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Answer by Tikhon Jelvis:
I think there’s a good—albeit not terribly popular—case for rejecting the law of the excluded middle:
This law states that every proposition is either true or false. Another way of thinking about it is that if leads to a contradiction, cannot hold. It embodies proof by contradiction and, more generally, “existence proofs” where we prove that something has to exist without actually providing an example.
The alternative to existence proofs are constructive proofs where we prove that something exists by actually constructing (or at least providing a procedure to construct) an example. Taking away the law of the excluded middle gives rise to constructive logic, also called intuitionistic logic, which only permits constructive proofs. We cannot prove something exists without constructing—or at least giving a procedure to construct—an example.
Now, existence proofs are very useful. But, in a very deep sense, they are not as satisfying as constructive proofs. For example, you could prove that something exists but, at the same time, be completely unable to actually ever find an example! That feels like a weaker form of “existing” than actually having an example in your hand. This is why constructive logic is preferable in a philosophical sense.
However, constructive logic is strictly weaker than classical logic. There are classical theorems you cannot prove constructively but not vice-versa. This is an important practical consideration: many useful results would not be possible purely constructively. I think this is the main reason constructive logic is less popular.
However, there’s a different way of thinking about it that actually makes constructive logic to be more powerful—not in the theorems it can prove, but in the way it can distinguish certain theorems. Let me elaborate.
Something to note about constructive logic: double negation elimination is not a theorm. In other words, it is not always true that
However, you can go the other way:
The intuition here is that is a constructive proof—you provide an example of . On the other hand, is a proof that “ does not lead to a contradiction”, which is weaker because it does not necessarily involve producing an example of .
This is the key way in which constructive logic is actually “more powerful” than classical logic: we can differentiate from . Or, put another way, we can differentiate a proof of from a weaker proof that does not lead to any contradictions.
This intuition can be formalized by actually embedding classical logic in constructive logic. We can do this by mapping every classical theorem to a constructive theorem . It turns out that if we can prove classically, we can always prove constructively. Intuitively, we can think of the classical proof as a constructed example of how does not lead to a contradiction!
I’m not sure how mathematicians prove this translation is valid. However, it turns out to be really easy by dropping into CS via the Curry-Howard isomorphism. This “double negation translation” corresponds to transforming code into “continuation-passing style” (CPS). A CPS transform is pretty easy to implement, which shows the translation works.
So it turns out we can use constructive logic just like a more precise classical logic. Whenever we want to prove classically, we can try to prove either or constructively. Either result is equal to a classical proof, but we also know that a proof for is weaker in a very specific way.
Recursive languages or formal systems or automata are generally decidable; the set of such languages, formal systems, or automata are generally undecidable. In my work, I am looking for the logical description or model of a universal Turing machine physically encoded, and I believe I know what the physical encoding is. The characteristic of a universal Turing machine is the acceptance of every input and the transformation of state for every input as output; physically, this corresponds pretty neatly to the characteristics of a blackhole which absorbs all matter and energy and transforms state for everything absorbed.
A universal Turing machine can simulate any other machine if supplied with the equivalent encoding of that Turing machine. If we correspond this behavior to blackholes then we get an interesting potential consequence; feeding a blackhole a human being gives it the ability to simulate a human being and likewise for fundamental particles. This correspondence works the other way too. Theoretically, blackholes (totally absorbative) are joined to white holes (totally emitting) via a Einstein-Rosen bridge (wormhole aka bifrost), so this would seem to correspond to how the internal parts of a universal Turing machine connect with the external parts; the internal memory maps to the external memory via a negatively curved space.
What does this tell us? For one, if we live in this kind of universe then we can move from one universal Turing machine to another via bridges. There’s additional complexity involved in transforming from one universal Turing machine to another and in escaping one universe for another, but it is in principle possible to do it. For two, it tells us it is plausible that we are some kind of Turing machine operating within a universal Turing machine. It tells us it is plausible that we might generate a table for programming ourselves and our environment.
It tells us that if we live in this kind of universe then we don’t need to develop infinitely better machines because the smallest components of our universe are logical circuits corresponding to finite state machines and regular languages. All the mechanism are there waiting for us to figure out the programming codes to make them perform their functions. We need to develop the interfaces between our semi-classical first hand experience and the quantum mechanical Planck state transformations of the information physically encoding us. It is akin to the work which was done to convert special relativity and classical physical theory to quantum mechanics; special relativity tells us about the behavior of two-state systems with zero rest mass aka photons and gluons aka Qubits. Chromodynamics by virtue of its finite extent probably corresponds to regular physical languages and finite state machines; spin mechanics likely corresponds similarly, but we know that electromagnetics join with weak force mechanics to form electroweak dynamics, so we know that weak forces border on infinite extents. My inkling is that electromagnetics are countably infinite in extent whereas gravitation is uncountably infinite in extent. Which means that electromagnetics, weak, and strong forces correspond to regular, context-free, context-sensitive, and recursive physical languages whereas gravitation corresponds to recursively enumerable and analytical physical languages.
My primary research at this time considers the possibility of contradiction tolerant universes and physical systems. I feel it is necessary to pursue contradiction tolerant alternative methods and theories in order to explicitly account for things such as hallucinations and dream logics. We tell stories and assign meaning and purposes to people and things in ways which are arguable not consistent. The very fact that we can identify and discuss contradictions ought to be a significant clue that contradictions do in fact “exist” in our universe in so far as we exist in our universe and our discourse about contradictions exists in our universe. This is oddly disputed by some otherwise very bright people; it is an untenable position for a materialist to have as it leads readily to a supermaterialistic dualism which places human consciousness outside of the domain of physics. While I will humor spiritual and religious arguments about the supernatural properties of consciousness, when it comes to physical theory my loyalties are with attempting to disprove supernatural and unphysical characterizations of human consciousness. Counter-intuitively to many, this means theorizing human consciousness totally as a materialistic process and creating reductionist and instrumentalist experiments based on that theory. The point of such scientific inquiry is almost never to confirm the theory and succeed but to fail while doing your damnedest to succeed. Failure is interesting theoretically speaking whereas success is not.
From the purely materialistic view point, everything we experience has to have a material origin. My thoughts may seem ephemeral, but they are material just as energy and electrical charge are materially mediated. My thoughts exist as a mixture of molecules and electromagnetic signals communicated between the medium of my body and the environment within which my body resides. When we connect the material to the computational, what is material origins becomes information origins; my thoughts simplify somewhat to sequences or configurations of various kinds of states, so what I experience—even dreams and stories and hallucinations—is a direct experience of information materially encoded in the medium of me. In that sense, what I experience are programs. Mechanics simulated in the subspace of the machine that is me; mechanical interactivity.
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