Abstract
In the paper I discuss the importance of relativistic hypercomputation for the philosophy of mathematics, in particular for our understanding of mathematical knowledge. I also discuss the problem of the explanatory role of mathematics in physics and argue that relativistic computation fits very well into the so-called programming account. Relativistic computation reveals an interesting interplay between the empirical realm and the realm of very abstract mathematical principles that even exceed standard mathematics and suggests, that such principles might play an explanatory role. I also argue that relativistic computation does not have some of the weaknesses of other hypercomputational models, thus it is particularly attractive for the philosophy of mathematics.
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Notes
- 1.
- 2.
The best-known and most discussed example is the four-color theorem, proved in 1976 [7, 8]. A more contemporary example is Kepler’s conjecture concerning optimal packing of spheres in space, which was proved by Hales [39, 40]. There is an intense discussion concerning the status of computer proofs, in particular concerning the status of “empirical input”, which is present as we have to rely on strong and sophisticated physical theories (like quantum mechanics) in order to believe that our computer performs what it is supposed to so that it is reliable. This shows that the problem of the “empirical ingredient” in mathematics cannot be simply dismissed.
- 3.
In some cases, quantum computing provides a speed-up effect, the most spectacular example being Shor’s quantum factoring algorithm [88, 89]. So, if quantum computers existed, they could perhaps speedup at least some of the proofs. The importance of quantum computation for the philosophy of mathematics is discussed in Wójtowicz [106, 108]. For an introduction to quantum computing, see for example Nielsen and Chuang [70]; a popular survey on quantum algorithms is Montanaro [68]. An important paper also discussing the philosophical issues is Deutsch et al. [28].
- 4.
Cf. e.g. the special issues of Minds and Machines (12, 2002 13, 2003), Applied Mathematics and Computation (178, 2006), Theoretical Computer Science (317, 2004), Parallel Processing Letters (3, 2012). Syropoulos [97] is a survey of the hypercomputational models; a recent survey of analogue models of computation is given in Bournez and Pouly [14]; a general survey of physical models is Piccinini [75]; for a general discussion on the physical Church–Turing thesis see e.g. Cotogno [21] or Piccinini [74].
- 5.
- 6.
A computer performing 1010000 computational steps needs energy, it might break down, the black hole might evaporate etc. Another difficulty is the communication problem addressed in Németi and Dávid [71]. Wüthrich [109] discusses it in the context of quantum mechanics, considering viable solutions exploiting quantum entanglement. As getting into a black hole is not particularly attractive, in Andréka et al. [6] a scenario is discussed that is more optimistic for the observer and is based on the wormhole hypothesis. It is also interesting to observe that the possibility of performing a hypercomputational task (like checking the consistency of ZFC) within Special Relativity is equivalent to the existence of superluminal signals [73].
- 7.
In general, this is a subset of Δ2: equality holds under the assumption that there is no fixed upper bound on the number of signals sent on a finite path. Theoretical results of a different type (concerning ordinal computation times) are given in Hamkins and Lewis [44] (Hamkins [42] is a more popular presentation). It is known Stannett [95] that exactly the countable ordinal times (i.e. times < ω1) can be imbedded into real numbers. This opens space for investigations concerning the relationships between general notions of analogue computability and ordinal computability.
- 8.
We might also think of two RTMs, one of which tries to prove TPC, while the other tries to prove its negation. Then we will either obtain (i) no signal (TPC is independent of ZFC), (ii) one signal (we will learn whether it is TPC or its negation which is provable in ZFC), or (iii) two signals, which means that ZFC is inconsistent. So, this scenario involves an implicit consistency check.
- 9.
A clear expression of the thesis (in the context of computer-assisted proofs) is given by Rota: “Verification is proof, but verification may not give the reason. What, then, are we to mean by a reason” [84, 187]. An eminent mathematician, Mordell, claims that “Even when a proof has been mastered, there may be a feeling of dissatisfaction with it, though it may be strictly logical and convincing: such as, for example, the proof of a proposition in Euclid. The reader may feel that something is missing. The argument may have been presented in such a way as to throw no light on the why and wherefore of the procedure or on the origin of the proof or why it succeeds” [69, 11] (quotation from Mancosu [66, 142]). For an introduction to the problem of explanation within mathematics, see Mancosu [65–67].
- 10.
This claim might be considered controversial (I am indebted to the anonymous referee for the comments which inspired this footnote). For sufficiently short proofs understanding would not pose a problem—but I think that our understanding of even a short, one-page long formal proof in ZFC would consist in appealing to semantics. We would not treat the string “∀x,y∃z∀t (t∈z⇔(t∈x ∨ t∈y))” as meaningless, but rather would recognize it as the Union Axiom. Similarly we would try to identify the meaning of the formulas in the proof—and we would grasp the essence of the formal manipulations because we would interpret them in some way. Even understanding of hypothetical 1000-pages long formal proofs would consist in “translating” them into the “everyday mathematical concepts” (i.e. for instance we would think of lines 3235-5298 of the proof as of the application of the Central Limit Theorem—and this would constitute our understanding). However the relationship between the expertise in manipulating symbols and understanding is intricate as the never-ending discussions concerning Chinese room and its numerous variants show.
- 11.
An oracle, “Pythiagora”, is described in Rav [82]; the author claims that it would have a destructive impact on mathematics as proof lies at the heart of mathematics, not theorems. A comparable situation is also considered in Tymoczko [101] (the first philosophical paper on the proof of the four-color theorem): Simon is a mathematical genius who solves the most difficult open problems in mathematics. But after some time, Simon refuses to present any proofs as he considers them to be too complex for us to understand. Should we trust him, and replace the standard proof (being a warrant for the truth of mathematical claims) by the phrase “Simon says”? The difference between Pythiagora, Simon, and the RTM is substantial: we know how the RTM works, so perhaps the oracle metaphor is misguided? However, from the purely mathematical point of view, the difference seems to be not so big: we only get the output, without any information concerning the proof.
- 12.
- 13.
See for instance Hauser [45] for a philosophical discussion and arguments in favor of the thesis that CH can be considered a legitimate, meaningful mathematical question.
- 14.
Large cardinals are not a remedy: Levy and Solovay showed that assuming the existence of measurable, compact, Ramsey cardinals is consistent with both CH and ¬CH [59]. Moreover‚ Easton [30] proves that the value of the continuum can be almost arbitrary as the κ → 2κ function can behave in hugely different ways.
- 15.
Such a discussion is important in modern set theory; cf. the discussion Feferman [33], Friedman [34], Maddy [62], Steel [96] (or more recent Maddy [63]). The recent discussion on the multiverse and hyperuniverse approach in set theory or the inner model program has a similar flavor: the aim is to find some axioms that give a true description of the set-theoretic world. See Hamkins [43], Arrigoni and Friedman [9].
- 16.
In Szabó [98] a radical view is defended according to which formal systems can be treated as (embodied in) physical objects. This goes with a formalistic and physicalistic philosophy of mathematics, therefore my claim concerning the unnaturalness of (2) could not be accepted from his point of view.
- 17.
- 18.
- 19.
There are many examples discussed in the literature; one of the simplest is Euler’s theorem, which explains why it is not possible to cross the famous system of bridges in Königsberg (crossing every bridge exactly once). No physical details of this system are relevant for the explanation. There are of course much more advanced examples, to mention a few: (i) The Borsuk-Ulam theorem explains why there must be two antipodal points on the surface of the Earth where two physical parameters (say: temperature and pressure) are equal (cf. Baker [10, 11], Baker and Colyvan [12]); (ii) Lyon and Colyvan [60] discuss the explanatory role of phase spaces in physics, in which theorems on these spaces explain some phenomena; (iii) Baron [13] discusses a theorem on stochastic processes which can be considered to provide an explanation of the behavior of predators.
- 20.
A recent survey of non-causal (in particular: mathematical) explanations is for instance [83].
- 21.
The anonymous referee pointed out that this is really a case-by-case check and the result of this procedure is that it is not the case that we can prove 0 = 1 in PA. In order to justify Con(PA) we need one more step, i.e. the double-negation elimination. It is therefore non-constructive—and indirect.
- 22.
This is true under the assumption of consistency of ZFC (if ZFC is inconsistent, we can prove everything, in particular Con(PA)). If we already know that ZFC is consistent (for instance by proving it in a stronger theory), we can use this fact as an explanation for the fact, that our procedure ends in the expected way.
- 23.
In fact, our two RTMs from footnote 8 that check the truth/falsehood of the twin prime conjecture are performing (as a nice “byproduct” of the primary task) an “indirect Con(ZFC) RTM check”.
- 24.
In fact, this is also the best explanation of the fact that no (human) mathematician has ever found an inconsistency. You cannot find something which does not exist, and the detailed story of your search does not contribute to the explanation.
- 25.
- 26.
Trivially, ZFC + Con(ZFC) proves Con(ZFC) (and T + Con(T) proves Con(T)…), but this choice of theory explaining the outcome is ad hoc.
- 27.
It is natural to think of such progressions in foundational studies. Gödel considered the reasonableness of formulating increasingly stronger axioms of infinity (which is often called “Gödel’s program”) and the research on large cardinals is one of the central parts of contemporary set theory. We can also think of reflection principles that express beliefs about the similarity between the universe and its parts. All these new axioms exceed ZFC.
- 28.
In Reverse Mathematics, mathematical notions are encoded in the language of second-order arithmetic Z2, and the main focus of study is identifying the strength of axioms (often: set existence axioms) necessary to prove ordinary mathematical theorems. Simpson [92] is the main reference.
- 29.
“Further, we intend to analyze the logical structure of the theory: which assumptions are responsible for which predictions; what happens if we weaken/fine-tune the assumptions, what we could have done differently. We seek insights, a deeper understanding. We could call this approach ‘reverse relativity’ in analogy with ‘reverse mathematics’” Andreka et al. [1, 608]. An elaboration of these ideas in the context of why-questions within axiomatic relativity is given in Székely [99]. An important and interesting feature of this approach is that theories – not sentences – are considered to be answers to particular why-questions. So, loosely speaking, the content necessary to present an explanation is presented as a theory (which seems to suit scientific needs better than just presenting a single principle/axiom).
- 30.
A topic I find particularly interesting is the choice of first-order logic as the framework (motivations were given for instance in Andréka et al. [4]) and the impact of this choice on the overall philosophical picture.
- 31.
Performing an RTM check is not performing a supertask (i.e. a computation with, informally speaking, unbounded acceleration). The computation proceeds in an ordinary way, the hypercomputational effect appears due to the relationship between the observer’s and the computer’s time. We might view this effect due to an imbedding of the computer’s time into the observer’s time without preserving cofinality, which allows the outcome of the computational process within a bounded time to be ‘observed’ (cf. Stannett [95, 14]). This is not an inherent feature of the computing subject (computer/computer), in particular we do not have to resolve puzzles like “What happens just after the infinite accelerating computation terminates?”, as in Thompson-lamp-like examples. (For supertasks see Manchak and Roberts [64] or the former entry Laraudagoitia [58].
- 32.
Such doubts concern, for example, the ARNN model [90, 91], which—in cases in which the coefficients are computable—is equivalent to the standard Turing model. Davis [26] claims therefore that the model does not really go beyond the Turing limit in any reasonable sense. See also Davis [25] for a forceful critique of the very idea of hypercomputation. See also Copeland [16–19] for a general discussion of the notion of computation.
- 33.
In a series of papers, Da Costa and Doria [20, 23, 24] provide interesting examples of sentences that are undecidable within ZFC and that seem to have a natural physical interpretation. (In Chaitin et al. [15] the results are put together and given a popular presentation.) Independence results in the language of relativity theory (involving models based on Archimedean fields) are presented in Andréka et al. [3].
- 34.
It is also not clear whether there might be hypercomputation on the quantum level. Kieu [51, 52] defined such a quantum system: a retort is given in Smith [93]. The status of quantum hypercomputation is not settled. In Cubitt et al. [22] a quantum system is constructed for which the spectral gap problem is undecidable; so, if it were possible to verify it by empirical methods, it would yield an “empirical solution” to an undecidable problem.
- 35.
“Ohne ein Quentchen Metaphysik läßt sich, meiner Überzeugung nach, keine exacte Wissenschaft begründen.” (from Cantor’s Nachlass).
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Acknowledgements
I would like to express my gratitude to the Editors (especially to Gergely Székely) for many helpful comments and bibliographic hints during the preparation of the paper. I am also indebted to Istvan Németi and Hajnal Andréka for their friendly support and feedback. Finally I would like to thank the anonymous referee for insightful and stimulating comments. The preparation of this paper was supported by an National Science Center (NCN) grant: 2016/21/B/HS1/01955.
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Wójtowicz, K. (2021). The Significance of Relativistic Computation for the Philosophy of Mathematics. In: Madarász, J., Székely, G. (eds) Hajnal Andréka and István Németi on Unity of Science. Outstanding Contributions to Logic, vol 19. Springer, Cham. https://doi.org/10.1007/978-3-030-64187-0_7
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