The reason why we find it possible to construct, say, electronic calculators, and indeed why we can perform mental arithmetic, cannot be found in mathematics or logic. The reason is that the laws of physics “happen to” permit the existence of physical models for the operations of arithmetic such as addition, subtraction and multiplication.
This suggests the primacy of physical reality over mathematics, whereas usually the opposite assumption is made. From this perspective, 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, the continuum) are prone to problems in formalization or mechanization. Physics – i.e., models which can be compared to experimental observation, actual “effective procedures” – does not ever require infinity, although it may be of some conceptual convenience. Hence it seems possible, and the finitists believe, that the Axiom of Infinity and its equivalents do not provide a sound foundation for mathematics.See also
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.Paul Cohen: I will ascribe to Skolem a view, not explicitly stated by him, that there is a reality to mathematics, but axioms cannot describe it. Indeed one goes further and says that there is no reason to think that any axiom system can adequately describe it.This "it" (mathematics) that Cohen describes may be the set of idealizations constructed by our brains extrapolating from physical reality. But there is no guarantee that these idealizations have a strong kind of internal consistency and indeed they cannot be adequately described by any axiom system.
There was a concerted effort beginning in the 20th century to place infinity and the continuum on a rigorous foundation using logic and set theory. However, these efforts have not been successful. ...This refers to Hilbert's Program:
In the early 1920s, the German mathematician David Hilbert (1862–1943) put forward a new proposal for the foundation of classical mathematics which has come to be known as Hilbert’s Program. It calls for a formalization of all of mathematics in axiomatic form, together with a proof that this axiomatization of mathematics is consistent. The consistency proof itself was to be carried out using only what Hilbert called “finitary” methods. The special epistemological character of finitary reasoning then yields the required justification of classical mathematics. Although Hilbert proposed his program in this form only in 1921, various facets of it are rooted in foundational work of his going back until around 1900, when he first pointed out the necessity of giving a direct consistency proof of analysis. ...which Godel showed is not possible to carry out. Note that one of Hilbert's main motivations was the continuum (e.g., construction of the Reals in analysis). What has subsequently been adopted as the rigorous basis for analysis does not satisfy Hilbert's desire for axiomatic, finitary methods.
The remaining sentences in the paragraph are meant to elucidate aspects of the modern treatment that its critics find unappealing. Of course, judgements of this type are philosophical in nature.
... For example, the standard axioms of Zermelo-Fraenkel (ZFC) set theory applied to infinite sets lead to many counterintuitive results such as the Banach-Tarski Paradox: given any two solid objects, the cut pieces of either one can be reassembled into the other [23]. When examined closely all of the axioms of ZFC (e.g., Axiom of Choice) are intuitively obvious if applied to finite sets, with the exception of the Axiom of Infinity, which admits infinite sets. (Infinite sets are inexhaustible, so application of the Axiom of Choice leads to pathological results.) The Continuum Hypothesis, which proposes that there is no cardinality strictly between that of the integers and reals, has been shown to be independent (neither provable nor disprovable) in ZFC [24]. Finitists assert that this illustrates how little control rigorous mathematics has on even the most fundamental properties of the continuum.
Skolem and pessimism about proof in mathematics
Abstract: Attitudes towards formalization and proof have gone through large swings during the last 150 years. We sketch the development from Frege’s first formalization, to the debates over intuitionism and other schools, through Hilbert’s program and the decisive blow of the Go¨del Incompleteness Theorem. A critical role is played by the Skolem–Lowenheim Theorem, which showed that no first-order axiom system can characterize a unique infinite model. Skolem himself regarded this as a body blow to the belief that mathematics can be reliably founded only on formal axiomatic systems. In a remarkably prescient paper, he even sketches the possibility of interesting new models for set theory itself, something later realized by the method of forcing. This is in contrast to Hilbert’s belief that mathematics could resolve all its questions. We discuss the role of new axioms for set theory, questions in set theory itself, and their relevance for number theory. We then look in detail at what the methods of the predicate calculus, i.e. mathematical reasoning, really entail. The conclusion is that there is no reasonable basis for Hilbert’s assumption. The vast majority of questions even in elementary number theory, of reasonable complexity, are beyond the reach of any such reasoning ...
... The startling conclusion that Skolem drew is the famous Skolem Paradox, that any of the usual axiom systems for set theory will have countable models, unless they are contradictory. Since I will not assume that my audience are all trained logicians, I point out that though the set of reals from the countable model is countable seen from outside, there is no function ‘living in the model’ which puts it in one-to-one correspondence with the set of integers of the model. This fact and other considerations led Skolem to this viewpoint:
I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics, that mathematicians would, for the most part, not be very much concerned by it.
The view that I shall present differs somewhat from this, and is in a sense more radical, namely that it is unreasonable to expect that any reasoning of the type we call rigorous mathematics can hope to resolve all but the tiniest fraction of possible mathematical questions.
The theorem of Lowenheim–Skolem was the first truly important discovery about formal systems in general, and it remains probably the most basic. ...
Conclusion: ...Therefore, my conclusion is the following. I believe that the vast majority of statements about the integers are totally and permanently beyond proof in any reasonable system. Here I am using proof in the sense that mathematicians use that word. Can statistical evidence be regarded as proof ? I would like to have an open mind, and say ‘Why not?’. If the first ten billion zeros of the zeta function lie on the line whose real part is 1/2, what conclusion shall we draw? I feel incompetent even to speculate on how future generations will regard numerical evidence of this kind.
In this pessimistic spirit, I may conclude by asking if we are witnessing the end of the era of pure proof, begun so gloriously by the Greeks. I hope that mathematics lives for a very long time, and that we do not reach that dead end for many generations to come.
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