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khafra comments on Beyond the Reach of God - Less Wrong
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How exactly can you construct a disproof of X without using premises that rule out X? That's what disproving is.
Non-infinite mathematics: also known as finite mathematics, also known as discrete mathematics. Non-continuum. Not requiring the existence of the real numbers.
To the best of our knowledge, reality only seems to *require* the integers, although constructing models that way is a massive pain. If you can give us an example of a physical phenomenon that cannot be generated by a Turing Machine's output -- just one example -- then I will grant that we have no grounds for presuming human cognition can be so constructed. Also, you'll win several Nobel Prizes and go down in history as one of the greatest scientist-thinkers ever.
I'm not holding my breath.
You can accept X as a premise and come to a contradiction of X with other accepted premises. Coming to something that seems absurd may also be grounds for doubting X, but doesn't disprove X. It might also be possible to prove that X and ~X are consistent with the other premises, which if the desire is to disprove X should be enough to safely ignore the possibility X is correct without further information.
I think for the Turing Machine part of this P/NP would need to be resolved first so he would also win 1 million dollars (or if P=NP then depending on his preferences he might not want to publish and use his code to solve every open question out there and get himself pretty much as much money as he wished)
Caledonian was a combination contrarian and curmudgeon back in the OvercomingBias days, and hasn't been around in years; so you probably won't get a direct reply.
However, if I understand this comment correctly as a follow-up to this one, you may want to look into the Church-Turing Thesis. The theory "physics is computable" is still somewhat controversial, but it has a great deal of support. If physics is computable, and humans are made out of physics, then by the Church-Turing Thesis, humans are Turing Machines.
I am actually familiar with the Church-Turing Thesis, as well as both Godel's incompleteness proof and the Halting problem. The theory that humans are Turing machines is one that needs to be investigated.
'The theory that humans are Turing machines is one that needs to be investigated."
Yes, but that question isn't where we need to start necessarily. It is a subset of a possibly much simpler problem: are the laws of physics Turing computable? If so, then humans cannot do anything Turing machines cannot do. if not, then the human-specific question remains open.
We don't know, and there are many relevant-but-not-too-convincing arguments either way.
The laws of physics as generally taught are both continuous (symmetries, calculus in QM), and quantum (discrete allowed particle states, Planck length, linear algebra in QM).
No one has ever observed nature calculating with an uncomputable general real (or complex) variable (how could we with human minds and finitely precise instrumentation?), while any computable algebraic or transcendental number seems to be fair game. But, building a model of physics that rules out general real variables is apparently much more difficult.
Even if there are general real variables in physics, they may only arise as a result of previous general real variables, in which case whatever-the-universe-runs-on may be able to handle them symbolically instead of explicitly. Anyone here who decides to read my many-years-late wall of text have any idea what the implications would be of this one? Possibly may or may not allow construction of architectures that are not Turing computing but also not fully general, limited by whatever non-Turing-computable stuff happens to have always existed?
If space and time are quantized (digital), that makes for more even trouble with special and general relativity - are the Planck length/time/etc.somehow reference frame dependent?
Also, see http://lesswrong.com/lw/h9c/can_somebody_explain_this_to_me_the_computability/