Tuesday, September 13, 2016
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Jeffrey Shallit
Recurrent thoughts about mathematics, science, politics, music, religion, and
Recurrent thoughts about mathematics, science, politics, music, religion, and
Recurrent thoughts about mathematics, science, politics, music, religion, and
Recurrent thoughts about ....
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Jeffrey Shallit
American mathematician, professor of computer science at a major Canadian university, & skeptic.
All comments are moderated to remove spam and duplication. Please be patient; your comment will usually appear within 12 hours.
Recursivity
Blog Archive
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2016
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September
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- Local Pastor Wants More Internet Censorship
- For Good People to Do Evil, that Requires....
- A Visit to Nari
- Was the Moose Hurt?
- World's Dumbest Journal Name
- Berlinski Exposed
- The Brain and Computation
- Beating the Dead Horse of Intelligent Design
- Robert Marks: Two Years Later, Still No Answer
- The Grammatical Rule that Isn't
- Anonymous Theist Coward Tries to Get Me Fired
- Friday Moose Blogging: Colorado Moose!
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September
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11 comments:
Here computation is not limited to algorithmic computation. I mention this because creationists like Eric Holloway (now a doctoral student of Robert Marks at Baylor) and Jonathan Bartlett stipulate that nature operates according to algorithms, and then treat (highly dubious) evidence that humans perform nonalgorithmic calculations as evidence that human minds are supernatural.
What is an example of a computation that is not algorithmic?
Jeff,
Sorry for the slow response. I've now checked "Email follow-up."
Connectionism has long been important in the computational theory of mind. Some recurrent neural nets are chaotic systems. If I build a recurrent neural net into the head of a robot, I will say that the robot computes responses to stimuli, irrespective of whether the neural net is a continuous system or a discrete system approximating a continuous system. If continuous quantities exist in physical reality, then a recurrent neural net that I implement, e.g., in "analog" VLSI, may in fact diverge exponentially in phase space from all discrete approximations.
There is no standard language for talking about finite-time computations ranging over uncountable sets. I didn't originate the distinction of algorithmic and nonalgorithmic computation. I'm not satisfied with it, and would be glad to have your advice on terminology.
I'd say (a) there is no evidence that "continuous quantities exist in physical reality", and there's evidence suggesting just the opposite and (b) even if there is, there exists is a good computational theory of computing with real numbers (e.g., Blum-Shub-Smale) whose limits are reasonably well understood.
So I'd dispute both of your points. But it would be very interesting if someone could show the brain had access to computation in the Blum-Shub-Smale model.
I analyzed the "real world plausibility" of one continuous neural network model (that of H. Siegelmann) for my MS at CMU and later in a followup paper for the volume _Computing Nature_.
Short version result (dogmatic, but see the work for the reasons) is that the idealizations involved are ridiculously underjustified. So, Turing really did discover something very interesting in 1936.
I presume another example of nonalgorithmic computation is a hypercomputer with access to some sort of oracle. I know a few people who believe that this is exactly how the brain works.
~~ Paul
Jeff,
You apparently believe that physicists can obtain evidence in support of a claim that all elements of a theoretical continuum are physically possible. Otherwise, you would not weigh evidence for and against the claim. So please explain to me how to provide supporting evidence. I've gotten the impression that physicists regard continuity as falsifiable (only), not verifiable.
Each theoretical continuum must be addressed individually. I'm aware of limited evidence of discontinuity in space-time (particularly, space). Time and the energy spectrum are continuous in quantum mechanics. I'm unaware of evidence to the contrary. If I've missed something, please let me know.
But it would be very interesting if someone could show the brain had access to computation in the Blum-Shub-Smale model.
How would a scientist show such a thing?
You apparently believe that physicists can obtain evidence in support of a claim that all elements of a theoretical continuum are physically possible.
Sure. Describe an experiment that could, in principle, give us a billion bits of the base-2 expansion of the fine-structure constant.
"there is no evidence that "continuous quantities exist in physical reality", and there's evidence suggesting just the opposite"
I wouldn't have dared to put it that strongly, at least in the first clause, but that's where I would put my money, if a bet were feasible. In other words, calculus was a great human achievement and very useful, but in fact it may only work as a (very good) approximation to discrete systems with very fine increments. In other other words, Zeno was probably correct (infinitesimal continuity is an illusion).
After all, all of our calculations are done using discrete numbers - every computer has a minimum floating point value. (I used a slide rule in college but I could only read three digits off it.)
In the model I mentioned above it doesn't matter (implementationally/hypercomputationally) if there really are continuous quantities (or other sources of "real reals"), because even the *slightest* bit of noise renders the model sub-Turing. (As the author of the model herself proves, no less.) Worse, the noise has to be absent in the "sensors" of the system, not just the property itself, otherwise the property is not causally connected to the relevant part of the rest of the universe! (This, in outline, is my biggest complaint with the whole proposal.)
In another model, which I don't claim to fully understand - my GR is weak, to say the least - the Malament-Hogarth spacetime idea, you need a sensor which can receive signals which are arbitrarily blueshifted / high energy. Not going to happen. (Especially if somehow you think there's an appropriate spacetime in your head or something bizarre like that.)
Alas, the Simons Institute page has been down for a few days.
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