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There is a programme of inconsistent mathematics which revives and completes Hilberts logicist programme by defanging Russells Paradox and the incompleteness theorems of Godels.

Can inconsistency/paraconsistency provide insights into Physics as well? And what would this mean for the philosophy of Physics? Has there been any research done along these lines?

The SEP/IEP has nothing about the possible physical ramifications of inconsistency. Though given the symbiotic relationship between mathematics and physics one expects there to be some.

It does mention a couple of geometric incarnations of paraconsistency that may have fruitful physical consequences:

a. gluing of geometries along boundaries - the boundary belongs to the first & not the second and not to the first but to the second (and to both?). Gluing is important in toy theories of QG such as TQFTs.

b. Open set logic is intuitionistic; its dual closed set logic is paraconsistent.

Perhaps when a state collapses phenomenologically when provoked by a measurement (but ontologically not) this could instead be instead be interpreted as a state not having a specific value and having a specific value - which gets rid of the phenomenology - perhaps, as this is all pure speculation.

EDIT

Andreas Doring, who works on Topos interpretation of Physics as initiated by Chris Isham has the following paper where he explains:

The main thrust of this article is to replace the standard algebraic representation of quantum logic...by a better behaved form based on bi-Heyting algebras. Instead of having a non-distributive orthomodular lattice of projections, which comes with a host of well-known conceptual and interpretational problems, one can consider a complete bi-Heyting algebra of propositions. In particular, this provides a distributive form of quantum logic. Roughly speaking, a non-distributive lattice with an orthocomplement has been traded for a distributive one with two different negations.

Now what this means is that this 'new' logic is both intuitionistic & paraconsistent. Of course this is to a large extent a formal result. Its Physical meaning of paraconsistency here remains tentative.

  • When one speaks of paraconsistent mathematics, one has a clear formal definition of what "inconsistent" means. What is this supposed to be in your quesiton for physics? Do you speak of a mathematical inconsistency in the formulation of a physical theory? A positivist will not care about formalities under the hood, as long as a principle of explosion doesn't spoil the predicitive power of the theory. Do you only consider theories which are intended to be realist and fundamental? Otherwise, from the practical side, people don't require their theories to be valid for every application anyway. – Nikolaj-K Jun 21 '13 at 8:18
  • @Kidman: I think its very early to look at paraconsistentcy in physics, so formal ideas won't have much purchase. We have a clear formal definition of what 'inconsistency' means after a fair bit of work & development. I'm not interested in postivistic ontologies. It depends on what you mean by realist. I think I do, yes. I agree that theories have an effective range; and sometimes you can take low-energy limits etc for relative consistency arguments. – Mozibur Ullah Jun 21 '13 at 13:40
  • "We have a clear formal definition of what 'inconsistency' means after a fair bit of work & development" - I mean in mathematics, not physics. – Mozibur Ullah Jun 21 '13 at 16:40
  • No interest in positivistic ontologies? Why? In any case, it's funny I encounter your physics+inconsistency questions here. I'm a physicist but asked about inconsistent mathematical theories here, on the MathSE site, only two days ago. – Nikolaj-K Jun 21 '13 at 18:54
  • @Kidman: It's an interesting perspective - particularly in (scientific) physics, but I can't think of it as a full philosophy - in the sense of taking ethics, aesthetics, subjectivity & the social into proper account. Although it does have effects in those arena. For example 'behaviouralism' - the study of man by focusing simply on external behaviour and ignoring inner life & the intersubjectivity. Again an interesting partial perspective. But there have been people who have argued for it as a total explanation. – Mozibur Ullah Jun 21 '13 at 19:29
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At least our physical theory about the physical world can be inconsistent. We can assume that general relativity and quantum mechanics both correctly describe the physical world, and use this assumption to compute things like the Hawking radiation of black holes.

What is interesting here for me is that it shows one way how humans manage to work with inconsistent theories. (I wondered about this question in the context of AI, because if AI wants to ever match human reasoning skills, then it also needs the ability to handle inconsistencies.) By splitting the single inconsistent theory into two independent consistent theories, we are able to construct mathematical models for both theories separately, which allows to exploit the usual mathematical tools for deriving predictions from the theories. The interactions between the two theories on the other hand are specified by more informal and less mathematically rigorous rules, allowing flexible application of the combined theory to come to prediction about the physical world even in scenarios where both theories have to interact.

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Has there been any research done along these lines?

A (the?) name that pops up repeatedly in paraconsistency and physics is that of Newton da Costa (with collaborators). Here are some references to his and others' work.

  1. Remarks on the applications of paraconsistent logic to physics

  2. The Paraconsistent Logic of Quantum Superpositions, submitted just a week ago!

  3. The future of paraconsistent logic

  4. E.G. de Souza, 1995, Destouchess problem and heterodoxical logics : essay on the use of non classical logics in the treatement of inconsistencies in physics, PhD, University of Sao Paulo, Sao Paulo.

  • Thanks for these references. I'd notice da Costa coming up with regard to paraconsistent mathematics - but I didn't notice he'd worked on physical theories too. Its also called Brazilian mathematics for that reason! – Mozibur Ullah Jun 21 '13 at 13:33
  • @MoziburUllah I too noticed that there is a whole Brazilian contingent working with paraconsistency. Up until today I thought it was an Australian thing. Something else: Can you have a look at the Priest book and tell me the number on its very last page (not the cover)? I have 3677860R00186 and I was wondering if it was printed on demand. – user3164 Jun 21 '13 at 13:38
  • Da Costa calls the Australian school relevant logic which has a certain overlap with paraconsistent logic. His paper on quantum super-position touches on roughly the speculation I mentioned! Which Priest book - I think you mentioned one before? – Mozibur Ullah Jun 21 '13 at 13:55
  • @MoziburUllah Well, since the paper is from last week, that must mean you are now at or beyond the frontiers of knowledge! I think it was you who mentioned Priest's Beyond the Limits of Thought, so I started reading it. If you have it at hand, I would appreciate you having a look at the number on the very last page. – user3164 Jun 21 '13 at 15:19
  • There isn't one on the last page. You're not talking about the ISBN number? – Mozibur Ullah Jun 21 '13 at 16:39
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Not sure if it is what your looking for, but since you tagged it as philosophy of science...:

Imre Lakatos developed a theory to weaken Poppers falsificationism. According to Lakatos, theories are never isolated, but part of chains of theories. Each part of a chain is a theory that shares with the other parts its hard core, consisting of theoretical assumptions that cannot be abandoned or altered without abandoning the programme altogether. To the hard core comes a weak hull, consisting of auxiliary hypotheses. Auxiliary hypotheses are considered expendable by the adherents of the research programme - they may be altered or abandoned as empirical discoveries require in order to 'protect' the 'hard core'.

One feature of the theory of lakatos was that he allowed contradictions in the weak hull, as long as they are not permanent. In this sense he would allow an inconsistent physics.

  • See also the wiki-article, of which some parts already appear in my answer. The relevant chapter is 'research programes'
  • What you write reminds of the Duhem–Quine thesis. – Nikolaj-K Jun 21 '13 at 8:13
  • Thanks for the answer. It wasn't what I was looking for - but using a paraconsistent logic to elaborate the relationships between effective theories is an interesting insight. – Mozibur Ullah Jun 21 '13 at 16:35
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The answer is a clear NO.

The base of all physics are phenomena. If two theories contradict each other there will be an experiment to resolve this contradiction i.e. a phenomenon that falsifies one of the theories.

Phenomena by them self cannot be contradictory - if there are two phenomena for the exact same setup/environment, this is another phenomenon (which I call metaphenomenon). Some causes to phenomena might be hidden or unobservable (or as some claim purely probabilistic) but at some point the metaphenomenon can be described as a causal reaction of a setup/environment and this is what physics is all about.

  • This is what I expect the mainstream opinion to be. But whats your considered opinion about inconsistent mathematics? Its seems to get some very surprising results by embracing inconsistency rather than incompleteness. – Mozibur Ullah Jun 21 '13 at 13:35
  • what if we cannot perform such an experiment? for example, it was possible only in the first second of Universe existance? – Bulat Jun 24 '13 at 19:25
  • If there is no experiment to perform - the theory is not of relevance. – Brandli Jun 25 '13 at 12:01
  • @Bulat: If there isn't a direct experiment that one can perform then people look for indirect means of validation. That of course doesn't mean there will always be one. For example string theory is outside the possibility of experiment for now, and probably for quite some time in the future. – Mozibur Ullah Jun 29 '13 at 12:44

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